Intelligent and Fuzzy Systems: Intelligence and Sustainable Future Proceedings of the INFUS 2023 Conference, Volume 1 (Lecture Notes in Networks and Systems, 758) 3031397738, 9783031397738

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Lecture Notes in Networks and Systems 758

Cengiz Kahraman · Irem Ucal Sari · Basar Oztaysi · Selcuk Cebi · Sezi Cevik Onar · A. Çağrı Tolga   Editors

Intelligent and Fuzzy Systems Intelligence and Sustainable Future Proceedings of the INFUS 2023 Conference, Volume 1

Lecture Notes in Networks and Systems

758

Series Editor Janusz Kacprzyk , Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

Advisory Editors Fernando Gomide, Department of Computer Engineering and Automation—DCA, School of Electrical and Computer Engineering—FEEC, University of Campinas—UNICAMP, São Paulo, Brazil Okyay Kaynak, Department of Electrical and Electronic Engineering, Bogazici University, Istanbul, Türkiye Derong Liu, Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, USA Institute of Automation, Chinese Academy of Sciences, Beijing, China Witold Pedrycz, Department of Electrical and Computer Engineering, University of Alberta, Alberta, Canada Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Marios M. Polycarpou, Department of Electrical and Computer Engineering, KIOS Research Center for Intelligent Systems and Networks, University of Cyprus, Nicosia, Cyprus Imre J. Rudas, Óbuda University, Budapest, Hungary Jun Wang, Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong

The series “Lecture Notes in Networks and Systems” publishes the latest developments in Networks and Systems—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNNS. Volumes published in LNNS embrace all aspects and subfields of, as well as new challenges in, Networks and Systems. The series contains proceedings and edited volumes in systems and networks, spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. The series covers the theory, applications, and perspectives on the state of the art and future developments relevant to systems and networks, decision making, control, complex processes and related areas, as embedded in the fields of interdisciplinary and applied sciences, engineering, computer science, physics, economics, social, and life sciences, as well as the paradigms and methodologies behind them. Indexed by SCOPUS, INSPEC, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science. For proposals from Asia please contact Aninda Bose ([email protected]).

Cengiz Kahraman · Irem Ucal Sari · Basar Oztaysi · Selcuk Cebi · Sezi Cevik Onar · A. Ça˘grı Tolga Editors

Intelligent and Fuzzy Systems Intelligence and Sustainable Future Proceedings of the INFUS 2023 Conference, Volume 1

Editors Cengiz Kahraman Department of Industrial Engineering Istanbul Technical University Istanbul, Türkiye

Irem Ucal Sari Department of Industrial Engineering Istanbul Technical University Istanbul, Türkiye

Basar Oztaysi Department of Industrial Engineering Istanbul Technical University ˙Istanbul, Türkiye

Selcuk Cebi Department of Industrial Engineering Yildiz Technical University Istanbul, Türkiye

Sezi Cevik Onar Department of Industrial Engineering Istanbul Technical University Istanbul, Türkiye

A. Ça˘grı Tolga Department of Industrial Engineering Galatasaray University Istanbul, Türkiye

ISSN 2367-3370 ISSN 2367-3389 (electronic) Lecture Notes in Networks and Systems ISBN 978-3-031-39773-8 ISBN 978-3-031-39774-5 (eBook) https://doi.org/10.1007/978-3-031-39774-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023, corrected publication 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

INFUS is an acronym for intelligent and fuzzy systems. INFUS 2019 was an on-site conference organized in Istanbul, Türkiye. INFUS 2020 and INFUS 2021 conferences were organized as online conferences because of pandemic conditions. INFUS 2022 conference was organized as both online and on-site conference in Izmir with the cooperation of Yasar University and Izmir Bakircay University. INFUS 2023 is the fifth conference of this series organized by Istanbul Technical University. The theme of INFUS 2023 conference this year is Intelligent and Sustainable Future. Intelligence can be used in a wide variety of ways to manage environmental impacts and climate change such as clean sustainable supply chains, environmental monitoring and enforcement, and advanced weather and disaster forecasting. The emergence of artificial intelligence (AI) and its growing impact on many industries require research into how it can be used to achieve the Sustainable Development Goals. Applications of AI can create a sustainable and eco-friendly future. AI is a promising tool for the production of new materials that help in building a sustainable environment. The sustainability of biological diversity is another very important problem since many animal species are extinct or endangered. Therefore, intelligence can be used to study animal behavior patterns. Soil pollution is another important problem today as population growth, intensive farming, and other activities increase day by day. Since food production is the key to sustain human life, we can maintain environmental sustainability by monitoring crops and soils and maximize the crop yields, while having less impact on the environment through AI-augmented agriculture. The excessive consumption of natural resources by humans has a detrimental effect on water resources. The level of garbage accumulating in the oceans is higher than ever before. Artificial intelligence tools should be used to ensure environmental sustainability. Artificial intelligence can be used in automated garbage collection vehicles; it can help solve problems such as illegal fishing and discharge of industrial wastewater into water bodies and illegal dumping of solid wastes into the seas. The use of intelligence for a livable future has become a necessity. A program focusing on intelligence and sustainability future, which is the theme of this year’s INFUS 2023 conference, is foreseen. INFUS 2023 aims to bring together the latest theoretical and practical intelligent and fuzzy studies on sustainable future in order to create a discussion environment. Researchers from more than 30 countries such as Türkiye, Russia, China, Iran, Poland, India, Azerbaijan, Bulgaria, Spain, Ukraine, Pakistan, South Korea, UK, Indonesia, USA, Vietnam, Finland, Romania, France, Uzbekistan, Italy, and Austria contributed to INFUS 2023. Our invited speakers this year are Prof. Krassimir Atanassov, Prof. Vicenc Torra, Prof. Janusz Kacprzyk, Prof. Ahmet Fahri Özok, and Prof. Ajith Abraham, and Prof. Irina Perfilieva. It is an honor to include their invaluable speeches in our conference program. We appreciate their voluntary contributions to INFUS 2023, and we hope to see them at INFUS conferences for many years. This year, the number of submitted papers became 291. After the review process, about 40% of these papers have

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Preface

been rejected. More than 50% of the accepted papers are from other countries outside Türkiye. We again thank all the representatives of their countries for selecting INFUS 2023 as an international scientific arena to present their valuable research results. We are honored and aware of our responsibility that our participants have chosen us in a highly competitive environment with hundreds of conferences in the same field and organized in close dates to each other. INFUS conference manages high-cost international conference participation processes for the benefit of the participants, with lower registration fees but more well-known expert invitations and rich social activities. We also thank the anonymous reviewers for their hard works in selecting high-quality papers of INFUS 2023. Each of the organizing committee members provided invaluable contributions to INFUS 2023. INFUS conferences would be impossible without their efforts. We hope meeting all of our participants next year in Türkiye one more time with a new research theme at a new city and new social activities. We would like to thank our publisher Springer Publishing Company, Series Editor Prof. Janusz Kacprzyk, Interdisciplinary and Applied Sciences and Engineering, and Editorial Director Thomas Ditzinger, last but not least, Project Coordinator Nareshkumar Mani for their supportive, patient, and helpful roles during the preparation of this book. Cengiz Kahraman Irem Ucal Sari Basar Oztaysi Selcuk Cebi Sezi Cevik Onar A. Ça˘grı Tolga

Organization

Program Committee Chairs Cebi, Selcuk Kahraman, Cengiz Cevik Onar, Sezi Oztaysi, Basar Tolga, Ça˘grı Ucal Sari, Irem

Yildiz Technical University, Industrial Engineering, Istanbul, Türkiye ITU, Industrial Engineering Department, Istanbul, Türkiye Istanbul Technical University, Istanbul, Türkiye Istanbul Technical University, Istanbul, Türkiye Galatasaray University, Department of Industrial Engineering, Istanbul, Türkiye Istanbul Technical University, Industrial Engineering, Istanbul, Türkiye

Program Committee Members Alkan, Nur¸sah

Aydin, Serhat Boltürk, Eda Cebi, Selcuk Dogan, Onur Haktanır Akta¸s, Elif Kahraman, Cengiz Kutlu Gündo˘gdu, Fatma Otay, Irem Cevik Onar, Sezi

Istanbul Technical University, Department of Industrial Engineering, Maçka, Be¸sikta¸s, Türkiye National Defence University, Industrial Engineering Department, Istanbul, Türkiye Istanbul Technical University, Istanbul, Türkiye Yildiz Technical University, Industrial Engineering, Istanbul, Türkiye Izmir Bakircay University, Department of Industrial Engineering, ˙Izmir, Türkiye Bahcesehir University, Industrial Engineering, Istanbul, Türkiye ITU, Industrial Engineering Department, Istanbul, Türkiye National Defence University, Industrial Engineering Department, Istanbul, Türkiye Istanbul Bilgi University, Department of Industrial Engineering, Istanbul, Türkiye Istanbul Technical University, Istanbul, Türkiye

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Organization

Oztaysi, Basar Tolga, Ça˘grı Ucal Sari, Irem

˙Istanbul Technical University, Istanbul, Türkiye Galatasaray University, Department of Industrial Engineering, Istanbul, Türkiye Istanbul Technical University, Industrial Engineering, Istanbul, Türkiye

Contents

Keynote Speeches Contextual Bipolar Database Queries: A Conjunctive and Disjunctive Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Janusz Kacprzyk and Sławomir Zadro˙zny

3

Intuitionistic Fuzzy Modal Topological Structures Based on Two New Intuitionistic Fuzzy Modal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krassimir Atanassov

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Multi-scale Dimensionality Reduction with F-Transforms in Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irina Perfilieva

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Mathematics, Scientific Reasoning and Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . Ahmet Fahri Özok Optimal Transport and the Wasserstein Distance for Fuzzy Measures: An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vicenç Torra

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Theoretical Improvements in Fuzzy Sets Some Properties of a New Concept of Fractional Derivative for Fuzzy Functions on Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mina Shahidi and Estevão Esmi

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On Monotonic Function Method for Generating Fuzzy Similarity Measures . . . . Surender Singh and Koushal Singh

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Regularity and Paracompactness: Relation in the Field of Fuzziness . . . . . . . . . . Francisco Gallego Lupiáñez

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Fuzzy Threshold Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Lepskiy

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A Novel Bivariate Elliptic Fuzzy Membership Function: A Modeling and Decision-Making Tool for Bike Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alparslan Abdurrahman Basaran, Murat Alper Basaran, and Mehmet Ozer Demir

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Cryptography Based on Fuzzy Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mariana Durcheva and Malinka Ivanova Aggregating Distances with Uncertainty: The Modular (pseudo-)metric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. D. M. Bibiloni-Femenias, J.-J. Miñana, and O. Valero

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Peterson’s Rules Based on Grades for Fuzzy Logical Syllogisms . . . . . . . . . . . . . 102 Petra Murinová, Michal Burda, and Viktor Pavliska A New Construction of Uninorms in Bounded Lattices Derived from T-Norms and T-Conorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Gül Deniz Çaylı Aggregation Operators for Face Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Nebojša Ralevi´c, Andrija Blesi´c, Vladimir Ili´c, Marija Paunovi´c, ˇ and Lidija Comi´ c First-Order Representations and Calculi of Categorical Propositions . . . . . . . . . . 128 Yinsheng Zhang Properties of General Extensional Fuzzy Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Jiˇrí Moˇckoˇr Reverse Order Pentagonal Fuzzy Numbers and Its Application in Game Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 V. Kamal Nasir and A. Jamal Barakath Similarity Analysis of Means-End Chain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Umut Asan and Hatice Kocaman Unfolding Computation Graph for Dynamic Planning Under Uncertainty . . . . . . 160 Margarita Knyazeva, Alexander Bozhenyuk, and Leontiy Samoylov Strong Connectivity Definition of Periodic Fuzzy Graph . . . . . . . . . . . . . . . . . . . . 168 Alexander Bozhenyuk, Margarita Knyazeva, Olesiya Kosenko, and Igor Rozenberg Intuitionistic Fuzzy Sets An Automatic Rating System Based on Review Sentiments and Intuitionistic Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Mustafa Ünver

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Circular Intuitionistic Fuzzy Analysis of Variance on the Factor Season of Apple Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Velichka Traneva and Stoyan Tranev Petrol Station Franchisor Selection Through Circular Intuitionistic Fuzzy Multicriteria Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Velichka Traneva and Stoyan Tranev On Characterizations of Watson Crick Intuitionistic Fuzzy Automata . . . . . . . . . 204 N. Jansirani, N. Vijayaraghavan, and V. R. Dare An Intelligent Data Analysis Approach for Women with Menopausal Genitourinary Syndrome with Intuitionistic Fuzzy Logic . . . . . . . . . . . . . . . . . . . . 212 Pavel Dobrev and Evdokia Sotirova Measuring Happiness: Evaluation of Elementary School Students’ Perception of Happiness Assessed by Intuitionistic Fuzzy Logic . . . . . . . . . . . . . . 220 Gergana Avramova-Todorova and Veselina Bureva Selecting an Employer: Evaluation of University Students’ Perception About Business Companies Assessed by Intuitionistic Fuzzy Logic . . . . . . . . . . . 227 Milen Todorov and Veselina Bureva A Novel Intuitionistic Fuzzy Grey Model for Forecasting Electricity Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Bahadır Yörür, Nihal Erginel, and Sevil S¸ entürk Generalized Net Model with Intuitionistic Fuzzy Estimations of the Humanoid Robot Behavior During Navigation Tasks . . . . . . . . . . . . . . . . . . 243 Simeon Ribagin, Sotir Sotirov, and Krassimir Atanassov Intuitionistic Fuzzy Generalized Net Model of a Human-Robot Interaction . . . . . 252 Simeon Ribagin, Sotir Sotirov, and Evdokia Sotirova Intuitionistic Fuzzy Evaluations of Garbage Sorting Using a Robotic Arm . . . . . 259 Petar Petrov, Veselina Bureva, and Krassimir Atanassov Picture Fuzzy Sets Picture Fuzzy Internal Rate of Return Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Elif Haktanır LR-Type Spherical Fuzzy Numbers and Their Usage in MCDM Problems . . . . . 275 Cengiz Kahraman, Sezi Cevik Onar, and Basar Öztaysi

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Contents

Analyzing Customer Requirements Based on Text Mining via Spherical Fuzzy QFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Sezen Ayber, Nihal Erginel, Mustafa Ünver, Gökhan Göksel, and Ahmet Aydın A Method Based on Picture Fuzzy Graph Coloring for Determining Traffic Signal Phasing on an Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Isnaini Rosyida, Ch. Rini Indrati, and Sunny Joseph Kalayathankal Type-2 Fuzzy Sets Interval Type-2 Fuzzy CODAS: An Application in Flight Selection Problem . . . 307 Erdem Akın and Ba¸sar Öztay¸si Stabilization of a D.C Motor Controller Using an Interval Type-2 Fuzzy Logic System Designed with the Bee Colony Optimization Algorithm . . . . . . . . 318 Leticia Amador-Angulo and Oscar Castillo Requirement Prioritization by Using Type-2 Fuzzy TOPSIS . . . . . . . . . . . . . . . . . 326 Basar Oztaysi, Sezi Cevik Onar, and Cengiz Kahraman Fuzzy Z-Numbers Picture Fuzzy Z-AHP: Application to Panel Selection of Solar Energy . . . . . . . . 337 Nurdan Tüysüz and Cengiz Kahraman LR-Type Z Fuzzy Numbers and Their Usage in MCDM Problems . . . . . . . . . . . . 346 Cengiz Kahraman, Sezi Cevik Onar, and Basar Öztaysi The Use of Z-numbers to Assess the Level of Motivation of Employees, Taking into Account Non-formalized Motivational Factors . . . . . . . . . . . . . . . . . . 354 Alekperov Ramiz Balashirin Literature Review A Literature Review on Fuzzy ELECTRE Methods . . . . . . . . . . . . . . . . . . . . . . . . . 365 Beril Akkaya and Cengiz Kahraman Evolution of Fuzzy Sets: A Comprehensive Literature Review . . . . . . . . . . . . . . . 376 Murat Gülbay and Cengiz Kahraman Fuzzy Multicriteria Decision Making in Earthquake Supply Chain Management: A Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Sezi Cevik Onar, Cengiz Kahraman, and Basar Oztaysi

Contents

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Artificial Intelligence and Household Energy-Saving Policies: A Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Samrand Toufani, Irem Ucal Sari, and Gizem Intepe Optimization Optimization and Intellectualization of Adaptive Control of Investment Projects of Multi-agent Network Industrial Complexes with Fuzzy Data . . . . . . . 407 Andrey Shorikov and Elena Butsenko Fuzzy Model for Multi-objective Airport Gate Assignment Problem . . . . . . . . . . 415 Mert Paldrak, Melis Tan Tacoglu, and Mustafa Arslan Örnek Two-Stage Transportation Model for Distributing Relief Aids to the Affected Regions in an Emergency Response Under Uncertainty . . . . . . . . 426 Jency Leona Edward and K. Palanivel A Fuzzy Based Optimization Model for Nonlinear Programming with Lagrangian Multiplier Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 K. Palanivel and Selcuk Cebi Biometric Identification of Hand by Particle Swarm Optimization (PSO) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Hatem Ghodbane, Hichem Amar, Monir Amir, Badreddine Babes, and Noureddine Hamouda Optimization of Alkaline Zinc Plating Process in a Company Using Taguchi Model Based on Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Furkan Atik and Ahmet Sarucan Study of Multiverse Optimizer Variations with Chaos Theory and Fuzzy Logic Over Benchmark Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Lucio Amézquita, Oscar Castillo, Jose Soria, and Prometeo Cortes-Antonio Fuzzy Goal Programming Model for Sequencing Multi-model Assembly Line with Sequence Dependent Setup Times in Garment Industry . . . . . . . . . . . . 480 Elvin Sarı, Mert Paldrak, Tunahan Kuzu, Devin Duran, Yaren Can, Sude Dila Ceylan, Mustafa Arslan Örnek, and Ba¸sak Erol Fuzzy Color Computing Based on Optical Logical Architecture . . . . . . . . . . . . . . 491 Victor Timchenko, Yuriy Kondratenko, Oleksiy Kozlov, and Vladik Kreinovich

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A Fuzzy Mixed-Integer Linear Programming Model for Aircraft Maintenance Workforce Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Safacan Hasancebi, Gulfem Tuzkaya, and Huseyin Selcuk Kilic An Optimal Model for Integer Programming Problems Based on Fuzzy Membership Functions with Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 507 K. Palanivel Fuzzy Goal Programming Approach to Multi-objective Facility Location Problem for Emergency Goods and Services Distribution . . . . . . . . . . . . . . . . . . . . 521 Mert Paldrak, Simge Güçlükol Ergin, Gamze Erdem, and Melis Tan Taco˘glu Social Spider Optimization for Text Classification Enhancement . . . . . . . . . . . . . 532 Fawaz S. Al-Anzi and Sumi Sarath An Algorithm for Fully Intuitionistic Fuzzy Multiobjective Transportation Problem with a Goal Programming Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Sakshi Dhruv, Ritu Arora, and Shalini Arora Application of the Crow Search Algorithm for Dynamic Route Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 Hubert Zarzycki Neuro-Fuzzy Modeling PD-Type-2 Fuzzy Neural Network Based Control of a Super-Lift Luo Converter Designed for Sustainable Future Energy Applications . . . . . . . . . . . . . . 561 Ahmet Gani A Novel Fuzzy-Clustering-Based Deep Learning Approach for Spatio-Temporal Traffic Speed Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Jiyao An, Ju Fang, Xuan Zhang, and Qingqin Liu Structured Neural Network Based Quadcopter Control Under Overland Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Ali Abbasov, Ramin Rzayev, Tunjay Habibbayli, and Murad Aliyev Drug Delivery in Chemotherapy Using an Online Wavelet-Based Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 Pariya Khalili, Mansour Ansari, Ali Akbar Safavi, and Ramin Vatankhah

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Heuristics Fault-Tolerant Control Using Optimized Neurons in Feed-Forward Backpropagation Neural Network-For MIMO Uncertain System: A Metaheuristic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Sejal Raval, Himanshukumar R. Patel, Vipul Shah, Umesh C. Rathore, and Paresh P. Kotak Matheuristic Algorithm for Automated Guided Vehicle (AGV) Assisted Intelligent Order Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Simge Güçlükol Ergin and Mahmut Ali Gökçe A Study of Future Life Satisfaction Using Fuzzy Partition . . . . . . . . . . . . . . . . . . . 619 Seung Hoe Choi, Nan-Hi Lee, and Mi Young Kim Software Test Suite Minimization Using Hybrid Metaheuristics . . . . . . . . . . . . . . 626 Anu Bajaj, Ajith Abraham, and Nitigya Sambyal Decision Making Net-Zero Policy Performance Assessment South America Countries Through DEA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Mehtap Dursun and Rana Duygu Alkurt Sustainable Supplier Selection in Fuzzy Environment: A Case Study in Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 Ilgaz Cerit and Tuncay Gürbüz Evaluation of Mechanical Energy Storage Technologies in the Context of a Fuzzy Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Ahmet Sarucan, Mehmet Emin Baysal, and Orhan Engin Selection of Fighter Aircraft for Turkish Air Forces Under Uncertain Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 Yusuf Çifçi, Akın Metin, Tarık Tu˘gra Arslan, and Fatma Kutlu Gündo˘gdu Analysis of Suppliers’ Resilience Factors Under Uncertainty . . . . . . . . . . . . . . . . 670 Fatma Cayvaz Parlak, Huseyin Selcuk Kilic, and Gulfem Tuzkaya A Framework of Directed Network Based Influence-Trust Fuzzy Group Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 Nor Hanimah Kamis, Adem Kilicman, Norhidayah A Kadir, and Francisco Chiclana

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Rough Data Envelopment Analysis: An Application to Indian Agriculture . . . . . 689 Alka Arya, Adel Hatami-Marbini, and Pegah Khoshnevis Finance Analysis of Various Portfolio Allocation Decision-Making Techniques in Crypto Assets Using Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 Murat Levent Demircan and Tayfun Dirinda Intelligent Software for Optimizing Adaptive Control of Regional Investment Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 Andrey Shorikov and Elena Butsenko Q-Rung Orthopair Fuzzy Benefit/Cost Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 Eda Boltürk Cash Replenishment and Vehicle Routing Improvement for Automated Teller Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 Deniz Orhan and Müjde Erol Genevois A Dynamic Feature Selection Technique for the Stock Price Forecasting . . . . . . 730 Mahmut Sami Sivri, Ahmet Berkay Gultekin, Alp Ustundag, Omer Faruk Beyca, Omer Faruk Gurcan, and Emre Ari Blockchain: Architecture, Security and Consensus Algorithms . . . . . . . . . . . . . . . 738 Taher Abouzaid Abdel Aty Abdel Bary, Basem Mohamed Elomda, and Hesham Ahmed Hassan Risk Assessment Fuzzy Logic Approach in Failure Mode and Effects Analysis: Glass Industry Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 Irem Düzdar Argun and Tugce Ozdemir Market Risk Assessment by Expert Knowledge Compilation Using a Fuzzy Maximin Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 Ramin Rzayev, Elmar Aliev, Jamirza Aghajanov, and Inara Rzayeva Decomposed Fuzzy Set-Based Failure Mode and Effects Analysis for Occupational Health and Safety Risk Assessment . . . . . . . . . . . . . . . . . . . . . . . 776 Selcuk Cebi

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A Novel Risk Assessment Approach: Decomposed Fuzzy Set-Based Fine-Kinney Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 Selcuk Cebi and Palanivel Kaliyaperuma Risk Assessment on the Grinding Machine with SWARA/ARAS and Visual PROMETHEE Based on Unstable Fuzzy Linguistic Terms . . . . . . . . . 798 Turgay Duruel and Bahadır Gülsün An Intelligent Fuzzy Functional Resonance Analysis Model on System Safety and Human Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Esmaeil Zarei, Mohammad Yazdi, Brian J. Roggow, and Ahmad BahooToroody Vehicle and Pedestrian Crash Risk Modeling in Arabian Gulf Region . . . . . . . . . 816 Sharaf AlKheder Correction to: The Use of Z-numbers to Assess the Level of Motivation of Employees, Taking into Account Non-formalized Motivational Factors . . . . . Alekperov Ramiz Balashirin

C1

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

Keynote Speeches

Contextual Bipolar Database Queries: A Conjunctive and Disjunctive Perspective Janusz Kacprzyk1,2(B) and Slawomir Zadro˙zny1 1

2

Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01–447 Warsaw, Poland WIT – Warsaw School of Information Technology, Newelska 6, 01–447 Warsaw, Poland [email protected]

Abstract. We extend our approach to bipolar database queries which makes it possible to use a necessary (required, mndatory, obligatory) and optional (desired) condition connected with a non-conventional aggregation operator “and, if possible” exemplified by “find a house which is “inexpensive” and, if possible, located possibly ‘close to public transportation”’ in which “and, if possible,...” is not the traditional “and”. We extend first this formulation by context exemplified by “find houses which are inexpensive and – if possible, with respect to other houses in town – are possibly close to public transportation”. We use a logical representation of the “and, if possible...” operator but also mention some other ones, notably based on the winnow operator. Moreover, we present a new formulation of a contextual bipolar query, “find a house which is inexpensive or - if impossible, with respect to other houses in own – is new”. We mention an inclusive and exclusive character of these two formulations. We present a context awareness related view of these bipolar queries, and provide some remarks on intention awareness. Keywords: database query · bipolar query · context · fuzzy logic user intention · user preference · context awareness · intention awareness

1

·

Introduction

We are concerned with database querying in which a human user intends to find information, here database records (tuples), which represent his/her intentions and preferences. Natural language, which is the only fully natural way of articulation and communication for the humans, is usually employed to pose questions, queries, requests, etc. Here, using examples from real estate, such a query may be “find all houses that are inexpensive and close to public transportation”. Such queries with imprecisely specified (via fuzzy sets) terms have a long tradition, and among more advanced versions one camn mention our query c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 3–10, 2023. https://doi.org/10.1007/978-3-031-39774-5_1

4

J. Kacprzyk and S. Zadro˙zny

based on fuzzy logic with linguistic quantifiers (cf. Kacprzyk and Zi´ olkowski [21], Kacprzyk, Zi´ olkowski and Zadro˙zny [22]). One of the next relevant developments in our contexts is the bipolar queries which are ment to explicitly account for the bipolarity (yes-no, pro-con, . . . ) in human judgments, intentions and preferences (cf. Zadro˙zny and Kacprzyk [32], and later papers). Then, atop of the bipolar queries we include context, in two versions, (cf. Zadro˙zny, Kacprzyk and Dziedzic [35]. This yields what may be termed a context-aware perspective of bipolar querying. Then, we extend it to what may be termed an intention-aware perspective.

2

Bipolar Fuzzy Queries

Historically, the term “bipolar query” was proposed, in our perspective, by Dubois and Prade [7,9]. The very essence is that in a query two types of query conditions are used to express negative and positive user preferences which commonly happen in human discourse and appear through a bipolar scale which specifies: – some degree of being negative, i.e., to be rejected, – some degree of being positive, i.e., to be accepted. noindent and in practice two such bipolar scales are used: – bipolar univariate and – unipolar bivariate, and the in the former there is one scale with three main levels of, respectively, negative, neutral and positive evaluation, gradually changing from one end of the scale to another, usually represented by [−1, 1], and in the latter two independent scales for a positive and negative evaluation, usually represented by [0, 1] are used. We will use the latter one. In our field of fuzzy logic and possibility theory, this topic was considered by many authors, e.g. Dubois and Prade and their collaborators, e.g.: Benferhat, Dubois, Kaci, Prade [8], Dubois and Prade [7,9], cf. also Dubois and Prade [10], Dziedzic, Kacprzyk and Zadrozny [11], Hadjali, Kaci and Prade [12], Lietard and Rocacher [24], Matth´e, De Tr´e, Zadro˙zny, Kacprzyk and Bronselaer [26], etc. The key problem is to properly define the semantics of the negative and positive evaluations (gradually given). In our works we assume that the objects (tuples) with the negative evaluation are rejected and the positive evaluation contributes to the overall evaluation of an object only if it is not rejected. Therefore, even if the positive evaluations play a weaker role, they are equally important as the negative evaluations in the case when there are not rejected objects through the positive evaluations. A good representation is here via a special aggregation operator “and, if possible...”. A bipolar query may generally be written as: C and possibly (i.e. and, if possible)P

(1)

Contextual Bipolar Database Queries

5

exemplified by “find a house which is inexpensive (C) and possibly close to public transportation (P )” meant as that this query is satisfied by a tuple t only if either one of the two conditions holds: 1. it satisfies (of course, possibly to a high degree) both conditions C and P , or 2. it satisfies only C and there is no tuple which satisfies both conditions. The key problem is a proper aggregation method to reflect this “and, if possible . . . ” which is clearly not the traditional “and”. The concept of such a bipolar query appeared first in Lacroix and Lavency [23] who used in the query (C, P ) two conditions: C which stands for what is required (mandatory) and P which stands for what is preferred (desired), meant as: if at least one tuple satisfies both the mandatory and desired condition then the “and, if possible” operator is interpreted as the standard conjunction (“and”) and otherwise only the mandatory condition is taken into account. Such an aggregation operator has been later proposed independently by Dubois and Prade [6] in default reasoning and by Yager [27,28] in multicriteria decision making, cf. also Bordogna and Pasi [2] in information retrieval. Moreover, the bipolar queries with the “and, if possible” operator may be also be viewed as a special case of Chomicki’s [5] queries with preferences which are based on an extra relational algebra operator, the winnow. (cf. Zadro˙zny and Kacprzyk [32]) but this will not be considered here. In Lacroix and Lavency [23], with the crisp (nonfuzzy) conditions C and P , a bipolar query (C, P ) can be processed via the “first select using C then order using P ” strategy, i.e., by finding tuples satisfying C and, second, choosing from among them those satisfying P , if any. Some fuzzifications of the original Lacroix and Lavency’s approach are proposed by Zadro˙zny [29], and Zadro˙zny and Kacprzyk [31,32]; for some other approaches, see Bosc et al. [3], or Lietard. Rocacher and Bosc [25], etc. In our general form of a bipolar query (1), C is the complement of the negative assessment (e.g., “price is inexpensive”), and P – the positive assessment (e.g., located “near public transportation”). Then, the semantics of the bipolar query (1) is: – a tuple t belongs to the answer set of the query (1) if it satisfies (P (t) and C(t) clearly are binary predicates): C(t) and possibly P (t) ≡ C(t) ∧ ∃s(C(s) ∧ P (s)) ⇒ P (t)

(2)

– and if there are tuples satisfying both P and C, then (2) boils down to C ∧ P while otherwise it boils down to C alone. The fuzzification proposed by the authors (cf. Zadro˙zny and Kacprzyk [32] for a comprehensive account) can be done via a direct fuzzification of (2) (with fuzzy predicates): C(t) and possibly P (t) ≡ C(t) ∧ ∧∃s (C(s) ∧ P (s)) ⇒ P (t)

(3)

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J. Kacprzyk and S. Zadro˙zny

and via a direct fuzzification of the winnow operator (cf. Chomicki [5]) but it will not be used here (cf. Zadro˙zny and Kacprzyk [32]). Then, in this formulation one can use a specific form of the conjunction and disjunction, i.e. a t-norm and t-conorm (s-norm), and the negation, i.e. a specific De Morgan Triple. In addition to the “and, if possible...”, which has a clear conjunctive character, in Kacprzyk, Zadro˙zny and Dziedzic [34] w new operator, “or, if impossible...” is proposed, which has a clear disjunctive character. Then, one can define a bipolar query with the “or, if impossible” operator to be written as: P or, if impossible C

(4)

Find houses which are inexpensive or, if impossible, are large

(5)

and exemplified by:

to be interpreted as: if there is no house which is innexpensive, then and only then it is impossible to satisfy the first condition and the answer set comprises houses satisfying just the second condition, i.e., who are large, if any. On the other hand, if there is such a house which is inexpensive, then only such houses are retrieved, and if they are is not important. Therefore, in the former case the query (4) reduces to the condition C alone while in the latter case it reduces to the P a condition lone. This can formally be written, for the fuzzy C and P , with the minimum and maximum operators standing for the, respectively, the conjunction and disjunction, with T (.) denoting the truth values from [0, 1], as: T (P (t) or, if impossible C(t)) = max(µP (t), min(1 − max µP (s), µC (t))) (6) s∈R

Putting it in a different way, the use of “or, if impossible” is meant as to aggregate the negative and positive conditions of a bipolar query (C, P ). That is, for “and, if possible...” the (complement of) negative condition C is a constraint to be satisfied, i.e. mandatory (necessary, obligatory), while the positive condition P is to be satisfied only if possible, i.e., if its satisfaction does not imply a conflict with the satisfaction of C. On the other hand, for the “or, if impossible...” as in (4) we want the positive condition P to be satisfied, and only if this is not possible, then the satisfaction of C matters. Technically, the “and, if possible...” and “or, if impossible...” operators are related to each other (cf. Kacprzyk, Zadro˙zny and Dziedzic [34]).

3

Bipolar Queries Under a Context: An Inclusive and Exclusive Perspective

In general, when we deal with all kinds of human activities, judgements, assessments, etc. context is of utmost importance and a prerequicite for meaningful analyses. In the field of bipolar queries to databases the inclusion of context has

Contextual Bipolar Database Queries

7

provided much momentum and triggered a new class of contextual bipolar query, proposed by Zadro˙zny, Kacprzyk and Dziedzic [34,35]; cf. also Kacprzyk and Zadro˙zny [17], and Kacprzyk and Zadro˙zny [19]. Briefly speaking, for the bipolar query with the required/desired semantics as considered here, i.e. due to (1), in virtually all realistic cases the “and possibly” in (1) is specified with the satisfaction of both C and P to be meant in a certain context. For instance, usually while looking for an inexpensive house it should be taken into account to which part of a city this applies which implies a contextual bipolar query: Find an inexpensive house and possibly

(7)

– with respect to the hotels located in the same region – close to public transportation to be meant to be satisfied (all to a possibly high degree) by a house if: 1. it is inexpensive and close to public transportation, or 2. it is inexpensive and there is no other hotel located in the same region which is both inexpensive and close to public transportation. This new “and possibly + context” operator may be formalized as that the context is equated with a part of the database defined by an additional binary predicate W , i.e., Context(t) = {s ∈ R : W (t, s)} (8) where R denotes the whole database (relation). Therefore, the “and possibly + context” has three arguments: C and possibly P with respect to W with C and P meant as the required and desired conditions, respectively, and W standing for the context. Then, 8 is interpreted as: C(t) and possibly P (t) with respect to W ≡

(9)

≡ C(t) ∧ (∃s(W (t, s) ∧ C(s) ∧ P (s)) ⇒ P (t)) This form is used here though an equivalent winnow operator based form can also be employed (cf. Kacprzyk and Zadro˙zny [19]. Following the reasoning on the bipolar queries with the “and, if possible...” and “or, if impossible...” operators, we can apply the similar arguments now in the case of the contextual bipolar queries. We have shown above how the conjunctive case of “and, if possible...” can be dealt with, and below we will show how to deal with the disjunctive type, i.e. “or, if impossible...”. The generic form of the contextual bipolar query with the “or, if impossible...” operator is: P or, if impossible C with respect to W

(10)

8

J. Kacprzyk and S. Zadro˙zny

and its semantics can be expressed as: P (t) or, if impossible C(t) with respect to W ≡ P (t) ∨ (¬∃s∈R (P (s) ∧ W (t, s)) ∧ C(t))

(11)

and for the minimum and maximum operators standing for, respectively, the conjunction and disjunction, we have the degree of truth T (.): T (P (t) or, if impossible C(t) with respect to W ) = max(µP (t),

(12)

min(1 − max min(µP (s), µW (t, s)), µC (t)))

(13)

s∈R

4

Remarks on a Way from Context Aware to Intention Aware Bipolar Queries

The extension of the concept of a bipolar database query, assumed here to be in a fuzzy logic and possibility theory based setting, can clearly be considered to be an example of a context-aware solution in which a new quality is obtained by the introduction of context as a crucial element of the problem formulation. As a further step, we can extend our solution to the human centric and friendly database querying, which can already provide very powerful tools and techniques to reflect human preferences and intentions. A promising option is here to extend the present solution, which could be termed context-aware, to a qualitatively new setting in which the context awareness is extended into intention awareness that is intended to reduce and facilitate the interaction between the computer system and its user that is of a crucial importance in the case of database querying, exemplified here by a synergistic and proactive interaction between a real estate agent and a customer. More specifically, the challenge is here to try to predict which action the user will take next knowing what he/she has done in the past and now, as well as his/her personal characteristics, cognitive biases, etc. These problems constitute real challenges and may need the use of, for instance, neural networks, decision trees, Bayesian analyses, etc.

5

Conclusions

The bipolar database queries are first presented, mainly from the perspective of the authors’ works, using a fuzzy logic and possibility theory based perspective. Then, a new extension of the bipolar database queries, the contextual bipolar queries are introduced, based on the authors’ settings, in which the elements (conditions) of the bipolar database queries are semantically related to context. Two forms of the introduction of context are proposed, the so called conjunctive one which corresponds to the aggregation of query conditions via “and, if possible...”, to the so called disjunctive one which correspond to the aggregation of query conditions via “or, if impossible...’. finally, a possibility of an extension of the above context awareness of the bipolar queries considered to the intention awareness is mentioned.

Contextual Bipolar Database Queries

9

References 1. Benferhat, S., Dubois, D., Kaci, S., Prade, H.: Modeling positive and negative information in possibility theory. Int. J. Intell. Syst. 23, 1094–1118 (2008) 2. Bordogna, G., Pasi, G.: Linguistic aggregation operators of selection criteria in fuzzy information retrieval. Int. J. Intell. Syst. 10(2), 233–248 (1995) 3. Bosc, P., Pivert, O., Mokhtari, A., Lietard, L.: Extending relational algebra to handle bipolarity. In: Shin, S.Y., Ossowski, S., Schumacher, M., Palakal, M.J., Hung, Ch.-Ch. (eds.) Proceedings of the 2010 ACM Symposium on Applied Computing (SAC), Sierre, Switzerland, pp. 1718–1722. ACM (2010) 4. Bosc, P., Pivert, O.: On four noncommutative fuzzy connectives and their axiomatization. Fuzzy Sets Syst. 202, 42–60 (2012) 5. Chomicki, J.: Preference formulas in relational queries. ACM Trans. Database Syst. 28(4), 427–466 (2003) 6. Dubois, D., Prade, H.: Default reasoning and possibility theory. Artif. Intell. 25(2), 243–257 (1988) 7. Dubois, D., Prade, H.: Bipolarity in flexible querying. In: Carbonell, J.G., Siekmann, J., Andreasen, T., Christiansen, H., Motro, A., Legind Larsen, H. (eds.) FQAS 2002. LNCS (LNAI), vol. 2522, pp. 174–182. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36109-X 14 8. Dubois, D., Prade, H.: An introduction to bipolar representations of information and preference. Int. J. Intell. Syst. 23, 866–877 (2008) 9. Dubois, D., Prade, H.: An overview of the asymmetric bipolar representation of positive and negative information in possibility theory. Fuzzy Sets Syst. 160(10), 1355–1366 (2009) 10. Dubois, D., Prade, H.: Modeling “and if possible” and “or at least”: Different forms of bipolarity in flexible querying. In: Pivert, O., Zadro˙zny, S. (eds.) Flexible Approaches to Data Management, a Volume Dedicated to Patrick Bosc. SCI, vol. 497, pp. 3–19. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-0095441 11. Dziedzic, M., Kacprzyk, J., Zadrozny, S.: Contextual bipolarity and its quality criteria in bipolar linguistic summaries. Tech. Trans. Autom. Control, vol. 4-AC, pp. 117–127 (2014) 12. Hadjali, A., Kaci, S., Prade, H.: Database preference queries - a possibilistic logic approach with symbolic priorities. Ann. Math. Artif. Intell. 63(3–4), 357–383 (2011) 13. Kacprzyk, J., Zadrozny, S.: Hierarchical bipolar fuzzy queries: towards more human consistent flexible queries, FUZZ-IEEE 2013. In: International Conference on Fuzzy Systems (FUZZ-IEEE’2013), pp.1-8 (2013) 14. Kacprzyk, J., Zadrozny, S.: Compound bipolar queries: combining bipolar queries and queries with fuzzy linguistic quantifiers. In: Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-13), University of Milano-Bicocca, Milan, Italy, September 11-13. Atlantis Press, pp. 848–855 (2013) 15. Kacprzyk, J., Zadro˙zny, S.: Compound Bipolar Queries: A Step Towards an Enhanced Human Consistency and Human Friendlinepp. In: Matwin, S., Mielniczuk, J. (eds.) Challenges in Computational Statistics and Data Mining, pp. 93–111. Springer, Cham (2016) 16. Kacprzyk, J., Zadro˙zzny, S.: Compound bipolar queries: the case of data with a variable quality. In: Proceedings of FUZZ-IEEE 2017, IEEE International Conference on Fuzzy Systems, Naples, Italy, 9–12 July, 2017, pp. 1–6. IEEE Press (2017)

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17. Kacprzyk, J., Zadro˙zny, S.: Towards a hierarchical extension of contextual bipolar queries. In: Medina, J., Ojeda-Aciego, M., Verdegay, J.L., Pelta, D.A., Cabrera, I.P., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2018. CCIS, vol. 854, pp. 63–74. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91476-3 6 18. Kacprzyk, J., Zadro˙zzny, S.: Compound bipolar queries: the case of data with a variable quality. In: Proceedings of FUZZ-IEEE 2019, IEEE International Conference on Fuzzy Systems, New Orleans, USA, 23–26 June, 2019, pp. 1–6. IEEE (2019) 19. Kacprzyk, J., Zadrozny, S.: A novel approach to hierarchical contextual bipolar queries: a winnow operator approach. Control. Cybern. 51(2), 267–283 (2022) 20. Kacprzyk, J., Zadro˙zny, S., Dziedzic, M.: A novel view of bipolarity in linguistic data summaries. In: K´ oczy, L.T., Pozna, C.R., Kacprzyk, J. (eds.) Issues and Challenges of Intelligent Systems and Computational Intelligence. SCI, vol. 530, pp. 215–229. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-03206-1 16 21. Kacprzyk, J., Zi´ olkowski: Database queries with fuzzy linguistic quantifiers. IEEE Trans. Syst. Man Cybern. SMC 16, 474 479 (1986) 22. Kacprzyk, J., Zadro˙zny, S., Zi´ olkowski, A.: FQUERY III+: a “human consistent” database querying system based on fuzzy logic with linguistic quantifiers. Inf. Syst.14(6), 443–453 (1989) 23. Lacroix, M., Lavency, P.: Preferences: putting more knowledge into queries. In: Proceedings of the 13 Int. Conf. on Very Large Databases, pp. 217–225 (1987) 24. Lietard, L., Rocacher, D.: On the definition of extended norms and co-norms to aggregate fuzzy bipolar conditions. In: Carvalho, J.P., Dubois, D., Kaymak, U., da Costa Sousa, J.M. (eds.) Proceedings of the Joint 2009 IFSA/EUSFLAT, Lisbon, Portugal, pp. 513–518 (2009) 25. Lietard, L., Rocacher, D., Bosc, P.: On the extension of SQL to fuzzy bipolar conditions. In: Proceedings of NAFIPS-2009 Conference, pp. 1–6 (2009) 26. Matth´e, T., De Tr´e, G., Zadro˙zny, S., Kacprzyk, J., Bronselaer, A.: Bipolar database querying using bipolar satisfaction degrees. Int. J. Intell. Syst. 26(10), 890–910 (2011) 27. Yager, R.: Higher structures in multi-criteria decision making. Int. J. Man Mach. Stud. 36, 553–570 (1992) 28. Yager, R.: Fuzzy logic in the formulation of decision functions from linguistic specifications. Kybernetes 25(4), 119–130 (1996) 29. Zadro˙zny, S.: Bipolar queries revisited. In: Torra, V., Narukawa, Y., Miyamoto, S. (eds.) MDAI 2005. LNCS (LNAI), vol. 3558, pp. 387–398. Springer, Heidelberg (2005). https://doi.org/10.1007/11526018 38 30. Zadro˙zny, S., De Tr´e, G., Kacprzyk, J.: Remarks on various aspects of bipolarity in database querying. In: Proceedings of DEXA’10, Bilbao, Spain, pp. 323–327. IEEE (2010) 31. Zadro˙zny, S., Kacprzyk, J.: Bipolar queries and queries with preferences. In: Proceedings of DEXA’06, pp. 415–419 (2006) 32. Zadro˙zny, S., Kacprzyk, J.: Bipolar queries: an aggregation operator focused perspective. Fuzzy Sets Syst. 196, 69–81 (2012) 33. Zadro˙zny, S., Kacprzyk, J., De Tr´e, G.: Bipolar queries in textual information retrieval: a new perspective. Inf. Process. Manage. 48(3), 390–398 (2012) 34. Zadro˙zny, S., Kacprzyk, J., Dziedzic, M.: Contextual bipolar queries. In: Proceedings of IFSA/EUSFLAT 2015, Gijon, Spain, pp. 1266–1273. Atlantis Press 35. Zadro˙zny, S., Kacprzyk, J., Dziedzic, M., De Tr´e, G.: Contextual bipolar queries. In: Jamshidi, M., Kreinovich, V., Kacprzyk, J. (eds.) Advance Trends in Soft Computing. SFSC, vol. 312, pp. 421–428. Springer, Cham (2014). https://doi.org/10. 1007/978-3-319-03674-8 40

Intuitionistic Fuzzy Modal Topological Structures Based on Two New Intuitionistic Fuzzy Modal Operators Krassimir Atanassov1,2(B) 1 Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, 105 “Acad. Georgi Bonchev” Str., 1113 Sofia, Bulgaria 2 Intelligent Systems Laboratory, Prof. Dr. Assen Zlatarov University, 1 “Prof. Yakimov” Blvd., 8010 Burgas, Bulgaria [email protected]

Abstract. In the present paper, four Intuitionistic Fuzzy Modal Topological Structures (IFMTSs) are described and some of their properties are discussed. While in the first paper over an IFMTS the modal operators were the simplest (standard) ones, here particular cases of three of the extended modal operators over intuitionistic fuzzy sets are used. Some properties of the new operators are discussed. Keywords: intuitionistic fuzzy modal operator · intuitionistic fuzzy set · intuitionistic fuzzy topological operator · intuitionistic fuzzy topological structure AMS: 03E72

1

Introduction

The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced in [2]. It was based on the definitions by Kazimierz Kuratowski (1896–1980) proposed in [6] for topological structures that satisfy the conditions: C1 C2 C3 C4

cl(AΔB) = cl(A)Δcl(B), A ⊆ cl(A), cl(cl(A)) = cl(A), cl(O) = O,

where A, B ∈ X, X is some fixed set of sets with a minimal element O, cl is the topological operator “closure” and Δ : X × X → X is the operation that generates cl; and I1 I2 I3 I4

in(A∇B) = in(A)∇in(B), in(A) ⊆ A, in(in(A)) = in(A), in(I) = I,

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 11–21, 2023. https://doi.org/10.1007/978-3-031-39774-5_2

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K. Atanassov

where A, B ∈ X, X is the same set and I is its maximal element, in is the topological operator “interior” and ∇ : X × X → X is the operation that generates in (see also [10]). In the Intuitionistic Fuzzy Sets (IFSs) theory there are some types of operators – modal, topological, level and others. The present paper is a modification of the ideas of [2]. Here, the standard modal operators are changed with new intuitionistic fuzzy modal operators that are particular cases of three of the extended intuitionistic fuzzy modal operators. Later, the operators that satisfy the conditions C1–C4 will be termed as “operators of closure type” and those satisfying the conditions I1–I4 will be termed as “operators of interior type”.

2

Short Remarks on Intuitionistic Fuzzy Sets

IFSs are one of the early extensions of fuzzy sets proposed by Lotfi Zadeh (1921– 2017) [14]. When a set E, called “universum” is fixed and A is its subset, each IFS in E has the form: A∗ = {x, μA (x), νA (x)|x ∈ E}, where the functions μA : E → [0, 1] and νA : E → [0, 1] define, respectively, the degree of membership and the degree of non-membership of the element x ∈ E to the set A ⊆ E, and for each x ∈ E: 0 ≤ μA (x) + νA (x) ≤ 1. As usual, instead of A∗ for brevity, the notation A is used. Over the IFSs a lot of operations, relations and operators are defined (see [1]). The most utilized among them, which we will need below, are A⊆B A⊇B A=B ¬A A∩B A∪B

iff iff iff = = =

(∀x ∈ E)(μA (x) ≤ μB (x) & νA (x) ≥ νB (x)); B ⊆ A; (∀x ∈ E)(μA (x) = μB (x) & νA (x) = νB (x)); {x, νA (x), μA (x)|x ∈ E}; {x, min(μA (x), μB (x)), max(νA (x), νB (x))|x ∈ E}; {x, max(μA (x), μB (x)), min(νA (x), νB (x))|x ∈ E}.

The first two (simplest) analogues of the topological operators “closure” and “interior” (defined over IFSs) are introduced in [1] as follows: C(A) = {x, sup μA (y), inf νA (y)|x ∈ E}, y∈E

y∈E

y∈E

y∈E

I(A) = {x, inf μA (y), sup νA (y)|x ∈ E}. Now, over IFSs a lot of modal operators are defined. Three of them are the following (see [1]). Gα,β (A) = {x, αμA (x), βνA (x)|x ∈ E}, Hα,β (A) = {x, αμA (x), νA (x) + βπA (x)|x ∈ E}, Jα,β (A) = {x, μA (x) + απA (x), βνA (x)|x ∈ E},

Intuitionistic Fuzzy Modal Topological Structures

13

where A is an IFS and α, β ∈ [0, 1]. For the needs of the present research, we will use particular cases of operators Hα,β and Jα,β that we will introduce below and in the next Section we will study their behaviour.

3

Two New Modal Operators

Here, we define the two (particular) modal operators Hα# (A) = {x, α.μA (x), νA (x)|x ∈ E}, Jα# (A) = {x, μA (x), α.νA (x)|x ∈ E}, where A is an IFS and α ∈ [0, 1]. We see immediately that for each IFS A and for each α ∈ [0, 1]: Hα# (A) = Hα,0 (A), Jα# (A) = J0,α (A). The geometrical interpretation of both operators is shown on Fig. 1., where Hα# (x) and Jβ# (x) are the results of the applying of the two operators over element x ∈ E.

0, 1

Hα# (x) νA (x)

x Jβ# (x)

βνA (x)

0, 0

μA (x)

1, 0

αμA (x) Fig. 1. The geometrical interpretation of Hα# (x) and Jβ# (x).

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K. Atanassov

For both operators we see that the following equalities are valid for each IFS A and for each α ∈ [0, 1]: ¬Jα# (¬A) = ¬Jα# ({x, νA (x), μA (x)|x ∈ E}) = ¬{x, νA (x), αμA (x)|x ∈ E}) = {x, αμA (x), νA (x)|x ∈ E}) = Hα# (A) and by analogy, ¬Hα# (¬A) = Jα# (A). Moreover, for each IFS A and for each α ∈ [0, 1]: Hα# (A) = Gα,1 (A), Jα# (A) = G1,α (A).

4

Four New Intuitionistic Fuzzy Modal Topological Structures

Following [1], let us define: O∗ = {x, 0, 1|x ∈ E}, E ∗ = {x, 1, 0|x ∈ E}. Therefore, for each IFS A: O∗ ⊆ A ⊆ E ∗ . If for each set X

P(X) = {Y |Y ⊆ X},

then for each IFS A over the universe E: P(O∗ ) = O∗ , P(E ∗ ) = {A|A ⊆ E ∗ }. Let O and Q be topological operators such that for each IFS A ∈ P(E ∗ ): O(A) = ¬Q(¬A), Q(A) = ¬O(¬A). Let Δ, ∇ : P(E ∗ ) × P(E ∗ ) → P(E ∗ ) be operations that for every two IFSs A, B ∈ P(E ∗ ): A∇B = ¬(¬AΔ¬B), AΔB = ¬(¬A∇¬B).

Intuitionistic Fuzzy Modal Topological Structures

15

Let ◦ and • be modal operators such that for each IFS A ∈ P(E ∗ ): ◦A = ¬ • ¬A, •A = ¬ ◦ ¬A. By analogy with [2], and extending the definitions from there, we will introduce four new Intuitionistic Fuzzy Modal Topological Structures that use the two new intuitionistic fuzzy modal operators: cl-cl-IFMTS, in-in-IFMTS, cl-inIFMTS and in-cl-IFMTS. 4.1

cl-cl-IFMTS

The cl-cl-IFMTS is the object P(E ∗ ), O, Δ, ◦, where E is a fixed universe, O : P(E ∗ ) → P(E ∗ ) is an operator of a closure type related to operation Δ; ◦ : P(E ∗ ) → P(E ∗ ) is a modal operator and for every two IFSs A, B ∈ P(E ∗ ) the following nine conditions hold: CC1 CC2 CC3 CC4 CC5 CC6 CC7 CC8 CC9

O(AΔB) = O(A)ΔO(B), A ⊆ O(A), O(O∗ ) = O∗ , O(O(A)) = O(A), ◦(A∇B) = ◦A∇ ◦ B, A ⊆ ◦A, ◦E ∗ = E ∗ , ◦ ◦ A = ◦A, ◦ O(A) = O(◦A).

Now, we see that the first four conditions correspond to the conditions C1 – C4 for the topological operator “closure”, that satisfies the condition CC2, next four conditions correspond to the conditions C1 – C4, too, but for a modal operator that satisfies the condition CC6, i.e., it is in some sense from a “closure” type and by this reason we use letters “CC” in the notation of these conditions; and condition CC9 determines the relation between the two types of operators. Theorem 1. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), C, ∪, Jα#  is a cl-cl-IFMTS. Proof. Let the IFSs A, B ∈ P(E ∗ ) and α ∈ [0, 1], be given. The checks of conditions CC1 – CC4 are given in [2], but we will give them only for completeness of the present proof.

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CC1. C(A ∪ B) = C({x, μA (x), νA (x)|x ∈ E} ∪ {x, μB (x), νB (x)|x ∈ E}) = C({x, max(μA (x), μB (x)), min(νA (x), νB (x))|x ∈ E}) = {x, sup max(μA (y), μB (y)), inf min(νA (y), νB (y))|x ∈ E} y∈E

y∈E

= {x, max( sup μA (y), sup μB (y)), min( inf νA (y), inf νB (y))|x ∈ E} y∈E

CC2.

y∈E

y∈E

= C(A) ∪ C(B);

y∈E

A = {x, μA (x), νA (x)|x ∈ E} ⊆ {x, sup μA (y), inf νA (y)|x ∈ E} y∈E

y∈E

= C(A); CC3.

C(O∗ ) = C({x, 0, 1|x ∈ E}) = {x, sup 0, inf 1|x ∈ E} y∈E

y∈E

= {x, 0, 1|x ∈ E} = O∗ ; CC4. Having in mind that sup μA (y) and inf νA (y) are constants, we obtain y∈E

y∈E

that: C(C(A)) = C({x, sup μA (y), inf νA (y)|x ∈ E}) y∈E

y∈E

= {x, supz∈E sup μA (y), inf z∈E inf νA (y)|x ∈ E} y∈E

y∈E

= {x, sup μA (y), inf νA (y)|x ∈ E} y∈E

y∈E

= C(A); CC5.

Jα# (A ∩ B) = Jα# ({x, min(μA (x), μB (x)), max(νA (x), νB (x))|x ∈ E}) = {x, min(μA (x), μB (x)), α max(νA (x), νB (x))|x ∈ E} = {x, min(μA (x), μB (x)), max(ανA (x), ανB (x))|x ∈ E} = {x, μA (x), ανA (x)|x ∈ E} ∩ {x, μA (x), ανA (x)|x ∈ E} = Jα# (A) ∩ Jα# (B); CC6.

CC7.

A = {x, μA (x), νA (x)|x ∈ E} ⊆ {x, μA (x), ανA (x)|x ∈ E} = Jα# (A); Jα# (E ∗ ) = Jα# ({x, 1, 0|x ∈ E}) = {x, 1, α.0|x ∈ E} = ({x, 1, 0|x ∈ E}) = E∗;

Intuitionistic Fuzzy Modal Topological Structures

17

CC8. Let α, β ∈ [0, 1]. Then Jα# (Jβ# (A) = Jα# ({x, μA (x), βνA (x)|x ∈ E}) = {x, μA (x), αβνA (x)|x ∈ E} # (A); = Jαβ CC9.

Jα# (C(A)) = Jα# ({x, sup μA (y), inf νA (y)|x ∈ E}) y∈E

y∈E

= {x, sup μA (y), α inf νA (y)|x ∈ E} y∈E

y∈E

= {x, sup μA (y), inf ανA (y)|x ∈ E} y∈E

y∈E

= C({x, μA (y), ανA (y)|x ∈ E}) = C(Jα# (A)).  

This completes the proof.

Corollary 1. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), C, ∪, G1,α  is a cl-cl-IFMTS. 4.2

in-in-IFMTS

The in-in-IFMTS is the object P(E ∗ ), Q, ∇, •, where E is a fixed universe, Q : P(E ∗ ) → P(E ∗ ) is an operator of interior type that satisfies the condition II2, related to the operation ∇; • : P(E ∗ ) → P(E ∗ ) is a modal operator satisfying the condition II6, i.e., it is from an “interior” type (and by this reason we use letters “II”), and for every two IFSs A, B ∈ P(E ∗ ) the following nine conditions hold: II1 II2 II3 II4 II5 II6 II7 II8 II9

Q(A∇B) = Q(A)∇Q(B), Q(A) ⊆ A, Q(E ∗ ) = E ∗ , Q(Q(A)) = Q(A), •(AΔB) = •AΔ • B, •A ⊆ A, •O∗ = O∗ , • • A = •A, • Q(A) = Q(•A).

We see that the first four conditions correspond to the conditions I1 – I4 for the topological operator “interior”, next four conditions correspond to the conditions I1 – I4, too, but for a modal operator of interior type, and condition II9 determines the relation between the two types of operators.

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K. Atanassov

Theorem 2. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), I, ∩, Hα#  is an in-in-IFMTS. Proof. Let the IFSs A, B ∈ P(E ∗ ) be given. The checks of conditions II1 – II4 are given in [2] and are similar to the above ones. So, we will check only the validity of conditions II5 – II9. II5. Hα# (A ∪ B) = Hα# ({x, max(μA (x), μB (x)), min(νA (x), νB (x))|x ∈ E}) = {x, α max(μA (x), μB (x)), min(νA (x), νB (x))|x ∈ E} = {x, max(αμA (x), αμB (x)), min(νA (x), νB (x))|x ∈ E} = {x, αμA (x), νA (x)|x ∈ E} ∪ {x, αμA (x), νA (x)|x ∈ E} = Hα# (A) ∪ Hα# (B); II6.

II7.

Hα# (A) = {x, αμA (x), νA (x)|x ∈ E} ⊆ {x, μA (x), νA (x)|x ∈ E} = A; Hα# (O∗ ) = Jα# ({x, 0, 1|x ∈ E}) = {x, α.0, 1|x ∈ E} = ({x, 0, 1|x ∈ E}) = O∗ ;

II8. Let α, β ∈ [0, 1]. Then Hα# (Hβ# (A) = Hα# ({x, βμA (x), νA (x)|x ∈ E}) = {x, αβμA (x), νA (x)|x ∈ E} # (A); = Hαβ II9.

Jα# (C(A)) = Jα# ({x, sup μA (y), inf νA (y)|x ∈ E}) y∈E

y∈E

= {x, sup μA (y), α inf νA (y)|x ∈ E} y∈E

y∈E

= {x, sup μA (y), inf ανA (y)|x ∈ E} y∈E

y∈E

= C({x, μA (y), ανA (y)|x ∈ E}) = C(Jα# (A)). This completes the proof.

 

Corollary 2. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), I, ∩, Gα,1  is an in-in-IFMTS.

Intuitionistic Fuzzy Modal Topological Structures

4.3

19

cl-in-IFFM2TS

The cl-in-IFFM2TS is the object P(E ∗ ), O, Δ, •, where E is a fixed universe, O : P(E ∗ ) → P(E ∗ ) is an operator of closure type related to operation Δ; • : P(E ∗ ) → P(E ∗ ) is a modal operator of interior type, and for every two IFSs A, B ∈ P(E ∗ ) the following nine conditions hold: CI1 CI2 CI3 CI4 CI5 CI6 CI7 CI8 CI9

O(AΔB) = O(A)ΔO(B), A ⊆ O(A), O(O∗ ) = O∗ , O(O(A)) = O(A), •(A∇B) = •A∇ • B, •A ⊆ A, •O∗ = O∗ , • • A = •A, • O(A) = O(•A).

Now, we see that the first four conditions coincide with the conditions CC1 – CC4 from Subsect. 4.1 and the conditions CI6 – CI8 coincide with conditions II6 – II8 from Subsect. 4.2. Theorem 3. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), C, ∪, Hβ#  is a cl-in-IFMTS. Proof. Let the IFS A ∈ P(E ∗ ) be given. The checks of the conditions CI1–CI4 coincide with the proofs of conditions CC1–CC4 in Theorem 1. The checks of the conditions CI6 – CI8 coincide with the proofs of conditions II6–II8. Hence, it remains that we check only the validity of the conditions CI5 and CI9: Hα# (A ∩ B) = Hα# ({x, min(μA (x), μB (x)), max(νA (x), νB (x))|x ∈ E}) = {x, α min(μA (x), μB (x)), max(νA (x), νB (x))|x ∈ E} = {x, min(αμA (x), αμB (x)), max(νA (x), νB (x))|x ∈ E} = {x, αμA (x), νA (x)|x ∈ E} ∩ {x, αμA (x), νA (x)|x ∈ E} = Hα# (A) ∩ Hα# (B); and

Hα# (C(A)) = Hα# ({x, sup μA (y), inf νA (y)|x ∈ E}) y∈E

y∈E

= {x, α sup μA (y), inf νA (y)|x ∈ E} y∈E

y∈E

= {x, sup αμA (y), inf νA (y)|x ∈ E} y∈E

y∈E

= C({x, αμA (y), νA (y)|x ∈ E}) = C(Hα# (A)). This completes the proof.

  ∗

Corollary 3. For each universe E and for each real number α ∈ [0, 1], P(E ), C, ∩, Gα,1  is an cl-in-IFMTS.

20

4.4

K. Atanassov

in-cl-Intuitionistic Fuzzy Level Topological Structure

The in-cl-IFLTS is the object P(E ∗ ), Q, ∇, ◦, where E is a fixed universe, Q : P(E ∗ ) → P(E ∗ ) is an operator of interior type, related to the operation ∇, ◦ : P(E ∗ ) → P(E ∗ ) is a level operator of closure type, and for every two IFSs A, B ∈ P(E ∗ ) the following nine conditions hold: IC1 IC2 IC3 IC4 IC5 IC6 IC7 IC8 IC9

Q(A∇B) = Q(A)∇Q(B), Q(A) ⊆ A, Q(E ∗ ) = E ∗ , Q(Q(A)) = Q(A), ◦(AΔB) = ◦AΔ ◦ B, A ⊆ ◦A, ◦E ∗ = E ∗ , ◦ ◦ A = ◦A, ◦ Q(A) = Q(◦A).

Now, we see that the first four conditions correspond to the conditions I1 – I4 for the topological operator “interior”, the next four conditions correspond to the conditions C1 – C4, but for a modal topological operator of closure type, and condition IC9 determines the relation between the two types of operators. Theorem 4. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), I, ∩, Jα#  is an in-cl-IFLTS. The proof is similar to the previous ones. Corollary 4. For each universe E and for each real number α ∈ [0, 1], P(E ∗ ), I, ∩, G1,α  is an in-cl-IFLTS.

5

Conclusion

As it was mentioned in [2], the idea for IFMTSs opens some directions for future research, that are extensions of the IFTSs. The standard research of the IFTSs started with [4,5,7–9,11–13], in reality, the first step in the direction of intuitionistic fuzzy temporal topological structures was published in [3]. Another direction is related to intuitionistic fuzzy level topological structures, based on the intuitionistic fuzzy level operators. Already there are applications of the discussed topological structures in the area of decision making procedures. Other possible applications, that will be object of separate research will be directed to the areas of graph theory, intercriteria analysis, artificial intelligence, etc. Acknowledgements. This research was realized in the frames of project KP-06-N221/2018 “Theoretical research and applications of InterCriteria Analysis”.

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References 1. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 2. Atanassov, K.: Intuitionistic fuzzy modal topological structure. Mathematics, 10, 3313 (2022). https://doi.org/10.3390/math10183313 3. Atanassov, K.: On intuitionistic fuzzy temporal topological structures. Axioms 12, 182 (2023). https://doi.org/10.3390/axioms12020182 4. C ¸ oker, D.: On topological structures using intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets 3(5), 138–142 (1997) 5. C ¸ oker, D.: An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets Syst. 88, 81–89 (1997) 6. Kuratowski, K.: Topology, vol. 1. Academic Press, New York (1966) 7. Lupia˜ nez, F.G.: Separation in intuitionistic fuzzy topological spaces. Int. J. Pure Appl. Math. 17(1), 29–34 (2004) 8. Lupia˜ nez, F.G.: On intuitionistic fuzzy topological spaces. Kybernetes 35(5–6), 743–747 (2006) 9. Lee, S.J., Lee, E.P.: The category of intuitionistic fuzzy topological spaces. Bull. Korean Math. Soc. 37, 63–76 (2000) 10. Munkres, J.: Topology. Prentice Hall Inc., New Jersey (2000) ¨ 11. Ozbakir, O., C ¸ oker, D.: Fuzzy multifunctions in intuitionistic fuzzy topological spaces. Notes Intuitionistic Fuzzy Sets 5(3), 1–5 (1999) 12. Saadati, R., Park, J.H.: On the intuitionistic fuzzy topological spaces. Chaos, Solitons Fractals 27, 331–334 (2006) 13. Thakur, S., Chaturvedi, R.: Generalized continuity in intuitionistic fuzzy topological spaces. Notes Intuitionistic Fuzzy Sets 12(1), 38–44 (2006) 14. Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

Multi-scale Dimensionality Reduction with F-Transforms in Time Series Analysis Irina Perfilieva(B) Institute for Research and Applications of Fuzzy Modeling, Centre of Excellence IT4Innovations, University of Ostrava, 30. dubna 22, Ostrava, Czech Republic [email protected] Abstract. Our first contribution to this topic is as follows: we show that in the case of large datasets, dimensionality reduction should be divided into several subtasks, determined by the choice of keypoints as centers corresponding to clusters. For specific time series datasets, we connect keypoints to centers that maximize the values of the non-local Laplacians. Moreover, we propose to use the scale space approach and consider a scale-dependent sequence of non-local Laplacians. As a second contribution, we use non-traditional kernels obtained from the theory of F-transforms [11]. This allows to simplify the scaling and selection of keypoints, reduce their number and increase reliability. We also propose a new keypoint descriptor and test it against high volatility financial time series.

Keywords: Multi-scale representation Fuzzy transform

1

· Keypoint · Fuzzy partition ·

Introduction

Modeling and processing large data bases (texts, images, video signals, cash flows, etc.) motivate applying machine learning theory and algorithms. “Big data” mining is based on discovering structured knowledge from spatiotemporally correlated data. The first step is a certain granulation of data, which means developing a low dimensional representation of data that arises from sampling a complex high dimensional data. The generic problem of dimensionality reduction is to find a set of points y1 , . . . , yk in the space Rm such that yi “represents” a point xi from the given dataset x1 , . . . , xk that belongs to the space Rl with the substantially larger dimension so that m  l. Let us give some remarks regarding the history of the problem of dimensionality reduction, see [1]. Classical approaches include principal components analysis (PCA) and multidimensional scaling. Various methods that generate nonlinear maps have also been considered. Most of them, such as self-organizing maps and other neural network-based approaches (e.g., [5]), set up a nonlinear c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 22–34, 2023. https://doi.org/10.1007/978-3-031-39774-5_3

FzT in Dimensionality Reduction

23

optimization problem whose solution is typically obtained by gradient descent that is guaranteed only to produce a local optimum; global optima are difficult to attain by efficient means. To our knowledge, the approach of generalizing the PCA through kernel-based techniques in [15] does not have this shortcoming. However, most of methods in [15] do not explicitly consider the structure of the manifold (space Rl ) on which the data may possibly lie. In [1], an approach that builds a graph incorporating neighborhood information of the data set is proposed. Using the notion of the Laplacian of the graph, a low-dimensional representation of the data set that optimally preserves local neighborhood information is computed. The representation map generated by the proposed algorithm may be viewed as a discrete approximation to a continuous map that naturally arises from the geometry of the manifold. The most important feature of the solution consists in reflecting the intrinsic geometric structure of the manifold. The latter is approximated by the adjacency graph computed from the data points. In fuzzy literature, the dimensionality reduction was hidden under the notions and techniques of granulation, clustering and fuzzy partition, [17]. The results were used in the form of collections of fuzzy sets and after that in fuzzy rules databases. Actually, the main advantage of modeling with fuzzy IF-THEN rules is in transforming a problem from an initial complex high dimensional data space to the low dimensional space of fuzzy sets that are new atomic units. However, despite of this obvious similarity, the dimensionality reduction in the sense of machine learning is different. The difference is in the way of representation. Instead of center-shape (clustering) or membership function (fuzzy sets) representation, low-dimensional images are characterized in terms of features (eigenvectors). Therefore, a cluster (granule) is characterized as a collection of common features that are extracted from an initial data embedded into a particular manifold. In this respect, the only fuzzy technique which is similar to the machinelearning-based dimensionality reduction is the F -(fuzzy) transform. The distinguished property of the F -transform is that it uses a fuzzy partition as a space characterization of data where the notion of closeness is determined by membership functions of partition units – basic functions. The extracted features are the so called (functionally expressed) F -transform components that are weighed projections on elementary functions (polynomials) where weights are fuzzy partition units. Below, we show how this is connected with eigenvectors that arise in the transformation of high to low-dimensional representation. In the theoretical part of this contribution (Sects. 3–5), we show that the technique of F -transforms (“F ” stands for fuzzy) fully agrees with the technique in [1]. The theory of fuzzy (F)-transforms provides a (dimensional) reduced and robust representation of original data. The main characteristics with respect to input data: size reduction, noise removal, invariance to geometrical transformations, knowledge transfer from conventional mathematics, fast computation. The F -transform has been applied to: image processing, computer vision, online pattern recognition in big data bases, time series analysis and forecasting, mathematical finance, numerical methods for differential equations, deep learning neural networks.

24

I. Perfilieva

To justify the main claim, we characterize the processed by the F -transform data in terms of the adjacency graph that reflects their intrinsic geometry. In the application part (Sect. 6), we consider the problem of time series analysis and explain how it can be properly formulated and solved in the language of F -transform and using the proposed technique of dimensionality reduction. Our first contribution to this topic is as follows: we show that in the case of large datasets, dimensionality reduction should be divided into several subtasks, determined by the choice of key points as centers corresponding to clusters. For specific time series datasets, we connect key points to centers that maximize the values of the non-local Laplacians. Moreover, we propose to use the scale space approach and consider a scale-dependent sequence of non-local Laplacians. As a second contribution, we use non-traditional kernels obtained from the theory of F-transforms [11]. This allows to simplify the scaling and selection of key points, reduce their number and increase reliability. We also propose a new key point descriptor and test it against high volatility financial time series. The main theoretical result that we arrive at here is that the Gaussian kernel as the predominant in the scale-space theory can be replaced with the same success by a special symmetric positive semi-definite kernel with a local support. In particular, we show that generating function of a triangular-based uniform fuzzy partition of R can be used for determining such kernel. This fact allows us to base upon the theory of F-transforms and its ability to extract features (keypoints) with a clear understanding of their semantic meaning [12].

2

Briefly About the Theory of Scale-Space Representations

We start with a brief overview of the mentioned theory because it explains the proposed methods. Perhaps, the quad-tree methodology [6] is the first type of multi-scale representation of image data. It focuses on recursively dividing an image into smaller areas controlled by the intensity range. A low-pass pyramid representation was proposed in [4], where the added benefit to multi-scaling was that the image size decreased exponentially compared to scale level. Koenderink [7] emphasized that scaling up and down the internal scope of observations and handling image structures at all scales (in accordance with the task) contribute to a successful image analysis. The challenge is to understand the image at all relevant scales at the same time, but not as an unrelated set of derived images at different levels of blur. The basic idea (in Lindeberg [8]) how to obtain a multi-scale representation of an object is to embed it into a one-parameter family of gradually smoothed ones where fine-scale details are sequentially suppressed. Under fairly general conditions, the author showed that the Gaussian kernel and its derivatives are the only possible smoothing kernels. These conditions are mainly linearity and shift invariance, combined with various ways of formalizing the notion that structures on a coarse scale should correspond to simplifications of corresponding structures on a fine scale.

FzT in Dimensionality Reduction

25

A scale-space representation differs from a multi-scale representation in that it uses the same spatial sampling at all scales and one continuous scale parameter as the generator. By the construction in Witkin [16], a scale-space representation is a one-parameter family of derived signals constructed using convolution with a one-parameter family of Gaussian kernels of increasing width. Formally, a scale-space family of a continuous signal is constructed as follows. For a signal f : RN → R, the scale-space representation L : RN × R+ → R is defined by: L(·, 0) =f (·), L(·, t) =g(·, t)  f,

(1)

where t ∈ R+ is the scale parameter and g : RN × R+ → R is the Gaussian kernel as follows: N  x2i 1 . g(x, t) = exp − N/2 2t (2πt) i=1 The scale parameter t relates to the standard deviation of the kernel g, and is a natural measure of spatial scale at the level t. As an important remark, we note that the scale-space family L can be defined as the solution to the diffusion (heat) equation ∂t L =

1 T ∇ ∇L, 2

(2)

with initial condition L(·, 0) = f . The Laplace operator, ∇T ∇ or Δ, the divergence of the gradient, is taken in the spatial variables. The solution to (2) in one-dimension and in the case where the spatial domain is R is known as the convolution () of f (initial condition) and the fundamental solution: L(·, t) =g(·, t)  f,

(3) 2

x 1 exp − . g(x, t) = √ 2t ( 2πt)

(4)

The following two questions arise: is this approach the only reasonable way to perform low-level processing, and are Gaussian kernels and their derivatives the only smoothing kernels that can be used? Many authors [7,8,16] answer these questions positively, which leads to the default choice of Gaussian kernels in most image processing tasks. In this article, we want to expand on the set of useful kernels suitable for performing scale-space representations. In particular, we propose to use kernels arising from generating functions of fuzzy partitioning.

3

Space with a Fuzzy Partition

In this section, we introduce space that plays an important role in our research. A space with a fuzzy partition is considered as a space with a proximity (closeness)

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relation, which is a weak version of a metric space. Our goal is to show that the diffusion (heat conduction) equation in (2) can be extended to spaces with closeness, where the concepts of derivatives are adapted to nonlocal cases. Let us first recall the basic definitions. As we indicated at the beginning, our goal is to extend the Laplace operators to those that take into account the specifics of spaces with fuzzy partitions. For this reason, in the following sections, we recall the basic concepts on this topic. 3.1

Fuzzy Partition

Definition 1: Fuzzy sets A1 , . . . , An : [a, b] → R, establish a fuzzy partition of the real interval [a, b] with nodes a = x1 < . . . < xn = b, if for all k = 1, . . . , n, the following conditions are valid (we assume x0 = a, xn+1 = b): 1. 2. 3. 4.

Ak (xk ) = 1, Ak (x) > 0 if x ∈ (xk−1 , xk+1 ); Ak (x) = 0 if x ∈ (xk−1 , xk+1 ); Ak (x) is continuous, Ak (x), for k = 2, . . . , n, strictly increases on [xk−1 , xk ] and Ak (x) strictly decreases on [xk , xk+1 ] for k = 1, . . . , n − 1,

The membership functions A1 , . . . , An are called basic functions [11]. Definition 2: The fuzzy partition A1 , . . . , An , where n ≥ 2, is h-uniform if nodes x1 < · · · < xn are h-equidistant, i.e. for all k = 1, . . . , n−1, xk+1 = xk +h, where h = (b − a)/(n − 1), and the following additional properties are fulfilled [11]: 1. for all k = 2, . . . , n − 1 and for all x ∈ [0, h], Ak (xk − x) = Ak (xk + x), 2. for all k = 2, . . . , n − 1, and for all x ∈ [xk , xk+1 ], Ak (x) = Ak−1 (x − h), and Ak+1 (x) = Ak (x − h). Proposition 1: If the fuzzy partition A1 , . . . , An of [a, b] is h-uniform, then there exists an even function A0 : [−1, 1] → [0, 1], such that for all k = 1, . . . , n:   x − xk Ak (x) = A0 , x ∈ [xk−1 , xk+1 ]. h A0 is called a generating function of uniform fuzzy partition [11]. Remark 1. Generating function Ah (x) = A0 (x/h) of an h-uniform fuzzy partition produces the corresponding to it kernel Ah (x−y) and the normalized kernel 1 h Ah (x − y), so that for all x ∈ R,  1 ∞ Ah (x − y)dy = 1. h −∞ Remark 2. A fuzzy partition of an interval can be easily generalized to any (finite) direct product of intervals and by this, to an arbitrary n-dimensional region. As an example, we take two intervals [a, b] and [c, d] and consider [a, b] × [c, d] as a rectangular area in the 2D space. If fuzzy sets A1 , . . . , An : [a, b] → R, and B1 , . . . , Bm : [c, d] → R, establish fuzzy partitions of the corresponding intervals [a, b] and [c, d], then their products Ai Bj , i = 1, . . . , n; j = 1, . . . , m, establish a fuzzy partition of [a, b] × [c, d].

FzT in Dimensionality Reduction

3.2

27

Discrete Universe and Its Fuzzy Partition

From the point of view of image/signal processing, we assume that the domain of the corresponding functions is finite, i.e. finitely sampled in R, and the functions are identified with high-dimensional vectors of their values at the selected samples in the discretized domain. Moreover, we assume that the domain and the range of all considered functions are equipped with the corresponding relations of closeness. The best formal model of all these assumptions is a weighted graph G = (V, E, w) where V = {v1 , . . . , v } is a finite set of vertices, and E (E ⊂ V × V ) is a set of weighted edges so that w : E → R+ . The edge e = (vi , vj ) connects two vertices vi and vj , and then the weight of e is w(vi , vj ) or just wij . Weights are set using the function w : V × V → R+ , which is symmetric (wij = wji , ∀ 1 ≤ i, j ≤ ), non-negative (wij ≥ 0) and wij = 0 if (vi , vj ) ∈ E. The notation vi ∼ vj denotes two adjacent vertices vi and vj with an existing edge connecting them. Let H(V ) denote the Hilbert space of real-valued functions on the set of vertices V of the graph, where  if f, h ∈ H(V ) and f, h : V → R, then the inner product f, hH(V ) = v∈V f (v)h(v). Similarly, H(E) denotes the space of real-valued functions defined on the set E  of edges of a graph G. This space has the inner product F, HH(E) = (u,v)∈E F (u, v)H(u, v) =   F (u, v)H(u, v), where F, H : E → R are two functions on H(E). u∈V v∼u We assume that the set of vertices V is identified with the set of indices V = {1, . . . , } and that [1, ] is h-uniform fuzzy partitioned with normalized basic functions Ah1 , . . . Ah , so that Ahk (x) = Ah (x − k)/h, k = 1, . . . , , Ah (x) = A0 (x/h) and A0 is the generating function. Definition 3: A weighted graph G = (V, E, w) is fuzzy weighted, if V = {1, . . . , }, Ah1 , . . . Ah is an h-uniform fuzzy partition, generated by A0 , and wij = Ahi (j), i, j = 1, . . . , . The fuzzy weighted graph G = (V, E, w), corresponding to the h-uniform fuzzy partition, will be denoted Gh = (V, E, Ah ).

4

Discrete Laplace Operator

In this section, we recall the definition of (non-local) Laplace operator as a differential operator given by the divergence of the gradient of a function (see [3]). Let G = (V, E, w) be a weighted graph, and let f : V → R be a function in H(V ). The difference operator d : H(V ) → H(E) of f , is defined on (u, v) ∈ E by  (5) (df )(u, v) = w(u, v) (f (v) − f (u)) . The directional derivative of f , at vertex v ∈ V , along the edge e = (u, v), is defined as: ∂v f (u) = (df )(u, v). (6)

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The adjoint to the difference operator d∗ : H(E) → H(V ), is a linear operator defined by: (7)

df, HH(E) = f, d∗ HH(V ) , for any function H ∈ H(E) and function f ∈ H(V ). Proposition 1: The adjoint operator d∗ can be expressed at a vertex u ∈ V by the following formula:  w(u, v) (H(v, u) − H(u, v)) . (8) (d∗ H)(u) = v∼u

The divergence operator, defined by −d∗ , measures the network outflow of a function in H(E), at each vertex of the graph. The weighted gradient operator of f ∈ H(V ), at vertex u ∈ V, ∀(u, vi ) ∈ E, is a column vector: ∇w f (u) = (∂v f (u) : v ∼ u)T = (∂v1 f (u), . . . , ∂vk f (u))T . The weighted Laplace operator Δw : H(V ) → H(V ), is defined by: 1 Δw f = − d∗ (df ). 2

(9)

Proposition 2 [3]: The weighted Laplace operator Δw at f ∈ H(V ) acts as follows:  w(u, v)(f (v) − f (u)). (Δw f )(u) = − v∼u

This Laplace operator is linear and corresponds to the graph Laplacian. Proposition 3 [13]: Let Gh = (V, E, Ah ) be a fuzzy weighted graph, corresponding to the h-uniform fuzzy partition of V = {1, . . . }. Then, the weighted Laplace operator Δh at f ∈ H(V ) acts as follows:  (Δh f )(i) = − Ahi (j)(f (j) − f (i)) = f (i) − F h [f ]i , i∼j

where F h [f ]i , i = 1, . . . , , is the i-th discrete F-transform component of f , cf. [11].

5

Multi-scale Representation in a Space with a Fuzzy Partition

Taking into account the introduced notation, we propose the following scheme for the multi-scale representation LF P of the signal f : V → R, where V = {1, . . . , } and subscript “FP” stands for an h-uniform fuzzy partition determined by parameter h ∈ N, h ≥ 1: LF P (·, 0) =f (·), t

LF P (·, t) =F 2 h [f ],

(10)

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t

where t ∈ N is the scale parameter and F 2 h [f ] is the whole vector of F-transform components of f . The scale parameter t relates to the length of the support of the corresponding basic function. As in the case of (1), it is a natural measure of spatial scale at level t. To show the relationship to the diffusion equation, we formulate the following general result. Proposition 4 : Assume that two time continuously differentiated real function f : [a, b] → R, and [a, b] is h- and 2h-uniform fuzzy partitioned by Ah1 , . . . , Ahn 2h h 2h and A2h 1 , . . . , An , where basic functions Ai (Ai ), i = 1, . . . , n, are generated b−a by A0 (x) = 1 − |x| with the node at xi = a + n−1 (i − 1). Then, F 2h [f ]i − F h [f ]i ≈

h2  f (xi ). 4

(11)

The semantic meaning of this proposition in relation to the proposed scheme (10) of multi-scale representation LF P of f is as follows: The FT-based Laplacian of f (11) can be approximated by the (weighted) differences of two adjacent convolutions determined by the triangular-shaped generating function of a fuzzy partition.

6 6.1

Experiments with Time Series Reconstruction from FT-Based Laplacians

To demonstrate the effectiveness of the proposed representation, we first show that an initial time series can be (with a sufficient precision) reconstructed from a sequence of FT-based Laplacians. Below, we illustrate this claim on a financial time series with high volatility. With each value of t = 1, 2, . . . we obtain the corresponding FT-based Laplacian as the difference between two adjacent convolutions (vectors with F-transform components), so that we obtain the sequence {LF P (·, t + 1) − LF P (·, t) | t = 1, 2, . . .} The stop criterion is closeness to zero of the current difference. We then compute the reconstruction by summing all the elements in the sequence. Figure 1 shows the step-by-step reconstruction and the final reconstructed time series. The latter is plotted on the bottom image along with the original time series to give confidence in a perfect fit. The estimated Euclidean distance is 89.6. In the same Fig. 1, we show one MLP reconstructions of the same time series with the following configurations: 4 hidden layers with 4086 neurons in each layer (common setting) and learning rate 0.001. It is obvious that the proposed multi-scale representation and subsequent reconstruction are computationally cheaper and give results with better reconstruction quality. To confirm, we give estimates of the Euclidean distances between the original time series and its reconstructions: (from a sequence of FT-based Laplacians) against 159.3 (using MLP).

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Keypoint Localization and Description

Keypoint Localization. The localization accuracy of key points depends on the problem being solved. When analyzing time series, the accuracy requirements are different from those used in computer vision to match or register images. Time series focuses on comparing the target and reference series in order to detect similarities and use them to make a forecast. Therefore, the spatial coordinate is not so important in contrast to the comparative analysis of local trends and their changes in time intervals with adjacent key points as boundaries. Taking into account the above arguments, we propose to localize and identify keypoints from the second-to-last scaled representation of the Laplacian before the latter meets the stopping criterion. We then follow the technique suggested in [9,10] and identify the keypoint with the local extremum point of the Laplacian corresponding to the selected scale. As in the cited above works, we faced a number of technical problems related to the stability of local extrema, sampling frequency in a scale, etc. Due to the different spatial organization of the analyzed objects (time series versus images), we found simpler solutions to the problems raised. For example, in order to exclude extrema close to each other (and therefore they are very unstable), we leave only one representative, the value of which gives the best semantic correlation with the characteristic of this particular extremum. Below, we give illustrations to some processed by us time series. They were selected from the cite with historical data in Yahoo Finance. We analyzed the 2016 daily adjusting closing prices using international stock indices, namely Prague (PX), Paris (FCHI), Frankfurt (GDAXI) and Moscow (MOEX). Due to the daily nature of the time series, they all have high volatility, which is additional support for the proposed method. In Fig. 2, we show the time series with stock indices PX (Prague) and its last three scaled representations of the Laplacian, where the latter satisfies the stopping criterion. Selected (filtered out) keypoints are marked with red (blue) dots. Keypoint Description. Due to the specificity of time series with high volatility, we propose a keypoint descriptor as a vector that includes only the Laplacian values at keypoints from two adjacent scales and in an area bounded by an interval with boundaries set by adjacent left/right keypoints from the same scale. In addition, we normalize the keypoint descriptor coordinates by the Laplacian value of the principal keypoint. As our experiments with matching keypoint descriptors of different time series show, the proposed keypoint descriptor is robust to noise and invariant with respect to spatial shifts and time series ranges. The last remark is that the quality of matching is estimated by the Euclidean distance between keypoint descriptors.

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31

Fig. 1. Top. The sequence of reconstruction steps, where with each value t = 1, 2, . . . we improve the quality of the reconstruction by adding the corresponding Laplacian to the previous one. Middle. The original time series (blue) against its reconstruction (red) from a sequence of FT-based Laplacians with the distance 89.6. Bottom. The original time series (blue) against its MLP reconstruction (in red) with the distance 159.3.

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Fig. 2. The time series with stock indices PX (Prague) and its last three scaled representations of the Laplacian, where the latter bottom image satisfies the stopping criterion. Selected (filtered out) keypoints are marked with red (blue) dots.

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7

33

Conclusion

We were focused on dimensionality reduction in the case of large datasets and time series. We have contributed to this topic by showing that – dimensionality reduction should be divided into several subtasks, determined by the choice of key points as centers corresponding to clusters; – the use of non-traditional kernels derived from the theory of F-transforms leads to simplified algorithms and comparable efficiency in the selection of keypoints. We also proposed a new keypoint descriptor and tested it on matching financial time series with high volatility. Acknowledgment. Experiments and illustrations were performed by Mgr. David Adamczyk under the direction of Irina Perfilieva.

References 1. Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003) 2. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of 27th Annual Conference Computer Graphics and Interactive Techniques, pp. 417–424. ACM Press/Addison-Wesley Publishing Co. (2000) 3. Elmoataz, A., Lezoray, O., Bougleux, S.: Discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17, 1047–1060 (2008) 4. Burt, P.J.: Fast filter transform for image processing. Comput. Graphics Image Process. 16, 20–51 (1981) 5. Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River (1999) 6. Klinger, A.: Pattern and search statistics. In: Proceedings of Symposium Held at the Center for Tomorrow, the Ohio State University, 14–16 June, 1971, pp. 303–337 (1971) 7. Koenderink, J.J.: The structure of images. Biol. Cybern. 50, 363–370 (1984) 8. Lindeberg, T.: Scale-space theory: a basic tool for analyzing structures at different scales. J. Appl. Stat. 21, 225–270 (1994) 9. Lindeberg, T.: Image matching using generalized scale-space interest points. J. Math. Imaging Vision 52(1), 3–36 (2015) 10. Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vision 60(2), 91–110 (2004) 11. Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets Syst. 157(8), 993–1023 (2006) 12. Molek, V., Perfilieva, I.: Deep learning and higher degree F-transforms: interpretable kernels before and after learning. Int. J. Comput. Intell. Syst. 13(1), 1404–1414 (2020) 13. Perfilieva, I., Vlasanek, P.: Total variation with nonlocal FT-Laplacian for patchbased inpainting. Soft. Comput. 23, 1833–1841 (2019)

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14. Schmid, C., Mohr, R.: Local gray value invariants for image retrieval. IEEE Trans. Pattern Anal. Mach. Intell. 19(5), 530–535 (1997) 15. Scholkopf, B., Smola, A., Mulller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998) 16. Witkin, A.P.: Scale-space filtering. In: Proceedings of 8th International Joint Conference on Artifcial Intelligence, IJCAI 2083, v.2, 1983, pp. 1019–1022 (1983) 17. Zadeh, L.A.: Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst. 90, 111–127 (1997)

Mathematics, Scientific Reasoning and Fuzzy Logic Ahmet Fahri Özok(B) Industrial Engineering Department, Istanbul Okan University, Akfırat, 34959 Istanbul, Turkey [email protected]

Abstract. This article tries to show the relationship among general principles of mathematics, human scientific reasoning and Fuzzy Logic (FL). In applied sciences, in engineering and in social sciences Fuzzy Logic can be used in solution of real life problems. Two main types of mathematical reasoning; induction and deduction and also abduction are essential ways of FL. If we take into consideration dedication we can assume it is always a valid process, but it is not an infallible method. Before the concept FL, in Western World, logic was based on the principle of the bivalent Logic. Ideas of multivalued or even infinite valued logic is based on the mathematical theory of Fuzzy Sets (FS). It means one can pass from bivalent logic to infinitely many values lying in the interval [0,1]. Philosophy of Mathematics and Scientific Reasoning gives us the opportunity to discuss the mathematical background of FL. Rather than being strictly on engineering problems, FL provides a number of broader applications; Artificial intelligence, neural networks, genetic algorithms, biological processes are some of these applications. What kind of mathematical tool we use, in application? We have to show that instead of deterministic or probabilistic solution, FL gives better result. Keywords: Scientific reasoning · fuzzy logic · linguistic · abstraction · fuzzy modeling

1 Introduction As we know FL was first confronted with some distrust and reservation by the most academicians. As an important capability our brain can grasp and understand the main properties of definitive, probabilistic and fuzzy modelling. In fact, Human Scientific Reasoning is characterized by inaccuracies and uncertainties which stem from the conditions of real world [1]. Generally, we don’t live and opposed to discrete values. Nowadays FL has a sound mathematical background and has found many and important applications in many areas of human activities. Because of its mathematical background introducing the concept of membership degree, which makes it possible a condition to be a state other than true or false, provides a flexibility to form human scientific reasoning [1]. On the other hand, the rules of FL are set in natural language with the help of linguistic. It means in FL human reasoning process can be made by using natural language. This is essential to understand the structure of expert systems. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 35–38, 2023. https://doi.org/10.1007/978-3-031-39774-5_4

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The art of FL Modelling is certainly and closely related with Mathematical Art. Generalizing, creating and improving scientific knowledge are also important in FL Modelling.

2 Mathematics and Human Reasoning We know that from beginning of mathematical history raised a wealth of scientific and philosophical questions. For example, mathematics where does it come from? Mathematics, as a base of FL, is one of the most successful, but in the same time most puzzling human endeavors. It is of course an autonomous discipline and a common language of different sciences [2]. Mathematics uses a meaningful and a special language to express the truths. Empirical sciences, like in engineering, it differs profoundly what we assume as “real”. For example, Number Theory and Set Theory are two kinds of object are equally “real”. If we are dealing with FL, how we can explain our way of reasoning, using objective mathematical truths. But in the same time, we have to express our propositions using linguistic terms. They have to logical and conceptual or of some indirect form that related directly to mathematical principles. If we think any problem in FL we know that according to some degree one form of evidence need not exclude another. If we the into consideration the relationship Mathematics and Human Reasoning, we can claim that our mathematical knowledge has three distinctive characteristics [3]. First, it is a priori, namely it doesn’t rely on sense experience or on experimentation. Truths are arrived at by reflection alone and not with any sensory observation. Secondly the knowledge is concerned with bare truths that are necessary. Third our knowledge is concerned with some concepts which are not located in space or time [3]. Mathematical truths do not need any justification. Concepts in mathematics are innate, it means they are not acquired but from part of the mind’s inborn endowment [3]. For satisfactory and scientific good answers, we need to have good questions. Like in pure mathematics, the truths of FL are counterfactually independent of human being. It is not wrong to assume that if we had no intelligent concepts and life, these truths would still have remained the same. By the way, we have to remember that, the mathematical truths are different from our ordinary daily life. Pure mathematics is not valid exactly in various social activities. The third distinctive feature of mathematics is abstraction. If we try to define an object as abstract object we can say; an object is said to be abstract if it lacks spatiotemporal location is causally inefficacious, otherwise it is said to be concrete. It is wrong, if we claim that all mathematical objects are abstract. If mathematical objects had spatiotemporal location, the our mathematical practice, what we use in our daily life would be misguided and inadequate [3]. In fact, we have an abstract world of mathematics and the world of reality. The real world is just what we live in and what we face all the time. The world sense experience, the world we sense and feel, have numerous problems to solve. In this world we always try to handle with problems and have good solutions. We have necessity to understand and to control this world. We know that science sprang

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37

from this necessity. Curiosity of human being created science… Just like magic and religion. In FL we have to be able to convert the problem parameters, what we face, into Mathematical parameters. And then we can use any one of Fuzzy Mathematical tools, Mathematics is world of abstractions and ideas. The things in this world. Are mathematical objects and topics; like; functions, numbers, differential equations sequences and topological spaces, etc.

3 Scientific Reasoning and Fuzzy Logic Fuzzy mathematics is always trying to cover and complete the inherent weakness of crisp definitive mathematics [1]. Of course FL is not a final solution for represanting human knowledge about the very different and complex problems, we face. Generally speaking scientific reasoning Fuzzy Modelling can easily be grasped by scientist and researchers alike to form human reasoning. In various generalisations and formalising for different kind of problems, we have different Fuzzy Modelling: Type -2 FS, Interval valued FS, Intuitionistic FS, Hesitant FS, Pythagorean FS, Complex FS, Neutrosophic FS, Fuzzy Hybrid Method etc. Human Reasoning can be definitive, probabilistic and Fuzzy [4]. Because of the nature of human being our senses and observation tools do not allow us also to reach an absolute precision in a world which is based on the mathematical continuity and logical correctness, Consequently FL, using the concept of membership degree makes it possible not only bivalent evaluation like to be a state true or false, also a flexibility to form human reasoning [1]. We have to be careful in FL that our premises are true. Because a deductive argument is always valid if its premisses are true and logical result its inference is also true. The common and typical logic is interested only for the validity and not for the truth of an argument against to the principles of the bivalent logic, not taking into consideration its inference is true.

4 Conclusion If we grasp the potential power of FLwe can use it as a very strong tool to solve our problems in every kind and in every areas we know that in different areas, we are faced with problems for which we are not concious thoroughly, what kind of mathematical tool we have to use. It has to be mentioned that before the development of FL by Zadeh the unique tool dealing with the problems, which had vague or character was probability theory. It is essential to bring together main ideas of philosophy of mathematics, human reasoning and FL. Nowadays Fuzzy Set Theory, as formal theory becomes in many cases, more and more developed, specified, enlarged, sophisticated and created new scientific ideas and concepts. Sometimes if we are in the situation to have incomplete data and if we can not use induction or deduction, it is always possible to use FL in abduction. Mathematics,

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Scientific Reasoning and Fuzzy Logic, all together becoming a challenge for for further academic, scientific and practical discussions [4].

References 1. Athanassopoulos, E., Voskoglou, M.: G: A philosophical treatise on the connection of scientific reasoning with fuzzy logic.  mathematics 8(875), 2–15 (2020) 2. King, J.P.: The Art of Mathematics. Fawcett Columbine, New York, pp. 27–28 (1993) 3. Linnebo: Philosophy of Mathematics: Princeton Foundations of Contemporary Philosopy. Princeton University Press, pp. 6–12 (2020) 4. Ozok, A.F.: Fuzzy set theory and the social sciences, In: Kahraman, C., Cebi, S., Onar, S.C., Oztaysi, B., Tolga, A.C., Sari, I,U. (eds.) INFUS 2021. AISC, vol. 307, pp. 182–184 (2022)

Optimal Transport and the Wasserstein Distance for Fuzzy Measures: An Example Vicen¸c Torra(B) Department of Computing Science, Ume˚ a University, Ume˚ a, Sweden [email protected] Abstract. Probabilities and, in general, additive measures are extensively used in all kind of applications. A key concept in mathematics is the one of a distance. Different distances provide different implementations of what means to be near. Wasserstein distance is one of them for probabilities, with interesting properties and a large number of applications. It is based on the optimal transport problem. Non-additive measures also known as fuzzy measures, capacities and monotonic games, generalize probabilities replacing the additivity axiom by a monotonicity condition. Applications have been developed for this type of measures. In a recent paper we have introduced the optimal transport problem for non-additive measures. This permits to define the Wasserstein distance for non-additive measures. It is based on the (max, +)-transform. We review in this paper this definition, and provide some examples. Examples have been computed with an implementation we have provided in Python. Keywords: Fuzzy measure · Non-additive measure transport · Wasserstein distance

1

· Optimal

Introduction

Additive measures and probabilities are commonly used in all kind of applications. A key property is the additivity axiom which states that the measure of the union of two disjoint sets is the addition of the measures of these sets. This property naturally appears when we consider measure as lengths, areas, and volumes. Non-additive measures [4,16] have been introduced in which this axiom is replaced by a monotonicity condition. That is, the larger the set, the larger the measure. Then, the additivity condition is no longer a requirement. It may hold, but it may not hold. These more general measures allow us to represent interactions present in the set. We have positive interactions when the measure of the union is larger than the addition of the measure of the sets. In contrast, This study was partially funded by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 39–44, 2023. https://doi.org/10.1007/978-3-031-39774-5_5

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we have negative interactions when the measure of the union is smaller than the addition of the measure of the sets. These interactions can depend on the presence or absence of an element into the set [15]. Non-additive measures appear with different names in the literature. They are known as fuzzy measures within the fuzzy community. They were introduced by Sugeno in 1972 [13] and in his PhD in 1974 [14]. They are also known as capacities (see e.g. Choquet and his definition of the integral [3]) in potential theory, and monotonic games in game theory [23]. Non-additive measures are often applied with fuzzy integrals. Examples include the Choquet [3] and Sugeno [14] integrals, but other integrals exist [8–11]. See e.g. [2,12] for latest results and an overview of generalizations of the Choquet integral. A basic concept in mathematics is the one of distance. Metric spaces are then defined in terms of them. In probabilities, a large literature exists on alternative definitions for distances, their properties, and applications of them. Among them, we can distinguish the Wasserstein distance which is based on the optimal transport problem. Both the optimal transport problem and the Wasserstein distance have been extensively studied for the theoretical properties, and have been used for building applications. In a recent paper, we have introduced the optimal transport problem for non-additive measures. In this paper we review its definition and provide some actual examples of their computation. We have developed software in python to compute these solutions. This software has been used to compute the examples provided in this paper. The structure of the paper is as follows. In Sect. 2 we provide some definitions we need later in this paper. Then, in Sect. 3, we review the optimal transport problem for non-additive measures. In Sect. 4 we provide some examples. The paper is finished with some conclusions.

2

Preliminaries

Non-additive measures, also known as capacities, fuzzy measures, and monotonic games are set functions which are zero in the empty set and monotonic with respect to set inclusion. Let us consider a finite reference set X. Then, a non-additive measure μ is a set function μ : 2X → R+ that satisfies the following axioms: – μ(∅) = 0 – if A ⊂ B ⊆ X, then μ(A) ≤ μ(B). We say that μ is a normalized measure if μ(X) = 1. Given a non-additive measure μ, there are several transforms. They are set functions that permit us to reconstruct μ from them. The M¨ obius transform is the most well known one. Nevertheless, there are a few alternatives (see e.g. [5,6]). In a recent work [20] we introduced the (max,+)-transform that is appropriate for our work. We review it below.

Optimal Transport and the Wasserstein Distance

41

Definition 1. [20] Let μ be a normalized fuzzy measure on 2X . Then, we define the (max, +)-transform as the set function m : 2X → [0, 1] such that: m(B) = μ(B) − max μ(A) A⊂B

(1)

Here we understand maxA⊂∅ μ(A) = 0. The following expression is equivalent to Eq. 1: m(B) = μ(B) − max μ(B \ {x0 }). x0 ∈B

(2)

Among the properties of the (max,+)-transform, we have that [20] given a measure μ and its (max, +)-transform m, we have that we can reconstruct μ as follows:   (3) μ(B) = max μ(A) + m(B), A⊂B

Moreover, if m is a set function in [0,1] with m(∅) = 0, then μ constructed from m using Eq. 3 is a non-additive measure, and its transform is m. Note, however, that this measure does not need, in general, to be normalized.

3

Optimal Transport for Non-additive Measures

Given two measures μ and ν defined on the reference set X, the transport problem [22] can be seen as follows: find a function on the pairs (A, B) so that the marginals on each dimension correspond to μ(A) and ν(B), respectively. We will call such assignment assg. In other words, we assign μ(A) for A ⊆ X into sets B ⊆ X so that the assignwe need that ment adds up to μ(A). That is, μ(A) = B  assg(A, B  ). Similarly,  the assignment for all A ⊆ X add to ν(B). That is, ν(B) = A assg(A , B). For additive measures such assignment always exist. For non-additive measures, we introduced an alternative definition in which the assignment is based on the (max,+)-transforms of μ and ν. Given a cost function c, we define the optimal transport as the assignment that has a minimum cost. For additive measures, the cost function is a function c : X × X → R+ . In our extension for non-additive measures, we consider cost functions c : 2X × 2X → R+ . The definition of the transport problem follows. Definition 2. [21] Let μ and ν be non-additive measures on X, with (max, +)transforms τμ and τν , respectively. Then, the transport problem between μ and ν is a function assg : 2X × 2X → [0, 1] that satisfies assg(∅, ∅) = 0  assg(A, B  ) for all A = ∅ τμ (A) = B  ⊆X

42

V. Torra

τν (B) =



assg(A , B) for all B = ∅

A ⊆X

Then, given the cost function ca : 2X ×2X → [0, 1], the cost of the assignment assg is:   cost(ca , assg) = ca (A, B)assg(A, B). A⊆X B⊆X

Then, the optimal transport problem is the one that minimizes the cost. This is formalized in the next definition. Definition 3. [21] Let μ, ν, τμ , τν , and assg as in Definition 2. Then, the optimal problem is to find an assignment assg that minimizes cost(ca , assg). The cost of the optimal transport problem corresponds in the classical setting to the Wasserstein distance for two probabilities. Because of that, we introduced in [21] the Wasserstein distance for non-additive measures in the same way. That is, as the cost of the assignment that is the solution of the optimal transport problem (Definition 3). In other words, if assg ∗ is the optimal assignment for cost ca , then, the Wasserstein distance corresponds to cost(ca , assg ∗ ).

4

Examples

We have developed software in Python to compute the optimal transport of two measures given a cost function. This software is provided in our web page [24]. To illustrate the computation of the optimal transport and the Wasserstein distance we have considered two measures, each defined on X with three elements. The measures are additive, μ with probabilities (0.2, 0.3, 0.5) and ν with probabilities (0.2, 0.2, 0.6). Then, we have done two examples. First, we compare μ and noisy versions of μ. Second, we compare μ and noisy versions of ν. For example, for the later, we have perturbed ν adding increasing levels of noise, and computed the Wasserstein distance between μ and these variations of ν. In this way, we can study how the distance changes with respect to the noise. In order to define noisy versions of ν, we have proceeded as follows. First, we compute the (max,+)-transform and added a noise N (0, sd) to each element of the transform. If any of these values were negatives, we have just assigned it to zero. Recall that the (max, +)-transform is always positive. Then, we have normalized the resulting measure so that ν  (X) = 1. For different values of sd we have different measures νsd . Observe that while the original measures μ and ν are additive, their distorted versions are not. That is, they are proper fuzzy measures. Therefore, the computation of their distance cannot be done using the standard optimal transport problem but needs to be done using our definition. In Fig. 1 we represent the distance with respect to the parameter sd. For each value of sd, we have computed the mean of ten executions. In Fig. 1 (left) we compute the distance between μ and distorted versions of the same μ, and in Fig. 1 (right) we compute the distance between μ and distorted versions of ν. The computation uses a cost function ca built according to Proposition 21 in our previous work [21] from the cost function c : X × X → R+ such that c(x, x) = 0 and c(x, y) = 5 for x = y.

Optimal Transport and the Wasserstein Distance

43

Fig. 1. Wasserstein distance of μ and distorted versions of μ (left) and distorted versions of ν (right).

5

Conclusions

In this paper we have given an overview of the optimal transport problem for nonadditive (fuzzy) measures, and of the Wasserstein distance. We have illustrated its computation with an example, where an additive measure is distorted in a way that produces a non-additive measure. We can observe that, as expected, the larger the distortion, the larger the distance. These are not the only type of distances for fuzzy measures. We contributed with distances [17,18] based on the Radon-Nikodym derivative. That is a completely different type of distance, which permits to define f-divergence and KLdivergence, as well as entropy. That research direction has been followed by other authors [1,7]. As future work, we plan to consider further mathematical properties of our definitions, as well as extend its applicability. A key element of the Wasserstein distance is the cost function. We plan to study different cost functions and its effects on the distance. We had some discussion on suitable cost functions in our previous work [21], and did some research on cost functions for pairs of sets A, B [19] that can be useful in this context. Comparison and analysis with f-divergence is another open research direction.

References 1. Agahi, H.: A generalized Hellinger distance for Choquet integral. Fuzzy Sets Syst. 396, 42–50 (2020) 2. Bustince, H., et al.: d-Choquet integrals: Choquet integrals based on dissimilarities. Fuzzy Sets Syst. 414, 1–27 (2021) 3. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953/54) 4. Denneberg, D.: Non Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994) 5. Mesiar, R.: Maxitive and k-order maxitive measures. In: Proceedings of IFAC (2001) 6. Mesiar, R.: Generalizations of k-order additive discrete fuzzy measures. Fuzzy Sets Syst. 102(3), 423–428 (1999)

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7. Ontkoviˇcov´ a, Z., Kisel´ ak, J.: A way to proper generalization of φ-divergence based on Choquet derivatives. Soft. Comput. 26, 11295–11314 (2022) 8. Pap, E.: Integral generated by decomposable measure, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20(1), 135–144 (1990) 9. Pap, E.: Pseudo-analysis as a mathematical base for soft computing. Soft. Comput. 1, 61–68 (1997) 10. Pap, E.: Pseudo-Additive Measures and Their Applications, Handbook of Measure Theory, vol. II, pp. 1403–1468. North-Holland, Amsterdam (2002) 11. Pap, E.: Generalized real analysis and its applications. Int. J. Approximative Reasoning 47, 368–386 (2008) 12. Pereira Dimuro, G., et al.: The state-of-art of the generalizations of the Choquet integral: from aggregation and pre-aggregation to ordered directionally monotone functions. Inf. Fusion 57, 27–43 (2020) 13. Sugeno, M.: Fuzzy measures and fuzzy integrals (in Japanese). Trans. Soc. Instrument Control Eng. 8, 2 (1972) 14. Sugeno, M.: Theory of Fuzzy Integrals and its Applications, Ph.D. Dissertation, Tokyo Institute of Technology, Tokyo, Japan (1974) 15. Torra, V., Guillen, M., Santolino, M.: Continuous m-dimensional distorted probabilities. Inf. Fusion 44, 97–102 (2018) 16. Torra, V., Narukawa, Y., Sugeno, M. (eds.) Non-additive measures: theory and applications. Springer (2013) 17. Torra, V., Narukawa, Y., Sugeno, M.: On the f -divergence for non-additive measures. Fuzzy Sets Syst. 292, 364–379 (2016) 18. Torra, V., Narukawa, Y., Sugeno, M.: On the f-divergence for discrete non-additive measures. Inf. Sci. 512, 50–63 (2020) 19. Torra, V.: Non-additive measures, set distances and cost functions on sets: a Fr´echet-Nikodym-Aronszajn distance and cost function. In: Proceedings of INFUS 2022, vol. 1 (2022) 20. Torra, V.: (M ax, ⊕)-transforms and genetic algorithms for fuzzy measure identification. Fuzzy Sets Syst. 451, 253–265 (2022). https://doi.org/10.1016/j.fss.2022. 09.008 21. Torra, V.: The transport problem for non-additive measures. Europ. J. Oper. Res. 311(2), 679–689 (2023). https://doi.org/10.1016/j.ejor.2023.03.016 22. Villani, C.: Optimal Transport: Old and New. Springer, Cham (2008) 23. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press (1944) 24. http://www.mdai.cat/code

Theoretical Improvements in Fuzzy Sets

Some Properties of a New Concept of Fractional Derivative for Fuzzy Functions on Time Scales Mina Shahidi(B)

and Estev˜ ao Esmi

Department of Mathematics, Statistic and Scientific Computing, Unicamp, Campinas, Brazil [email protected] Abstract. In this paper, we propose a new concept of fractional derivative for fuzzy functions on time scales. The presented fuzzy fractional derivative is a natural extension of the generalized Hukuhara derivative. Furthermore, several properties of the introduced derivative are studied. Also, we present the characterization theorem of the new derivative in terms of the differentiability of its endpoint functions. Keywords: Fuzzy functions

1

· Fractional derivative · Time scales

Introduction

The theory of time scales was originated by Hilger in 1988 [8]. By its general foundation, the calculus on time scales can be considered as a unification of the discrete and continuous [4]. Furthermore, time scale calculus provides a new framework for modelling and solving real-world phenomena on general domains, not only restricted on real intervals or integers but also extended to more general time scales, for example, the sets of quantum numbers or the Cantor set. Particularly, one can find several applications of the time scale theory in diverse fields of science, such as pure and applied mathematics, economics, neural networks, and so forth. In a real-world problem, we face terms that are inherently vague or uncertain. For this reason, Zadeh [16] proposed fuzzy set theory, which provided an important theoretical basis for dealing with uncertain problems. Since then, fuzzy mathematics has made great progress in both theory and application. In light of this, in recent years, the authors studied calculus on time scales for fuzzy functions deeply [7,13,14]. Fractional calculus that leads to the examination and applications of integrals and derivatives of arbitrary order has a long history [9]. The theory has gained popularity due to the various applications in science, engineering, electrochemistry, biology, and etc. [5,6,10,12,15]. The investigation of fractional derivatives on time scales is a well-known matter provided an excellent tool for modeling real problems in scientific fields and diffusion processes and so on [3]. Recently, in [11], Shahidi and Khastan introduced a fractional derivative based on the Hukuhara difference for fuzzy functions on time scales. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 47–53, 2023. https://doi.org/10.1007/978-3-031-39774-5_6

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Applying the generalized Hukuhara difference [1] and the differentiability concept proposed in [3], this paper introduces and studies a new concept of fuzzy fractional derivative on time scales. Therefore, this study extends the result presented in [11]. After providing several main concepts and results, we establish the new definition of fractional derivative, called (β)-derivative on time scales. Also, we provide sufficient conditions for β-derivative to preserve the addition and the scalar multiplication. We also present the characterization theorem of the introduced derivative in terms of the differentiability of the endpoint functions.

2

Preliminaries

In what follows, we provide several basic concepts and required notions that will be used in the rest of the paper. Definition 1. Let Y be a nonempty set. A fuzzy set v of Y is a map v : Y → [0, 1], such that v(t) is the membership degree of t ∈ Y to the fuzzy set v. For each r ∈ (0, 1], the r-cut is represented by [v]r = {t ∈ R : v(t) ≥ r} and for r = 0 by [v]0 = cl{t ∈ R : v(t) > 0} in which cl(A) states the closure of the subset A ⊆ R. The set of fuzzy convex, normal, upper semi-continuous, and compact supported fuzzy sets is named the space of fuzzy numbers and it is represented by RF . The notation [v]r = [vr− , vr+ ], denotes the r-level set of v. The addition and the scalar multiplication in RF are given levelwise by [v + w]r = [v]r + [w]r , [λv]r = λ[v]r , ∀r ∈ [0, 1], for λ ∈ R and for all v, w ∈ RF . Proposition 1. [1] Let the functions vr− , vr+ : [0, 1] → R satisfy the conditions below: • vr− ∈ R is non-decreasing, bounded, left-continuous in (0, 1], and right continuous at 0. • vr+ ∈ R is non-increasing, bounded, left-continuous in (0, 1], and right continuous at 0. • v1− ≤ v1+ . Then, there is a fuzzy number v ∈ RF that has vr− , vr+ as endpoints of its r-cuts. The Hausdorff distance on RF is shown by D(v, w) = sup max{|vr− − wr− |, |vr+ − wr+ |}, v, w ∈ RF . r∈[0,1]

The well known features of the metric D are given, as follows: • D(v + l, w + l) = D(v, w), ∀v, w, l ∈ RF , • D(μv, μl) = |μ|D(v, l), ∀v, l ∈ RF , μ ∈ R,

Some Properties of a New Concept of Fractional Derivative

49

• D(v + l, z + e) ≤ D(v, z) + D(l, e), ∀v, l, z, e ∈ RF , • D(λv, μv) = |λ − μ|D(v, ˜0), for λμ > 0, and ˜ 0 = χ{0} , and metric space (RF , D) is complete. Definition 2. [1] Let v, w ∈ RF . The generalized Hukuhara difference (gHdifference, for short) is defined by the fuzzy number z ∈ RF , if it exists, such that  (i)v = w + z, v gH w = z ⇐⇒ or(ii)w = v + (−1)z. Theorem 1. [1] Let v, w ∈ RF . The gH-difference satisfies the following properties: (i) (v + w) gH w = v and v gH (v + w) = −w. (ii) v gH w exists, iff, w gH v and (−w) gH (−v) exist and we get v gH w = (−w) gH (−v) = −(w gH v). Definition 3. [4] An arbitrary nonempty closed subset of the real numbers is named a time scale T. Definition 4. [4] For t ∈ T, the backward jump operator ρ : T → T is defined by ρ(t) := sup{s ∈ T : s < t} and the forward jump operator σ : T → T is given by σ(t) := inf{s ∈ T : s > t}. In addition, the graininess function μ : T → [0, ∞) is defined by μ(t) := σ(t) − t. Definition 5. [4] If σ(t) > t, then the point t is called right-scattered. Furthermore, if ρ(t) < t, then t is left-scattered. If ρ(t) = t, then t is left-dense and if σ(t) = t, then t is right-dense. Note that if T has a left-scattered maximum m, then Tk = T − {m}, if not, Tk = T. Definition 6. [3] Let g : T → R, t ∈ Tk and β ∈ (0, 1]. For t > 0, the function g is named fractional differentiable at t if there is Tβ g(t) ∈ R such that for all  > 0, there exists δ > 0 and   [g(σ(t)) − g(s)] t1−β − Tβ g(t) [σ(t) − s] ≤  |σ(t) − s| , for all s ∈ UT (t, δ) = (t − δ, t + δ) ∩ T. The function Tβ g(t) is named the conformable fractional derivative of g of order β at t.

3

Fuzzy Fractional Differentiability on Time Scales

In this part, we propose a novel notion of the fractional differentiability of order β ∈ (0, 1] for fuzzy functions on time scales.

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Definition 7. Let v : T → RF and t ∈ Tk . Then, v is named (β)-differentiable at t > 0 if for any  > 0, there exists a neighborhood UT of t such that t ∓ h ∈ UT such that v(t + h) gH v(σ(t)) and v(σ(t)) gH v(t − h) exist and   D (v(t + h) gH v(σ(t)))t1−β , v (β) (t)(h − μ(t)) ≤  |h − μ(t)| ,   D (v(σ(t)) gH v(t − h))t1−β , v (β) (t)(h + μ(t)) ≤ (h + μ(t)), with 0 ≤ h < δ. Theorem 2. Let v : T → RF and t ∈ Tk , then the properties hold, as follows: (i) Let t be right-scattered. If v is continuous at t, then v is (β)-differentiable at t > 0 and v(σ(t)) gH v(t) 1−β v (β) (t) = t . μ(t) (ii) Let t be right-dense. Then, v is (β)-differentiable at t, iff, lim

h→0+

1 1 (v(t + h) gH v(t))t1−β = lim+ (v(t) gH v(t − h))t1−β = v (β) (t), h h→0 h

provided that the corresponding limits exist for h > 0 sufficiently small. (iii) If v is (β)-differentiable at t > 0, then v(σ(t)) = v(t) + μ(t)v (β) (t)tβ−1 , or v(t) = v(σ(t)) + (−1)μ(t)v (β) (t)tβ−1 . Proof. We only prove case (i) and the other cases are similar to Theorem 2 in [7]. Indeed, suppose that v is continuous at t right-scattered. Therefore, it implies that v(t + h) gH v(σ(t)) 1−β v(t) gH v(σ(t)) 1−β t t lim+ = , h − μ(t) −μ(t) h→0 lim

h→0+

v(σ(t)) gH v(t − h) 1−β v(σ(t)) gH v(t) 1−β t t = . μ(t) + h μ(t)

In addition, by Theorem 1, we get   v(t) gH v(σ(t)) 1−β v(σ(t)) gH v(t) 1−β t t D , = 0. −μ(t) μ(t) Thus, given  > 0, there is a neighborhood UT of t such that   v(t + h) gH v(σ(t)) 1−β v(σ(t)) gH v(t) 1−β t t , D ≤ , h − μ(t) μ(t)   v(σ(t)) gH v(t − h) 1−β v(σ(t)) gH v(t) 1−β t t D , ≤ , μ(t) + h μ(t) for all t ∓ h ∈ UT assuming 0 ≤ h < δ. Hence, it implies that v (β) (t) =

v(σ(t)) gH v(t) 1−β t . μ(t)

Some Properties of a New Concept of Fractional Derivative

51

Corollary 1. Let t be right-scattered. If v : T → RF is Δ-differentiable and continuous at t, then it is (β)-differentiable at t and v (β) (t) = t1−β v Δ (t). Example 1. Let v(t) = (1, 2, 3)t2 . If T = hN := {hk : k ∈ N}, then v is (β)differentiable at t and v (β) (t) = (2t + h)t1−β (1, 2, 3). Indeed, by Theorem 2 (i), we have v (β) (t) =

(1, 2, 3)(t + h)2 gH (1, 2, 3)t2 1−β t h (1, 2, 3)((t + h)2 − t2 ) 1−β t = h = (2t + h)t1−β (1, 2, 3).

If T = R, according to Theorem 2 (ii), we have v (β) (t) = 2t2−β (1, 2, 3). Besides, if T = q N0 := {q n |n ∈ N0 } with q > 1. So, σ(t) = qt and μ(t) = (q − 1)t. Then, by Theorem 2 (i), we have v (β) (t) = (q + 1)t2−β (1, 2, 3) for all t ∈ T. Theorem 3. Let v : T → RF be continuous at t with v(t, r) = [v − (t, r), v + (t, r)], for all r ∈ [0, 1]. Suppose that v − (t, r) and v + (t, r) are conformable differentiable at t > 0, uniformly in r ∈ [0, 1]. Then, v is (β)-differentiable at t, iff, the following cases hold: (i) Tβ v + (t, r) is decreasing and Tβ v − (t, r) is increasing with respect to r, and Tβ v − (t, 1) ≤ Tβ v + (t, 1), or (ii) Tβ v − (t, r) is decreasing and Tβ v + (t, r) is increasing with respect to r, and Tβ v + (t, 1) ≤ Tβ v − (t, 1). Furthermore, for all r ∈ [0, 1], the following expression holds v (β) (t, r) = [min{Tβ v − (t, r), Tβ v + (t, r)}, max{Tβ v − (t, r), Tβ v + (t, r)}].

(1)

Proof. Let v be (β)-differentiable and continuous at t. If Tβ v − (t, r) and Tβ v + (t, r) are differentiable at t, then two cases are obtained, as follows: Case (1). Let t be right-dense. The proof of (1) is followed immediately from Theorem 24 in [2] and Theorem 2 (ii). Case (2): Let t be right-scattered. By Definition 2 and Theorem 2 (i), we achieve  ⎧ − v (σ(t),r)−v − (t,r) 1−β v + (σ(t),r)−v + (t,r) 1−β ⎪ , t , t ⎪  μ(t) μ(t) ⎨ v(σ(t)) gH v(t) 1−β t = or   ⎪ μ(t) ⎪ r ⎩ v+ (σ(t),r)−v+ (t,r) t1−β , v− (σ(t),r)−v− (t,r) t1−β , μ(t)

μ(t)

which is the required result. Since v is (β)-differentiable at t, v (β) (t) is a fuzzy number. Therefore, it follows that (i) or (ii) holds. Reciprocally, we prove that if case (i) or (ii) holds, then v is (β)-differentiable at t. Indeed, for example, let case (i) be valid. So, by Proposition 1, intervals [Tβ v − (t, r), Tβ v + (t, r)] determine a fuzzy number and it is obvious to see that v is (β)-differentiable at t.

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Based on Theorem 3, the (β)-differentiability is characterized by two cases, as follows. Definition 8. Let v : T → RF with [v(t)]r = [v − (t, r), v + (t, r)], for all r ∈ [0, 1]. Let v − (t, r) and v + (t, r) be conformable differentiable at t. I. Let v be (β, 1)-differentiable at t if v (β) (t, r) = [Tβ v − (t, r), Tβ v + (t, r)], ∀r ∈ [0, 1]. II. Let v be (β, 2)-differentiable at t if v (β) (t, r) = [Tβ v + (t, r), Tβ v − (t, r)], ∀r ∈ [0, 1]. Theorem 4.(1) Let v, w : T → RF be (β, 1)-differentiable (or (β, 2)-differentiable) at t ∈ Tk , then v + w : T → RF is (β)-differentiable at t and (v + w)(β) (t) = v (β) (t) + w(β) (t). (2) Let μ ∈ R and v be (β)-differentiable at t, then μv : T → RF is (β)differentiable at t and (μv)(β) (t) = μv (β) (t). Proof.(1) Let v, w be (β, 1)-differentiable, then we get −

+

[(v + w)(β) (t)]r = [Tβ (v + w) (t, r), Tβ (v + w) (t, r)] = [Tβ v − (t, r) + Tβ w− (t, r), Tβ v + (t, r) + Tβ w+ (t, r)] = [Tβ v − (t, r), Tβ v + (t, r)] + [Tβ w− (t, r), Tβ w+ (t, r)] = [v (β) (t)]r + [w(β) (t)]r , ∀r ∈ [0, 1]. The proof of another case is analogous. (2) It is an immediate consequence of Definition 8.

4

Conclusions

In this study, we have introduced the new notion of fractional derivative for fuzzy functions on time scales. We have also proved the characterization theorem for the introduced derivative in terms of the differentiability of the endpoint functions. The obtained results can be applied in further research for the investigation of fractional fuzzy differential equations on time scales. Acknowledgements. This article was partially supported by FAPESP under grants no. 2022/00196-1 and 2020/09838-0 and by CNPq under grant no. 313313/2020-2.

Some Properties of a New Concept of Fractional Derivative

53

References 1. Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Studies in Fuzziness and Soft Computing, Springer, London (2013). https://doi.org/10.1007/978-3-642-35221-8 2. Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013) 3. Benkhettou, N., Hassani, S., Torres, D.F.M.: A conformable fractional calculus on arbitrary time scales. J. King Saud Univ.-Sci. 28, 93–98 (2016) 4. Bohner, M., Peterson, A.: Dynamic Equations on Time Scale: An Introduction with Applications. Birkhauser, Boston (2001) 5. Caputo, M., Fabrizio, M.: Damage and fatigue described by a fractional derivative model. J. Comput. Phys. 293, 400–408 (2015) 6. Caputo, M., Cametti, C.: Memory diffusion in two cases of biological interest. J. Theor. Biol. 254, 697–703 (2008) 7. Fard, O.S., Bidgoli, T.A.: Calculus of fuzzy functions on time scales (I). Soft. Comput. 19, 293–305 (2014) 8. Hilger, S.: Ein Maßkettenkalk¨ ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. dissertation, University of W¨ urzburg (1988) 9. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) 10. Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, vol. 84. Springer, Cham (2011). https://doi.org/10.1007/ 978-94-007-0747-4 11. Shahidi, M., Khastan, A.: New fractional derivative for fuzzy functions and its applications on time scale. In: Pinto, C.M. (ed.) Nonlinear Dynamics and Complexity: Mathematical Modelling of Real-World Problems. Nonlinear Systems and Complexity, vol. 36, pp. 337–354. Springer, Cham (2022). https://doi.org/10.1007/ 978-3-031-06632-0 16 12. Si, G., Sun, Z., Zhang, H., Zhang, Y.: Parameter estimation and topology identification of uncertain fractional order complex networks. Commun. Nonlinear Sci. Numer. Simul. 17, 5158–5171 (2012) 13. Vasavi, C., Kumar, G.S., Murty, M.S.N.: Fuzzy Hukuhara delta differential and applications to fuzzy dynamic equations on time scales. J. Uncertain Syst. 83, 163–180 (2016) 14. Vasavi, C., Kumar, G.S., Murty, M.S.N.: Generalized differentiability and integrability for fuzzy set-valued functions on time scales. Soft. Comput. 20, 1093–1104 (2016) 15. Wang, Z., Huang, X., Shen, H.: Control of an uncertain fractional order economic system via adaptive sliding mode. Neurocomputing 83, 83–88 (2012) 16. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

On Monotonic Function Method for Generating Fuzzy Similarity Measures Surender Singh(B)

and Koushal Singh

School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, India [email protected]

Abstract. The concept of a fuzzy set covers the indefiniteness of a fuzzy term using a set-theoretic representation. The fuzzy similarity metrics quantify the proximity of two fuzzy sets. Such computations are unavoidable in human-centered computation covering a wide range of applications in interdisciplinary research, including clustering, diagnostic problems, pattern classification, decision-making, etc. In this article, we derive a fuzzy similarity measure using some monotonic functions. Considering some shortcomings of existing similarity measures, we compare the proposed measures with known compatibility measures using some traditional tools such as linguistic structured variables and classification examples. A comparative study demonstrates the purpose of the suggested fuzzy similarity measure. Keywords: Monotonic function · Fuzzy set linguistic variables · Pattern classification

1

· Similarity measure ·

Introduction

In 1965, Zadeh [1] developed the notion of fuzzy sets (FSs) as a means of coping with ambiguity and vagueness in real-life scenarios. A measure of similarity is crucial to various studies concerning behavioral analysis, pattern classification, computer vision, and machine learning. Tversky [2] considered features as objects in a set and conceptualized a set-theoretic approach to measure the similarity of two feature spaces. The quantification of similarity using various kinds of categorical data in different paradigms may give different results due to the usage of different similarity measures. For example, the matching of two images is qualitatively different from the matching of the behavior of two persons, thereby needing to be dealt with using different measures of comparison. So, formulation of a novel similarity measure in any framework is always a theoretical and practical contribution. Various studies investigated the implications of different similarity measures. Some classical approaches are Hamming distance, the closeness of sets, similarity as dual of distance, etc. In 1983, Wang [3] applies the idea to fuzzy theory and provided a precise formula to compute the similarity level of pair of FSs. Pappis [4] introduced the notions of closeness measure and approximate equal fuzzy sets as well as addressed the idea of approximating c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 54–64, 2023. https://doi.org/10.1007/978-3-031-39774-5_7

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55

fuzzy values. Chen et al. [5] extend the work of Pappis and Karacapilidis [6] to show and contrast the characteristics of various fuzzy value similarity measures. In this article, we develop an approach to derive fuzzy similarity measures using monotonic functions. The remaining content of the paper is divided into 7 sections. Section 2 presents the related studies of the present research work. Section 3 of this paper includes some basic definitions and some existing measures of comparison. Section 4 presents a new similarity measure using a novel result. Some numerical experiments concerning the proposed similarity measure are presented in Sect. 5. Section 6 offers the theoretical investigations of some existing and proposed similarity measures. Section 7 presents the application of the suggested similarity measure in the pattern classification problem. Finally, Sect. 8 presents the concluding remarks.

2

Literature Review

In this section, we present a brief review of the relevant literature. After initial developments ([2,3], and [4]) concerning fuzzy similarity measures, Kwang et al. [7] introduced two similarity metrics; the similarity between fuzzy sets and between elements in the fuzzy set with an example to show that the proposed measure can be used in the behavior analysis of an organization. Zwick et al. [8] in an experiment assessed nineteen metrics and compared how well they performed. Wang [9] introduced new fuzzy similarity measures and illustrate some examples to draw comparisons between existing and proposed measures. Xucheng [10] studied axiomatic definitions of similarity, distance, and entropy measures of fuzzy sets and their basic relationships. Adlassing [11], Chen [12], Tong and Bonissone [13], Chen and Tan [14], and Chen [15] also present significant interdisciplinary applications of the fuzzy similarity measures. Bustince et al. [16] studied the concept of restricted equivalence functions and show that how restricted equivalence functions are utilized to generate similarity measures for fuzzy sets. Bustince et al. [17] established the interrelationships among the concepts, restricted equivalence, restricted dissimilarity, and normal E N functions. Couso et al. [18] reviewed important axiomatic definitions of similarity and dissimilarity measures by referring to the axioms of each specific definition. The present study is mainly concerned with the similarity measure of fuzzy sets. The primary motivation behind considering the present study is that the performance of commonly used similarity metrics is sometimes not satisfactory. We point out some pitfalls of the existing fuzzy compatibility measures. • Some fail to satisfy the necessary mathematical properties. • Same similarity value is obtained for different pairs of fuzzy sets. This also leads to unsatisfactory results in pattern classification problems. • Unable to capture the structured linguistic variables in the desired theoretical and practical sense. The existing similarity measures encounter one or more above-mentioned shortcomings. The proposed measure is expected to remove the above shortcomings.

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In this article, we introduce a novel result that enables us to formulate new similarity measures for fuzzy sets by the transformation of a monotonic function of a single variable. Using this result, we formulate a fuzzy similarity measure and investigate its implications and advantages.

3

Preliminaries

In this section, we present some definitions along with some fundamental operations that are related to the current research. A precise definition of a fuzzy set is as follows. Definition 1 ([1]): Let U be a universal set. Then a fuzzy set J in U is defined and represented as follows J = {(ui , μJ (ui )) | ui ∈ U } , where μJ : U → [0, 1] represents membership function. Definition 2 ([12]): Let S : F S (U ) × F S (U ) → [0, 1] be a real function. S is said to be a similarity measure if for all J, K and L ∈ F S (U ) it satisfies the following properties: (S1) : S (J, K) = 1 ⇔ J = K; ( S2): S (J, J c ) = 0 iff J is a crisp set; (S3) : S (J, K) = S (K, J) ; (S4) : If J ⊆ K ⊆ L, then S (J, L) ≤ S (J, K) and S (J, L) ≤ S (K, L) . In the following, we include some existing and prominent fuzzy compatibility measures. 3.1

Some Existing Similarity/correlation Measures

Similarity measures due to Wang [9]  n  1  min (μJ (ui ) , μK (ui )) S 1 (J, K) = n i=1 max (μJ (ui ) , μK (ui )) S 2 (J, K) =

n  [1 − |μJ (ui ) − μK (ui )|] i=1

n

(1)

(2)

Similarity measure due to Hyung et al. [7] S 3 (J, K) = max min (μJ (ui ) , μK (ui )) ui ∈U

(3)

On Monotonic Function Method for Generating Fuzzy Similarity Measures

Similarity measures due to Pappis and Karacapilidis [6] n min (μJ (ui ) , μK (ui )) S 4 (J, K) = ni=1 max (μJ (ui ) , μK (ui )) i=1 S 5 (J, K) = 1 − maxi (|μJ (ui ) − μK (ui )|) n |μJ (ui ) − μK (ui )| S 6 (J, K) = 1 − i=1 n i=1 |μJ (ui ) + μK (ui )|

57

(4) (5) (6)

Correlation measure due to Gerstenkorn and Manko [19] C 1 (J, K) = 

K (J, K)

, (7) [T (J) · T (K)]  n  2 2 where T (J) = i=1 (μJ (ui )) + (νJ (ui )) , νJ (ui ) = 1 − μJ (ui ) and n K (J, K) = i=1 [μJ (ui ) μK (ui ) + νJ (ui ) νK (ui )]. In the following section, we propose a similarity measure of fuzzy sets that strictly follows the requirements in Definition 2.

4

New Fuzzy Similarity Measure

In this section, first, we prove a result to derive new fuzzy similarity measures using monotonic functions. Theorem 1. Let f : [0, 1] → [0, 1] be a monotonically decreasing function with boundary conditions f (0) = 1 and f (1) = 0. Let J and K be two fuzzy sets in a finite universe U with membership function μJ and μK respectively, then n

S(J, K) =

1 f (|μJ (ui ) − μK (ui )|), ∀ ui ∈ U n i=1

(8)

is a fuzzy similarity measure. Proof. We examine the axioms S1 -S4. (S1) Sufficient: Suppose J = K implies that μJ (ui ) = μK (ui ) the from Eq. (8) we have n S(J, K) = n1  i=1 f (0) n ⇒ S(J, K) = n1 i=1 1 ⇒ S(J, K) = 1. Necessary : Suppose S(J, K) = 1 then we have n 1 f (| μ J (ui ) − μK (ui ) |) = 1 i=1 n ⇒ f (| μJ (ui ) − μK (ui ) |) = 1 ⇒| μJ (ui ) − μK (ui ) |= 0 ⇒ μJ (ui ) = μK (ui ).

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Now, we prove S2. (S2) Sufficient: Suppose J is a crisp set, that is, J = 0 or J = 1. J = 0 implies μJ (ui ) = 0 and μJ c (ui ) = 1 then from Eq. (8), we have n S(J, J c ) = n1 i=1 f (| 1 |) c ⇒ S(J, J ) = 0. Similarly, we can prove for J = 1. Necessary : Suppose S(J, J c ) = 0 then we have n 1 i=1 f (| μJ (ui ) − μJ c (ui ) |) = 0 n ⇒ f (| μJ (ui ) − μJ c (ui ) |) = 0 ⇒| μJ (ui ) − μJ c (ui ) |= 1 The above equation hold only if μJ (ui ) = 0 and μJ c (ui ) = 1 or μJ (ui ) = 1 and μJ c (ui ) = 0. ⇒ J is a crisp set. Now, we prove S3. (S3): From the expression in Eq. (8) it is very easy to prove this property. Now, we prove S4 (S4): Suppose J ⊆ K ⊆ L then we have μJ (ui ) ≤ μK (ui ) ≤ μL (ui ). ⇒| μJ (ui ) − μL (ui ) |≥| μK (ui ) − μL (ui ) |. Since f is monotonically decreasing, we have (ui ) − μL (ui ) |) ≤ f (| μK (ui ) −  μL (ui ) |) f (| μJ n n ⇒ n1 i=1 f (| μJ (ui ) − μL (ui ) |) ≤ n1 i=1 f (| μK (ui ) − μL (ui ) |), ⇒ S(J, L) ≤ S(K, L). Similarly, we can prove that S(J, L) ≤ S(J, K). This proves S4 and hence the Theorem 1. Now, we suggest the new fuzzy similarity measure with the help of Theorem 1. Suppose 1 1 π 1 π π π 1 (9) f (x) = (Sin( x)e(1− 2 Sin( 2 x)) − (1 − (Sin( x)e 2 (Sin( 2 x) ) 2 2 2 2 The function given in Eq. (9) is a monotonically decreasing function with f (0) = 1 and f (1) = 0 then by using Theorem 1, we have the following similarity measure defined in Eq. (10). n 1 π π 1  1 1 (1− 12 Sin( π x)) (Sin( π x) 2 2 2 (Sin( x)e S(J, K) = − (1 − (Sin( x)e ) , n i=1 2 2 2 2 (10) where x =| μJ (ui ) − μK (ui ) | . With the aid of various numerical experiments, we analyze the advantages of the proposed similarity measure given in Eq. (10) as well as some pitfalls of the existing similarity metrics in the following section.

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5

59

Numerical Findings

Several studies employ numerical experiments as primary tool to assess the suitability of the similarity measures. Moreover, an effective similarity measure strictly follows the axiomatic requirements S1−S4. We numerically compare the efficiency of our suggested similarity measure with some existing measures presented in Sect. 3.1. The numerical experiment uses nine different fuzzy instances and is shown in Table 1.

Table 1. Different fuzzy instances. Different Instances Universe of discourse

Fuzzy sets

Instance 1

One element

{0.0}

{0.0}

Instance 2

One element

{0.1}

{0.1}

Instance 3

One element

{0.591}

{0.444}

Instance 4

One element

{0.612}

{0.781}

Instance 5

One element

{1.0}

{0.5}

Instance 6

One element

{0.1}

{0.2}

Instance 7

Three elements

{0.440, 0.690, 0.722} { 0.201, 0.341, 0.450}

Instance 8

Three elements

{0.443, 0.688, 0.721} {0.201, 0.341, 0.450}

Instance 9

Three elements

{0.443, 0.688, 0.721}

{0.201, 0.341, 0.450}

Table 2 contains a list of the experimental findings of the data in Table 1, where bold values show the counter-intuitive findings. The counter-intuitive findings in Table 2 are explained in the following remarks. Remark 1: The similarity measures S 1 , S 4 , and S 6 creates output as not a number (NaN ) for the fuzzy sets (Instance 1) whereas the similarity measure S 3 gives output as zero. This violates the axiom S1 of the definition of the similarity measure. Remark 2: For two equal fuzzy sets (Instance 2), the value of the similarity is equal to 1 but the measure S 3 provides results as 0.1. Remark 3: The similarity measures S 1 , S 4 and S 6 assign the same level of similarity Instance 5 (normalized fuzzy set)and Instance 6 but the proposed similarity differentiate these two instances distinctively. Remark 4: The correlation measure C 1 calculates the same level of similarity to Instance 3 and Instance 4 that might not be appropriate in various real-life applications. Remark 5: The similarity measures S 2 , S 3 , S 4 , S 5 , and S 6 cannot differentiate Instance 7, Instance 8, and Instance 9. In contrast, the proposed measure faced no counter-intuitive situations. This reflects the significance and advantage of the suggested similarity metric. In the next section, we carry out some theoretical investigation and comparison to test the effectiveness of the suggested similarity metric because of structured linguistic variables.

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S. Singh and K. Singh Table 2. Comparative findings using different measures.

Instances→ Measures↓

1

2

3

4

5

6

7

8

9

S1

NaN

1.0000

0.7513

0.7836

0.5000 0.5000 0.5248

S2

1.0000

1.0000

0.8530

0.8310

0.5000

0.9000

0.7133 0.7133 0.7010

0.5245

0.5145

S3

0.0000 0.1000 0.4440

0.1000

0.4500 0.4500 0.4500

S4

NaN

S5 S6 C7

0.6120

0.5000

1.0000

0.7513

0.7836

0.5000 0.5000 0.5356 0.5356 0.5251

1.0000

1.0000

0.8530

0.8310

0.5000

NaN

1.0000

0.8580

0.8787

0.6667 0.6667 0.6976 0.6976 0.6886

1.0000

1.0000

0.9578 0.9578 0.7071

0.9910

1.4820

1.4823

1.4649

S(Proposed) 1.0000

1.0000

0.7155

0.8002

0.4991

0.4989

0.4818

6

0.6779

0.2458

0.9000

0.6510

0.6530 0.6530

Comparative Analysis Based on structured linguistic variables

Zadeh [20] proposed the idea of structural linguistic variables. The number of structured linguistic variables that must be defined for analysis depends on the system’s requirements. The modifier Z s of a FS Z in a universe U is defined as s

Z s = {(ui , (μZ (ui )) ) | ui ε U, i = 1, 2, 3, . . . , m } ,

(11)

where ‘s’ is a positive real number. We define a number of linguistic variables with the aid of Eq. (11), including More or Less Large (M.L.L), Large (L.), Very Large (V.L.), Very-Very, and Large (V.V.L.). Now, we use several fuzzy similarity metrics to determine how similar these variables are in the following example. Example 1: Assume an FS Z in a finite universal set U = {u1 , u2 , u3 , u4 } given as; Z = {(u1 , 0.0002) , (u2 , 0.0001) , (u3 , 0.9310) , (u4 , 0.8709)} . 1

The FSs Z 2 , Z 2 and Z 4 is computed using Eq. (11). Consider Z as ‘Large’, 1 Z 2 as ‘More or Less Large’, Z 2 means ‘Very Large’, while Z 4 means ‘Very-Very Large’. In general, (M.L.L.) will be more similar to the (L.) than that of (V.L.). The following inequalities are described mathematically and should be met by a reliable fuzzy similarity/correlation measure. S(M.L.L., L.) > S(M.L.L., V.L.) > S(M.L.L., V.V.L.)

(12)

S(L., M.L.L.) > S(L., V.L.) > S(L., V.V.L.)

(13)

S (V.L., L.) > S (V.L., V.V.L.) > S (V.L, M.L.L.)

(14)

S (V.V.L., V.L.) > S (V.V.L., L.) > S (V.V.L., M.L.L.)

(15)

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Table 3. Results generated due to various measures.

Fuzzy comparison measures Remark S1

Fail to satisfy the inequality (14)

S2

Fail to satisfy the inequality (14)

S3

Fail to satisfy the inequalities (14) and (15)

S4

Fail to satisfy the inequality (14)

S5

Fail to satisfy the inequality (14)

S6

Fail to satisfy the inequality (14)

C1

Fail to satisfy the inequality (14)

S(Proposed)

Satisfy all the inequalities (12)-(15)

Table 3 presents the computational results concerning inequalities (12)-(15). We observe that the proposed similarity measure is more reliable than existing measures S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , and C 1 listed in Sect. 3.1 while dealing with linguistic variables.

7

Application in Pattern Classification Problem

In this section, we discuss an application of the suggested similarity metric (Eq. (10)) in pattern identification problems, and the outcomes are compared with some existing comparison measures. The following is a description of a pattern recognition problem in a fuzzy environment: Circumstance: Let Jk , k = 1, 2, 3, . . . , m be some known patterns and I be an unknown pattern as Jk = {(ui , μJk (ui ) ) | ui ∈ U, i = 1, 2, 3, . . . , n } , I = {(uk , μI (uk )) | uk ∈ U, k = 1, 2, 3, . . . , n } . Goal: To recognize the specified (known) pattern Jk , k = 1, 2, 3, . . . , m with which the unspecified (unknown) pattern I has the greatest closeness. Recognition principle:The following approaches can be used for the identification of the unknown pattern I to one the known patterns Jk , k = 1, 2, 3, . . . , m. (a) Similarity/correlation method: Consider S(Jk , I) be the fuzzy similarity/correlation measure between the specified pattern Jk , k = 1, 2, 3, . . . , m and the unspecified pattern I, then I is assigned to Jk , where Jk = max S (Jk , I) , k = 1, 2, . . . , m.

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Example 2: Let the five known patterns be J1 , J2 , J3 , J4 , J5 and an unidentified pattern I represented in the form of FSs as shown in Table 4. Now, we use the above-mentioned method to compute the value between the known and the unknown patterns. The calculated values between the known and the unknown patterns using the proposed similarity measure and existing compatibility measures are shown in Table 5.

Table 4. Fuzzy data related to Example 2. Known patterns J1 J2 J3 J4 J5

= {u1 , = {u1 , = {u1 , = {u1 , = {u1 ,

0.832 , 0.441 , 0.440 , 0.443 , 0.835 ,

Unknown pattern u2 , u2 , u2 , u2 , u2 ,

0.686 , u3 , 0.688 , u3 , 0.690 , u3 , 0.688 , u3 , 0.688 , u3 ,

0.412} 0.760} 0.722} I = {u1 , 0.201 , u2 , 0.341 , u3 , 0.450} 0.721} 0.413}

From Table 5, we see that the existing measures failed to recognize the unknown pattern I to any of the known pattern but the proposed similarity measure classified the unknown pattern I to known patterns J3 . As a result, the pattern recognition problem that cannot be resolved by the existing measures has been effectively resolved by the proposed similarity measure. This demonstrates the effectiveness of the suggested similarity measures. Table 5. Computed values using different measures regarding Example 2. Fuzzy comparison measures (J1 , I) (J2 , I) (J3 , I) (J4 , I) (J5 , I) Result S1 S2 S3 S4 S5 S6 C1 S(Proposed)

8

0.5514 0.6620 0.4120 0.4848 0.3690 0.6530 1.2136 0.4917

0.5145 0.7010 0.4500 0.5251 0.6530 0.6886 1.4649 0.4818

0.5248 0.7133 0.4500 0.5356 0.6510 0.6976 1.4820 0.4991

0.5245 0.7133 0.4500 0.5356 0.6530 0.6976 1.4820 0.4989

0.5514 0.6607 0.4130 0.4840 0.3660 0.6523 1.2093 0.4908

Unclassified Unclassified Unclassified Unclassified Unclassified Unclassified Unclassified J3

Concluding Remarks and Future Work

This article introduced a new similarity measure of fuzzy sets using a novel approach. The suggested similarity measure demonstrated an excellent ability to

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compare two distinct FSs, and improved performance in identification of structured linguistic variables. Furthermore, the suggested fuzzy similarity metric provides better classification results in recognizing an unknown pattern. Our future studies includes the investigation regarding the applications of suggested similarity metrices in clustering, MADM, and image thresholding, and extension of the proposed method to other non-standard fuzzy environments.

Acknowledgments. Authors are highly thankful to anonymous reviewers and Editor for their constructive suggestions for improvement of the paper.

References 1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 2. Tversky, A.: Features of similarity. Psychol. Rev. 84, 327–352 (1977) 3. Wang, P.Z.: Fuzzy sets and its applications. Shangai Science and Technology Press vol. 10 (1983) 4. Pappis, C.P.: Value approximation of fuzzy systems variables. Fuzzy Sets Syst. 39, 111–115 (1991) 5. Chen, S.M., Yeh, M.S., Hsiao, M.S.: A comparison of similarity measures of fuzzy values. Fuzzy Sets Syst. 72, 79–89 (1995) 6. Pappis, C.P., Karacapilidis, N.I.: A comparative assessment of measures of similarity of fuzzy values. Fuzzy Sets Syst. 56, 171–174 (1993) 7. Kwang, H.L., Song, Y.S., Lee, K.M.: Similarity measure between fuzzy sets and between elements. Fuzzy Sets Syst. 62, 291–293 (1994) 8. Zwick, R., Carlstein, E., Budescu, D.V.: Measures of similarity among fuzzy concepts: a comparative analysis. Int. J. Sci. Technol. 1, 221–242 (1987) 9. Wang, W.J.: New similarity measures on fuzzy sets and on elements. Fuzzy Sets Syst. 85, 305–309 (1997) 10. Xuecheng, L.: Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy Sets Syst. 52, 305–318 (1992) 11. Adlassnig, K.P.: Fuzzy set theory in medical diagnosis. IEEE Trans. Syst. Man Cybern. 16, 260–265 (1986) 12. Chen, S.M.: Similarity measures between vague sets and between elements. IEEE Trans. Syst. Man Cybern. 27, 153–158 (1997) 13. Tong, R.M., Bonissone, P.P.: A linguistic approach to decision making with fuzzy sets. IEEE Trans. Syst. Man Cybern. 10, 716–723 (1980) 14. Chen, S.M., Tan, J.M.: Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 67, 163–172 (1994) 15. Chen, S.M.: A weighted fuzzy reasoning algorithm for medical diagnosis. Decis. Support Syst. 11, 37–43 (1994) 16. Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets Syst. 157, 2333–2346 (2006). https://doi.org/10.1016/j.fss.2006.03.018 17. Bustince, H., Barrenechea, E., Pagola, M.: Relationship between restricted dissimilarity functions, restricted equivalence functions and normal EN-functions: image thresholding invariant. Pattern Recogn. Lett. 29, 525–536 (2008) 18. Couso, I., Garrido, L., S´ anchez, L.: Similarity and dissimilarity measures between fuzzy sets: a formal relational study. Inf. Sci. (Ny) 229, 122–141 (2013)

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19. Gerstenkorn, T., Ma´ nko, J.: Correlation of intuitionistic fuzzy sets. Fuzzy Sets Syst. 44, 39–43 (1991) 20. Zadeh, L.A.: A fuzzy-set-theoretic interpretation of linguistic hedges. J. Cybern. 2, 4–34 (1972)

Regularity and Paracompactness: Relation in the Field of Fuzziness Francisco Gallego Lupiáñez(B) Interdisciplinary Mathematics Institute (IMI) and Department of Mathematics, Universidad Complutense, 28040 Madrid, Spain [email protected]

Abstract. The importance of paracompactness (and the concepts related to it) in the field of Topology is well known. In this paper we obtain two characterizations of regular fuzzy topological spaces using Luo’s and Abd El-Monsef and others’ paracompact fuzzy topological spaces. Thus, it shows that regularity of fuzzy topological spaces can be considered as a paracompact-type property with several kinds of paracompact fuzzy topological spaces. Indeed, we prove that for a fuzzy Hausdorff fuzzy topological space (in any of Wuyts and Lowen’s definitions that are good extensions of Hausdorffness) there is a characterization using Luo’s paracompact fuzzy topological spaces, and also another result with a characterization using definition due to Abd El-Monsef, Zeyada, El-Deeb, and Hanafy. This supposes a stimulus for further investigations, tending to obtain characterizations of other fuzzy separation properties (for example fuzzy normality, fuzzy complete regularity,..) as fuzzy covering properties. Keywords: Topology · Fuzzy sets · Separation properties · Covering properties · Fuzzy paracompactness

1 Introduction Regularity is a very useful separation property of topological spaces. Some authors obtained characterizations of regularity as a covering property. In this paper we work on fuzziness, obtaining two characterizations of fuzzy regularity as a fuzzy covering property. Indeed, we show that one can characterize fuzzy regularity as a paracompacttype fuzzy property in Luo’s and Abd El-Monsef and others’ both senses. In Sect. 3 we give previous definitions to provide better readership and make the material more accessible to reader, and main results of this paper. Suggestions for future research are in the Conclusion section.

2 Literature Review There exist various different definitions of paracompactness of fuzzy topological spaces due to Luo [1] and Abd El-Monsef, Zeyada, El-Deeb, and Hanafy [2], On the other hand, various authors (Abdelhay [5], Boyte [6], Chew [7]) obtained characterization © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 65–68, 2023. https://doi.org/10.1007/978-3-031-39774-5_8

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of regularity as a covering property of Hausdorff topological spaces. A notion of a Hausdorff fuzzy property that is a good extension of classical topological property can be seen in [3]. Many other definitions and results on topological fuzziness can be found in book [4].

3 Definition and Main Results Definition 1 [1] . Let μ be a set in a fts (X , τ ) and let r ∈ (0, 1] , s ∈ [0, 1); we define. μ[r] = χ{x∈X :μ(x)r} μ(s) = χ{x∈X :μ(x)>s} μ = rμ[r] Definition 2 [1] . Let be a family of sets and μ be a set in a fts (X , τ ). We say is locally finite (resp. ∗ -locally finite) in μ for each point e in μ, there exists that ν ∈ Q(e) such that ν is quasi-coincident (resp. intersects) with at most a finite number ; we often omit the word “in μ” when μ = X . of sets of is called a Q-cover of a set µ if for each Definition 3 [1] . A family of sets such that ν and μ are quasi-coincident at x. Let x ∈ supp(μ), there exist a ν ∈ is called a r− Q cover of μ if is a Q-cover of μr. r ∈ (0, 1]. Definition 4 [1] . Let r ∈ (0, 1] , μ be a set in a fts (X , τ ). We say that μ is r-paracompact (resp. r ∗ -paracompact) if for each r-open Q-cover of μ there exists an open refinement of it which is both locally finite (resp. ∗ -locally finite) in μ and a r-Q-cover of μ. The fuzzy set μ is called S-paracompact (resp. S*-paracompact) if for every r ∈ (0, 1], μ is r-paracompact (resp. r ∗ -paracompact). Definition 5 [2] . A family of fuzzy sets .

is called an L-cover of a fuzzy set μ if

Definition 6 [2] . Let μ be a fuzzy set in a fts (X , τ ). We say that μ is fuzzy paracompact of μ and for each ξ ∈ (0, 1], (resp. ∗ -fuzzy paracompact) if for each open L-cover of which is both locally finite (resp. ∗ -locally there exists an open refinement finite) in μ and L-cover of μ − ξ . We say that a fts (X , τ ) is fuzzy paracompact (resp. ∗ -fuzzy paracompact) if each constant set in X is fuzzy paracompact (resp. ∗ -fuzzy paracompact). Theorem 1. Let (X , τ ) a fuzzy Hausdorff fts (in any of Wuyts and Lowen’s definitions that are good extensions of Hausdorffness). Then (X , τ ) is fuzzy regular if and only if for each r ∈ (0, 1], for each r-open Q-cover of (X , τ ) and for each fuzzy point xλ of X there exists an open refinement of it which is both ∗ -locally finite in xλ and a r-Q-cover of (X , τ ).

Regularity and Paracompactness

67

Proof. (⇒) For each r ∈ (0, 1], let be a r-open Q-cover of (X , τ ), and xλ be a fuzzy , is an open point of X. Then, we have that the family of crisp sets cover of (X , [τ ]), which is Hausdorff and regular ([3, 4]). Then ([5–7]), it has an open which is a cover of X , and is locally finite in x. For each refinement with V ⊂ (UV )(1−r) .

we have an

Then,

Let

, is both an open refinement of

and a r − Q-cover of (X , τ ), and also is ∗ -locally finite in xλ , indeed, because is locally finite in x, we have an open neighbourhood G of x that G intersects with only finite number of members of . Then χG ∈ Q(xλ ) intersects with only a finite number of members of

. ⊂ [τ ] be an open cover of (X , [τ ]); then

is an open

Q-cover of 1X , and, for each x ∈ X , it has an open refinement

which is a Q-cover of

(⇐) Let

1X and also locally finite in x1−r . Let ; then is both a refinement of and a cover of (X , [τ ]). Also, is locally finite in x. Indeed: we take O1 ∈ Q(x1−r ) such that O1 , is quasi-coincident with only a finite number of members V1 , ..., Vn , of . Let , then x ∈ O ∈ [τ ]. For each , if O ∧V(1−r) = ∅, we have a crisp point y ∈ X , such that O1 (y) > r, V (y) > 1 − r, O1 (y) + V (y) > 1, then O1 qV and V ∈ {V1 , ..., Vn }. Hence the neighborhood O of x intersects with only a . finite number of members Theorem 2. Let (X , τ ) be a fuzzy Hausdorff fts (in any of Wuyts and Lowen’s definitions that are good extensions of Hausdorffness [3]). Then (X , τ ) is fuzzy regular if and only if for each r ∈ I , and for each open L-cover of r, for each ξ ∈ (0, 1], and for each of it which is both ∗ -locally fuzzy point xλ of X , there exists an open refinement finite in xλ and L-cover of r − ξ. of r, for each ξ ∈ Proof (⇒) For each r ∈ I , and for each open L-cover (0, 1], and for each fuzzy point xλ of X , we have that the family of crisp sets is an open cover of (X , [τ ]) which is Hausdorff and regular ([3, 4]). Then ([5–7]), it has an open refinement which is a cover of , there exists , such that X and is locally finite in x. For each V ⊂ GV−1 ((r − ξ, 1]). So there exists and

is refinement of

. Then,

such that x ∈ V and GV (x) > r − ξ. So, (χV ∧ GV )(x) ≥ r − ξ, Since

is locally finite in x, there exists A ∈ [τ ]

with x ∈ A, such that intersects with at most a finite number of members of . Then, there exists χA ∈ τ such that xλ qχA and χA intersects with a finite number of fuzzy sets . of be an open cover of X and x ∈ X , then (⇐) Let is an open L-cover of (X , τ ) and for each, r ∈ I is

. For each

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ξ ∈ (0, 1], there exists an open refinement xλ and L-cover of r − ξ . This implies that , then for each Since

there exists , then

of

which is both locally finite in for all ξ1 > ξ. Let

is an open refinement of

(indeed,

such that G − ((r − ξ, 1]) ⊂ VG ). is an open refinement of

. And, since x ∈

X there exists A ∈ Q(xλ ) which intersects with only Since A(x)+λ > 1, we have A(x) > 1 − λ, then x ∈ A−1 ((1 − λ, 1]) ∈ [τ ]. If A−1 ((1 − λ, 1])∩G −1 ((r − ξ, 1]) = ∅, there exists some point z, such that A(z) > 1 − λ and G(z) > r − ξ , so A ∧ G = ∅. Then, if the neighbourhood A−1 ((1 − λ, 1]) of , A intersects with infinite members of . x intersects with infinite members of is locally finite in x. Thus This yields that the Hausdorff topological space (X , [τ ]) is regular ([5–7]) and (X , τ ) is fuzzy regular ([4]).

4 Usability and Applications This a paper on fuzzy topology, therefore it is a theoretical paper. However, many papers on fuzzy mathematics are having applications to engineering, computing, artificial intelligence, medicine, and other sciences, in particular, some papers about fuzzy topology authored by ours.

5 Conclusion In this paper, fuzzy regularity is characterized as a fuzzy covering property. Future research could obtain characterization of other fuzzy separation properties (fuzzy normality, fuzzy completely regularity) as fuzzy covering properties. Obviously, these are all theoretical problems, but this is not a limitation, because many theoretical findings have further practical applications.

References 1. Luo, M.K.: Paracompactness in fuzzy topological spaces. J. Math. Anal. Appl. 130, 55–77 (1988) 2. Abd El-Monsef, M.E., Zeyada, F.M., El-Deeb, S.N., Hanafy, I.M.: Good extensions of paracompactness. Math. Japonica 37, 195–200 (1992) 3. Wuyts, P., Lowen, R.: On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces, and fuzzy uniform spaces. J. Math. Anal. Appl. 93, 27–41 (1983) 4. Liu, Y.-M., Luo, M.-K.: Fuzzy Topology. World Scientific Publishing, Singapore (1997) 5. Abdelhay, J.: Characterizaçao dos espaços topológicos regulares e normais por meio de coberturas, Gaz. Mat. (Lisboa) 9(37–38), 8–9 (1948) (in Portuguese) 6. Boyte, J.M.: Point (countable) paracompactness. J. Austral. Math. Soc. 15, 138–144 (1973) 7. Chew, J.: Regularity as a relaxation of paracompactness. Amer. Math. Monthly 79, 630–632 (1972)

Fuzzy Threshold Aggregation Alexander Lepskiy(B) Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia [email protected] https://www.hse.ru/en/org/persons/10586209 Abstract. The threshold aggregation rule used to rank alternatives that are evaluated against a set of criteria is known in decision theory. The generalization of the threshold aggregation rule to the case when the estimates in the alternatives are described by fuzzy numbers is considered in the paper. The fuzzy threshold aggregation procedure has been developed and studied. The procedure for blurring point data by information about the reliability of such data has been developed and studied as well. An example of using fuzzy threshold aggregation when making decisions on the admission of articles in conference management systems is considered (for example, EasyChair). Keywords: Threshold rule · Aggregation of alternatives criteria · Conference management systems

1

· Fuzzy

Introduction

The rules for aggregation of individual preferences are considered and investigated in the social choice theory [1]. Such rules are used in the problem of ranking alternatives, each of which is evaluated according to a certain set of criteria. In some cases, aggregation rules should be non-compensatory. This implies that low scores on one criterion cannot be compensated for by high scores on others. For example, such a rule is often used when making decisions about the publication of articles based on the feedback of several reviewers, when choosing products by characteristics, etc. The so-called threshold rule [2,3] is one of the popular aggregation rules that has a non-compensatory property. In some cases, some or all of the characteristics of alternatives may be fuzzy. Then the problem of generalizing the threshold aggregation rule to the case of fuzzy data is relevant. The present article is devoted to the solution of this problem. The application of the proposed generalization of the threshold aggregation rule to fuzzy data is illustrated by the example of ranking articles according The results of the project “Study of models and methods of decision-making under conditions of deep uncertainty: anticipating natural disasters and logistics challenges” carried out within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE University) in 2023, are presented in this work. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 69–76, 2023. https://doi.org/10.1007/978-3-031-39774-5_9

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to the results of (point) assessments of reviewers and considering the degrees of confidence in their assessments. These degrees of confidence are used to blur point estimates and generate fuzzy data. The general properties of such blurs are also discussed in the article. The rest of the article has the following structure. The axiomatics of the nonfuzzy threshold aggregation rule is discussed in Sect. 2. In Sect. 3, the problem of threshold aggregation with fuzzy data is formulated and a general approach to solving this problem is discussed. The formation of fuzzy estimates from point expert data and information about their reliability is discussed in Sect. 4. Section 5 gives a numerical example of applying the threshold aggregation rule with fuzzy data when reviewing articles in a conference management system. Finally, Sect. 6 draws some conclusions from the study.

2

Non-Fuzzy Formulation of the Threshold Aggregation Problem

The problem of ranking alternatives of a certain set X of evaluated by n criteria in a three-gradation scale is being considered. We can assume in this case that the alternatives are represented by n-dimensional vectors: x = (x1 , ..., xn ), where xi ∈ {1, 2, 3}. It is required to find such a transformation (aggregation operator) ϕn = ϕ : X → R that satisfies the conditions (axioms) [3]: 1) Pareto-domination: if x, y ∈ X and xi ≥ yi ∀i, ∃s : xs > ys , then ϕ(x) > ϕ(y); 2) pairwise compensability of criteria: if x, y ∈ X and vk (x) = vk (y) k = 1, 2, then ϕ(x) = ϕ(y), where vk (x) = |{i : xi = k}| is the number of estimates of k in the alternative x, k = 1, 2, 3; 3) threshold noncompensability: ϕ(2, . . . , 2) > ϕ(x) ∀x ∈ X: ∃s : xs = 1;    n

4) the reduction axiom: if ∀x, y ∈ X ∃s : xs = ys , then ϕn (x) > ϕn (y) ⇔ ϕn−1 (x−s ) > ϕn−1 (y−s ), where x−s = (x1 , . . . , xs−1 , xs+1 , . . . , xn ). This problem was formulated and studied in [3]. It is shown that the lexicographic aggregation rule is a solution to this problem: ϕ(x) > ϕ(y) ⇔ ∃j ∈ {1, 2} : vk (x) = vk (y) ∀k ≤ j and vk+1 (x) < vk+1 (y). This problem was generalized in [2] to the case of m-gradation scales, m ≥ 3.

3

Formulation and Solution of the Problem of Threshold Aggregation with Fuzzy Data

Let us now assume that the alternatives are represented by n-dimensional vectors  = ( n ). Each fuzzy number belongs to one of three of fuzzy numbers x x1 , . . . , x classes: the low score class L, the median score class M , or the high score class H. The distribution of fuzzy numbers by class can either be known in advance, or can be determined by the nearest neighbour method with respect to reference

Fuzzy Threshold Aggregation

numbers L0 , M0 and H0 , and each class xi ∈

71

arg min d(xi , S0 ), where d is

S∈{L,M,H}

some metric (or pseudometric) on the set of fuzzy numbers [5]. We will assume that the supports of all fuzzy estimators of the class L are located on the segment [−a, 0], a > 0, the supports of all fuzzy estimators of the class H are located on the segment [0, a], a > 0, and the supports of all fuzzy estimators of the class M are located on the segment [−b, b], 0 < b < a. Crisp numbers L0 = −a, M0 = 0 and H0 = a are reference elements. x∈M : We will consider a set of median positive estimates M + = { x, M0− ) and a set of median negative estimates M − = { x∈M : d( x, M0+ ) ≤ d(  − + d( x, M0 ) ≤ d( x, M0 ) , which are subsets of the set of median estimates M , where M0− = −b, M0+ = b are the reference estimates of subclasses M − and M + , respectively. x, M0+ ) holds Note that those estimates for which equality d( x, M0− ) = d( (and only they) fall into both subsets. General scheme of threshold fuzzy ranking.  = ( n ) and The following steps are performed for each alternative x x1 , . . . , x each class S ∈ {L, M − , M + , H}. 1. The distances d (xi , S0 ) between all estimates xi ∈ S of one class and the reference fuzzy (or crisp) number S0 of this class are determined (the values d (xi , M0 ) and d xi , M0± are calculated for the class M ). 2. The normalized function FS (xi ) = ϕ (d (xi , S0 )) of the proximity of the estimate to the reference number of the class is calculated, where nonincreasing function ϕ : [0, +∞) → [0, 1] satisfies the condition ϕ(0) = 1. The value FS (xi ) characterizes the normalized degree of confidence that the estimate xi ∈ S. It can be considered as a function of belonging to a subset S defined on a set of fuzzy numbers. 3. Let’s find values  x) = FS (xi ). (1) vS ( xi ∈S

 = ( for the alternative x x1 , . . . , x n ) and each class S ∈ {L, M − , M + , H}. The value vS (x) characterizes the cardinality of the set of fuzzy estimates of the class S. x) < vL ( y) or 4. Let’s apply the threshold aggregation rule: ϕ ( x) > ϕ ( y) ⇔ vL ( x) = vL ( y), vM − ( x) < vM − ( y) or vL ( x) = vL ( y), vM − ( x) = vM − ( y), vL ( x) < vM + ( y) or vL ( x) = vL ( y), vM − ( x) = vM − ( y), vM + ( x) = vM + ( y), vH ( x) < vH ( y). vM + ( Remark 1. If the fuzzy estimates are crisp numbers on a three-gradation x) coincide with the values ,vS (x) = scale {L0 , M0 , H0 }, then the values vS ( |{i : xi ∈ S}| S ∈ {L, M − , M + , H} (M − = M + = M ) and the threshold fuzzy aggregation will coincide with the usual threshold aggregation. Remark 2. We can use the robust ”soft” comparison ϕ(x(3) ) = ϕ(x(2) ) (k) vexp (x ) ϕ(x(1) ) > ϕ(x(4) ) = ϕ(x(3) ) > ϕ(x(2) ) v( x(k) ) ϕ(x(1) ) > ϕ(x(4) ) > ϕ(x(3) ) > ϕ(x(2) )

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6

Conclusion

The paper developed a procedure for threshold ranking of alternatives represented by vectors of fuzzy numbers. The procedure is described for the case of a three-grade fuzzy rating scale but can be generalized to an arbitrary case of m-gradation scales, m ≥ 3. In addition, the paper proposes and investigates a procedure for blurring point expert data on information about the degree of confidence of experts in their estimates. Both the general properties of such estimates and the properties in relation to specific blur models are studied. The specified procedures of fuzzy threshold aggregation and blurring are demonstrated on the example of ranking articles according to the recommendations of reviewers and the degree of their confidence in their recommendations. In the future, it is of interest to develop the axiomatic of fuzzy threshold aggregation.

References 1. Aleskerov, F.: Arrovian Aggregation Models. Kluwer, Dordrecht (1999) 2. Aleskerov, F., Chistyakov, V., Kalyagin, V.: Social threshold aggregations. Soc. Choice Wel. 35, 627–646 (2010) https://doi.org/10.1007/s00355-010-0454-9 3. Aleskerov, F.T., Yakuba, V.I.: A method for threshold aggregation of three-grade rankings. Doklady Math. 75(2), 322–324 (2007) 4. Delgado, M., Vila, M.A., Voxman, W.: On a canonical representation of fuzzy numbers. Fuzzy Sets Syst. 93, 125–135 (1998) 5. Grzegorzewski, P.: Metrics and orders in space of fuzzy numbers. Fuzzy Sets Syst. 97, 83–94 (1998) 6. Heilpern, S.: The expected value of a fuzzy number. Fuzzy Sets Syst. 47, 81–86 (1992) 7. Voxman, W.: Some remarks on distances between fuzzy numbers. Fuzzy Sets Syst. 100, 353–365 (1998)

A Novel Bivariate Elliptic Fuzzy Membership Function: A Modeling and Decision-Making Tool for Bike Sharing Alparslan Abdurrahman Basaran1 , Murat Alper Basaran2(B) and Mehmet Ozer Demir2

,

1 Hacettepe University, Ankara 06800, Turkey 2 Alanya Alaaddin Keykubat University, Antalya 07435, Turkey

[email protected]

Abstract. In this paper, the data set generated by Bike Sharing System (BSS) has been modeled by a proposed method called fuzzy bivariate elliptic membership function that generates membership values between an independent variable and dependent variable whose functional form follows an ellipse since all numerical variables following a cyclic pattern such as season, month, hour, weather situation and so on. Besides, each membership value corresponding to each independent and dependent variable is used to find an aggregate outcome of a dependent variable based on a new decision-making tool. Therefore, how both weather and time combinations have an impact on the dependent variable could be derived. Since there is no built-in membership function available, the data set is used to construct a data-driven elliptic fuzzy membership function. Thus, the Chebyshev inequality based on the correlated variables is used to determine both the a and b parameters of the elliptic function representing 95% of the whole dataset. Keywords: Elliptic fuzzy membership function · Chebyshev’s inequality · Bike-sharing · Decision making

1 Introduction In this manuscript, one of the sharing practices, namely the bike sharing system (BSS), pertinent to sustainable transportation is investigated based on a proposed fuzzy method that can function not only as a modeling tool but also as a practical tool in the process of decision-making. Sustainable transportation has been a concern due to several reasons that have been extensively underlined by local and national governmental bodies since transportation has been imposing a heavy burden on the environment globally, namely, global warming and its indirect effects. With 17% of global greenhouse gas emissions, transportation is dramatically the fastest growing sector, and this remarkable growth is, therefore, the most significant threat to sustainable development goals, known as the climate crisis. For example, transportation accounted for %27 emissions both in the USA and in the EU [1–3]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 77–84, 2023. https://doi.org/10.1007/978-3-031-39774-5_10

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BSS is a sustainable mode of transportation using public bicycles that can substitute for short-distance travel modes and seamlessly connect with public transportation [4]. BSS has several issues related to its operations. Just a few of them are planning and design, optimal distribution and repositioning, demand forecasting and bicycle relocation, maintenance, and environmental issues. Even though vehicle sharing especially BSS has been investigated based on several data analyses and statistical methods [5–7], fuzzy-based methods have been implemented relatively less. Deveci et al. [5] proposed Fuzzy Einstein WASPAS to assess the mitigation strategies of climate change on economic and social aspects when urban mobility planning is under consideration. Simic et al. [7] investigated transportation planning related to Covid-19 with a fuzzy model called Farmetean. Aydın et al. proposed a new method to select the best location utilizing MCMD containing both type-2 fuzzy AHP and type-2 fuzzy WASPAS [8]. Patel and Patel proposed a FAHP to assess the barriers and priorities related to the bike-sharing system [9]. Ataç et al. investigated the vehicle-sharing system covering a holistic framework by presenting the decision issues and fuzzy characters of the system [10]. Xian et al. proposed a MADM based on a Znumber examination of sharing car venture capital problem [11]. Bigerna et al. used a fuzzy set-based approach in which alternative-fueled vehicles are preferred by youngsters [12]. Kaya et al. proposed a method utilizing both FAHP and TOPSIS to determine the charging station site of electric taxis [13]. Sun et al. investigated clustering schemes in management and decision systems utilizing bike-sharing applications [14]. The underlined distribution of the data set is unknowledge, therefore, the classical statistical methods could not provide a full account of insights even though many useful and practical insights could still be derived. For example, the data in [15] was examined to search for the meteorological barriers to the bike-sharing system in Washington D.C. between 2011 and 2012 by employing necessary condition analysis (NCA), which is the only example [16]. Fuzzy set-based, or fuzzy logic-based approaches leading to more informative results to a certain extent are one of the widely implemented ones that fit the features of the data as much as possible since those methods are expected to implement both subjective and objective assessments of the data set by experts. The rest of the article is organized as follows: Sect. 2 presents the preliminary, and Sect. 3 presents the data set and data processing steps. Section 4 presents the proposed method. Section 5 presents the implementation. Section 6 concludes the research. The contribution of the manuscript can be summarized as follows: 1. Since bike-sharing data has a cyclic pattern it should be represented by an elliptic functional form. 2. A novel bivariate elliptic fuzzy membership function is proposed. 3. A decision-making approach is proposed to determine which attribute or attributes play a key role when both independent variables and dependent variables are a concern concurrently.

2 Preliminaries Fuzzy set theory and its related mathematical tool were first proposed by Zadeh [17] to deal with ambiguity in natural languages. The main motivation was to make computations with words to derive meanings that can be utilized in systems that mimic human

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reasoning. Especially when human interaction is dominant, for example, evaluation systems, decision-making systems, and expert knowledge systems that are represented mostly by verbal statements or mixed ones rather than numerical values, fuzzy-based methods overwhelmingly dominate the research field. We refer to some references that provide comprehensive treatment of the subject [17, 18].

3 Data Set and Data Preprocessing We used a benchmark data set in [16]. This data set contains causal, registered, and total numbers of the users of the bike-sharing system in Washington D.C covering 2011 and 2012 regarding attributes of the season (4 categories), hour (24 categories), month (12 categories), weekday (7 categories), the holiday (2 categories), working day (2 categories), weather situations (4 categories). Besides, the data set has meteorological attributes, namely, temperature (T), feeling temperature (FT), humidity (H), and wind speed (WS). While the variables of causal (C), the registered (R), and total numbers (TN) of users are called dependent variables, the rest of the lists containing both numerical and categorical variables are called independent variables. Hence, the motivation of this data set is to search and construct a model that accounts for those dependent variables by employing the set of independent variables. Firstly, all numeric variables are standardized by using the whole data set, which results in mean values of 0 and a standard deviation of 1. Besides, the new numerical variables are denoted by the Z prefix in front of the original variable, for example, while Z is represented by ZT, C is denoted by ZC. Since the whole data set contains 17379 observations and consists of both numerical and categorical variables, the assumption of a normal distribution or other probability distributions cannot be assumed so we employ the Chebyshev inequality to find the bounds of multivariate distribution of the whole variables. For example, when Zt and ZC are a concern, we use the Chebyshev inequality to find a 95% probability upper bound to determine the coefficients of the ellipse. Therefore, we employed the expression suggested by Lal [19]. For a general bivariate case k = (k1 , k2 )T , where ki > 0, i = 1, 2. δx (k) ≤

 2   2 1/2 k1 + k22 + [ k12 + k22 − 4ρ 2 k12 k22 ] 2k12 k22

(1)

where ρ is the correlation coefficient between ZT and ZC and k1 and k2 are the positive integers that show upper limits. Table 1 summarizes the standardized results of two variables. Other descriptive statistics related to numerical variables concerning categorical variables can be similarly attained. SPSS 26.0 version is utilized for computations. A generic ellipse at the origin of the XY-plane can be represented by Eq. (2) as follows: x2 y2 + =1 a2 b2 where a and b represent the vertexes of an ellipse

(2)

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A. A. Basaran et al. Table 1. Statistics of Variables based on standardized variables.

Var.

Stats

Winter

Spring

Summer

Fall

Corr

ZT

Mean

–1.02

0.25

1.08

–0.38

0.46

Std

0.62

0.72

0.63

0.64

Min-Max

–2.47 1.16

–1.75 2.3

–1.86 1.36

–1.85 1.36

N

4242

4409

4496

4232

Mean

–0.43

0.25

1.09

–1.02

Std

0.56

1.16

0.49

0.93

Min-Max

–0.72 6.72

–0.72 6.6

–0.72 6.38

–072 6,62

N

4242

4409

4496

4232

ZC

Therefore, the constructed ellipse function based on the data set covers 95.53% of the whole data when a and b are extracted, which are 10 and 15, respectively. Therefore, the constructed ellipse function based on the data set is denoted by Eq. (3) as follows: x2 (10)

2

+

y2 (15)2

≤1

(3)

4 Proposed Method We proposed a novel method called fuzzy elliptic bivariate membership function (BVFEMF) to model the data set that was previously presented in Sect. 3. Since the data follows a circular form, for example, the winter is followed by the spring, or a holiday is followed by a working day any model that can be used to represent the data should be in the elliptic form in general. The grand ellipse function represented in Eq. (3) is attained based on the available data and contains 95.53% of the data within this elliptic representation. Noted that 95% coverage is satisfied with 10 and 15 obtained by using the Chebyshev inequality. We proposed 4 BVFMF to represent the relationships between each dependent and independent variable concerning the season. So, we can easily adapt this to other fuzzy times and weather conditions such as season, month, hour, weekday, and so on. For example, BVFEMFs between ZT (standardized temperature) and ZC (standardized casual rent) concerning the season are denoted by μW −SPR (zt, zc), μSPR−S (zt, zc), μS−F (zt, zc), μF−W (zt, zc) The general elliptic functions between winter and spring and summer and fall are derived from the grand elliptic function based on their descriptive statistics by solving for variable zc. The others, namely, spring-summer and fall-winter are extracted by

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solving for zt. Consequently, we present two Algorithms. While Algorithm 1 presents the construction of the fuzzy bivariate elliptic fuzzy membership function, Algorithm 2 presents the proposed decision rule. Algorithm 1: The Model Construction. Step 1. Standardize all numerical attributes leading to zero mean and standard deviation. Step 2. Utilize the bivariate Chebyshev inequality expression to determine upper bound probability. Find a and b, namely, constants of an ellipse function. Step 3. Modify a and b if needed based on data. Step 4. Compute the grand ellipse function representing the whole data set. Step 5. Derive each ellipse function based on the partitioned fuzzy variable. Step 6. Construct elliptic bivariate fuzzy membership functions. Algorithm 2: The Decision-making. Step 1. For each pair of dependent and independent variables, Compute membership values concerning each fuzzy time and weather value. Step 2. Aggregate the attributes to find the decision.      μHOUR (z) μWEATHER (z) μHOLI˙ DAY (z) μWD ) μAD (z) = μS (μMONTH (z) where WD represents whether the given day is a working day or not.

5 Implementation Equations (4) through (7) are determined based on the data set to make computations. Equation (4) will be presented in detail then, the final representations of them will be provided directly. For winter and spring seasons, Eq. (4) is derived as follows:  (6.72)2 ZT 2 ZC = (6.72)2 − (4) (2.47)2  (6.72)2 ZT 2 ZC = − (6.72)2 − (5) (2.47)2  (2.47)2 ZC 2 ZT = (2.47)2 − (6) (6.72)2  (2.47)2 ZC 2 ZT = − (2.47)2 − (7) (6.72)2

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μS−F (zt, zc) =

⎧ ⎪ 0, zc < −2.61 and zt < −.072 ⎪ ⎪ ⎪ ⎪ orzc > 2.61 and zt > 6.62 ⎪  ⎪ ⎪ 2 2 ⎪ ⎪ ZC− (6.62)2 − (6.62) ZT ⎪ (2.61)2 ⎪ ⎪ 1− ⎪ ⎪ (6.62)2 ⎪ ⎪ ⎪ ⎪ zt ∈ −1.85) ∪ (01.36, 2.61), zc ∈ (−0.72, 0) ∪ (0, 6.38) (−0.61, ⎪  ⎪ ⎨ 2 (6.71)2 ZT 2 1−

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(2.61)2

(6.71)2

,

zt ∈ (−1.85, 0) ∪ (0, 1.36), zc ∈ (−0.72, 0) ∪ (0, 6.6)  1−

2 ZT 2 (2.61)2

ZC− (6.62)2 − (66) (6.6)2

zt ∈ (−1.85, 0) ∪ (0, 1.36)), zc ∈ (6.38, 6.62) 1, zc = zt = 0

μSPR−S (zt, zc) =

μF−W (zt, zc) =

ZC− (6.62) −

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎧ ⎪ 0, zc < −2.61 and zt < −.072 ⎪ ⎪ ⎪ ⎪ orzc > 2.61 and zt > 6.6 ⎪  ⎪ ⎪ 2 2 ⎪ ⎪ ZT− (6.6)2 − (6.6) ZC2 ⎪ (2.61) ⎪ ⎪ 1− ⎪ ⎪ (6.6)2 ⎪ ⎪ ⎪ ⎪ zt ∈ (−1.75, 0) ∪ (0, 2.3), zc ∈ (−0.72, 0) ∪ (0, 6.6) ⎪  ⎪ ⎨ 2 2

(8)

, 1− ⎪ ⎪ (6.38)2 ⎪ ⎪ ⎪ zt ∈ (−1.75, 0) ∪ (0, 2.3, zc ∈ (−0.72, 0) ∪ (0, 6.38) ⎪ ⎪  ⎪ ⎪ 2 2 ZC (6.6) ⎪ ZT− (6.6)2 − ⎪ ⎪ (2.61)2 ⎪ ⎪ 1− ⎪ 2 (6.6) ⎪ ⎪ ⎪ ⎪ zt ∈ (2.3., 2.61), zc ∈ (−0.72, 0) ∪ (0, 6.38) ⎪ ⎪ ⎩ 1, zc = zt = 0

(9)

ZT− (6.6)2 − (6.6)

ZC (2.61)2

0, zc < −2.47 and zt < −.072 orzc > 2.61 and zt > 6.72  1−

2 ZC 2 (2.61)2

ZT+ (6.72)2 − (6.72) (6.72)2

zt ∈ (−2.47, 0) ∪ (0, 1.16) ∪ (1.16, 2.61), zc ∈ (−0.72, 0) ∪ (0, 6.72)  (6.71)2 ZC 2

ZT+ (6.72) − (2.61)2 ⎪ ⎪ , 1 − ⎪ 2 ⎪ (6.72) ⎪ ⎪ ⎪ zt ∈ (−2.47, 0) ∪ (0, 1.16), zc ∈ (−0.72, 0) ∪ (0, 6.72) ⎪ ⎪  ⎪ ⎪ 2 2 ⎪ ⎪ ZC+ (6.72)2 − (6.72) ZT ⎪ (2.61)2 ⎪ ⎪ 1− ⎪ ⎪ (6.72)2 ⎪ ⎪ ⎪ zt ∈ 0) ∪ 1.16), zc ∈ (−0.72, 0) ∪ (0, 6.72) (−2.47, (0, ⎪ ⎪ ⎩ 1, zc = zt = 0 2

(10)

So, we constructed all season combinations regarding temperature, casual rent, registered rent, and combined rent as follows: for example, (zt = 1, zc = 1), zt = 1, zr =

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1), and (zt = 1, ztn = 1) are assumed, Algorithm 1 returns as follows: μW −SPR (zt, zc) = 0.86, μW −SPRPR (zt, zr) = 0.73, μW −SPR (zt, ztn) = 0.91 Algorithm 2 implies that for ZC and ZT, the winter-spring is preferred by the casual users with the largest membership value.

6 Conclusion In this manuscript, a novel BVFMF has been proposed that functions as not only a modeling tool but also a decision-making tool to determine which attributes play a key role in the implementation of bike-sharing programs. To compute how each attribute plays a key role in the decision-making process, Algorithm 2 is proposed to present the decision outcome. For example, when time and casual rent are a concern, spring-winter is the preferred season among other seasons. Since the data set has a cyclic pattern elliptic mathematical expression could fit better than other mathematical expressions. So, BVFMF is used to better present the relationship between dependent and independent variables based on time variables such as season, day, month, and holiday.

References 1. Critical Issues in Transportation 2019, Policy Snapshot, The National Academics of ScienceEngineering-Medicine, Transportation Research Board, 1–32 2. U.S. Environmental Protection Agency, Fast Facts on transportation, Greenhouse Gas Emissions-US EPA (2019). https://www.epa.gov/greenvehicles/fast-facts-transportation-gre ehouse-gas-emissions 3. Transport Emissions, A European Strategy for low-emission climate action. https://ec.europa. eu/clima/eu-action/trasport-emission 4. Kim, K.: Investigation of modal integration of bike-sharing and public transit in Seoul for the holders of 365-day passes. J. Transp. Geogr. 106, 103518 (2023) 5. Deveci, M., Pamucar, D., Gokasar, I., Isik, M., Coffman, D.: Fuzzy Einstein WASPAS approach for the economic and societal dynamics of the climate change mitigation strategies in urban mobility planning. Struct. Change Econ. Dyn. 61, 1–17 (2022).https://doi.org/10.1016/ j.strueco.2022.01.009 6. Simic, V., Ivanovic, I., Torkayesh, A.E.: Adapting urban transport planning to Covid-19 pandemic: an integrated fermatean fuzzy model. Sustain. Cities Soc. 79, 103669 (2022).https:// doi.org/10.1016/j.scs.2022.103669 7. Aydın, N., Seker, S., Özkan, B.: Planning location of mobility hub for sustainable urban mobility. Sustain. Cities Soc. 81, 103843 (2022). https://doi.org/10.1016/j.scs.2022.103843 8. Patel, S., Patel, C.R.: A stakeholder perspective on improving barriers in the implementation of public bicycle sharing system (PBSS). Transp. Res. Part A 138, 353–366 (2020). https:// doi.org/10.1016/j.tra.2020.06.007

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9. Ataç, S., Obrenoic, N., Bierlaire, M.: Vehicle sharing system: a review and holistic management framework. EURO J. Transp. Logistics 10, 100033 (2021). https://doi.org/10.1016/j. ejtl.2021.100033 10. Xian, S., Chai, J., Li, T., Huang, J.: A ranking model of Z-mixture - numbers based on the ideal degree and its application in multi-attribute decision making. Inf. Sci. 550, 145–165 (2021). https://doi.org/10.1016/j.ins.2020.10.038 11. Bigerna, S., Bollino, C.A., Micheli, S.: Italian youngsters’ perceptions of alternative fuel vehicles: a fuzzy set approach. J. Bus. Res. 69, 5426–5430 (2016).https://doi.org/10.1016/j. jbusres.2016.04.149 12. Kaya, Ö., Alemdar, K.D., Çodur, M.Y.: A novel two-stage approach for electric taxis charging stations site selection. Sustain. Cities Soc. 62, 102396 (2020).https://doi.org/10.1016/j.scs. 2020.102396 13. Sun, L., Chen, G., Xiong, H., Guo, C.: Cluster analysis in data-driven management and decisions. JMSE 2(4), 227–251 (2017). https://doi.org/10.3724/SP.J.1383.204011 14. Fanaee-T, H., Gama, J.: Event labeling combining ensemble detectors and background knowledge. Prog. Artif. Intell. 2(2–3), 113–127 (2013). https://doi.org/10.1007/s13748-0130040-3 15. Kumar, D.: Meteorological barriers to bike rental demands: a case of Washington D.C. using NCA approach. Case Stud. Transp. Policy 9, 830–841 (2021).https://doi.org/10.1016/j.cstp. 2021.04.002 16. Ruan, D., Huang, C.: “Fuzzy sets”, Fuzzy Sets and Fuzzy Information - Granulation Theory 3rd Volume: Key selected papers. In: Zadeh, L.A. (ed.) Advances in Fuzzy Mathematics and Engineering Series, pp. 3–62. Beijing Normal University Press, Beijing, China, ch. 1, sec 1 (2020) 17. Didier, D., Prade, H.: “Fuzzy Relations” Fuzzy Sets and Systems: Theory and Application, Mathematics in Science and Engineering, vol. 144, pp. 84–88. Academic Press, Newyork, USA, Ch. 3, Sec. E (1980) 18. Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall (1995) 19. Lal, D.N.: A note on a form of Tchebycheff’s inequality for two or more variables. Sankhyh 15, 317–320 (1955)

Cryptography Based on Fuzzy Graphs Mariana Durcheva1,2 and Malinka Ivanova2(B) 1 Department of Mathematics, Shamoon College of Engineering, 77245 Ashdod, Israel

[email protected], [email protected]

2 Department of Informatics, Faculty of Applied Mathematics and Informatics,

Technical University of Sofia, 1797 Sofia, Bulgaria [email protected]

Abstract. Cryptography is a scientific area, which is in continue progress. It is dictated by the technological development, as well as the complexity of cyberattacks. For this reason, existing cryptosystems are being improved and new ones are emerging. Recently, the apparatus of fuzzy graphs and its applicability in a wide variety of domains, including cryptography, is explored for achieving a high level of security. The aim of the paper is to present a discussion regarding properties of fuzzy graphs, which are suitable for usage in cryptography and an idea for implementation for a Diffie – Hellman like protocol to be argued. Machine learning models, predicting the man-in-the-middle attack are presented. Keywords: Asymmetric cryptography · Diffie-Hellman like protocol · M-polar fuzzy graph · Machine learning · Man-in-the-middle attack

1 Introduction Contemporary public-key based cryptography is well accepted approach for securing a wide variety of applications, which operates with important for users’ information. The users must feel comfortable and secure when they provide their data to known or unknown web-based sites. Cryptography is used for different purposes: from generation of session keys through securing information in its transfer and storage in databases to generation of digital certificates and digital signatures [1–4]. Multiple cryptosystems are standardized and intensively applied in practice. However, the existing ones are being improved as well as researchers are coming up with new ideas and solutions [5, 6]. This is dictated by the development of technologies, the ever-increasing number of attacks, as well as their increasing complexity. Fuzzy graphs combine properties typical for fuzzy sets and graph theories to describe some events or systems. The benefits of fuzzy graphs are seen at solving variety of problems like: better understanding of traffic flow and avoiding accidents [7], segmentation of images with high accuracy [8], exploration the symptoms and diabetic diseases [9], in assisting decision making when a selection has to be done [10]. Multiple surveys show increased interest by researchers to other different application areas [11, 12]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 85–93, 2023. https://doi.org/10.1007/978-3-031-39774-5_11

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Several cryptography issues resolved through fuzzy graphs are also explored by Qaid et al. to secure images [13], by Ali et al. presenting symmetric cryptography system [14], as the authors prove the achieved high level of security. Further investigation regarding the possibility of fizzy graphs to be applied in cryptography should be conducted, taking into account beneficial properties for improving some security problems. The aim of the paper is to explore some features of fuzzy graphs that can be used in a cryptography system and an implementation in a Diffie-Hellman like protocol to be presented in combination of machine learning utilization for predicting a man-in-themiddle attack. It should be noted that this work is the first to propose the use of m-polar fuzzy graphs as building blocks for a public key encryption scheme. The organization of the paper is as follows: In Sect. 2, we give some notions about m-polar fuzzy graphs and their products that are relevant to our discussion in this paper; Sect. 3 is devoted to the semigroup structure in an m-polar fuzzy graph; the key result in this paper, namely the Diffie-Hellman like protocol based on m-polar fuzzy graph, is presented in Sect. 4; Sect. 5 proposes predictive machine learning-driven models; finally, the last section (Sect. 6) draws conclusion.

2 Preliminaries 2.1 Definitions As usual, a graph is a pair G = (V, E) of a nonempty set of vertices V (or nodes) and a set of edges E. Definition 1 [15] . A fuzzy graph G = (V , σ, μ) is a triple, where V is a nonempty set, together with functions σ : V → [0, 1] and μ : E → [0, 1] such that for all x, y ∈ V, μ(xy) ≤ σ (x) ∧ σ (y). Here ∧ denotes the minimum. Definition 2 [16] . An m-polar fuzzy set ([0, 1]m -set) on the set V is a mapping M : V → [0, 1]m . The [0, 1]m is a poset with point-wise order ≤ (m is natural), where ≤ is defined by: x ≤ y ⇐⇒ Pi (x) ≤ Pi (y) for eachi = 1, 2, . . . , m,x, y ∈ [0, 1]m , and Pi : [0, 1]m → [0, 1] is the i − th projection mapping (i = 1, 2, . . . , m). Of course, 1 = (1, 1,…, 1) is the greatest value and 0 = (0, 0,…, 0) is the smallest value in[0, 1]m . Definition 3 [16] . For an m-polar fuzzy set σ on a set V, an m-polar fuzzy relation on σ is defined as an m-polar fuzzy set μ of V × V such that μ(xy) ≤ σ (x) ∧ σ (y) for all x, y ∈ V ( i = 1, 2, . . . , m): Pi ◦ μ(xy) ≤ Pi ◦ σ (x) ∧ Pi ◦ σ (y). Definition 4 [17] . A triple G = (V , σ, μ) is called an m-polar fuzzy graph. Here V is a nonempty set, together with functions σ : V → [0, 1]m and μ : E = V × V → [0, 1]m ;

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σ is an m-polar fuzzy set on the set of vertices V and μ is an m-polar fuzzy relation in V such that for all x, y ∈ V, μ(xy) ≤ σ (x) ∧ σ (y).

2.2 Products of m-Polar Fuzzy Graphs Here, we present the main products of m-polar fuzzy graphs. It should be noted that all products mentioned below lead to m-polar fuzzy graphs. Throughout this paper we mean that Gi = (Vi , σi , μi ) is an m-polar fuzzy graph of a crisp graph Gi ∗ = (Vi , Ei ). Cartesian Product. The Cartesian product of G1 and G2 [18] (V1 ∩ V2 = ∅) is an m-polar fuzzy graph G1 × G2 = (V1 × V2 , σ1 × σ2 , μ1 × μ2 ) of G ∗ = (V1 × V2 , E), where E = {(x, y1 )(x, y2 )|x ∈ V1 , y1 y2 ∈ E2 } ∪ {(x1 , y)(x2 , y)|x1 x2 ∈ E1 , y ∈ V2 } such that for each i = 1, 2, . . . , m: • Pi ◦ (σ1 × σ2 )(x1 , y1 ) = Pi ◦ σ1 (x1 ) ∧ Pi ◦ σ2 (y1 ) for all (x1 , y1 ) ∈ V1 × V2 ; • Pi ◦ (μ1 × μ2 )((x, y1 )(x, y2 )) = Pi ◦ σ1 (x) ∧ Pi ◦ μ2 (y1 y2 ) for all x ∈ V1 and y1 y2 ∈ E2 ; • Pi ◦ (μ1 ×μ2 )((x1 , y), (x2 , y)) = Pi ◦ μ1 (x1 x2 ) ∧ Pi ◦ σ2 (y) for all x1 x2 ∈ E1 and y ∈ V2 ; Proposition 5. The Cartesian product of m-polar fuzzy graphs is commutative and associative operation. Proof. Follows from the definition of the Cartesian product. Direct Product.The direct  (V1 ∩ V2 = ∅) is an m-polar  product ofG1 and G 2 [18] fuzzy graph G1 G2 = V1 × V2 , σ1 σ2 , μ1 μ2 of G ∗ = (V1 × V2 , E), where E = {(x1 , y1 )(x2 , y2 )|x1 x2 ∈ E1 , y1 y2 ∈ E2 } such that for each i = 1, 2, . . . , m:    • Pi ◦ σ1 σ2 (x  1 , y1 ) = Pi ◦ σ1 (x1 ) ∧ Pi ◦ σ2 (y1 ) for all (x1 , y1 ) ∈ V1 × V2 ; • Pi ◦ μ1 μ2 ((x1 , y1 )(x2 , y2 )) = Pi ◦ μ1 (x1 x2 ) ∧ Pi ◦ μ2 (y1 y2 ) for all x1 x2 ∈ E1 , y1 y2 ∈ E2 . Proposition 6. The direct product of m-polar fuzzy graphs is commutative and associative operation. Proof. Follows from the definition of the direct product. Semi-Strong Product. The semi-strong product [18] of G1 and G2 (V1 ∩ V2 = ∅) is a graph G1 G2 = (V1 × V2 , σ1 σ2 , μ1 μ2 ) of G ∗ = (V1 × V2 , E), where E = {(x1 , y1 )(x1 , y2 )|x1 ∈ V1 , y1 y2 ∈ E2 }∪{(x1 , y1 )(x2 , y2 )|x1 x2 ∈ E1 , y1 y2 ∈ E2 }, such that for each i = 1, 2, . . . , m: • Pi ◦ (σ1 σ2 )(x, y) = Pi ◦ σ1 (x) ∧ Pi ◦ σ2 (y) for all (x, y) ∈ V1 × V2 ; • Pi ◦ (μ1 μ2 )((x1 , x2 )(x1 , y2 )) = Pi ◦ σ1 (x1 ) ∧ Pi ◦ μ2 (x2 y2 ) for all x1 ∈ V1 and x2 y2 ∈ E2 ; • Pi ◦ (μ1 μ2 )((x1 , x2 )(y1 , y2 )) = Pi ◦ μ1 (x1 y1 ) ∧ Pi ◦ μ2 (x2 y2 ) for all x1 x2 ∈ E1 and x2 y2 ∈ E2 .

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Proposition 7. The semi-strong product of m-polar fuzzy graphs is neither commutative nor associative operation. Proof. Follows from the definition of the semi-strong product. Strong Product. The strong product [18] of G1 and G2 (V1 ∩ V2 = ∅) is a graph G1 ⊗ G2 = (V1 × V2 , σ1 ⊗ σ2 , μ1 ⊗ μ2 ) of G ∗ = (V1 × V2 , E), where E = {(x, y1 )(x, y2 )|x ∈ V1 , y1 y2 ∈ E2 } ∪ {(x1 , y)(x2 , y)|x1 x2 ∈ E1 , y ∈ V2 } ∪ {(x1 , y1 )(x2 , y2 )|x1 x2 ∈ E1 , y1 y2 ∈ E2 } such that for each i = 1, 2, . . . , m: • Pi ◦ (σ1 ⊗ σ2 )(x, y) = Pi ◦ σ1 (x) ∧ Pi ◦ σ2 (y) for all (x, y) ∈ V1 × V2 ; • Pi ◦ (μ1 ⊗ μ2 )((x, y1 )(x, y2 )) = Pi ◦ σ1 (x) ∧ Pi ◦ μ2 (y1 y2 ) for all x ∈ V1 and y1 y2 ∈ E2 ; • Pi ◦ (μ1 ⊗ μ2 )((x1 , y), (x2 , y)) = Pi ◦ μ1 (x1 x2 ) ∧ Pi ◦ σ2 (y) for all x1 x2 ∈ E1 and y ∈ V2 ; • Pi ◦ (μ1 ⊗ μ2 )((x1 , y1 ), (x2 , y2 )) = Pi ◦ μ1 (x1 x2 ) ∧ Pi ◦ μ2 (y1 y2 ) for all x1 x2 ∈ E1 and y1 y2 ∈ E2 . Proposition 8. The strong product of m-polar fuzzy graphs is commutative and associative operation. Proof. Follows from the definition of the strong product. It can be noted that Cartesian, direct, and strong products have a potential to be used for cryptographic purposes. In this paper, we employ another operation, namely, the union of fuzzy graphs for our protocol.

3 Semigroup Structure in m-Polar Fuzzy Graph Some group, semigroup, ring and semiring structures for graphs (including fuzzy graphs) has been studied in [19–22]. However, as far as we know, there is no such study specifically for m-polar fuzzy graphs. Definition 9 [17] . The union of G1 and G2 (denoted by G1 ∪ G2 ) is a graph G1 ∪ G2 = (V1 ∪ V2 , E1 ∪ E2 , σ1 ∪ σ2 , μ1 ∪ μ2 ) such that for each i = 1, 2, . . . , m: ⎧ ⎪ ⎨

Pi ◦ σ1 (x) if x ∈ V1 − V2 • Pi ◦ (σ1 ∪ σ2 )(x) = Pi ◦ σ2 (x) if x ∈ V2 − V1 ; ⎪ ⎩ P ◦ σ (x) ∨ P ◦ σ (x) if x ∈ V ∩ V i 1 i 2 1 2 ⎧ ⎪ Pi ◦ μ1 (xy) if xy ∈ E1 − E2 ⎨ Pi ◦ μ2 (x) if xy ∈ E2 − E1 . • Pi ◦ (μ1 ∪ μ2 )(xy) = ⎪ ⎩ P ◦ μ (xy) ∨ P ◦ μ (xy) if xy ∈ E ∩ E i 1 i 2 1 2 Here, the symbol ∨ denotes maximum. Example 10. The union of two 3-polar fuzzy graphs is shown in Fig. 1.

Cryptography Based on Fuzzy Graphs a (0.3,0.7,0.8)

a (0.3,0.5,0.7)

(0.3,0.7,0.5)

a (0.3,0.7,0.8)

d (0.4,0.3,0.6)

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b (0.2,0.3,0.8)

(0.1,0.3,0.4)

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G2

c (0.3,0.5,0.6)

G1 U G 2

Fig. 1. Union of G1 and G2

Proposition 11 [23] . The union of two m-polar fuzzy graphs is an m-polar fuzzy graph. Proposition 12. Let G1 and G2 be two m-polar fuzzy graphs. The operation union of G1 and G2 is commutative, i.e., G1 ∪ G2 = G2 ∪ G1 . Proof. Follows from the definition of the union. Proposition 13. Let G1 , G2 and G3 be three m-polar fuzzy graphs. The operation union of G1 , G2 and G3 is associative, i.e., G1 ∪ (G2 ∪ G3 ) = (G1 ∪ G2 ) ∪ G3 . Proof. Follows from the definition of the union. Theorem 14. Let S m be the set of all m-polar fuzzy graphs, then the structure (Sm , ∪) is commutative semigroup. Proof. According to Proposition 11, for any m-polar fuzzy graphs G1 and G2 , the union G1 ∪ G2 is an m-polar fuzzy graph, i.e., G1 ∪ G2 ∈ Sm . From the associativity of the union (Proposition 13) follows that (Sm , ∪) is a semigroup. Proposition 12 gives the commutativity. Remark 15. An empty graph G0 of G ∗ = (∅, ∅) is a neutral element for the commutative idempotent semigroup (Sm , ∪).

4 The Diffie-Hellman Like Protocol Here we propose a public key-exchange protocol which is an analogue to the DiffieHellman key-exchange protocol [24]. The protocol works as follows: Users agree on the set of the vertices V, the size m of the used m-polar fuzzy graphs, as well as the m-polar fuzzy graph C = (Vc ⊂ V , Ec , σc , μc ). 1. Alice chooses as her secret key an m-polar fuzzy graph A = (V1 ⊂ V , E1 , σ1 , μ1 ). She computes: D = A ∪ C using the union operation and sends D to Bob. 2. Bob chooses as his secret key an m-polar fuzzy graph B = (V2 ⊂ V , E2 , σ2 , μ2 ). He computes: E = B ∪ C using the union operation and sends E to Alice. 3. Alice computes kA = A ∪ E = A ∪ (B ∪ C). 4. Bob computes kB = B ∪ D = B ∪ (A ∪ C).

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Theorem 16. At the end of the protocol, Alice and Bob obtain the same shared key kA = kB . Proof. According to Proposition 13, the union of two m-polar fuzzy graph is an associative operation, i.e.: kA = A ∪ (B ∪ C) = (A ∪ B) ∪ C, kB = B ∪ (A ∪ C) = (B ∪ A) ∪ C. In addition, according to Proposition 12, the union of two m-polar fuzzy graphs is a commutative operation, i.e.: A ∪ B = B ∪ A. It follows that kA = kB . The security of the protocol is due to the fact that the operation union includes only max-operations. As it is well known, the max-operation is non-invertible. Users can choose any real number from the set [0,1] as the parameters in their secret m-polar fuzzy graphs. If the size of the m-polar graphs is large enough and the users’ secret graphs are well chosen, we consider this scheme safe enough. Of course, further research will be required to establish acceptable graph parameters. It should be noted that, like any Diffie-Hellman-type scheme, this scheme is also vulnerable to a man-in-the-middle attack.

5 Predicting Man-in-the-Middle Attack As mentioned above, one of the most common attacks on Diffie-Hellman-like protocols is the man-in-the-middle (MIM) attack. The proposed protocol will work more securely if such attack can be predicted to take preventive measures. To do this, the research conducted to date is briefly summarized and a generalized model is proposed to predict the occurrence of this attack using machine learning. The predictive model of Dong et al. is based on the assessment of the round-trip time and the strength of the received signal [25]. Chowdary et al. gather knowledge about IP address entropy, location of a port and rate of parcel entry in a cloud environment to identify a MIM attack [26]. Mantooa and Kaur use data from UDP (User Datagram Protocol) and TCP (Transmission Control Protocol) header to detect a MIM attack [27]. A set with predictors (including characteristics related to time for packets obtaining, IP payload length, source port, etc.) is created by Al-Juboori et al. and four machine learning algorithms are used to MIM detection [28]. In this work, the predictive model is created based on published predictors in the explored literature sources as the set with important parameters is formed (time for packet delivery, IP payload, port delivery, frame length, message topic, signal strength, whether the traffic is in the borders, TCP packet header). It generalizes availability of possible conditions and occurrence of certain events to predict the coming MIM attack.

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100% 80% 60% 40% 20% 0% Accuracy

Absolute Root Mean Squared error Squared error Error

Random Forest

Kappa

Deep Learning

Fig. 2. Performance of predictive models

The predictive model is created in Rapid Miner Studio through utilization of two algorithms: Random Forest (RF) and Deep Learning (DL). Two models are compared following standard for machine learning metrics (Fig. 2).

6 Conclusion In this paper we have presented a Diffie-Hellman like key exchange protocol based on mpolar fuzzy graphs. The protocol uses the semigroup structure of the operation union in m-polar fuzzy graphs. Although some cryptography issues have been solved with fuzzy graphs, they belong to the field of symmetric key cryptography. To our best knowledge, there is not asymmetric key cryptographic protocol employing (m-polar) fuzzy graphs as building blocks. Models, predicting with high performance the occurrence of MIM attack, are also presented. As a future work, we plan to further explore usage of m-polar fuzzy graphs for cryptographic purposes. M-polar fuzzy graphs are a promising new approach to cryptography. They are difficult to analyze and break, which makes them a good choice for protecting sensitive data. However, more research is needed to develop practical m-polar fuzzy graph cryptosystems.

References 1. Ahmad, J.I., Din, R., Ahmad, M.: Analysis review on public key cryptography algorithms. Indonesian J. Elec. Eng. Comput. Sci. 12(2), 447–454 (2018). https://doi.org/10.11591/ije ecs.v12.i2 2. Pavani, K., Sriramya, P.: Enhancing public key cryptography using RSA, RSA-CRT and N-Prime RSA with multiple keys. In: Third International Conference on Intelligent Communication Technologies and Virtual Mobile Networks (ICICV), Tirunelveli, India, pp. 1–6 (2021). https://doi.org/10.1109/ICICV50876.2021.9388621 3. Peeran, A.M., Shanavas, A.R.M.: E-governance security via public key cryptography using elliptic curve cryptography. In: Materials Today, vol. 49, Part 8, pp. 3568–3573 (2022). https:// doi.org/10.1016/j.matpr.2021.08.090 4. Boyd, C., Mathuria, A., Stebila, D.: Authentication and key transport using public key cryptography. In: Protocols for Authentication and Key Establishment. Information Security and Cryptography. Springer, Heidelberg (2020). https://doi.org/10.1007/978-3-662-58146-9_4

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5. Raghunandan, K.R., Ganesh, A., Surendra, S., Bhavya, K.: Key generation using generalized Pell’s equation in public key cryptography based on the prime fake modulus principle to image encryption and its security analysis. Cybern. Inf. Technol. 20(3), 86–101 (2020). https://doi. org/10.2478/cait-2020-0030 6. Wang, S., Zhang, B.: Research on security protocol of RFID system based on public key cryptography. IOP Conf. Ser. J. Phys. Conf. Ser. 1237 (2019). https://doi.org/10.1088/17426596/1237/2/022134 7. Myna, R.: Application of fuzzy graph in traffic. Int. J. Sci. Eng. Res. 6(2), 1692–1696 (2015) 8. Thakur, G.K., Priya, B., Kumar, S.P.: A novel fuzzy graph theory-based approach for image representation and segmentation via graph coloring. J. Appl. Secur. Res. 14(1), 74–87 (2019). https://doi.org/10.1080/19361610.2019.1545273 9. Sudha, T., Jayalalitha, G.: Application of fuzzy network graph in hospital. Adv. Appl. Math. Sci. 20(9), 1823–1829 (2021) 10. Akram, M., Arshad, M., Shumaiza: Fuzzy rough graph theory with applications. Int. J. Comput. Intell. Syst. 12, 90–107 (2018) 11. Sitara, M., Akram, M., Yousaf Bhatti, M.: Fuzzy graph structures with application. Mathematics 7(1), 63 (2019). https://doi.org/10.3390/math7010063 12. Muhiuddin, G., Mahapatra, T., Pal, M., Alshahrani, O., Mahboob, A.: Integrity on m-Polar fuzzy graphs and its application. Mathematics 11(6), 1398 (2023). https://doi.org/10.3390/ math11061398 13. Qaid, G.R.S., Talbar, S.N., AL – Kubati, A.A.M.: Image security by using fuzzy graph. Int. J. Eng. Innov. Technol. (IJEIT) 3(12), 154–157 (2014) 14. Ali, M.A.H., Omran, A.A., Ajeena, R.K.K.: The cartesian product graph for encryption schemes. In: 2nd International Conference on Modern Applications of Information and Communication Technology, AIP Conference Proceedings, vol. 2591 (2023). https://doi.org/10. 1063/5.0128662 15. Mordeson, J., Mathew, S., Malik, D.: Fuzzy Graph Theory with Applications to Human Trafficking. Fuzzy Graph Theory. Springer (2018) 16. Chen, J., Li, S., Ma, S., Wang, X.: In m-polar fuzzy sets: an extension of bipolar fuzzy sets. Sci. World J. 2014, Article Id 416530 (2014). Hindawi Publishing Corporation 17. Akram, M.: m-Polar Fuzzy Graphs, Theory, Methods and Applications. Springer (2019) 18. Pal, M., Samanta, S., Ghorai, G.: Modern Trends in Fuzzy Graph Theory. Springer (2020) 19. Mordeson, J., Malik, D., Kuroki, N.: Fuzzy Semigroups. Studies in Fuzziness and Soft Computing, vol. 131. Springer Science and Business Media, Berlin (2003) 20. Knill, O.: On Graphs, Groups and Geometry. Preprint, arXiv:2205.14097v1. math, [2205.14097] (2022) 21. Knill, O.: Remarks about the Arithmetic of Graphs. Preprint (2021). https://arxiv.org/abs/ 2106.10093v2 22. Rahman, S., Umbrey, G.: Semirings of graphs: homomorphisms and applications in network problems. Proyecciones J. Math. 41(6), 1273–1296 (2022) 23. Ghorai, G., Pal, M.: Some isomorphic properties of m-polar fuzzy graphs with applications. Springer Plus 5, 2104 (2016) 24. Diffie, W., Hellman, M.: New directions in cryptography (PDF). IEEE Trans. Inf. Theory 22(6), 644–654 (1976) 25. Dong, Z., Espejo, R., Wan, Y., Zhuang, W.: Detecting and locating man-in-the-middle attacks in fixed wireless networks. CIT 23(4), 283–293 (2015). https://doi.org/10.2498/cit.1002530

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Aggregating Distances with Uncertainty: The Modular (pseudo-)metric Case M. D. M. Bibiloni-Femenias1,2 , J.-J. Mi˜ nana1,2(B) , and O. Valero1,2 1

Mathematics and Computer Science Department, Universitat de les Illes Balears, Ctra. Valldemossa km. 7.5, 07122 Palma, Spain {m.bibiloni,o.valero}@uib.es 2 Institut d’ Investigaci´ o Sanit` aria Illes Balears (IdISBa), Hospital Universitari Son Espases, 07120 Palma, Illes Balears, Spain [email protected] Abstract. Many real problems involve the use of metrics to be solved. In such problems to merge some measurements obtained through the used metric is often needed in order to make a working decision. So functions that merge a collection of metrics providing a new one as output have roused interest of different authors. Nevertheless, the axiomatic that defines the concept of metric limits their applicability in some situations. Motivated by this fact, different generalizations of the concept of metric can be found in the literature. Among them, it is worth mentioning the notion of modular metric introduced by Chistyakov. This kind of metric involves uncertainty due to the inclusion of a parameter in the distance evaluation. The aim of this paper consists in characterizing those functions that merge a collection of modular (pseudo-)metrics into a new modular (pseudo-)metric, all of them defined on the same set. Keywords: Modular (pseudo-)metric Monotonicity · Subadditivity

1

· Aggregation · Triangle triplet ·

An Introduction to the Aggregation of Metrics

The problem of how to aggregate information is often studied in applied sciences. The idea is to take a collection of numerical data, possibly coming from different sources, and merge them through a function which combines all the numerical information content with the aim of making a working decision. Sometimes all sources provide data sharing a common property. Now the aggregation problem consists in finding a function that merges all the information content preserving the aforementioned property. A particular instance of this problem is the socalled (pseudo-)metric aggregation problem. This problem consists in finding the This research was funded by project BUGWRIGHT2. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreements No. 871260. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 94–101, 2023. https://doi.org/10.1007/978-3-031-39774-5_12

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conditions which must be satisfied by a function with the aim of merging each family of (pseudo-)metrics defined on the same set into a single one. Formally, given n ∈ N (N will denote the set of positive integers throughout the paper), a function F : [0, +∞]n → [0, +∞] is said to aggregate (pseudo-)metrics if, for each family of (pseudo-)metrics {di }ni=1 on the same (non-empty) set X, the function F ◦ d is a (pseudo-)metric on X, where d : X × X → [0, +∞[n is defined  y) = (d1 (x, y), . . . , dn (x, y)) for all x, y ∈ X. In such a case, F is said to by d(x, be a (pseudo-)metric aggregation function. In [2,4], both problems (the pseudo-metric and the metric one) were studied in depth and functions aggregating metrics and pseudo-metrics were characterized, respectively. Below we can find these characterizations. With the aim of stating them, let us recall that a, b, c ∈ [0, +∞]n forms an n-triangular triplet whenever a  b + c, b  a + c and c  a + b. Notice that ≤ denotes the usual partial order defined on [0, +∞] and that a  b ⇔ ai ≤ bi for all i = 1, . . . , n. Theorem 1 ([4]). Let n ∈ N and let F : [0, +∞[n → [0, +∞[ be a function. Then the statements below are equivalent: (1) F is a pseudo-metric aggregation function. (2) F has the below properties: (2.1) F (0n ) = 0, (2.2) F transforms n-triangular triplets into 1-triangular triplet. Theorem 2 ([2]). Let n ∈ N and let F : [0, +∞[n → [0, +∞[ be a function. Then the statements below are equivalent: (1) F is a metric aggregation function. (2) F has the below properties: (2.1) F (0n ) = 0, (2.2) F transforms n-triangular triplets into 1-triangular triplet, (2.3) If a ∈ [0, +∞[n and F (a) = 0, then min{a1 , . . . , an } = 0. Different generalizations of the concept of metric can be found in the literature. For instance, in [1] the notion of modular (pseudo-)metric were introduced. In the new axiomatic the metric can take the value +∞ and it depends on a positive parameter such as the next definition shows. Definition 1. A function w :]0, +∞[×X ×X → [0, +∞] is said to be a modular metric on a non-empty set X provided that, for each x, y, z ∈ X and λ, μ > 0, the following conditions are satisfied: (MM1) w(λ, x, y) = 0 for all λ > 0 if and only if x = y, (MM2) w(λ, x, y) = w(λ, y, x), (MM3) w(λ + μ, x, z) ≤ w(λ, x, y) + w(λ, y, z). When condition (MM1) is exchanged by condition (MM1’), we say that w is a modular pseudo-metric on X, where (MM1’) w(λ, x, x) = 0 for all λ > 0.

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The value provided by a modular (pseudo-)metric can be interpreted as a measurement of the dissimilarity relative to the parameter λ. Then, this kind of generalized metric can show more robustness in problems in which objects of different scale are under consideration. On account of the studies carried out in [1], our target is to characterize modular (pseudo-)metric aggregation functions. The remaining of the paper is organized as follows. Section 2 is focused on the aggregation of modular pseudo-metrics. Concretely, we provide an appropriate characterization of modular (pseudo-)metric aggregation functions in terms of triangle triplets. Next, in Sect. 3, we will extend the study exposed in Sect. 2 to the framework of modular metric aggregation functions. Finally, Sect. 4 exposes the conclusions of the study carried out.

2

The Aggregation of Modular Pseudo-Metrics

First of all, let us introduce the following definition in the spirit of [3,5]. Definition 2. Let n ∈ N. A function F : [0, +∞]n → [0, +∞] is said to aggregate modular (pseudo-)metrics if, for each family of modular (pseudo-)metrics  is a modular (pseudo{wi }ni=1 defined on the same set X, the function F ◦ w )metric on X, where w  :]0, +∞[×X × X → [0, +∞]n is given, for all x, y ∈ X, λ > 0, by w(λ,  x, y) = (w1 (λ, x, y), . . . , wn (λ, x, y)) . In such a case, F is said to be a modular (pseudo-)metric aggregation function. Taking into consideration the above notion we characterize functions aggregating modular pseudo-metrics inthe following. With this aim, we will consider the following result which was proved in [1]. Proposition 1. Let w be a modular pseudo-metric on a non-empty set X. For each x, y ∈ X, the function wxy :]0, +∞[ → [0, +∞] defined by wxy (λ) = w(λ, x, y) is decreasing. Now, we will provide necessary conditions for the functions which aggregate modular pseudo-metrics. With this aim, given n ∈ N, we will denote by 0n the element (0, . . . , 0) ∈ [0, +∞]n . Also, a function F : [0, +∞]n → [0, +∞] is said to be monotone provided that F (a) ≤ F (b) whenever a  b and, in addition, F is said to be subadditive when F (a + b) ≤ F (a) + F (b) for each a, b ∈ [0, +∞]n . Theorem 3. Let n ∈ N. If F : [0, +∞]n → [0, +∞] is a modular pseudo-metric aggregation function, then F (0n ) = 0, F is monotone and subadditive. Proof. Let F be a modular pseudo-metric aggregation function. First, we show that F (0n ) = 0. Given an arbitrary family of modular pseudo-metrics {wi }ni=1 on a non-empty set X. Since F is a modular pseudo-metric aggregation function we have that F ◦ w  is a modular pseudo-metric. Then, given x ∈ X we get, by  x, x) = 0. (MM1’), that F (0n ) = F (w1 (λ, x, x), . . . , wn (λ, x, x)) = F ◦ w(λ, Now, we are showing the monotonicity of F . Let a, b ∈ [0, +∞]n with a  b. Next we prove that F (a) ≤ F (b).

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Set X = {x, y}, where x = y, and let {wi }ni=1 be a family of functions defined on ]0, +∞[×X × X by wi (λ, x, y) = wi (λ, y, x) = bi if 0 < λ < 1, wi (λ, x, y) = wi (λ, y, x) = ai if λ ≥ 1 and wi (λ, y, y) = wi (λ, x, x) = 0 for all λ > 0. Next we will see that each wi defines a modular pseudo-metric on X. Fix an arbitrary i ∈ {1, . . . , n}. By its construction, wi satisfies axioms (MM1’) and (MM2). So, it remains to show that (MM3) is also satisfied. Let u, v, w ∈ X and λ, μ > 0. We are showing that wi (λ + μ, u, w) ≤ wi (λ, u, v) + wi (μ, v, w). The case u = w is obvious, so assume u = w. We just need to distinguish two cases. – Suppose λ + μ ≤ 1. In this case, λ, μ ≤ 1 and so wi (λ + μ, x, y) = bi ≤ bi + 0 = wi (λ, x, y) + wi (μ, y, y), wi (λ + μ, x, y) = bi ≤ 0 + bi = wi (λ, x, x) + wi (μ, x, y). – Suppose now that λ + μ ≥ 1. In this case, by Proposition 1, we have wi (λ + μ, x, y) = ai ≤ wi (λ, x, y) = wi (λ, x, y) + wi (μ, y, y), wi (λ + μ, x, y) = ai ≤ wi (μ, x, y) = wi (λ, x, x) + wi (μ, x, y). Thus, {wi }ni=1 is a collection of modular pseudo-metrics on X. Since F aggregates modular pseudo-metrics we have that F ◦ w  is a modular pseudo-metric on X. By (MM1’) and (MM3) and, on account that F (0n ) = 0, we get: F (b)        = F (b) + F (0n ) =  F w1  12 , x, y , . . . , wn 12 , x, y + F w1 12 , y, y , . . . , wn 12 , y, y = F ◦w  12 , x, y + F ◦ w  12 , y, y ≥ F ◦ w  (1, x, y) = F (w1 (1, x, y) , . . . , wn (1, y, y)) = F (a).

Then F is monotone. Finally, we will show that F is subadditive. Let a, b ∈ [0, +∞]n and set X = {x, y, z} with different x, y, z. Let {wi }ni=1 be the family of functions on X defined, for each i ∈ {1, . . . , n} and for all λ > 0, as follows: ccwi (λ, x, y) = wi (λ, y, x) = ai , wi (λ, x, z) = wi (λ, z, x) = ai + bi ,

wi (λ, y, z) = wi (λ, z, y) = bi , wi (λ, y, y) = wi (λ, x, x) = wi (λ, z, z) = 0.

In the following, let us show that each wi is a modular pseudo-metric. Fix i ∈ {1, . . . , n}. Again, by construction, conditions (MM1’) and (MM2) are hold by wi . Now, we will show that (MM3) is also fulfilled. Let λ, μ > 0, then wi (λ, x, y) + wi (μ, y, z) = ai + bi ≥ ai + bi = wi (λ + μ, x, z), wi (λ, x, z) + wi (μ, z, y) = ai + 2bi ≥ ai = wi (λ + μ, x, y), wi (λ, y, x) + wi (μ, x, z) = 2ai + bi ≥ bi = wi (λ + μ, y, z). Hence, for all u, v ∈ X with u = v, we get wi (λ, u, u) + wi (μ, u, v) ≥ wi (λ + μ, u, v). Moreover, for all u, v ∈ X, we have wi (λ, u, v) + wi (μ, v, u) ≥ 0 = wi (λ + μ, u, u).

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The remaining cases are obtained by the symmetry of wi . Therefore, {wi }ni=1 is a collection of modular pseudo-metrics on X. Since F is a modular pseudo-metric aggregation function we obtain that F (a + b) = F (a1 + b1 , . . . , an + bn ) = F (w1 (λ + μ, x, z), . . . , wn (λ + μ, x, z)) = F ◦ w(λ  + μ, x, z) ≤ F ◦ w(λ,  x, y) + F ◦ w(μ,  y, z) = F (w1 (λ, x, y), . . . , wn (λ, x, y)) + F (w1 (μ, y, z), . . . , wn (μ, y, z)) = F (a) + F (b).

According to Theorem 3, we obtain more necessary conditions for modular pseudo-metric aggregation functions. Theorem 4. Let n ∈ N. If F : [0, +∞]n → [0, +∞] is a function such that F (0n ) = 0, then the statements below are equivalent: (1) F is monotone and subadditive. (2) F (a) + F (b) ≥ F (c) for all a, b, c ∈ [0, +∞]n with c  a + b. (3) F is monotone and transforms n-triangular triplets into a 1-triangular triplet. Proof. (1) ⇒ (2). Let a, b ∈ [0, +∞]n with c  a + b. Then F (a) + F (b) ≥ F (a + b) ≥ F (c), the first inequality is due to the subadditivity of F and the second one follows from the monotony of F . (2) ⇒ (3). Obviously, condition (2) gives that F transforms n-triangular triplets into 1-triangular triplet. Moreover, if b  a, then b  a+0n and condition (2) ensures that F (a) + F (0n ) ≥ F (b). Taking into account that F (0n ) = 0 we obtain that F (a) ≥ F (b) and so F turns out to be monotone. (3) ⇒ (1). We just need to prove that F is subadditve. Let a, b ∈ [0, +∞]n and consider c = a + b. It is clear that a, b, c forms an n-dimensional triangular triplet, then F (a) + F (b) ≥ F (c) = F (a + b). In the following we provide a characterization of modular pseudo-metric aggregation functions. Theorem 5. Let n ∈ N and let F : [0, +∞]n → [0, +∞] be a function. Then the statements below are equivalent: (1) (2) (3) (4)

F is a modular pseudo-metric aggregation function. F (0n ) = 0, F is monotone and subadditive. F (0n ) = 0 and F (a) + F (b) ≥ F (c) for a, b ∈ [0, +∞]n with c  a + b. F (0n ) = 0, F is monotone and transforms n-triangular triplets into 1triangular triplet.

Proof. (1) ⇒ (2) This implication is guaranteed by Theorem 3. (2) ⇒ (1) Let {wi }ni=1 be a collection of modular pseudo-metrics on X. Let us see that if F satisfies (2), then it aggregates {wi }ni=1 . (MM1’) Let x ∈ X, then F (w1 (λ, x, x), . . . wn (λ, x, x)) = F (0n ) = 0.

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(MM3) Let x, y, z ∈ X and λ, μ > 0. Taking into account that each wi is a modular pseudo-metric on X we have wi (λ, x, y) + wi (μ, y, z) ≥ wi (λ + μ, x, z). Then, F (w1 (λ, x, y), . . . , wn (λ, x, y)) + F (w1 (μ, y, z), . . . , wn (μ, y, z)) ≥ F (w1 (λ, x, y) + w1 (μ, y, z), . . . , wn (λ, x, y) + wn (μ, y, z)) ≥ F (w1 (λ + μ, x, z), . . . , wn (λ + μ, x, z)) . Notice that the first inequality is provided by the subadditivity of F and that the second one is due to the fact that F is monotone. (2) ⇔ (3) ⇔ (4) are provided by Theorem 4.

3

The Aggregation of Modular Metrics

Next we will characterize modular metric aggregation functions. We begin by proving the next result that provides necessary conditions. Theorem 6. Let n ∈ N. If F : [0, +∞]n → [0, +∞] is a modular metric aggregation function, then F (0n ) = 0, F is monotone and subadditive. Besides, if a ∈ [0, +∞]n and F (a) = 0, then ai = 0 for some i = 1, . . . , n. Proof. The same proof as in Theorem 3 remains valid to provide that F (0n ) = 0 and to show the monotony and subaddtivity of F . So, we will prove that if a ∈ [0, +∞]n and F (a) = 0, then ai = 0 for some i = 1, . . . , n. Suppose, contrary to our claim, that a ∈ [0, +∞]n , F (a) = 0 and, in addition, that ai = 0 for all i = 1, . . . , n. Next consider the family of modular metrics {wi }ni=1 defined on X = {x, y} (with x = y) as follows: wi (λ, x, y) = wi (λ, y, x) = ai and wi (λ, x, x) = wi (λ, y, y) = 0, for all λ > 0. Next we see that each wi is a modular metric. Obviously, for all i ∈ {1, . . . , n}, wi (λ, x, y) = wi (λ, y, x) for each x, y ∈ X and λ > 0. So, (MM2) is fulfilled. Moreover, by construction of each wi , we have that wi (λ, x, y) = 0 for all λ > 0 if, and only if, x = y. Thus, (MM1) is also held. It remains to show that (MM3) is also satisfied. Fix i ∈ {1, . . . , n} and let λ, μ > 0, wi (λ + μ, x, y) = ai ≤ ai + 0 = wi (λ, x, y) + wi (μ, y, y), wi (λ + μ, x, y) = ai ≤ 0 + ai = wi (λ, x, x) + wi (μ, x, y). The other cases are analogous by the symmetry of wi . Hence, for all u, v ∈ X with u = v, we have that wi (λ + μ, u, v) ≤ wi (λ, u, u) + wi (μ, u, v). Moreover, for all u, v ∈ X, wi (λ + μ, u, u) = 0 ≤ wi (λ, u, v) + wi (μ, v, u).

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The fact that F is a modular metric aggregation function yields, for all λ > 0, that F ◦ w(λ,  x, y) = F (w1 (λ, x, y), . . . , wn (λ, x, y)) = F (a) = 0. This yields a contradiction because x = y. Let us recall that a function F : [0, +∞]n → [0, +∞] is said to be positive provided that F (a) ∈]0, +∞] whenever a ∈]0, +∞]n . Notice that Theorem 6 warranties that every modular pseudo-metric aggregation function F is positive. Indeed, if a ∈]0, +∞]n then F (a) = 0 because otherwise ai = 0 for any i ∈ {1, . . . , n}. Theorem 6 allows us to characterize modular metric aggregation functions. Theorem 7. Let n ∈ N and let F : [0, +∞]n → [0, +∞] be a function. The statements below are equivalent: (1) F is a modular metric aggregation function. (2) F (0n ) = 0, F is monotone and subadditive. Moreover, if a ∈ [0, +∞]n and F (a) = 0, then ai = 0 for some i = 1, . . . , n. (3) F (0n ) = 0, F is monotone and, in addition, F (a) + F (b) ≥ F (c) for all a, b, c ∈ [0, +∞]n with c  a + b. Moreover, if a ∈ [0, +∞]n and F (a) = 0, then ai = 0 for some i = 1, . . . , n. (4) F (0n ) = 0, F is monotone and transforms n-triangular triplets into a 1triangular triplet. Moreover, if a ∈ [0, +∞]n and F (a) = 0, then ai = 0 for some i = 1, . . . , n. Proof. (1) ⇒ (2). It follows from Theorem 6. (2) ⇒ (1) Let {wi }ni=1 be a collection of modular metrics on X. We need to show that F ◦ w  is a modular metric on X. On the one hand, F ◦ w(λ,  x, y) = F ◦ w(λ,  y, x) for each x, y ∈ X and λ > 0. Whence we get that (MM2) is held. On the other hand, fix λ > 0. If F ◦ w(λ,  x, y) = F (w1 (λ, x, y), . . . , wn (λ, x, y)) = 0, then wi (λ, x, y) = 0 for some i = 1, . . . , n and, thus, x = y. Moreover, F ◦ w(λ,  x, x) = F (w1 (λ, x, x), . . . , wn (λ, x, x)) = F (0n ) = 0. Hence, (MM1) is also fulfilled. Next we show that (MM3) also holds. By Theorem 5 we deduce that F (a) + F (b) ≥ F (c) for all a, b, c ∈ [0, +∞]n with c  a + b. Therefore, given x, y, z ∈ X and λ, μ > 0, we have F ◦ w(λ,  x, y) + F ◦ w(μ,  y, z) = F (w1 (λ, x, y), . . . , wn (λ, x, y)) + F (w1 (μ, y, z), . . . , wn (μ, y, z)) ≥ F (w1 (λ, x, y) + w1 (μ, y, z), . . . , wn (λ, x, y) + wn (μ, y, z)) ≥ F (w1 (λ + μ, x, z), . . . , wn (λ + μ, x, z)) = F ◦ w(λ  + μ, x, z), since wi (λ, x, y) + wi (μ, y, z) ≥ wi (λ + μ, x, z). Thus, F aggregates modular metrics. (2) ⇔ (3) ⇔ (4) follow from Theorem 4.

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On account of Theorems 5 and 7 we obtain the following corollary, which establishes a relationship between modular metric aggregation functions and modular pseudo-metric aggregation functions. Corollary 1. Let n ∈ N and F : [0, +∞]n → [0, +∞] be a function. If F is a modular metric aggregation function, then F is a modular pseudo-metric aggregation function. Next example clarifies that the converse of the previous corollary it not true, in general. Example 1. Let F : [0, +∞]n → [0, +∞] be the function defined by F (a) = 0 for all a ∈ [0, +∞]n . Clearly, F is monotone, F (0n ) = 0 and, given a, b ∈ [0, +∞]n , we have that F (a) + F (b) = 0 ≥ 0 = F (a + b). Theorem 5 gives that F is a modular pseudo-metric aggregation function. However, F is not a modular pseudo-metric aggregation function, since F does not satisfy (2) in Theorem 7.

4

Conclusions

The aggregation problem of modular (pseudo-)metrics has been studied in detail. In particular, the concept of modular (pseudo-)metric aggregation function has been introduced and, in addition, we have obtained appropriate characterizations in terms of triangle triplets. As a further work we will explore the possible kinship between modular (pseudo-)metric aggregation functions and (pseudo)metric aggregation functions. Moreover we will consider to apply the theory exposed throughout the paper in problems as clustering where the objects to classify are defined in different scale.

References 1. Chistyakov, V.V.: Modular metric spaces, I: basic concepts. Nonlinear Anal. Theory Methods Appl. 72(1), 1–14 (2010) 2. Mayor, G., Valero, O.: Metric aggregation functions revisited. Eur. J. Combin. 80, 390–400 (2019) 3. Pedraza, T., Rodr´ıguez-L´ opez, J., Valero, O.: Aggregation of fuzzy quasi-metrics. Inf. Sci. 581, 362–389 (2021) 4. Pradera, A., Trillas, E.: A note on pseudometrics aggregation. Int. J. General Syst. 31(1), 51–51 (2002) 5. Calvo S´ anchez, T., Fuster-Parra, P., Valero, O.: The aggregation of transitive fuzzy relations revisited. Fuzzy Sets Syst. 446, 243–260 (2022)

Peterson’s Rules Based on Grades for Fuzzy Logical Syllogisms Petra Murinov´ a(B) , Michal Burda, and Viktor Pavliska Institute for Research and Application of Fuzzy Modeling, University of Ostrava, Ostrava, Czech Republic [email protected] https://ifm.osu.cz/ Abstract. In this publication, we continue the study of fuzzy Peterson’s syllogisms. While in the previous publication, we focused on verifying the validity of these syllogisms using the construction of formal proofs and semantic verification, in this publication we focus on verifying the validity of syllogisms using Peterson’s rules based on grades. The main goal will be to mathematically formulate Peterson’s rules for verifying the validity and invalidity of logical syllogisms with intermediate quantifiers. Keywords: Peterson’s syllogisms · Peterson’s rules of opposition · Peterson’s rules based on grades

1

· Peterson’s square

Introduction

The theory of generalized syllogisms was studied by several authors as a generalization of classical Aristotle’s syllogisms [12] The extended syllogistic reasoning (see, e.g., [1,13]) adds new quantifiers. Other mathematical models of some generalized quantifiers and its corresponding syllogisms were suggested by, for example, H´ajek, Pereira and others [3,4,10,11]. In this publication we will focus on verifying the validity of syllogisms using Peterson’s rules based on grades, the main idea of which is built on the position of intermediate quantifiers in graded Peterson’s square of opposition [5,6]. We will follow up on the previous results, where we mathematically formulated Peterson’s rules based on grades (see [7]). We suggested formal definitions of distributivity, quality and quantity rules, respectively. We formally proved that the rules of quantity can inferred from the rules of distributivity and quality. In this paper we verify validity of selected forms of syllogisms with intermediate quantifiers using Peterson’s rules based on grades, see [7]. We prove that a logical syllogism of Figure I with intermediate quantifiers is valid iff it satisfies all the extended Peterson’s rules quality, quantity and distributivity. OP PIK CZ.01.1.02/0.0/0.0/17147/0020575 AI-Met4Laser: Consortium for industrial research and development of new applications of laser technologies using artificial intelligence methods (10/2020 - 6/2023), of the Ministry of Industry and Trade of the Czech Republic. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 102–109, 2023. https://doi.org/10.1007/978-3-031-39774-5_13

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Another approach of proposed extended Peterson rules is given in [9]. This approach is related to the mathematical definitions of intermediate quantifiers, which are based on the theory of evaluative linguistic expressions [8]. Our approach is much more general, as it is based only on the position of intermediate quantifiers in a graded Peterson square of opposition.

2 2.1

Background Peterson’s Syllogisms and Peterson’s Square

Syllogisms with a single middle term yield the following four figures: Figure Q1 M is Q2 S is Q3 S is

I P M P

Figure Q1 P is Q2 S is Q3 S is

II M M P

Figure III Q1 M is P Q2 M is S Q3 S is P

Figure IV Q1 P is M Q2 M is S Q3 S is P

As we stated in the introduction to this article, Peterson’s syllogisms belong to the group of generalized syllogisms. The explanation of relationships which follow from graded Peterson’s square of opposition is demonstrated on the following example: If “Almost all children like chocolate” is valid, then according to the contrary relationship, “A few children like chocolate” cannot be valid. The main objective of this subsection is to remind what distribution is. As we are limited by space, we will list Peterson’s rules as part of Peterson’s rules based on grades. For the detail see [2]. Definition 1 (Distribution). A formula in the position of subject or predicate is a claim is distributed when it says something definite about all members of that category. Definition 2 (Undistribution). A formula in the position of subject or predicate is a claim is undistributed when it does not say something definite about all members of that category.

3

Peterson’s Rules Based on Grades

A set of all considered quantifiers will be denoted by Q. For Peterson’s framework of generalized intermediate syllogisms, we have Q = {“all”, “almost all”, “most”, “many”, “some”.} Generally we will work with affirmative and negative propositions as follows: Definition 3 (Proposition). Proposition is an expression in the form either Q(B, A)

or

Q(B, ¬A)

where Q is a 1, 1 quantifier and A, B are terms. Q(B, A) is an affirmative and Q(B, ¬A) is a negative proposition.

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We continue with the mathematical definition of the validity of logical syllogisms. Definition 4 (Valid Syllogism). Let S = P1 , P2 , C be a syllogism of three propositions. P1 , P2 are called premises (P1 is major and P2 is minor). A syllogism is valid if M(P1 ) ⊗ M(P2 ) ≤ M(C) for any model M. Naturally speaking the syllogism is valid if the L  ukasiewicz conjunction of the degrees of both premises is less than or equal to the degree of the conclusion. Definition 5. Let M |= T be a model and Q be quantifier. We assume that the following inequality holds: (A1) M((∀x)(A(x), B(x))) ⊗ M(Q(x)(C(x), A(x))) ≤ M(Q(x)(C(x), B(x))) The assumed property is very natural. The property is some generalization of the very well-known tautologies T  (A(x) ⇒ B(x) & C(x) ⇒ A(x)) ⇒ (C(x) ⇒ B(x)) and T  (A(x) ⇒ B(x)) & (C(x) ∧ A(x)) ⇒ (C(x) ∧ B(x)). Quantity. Each quantifier Q ∈ Q must have assigned a quantity(Q) which is explained in the subsequent definition. By quantity(Qalmost all ) we, for example, denote the quantity of the quantifier “Almost all”. If the kind of quantifier does not matter in the construction of the proof, we will write in general quantity(QGeneral ). Definition 6 (Quantity). Let Q be a quantifier. We say that1 (a) 0 < quantity(Q) ≤ 1 for all Q ∈ Q; (b) quantity(Q1 ) ≤ quantity(Q2 ) iff for Q1 , Q2 ∈ Q, proposition Q2 (B, A) is superaltern of proposition Q1 (B, A), i.e., M(Q2 (B, A)) ≤ M(Q1 (B, A)); (c) quantity(Q) > 0.5 iff for Q ∈ Q, proposition Q(B, A) is contrary to Q(B, ¬A); (d) quantity(Q1 ) + quantity(Q2 ) > 1 if Q1 (B, A) and Q2 (B, ¬A) form a contradictory pair. Grade. In Peterson’s approach, the distribution index is based on the number of intermediate quantifiers. It means that the maximal value is 5 and the other values depend on the position in the Peterson’s square of opposition. Our approach is based on the size that the quantifier represents in the Peterson square. In our approach, we represent this size by a grade, which is a value from an interval [0, 1]. Definition 7. Let us assume five basic Peterson’s quantifiers. Then quantity of quantifier is represented by grade as follows: 1

The reader can find a detailed definition of the properties mentioned in this definition in [5].

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quantity(QAll ) = 1, quantity(QAlmost all ) = p, quantity(QMost ) = t, quantity(QMany ) = k, quantity(QSome ) = 

such that (a) 0 <  < k < 0.5 < t < p < 1 −  < 1, (b) k + p > 1, (c) t + k ≤ 1. The property (a) guarantees a monotonous position of quantifies in Peterson’s square. Specially, the condition k < 0.5 < t < p means that P, B have predominant position and T, D have majority position. Moreover all positive as we´ll as negative quantifiers A, E, P, B, T, D are preponderance quantifiers while K, G, I, O are not. The condition (b) express the property of a contradictory of the pair [P, G] as well as of [B, K] while the condition (c) express the fact that [K, D] and [G, T] do not constitute a contradictory pair. For the sake of brevity, we put quantity(P) = quantity(Q), for any proposition P = Q(., .). Signum Definition 8 (Signum). The signum of proposition P is defined as  1 if P is affirmative, signum(P) = 0 if P is negative. Distribution Definition 9 (Distribution). The distribution of term T in proposition P, dist(T, P), equals to 1. quantity(Q) if P = Q(T, X); 2.  if P = Q(X, T ) and P is affirmative; 3. 1 if P = Q(X, T ) and P is negative. 3.1

Graded Peterson’s Rules

Below we recall Peterson’s rules and also we introduce all the six Peterson’s rules of distributivity, quality and quantity based on grades defined using generalized logical operations. By (R1)-(R6) we denote classical Peterson’s rules and by (GR1)-(GR6) we denote Peterson’s rules based on grades. Definition 10 (Peterson’s rules based on grades). Let ⊗ be L  ukasiewicz t-norm, ∧ be G¨ odel t-norm (minimum), ∨ be G¨ odel t-conorm (maximum), S = P1 , P2 , C be a syllogism such that S is the first term of conclusion C, P is the second term of conclusion C and M is the middle term.

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1. Rules of Distribution (R1) In a valid syllogism, the sum of distribution indices for the middle formula must exceed 5. (GR1) dist(M, P1 ) ⊗ dist(M, P2 ) > 0; (R2) No formula may be more nearly distributed in the conclusion than it is in the premises. (GR2a) dist(S, C) ≤ dist(S, P2 ); (GR2b) dist(P, C) ≤ dist(P, P1 ); 2. Rules of Quality (R3) At least one premise must be affirmative. (GR3) signum(P1 ) ∨ signum(P2 ) = 1; (R4) The conclusion is negative if and only if one of the premises is negative. (GR4) signum(P1 ) ∧ signum(P2 ) = signum(C); 3. Rules of Quantity (R5) At least one premise must have a quantity of majority (T or D) or higher. (GR5) quantity(P1 ) ∨ quantity(P2 ) > 0.5; (R6) If any premise is non-universal, the conclusion must have a quantity that is less than or equal to that premise. (GR6) quantity(P1 ) ∧ quantity(P2 ) ≥ quantity(C). Lemma 1 ([7]). Let rules (GR1-GR4) be satisfied. Then the rule (GR5) and (GR6) hold. 3.2

Syllogisms of Figure-I Affirmative

Lemma 2. Syllogisms AAA, AAP, AAT, AAK, AAI of Figure-I satisfy Rules (GR1)–(GR4). Proof. Let us consider syllogism AAA. Then P1 = QAll (M, P ), P2 = QAll (S, M ), C = QAll (S, P ) and QAll (M, P ) ⊗ QAll (S, M ) ≤ QAll (S, P ) holds. Then dist(M, QAll (M, P )) = quantity(QAll ) = 1 and dist(M, QAll (S, M )) = . Therefore, Rule (GR1) is satisfied. The same argument can be applied also to syllogisms AAP, AAT, AAK, AAI of Figure-I and, hence, they also satisfy Rule (GR1). Furthermore, dist(S, QAll (S, M )) = quantity(QAll ) = 1 and dist(S, QAll (S, P )) = quantity(QAll ) = 1. This means that syllogism AAA satisfies Rule (GR2a). The inequality dist(S, QMany (S, P )) ≤ dist(S, QMost (S, P )) ≤ dist(S, QAlmost all (S, P )) ≤ dist(S, QAll (S, M )) = 1, implies that Rule (GR2a) is satisfied by syllogisms AAP, AAT, AAK.

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Because dist(S, QSome (S, P )) = quantity(QSome ) =  then Rule (GR2a) is satisfied by syllogism AAI. Rule (GR2b) is satisfied by all the considered syllogisms as well because  = dist(P, QGeneral (S, P )) ≤ dist(P, QAll (M, P )). Rule (GR3) is obviously satisfied and Rule (GR4) does not apply. Lemma 3. Syllogisms APP, APT, APK, API of Figure-I satisfy Rules (GR1)–(GR4). Proof. Let us consider syllogism APP. Then P1 = QAll (M, P ), P2 = QAlmost all (S, M ), C = QAlmost all (S, P ) and QAll (M, P ) ⊗ QAlmost all (S, M ) ≤ QAlmost all (S, P ). Then dist(M, QAll (M, P )) = quantity(All) = 1 and dist(M, QAlmost all (S, M )) = . Therefore, Rule (GR1) is satisfied by APP and, consequently, also by APT, APK, API because the main role plays the universal quantifier in the major premise. Furthermore, quantity(QAlmost all ) ≤ quantity(QAlmost all ) which means that dist(S, QAlmost all (S, P )) ≤ dist(S, QAlmost all (S, M )) and, therefore, Rule (GR2a) is satisfied. Furthermore, since  = dist(P, QAll (M, P )) = dist(P, QAlmost all (S, P )) = . Thus syllogisms APP, APT, APK, API satisfy Rule (GR2b). Rule (GR3) is obviously satisfied and (GR4) does not apply. Based on the lemmas above, the following theorem follows. Theorem 1. All valid syllogisms of Figure-I satisfy Rules (GR1)–(GR4). Lemma 4. Let Rules (GR1)–(GR4) of distributivity and quality be satisfied. Then all Peterson’s syllogisms of Figure-I hold. Proof. Let us assume that the rules (GR1) to (GR4) are satisfied. We are going to prove that the syllogism S is valid, which means to prove the following inequality: M(P1 ) ⊗ M(P2 ) ≤ M(C). Rules (GR3) and (GR4) yield in three variants of four figures of syllogisms. Let us examine each variant: Figure-I a): P1 = Q1 (M, P ), P2 = Q2 (S, M ), C = QC (S, P ): Evidently, dist(M, P1 ) = quantity(Q1 ), dist(M, P2 ) = ε and because of (GR1), quantity(Q1 ) = 1, i.e., Q1 is “All”. From (A1), by putting A = M , B = P , C = S, we obtain

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M(QAll (M, P )) ⊗ M(Q2 (S, M )) ≤ M(Q2 (S, P )). At the same time, dist(S, C) = quantity(QC ) and dist(S, P2 ) = quantity(Q2 ). From (GR2a) we obtain quantity(QC ) ≤ quantity(Q2 ), i.e., Q2 is equal to or super-altern of QC , i.e., M(Q2 (S, P )) ≤ M(QC (S, P )), which yields to M(QAll (M, P )) ⊗ M(Q2 (S, M )) ≤ M(QC (S, P )). Figure-I b): P1 = Q1 (M, ¬P ), P2 = Q2 (S, M ), C = QC (S, ¬P ): by applying (A1) (putting A = M, B = ¬P, C = S), the proof is analogous to Figure-I (a). Figure-I c): P1 = Q1 (M, P ), P2 = Q2 (S, ¬M ), C = QC (S, ¬P ): dist(P, C) = 1 and dist(P, P1 ) = ε, which violates (GR2b) (dist(P, C) ≤ dist(P, P1 )), i.e., this variant of syllogisms always violates the rules. Based on Lemmas above we conclude theorem below. Theorem 2. Let (GR1)–(GR4) be satisfied. Then the syllogisms of Figure-I are valid. Theorem 3. All the syllogisms of Figure-I are valid iff (GR1)–(GR4) are satisfied.

4

Examples and Discussion About Generalized Rules

In this section we will focus on the explanation of individual rules on specific example of logical syllogism. Let us assume the example of Figure-I while the majority premise will be negative. EAE-I All M are ¬ P All S are M All S are ¬ P

M S

P

We can observe that this syllogism is valid in every model. Below we will also verify the validity using generalized Peterson rules. Rule (GR1): dist(M, P1 ) = quantity(QAll ) = 1 and dist(M, P2 ) = . So the Rule (GR1) is satisfied.

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Rule (GR2a): dist(S, P2 ) = quantity(QAll ) = 1 and dist(S, C) = quantity(QAll ) = 1. So the Rule (GR2a) is fulfilled. Rule (GR2b): dist(P, P1 ) = 1 and dist(P, P1 ) = 1. So the Rule (GR2b) is fulfilled. Rule (GR3): signum(P1 ) = 0 and signum(P2 ) = 1. So the Rule (GR3) is fulfilled (signum(P1 ) ∨ signum(P2 ) = 1). Rule (GR3): signum(P1 ) = 0, signum(P2 ) = 1 and signum(C) = 0. We have signum(P1 ) ∧ signum(P2 ) = signum(C). So the Rule (GR4) is fulfilled. Finally, we conclude that using Theorem 3 the syllogism is valid.

5

Conclusion

In this publication, we have devoted ourselves to studying generalized Peterson syllogisms and verifying their validity using Peterson’s rules using grades. We first recalled the mathematically designed Peterson’s rules using grades. We demonstrated the proposed rules on a concrete example of a logical syllogism and showed how these rules work. The main result of this publication was to show that a logical syllogism of Figure I with intermediate quantifiers is valid iff it satisfies all the extended Peterson’s rules quality, quantity and distributivity.

References 1. Dubois, D., Prade, H.: On fuzzy syllogisms. Comput. Intell. 4, 171–179 (1988) 2. Gainor, J.: What is distribution in categorical syllogisms (2011). https://www. youtube.com/watch?v=7 Y-Bxr4apQ 3. Gl¨ ockner, I.: Fuzzy Quantifiers: A Computational Theory. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-32503-4 4. H´ ajek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998) 5. Murinov´ a, P.: Graded structures of opposition in fuzzy natural logic. Log. Univers. 265, 495–522 (2020) 6. Murinov´ a, P., Nov´ ak, V.: Analysis of generalized square of opposition with intermediate quantifiers. Fuzzy Sets Syst. 242, 89–113 (2014) 7. Murinov´ a, P., Pavliska, V., Burda, M.: Generalized Peterson’s rules in fuzzy natural logic. In: Atlantis Studies in Uncertainty Modelling 2021-09-19 Bratislava, pp. 383– 390 (2021) 8. Nov´ ak, V.: A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets Syst. 159(22), 2939–2969 (2008) 9. Nov´ ak, V., Murinov´ a, P., Ferbas, P.: Formal analysis of Peterson’s rules for checking validity of syllogisms with intermediate quantifiers. Int. J. Approx. Reason. 150, 122–138 (2022) 10. Pereira-Fari˜ na, M., D´ıaz-Hermida, F., Bugar´ın, A.: On the analysis of set-based fuzzy quantified reasoning using classical syllogistics. Fuzzy Sets Syst. 214, 83–94 (2013) 11. Pereira-Fari˜ na, M., Vidal, J.C., D´ıaz-Hermida, F., Bugar´ın, A.: A fuzzy syllogistic reasoning schema for generalized quantifiers. Fuzzy Sets Syst. 234, 79–96 (2014) 12. Ross, W.D.: Aristotle’s Prior and Posterior Analytics. Clarendom Press, Oxford (1949) 13. Schwartz, D.G.: Dynamic reasoning with qualified syllogisms. Artif. Intell. 93, 103–167 (1997)

A New Construction of Uninorms in Bounded Lattices Derived from T-Norms and T-Conorms G¨ ul Deniz C ¸ aylı(B) Department of Mathematics, Faculty of Science, Karadeniz Technical University, 61080 Trabzon, Turkey [email protected] Abstract. The derivation of uninorms in bounded lattices has been a charming research interest. This article aims to enhance the generation of uninorms in bounded lattices on the basis of t-norms and t-conorms. We present a mean for getting uninorms in a bounded lattice possessing an identity element chosen as an arbitrary element of the lattice by virtue of a closure operator and a t-norm. Moreover, we propose a different technique to obtain uninorms in a bounded lattice derived from an interior operator and a t-conorm. We also provide an illustrative example to help understand these new families of uninorms in bounded lattices. Keywords: Bounded lattice Triangular norm · Uninorm

1

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Introduction

Uninorms in the real unit interval [0, 1] were presented by Yager and Rybalov [31] as a significant type of aggregation functions combining triangular norms (tnorms, in brief) and triangular conorms (t-conorms, in brief) [23,29]. Thereafter, Fodor et al. [19] consistently investigated these functions which permit for their identity element being anyplace in [0, 1] rather than at point 1 (as in t-norms) or point 0 (as in t-conorms). They also gave the characterization of uninorms originated in t-norms and t-conorms through an ordinal sum arrangement. Uninorms have been determined to be advantageous several remarkable disciplines including expert systems, neural networks, aggregation of information, fuzzy logics and fuzzy system modeling [1,2,13,14,16,25,32]. Uninorms in bounded lattices were firstly explored by Kara¸cal and Mesiar [22]. They proved their presence by identifying the least and greatest uninorms in bounded lattices. After that, the tools for getting uninorms in bounded lattices have received a wide attention, in especial, the sufficient conditions for their structure have presented. By virtue of the presentiality of a t-norm and a tconorm in bounded lattices, various investigations concerning uninorms possessing an identity element that can be an arbitrary element of a bounded lattice have been described [4,5,12]. Many authors presented the means to procure uninorms in bounded lattices under several additional hypotheses [6–11,21,27,30,33]. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 110–118, 2023. https://doi.org/10.1007/978-3-031-39774-5_14

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Recently, Ouyang and Zhang [26] studied the building methods of uninorms through closure and interior operators in bounded lattices. Their methods, which are derived from only one of t-norms and t-conorms, encompass those introduced in [22] as a particular case. Based on the fact that a t-norm and a t-conorm always exist a bounded lattice, could we generate novel classes of uninorms in bounded lattices by way of closure and interior operators? In this work, we shall propose two different procedure for generating uninorms with the help of closure and interior operators in bounded lattices. In that case, the novel tools contribute to obtaining uninorms in bounded lattices and generalizing some generation techniques in the literature. Accordingly, they have an effect on the progress of the families of uninorms in bounded lattices and the determination of their structure. The rest of this article is arranged as indicated below. In Sect. 2, we provide some fundamental definitions and characteristics. In Sect. 3, we develop two ways for getting uninorms in a bounded lattice possessing an identity element, which can be an arbitrary element of a bounded lattice, using closure and interior operators. These methods also exploit the existence of a t-norm and a t-conorm in a bounded lattice. Furthermore, we supply an example to explain that our constructing means differs from those already in use. Conclusions are followed in Sect. 4.

2

Preliminaries

In this part, we bear in mind some fundamental concepts and outcomes concerning bounded lattices (for more information, see, e.g., [3]) and uninorms on them. A lattice B is a nonempty set equipped with a partial order relation  such that any two elements u, v ∈ B have a greatest lower bound (called meet or infimum), indicated by u ∧ v, as well as a smallest upper bound (called join or supremum), indicated by u ∨ v. For u, v ∈ B, the symbol u < v implies that u  v and u = v. The symbol u  v implies that u and v are incomparable, i.e., neither u  v nor v < u. Iu denotes the set of all elements incomparable with u, i.e., Iu = {w ∈ B : wu}. A lattice (B, ) is bounded if it has a greatest (also known as top) element and a smallest (also known as bottom) element, which are indicated by 1B and 0B , respectively (i.e., two elements 1B , 0B ∈ B exist such that 0B  u  1B for all u ∈ B). In this article, unless otherwise stated, B indicates a bounded lattice (B, , ∧, ∨) possessing a top element 1B and a bottom element 0B . For u, v ∈ B with u  v, the subinterval [u, v] of B is delineated such that [u, v] = {w ∈ B : u  w  v}. The subintervals [u, v[, ]u, v], and ]u, v[ of B is delineated similarly. ([u, v], , ∧, ∨) is a bounded lattice possessing a top element v and a bottom element u. Definition 1 ([20,22]). A binary function U : B × B → B is said to be a uninorm if it is commutative, associative, increasing in regard to both variables and there is an element e ∈ B, called an identity element, i.e., U (u, e) = u for all u ∈ B. Accordingly, a uninorm is a t-norm (resp. t-conorm) if e = 1B (resp. e = 0B ) (for more information about t-norms and t-conorms, see, e.g., [24,28]).

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Definition 2 ([15,17,18]). A unary function cl : B → B is said to be a closure operator if, for any u, v ∈ B, the undermentioned circumstances are fulfilled: (1) Expansion: v  cl(v). (2) Preservation of join: cl(u ∨ v) = cl(u) ∨ cl(v). (3) Idempotence: cl (cl(v)) = cl(v). By (1), the statement (3) equals to cl (cl(v))  cl(v). Additionally, (2) refers to (2) : cl(u)  cl(v) if u  v. Birkhoff [3] establishes a closure operator by (1), (2) and (3). Definition 3 ([15,17,18]). A unary function int : B → B is said to be an interior operator if, for any u, v ∈ B, the undermentioned circumstances are fulfilled: (1) Contraction: int(v)  v. (2) Preservation of meet: int(u ∧ v) = int(u) ∧ int(v). (3) Idempotence: int (int(v)) = int(v). By (1), the statement (3) equals to int(v)  int (int(v)). Additionally, (2) refers to (2) : int(u)  int(v) if u  v. Birkhoff [3] establishes an interior operator by (1), (2) and (3).

3

Constructions of Uninorms

In this part, we develop in Theorem 1 a novel approach to construct the class of uninorm U(Te ,cl) in B possessing an identity element e, which exploits a t-norm Te in [0B , e]2 and a closure operator cl in B. Moreover, we suggest in Theorem 2 a different generation method for the family of uninorm U(S,int) in B possessing an identity element e, which is originated in a t-conorm Se in [e, 1B ]2 and an interior operator int in B. Theorem 1. Let e ∈ B\{0B , 1B }, Te : [0B , e]2 → [0B , e] be a t-norm and cl : B → B be a closure operator. The binary function U(Te ,cl) : B × B → B, presented by the formula (1), is a uninorm in B possessing an identity element e iff cl(x) ∨ cl (y) ∈ Ie ∪ {1B } for all x, y ∈ Ie . ⎧ Te (u, v) if (u, v) ∈ [0B , e]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈]e, 1B ]2 , 1B ⎪ ⎪ ⎨ u∨v if (u, v) ∈ [0B , e]×]e, 1B ]∪]e, 1B ] × [0B , e], U(Te ,cl) (u, v) = v if (u, v) ∈ [0B , e] × Ie , ⎪ ⎪ ⎪ ⎪ u if (u, v) ∈ Ie × [0B , e], ⎪ ⎪ ⎩ cl (u) ∨ cl (v) if (u, v) ∈ Ie ×]e, 1B ]∪]e, 1B ] × Ie ∪ Ie × Ie . (1) If we consider in Theorem 1 the closure operator cl : B → B determined as  cl(x) = 1B for all x ∈ B, we get the undermentioned uninorm U(Te ,cl) in B, which has been introduced already in [22, Theorem 1].

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Corollary 1. Let e ∈ B\{0B , 1B }. In this case, the binary function U(Te ,cl) : B × B → B, presented by the formula 2), is a uninorm in B possessing an identity element e. ⎧ Te (u, v) if (u, v) ∈ [0B , e]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈ [0B , e]×]e, 1B ]∪]e, 1B ] × [0B , e], ⎨u ∨ v  if (u, v) ∈ [0B , e] × Ie , U(Te ,cl) (u, v) = v (2) ⎪ ⎪ if (u, v) ∈ Ie × [0B , e], ⎪u ⎪ ⎩ otherwise. 1B If we consider in Theorem 1 the closure operator cl : B → B determined as  cl(x) = x for all x ∈ B, we obtain the undermentioned uninorm U(Te ,cl) in B, which has been introduced already in [9, Theorem 3.9]. 

Corollary 2. Let e ∈ B\{0B , 1B }. The binary function U(Te ,cl) : B × B → B, presented by the formula (3), is a uninorm in B possessing an identity element e iff x ∨ y ∈ Ie ∪ {1B } for all x, y ∈ Ie . ⎧ Te (u, v) if (u, v) ∈ [0B , e]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈]e, 1B ]2 , ⎨ 1B  if (u, v) ∈ [0B , e] × Ie , U(Te ,cl) (u, v) = v (3) ⎪ ⎪ if (u, v) ∈ Ie × [0B , e], ⎪u ⎪ ⎩ u∨v otherwise. If we allow in Theorem 1 to be a coatom of the element e ∈ B\{0B , 1B }, we define the corresponding uninorm as the undermentioned structure: Corollary 3. Let e ∈ B\{0B , 1B } be a coatom and cl : B → B be a closure  operator. In this case, the binary function U(Te ,cl) : B × B → B, presented by the formula (4), is a uninorm in B possessing an identity element e. ⎧ Te (u, v) if (u, v) ∈ [0B , e]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈ [0B , e] × Ie , ⎨v  if (u, v) ∈ Ie × [0B , e], U(Te ,cl) = u (4) ⎪ ⎪ × I , cl (u) ∨ cl (v) if (u, v) ∈ I ⎪ e e ⎪ ⎩ otherwise. 1B Remark 1. Let e ∈ B\{0B , 1B }, Se : [e, 1B ]2 → [e, 1B ] be a t-conorm and cl : B → B be a closure operator. We describe in Theorem 1 a novel building approach for uninorms in bounded lattices. To be more precise, (i) For (u, v) ∈ Ie ×]e, 1B ]∪]e, 1B ] × Ie ∪ Ie × Ie , Ut (u, v) in the method in [22, Theorem 1] takes the value 1B while U(Te ,cl) (u, v) in our construction takes the value of cl (u) ∨ cl (v). However, both constructions match up in the remaining domains. (ii) For (u, v) ∈ Ie ×]e, 1B ]∪]e, 1B ] × Ie ∪ Ie × Ie , UT (u, v) in the method in [9, Theorem 3.9] takes the value of u ∨ v while U(Te ,cl) (u, v) in our construction takes the value of cl (u) ∨ cl (v). However, both constructions match up in the remaining domains.

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G. D. C ¸ aylı 1B n s m

e

k

0B

Fig. 1. The lattice B

(iii) For (u, v) ∈ Ie ×]e, 1B ]∪]e, 1B ]×Ie ∪Ie ×Ie (resp. (u, v) ∈]e, 1B ]2 ), U(S,T ) (u, v) in the method in [11, Corollary 3.2] takes the value of u ∨ v (resp. Se (u, v)) while U(Te ,cl) (u, v) in our generation way takes the value of cl (u) ∨ cl (v) (resp. 1B ). However, both constructions match up in the remaining domains. (iv) For (u, v) ∈]e, 1B ]2 , U (u, v) in the method in [26, Theorem 4.1] takes the value of cl (u) ∨ cl (v) while U(Te ,cl) (u, v) in our construction takes the value 1B . However, both constructions match up in the remaining domains. Remark 2. Let e ∈ B\{0B , 1B }. In that case, we get the undermentioned statements: (i) If the closure operator cl : B → B is determined as cl(x) = 1B for all x ∈ B, then the uninorm U(Te ,cl) in Theorem 1 matches up with Ut in [22, Theorem 1] and U in [26, Theorem 4.1]. (ii) If the closure operator cl : B → B is determined as cl(x) = x for all x ∈ B, then (ii-1) the uninorm U(Te ,cl) in Theorem 1 matches up with UT in [9, Theorem 3.9]; (ii-2) the uninorm U(Te ,cl) in Theorem 1 matches up with U(S,T ) in [11, Corollary 3.2], where the t-conorm Se : [e, 1B ]2 → [e, 1B ] is Se = SW ; (ii-3) the uninorm U(Te ,cl) in Theorem 1 matches up with U(S,T ) in [11, Corollary 3.2], where e is a coatom. Notice that the uninorm obtained by the method in Theorem 1 does not have to match up with those introduced in [9,11,22,26]. In the undermentioned example, we demonstrate this examination. Example 1. Given the lattice B described by Hasse diagram in Fig. 1, the closure operator cl : B → B is determined as cl(0B ) = 0B , cl (e) = cl (m) = cl (s) = cl (n) = n, cl (k) = k and cl(1B ) = 1B . Then we obtain the undermentioned facts: (i) the uninorms UT and U(S,T ) in [9, Theorem 3.9] and [11, Corollary 3.2], respectively, satisfy that UT (s, m) = U(S,T ) (s, m) = s; (ii) the uninorm Ut in [22, Theorem 1] satisfies that Ut (m, m) = 1B ;

A New Construction of Uninorms in Bounded Lattices

(iii) the uninorm U in [26, Theorem 4.1] satisfies that U (s, n) = n; (iv) the uninorm U(Te ,cl) in Theorem 1 satisfies that U(Te ,cl) (s, m) U(Te ,cl) (m, m) = n, and U(Te ,cl) (s, n) = 1B . Hence, U(Te ,cl) differs from the uninorms UT , U(S,T ) , Ut and U in B.

115

=

We propose in Theorem 2 a dual generation method for uninorms in bounded lattices. To be more precise, we suggest the family of uninorm U(Se ,int) in B possessing an identity element e via a t-conorm Se in [e, 1B ]2 and an interior operator int in B. Theorem 2. Let e ∈ B\{0B , 1B }, Se : [e, 1B ]2 → [e, 1B ] be a t-conorm and int : B → B be an interior operator. Then the binary function U(Se ,int) : B × B → B, presented by the formula (5 ), is a uninorm in B possessing an identity element e iff int(x) ∧ int (y) ∈ Ie ∪ {0B } for all x, y ∈ Ie . ⎧ Se (u, v) if (u, v) ∈ [e, 1B ]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈ [0B , e[2 , 0B ⎪ ⎪ ⎨ u∧v if (u, v) ∈ [0B , e[×[e, 1B ] ∪ [e, 1B ] × [0B , e[, U(Se ,int) (u, v) = if (u, v) ∈ [e, 1B ] × Ie , ⎪v ⎪ ⎪ ⎪ if (u, v) ∈ Ie × [e, 1B ], ⎪u ⎪ ⎩ int (u) ∧ int (v) if (u, v) ∈ [0B , e[×Ie ∪ Ie × [0B , e[∪Ie × Ie . (5)

v||e

cl(u) ∨ cl(v) cl(u) ∨ cl(v)

v

1B

v

int(u) ∧ int(v)

u

Se (u, v)

u

0B

v

int(u) ∧ int(v)

1B v

1B

cl(u) ∨ cl(v)

Te (u, v)

u

u

e

0B

v||e int(u) ∧ int(v)

e

e

1B u e

Fig. 2. Uninorm U(Te ,cl) on B

0B

e

1B u e

Fig. 3. Uninorm U(Se ,int) on B

Remark 3. The forms of the uninorms U(Te ,cl) : B × B → B and U(Se ,int) : B × B → B are illustrated in Figs. 2 and 3, respectively. If we consider in Theorem 2 the interior operator int : B → B determined as  int(x) = 0B for all x ∈ B, we obtain the undermentioned uninorm U(Se ,int) in B, which has been introduced already in [22, Theorem 1].

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Corollary 4. Let e ∈ B\{0B , 1B }. In this case, the binary function U(Se ,int) : B × B → B, presented by the formula (6), is a uninorm in B possessing an identity element e. ⎧ Se (u, v) if (u, v) ∈ [e, 1B ]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈ [0B , e[×[e, 1B ] ∪ [e, 1B ] × [0B , e[, ⎨u ∧ v  if (u, v) ∈ [e, 1B ] × Ie , U(Se ,int) (u, v) = v (6) ⎪ ⎪ × [e, 1 ], u if (u, v) ∈ I ⎪ e B ⎪ ⎩ otherwise. 0B If we consider in Theorem 2 the interior operator int : B → B determined as  int(x) = x for all x ∈ B, we obtain the undermentioned uninorm U(Se ,int) in B, which has been introduced already in [9, Theorem 3.12]. 

Corollary 5. Let e ∈ B\{0B , 1B }. The binary function U(Se ,int) : B × B → B, presented by the formula (7), is a uninorm in B possessing an identity element e iff x ∧ y ∈ Ie ∪ {0B } for all x, y ∈ Ie . ⎧ Se (u, v) if (u, v) ∈ [e, 1B ]2 , ⎪ ⎪ ⎪ ⎪ if (u, v) ∈ [0B , e[2 , ⎨ 0B  if (u, v) ∈ [e, 1B ] × Ie , U(Se ,int) (u, v) = v (7) ⎪ ⎪ × [e, 1 ], u if (u, v) ∈ I ⎪ e B ⎪ ⎩ u∧v otherwise. 

We should note that there is a little difference between the uninorm U(Se ,int) in Corollary 5 and the uninorm US in [9, Theorem 3.12] defined as dual of [9, Theorem 3.9] from the perspective of their particular structures. Although it is stated that [9, Theorem 3.12] was deduced from the dual structure of [9, Theorem 3.9], the binary function US in B, for e ∈ B\{0B , 1B }, was given together with the values of y (resp. x) for (x, y) ∈ [0B , e] × Ie (resp. Ie × [0B , e]) and x ∧ y for (x, y) ∈]e, 1B ]×Ie ∪Ie ×]e, 1B ]. It is explicit that the dual structure of [9, Theorem 3.9] takes the values of y (resp. x) for (x, y) ∈ [e, 1B ] × Ie (resp. Ie × [e, 1B ]) and x ∧ y for (x, y) ∈ [0B , e[×Ie ∪ Ie × [0B , e[. More precisely, the structure defined as dual of [9, Theorem 3.9] is identified by the formula (7). In addition, this structure yields a uninorm in B possessing the identity element e under the circumstance x ∧ y  e for all x  e and y  e. Indeed, in all results succeeding [9, Theorem 3.12] it was considered that US was given by the formula (7). Therefore, its related results are correct.

4

Conclusions

The generation of uninorms concerning algebraic forms in bounded lattices is still an influential discipline of survey. In this article, we focused on uninorms in bounded lattices relying on t-norms and t-conorms. We first introduced a production to get uninorms in a bounded lattice B possessing an identity element e ∈ B\{0B , 1B } by use of a t-norm in [0B , e]2 and a closure operator in B. Next,

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with the help of a t-conorm in [e, 1B ]2 and an interior operator in B, a method for building uninorms in B possessing an identity element e ∈ B\{0B , 1B } was discussed. Notice that we obtained the necessary and sufficient condition for such methods to yield a uninorm in B, namely, the underlying closure operators or interior ones must satisfy certain conditions. Our methods derived from closure and interior operators contribute to enriching the family structure of uninorms in bounded lattices. The perspective of novel methods can provide a basis for constructing other aggregation operators. For future work, it is meaningful to research this problem.

References 1. Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-73721-6 2. Ben´ıtez, J.M., Castro, J.L., Requena, I.: Are artificial neural networks black boxes? IEEE Trans. Neural Netw. 8, 1156–1163 (1997) 3. Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publishers, Providence (1967) 4. Bodjanova, S., Kalina, M.: Uninorms on bounded lattices – recent development. In: Kacprzyk, J., Szmidt, E., Zadro˙zny, S., Atanassov, K.T., Krawczak, M. (eds.) IWIFSGN/EUSFLAT -2017. AISC, vol. 641, pp. 224–234. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66830-7 21 5. Bodjanova, S., Kalina, M.: Uninorms on bounded lattices with given underlying operations. In: Halaˇs, R., Gagolewski, M., Mesiar, R. (eds.) AGOP 2019. AISC, vol. 981, pp. 183–194. Springer, Cham (2019). https://doi.org/10.1007/978-3-03019494-9 17 6. C ¸ aylı, G.D.: Alternative approaches for generating uninorms on bounded lattices. Inf. Sci. 488, 111–139 (2019) 7. C ¸ aylı, G.D.: New methods to construct uninorms on bounded lattices. Int. J. Approx. Reason. 115, 254–264 (2019) 8. C ¸ aylı, G.D.: Uninorms on bounded lattices with the underlying t-norms and tconorms. Fuzzy Sets Syst. 395, 107–129 (2020) 9. C ¸ aylı, G.D., Kara¸cal, F.: Construction of uninorms on bounded lattices. Kybernetika 53, 394–417 (2017) 10. C ¸ aylı, G.D., Kara¸cal, F., Mesiar, R.: On internal and locally internal uninorms on bounded lattices. Int. J. Gen. Syst. 48, 235–259 (2019) 11. Dan, Y., Hu, B.Q., Qiao, J.: New constructions of uninorms on bounded lattices. Int. J. Approx. Reason. 110, 185–209 (2019) 12. Dan, Y., Hu, B.Q.: A new structure for uninorms on bounded lattices. Fuzzy Sets Syst. 386, 77–94 (2020) 13. De Baets, B.: Idempotent uninorms. Eur. J. Oper. Res. 118, 631–642 (1999) 14. Drewniak, J., Dryga´s, P.: On a class of uninorms. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 10, 5–10 (2002) 15. Drossos, C.A., Navara, M.: Generalized t-conorms and closure operators. In: Proceedings of EUFIT ’96, Aachen, pp. 22–26 (1996) 16. Dubois, D., Prade, H.: Fundamentals of Fuzzy Sets. Kluwer Academic Publisher, Boston (2000) 17. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

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18. Everett, C.J.: Closure operators and Galois theory in lattices. Trans. Am. Math. Soc. 55, 514–525 (1944) 19. Fodor, J., Yager, R.R., Rybalov, A.: Structure of uninorms. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5, 411–427 (1997) 20. He, P., Wang, X.P.: Constructing uninorms on bounded lattices by using additive generators. Int. J. Approx. Reason. 136, 1–13 (2021) 21. Hua, X.J., Ji, W.: Uninorms on bounded lattices constructed by t-norms and tsubconorms. Fuzzy Sets Syst. 427, 109–131 (2022) 22. Kara¸cal, F., Mesiar, R.: Uninorms on bounded lattices. Fuzzy Sets Syst. 261, 33–43 (2015) 23. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000) 24. Medina, J.: Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices. Fuzzy Sets Syst. 202, 75–88 (2012) 25. Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Log. 72, 834–864 (2007) 26. Ouyang, Y., Zhang, H.P.: Constructing uninorms via closure operators on a bounded lattice. Fuzzy Sets Syst. 395, 93–106 (2020) 27. Sun, X.R., Liu, H.W.: Further characterization of uninorms on bounded lattices. Fuzzy Sets Syst. 427, 96–108 (2022) 28. Saminger, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets Syts. 157, 1403–1416 (2006) 29. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier North-Holland, New York (1983) 30. Xie, A., Li, S.: On constructing the largest and smallest uninorms on bounded lattices. Fuzzy Sets Syts. 386, 95–104 (2021) 31. Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80, 111–120 (1996) 32. Yager, R.R.: Uninorms in fuzzy systems modelling. Fuzzy Sets Syst. 122, 167–175 (2001) 33. Zhao, B., Wu, T.: Some further results about uninorms on bounded lattices. Int. J. Approx. Reason. 130, 22–49 (2021)

Aggregation Operators for Face Recognition Nebojˇsa Ralevi´c1(B) , Andrija Blesi´c1 , Vladimir Ili´c1 , ˇ c1 Marija Paunovi´c2 , and Lidija Comi´ 1

Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia {nralevic,andrija.blesic,vlada.mzsvi,comic}@uns.ac.rs 2 Faculty of Hotel Management and Tourism, University of Kragujevac, 11080 Vrnjaˇcka Banja, Serbia [email protected] Abstract. In this paper we consider the aggregation functions’ application to problems of face recognition. A wide range of aggregation function classes can play a crucial role in multi-criteria decision-making theory. Some of the most well-known aggregation functions include generalized means, integrals based on non-additive measures, etc. Their usage also saves memory resources and can lead to an increase in the percentage of correctly classified objects (in our case, we deal with object recognition). Biometry and face recognition represent important modern problems in computer science, and some solutions are widely-adopted and applied in mobile devices and identity confirmation, border control, ID and other document verification, etc. Other than classic methods for face recognition, such as PCA, linear discriminant analysis, as well as various local descriptors and other methods, methods based on sparse data representation and deep learning methods have recently been used. Each part of the face can be compared individually, and a conclusion regarding the matchoverlap of two faces should be drawn from all these comparisons. In our investigation, we analyze the quality of the decisions made in face recognition problems, with regards to the aggregation functions used, as well as the different methods used for recognizing certain parts of the face. Keywords: Aggregation function invariants · Shape recognition

1

· Face recognition · Moment

Introduction

Shape recognition is one of the most intensively studied areas of scientific research primarily in computer science. One of the reasons is the possibility of applying existing approaches and methods to problems of face recognition. Specifically, this is the case in the flow control on state roads borders [3], the recognition of traffic participants using surveillance system cameras [11], in various applications on mobile devices, etc. The choice of methods that can be used depends on time constraints, applications, hardware, memory resources, and other factors. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 119–127, 2023. https://doi.org/10.1007/978-3-031-39774-5_15

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Currently, modern approaches, primarily based on artificial intelligence (machine learning, deep learning, etc.), as well as classical methods such as principal component analysis [18], discriminant analysis [2], descriptors [1,6,14,17] are constantly being improved. In this paper, we address a new approach to face recognition tasks by applying the aggregation function on suitably selected shape-based descriptors of each of the isolated face parts, previously binarized, as described below. We chose the eyebrows, eyes, left and right cheeks, noses and mouths. Of all the shapebased descriptors we considered, anisotropy [14] and disconnectedness [17] (based on image moments) were shown to be very promising in face recognition tasks applied to AT &T image databases. The final answer about the degree of face overlap is given on the basis of weighted aggregation operators such as the generalized arithmetic mean and the degree of the root, as well as the Choquet integral and its generalization (with different functions used in its definition). It should be emphasized that in our experiments we did not look for the best possible result among all available in the literature so far, but rather to present how a new approach using shape descriptors can address tasks of this type. This paper is organized in four Sections. Section 2 gives a short description of two shape-based descriptors, as well as an overview of some aggregation operators (such as the Choquet integral). Section 3 illustrates the results of the classification of facial images, and Sect. 4 summarizes our conclusions.

2

Preliminaries

We give here a short description of two shape-based descriptors: anisotropy [14] and disconnectedness [17], which, together with the first and second Hu moment invariants [16], were used during testing. Examples of distances used in classification are also presented. In the second part, we give an overview of some aggregation operators used for decision making in experiments, such as the Choquet integral based on Sugeno’s λ-fuzzy measures. 2.1

Classification

As already stated, we will give here a brief reminder of two very useful moment invariants applied to multi-component shapes, that proved to be very useful in the tasks of classification presented in this paper. We list their definitions without going into greater detail. For more details on the motivation and methodology of their introduction, we refer the reader to [14,17]. Definition 1 [14]. Let S be a multi-component shape with components S1 , S2 , . . . , Sn . The anisotropy measure A(S) = A(S1 ∪ S2 ∪ . . . ∪ Sm ) is defined as: √ C + A2 + B 2 √ A(S) = A(S1 ∪ S2 ∪ . . . ∪ Sn ) = , C − A2 + B 2

Aggregation Operators for Face Recognition

where we have C = B=

n  i=1

n  i=1

[μ2,0 (Si ) + μ0,2 (Si )]μ0,0 (Si ), A = 2

n  i=1

121

μ1,1 (Si )μ0,0 (Si ),

[μ2,0 (Si ) − μ0,2 (Si )]μ0,0 (Si ), and μp,q (S) and μp,q (S) represent the ordi-

nary and corresponding centralized moments of the considered shape S. Definition 2 [17]. Given a multi-component shape S = S1 ∪ S2 ∪ · · · ∪ Sn the disconnectedness measure D(S) is defined as follows D(S) = H1 (S1 ∪ S2 ∪ · · · ∪ Sn ) −

1 μ30,0 (S1 ∪ S2 ∪ · · · ∪ Sn )

·

n 

μ30,0 (Si ) · H1 (Si ),

i=1

where H1 (S) = (μ0,2 (S) + μ2,0 (S))/μ20,0 (S) represents the first Hu moment invariant. In working with the data, the following distance functions between points x = (x1 , x2 , ..., xn ) and y = (y1 , y2 , ..., yn ) in Rn are commonly used for classification:  n  – Euclidean metric: dE (x, y) := (xi − yi )2 , i=1  – Mahalanobis distance: dM (x, y) := (x − y)A−1 (x − y)T , where A is positive-definite (covariance) matrix, t , – Fuzzy T -metric (with respect to multiplication): t(x, y, t) := t+d(x,y) and its dual (with respect to the standard fuzzy complement), the fuzzy Sd(x,y) metric (with respect to the algebraic sum) [13]: s(x, y, t) := t+d(x,y) , n where d is a metric on R . n  |xi − yi | – Canberra distance: dC (x, y) : := . |xi | + |yi | i=1 2.2

Aggregation Functions

Definition 3. An n-ary aggregation function is a monotonically nondecreasing function ℵ[n] : [0, 1]n → [0, 1], n ∈ N \ {1} in all its arguments, such that the following boundary conditions are satisfied: 1) xi ≤ yi , i ∈ {1, . . . , n} ⇒ ℵ[n] (x1 , . . . , xn ) ≤ ℵ[n] (y1 , . . . , yn ), 2) ℵ[n] (0, . . . , 0) = 0 and ℵ[n] (1, . . . , 1) = 1. If n = 1, ℵ[1] is the identity function, i.e., ℵ[1] (x) = i|[0,1] (x) = x, for all  x ∈ [0, 1]. An extended aggregation function is a function ℵ : [0, 1]n → [0, 1], n∈N

whose restrictions to [0, 1]n for each n ∈ N are n-ary aggregation functions ℵ[n] . Some examples of aggregation functions include the (weighted) arithmetic mean, ordered averaging function (OWA), quasi-arithmetics mean, minimum and maximum functions and the median (see, for example, [8]).

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In applications, some of the most well-known aggregation functions are fuzzy integrals, i.e., integrals based on fuzzy measures, among which the most frequently used is the Choquet integral. Definition 4 ([8]). Let X be a non empty set and Σ be a non empty class of subsets of X, such that ∅ ∈ Σ. The map m : Σ → [0, ∞] is called a fuzzy measure on Σ if it holds that m(∅) = 0, and that, for every A, B ∈ Σ, if A ⊂ B then m(A) ≤ m(B). If additionally X ∈ Σ and m(X) = 1, then m is called a regular fuzzy measure. Example 1 [15]. The set function g : 2X →: [0, 1] is called λ-fuzzy measure, λ > −1 if g(∅) = 0, g(X) = 1, and the following is valid: (∀A, B ⊂ X) g(A ∪ B) = g(A) + g(B) + λg(A)g(B). If X = {x1 , ..., xn } and λ = 0, then λ can be determined from the following [15]: 1+λ=

n 

(1 + λgk ),

(1)

k=1

where gk = g({xk }) are fixed. By denoting Ak = {x1 , ..., xk }, the above implies: g(Ak+1 ) = g(Ak ) + gk+1 + λg(Ak )gk+1 ,

(2)

where g(A1 ) = g1 . Definition 5. The Choquet integral of f : X → [0, 1], based on a given λ-fuzzy measure g, is defined as:  CH

f ◦g =

n 

[f (xk ) − f (xk+1 )] g(Ak ),

k=1

where X = {x1 , . . . , xn } and f (xn+1 ) = 0. The values f (xk ) are ordered in a nonincreasing order (i.e. f (xk ) ≥ f (xk+1 ), k = 1, . . . , n). In [7], the authors used the generalized Choquet integral defined in [10]:  GCH

f ◦ g(M ) =

n 



M f (xk ) − f (xk+1 ), g(Ak ) ,

k=1

where, as before, X = {x1 , ..., xn } and f (xn+1 ) = 0, and M is a (1, 0)-increasing function M : [0, 1]2 → [0, 1] which satisfies M (x, y) ≤ x, M (x, 1) = 1, M (0, y) = 0. For more details on aggregation functions see [4] or [5].

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3

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Experimental Validation

This section illustrates the results of the classification of facial images of 15 persons. Each class contains 11 facial images of a person whose images differ by different lighting conditions, expressed feelings etc. The seven classifiers we considered are: the global image, eyes, eyebrows, mouth, nose, left and right cheek. For each of the seven classifiers, a database of 165 images was used: 150 images for the training set and 15 for the testing set. For the global image classifier we used the original AT &T image database (converted to PNG format), and for the other classifiers we used cropped regions of each of the 150 original training images in order to isolate meaningful regions that are natural and easy to recognize using usual human perception (all 165 images used for a given classifier, both for the training and the testing images, are of the same dimensions). Since our input images are grayscale, we first applied the Otsu binarization multithresholding method [12], and divided the input image into several subparts according to the pixel intensity of the input image. From such obtained sub-parts, we looked at those that gave us a representation that corresponds to our natural perception, used mophological operations (opening and closing) as required, bearing in mind that preprocessing does not need to give us the best possible representation of the observed regions. This stems from the fact that this approach would inevitably lead us to the effect of overfitting, which is certainly not our intention, as we aim to present an approach that would be generic and applicable to a wide class of face images and not only the class that we observe in this paper. In this sense, we will emphasize once again that our goal was not to achieve the best results in face recognition tasks, but to give a novel approach to solving the challenges of face recognition using shape analysis (Fig. 1).

Fig. 1. Facial regions extracted from the AT &T database, and further preprocessed.

After preprocessing, we used the Leave-One-Out validation technique in order to establish the prior classification accuracy using only one classifier. Table 1 shows the accuracy value, together with the values (moments) that were calculated for each image and used to form a feature vector.

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Table 1. The results of applying classification using each individual classifier, along with the invariants used to perform the classification. D, A, Hu1, Hu2, Hu3, and Hu4 denote disconnectedness, anisotropy and the first, second, third and fourth Hu moment invariant, respectively. Index Classifiers

Invariants

Accuracy

1

global image Hu1, Hu3, Hu4 58%

2

eyes

D, A

33.33%

3

eyebrows

D, A

32.67%

4

mouth

Hu2, A

35.33%

5

nose

Hu3, Hu4

34.67%

6

left cheek

Hu2

38.67%

7

right cheek

Hu2

34.67%

The accuracies listed in Table 1 are used to calculate the values g({Ak }), see Eq. (2). Namely, if pi is the accuracy value for classifier i, then: g1 = g({x1 }) = βp1 , gi = g({xi }) = (1 − β)pi ,

for i = 2, 3, . . . , 7.

We use β = 0.6, following the procedure in [9], chosen specifically to form a balance between the influence of the global image classifier and all other classifiers. For our concrete accuracy values, we obtain: (g1 , g2 , g3 , g4 , g5 , g6 , g7 ) = (0.348, 0.133, 0.131, 0.141, 0.139, 0.155, 0.139), and, using Eq. (1), we determine that λ = −0.348. Note that in order to calculate the values g({A1 }), g({A2 }), . . . , g({A7 }), in accordance with the formula from Example 1, relabelling may be required, depending on the values of function f in Definition 5, as described below. Following the formula from [9], we calculate the values fik : fik =

1  μij , Nk

(3)

μij ∈Ck

where: – – – –

i = 1, 2, . . . , 7 is the ordinal number of the classifier, j = 1, 2, . . . , 15 is the index of the training image, Nk denotes the number of images in k-th class Ck , μij = 1+d1ij /di denotes the degree of belonging to class Ck , calculated using the average distance di within the i-th classifier, and the distance dij between the test image and the j-th face image within the i-th classifier.

For obtaining distances dij we used both Euclidean distances and Canberra distances between feature vectors (values of invariants, as per Table 1).

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In formula (3) above, k = 1, 2, . . . , 15 refers to the class number. In our case, we have 15 classes, each representing a person whose 11 images (10 training and one testing images) were used in the AT &T database. For each k, we order and possibly relabel x1 , x2 , . . . , x7 according to the values f1k , f2k , . . . , f7k , in order to fulfill the monotonicity condition from Definition 5. For example, if f2k > f1k > f3k > f4k > f6k > f5k > f7k , the feature x2 would be relabeled as x1 , x1 as x2 , x6 as x5 , and feature x5 relabeled as x6 : (x1 , x2 , x3 , x4 , x5 , x6 , x7 ) → (x2 , x1 , x3 , x4 , x6 , x5 , x7 ). For each test image, and each class number k = 1, 2, . . . , 15, a different Choquet integral is obtained using values g({A1 }), g({A2 }), . . . , g({A7 }), and (reordered) values f1k , f2k , . . . , f7k . Thus, for each test image, 15 different Choquet integrals are calculated, and the test image is then taken to belong to the class Ck for which to the largest value of the Choquet integral is obtained. We also calculated the values of generalized Choquet integrals for every test image, using the following t-norms: MH (a, b)

=

ab α+(1+α)(a+b−ab) , α

, MSS2 (a, b) = 1 − [(1 − a) + (1 − b)α − (1 − a)α (1 − b)α ] ab ab MSS4 (a, b) = [aα +bα −a , M (a, b) = , K α bα ]1/α [1+(1−aα )(1−bα )]1/α 1/α

where α > 0. We tested with values α ∈ (0, 1] which were the most promising. While evaluating the Choquet integrals using our particular setup and the above t-norms, we have seen that the final accuracy percentages consistently falls below 13.33%, regardless of whether the Canberra or Euclidean distance functions were used. On the other hand, using the Choquet integral rather than its generalizations gave a classification accuracy of 33.33%. This implies that an approach like ours would not benefit from using the t-norms above. We thus varied the values of β, calculated the classical Choquet integral and performed the classification as before. The results are listed in Table 2, and show that the best accuracy is obtained for β = 0.4 or β = 0.5, using Euclidean distances. Table 2. The accuracy percentages of classification using the classical Choquet integral, using both Euclidean and Canberra distances. Maximal values are shown in bold. β

0

λ

−0.894 −0.854 −0.802 −0.733 −0.642 −0.52 −0.348 −0.0825 0.412

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1.8155

accuracy [Eucl.] 26.67

33.33

40

40

53.33

53.33 33.33

46.67

40

accuracy [Canb.] 20

26.66

33.33

33.33

40

40

46.67

46.67 46.67

4

33.33

40

Conclusions and Remarks

In this paper, we approached face recognition by working with classifiers representing various parts of the face (eyebrows, eyes, nose, mouth, left and right

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cheek) and their descriptors based on image moment invariants. We showed that, for our setup, the classical Choquet integral, which serves as an aggregation operator for classification in face recognition, shows most promise, regardless of the two distances used. Further research could be oriented towards varying the t-norms used for generalized Choquet integrals, or even using other fuzzy integrals, such as the Sugeno, Pap, Shilkret integrals and other. Also, alternative distance functions could be used (such as the Mahalanobis distance or others usually used for correlated data), or alternative fuzzy metrics. Another area of reseach could focus on examining other aggregation function classes, which may help clarify the problem of identifying the parts of the face or face shape, etc. The inclusion of more classifiers (such as the forehead or chin) would likely lead to an increase in accuracy results. Acknowledgements. The authors has been supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project no. 451-03-47/2023-01/200156: “Innovative scientific and artistic research from the FTS (activity) domain”.

References 1. Ahonen, T., Hadid, A., Pietik¨ ainen, M.: Face recognition with local binary patterns. In: Pajdla, T., Matas, J. (eds.) ECCV 2004. LNCS, vol. 3021, pp. 469–481. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24670-1 36 2. Belhumeur, P.N., Hespanha, J.P., Kriegman, D.J.: Eigenfaces vs. fisherfaces: recognition using class specific linear projection. IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997) 3. del Rio, J.S., Moctezuma, D., Conde, C., de Diego, I.M., Cabello, E.: Automated border control e-gates and facial recognition systems. Comput. Secur. 62, 49–72 (2016) 4. Dubois, D., Prade, H.: On the use of aggregation operations in information fusion processes. Fuzzy Sets Syst. 142, 143–161 (2004) 5. Grabish, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009) 6. Ili´c, V., Ralevi´c, N.M.: Fuzzy squareness: a new approach for measuring a shape. Inf. Sci. 545, 537–554 (2021) 7. Karczmarek, P., Kiersztyn, A., Pedrycz, W.: Generalized choquet integral for face recognition. Int. J. Fuzzy Syst. 20(3), 1047–1055 (2018) 8. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, New Jersey (1995) 9. Kwak, K.C., Pedrycz, W.: Face recognition: a study in information fusion using fuzzy integral. Pattern Recognit. Lett. 26, 719–733 (2005) 10. Lucca, G., et al.: Preaggregation functions: construction and an application. IEEE Trans. Fuzzy Syst. 24, 260–272 (2016) 11. Miklasz, M., Olszewski, P., Nowosielski, A., Kawka, G.: Pedestrian traffic distribution analysis using face recognition technology. In: Mikulski, J. (ed.) TST 2013. CCIS, vol. 395, pp. 303–312. Springer, Heidelberg (2013). https://doi.org/10.1007/ 978-3-642-41647-7 37

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12. Otsu, N.: A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979) 13. Ralevi´c, N.M., Karakli´c, D., Piˇstinjat, N.: Fuzzy metric and its applications in removing the image noise. Soft. Comput. 23(22), 12049–12061 (2019) ˇ 14. Rhouma, M.B.H., Zunic, J., Younis, M.C.: Moment invariants for multi-component shapes with applications to leaf classification. Comput. Electron. Agric. 142, 326– 337 (2017) 15. Sugeno, M.: Theory of fuzzy integral and its applications. Dissertation, Tokyo Institute of Technology, Tokyo (1974) 16. Hu, M.K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8(2), 179–187 (1962) ˇ 17. Zunic, J., Rosin, P.L., Ili´c, V.: Disconnectedness: a new moment invariant for multicomponent shapes. Pattern Recogn. 78, 91–102 (2018) 18. Turk, M., Pentland, A.: Eigenfaces for recognition. J. Cogn. Neurosci. 3, 71–86 (1991)

First-Order Representations and Calculi of Categorical Propositions Yinsheng Zhang(B) Institute of Scientific and Technical Information of China, Beijing 100038, China [email protected]

Abstract. Against the three kinds of defaults of categorical propositions in traditional logic: (1) inconsistent interpretations by Euler circles, (2) monadic quantification, and (3) localization of representation and inference, the author advocated “fully expanded categorical propositions (FECPs)” with consistent, dyadic fuzzy quantifiers, and ubiquitous knowledge representations. Here, FECPs and more generally, multi-adic FECPs (FEMCPs) are transformed to be first-order logic (FOL) expressions. Further, the author has proved that FEMCPs are adequate in FOL calculi. The ramification of FEMCPs predicts that this kind of reformed categorical propositions can be written as languages of common mathematical logic and as formal prototype of logic programming languages, making the reformed categorical propositions well integrated in wider knowledge contexts and applications for FOL expressions’ intrinsic features. Keywords: fuzzy quantifier · generalized quantifier · partial quantifier · particular quantifier · first-order calculus · logic programming language · automated reasoning · categorical proposition

1 Introduction A categorical proposition, traditionally, refers to such a proposition of relation between two categories---the notation “category” here used is primarily meaningful, not necessarily mathematical---in this primary sense, categories usually degenerate into sets, and mappings (arrows) between the sets are allowed not to be strictly defined, or not as conventionally defined as the current sub-discipline nominated by the term of “category theory” in the mathematics. So, any two given nonempty sets X and Y composed to be a categorical proposition in terms of the quantifier-monadic structure Qx copula y, where Q denotes a universal or a particular quantifier, are called “Aristotelian categorical propositions” (“ACPs”). It is well known that ACPs are relegated to the four forms, they are the universally and particularly positive or negative propositions, which are respectively named as A, E, I, and O [1], like “all x is y”, “all x is not y” [2], etc. [3] has summarized three limitations, A1 to A3, of the forms of ACPs as follows:. A1: Inconsistent interpretations. For the universal quantifier ∀ in ACPs, some interpret it as of all the elements of a subset, but some as of a proper subset, to the counterpart set. That is, the former regards ∀x as of X in the relation X ⊆ Y , the latter © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 128–135, 2023. https://doi.org/10.1007/978-3-031-39774-5_16

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as X ⊂ Y . Moreover, for the particular quantifier, some interpret it as of the existential elements, viz., a subset, but some as of a proper subset, of the denoted set [4]. That is, as ∃X where reads “the particular”, the latter regards as the former regards where reads “the partial” or “part of”. A2: Monadic quantification. ACPs take the form Qx copula y, a formatted expression of a natural language, where there is only one quantifier binding the term x [4]. Although the modern reformers have been transforming it into quantifier-dyadic forms like Q1 x copula Q2 y, which have defaults still, for example, Q ∈ {Q1 , Q2 } has not been designated to be the existential, rather than the partial quantifier by early quantifierdyadic advocates, in contrast, not to be the partial quantifier rather than the existential quantifier by the later advocates. A3: Localization of representation and inference. Essentially, the forms and inner logic of ACPs, even of the latter transformed forms, have some limitations which cause localization of representation and inference of ACPs, limiting them to a narrow or sealing knowledge contexts. For satisfying first-order forms, an ACP is commonly represented as a function and a variable in an annexation, that is, to symbolize a functional predicate (a verbal like “is”) and a term (variable like “x”) in one notation like “X”, (meaning “is x”), as all x are y is transformed as ∀(a)X (a) → Y (a) in Frege’s expressions [5], which isolates its style from a global mathematical and logical convention----this convention respectively and independently symbolizes a function and a variable, like Q1 x copula Q2 y. Moreover, ideal forms of categorical propositions, which should be wider to be linked to ubiquitous contexts of knowledge, including mathematical and logical explanations infused in various knowledge contexts, seem to be short [6]. To overcome the three problems A1 to A3, [3] sets forth a categorical proposition system of fully expanded categorical propositions (FECPs), which features the form Q1 x copula Q2 y as instantiated in (1) and (2). (1) (2) Where, B ∈ {+, ¬} is a positive (commonly ignored) or negative symbol, and the quantifiers and ∃ with its negative forms are defined as follows. Definition 1. (“Fully expanded quantifier”) QFe . ∀x := UN(X ) be the universal set of xi , i = 1, 2, . . . , n ≥ 2 . i.e., UN(X) = {x i }; a proper subset of UN(X), viz., PT(X ) = {xi |i = 1, 2, . . . , m, m < n}; ∃x := ID(X ), a subset of UN(X), viz., ID(X ) = {xi |i = 1, 2, . . . , m, m ≤ n}. , are logically defined as shown And the negative quantifiers of in Table 1 [3], where ≡ denotes logic equivalence. is named as “fully expanded quantifier” [3]. Essentially, QFe is a fuzzy variable, hereafter simply as Q. Let the membership grade μ∀, μ¬∀, μ〒 , μ¬〒 , μ∃, and μ¬∃ respectively stand for credibility of xi ∈ UN(X ), xi ∈ ¬UN(X ), xi ∈ PT(X ), xi ∈ ¬PT(X ) and xi ∈ ID(X ), and xi ∈ ¬ID(X ), i =

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Y. Zhang Table 1. Negation of the quantifier ≡

¬

〒

∀|¬∃

∀

〒¬∃

∃

¬〒∧¬∀

1, . . . , n, thus, the fuzzy expressions of {Q} for x are given by Lemma 1 [3], which are cited as (3) to (8), where “|” denotes excusive Or, viz., Xor. ∀x = {1/xi }, if μ∀ (xi ) = 1

(3)

¬∀x = {(p|0)/xi }, 0 < p < 1, if μ¬∀ (xi ) = p|0

(4) (5) (6)

∃x = {q/xi }, 0 < q ≤ 1, if μ∃ (xi ) = q

(7)

¬∃x = {0/xi }, if μ¬∃ (xi ) = 0

(8)

The reform of FECPs generates a quest if FECPs can be rewritten in first-order language. As is well known that first-order logic (FOL) in a special language is a formal language serving as a prototype mathematics-integrated and of logic programming, so, the transform of FECPs into FOL would globalize FECPs in wider knowledge contexts. The solution of transforming FECPs into FOL expressions is strived to be made bellow. Moreover, relations with multiple categories bound by {Q}, named as FEMCPs, are considered to be similarly operated for generalizing the categorical proposition in integration with FOL expressions and calculi.

2 Calculi of FEMCPS Within FOL 2.1 Formal Representations of FEMCPS A FECP is formulized as Qx be Qy, also rewritten as (9). QxQy R(x, y)

(9)

Where, a negative relation will be defined indicates “Exin Subsect. 2.2. Take an example, istential x is not partial y”. Generally, the variables of (9) are set as vectors as in (10). Qx R(x), or Qx R

(10)

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Where, Qx is inner production of vector Q and vector x. When the terms are beyond two, (10) is same as (11) and (12), where n is a possitive integer. Q1 x1 Q2 x2 . . . Qn xn R(x1 , x2 . . . xn )

(11)

R(Q1 x1 , Q2 x2 . . . Qn xn

(12)

(9) to (12) are all FOL expressions of FEMCPs. A basic calculi of FOL formulae are considered as calculi ¬, →, ∨, and ∧, which are to be applied on (11) and (12) given in Subsect. 2.2 to 2.5. 2.2 Negation Definition 2. ¬R(x) : = RC (x). Here, the power C denotes complement. When R(x) and ¬R(x) do not require their interpretations on elements Cardinal production. Then, the two all become predicates of terms x with opposite truths, mostly expresssed, for example, like P(x) and¬P(x). Let < Q > be a series of qantifiers {Q} in an order. Define the distribution of negation symbol “¬” for an ordered serial of quantifiers in the power set ℘ < Q >, neglecting ℘(2n ) < Q >= ∅, as stated in (13). 2 −1 2 −1 → {¬℘i < Q >i=1 ¬℘ : {℘i < Q >}i=1 n

n

(13)

Next, we consider the semantics of negation of Q, viz., the semantics of [¬]Q, which refers to denying every element of power set (without ∅) of , viz., {¬℘i < Q >} fori = 1, 2, . . . , 2n − 1, to be in general Xor relation with the complemental set C of the denied elements, as formalized in Definition 3. 2 −1 ({¬℘ < Q >} ∪ {℘iC < Q >}) Definition 3. [¬]Q := |i=1 Negation of an elements-quantified relation refers to the complexes of [¬]Q and term vector x by dot production in general Xor, as formulally stated in Definition 4. n

  Definition 4. ¬(QxR(x)) = ¬(Qx)R(x) = ¬ Q xR(x) = R(([¬]Q) • x) = R([¬]Qx) Example 1. Negation of a FEMCP:

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Example 2. ¬∀ :

2.3 Implication, Exclusive Disjunction, and Conjunction Definition 5. Calculi of implication(→), exclusively-disjunction (|, viz.Xor) or disjunction(Or, ∨), and conjunction (∧) are called “logically-basic binary calculi (LBBC)”. A LBBC is symbolized by “:”. Namely, : ∈ {→, |, ∧}, or, :∈ {→, ∨, ∧}. Definition 6. Given two vectors of FECP quantifiers Qα and Qβ with same cadinality,   viz., #(Qα ) = # Qβ = n is a positive integer more than 2, a n-element ordered . operators of LBBC < .. > of Qα and Qβ are defined as the ordered serial (vector) of LBBC for every pair of quantifiers-elements from Qα and Qβ with a same adic place, as following. Qα : Qβ := (Qα1 : Qβ1 )(Qα2 : Qβ2 ) . . . (Qαn : Qβn ) =< Qαi : Qβi >ni=1 Theorem. LBBC of two FEMCPs with a same relation symbol and terms, remain the truths same with LBBC of the two vectors of the quantifiers for the relation, as stated in (14). B(Qα xR(x) : Qβ xR(x)) = B((Qα : Qβ )R(x))

(14)

Where B : is a truth function. For every pair of operands Qαi : Qβi , there exists the logic as shown in Tables 2, 3 and 4. . Proof. When the operator..takes its instances, (14) is instantiated as (15) to (17). B(Qα xR(x) → Qβ xR(x)) = B(Qα → Qβ )R(x)

(15)

B(Qα xR(x)|Qβ xR(x)) = B(Qα |Qβ )R(x)

(16)

B(Qα xR(x) ∧ Qβ xR(x)) = B(Qα ∧ Qβ )R(x)

(17)

(15) to (17) state LBBC on Q for R. As R is of FOL expression whose semantics is independent to the variables (terms, elements), namely, “R”, serving as the functor, does not contain any variables that R functions on, so the functor “R” is not semantically influenced by the variables and remaining the same meanings in various contexts of

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variables, irrelevant to quantifiers distribution from the right to the left of an equation among (15) to (17), and vice versa. Thus whether the left equals the right of equations depends on whether the calculi {→, |, ∧} on the left equal the results on the right of the equation. Note that Tables 3 to 4 can be derived from (3) to (8) in terms of fuzzy logic operations of the operands {Q}, for examples as taken below. QED.

Table 2. Implication “ →” of the fully expanded quantifiers. →

〒

∀

∃

¬〒

¬∀

¬∃

〒

T

F

T

F

T

F

∀

F

T

T

T

F

F

∃

F

F

T

T

T

F

¬〒

F

F

F

T

F

F

¬∀

F

F

F

F

T

F

¬∃

F

F

F

T

T

T

Table 3. Exclusive-disjunction “|” of the fully expanded quantifiers. “U” denotes the space of “|” ranges over [0,1] |

〒

∀

∃

¬〒

¬∀

¬∃

〒

〒

∃

∃

U

¬∀

¬∀

∀

∃

∀

∃

¬〒

U

¬〒

∃

∃

∃

∃

∃

∃

U

¬〒

U

¬〒

U

¬〒

U

¬〒

¬∀

¬∀

U

U

U

¬∀

¬∀

¬∃

¬∀

¬〒

U

¬〒

¬∀

¬∃

Example 3. Implication of exclusive-disjunction of FEMCPs: An example of An example of An example of

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Y. Zhang Table 4. Conjunction “∧” of the fully expanded quantifiers.

∧

〒

∀

∃

¬〒

¬∀

¬∃

〒

〒

¬∃

〒

¬∃

〒

¬∃

∀

¬∃

∀

∀

∀

¬∃

¬∃

∃

〒

∀

∃

∀

〒

¬∃

¬〒

¬∃

∀

∀

¬〒

¬∃

¬∃

¬∀

〒

¬∃

〒

¬∃

¬∀

¬∃

¬∃

¬∃

¬∃

¬∃

¬∃

¬∃

¬∃

Example 4. Exclusive-disjunction Xor of FEMCPs:

Example 5. ∧ calculi, with the formula < ¬∃ > ≡ < ∀ > ¬R :

3 Conclusion For a long term, there have been views, representatively [8–14], deeming integration or consistency between Aristotelian categorical propositions (ACPs) with its quantifier theories and classical (FOL-based) logic. In contrast, the present author has noted the conflicts between ACPs theories and FOL for the defaults of ACPs especially on inconsistent interpretations (A1) and monadic quantification (A2). [6,15,14] contemplate to fix the problems by setting and inserting the fully expanded quantifiers to bind full terms including the sencond term y in categorical propositions, and by giving various mathematical expressions with the reformed categorical propositions like FECPs. Consequently, it follows a question if the efforts are well compatible with modern logic and mathematics. The current study shows that FEMCPs are FOL-representative and deductible, hence should be wholly used in modern FOL-based logic and mathematics for meeting the FOL creterion of bearing basic logic calculi {¬, ∨,∧, →}, although some variations happen: negation “¬”complicates, Or “∨” is narrowed as Xor “|”, and the truth in traditional FOL are expanded and converted into multiple value as Tables 3 and 4 show that the quantifier logic in FEMCPs is transformed from traditional {0,1} → {0,1} to the mapping [0,1] → [0,1] in terms of the fuzzy models given by (3) to (8). As languages complying with FOL are mathematics-compitable and a prototype of logic programming languages like Prolog, Lisp etc., it should be believable that FEMCPs can be accepted by classical logic with its programming languages.

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References 1. Preece, W.E., (general ed.), Goetz, P.W., (exe.ed.): “Syllogistic”, The new Encyclopaedia Britannica (in 32 volumes, 15th Edition), London, Encyclopaedia Britannica Inc., 455,129 (2009) 2. Aristotle: “Prior Analytics”, Categories, On Interpretation, Prior Analysis, Translated by H. P. Cooke, Hugh Tredennick, Loeb Classical Library, Later printing Edition. Harvard University Press, Cambridge, London, pp. 198–199 (Book I/I 24a,17–27) (1938) 3. Zhang, Y.: Harmonization, dualization and globalization of categorical propositions. J. Multiple-Valued Logic Soft Comput. 39, 505–538 (2022) 4. Kneale, W., Kneale, M.: The Development of Logic, pp. 348–354. Oxford At the Clarendon Press, Oxford (1962) 5. Frege, G.: Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought/ Jean van Heijenoort ed. From Frege to Gödel, A Source Book in Mathematical Logic, pp. 1879–1931 (1977) 6. Hodges, W.: Traditional logic, modern logic and natural language. J. Philosophic Logic 38, 589–606 (2009) 7. Kant, I.: Preface to the second edition. In: Critique of Pure Reason. 2nd ed., Translator and Editor Marcus Weigelt. Penguin Group, New York, USA, pp. Bvii,viii–Bix,x (2007) 8. Gardner, F.: Logic Machines, Diagrams and Boolean Algebra, pp. 38–39. Dover, New York (1968) 9. Russinoff, S.: The syllogism’s final solution. Bull. Symb. Log. 5(4), 451–469 (1999) 10. Ross, W.D.: The discovery of the syllogism. Philos. Rev. 48(3), 251–272 (1939) 11. Shelton, H.S.: The syllogism and other logic forms. Mind, New Series 28(110), 180–202 (1919) 12. Westerstah, D.: Aristotelian syllogisms and generalized quantifiers, Studia Logica XLVIII(4), 577–585 (1989) 13. Plato, J.V.: Generality and existence: quantificational logic in historical perspective. Bull. Symb. Log. 20(4), 417–448 (2014) 14. Smiley, T.: Syllogism and quantification. J. Symbolic Logic 27(1), 58–72 (1962) 15. Zhang, Y.: Improvements of categorical propositions on consistency and computability. J. Multiple-Valued Logic Soft Comput. 33, 397–413 (2019) 16. Zhang, Y.: Harmonization and systematization of categorical propositions by fuzzy quantification, INFUS 2021. LNNS 307, 245–253 (2022)

Properties of General Extensional Fuzzy Cuts Jiˇr´ı Moˇckoˇr(B) Institute for Research and Applications of Fuzzy Modeling University of Ostrava, 30. dubna 22, 701 03 Ostrava 1 Ostrava, Czech Republic [email protected] http://irafm.osu.cz/ Abstract. Using the theory of dual pairs of semirings as value sets of fuzzy set structures, the concepts of relational and extensional fuzzy cuts are defined. The possibility of extending any cut system to the extensional fuzzy cut and the reflective nature of the category of these structures are investigated. Using dual pairs of semirings, these results can be immediately applied to new types of fuzzy structure, such as intuitionistic, neutrosophic, or fuzzy soft sets. Keywords: Dual pair of semirings cuts · Reflective category of cuts

1

· Relational cuts · Extensional

Introduction

It is well known that any L-fuzzy set s : X → L can be be expressed as a system of subsets (Cα )α∈L in a set X, satisfying some properties ([3,5,9]). These systems are called cut systems in X and play a significant role in fuzzy analysis, fuzzy topology, fuzzy algebra, and many other fuzzy systems. Between the cut systems and L-fuzzy sets there are many important relationships and the theory of Lfuzzy sets can be fully substituted by the theory of cut systems ([6,7]). The explicit advantage of using cut systems instead of L-fuzzy sets lies in the fact that they create a natural possibility to approximate with arbitrary accuracy fuzzy sets using classical sets, which increases the interpretation possibilities of this theory. Recently, completely new fuzzy structures have been introduced, generalizing classical L-fuzzy sets, such as intuitionistic L-fuzzy sets [1], L-fuzzy soft sets [11] or neutrosophic L-fuzzy sets [2] and their mutual combinations such as intuitionistic L-fuzzy soft sets [10] etc. It is therefore natural to deal with the question of whether a theory analogous to cut systems in classic L-fuzzy sets can be also defined for these new fuzzy structures. One of the obstacles that prevents these new fuzzy structures from using mechanical imitation of the construction of classic cut systems for L fuzzy sets is that, unlike the classical L-fuzzy sets, new L-fuzzy structures generally are not defined as mappings X → L. To solve this problem, in our previous paper [4] we defined a new algebraic structure for fuzzy set values, called the dual pair of semirings, based on a c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 136–143, 2023. https://doi.org/10.1007/978-3-031-39774-5_17

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pair (S, S ∗ ) of complete commutative idempotent semirings with the common underlying set S and with the involutive semiring isomorphism ¬ : S → S ∗ . Mappings X → S are then called (S, S ∗ )-fuzzy sets in X and operations with these (S, S ∗ )-fuzzy sets are defined by operations of S and S ∗ . For illustration, this algebraic structure (S, S ∗ ) is more special than complete residuated lattices, but more general than complete M V -algebras. Introducing the notion of (S, S ∗ )fuzzy sets has the following advantages: 1. It will allow transforming a whole range of new L-fuzzy sets, including neutrosophic, intuitionistic or L-fuzzy soft sets into the (S, S ∗ )-fuzzy sets, that is, into mappings X → S, analogous to the classic L-fuzzy sets, 2. It will makes it possible to define the general theory of (S, S ∗ )-fuzzy cut systems for (S, S ∗ )-fuzzy sets as an analogy of classic cuts for L-fuzzy sets, 3. It will makes it possible to apply the theory of these general (S, S ∗ )-fuzzy cut systems to new L-fuzzy structures. Using a dual pair of semirings, we can define a pair of monads (TS , TS ∗ ) in the category of sets with the set T (X) = S X representing a power set structure of all (S, S ∗ )-fuzzy sets. These monads allowed us to define in [8] a general theory of relational (S, S ∗ )-fuzzy cuts (Wr )r∈S , where Wr : X ⇒ X are special relations defined by monads and which are another representation of pairs (X, δ), where δ is a (S, S ∗ )-fuzzy equivalence relation. In this short paper, we continue to build another part of the theory of general cut systems, namely, we will focus on the issue of special (S, S ∗ )-fuzzy cuts, which are extensional with respect to the given relational (S, S ∗ )-fuzzy cut (Wr )r∈S . We prove that the arbitrary system of subsets (Cr )r∈S of X can be extended into an extensional (S, S ∗ )-fuzzy cut with respect to the given relational (S, S ∗ )fuzzy cut, and we investigate continuous properties of morphisms defined in the category of (S, S ∗ )-fuzzy cuts. Using these results, we investigate a reflective subcategory of these structures. These results can be immediately used to extend cut theory in new L-fuzzy systems where L is a complete M V -algebra, such as intuitionistic, neutrosophic or L-fuzzy soft sets.

2

Extensional (S, S ∗ ) -Fuzzy Cuts

Due to the limited scope of the pages of the paper, we do not repeat the basic definitions of dual pairs of semirings (S, S ∗ ) and monads TS = (T, η, ♦). All these notions are defined in full detail in [8]. Recall only that by S and S ∗ we understand the complete commutative idempotent semirings (S, +, ×, 0S , 1S ) and (S, +∗ , ×∗ , 0∗S , 1∗S ), respectively, with semiring involutive isomorphism ¬ : S → S ∗ , ordered by x ≤ y ⇔ x + y = y, and by R : X ⇒ Y we denote the S-relation R : X → S Y . The composition of the S-relation R and S-relation S : Y ⇒ Z is denoted by S♦R. Any S-relation R : X ⇒ Y can be extended to a mapping R→ : S X → S Y , such that R→ = R♦1S X . Recall that for {si : i ∈ I} ⊆  R S X , x ∈ X, i∈I si (x) = i∈I si (x) and for s ∈ S X , r ∈ S, (r  s)(x) = r × s(x). Let us recall the basic definition of relational S-fuzzy cuts and S-fuzzy cuts extensional with respect to a relational S-fuzzy cut.

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Definition 1. [8] Let (S, S ∗ ) be a dual pair of semirings. 1. A pair [X, W] is called relational S-fuzzy cut, if (a) W = (Wr )r∈S is a system of discrete S-relations Wr : X ⇒ X, (b) For arbitrary r, s ∈ S, Wr ♦Ws ≤ Wr×s , Wr−1 = Wr , Wr ≥ ηX , (c) Forarbitrary r ≤ s, Ws ≤ Wr , (d) If r∈S:Wr (x)(x )=1R r ≥ s then Ws (x)(x ) = 1S . 2. Let [X, W] be a relational S-fuzzy cut. A C = (Cr )r∈S is called [X, W]extensional S-fuzzy cut, if (a) C  = (Cr )r∈S is a system of subsets Cr ⊆ X, (b) If r∈S,x∈Cr r ≥ q then x ∈ Cq , (c) For arbitrary q, r ∈ S, x ∈ X, (q  Wq )→ (r  Γ (Cr )) ≤ (r × q)  Γ (Cr×q )  holds, where Γ (A) = a∈A ηX (a) ∈ S X , for arbitrary A ⊆ X. For an arbitrary dual pair of semirings (S, S ∗ ), instead of relational S-fuzzy cuts and [X, W]-extensional S-fuzzy cuts, we can analogously define relational S ∗ -fuzzy cuts and [X, D]-extensional S ∗ -fuzzy cuts. The relationships between these dual cut systems are described in the following proposition. Proposition 1. Let (S, S ∗ ) be a dual pair of semirings and let X be a set. 1. [X, (Wr )r∈S ] is a relational S-fuzzy cut system ⇔ [X, (¬W¬r )r∈S ] is a relational S ∗ -fuzzy cut system. 2. (Cr )r∈S is [X, (Wr )r∈S ]-extensional S-fuzzy cut ⇔ (C¬r )r∈S is [X, (¬W¬r )r∈S ]-extensional S ∗ -fuzzy cut. Proof. Proof. For illustration, we prove the =⇒ implication in Statement 2. Let (Cr )r be the [X, W]-extensional S-fuzzy cut. For r, q ∈ S we set Br = C¬r , Eq = ¬W¬q . Using the properties of operations of (S, S ∗ ) and properties of compositions ♦, ♦∗ , we obtain the following. (q ∗ Eq )← (r ∗ Γ ∗ (Br )) = (q ×∗ ¬W¬q )← (r ×∗ ¬Γ (C¬r )) = ¬(¬q × W¬q )→ (¬(¬r × Γ (C¬r ))) = ¬((¬q × W¬q )→ (¬r × Γ (C¬r )) ≤∗ ¬((¬r × ¬q)  Γ (C¬r׬q )) = ¬((¬r × ¬q) × Γ ((C¬r׬q ) = (r ×∗ q) ∗ ¬Γ (C¬r׬q ) = (r ×∗ q) ∗ Γ ∗ (C¬(r×∗ q) ) = (r ×∗ q) ∗ Γ ∗ (Br×∗ q ). Hence, (C¬r )r is [X, (¬W¬q )q ]-extensional S ∗ -fuzzy cut.  An important property of [X, W]-extensional S-fuzzy cuts is the fact that these structures can be defined by special closure operators. In fact, let us consider the following ordered sets. 1. G0 (X) is the set of all C = (Cr )r∈R , where Cr ⊆ X,

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2. G[X, W] is the set of all C, such that C is [X, W]-extensional S-fuzzy cut, where [X, W] is a relational S-fuzzy cut. The sets G0 (X), G[X, W ] can be ordered by the relation (Ar )r ≤ (Br )r ⇔ ∀r ∈ S, Ar ⊆ Br . Proposition 2. Let [X, W] be a relational S-fuzzy cut and let the mapping c : G0 (X) → G0 (X) be defined by C ∈ G0 (X), c(C) = (c(Cr ))r∈S , ⎛ ⎞  c(Cr ) = {x ∈ X : ⎝ (q  Wq )→ (s  Γ (Cs ))⎠ (x) ≥ r}. s,q∈S

Then 1. c : (G0 (X), ≤) → (G0 (X), ≤) is the closure operator, that is, (a) c(C) ≥ C, (b) C ≤ B ⇒ c(C) ≤ c(B), (c) c(c(C)) = c(C), 2. G[X, W] is the set of all fixed points of c. Proof. For illustration we prove only that c(C) is [X, W]-extensional S-fuzzy cut. For an arbitrary r, q ∈ S, we need to prove (q  Wq )→ (r  Γ (c(Cr )) ≤ (r × q)  Γ (c(Cr×q )). For x ∈ X we obtain r × q, ∃t ∈ c(Cr ), Wq (t)(x) = 1S , → (q  Wq ) (r  Γ (c(Cq ))(x) = otherwise, 0S , r × q, x ∈ c(Cr×q ), (r × q)  Γ (c(Cr×q ))(x) = otherwise. 0S , Let t ∈ c(Cr ), Wq (t)(x) = 1S . It follows that  (p  Wp )→ (s  Γ (Cs ))(t) = s,p∈S



s × q ≥ r.

(z,s,p):z∈Cs ,Wp (z)(t)=1S

From inequality Wp ♦Wq ≤ Wp×q it follows {(s, p, z) : z ∈ Cs , Wp (z)(t) = 1S } ⊆ {(s, p, z) : z ∈ Cs , Wp×q (z)(x) = 1S },   and we obtain r ≤ {s,p,z:z∈Cs ,Wp (z)(t)} s×p ≤ {s,p,z:z∈Cs ,Wp×q (z)(x)=1S } s×p, and multiplying this inequality by r we obtain



r×q ≤ s × (p × q) = s × u. {s,p,z:z∈Cz ,Wp×q (z)(x)=1S }

{s,u,z:z∈Cs ,Wu (z)(x)=1S }

Therefore, x ∈ c(Cr×q ) and c(C) is the [X, W]-extensional S-fuzzy cut. 

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In connection with the existence of a closure operator in the set G0 (X), the natural question arises whether the morphisms defined between relational S-cut systems are continuous mappings between the corresponding closures. To answer this question, we first define the concept of a morphism between relational S-cut systems. Definition 2. A morphism R : [X, W] ⇒ [Y, V] is a S-relation R : X ⇒ Y , such that for arbitrary r ∈ S, R♦(s  Wr ) ≤ R and (s  Vs )♦R ≤ R. The following proposition shows that morphisms R are continuous relations with respect to the closure operators c defined in Proposition 2. Proposition 3. Let R : [X, W] ⇒ [Y, V] be a morphism of relational S-fuzzy cuts. Then R→ : (G0 (X), cX ) → (G0 (Y ), cY ) is a continuous mapping, that is, for C = (Cs )s ∈ G0 (X), R→ (cX (C)) ≤ cY (R→ (C)), where for arbitrary E = (Es )s∈S ∈ G0 (X), R→ (E) := (R→ (Es ))s∈S ,

R→ (Es ) := {y ∈ Y :



R→ (q  Γ (Eq ))(y) ≥ s}.

q∈S

Proof (Sketch). For simplicity we assume that [X, W] = [Y, V]. Let x ∈ X. From Proposition 2 it follows that the relation x ∈ R→ (c(Cr )) is equivalent to the inequality,



R→ (s  Γ (c(Cs ))(x) = s × R(t)(x) = r≤ s

{(s,t):t∈c(Cs )}



s × R(t)(x),

{(s,t)∈A}

A = {(s, t) : s ≤

where

(I)



p × q}.

{(z,p,q):z∈Cp ,Wq (z)(t)=1S }}

Analogously, x ∈ c(R→ (Cr )) is equivalent to the inequality.

s × Γ (R→ (Cs )) × q × Wq (t)(x) = r≤ {t∈X,s,q∈S}



{t∈X,s,q∈S:t∈R→ (Cs ),Wq (t)(x)=1S }

s×q =



s × q,

where

(II)

{(t,s,q,z)∈B}

B = {(t, s, q) : t ∈ X, s, q ∈ S, Wq (t)(x) = 1S ,



p × R(z)(t) ≥ s}.

{z,p:z∈Cp }

Using the properties of the morphism R we can prove that from inequality (I) inequality (II) follows, therefore, R→ (cX (Cr )) ⊆ cY (R→ (Cr )) and R is the continuous morphism. 

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The continuity of morphisms presented in Proposition 3 has one important consequence concerning the property of the category of extensional S-fuzzy cuts. This category CUT and its full subcategory CUT E are introduced in the following definition. Definition 3. The category CUT is defined by 1. Objects are (X, W, C), such that (a) [X, W] is an relational S-fuzzy cut, (b) C = (Cr )r∈S is such that C satisfies conditions 2(a),(b) of Definition 1. 2. Morphisms R : (X, W, C) ⇒ (Y, V, B) are defined by (a) R : [X, W] ⇒ [Y, V] is a morphism defined in Definition 2, (b) R→ (Cr ) ⊆ Br , for arbitrary r ∈ S. (c) The composition of the morphisms R and R is defined by R ♦R, (d) ηX : (X, W, C) ⇒ (X, W, C) is the unit morphism. It is easy to prove that this definition is correct. Let CUT E be the full subcategory of the category CUT with objects (X, W, C), such that C is the [X, D]extensional S-fuzzy cut. The following theorem holds. Theorem 1. CUT E is the full reflective subcategory in the category CUT . Proof. Let (X, W, C) be an object of CUT . Then ηX : (X, W, C) ⇒ → (C) = C (X, W, c(C)) is the refection of (X, W, C). In fact, because ηX and C ≤ c(C), ηX is a morphism from (X, W, C) to (X, W, c(C)). Let R : (X, W, C) ⇒ (Y, V, B) be another morphism in category CUT , where (Y, V, B) be an object of subcategory CUT E. Then, the following diagram commutes: ηX

(X, W, C) ====⇒ (X, W, c(C)) == == == =R= = = R ⇒ = ⇐ (Y, V, B) In fact, we need to prove that R : (X, W, c(C)) ⇒ (Y, V, B) is a morphism in CUT . Because B is extensional [Y, V] S-fuzzy cut, we have cY (B) = B. From Proposition 3 it follows that R→ (cX (C)) ⊆ cY (R→ (C)) ⊆ cY (B) = B. Therefore, R is the unique morphism such that the diagram commutes, that is, R♦ηX = R. Therefore, CUT E is the full reflective subcategory of CUT . 

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Illustrative Examples

As we mentioned in Introduction, dual pairs of semirings allow us to unify many standard constructions for new variants of fuzzy set, such as neutrosophic, intuitionistic, or L-fuzzy soft sets. The advantage of this method lies in the fact that although these new L-fuzzy structures are not mappings X → L, using appropriate dual pairs of semirings they can be transformed into S-fuzzy sets X → S, provided that the algebraic systems of both fuzzy structures are isomorphism. To illustrate it, in the following example, we use this method to define relational neutrosophic L-fuzzy cuts and extensional neutrosophic L-fuzzy cuts with respect to relational neutrosophic L-fuzzy cuts. In this examples, L be the complete M V -algebra (L, ∧, ∨, ⊗, ⊕, 0L , 1L ). Example 1. Since neutrosophic L-fuzzy sets in X are not, unlike classic L-fuzzy sets, mappings X → L ([2]), we first transform the value set L into a dual pair of semirings (S, S ∗ ) such that the algebraic system N (X) of all L-neutrosophic fuzzy sets in X is isomorphic to the algebraic system S X of all (S, S ∗ )-fuzzy sets X → S. The appropriate dual pair of semirings (S, S ∗ ) is defined by 1. The semiring S = (S = L3 , +, ×, 0, 1) is defined by (a) (a, b, c) + (a1 , b1 , c1 ) := (a ∨ a1 , b ∨ b1 , c ∧ c1 ), (b) (a, b, c) × (a1 , b1 , c1 ) := (a ⊗ a1 , b ⊗ b1 , c ⊕ c1 ), (c) 0 = (0L , 0L , 1L ), 1 = (1L , 1L , 0L ), 2. The semiring S ∗ = (S = L3 , +∗ , ×∗ , 0∗ , 1∗ ) is defined by (a) (a, b, c) +∗ (a1 , b1 , c1 ) := (a ∧ a1 , b ∧ b1 , c ∨ c1 ), (b) (a, b, c) ×∗ (a1 , b1 , c1 ) := (a ⊕ a1 , b ⊕ b1 , c ⊗ c1 ), (c) 0∗ = (1L , 1L , 0L ), 1∗ = (0L , 0L , 1L ). 3. The involutive isomorphism ¬ : S → S ∗ is defined by ¬(α, β, γ) = (γ, ¬β, α). Using the transformation of L-neutrosophis fuzzy sets into (S, S ∗ )-fuzzy sets and using Definition 1, we can define relational neutrosophic L-fuzzy cuts and extensional neutrosophic L-fuzzy cuts with respect to relational neutrosophic L -fuzzy cuts by definition. Definition 4. 1. Relational neutrosophic L-fuzzy cut relational S-fuzzy cut

W in a set X is the

[X, W] = [X, (W(a,b,c) )(a,,b,c)∈S ]. 2. Let W be a relational neutrosophic L-fuzzy cut in a set X. An W-extensional neutrosophic L-fuzzy cut is an [X, W]-extensional S-fuzzy cut C = (C(a,b,c) )(a,b,c)∈S ). For example, as follows from Definition 1 and operations in S, for arbitrary x, t ∈ X and (a, b, c), (a , b , c ) ∈ S, the following implications hold, which are the transcription of Conditions 1 (b), 2 (c) from Definition 1, respectively. We mention that relations W(a,b,c) can be substituted by subsets W(a,b,c) ⊆ X × X. In that case,  (x, t) ∈ W(a,b,c) , (t, x ) ∈ W(a ,b ,c ) =⇒ (x, x ) ∈ W(a⊗a ,b⊗b ,c⊕c ) ,  t ∈ C(a,b,c) , (t, x) ∈ W(a ,b ,c )) =⇒ x ∈ C(a⊗a ,b⊗b ,c⊕c ) .

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Example 2. From Proposition 2 it seems that it is not easy to imagine what the closure c(C) looks like. On the other hand, we can estimate approximately which elements of X correspond to this cut with at least degree r ∈ S, that is, we can determine what the set c(C)r = c(Cr ) looks like. Let arbitrary element r = (a, b, c) ∈ S, Cr = C(a,b,c) be subsets in X and let W be a relational neutrosophic L-fuzzy cut in X, W = (W(p,q,r) )(p,q,r) . Then we obtain the following lower approximation of c(C(a,b,c) ): c(C(a,b,c) ) ⊇ {x ∈ X : (t ∈ C(u,v,w) , (t, x) ∈ W(p,q,r) ) =⇒ (u, v, w) × (p, q, r) ≥ (a, b, c)} = {x ∈ X : (t ∈ C(u,v,w) , (t, x) ∈ W(p,q,r) ) =⇒ (u ⊗ p, v ⊗ q, w ⊕ r) ≥ (a, b, c)} = {x ∈ X : (t ∈ C(u,v,w) , (t, x) ∈ W(p,q,r) ) =⇒ u ⊗ p ≥ a, v ⊗ q ≤ b, w ⊕ r ≤ c}.

3

Conclusion

In the paper, we dealt with the further development of the theory of general cut systems for (S, S ∗ )-fuzzy sets. The advantage of this theory lies in the fact that it can be immediately applied to a number of new M V -valued fuzzy structures. From the above results, it follows that the theory of general cut systems represents a fully equivalent form of the theory of (S, S ∗ )-fuzzy sets, and, in addition, it allows approximating (S, S ∗ )-fuzzy sets by classical sets. In further research, it is appropriate to focus on the transformation of other parts of the (S, S ∗ )-fuzzy sets theory.

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Reverse Order Pentagonal Fuzzy Numbers and Its Application in Game Problems V. Kamal Nasir1

and A. Jamal Barakath2(B)

1 The New College, Chennai, India 2 KCG College of Technology, Chennai, India

[email protected]

Abstract. In order to resolve the erroneous payoff matrix entries in game theory reverse order pentagonal fuzzy numbers are introduced in this research work. It is assumed that all these imprecise entries are reverse order pentagonal fuzzy numbers. The suggested method also offers a fuzzy optimal solution for transforming the fuzzy game’s values to crisp ones. The concepts are tested using a few numerical examples, and the results are subsequently validated using various rank methodologies. Keywords: Pentagonal Fuzzy Number · Reverse order Pentagonal Fuzzy Number · Centroid · Fuzzy Ranking · Method of Dominance

1 Introduction An approach for studying decision-making in conflict and occasionally cooperativeridden circumstances is game theory. Game theory offers a formula for choosing the best course of action. A player in a game is a self-sufficient decision-making entity. A strategy is a set of guidelines for making decisions that spell out the player’s course of action in every situation. John Von Neumann published their book “Theory of games and Economic behavior” in 1944, providing the mathematical treatment of game theory. The best out of the worst premise served as the foundation for John van Neumann’s method of approaching game theory problems. Each person in a game problem tries to make the best choice by choosing from a variety of strategies from the available methods. To resolve competitive situations, classical game theory presupposes the existence of precise payoffs. However, in real-world circumstances, the information at hand is of a vague or ambiguous character, and the system in question already contains some degree of uncertainty. Therefore, it is possible that real-world problems cannot be formulated or solved using traditional mathematical methods. The fuzzy sets established by Zadeh in 1965 offer practical and useful tools and approaches to deal with these issues. Fuzzy number ranking is crucial to the decision-making process. Zadeh made the initial suggestion. The idea of making decisions in a fuzzy environment was further developed by Bellman and Zadeh. Jain was the first to suggest a strategy for ranking fuzzy numbers in order to make decisions in uncertain circumstances. Yager ranked fuzzy numbers using the idea of centroids. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 144–151, 2023. https://doi.org/10.1007/978-3-031-39774-5_18

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1.1 Literature Review [6] The “Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numbers” were proposed by T. Pathinathan et al. in 2015 [1]. In 2018, Avishek Chakraborty et.al. Proposed the pentagonal fuzzy number: its different representations, Properties, Ranking, Defuzzification and Application in Game problems [4]. In 2021, we presented the solution of matrix games with level (K-lower, K-upper) interval valued trapezoidal fuzzy payoffs in 2022 and [5] the solution of matrix games with level (g-lower, g-upper) interval valued pentagonal fuzzy payoffs in 2022 using the signed distance ranking approach [2]. In 2022, Adilakshmi Siripurapu and Ravi Shankar Nowpada proposed Fuzzy Project Planning and Scheduling with Pentagonal Fuzzy Number [8]. In 2022, Onyenike K and Ojarikre proposed a Study on Fuzzy Inventory model with Fuzzy demand with no shortages allowed using Pentagonal Fuzzy Numbers [7]. In 2022, Rajeswari. N and W. Ritha proposed the article named “To scrutinizes finite source queuing models with pentagonal fuzzy numbers using centroid of centroids technique under imprecise environment.” [9] In 2022, Thangaraj Beaula and S.Saravanan proposed Analysis of Fully Fuzzy Critical Path in Project Network with a New Representation of Pentagonal Fuzzy Numbers [3] The concept of “Solving game theory utilizing Reverse order Pentagonal fuzzy numbers" was put up by R. Gajalakshmi et al. in 2022. We examined a few ranking methods, presented a novel strategy, and contrasted it with earlier strategies. The main objective of this article is to find the optimal solution of the game problems using reverse order pentagonal fuzzy numbers. The remaining structure of this article is divided into the following Sections. Section 2 has some preliminary definitions. Section 3 has the existing ranking methods and the proofs of Proposed ranking technique. Section 4 contains the Algorithm and Flow chart for solving the Game problems. Section 5 contains the numerical example. Finally, we have the conclusion and Future scope in Section 6.

2 Preliminaries 2.1 Definition: Fuzzy Set (Zadeh L.A. (1965)) [10] ˜ ˜ Let A be a crisp  setand X = {x}. A Fuzzy set denoted by A and is defined as A =  x, μ ∼ (x) ; xX . Here μA˜ (x) is said to be a membership function and μA˜ (x) = A [0, 1].

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2.2 Definition: Linear PFN with Symmetry [1] ˜ = (ζ1 , ζ2 , ζ3 , ζ4 , ζ5 ) is called a Linear PFN with symmetry if its A Fuzzy number A membership function is defined as follows:  ⎧  x−ζ1 ⎪ λ ζ1 ≤ x ≤ ζ2 ⎪ ζ2 −ζ1 ,  ⎪  ⎪ ⎪ x−ζ ⎪ 2 ⎪ 1 − (1 − λ) ζ3 −ζ2 , ζ2 ≤ x ≤ ζ3 ⎪ ⎪ ⎪ ⎨ 1,  x = ζ3  μA˜ (x) = ζ4 −x ⎪ 1 − (1 − λ) ζ4 −ζ3 , ζ3 ≤ x ≤ ζ4 ⎪ ⎪ ⎪   ⎪ ⎪ ζ −x ⎪ ζ4 ≤ x ≤ ζ5 ⎪ λ ζ55−ζ4 , ⎪ ⎪ ⎩ 0, otherwise

2.3 Definition: Reverse order PFN [6] 

˜ is defined as A ˜ Rpent = −ζ1 , −ζ 2 , 0, ζ4 , ζ5 said to be reverse order PFN A Fuzzy set A if its membership function is given by ⎧ ⎪ ⎪ ⎪ 1, x ≤ −ζ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ζ1 x, −ζ1 ≤ x ≤ −ζ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1 x, −ζ2 ≤ x ≤ 0 ζ2 μA˜ Rpent (x) = ⎪ 1 ⎪ ⎪ ⎪ ζ4 x, 0 ≤ x ≤ ζ4 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ζ5 x, ζ4 ≤ x ≤ ζ5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1, x ≥ ζ5

2.4 Geometric Representation of Reverse Order Pentagonal Fuzzy Number (See Fig. 1)

Fig. 1. Reverse order Pentagonal Fuzzy Number

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2.5 Some Arithmetic Operations of Reverse order PFNs Let A˜ Rpent = (−ζ 1 , −ζ 2 , 0, ζ4 , ζ5 ) and B˜ Rpent = (−η1 , −η2 , 0, η4 , η5 ) be two Reverse order PFNs. Then (i) A˜ Rpent + B˜ Rpent = (−ζ 1 − η1 , −ζ 2 − η2 , 0, −ζ 4 − η4 , −ζ 5 − η5 ) (ii) A˜ Rpent − B˜ Rpent = (−ζ1 − η5 , −ζ 2 − η4 , 0, ζ4 + η2 , ζ5 + η1 ) (iii) σ A˜ Rpent = (−σ ζ1 , −σ ζ2 , 0, σ ζ 4 , σ ζ 5 ) For any scalar σ . Two Reverse order PFNs are added, subtracted and scalar multiplication in (i), (ii) and (iii).

3 Existing Ranking Methods and Proposed Ranking of PFN In this section, we explained existing ranking techniques and proposed ranking technique in PFN.

3 +9ϑ4 +2ϑ5 11λ+2 1. [1] Existing Ranking (1): R(Apent ) = 2ϑ1 +9ϑ2 +2ϑ , 24 24

  5 2. [7] Existing Ranking (2):R Apent = ϑ1 +ϑ2 +5ϑ9 3 +ϑ4 +ϑ5 , 18

  3. [8] Existing Ranking (3): R Apent = ϑ1 +2ϑ2 +2ϑ8 3 +2ϑ4 +ϑ5

  4. [9] Existing Ranking (4): R Apent = ϑ1 +ϑ2 +ϑ53 +ϑ4 +ϑ5 5. Proposed Method    R Apent = (x02 + y02 )  (x0 , y0 ) =

4ϑ1 + 7ϑ2 + 14ϑ3 + 7ϑ4 + 4ϑ5 14λ + 3 , 36 36

3.1 Proposition In ranking 1, if we put λ =

1 2

then R(Apent ) =





2ϑ1 +9ϑ2 +2ϑ3 +9ϑ4 +2ϑ5 5 , 16 24

3.2 Proposition In the proposed ranking, if we put λ = 1/2 then   4ϑ1 + 7ϑ2 + 14ϑ3 + 7ϑ4 + 4ϑ5 5 (x0 , y0 ) = , 36 18

4 Algorithm for Solving Game Problems Here, we are considering a two Person zero sum game where all the components in the payoff matrix are Reverse order PFN Assume that player I is the maximisation player and that player II is the minimization player. Both players will have three strategies each. The set of pure tactics must be used to choose a strategy by both players.

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Player II ⎞ a˜ 12 a˜ 13 a ˜ 11 ˜A = Player I ⎝ a˜ 21 a˜ 22 a˜ 23 ⎠ a˜ 31 a˜ 32 a˜ 33 ⎛

4.1 Dominance Method Step 1: Each element in the ith column should be compared to its corresponding element in the jth column. When comparing, the ith Column is said to be dominated by the jth Column and is removed from the matrix if its components are more than or equal to its equivalent components in the jth Column. Step 2: Each element in the ith row should be compared to its corresponding element in the jth row. If all the ith row’s components are equal to or less than the corresponding components of the jth row, the ith row is deemed to be dominated by the jth row during comparison and is removed from the matrix. Step 3: This is where you may find the oddments for each row and column. Step 4: Verify the total of the row and column oddments. Proceed to step 5 if they are equal; else, the technique will fail. Step 5: The probability of the mixed strategy is calculated by dividing the corresponding oddments by the total number of oddments. Step 6: To determine the worth of the game, multiply the probabilities from step 5 by the relevant payoffs (Fig. 2).

Fig. 2. Flowchart for solving Game Problems

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5 Numerical Example Look at the Reverse order Pentagonal fuzzy game problem below, where I and II are the two players and they are playing maximisation and minimization tactics, respectively. The entries of payoff matrix A˜ are Reverse order PFN. The main objective is to identify each player’s best course of action so that they can maximise their chances of winning while minimising their chances of losing. Table 1. Reverse order Pentagonal fuzzy numbers Reverse Order Pentagonal Fuzzy Numbers ϑ˜ 11 = (−4, −3, 2, 3, 4) ϑ˜ 12 = (−5, −3, 1, 3, 5) ϑ˜ 21 = (−10, −9, 5, 9, 10) ϑ˜ 31 = (−7.25, −5.25, 4, 5.25, 7.25)

ϑ˜ 13 = (−6, −4, 3, 4, 6)

ϑ˜ 22 = (−6.5, −4, 2.5, 4, 6.5) ϑ˜ 23 = (−3.2, −3, 1.5, 3, 3.2) ϑ˜ 32 = (−9.4, −8, 7, 8, 9.4) ϑ˜ 33 = (−8.5, −7, 6, 7, 8.5)

After the application of the proposed ranking method will gives the corresponding payoff matrix ⎛

⎞ 0.8259 0.4779 1.1993 ⎝ 1.9642 1.0111 0.6461 ⎠ 1.5802 2.7364 2.3498 The suggested ranking matrix lacks a saddle point. The game value is 1.7217 units after the application of the dominance principle. 5.1 Comparative Analysis of Suggested Method with Existing Methods Table 2 shows the value of the games for the corresponding Defuzzification methods. Table 2. Defuzzification methods and its values of the Game S.No

Ranking Method

Value of the Game

Optimal Solution

1

[1] Avishek Chakraborty et.al. (2019)

0.4765

I (0, 0.4470,0.5530) II (0.8521, 0, 0.1479)

2

[7] Rajeswari. N and W. Ritha (2022)

2.4417

I (0, 0.3662,0.6338) II (0.8171, 0, 0.1829)

3

[8] Onyenike K and Ojarikre (2022)

1.0909

I (0, 0.3636,0.6364) II (0.8182, 0, 0.1818)

4

[9] Thangaraj Beaula and S. Saravanan (2022)

0.8727

I (0, 0.3636,0.6364) II (0.8182, 0, 0.1818)

5

Proposed Method

1.7217

I (0, 0.3686,0.6314) II (0.8161, 0, 0.1839)

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The result graph is presented in Fig. 3.

Fig. 3. Result Chart

6 Conclusion We applied the proposed method to rank the reverse order PFN. The proposed ranking function is derived from the centroid of PFN. In Game problems the payoff matrix elements are expressed in terms of reverse order PFN. By using the proposed ranking algorithm, the fuzzy numbers are transformed into normal numbers then it is solved by dominance method to get the optimum strategies. The results are correlated with other existing ranking methods. In future, the concept can be applied to other category of fuzzy numbers.

References 1. Chakraborty, A., et al.: Pentagonal fuzzy number: its different representations, properties, ranking, defuzzification and application in game problems. Symmetry 11(2), 248 (2019) 2. Siripurapu, A., Nowpada, R.S.: Fuzzy project planning and scheduling with pentagonal fuzzy number. Reliabil. Theory Appl. 17, 131–138 (2022) 3. Gajalakshmi, R., Rabinson, G.C.: Solving game theory using reverse order pentagonal fuzzy numbers. J. Algebr. Stat. 13(3), 1785–1790 (2022) 4. Nasir, V.K., Barakath, A.J: Solving matrix games involving the level (K-lower, K-upper) Interval Valued Trapezoidal Fuzzy Payoffs: Signed distance ranking approach. In: AIP Conference Proceedings, vol. 2385 , p. 130004 (2022) 5. Nasir, V.K., Barakath, A.J: Solving matrix games involving the level (g-lower, g-upper) interval valued pentagonal fuzzy payoffs: signed distance ranking approach. In: Intelligent and Fuzzy Systems: Digital Acceleration and the New Normal-Proceedings of the INFUS 2022 Conference, vol. 1, pp. 328–338 (2022) 6. Pathinathan, T., Ponnivalavan, K.: Reverse order triangular, trapezoidal and pentagonal fuzzy numbers. Ann. Pure Appl. Math. 9(1), 107–117 (2015) 7. Rajeswari, N., Ritha, W.: To scrutinizes finite source queuing models with pentagonal fuzzy numbers using centroid of centroids technique under imprecise environment. J. Algebr. Stat. 13(3), 2647–2660 (2022)

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8. Onyenike, K., Ojarikre, H.I.: A study on fuzzy inventory model with fuzzy demand with no shortages allowed using pentagonal fuzzy numbers. Int. J. Innov. Sci. Res. Technol. 7(3), 772–776 (2022) 9. Beaula, T., Saravanan, S.: Analysis of fully fuzzy critical path in project network with a new representation of pentagonal fuzzy numbers. Math. Stat. Eng. Appl. 71(4), 2383–2397 (2022) 10. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

Similarity Analysis of Means-End Chain Models Umut Asan(B)

and Hatice Kocaman

Department of Industrial Engineering, Istanbul Technical University, Macka, Istanbul, Turkey {asanu,Kocaman18}@itu.edu.tr

Abstract. The Means-End Chain Theory is a commonly used approach in marketing that aims to understand how consumers make decisions about marketing offerings by examining their thought processes. The theory is built on a three-level hierarchical structure that encompasses the attributes of a product or service, the benefits derived from these attributes, and the values associated with these benefits. Since the Means-End Chain Theory reflects the personal views of consumers, researchers need to combine and/or compare individuals’ hierarchical value maps to gain a better understanding of the market. The main issue here is how to measure the similarity of consumers’ means-end chain models and consequently how to compare them. There are a limited number of studies in the literature that address this issue, however they have notable shortcomings. In this study, a new method is proposed that considers all possible combinations of direct and indirect relationships between the abstraction levels (attributes, benefits, and values) and the weights of these combinations to calculate the distances (i.e. dissimilarities) between consumers’ hierarchical value maps. The applicability and effectiveness of the proposed method is demonstrated by an example. Keywords: Means-End Chain Theory · Hierarchical Value Map · Indirect Relationship Analysis · Similarity Analysis · Weighting

1 Introduction The Means-End Chain (MEC) Theory is a commonly used approach in marketing that helps to understand how consumers make decisions about marketing offerings by examining their thought processes [1]. The theory is built on a three-level hierarchical structure encompassing the attributes of a product or service, the benefits derived from these attributes, and the associated values. To reveal this structure, a qualitative research method called “Laddering”, which is a semi-structured in-depth interviewing technique, is used [2]. The uncovered ladders (i.e. cognitive connections between the three conceptual categories) are typically visualized using a tree diagram, termed a hierarchical value map (HVM). Although the MEC theory is oriented towards individuals and reflects the personal views of consumers, researchers often need to combine and/or compare individuals’ hierarchical value maps to gain a better understanding of the market. The main purpose could be either to obtain a common value map of the target audience or to divide the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 152–159, 2023. https://doi.org/10.1007/978-3-031-39774-5_19

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market into subgroups of consumers who share similar cognitive processes related to purchasing. Therefore, how to compare the cognitive processes associated with consumers’ purchasing decisions and utilize this information for consumer segmentation is an important question to be addressed. There are a limited number of studies in the literature that examines this issue, however they have notable shortcomings. For example, some studies perform clustering analyses using simple similarity measures that consider only the numbers of relationships between the chain’s items. Other studies attempt to identify dominant chains by clustering means-end chains of consumers instead of clustering consumers based on the similarity of their means-end structures. Another group of studies compare consumers’ cognitive processes by calculating the similarity between items (e.g. attributes) evaluated on predefined scales. Moreover, none of the studies propose a measure that allows comparing direct and indirect relationships between levels of abstraction. Thus, how to measure the similarity of consumers’ means-end chain models remains an unresolved issue. To address this issue, this study proposes a new method that considers all possible combinations of direct and indirect relationships between the abstraction levels (attributes, benefits, and values) and their respective weights in calculating the dissimilarity (i.e. distance ratio) between consumers’ hierarchical value maps. The new method adapts a distance ratio formula from previous literature, originally developed to measure differences between cognitive maps, to hierarchical value maps and extends the utilization of this formula to also account for indirect paths. Moreover, it offers a weighting approach for objectively combining the distance ratios obtained by comparing all possible direct and indirect relationships. By using the calculated (dis)similarities between consumers’ means-end chains, it will be possible to segment the market into homogeneous groups of consumers with greater effectiveness and completeness. The applicability and effectiveness of the proposed method is demonstrated by an illustrative example. The rest of the paper is organized as follows. In the next section a review of MEC theory and methods used to compare/combine MECs are provided. The proposed method is presented in the following section. In Sect. 4 an illustrative example and in Sect. 5 conclusions and further research directions are provided.

2 Literature Review 2.1 Means-End Chain Theory Consumers, like all other individuals, create a cognitive representation of their environment in order to make life understandable, clear, and easy. Means-End Chain theory aims to reveal the consumer decision-making process from a cognitive perspective, attempting to reveal why and for what purpose consumers purchase a particular product, as well as what they find valuable in products [1, 3]. Means-End Chain theory gained acceptance in the marketing field thanks to Gutman’s (1982) article [1] published in the Journal of Marketing. The theory explains how product selection helps consumers achieve their desired outcomes by considering the interrelatedness between product attributes, consequences (benefits), and values.

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The Means-End Chain theory assumes that product meanings form hierarchically related elements in memory and are stored in a chain-like structure. The chain begins with the product information (product attributes) component and creates connections that extend to personal values through perceived outcomes or benefits generated by the product. These connections form the “means-end chain,” in which attributes serve as means to ends, thus creating the desired outcome or value for the product. Therefore, in the Means-End theory, there are three levels of abstraction: attributes, consequences, and values. The structure of the levels indicates a hierarchical ordering, and the structure is composed of interconnected elements [4]. Attributes can be defined as tangible (e.g., packaging) and abstract (e.g., image) features related to products, services, and brands, and they are located at the bottom level. Consequences can be defined as any outcome, whether physiological (e.g. satisfying hunger) or psychological (e.g. looking confidently to the future), that arises directly or indirectly from consumer behavior [1]. Conceptually, consequences that indicate a higher level of abstraction is positioned at a middle level hierarchically within the chain. Values are the answer to the question of what product usage helps the consumer to achieve. Values are considered a type of outcome representing desired end-states for which an individual would have no further reason to choose. Values represent the highest level of abstraction. In an application of Means-End Chain theory, the following basic steps are followed: 1. Laddering is a qualitative research method that utilizes a semi-structured, in-depth interviewing technique to gain insights into how consumers relate the attributes of products to personal values and beliefs, based on the means-end theory [2]. 2. Content analysis involves creating, merging, and separating new categories from data until an optimal solution is reached. 3. Implication Matrix involves developing a numerical, square-shaped matrix that contains information about all the attribute-consequence-value chains generated by the participants, which the researcher is trying to represent visually. 4. Creating a Hierarchical Value Map involves graphically representing the relationships between the attributes, consequences, and values (as identified in the inference matrix) that emerged during the Laddering interviews for a specific product in the form of a tree diagram. The hierarchical value map makes the research results (the consumer’s cognitive structures regarding a product) interpretable. 2.2 Related Works Several approaches have been suggested in the literature to measure the similarity of means-end chains. One group of studies considers only the number of common items and/or direct paths in comparing individuals’ MEC models. For example, to segment users of social media platforms, Asan et al. [5] consider only the total number of common paths in the hierarchical value maps to measure similarity. To perform a benefit-based cluster analysis, Xiao et al. [6] used a benefit × respondent matrix where the matrix contained the frequency of mentions for each benefit by every respondent. The study by Zhang et al. [7] segmented luxury travelers by means and ends according to how often they chose luxury hotels and first- or business-class flights and how often they mentioned values obtained in a laddering interview. Another group of studies asks respondents to assess the importance of each MEC item on a predefined scale, which is then used to

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calculate the distance (i.e. dissimilarity) between the respondents. For example, in the study by Lin and Yeh [8], the importance assessments obtained by a questionnaire for all means-end chain items were used to cluster hierarchical value maps. Botschen et al. [9] used a similar method to assess the similarity of laddering data when exploring segments. Apart from these Aurifeille and Valette-Florence [10] attempted to identify dominant chains by deriving the hypergeometric probability of the link between pairs of laddering items. And Myrda [11] suggested using sequence dissimilarity measures for clustering laddering data.

3 Proposed Method In general, there are three categories of measures used to compare network graphs, which are classified based on their intended purposes: i) Content ii) Structure, and iii) Dynamic Behaviors [12]. In this study a new content-based method to calculate the differences (i.e. dissimilarities) between consumers’ hierarchical value maps is proposed. The steps of the proposed method are explained below. Step 1: Constructing the HVM/implication matrix of each individual The basic steps of an MEC analysis (attribute elicitation, laddering, content analysis, implication matrix, and hierarchical value map) is performed for each respondent. As the input for the next step, the implication matrix indicates the frequency with which each item leads to other item, specifically through direct links. Step 2: Determining the indirect relationships Once the implication matrix (Mk ) of respondent k is built, the indirect relationships between items are derived by raising the matrix to successive powers. Raising the impliq cation matrix to the qth power (Mk ) gives the indirect effect of all paths of length q from item i to item j. Since MEC theory distinguishes three abstraction levels each containing two sublevels, the longest paths in MEC models should not be longer than five. Mk is raised until all the elements in a resulting matrix are obtained as zero, indicating that all possible indirect paths have been examined. Step 3: Normalizing the indirect relationships Raising the matrix to successive powers may produce increasingly larger values. In order to avoid any bias towards higher orders, an appropriate normalization procedure is used as follows,     q∗ q q (1) mk,ij = max m1k,ij · mk,ij /max mk,ij , ∀i, ∀j q∗

where mk,ij denotes the normalized value of the ij th element of the implication matrix of order q and respondent k. Step 4: Calculating the distance ratios between two individuals’ means-end chains A formula for measuring differences between cognitive maps, which was initially proposed by Langfield-Smith and Wirth [13] and later generalized by Markíczy and Goldberg [14] is adapted for hierarchical value maps in MEC theory. By utilizing the adapted formula given in Eq. 2, a distance ratio (DR) is obtained, which falls within the range of 0 to 1, for comparing pairs of hierarchical value maps that are both expressed as extended

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implication matrices. A ratio of 0 indicates that the maps are identical, while a ratio of 1 suggests that the distance between the maps is at its maximum. DR(MA , MB ) p   p i=1

j=1 diff(i, j)

       (β)pc2 + γ  2pc puA + puB + pu2A + pu2B − α (β)pc + γ  puA + puB ⎧ ⎨0   (i) if i = j and α = 1; diff(i, j) =  aij , bij (ii) if either ior j ∈ / Pc and i, j ∈ NA or i, j ∈ NB; ⎩

aij − bij (iii) otherwise. ⎧   ⎨ 0 (a) if γ = 0;  0 if γ = 0;  aij , bij = 0 (b) if γ = 1 and aij = bij = 0; γ = ⎩ 1 otherwise. 1 (c) otherwise (γ = 2). =

(2)

(3)

(4)

where MA and MB are extended implication matrices of respondents A and B, respectively; aij is the value of the ith row and j th column of MA ; p is the total number of items; Pc is the set of item common to both maps; pc is the number of these items; puA (puB ) is the number of items unique to map A (B); NA and NB are the sets of items in A and B; α, β, and γ are parameters denoting direct self-influence, maximum strength, and mismatch of items, respectively. The basic idea behind the formula is to add up, node by node and arc by arc, all the differences between two maps (implication matrices), and subsequently divide the resulting sum by the maximum achievable difference. In prior research, the DR formula has been employed to measure differences (dissimilarities) between cognitive maps by solely taking into account direct connections. In this study, the DR is calculated for all direct and indirect relationship combinations. For example, respondent A and B have hierarchical value maps where the longest path is two and three, respectively. Then, the comparison between the respondents’ MEC models (i.e. hierarchical value of thefollowing six (2×   the distance  ratios     maps) involves 3) pairs: DR(MA , MB ), DR MA , MB2∗ , DR MA , MB3∗ , DR MA2∗ , MB , DR MA2∗ , MB2∗ ,  2∗ 3∗  and DR MA , MB . Step 5: Determining the weights Not all combination pairs have the same contribution to the (dis)similarity of the MEC models. Comparison of direct relationships are assumed to be more indicative of the (dis)similarity. The impact on the result will be lower when comparing longer indirect paths. Therefore, assigning weights to the combination pairs will help prioritize their contribution and improve the accuracy of the (dis)similarity measurement. The following weighting approach is suggested: w(q, r) = 1/(q + r − 1) where q and r denotes the length of the indirect paths compared.

(5)

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Step 6: Calculating the weighted total distance ratio between two individuals Once the weights of indirect paths are determined, the DRs are aggregated using the following weighted sum formula:   q∗ (6) w(q, r) · DR MA , MBr∗ TDR(A, B) = q

r

Step 7: Constructing the normalized total distance ratio matrix The TDRs obtained in the previous step are normalized into the range of 0 to 1 as follows: TDR∗ (A, B) = TDR(A, B)/maximum achievable TDR

(7)

Finally, TDR∗ values for each pairwise comparison are accommodated into a respondent dissimilarity matrix of size K × K to be used in cluster analysis.

4 Illustrative Example Notice that since the main concern of this example is to demonstrate the original contributions of the new approach, the initial steps which are identical with the classical MEC are omitted. The hierarchical value maps of each respondent to be compared in this illustrative example is illustrated in Fig. 1. 5

5 9

4

4

8

10

3

3 6

6 7

2

2

1

1

A

B

1

C

Fig. 1. The hierarchical value maps of the respondents.

The value of q for respondents A, B and C is 4, 4 and 3, respectively. The distance of all matrix combinations between the respondents and the respective weights are provided in Table 1. For example, DR(MA , MB ) = 0.013 indicates the distance ratio between respondent A and B’s implication matrices of order one. The weighted total distance ratio between two individuals is calculated, and the similarity results are illustrated in Table 2 based on these weighted distance ratios. Table 2 shows individual A and B are the most similar pair. However, the similarity of individuals A and C and individuals B and C is the least similar compared to Individuals A and B.

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i

j

Weights A − B

DR

A−C

DR

B−C

DR

1 1 1

DR(MA , MB )

0.013 DR(MA , MC )

0.520 DR(MB , MC )

0.520

1 2 0.5

2∗ ) DR(MA , MB

2∗ ) 0.067 DR(MA , MC

2∗ ) 0.460 DR(MB , MC

0.460

1 3 0.33

3∗ ) DR(MA , MB

3∗ ) 0.053 DR(MA , MC

3∗ ) 0.440 DR(MB , MC

0.440

1 4 0.25

4∗ ) DR(MA , MB

0.046

2 1 0.5

2∗ , M ) DR(MA B

2∗ , M ) 0.060 DR(MA C

2∗ , M ) 0.500 DR(MB C

0.480

2 2 0.33

2∗ , M2∗ ) 0.020 DR(M2∗ , M2∗ ) 0.440 DR(M2∗ , M2∗ ) 0.420 DR(MA B A C B C

2 3 0.25

2∗ , M3∗ ) 0.033 DR(M2∗ , M3∗ ) 0.420 DR(M2∗ , M3∗ ) 0.400 DR(MA B A C B C

2 4 0.2

2∗ , M4∗ ) 0.040 DR(MA B

3 1 0.3

3∗ , M ) DR(MA B

3 2 0.25

3∗ , M2∗ ) DR(MA B 3∗ , M3∗ ) DR(MA B 3∗ , M4∗ ) DR(MA B 4∗ , M ) DR(MA B 4∗ , M2∗ ) DR(MA B 4∗ , M3∗ ) DR(MA B 4∗ , M4∗ ) DR(MA B

3 3 0.2 3 4 0.167 4 1 0.25 4 2 0.2 4 3 0.167 4 4 0.143

3∗ , M ) 0.060 DR(MA C

0.033 0.007

3∗ , M2∗ ) DR(MA C 3∗ , M3∗ ) DR(MA C

3∗ , M ) 0.460 DR(MB C

0.380 0.380

3∗ , M2∗ ) DR(MB C 3∗ , M3∗ ) DR(MB C

0.460 0.400 0.380

0.013 4∗ , M ) 0.046 DR(MA C

0.020 0.020

4∗ , M2∗ ) DR(MA C 4∗ , M3∗ ) DR(MA C

4∗ , M ) 0.440 DR(MB C

0.380 0.360

4∗ , M2∗ ) DR(MB C 4∗ , M3∗ ) DR(MB C

0.440 0.380 0.360

0

Table 2. Dissimilarity Values (Normalized Total Distance Ratios) Comparison

Dissimilarity Value

TDR∗ (A, B)

0.035

TDR∗ (A, C)

0.456

TDR∗ (B, C)

0.452

5 Conclusion and Further Research The proposed method is fundamentally different from previous approaches which are solely based on the comparisons of items, direct links or whole chains rather than the comparison of all possible combinations of direct and indirect chains. It reveals similar cognitive processes related to consumption, thus resulting in a more effective segmentation of the market into homogeneous consumer groups. Due to its general structure, this new approach can be easily adapted to areas of application other than marketing. There are also some potential future directions that should be acknowledged. The proposed approach can be further improved by applying different types of objective

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weighting approaches. Moreover, examining the applicability and effectiveness of the suggested method through a real-world problem would be valuable. Acknowledgements. This research was supported by Scientific Research Projects Department of Istanbul Technical University. Project Number: SGA-2022–44093.

References 1. Gutman, J.: A means-end chain model based on consumer categorization processes. J. Mark. 46(2), 60–72 (1982) 2. Reynolds, T.J., Gutman, J.: Laddering theory, method, analysis and interpretation. J. Advert. Res. 28(1), 11–31 (1988) 3. Reynolds, T.J.: Implications for value research : a micro versus macro perspective. Psychol. Mark. 4, 297–305 (1985) 4. Gutman, J.: Means–end chains as goal hierarchies. Psychol. Mark. 14(6), 545–560 (1997) 5. Asan, U., Cetin, A., Soyer, A.: Segmentation of social media users: a means-end chain approach. In: Calisir, F., Korhan, O. (eds.) GJCIE 2019. LNMIE, pp. 243–255. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-42416-9_22 6. Xiao, L., Guo, Z., D’Ambra, J.: Benefit-based O2O commerce segmentation: a means-end chain approach. Electron. Commer. Res. 19(2), 409–449 (2018). https://doi.org/10.1007/s10 660-017-9286-3 7. Zhang, E.Y., McKercher, B., Tse, T.S.: Are luxury travelers alike? a qualitative means–end segmentation approach. J. Hosp. Tourism Res. 10963480221103224 (2022) 8. Lin, C.F., Yeh, M.Y.: Means-end chains and cluster analysis: an integrated approach to improving marketing strategy. J. Target. Meas. Anal. Mark. 9, 20–35 (2000) 9. Botschen, G., Thelen, E.M., Pieters, R.: Using means-end structures for benefit segmentation: an application to services. Eur. J. Mark. 33(1/2), 38–58 (1999) 10. Aurifeille, J.M., Valette-Florence, P.: Determination of the dominant means-end chains: a constrained clustering approach. Int. J. Res. Mark. 12(3), 267–278 (1995) 11. Myrda, A.: The means-end approach in market segmentation–clustering of laddering data. Ekonometria 54, 72–81 (2016) 12. Yoon, B.S., Jetter, A.J.: Comparative analysis for fuzzy cognitive mapping. In: 2016 Portland International Conference on Management of Engineering and Technology (PICMET), pp. 1897–1908. IEEE (2016) 13. Langfield-Smith, K., Wirth, A.: Measuring differences between cognitive maps. J. Oper. Res. Soc. 43(12), 1135–1150 (1992) 14. Markíczy, L., Goldberg, J.: A method for eliciting and comparing causal maps. J. Manag. 21(2), 305–333 (1995)

Unfolding Computation Graph for Dynamic Planning Under Uncertainty Margarita Knyazeva , Alexander Bozhenyuk(B)

, and Leontiy Samoylov

Southern Federal University, Nekrasovskiy Street, 44, 347928 Taganrog, Russia [email protected], [email protected]

Abstract. A computational graph represents the structure of a set of computations, mapping inputs, parameters to outputs and loss, corresponding to a chain of events or calculations. Classical form of a dynamical system presupposes states of the system and recurrence relation for temporal planning. Unfolded computational graph is described by nodes representing states at some time or temporal interval and transition function that maps state at time t to the state at t + 1. Another representation may treat a dynamical system driven by external (input) signal, where state accumulate information about the whole past input sequence. In this paper the problem of unfolding computational graph under uncertainty for the planning production problem instance with temporal intervals is considered. The estimate level of demand for products for each planning period is given and the level of production during for temporal interval t can be used to cover demand which is different for temporal intervals. The approach proposed in this work allows developing such a production program, for which the total cost of production is minimized, while the demand is fully and timely satisfied. Keywords: Computational graph · Temporal Modelling · Operation Planning · Uncertainty · Combinatorial Optimization

1 Introduction The problem of unfolding the set of recurrent events within a computational graph for the planning problems usually corresponds to a chain-event ordering in discrete time moments. In traditional resource-constrained scheduling problems with generalized precedence relations RCPSP-GRP [1] each vertex of the event graph corresponds to an activity and each graph edge reflects the ordering sequence or precedence relations of other events. Each edge may be additionally associated with Boolean condition and/or a temporal delay. In literature this problem type is a combinatorial optimization problem, NP-hard [2]. A precedence graph (conflict graph) is usually used for modelling when the constraints and resource consumption are limited to the concurrency of events: two operations cannot be processed simultaneously and therefore a precedence relation is to be defined between them [3, 4]. And finally, an integrated scheduling algorithm based on the priority constraint table for complex products with tree structure is discussed in [5], where the idea is to consider the assembly order constraints among jobs, which makes the processing structure of products tree-like. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 160–167, 2023. https://doi.org/10.1007/978-3-031-39774-5_20

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In this work, the idea to treat permanent system restrictions and dynamic structural dependencies of optimization models simultaneously under uncertainty is considered. Among enumeration and some other combinatorial problems, one can single out a class of problems that have one good property: having solutions to some subproblems (for example, for a smaller number n), one can find a solution to the original problem almost without full enumeration. Such problems are solved by the dynamic programming method, and dynamic programming itself is understood as the reduction of a problem to subtasks. A common feature of all dynamic programming models is the reduction of the decision-making problem to obtaining recursive relations. For example, if fn (s) – is the objective function under the given element (state) s and if there are n steps to the final state (dynamic component of the model), then on (s) – is such a decision (output) that allows reaching fn (s). However optimal solutions to resource-constrained scheduling and planning problems may be sensitive to the state transition uncertainty. The idea of robust control of markov decision processes with uncertain transition matrices is presented in [6], where uncertainty on the transition matrices is presented by possibly nonconvex sets. Temporal aspect of scheduling problem presupposes both time moments and periods (intervals) of operation execution that is associated with uncertainty. Temporal uncertainty can be treated as follows: for each operation topological order and a pre-defined time window for the start of operation is known, but during operations performance that predefined time window may change (See Fig. 1). These instances can be used within dynamic strategies, for example in the pickup-and-delivery problem with time-windows (PDPTW) [7, 8]. Another example of planning mechanism under uncertainty is discussed in [9], where the dynamical approach for unfolding fuzzy temporal computational graph for project scheduling problem is presented.

Fig. 1. An example of operation’s timeline instance.

In this work the approach for the production planning problem under N-periods is considered with respect to dynamical aspects and temporal uncertainty. Temporal uncertainty is related to the fact that the decision maker cannot accurately predict the number of planning intervals to obtain the optimal solution for the problem. The storage of the resulting stocks is associated with certain costs, so the problem is to develop such a production program, for which the total cost of production and stocks is minimized.

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The paper is organized as follows. Section 2 introduces planning problem formalizations converting its qualitative description into a mathematical model. The numerical example is given in Sect. 3. In conclusion we summarize the results.

2 Planning Problem Formulation for Production System A temporal production project consists of basic variable such as: ot – the production output during the interval t; lt – the stock (inventory) level at the end of the interval t; Dt – is the demand of products for interval t; cost t (ot , lt ) – is the total cost of production and storage for each interval t. This cost function is nonlinear. Then the objective function can be written as follows: N t=1

costt (ot , lt ) → min

(1)

where ot = 0, 1, 2, 3 . . . (t = 1, 2, . . . , N )

(2)

lN = 0

(3)

lt−1 + ot − lt = Dt

(4)

lt = 0, 1, 2, 3 . . . (t = 1, 2, . . . , N − 1)

(5)

Equation (3) shows that the final stock should be equal to 0, Eq. (4) illustrates that the demand should be satisfyed within the interval t. Let’s consider the inventory constraint matrix in Table 1. Table 1. The inventory constraint matrix. Periods

o1

l1

1

1

–1

2 3 4

1

o2

l2

o3

l3

o4 = D1 − l0

1

−1 1

= D2 1

−1 1

= D3 1

= D4

In order to develop the mathematical model for the problem when each of the quantities ct (ot , lt ) is nonlinear, let’s formulate it in terms of dynamic programming [10]. The constructed system is adequate to the network shown in Fig. 2.

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Fig. 2. A network representing production and inventory flow.

2.1 Basic Concepts and Notations for the Problem Formulation In the process of dynamic planning the computational process usually starts from the final state to the initial one, where “1” corresponds to the final state, and N to the initial state. Let’s consider the following notations: dn – is the demand for products in interval n, separated from the end of the planning period by n intervals (including the one under consideration); cn (o, j) – are the costs in temporal interval n associated with the release of o units of production and with the maintenance of stocks (inventory), the level of which at the end of the interval is equal to j units; fn (s) – is the cost corresponding to the minimum cost strategy for n remaining intervals at the initial stock level l; on (l) – is the product output that achieves fn (s). According to Eq. (3), the inventory level at the end of the planning period is equal to zero: f0 (0) = 0(n = 0)

(6)

For n = 1 the initial stock level l can be defined by any non-negative integer not greater than d1 ; so the volume of output should be equal to (d1 − 1) so that: f1 (l) = c1 (d1 − l, 0), l = 0, 1, . . . , d1

(7)

For n = 2 (the initial inventory level is l and the production output is o) the total cost for two periods is as follows: c2 (o, l + o − d2 ) + f1 (l + o − d2 ), where (l + o − d2 ) is the inventory level at the end of interval 2. The value l can take any non-negative integer value not exceeding (d1 + d2 ). The performed analysis of the situation for n = 2 can be expressed by the following general expression: f2 (l) = mino [c2 (o, l + o − d2 ) + f1 (l + o − d2 )], where l = 0, 1, . . . , d1 + d2 and d2 − l ≤ o ≤ d1 + d2 − l. The general recurrence relation can be written as follows: fn (l) = mino [cn (o, l + o − dn ) + fn−1 (l + o − dn )]

(8)

where n = 1, 2, . . . , N ; l = 0, 1, . . . , d1 +· · ·+dn and dn −l ≤ o ≤ d1 +d2 +· · ·+dn −l. The initial stock level l is considered as a variable that fully characterizes the state of the system, the only independent control variable in the recurrence relation (8) is o, since the stock level at the end of the interval is equal to (l + o − dn ).

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2.2 Dynamic Programming Formulation for the Optimal Production To find the optimal production program, we determine what volume of production output oN (l0 ) allows achieving the obtained value fN (l0 ); the corresponding decision is the optimal solution for the initial interval of the planning period [10]. The inventory level at the beginning of the next temporal interval is equal to l0 +oN (l0 )−dn ; let’s find the volume of output that allows us to achieve the value we have obtained fN −1 [l0 + o(l0 ) − dn ]. The decision-making process here is a multi-step process: n - is the number of steps (for this problem it is the number of intervals of the planning period) until the end of the process. Suppose dn = 1 for n = 1, 2, 3, 4 (demand is constant over time) and o = 0,1,2 (product output is limited). The network built for this example is shown in Fig. 3. The vertex, indicated by indices (l, n), corresponds to the possible value of the state variable l for n intervals before the end of the planning period. The five vertices on the left side of the network represent the different possibilities that arise when l0 is 0, 1, 2, 3, or 4.

Fig. 3. A graph unfolding representation for production and inventory states of the system.

Since j = l + o + dn , then for n temporal intervals before the end of the planned period, the variable o = j − l + dn . Each admissible value of the variable should be displayed by one of the arcs. In the recurrence relation (8), the sequence of operations is inverse to their actual sequence in time.

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3 Numerical Example The example below illustrates the possible complexity of combinatorial problems, even if their dimension is small [10]. Let the demand is constant in time and equal to Dt = 3 units, hl t – unit storage cost. Ct (ot , lt ) = C(ot ) + hlt

(9)

where C(0) = 0, C(1) = 15, C(2) = 17, C(3) = 19, C(4) = 21, C(5) = 23, h = 1.  ot = 0, 1, . . . , 5 (10) lt = 0, 1, . . . , 4 For n = 1, i = 0, 1, 2, 3 we obtain the following results:  f1 (l) = C(3 − l) o1 (l) = 3 − l

(11)

And the recurrence relation for n = 2,3,… and l = 0,1,2,3,4 can be written as follows: fn (l) = mino [C(o) + hlt (l + o − 3) + fn−1 (l + o − 3)]

(12)

The function values fn (l) are calculated in the following Tables 2–3. For each step n a table is constructed having one row for each possible initial stock level l and one column for each possible output o. Table 2. The function values fn (l) for n = 1 l

o1 (l)

f1 (l)

0

3

19

1

2

17

2

1

15

3

0

0

Table 3. The function values fn (l) for n = 2, C(o) + 1(l + o − 3) + f1 (l + o − 3). 1

2

3

0

–

–

–

19 + 0 + 19 21 + 1 + 17 23 + 2 + 15 3

1

–

–

17 + 0 + 19 19 + 1 + 17 21 + 2 + 15 23 + 3 + 0

2

–

15 + 0 + 19 17 + 1 + 17 19 + 2 + 15 21 + 3 + 0

3

0 + 0 + 19 15 + 1 + 17 17 + 2 + 15 13 + 3 + 0

–

4

0 + 1 + 17 15 + 2 + 15 17 + 3 + 0

–

–

4

5

o2 (l) f2 (l)

l\o 0

− –

38

5

26

4

24

0

19

0

18

The function values fn (l) for n = 3 is [C(o) + 1(l + o − 3)] + f2 (l + o − 3). Iterative calculation of indicators at subsequent steps up to n = 6 allows getting the final results as follows:

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n=2

n=3

n=4

n=5

n=6

l

01 (l)

f1 (l)

02 (l)

f2 (l)

03 (l)

f3 (l)

o4 (l)

f4 (l)

o5 (l)

f5 (l)

06 (l)

f6 (l)

0

3

19

3

38

4

48

3,4

67

5

79

4

96

1

2

17

5

26

5

45

5

64

5

74

5

93

2

1

15

4

24

4

43

5

54

4

72

4

91

3

0

0

0

19

0

38

0

48

0

67

0

79

4

−

−

0

18

0

27

0

46

0

65

0

75

The Number of Planning Period Problem. Assume that the planning period begins from n = 1, the questions is how the optimal monthly production output volumes will change with an increase in the number of months N in the planning period. The results of the analysis based on the data of Table 4 are shown in Table 5. Table 5. Product output program (t0 = 0). N

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 Total amount of costs Average monthly costs

1

3

–

–

–

–

–

19

19

2

3

3

–

–

–

–

38

19

3

4

5

0

–

–

–

48

16

4

3,4

4,5

5,0

0,3

–

–

67

16.75

5

5

5

0

5

0

–

79

15.8

6

4

5

0

4

5

0

96

16

For the case n = 1 output (3 units) for N = 1 is taken from the first row of Table 3 with n equal to 1. The case n = 1 output (3 units) for N = 2 is taken from the same row for n = 2, etc. The minimum total cost for N = 6 is (21 + 1) + (23 + 3) + (0 + 0) + (21 + 1) + (23 + 3) + (0 + 0) = 96; this sum is written in the cell of Table 3 corresponding to f6 (0). An analysis of the optimal options for the production program, shown in Table 5, indicates that in case n = 1 product output depends on the length of the planning period. As the number of months N increases from 1 to 5, the optimal n = 1 output increases. However, with N = 6, production for n = 1 instance should be only 4 units (with N = 5, this value was equal to 5 units). Thus, the lengthening of the planning period can cause both an increase and a decrease in case n = 1 output, and for N = 4 there are two alternative optimal programs. The recursive computation can also be done with a network representation of the dynamic planning problem.

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4 Conclusion In this paper the problem of constructing computation graph for the planning production problem instance with temporal intervals is discussed. It is shown that a common feature of all dynamic programming models is the reduction of the decision-making problem to obtaining recursive relations. One of the main subjects of consideration in the work is the analysis of the stability of solutions depending on temporal intervals of planning. For example, an increase in the duration of the planning horizon can affect the correctness and optimality of the current choice. Along with this issue, the impact of initial conditions and constraints are also explored in this paper. Acknowledgments. The research was funded by the Russian Science Foundation Project No. 23–21-00206, https://rscf.ru/en/project/23-21-00206/ implemented by the Southern Federal University.

References 1. Herroelen, W., De Reyck, B., Demeulemeester, E.: Resource-constrained project scheduling: Notation, classification, models and methods. Eur. J. Oper. Res. 128(3), 221–230 (2000) 2. Brucker, P., Knust, S.: Complex Scheduling. Springer-Verlag, Berlin Heidelberg (2012) 3. Tellache, N.E.H., Boudhar, M.: Flow shop scheduling problem with conflict graphs. Ann. Oper. Res. 261(1–2), 339–363 (2017). https://doi.org/10.1007/s10479-017-2560-x 4. Chen, Y., Goebel, R., Lin, G., Su, B., Zhang, A.: Open-shop scheduling for unit jobs under precedence constraints. Theor. Comput. Sci. 803, 144–151 (2020) 5. Gao, Y., Xie, Z., Liu, X., Zhou, W., Yu, X.: Integrated scheduling algorithm based on the priority constraint table for complex products with tree structure. Adv. Mech. Eng. 12(12), 1–15 (2020) 6. Nilim, A., El Ghaoui, L.: Robust control of Markov decision processes with uncertain transition matrices. Oper. Res. 53(5), 780–798 (2005) 7. Baldacci, R., Bartolini, E., Mingozzi, A.: An exact algorithm for the pickup and delivery problem with time windows. Oper. Res. 59(2), 414–426 (2011) 8. Srour, F.J., Agatz, N., Oppen, J.: Strategies for handling temporal uncertainty in pickup and delivery problems with time windows. Transp. Sci. 52(1), 3–19 (2016) 9. Knyazeva, M., Bozhenyuk, A., Bozheniuk, V.: Unfolding fuzzy temporal computational graph for project scheduling problem. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds.) INFUS 2021. LNNS, vol. 307, pp. 615–622. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-85626-7_72 10. Wagner, H.M.: Principles of Operations Research with Applications to Managerial Decisions. Prentice Hall, 2nd edition (1975)

Strong Connectivity Definition of Periodic Fuzzy Graph Alexander Bozhenyuk1(B)

, Margarita Knyazeva1 and Igor Rozenberg2

, Olesiya Kosenko1

,

1 Southern Federal University, Nekrasovsky Street 44, 347922 Taganrog, Russia

[email protected] 2 Russian University of Transport (MIIT), Obraztsova Street, 9/9, 127994 Moscow, Russia

Abstract. In this paper, the new periodic fuzzy graph model is considered. A periodic fuzzy graph is a fuzzy temporal graph whose fuzziness changes in discrete time (horizon), and discrete time itself has the property of cyclicity. Periodic fuzzy graphs can be used for modeling processes for complex systems where the elements are connected by fuzzy relationships that change in discrete time and have the property of periodicity. The paper introduces the concept of a fuzzy reachable set between the vertices of a periodic fuzzy graph. The fuzzy reachability set allows you to determine both the degree of reachability of vertices in the shortest time, and the reachability time with the greatest degree. A method for finding a fuzzy reachable set is proposed. The concept of strong connectivity of a periodic fuzzy graph is introduced. The relationship between the strong connectivity of a periodic fuzzy graph and the strong connectivity of a crisp graph is given. Keywords: Fuzzy Graph · Periodic Fuzzy Graph · Invariant · Transitive Closure · Strong Connectivity

1 Introduction Traditionally, graph theory is used to represent relationships between elements of complex structures of various nature [1, 2]. Traditional crisp graphs may not be effective tool and applicable for the description and modeling when the interconnection between the elements of a certain structure change over time. In this case, the concept of a temporal graph, that is, a graph of a model where the relations between elements (graph vertices) change in time is actual. Temporal graphs (dynamic, evolving, time-varying graphs) can be described as graphs that change with time [3–5]. The concept and idea to use temporal graphs in the literature is interpreted in a fairly wide range [6–8]. The case when in the temporal graph, the connections between the vertices are fuzzy, the notion of a fuzzy temporal graph is introduced [9, 10]. In this paper, new graph models are considered, namely, periodic fuzzy graphs. A periodic fuzzy graph (PFG) is a fuzzy temporal graph whose fuzziness changes in discrete time (horizon), and discrete time itself has the property of cyclicity. Models based on PFG will adequately reflect the types of uncertainty, reflect the specifics of relations between © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 168–174, 2023. https://doi.org/10.1007/978-3-031-39774-5_21

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the simulated objects, optimization constraints, solve cyclic flow transport problems, network planning and control problems [11–14]. The paper introduces the concept of a fuzzy reachable set between vertices of PFG. This set allows you to determine both the degree of reachability of the vertices in the shortest time, and the time of reachability with the greatest degree. A method for finding it is proposed. The notion of a strong connection of PFG is considered. This paper is structured as follows. The second section contains the basic concepts of fuzzy temporal and periodic graphs. Then we state the problem of finding a fuzzy reachable set between vertices in PFG. The algorithm for finding a fuzzy reachable set is introduced and described in Sect. 3. Section 4 introduces the concept of strong PFG connectivity. The relationship between PFG strong connectivity and crisp graph strong connectivity is considered. Finally, the last section contains a conclusion.

2 Basic Concepts and Definitions ˜ = (V , U˜ t , T ). Here V is the By a fuzzy temporal graph [9] we mean a triple of sets G set of graph vertices with n vertices; T = {1,2,…,N} is the set of natural numbers that ˜ define discrete time   (horizon); Ut = {} is fuzzy set of edges, xi , xj ∈ V , μt xi , xj ∈ [0, 1] - value of the membership function μt for the edge (xi , xj ) at times t ∈ T. ˜ = (X , U˜ t , T ), in the form of an Example 1. We define a fuzzy temporal graph G oriented graph shown in Fig. 1, the arcs of which have the certain membership function μt are indicated at time T:

Fig. 1. Fuzzy temporal graph with T = {1,2,3}.

Here the set of vertices is V = {x 1 , x 2 , x 3 , x 4 , x 5 }, time T = {1, 2, 3}, N = 3, n = 5. Definition 1. A fuzzy temporal graph whose has the property of cyclicity is called a periodic fuzzy graph (PFG). In other words, in PFG, after time «N» time«1» again occurs.

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Let’s call the transition from one moment of time (t) to the next moment (t + 1) one tact.   Definition 2. Vertex xj is called fuzzy adjacent xi if (∃t ∈ T )μt xi , xj >0 is true. ˜ is a sequence of fuzzy arcs going from ˜ i , xj ) in the PFG G Definition 3. A fuzzy path L(x vertex xi to vertex xj , where the end vertex of any arc (except the last one) is the initial   vertex of the next arc: L˜ xi , xj = ,…,, for which the conditions μt1 (xi , x1 )>0, μt2 (x2 , x3 )>0, …, μtk (xk−1 , xk )>0. Moreover, in contrast to the fuzzy (non-periodic) temporal graph, this sequence does not depend on time points t 1 , t 2 ,…, t k ∈ T. ˜ if there is a path The vertex xj will be considered fuzzy reachable from xi in PFG G ˜ i , xj ). L(x Definition 4. Reachability Time τ (xi , xj ) Is the number of tact required to reach vertex ˜ i , xj ). xj from vertex xi along the path L(x ˜ s , xf ) from vertex xs to vertex xf , Example 2. Let T = {1, 2, 3}. Consider the path L(x shown in Fig. 2:

˜ s , xf ). Fig. 2. Fuzzy path L(x

From the vertex xs the vertex x1 is reached at the moment t = 1 (τ = 1). Then time t = 2 (τ = 2) is skipped, and the vertex x2 is reached at time t = 3 (τ = 3). Further, the time t = 1 (τ = 4) is skipped, and the vertex x3 is reached at the moment t = 2 of the second cycle (τ = 5). Time t = 3 of the second cycle is skipped (τ = 6) and the vertex  xf is reached at time t = 1 of the third cycle (τ = 7). Thus, the reachability time τL xs , xf = 7.   ˜ the following To find the reachability time τL xs , xf from xs to xf by the path L, algorithm is proposed: ˜ s , xf ) is sequentially assigned two numbers (l1 , l2 ). – Each vertex of the fuzzy path L(x Here l1 is the number of reachability cycles from the initial vertex xs ; l2 is the reachability time in the cycle. The assignment goes like this: – the initial vertex xs is assigned a pair (0,0); ˜ s , xf ) is – if a vertex xi is assigned a pair (i1 , i2 ), then the next vertex xj in the path L(x assigned a pair (j1 , j2 ). Here j2 is the reachability time of the vertex xj , and j1 = i1 , if j2 > i2 (the number of cycles does not change), otherwise - j1 = i1 + 1 (the number of cycles increases by 1).

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  f f – If in this way the pair l1 , l2 is assigned to the vertex xf , then the reachability time is determined according to (1):   f f τL xs , xf = l1 N + l2 .

(1)

Example (continued). In Fig. 2. Assignment sequence will be as follows: the pair (0, 0) is assigned to the vertex xs , vertex x1 is assigned a pair (0, 1), vertex x2 is assigned a pair (0, 3), vertex  x3 is  assigned the pair (1, 2), and the vertex xf is assigned the pair (2, 1). Hence: τL xs , xf = 2 × 3 + 1 = 7. Definition 5. The reachability degree μL (xi , xj ) From the vertex xi to the vertex xj by ˜ i , xj ) is the value of the smallest membership function of the arc included the path L(x in this path:     μL xi , xj = μt1 (xi , x1 )&μt2 (x1 , x2 )& . . . &μtj xj−1 , xj . ˜ Example  3. The degree of reachability of the pat L, shown in Fig. 2 is equal to: μL xs , xf = 0.8 & 0.4 & 0.6 & 0.9 = 0.4.

3 Fuzzy Reachable Set in PFG In the example above, there was only one path from vertex xs to vertex xf . However, between two arbitrary vertices, there can be several different paths that differ in different edges, as well as paths passing along the same edges, but at different times t. In this case, the following optimization problems arise: – finding the path with the least time and the degree of reachability at this time; – finding the reachability time of the path with the highest degree of reachability. Definition 6. Let μτ be the greatest reachability degree of xf from xs in the number of tact τ . A fuzzy reachable set is a set:   τ˜ xs , xf = {}, μτ [0, 1], τ 1, 2, . . . , N (n − 1). If the fuzzy reachable set is found, then the above problems are solved automatically. To find the fuzzy reachable set, it is proposed to use an algorithm based on the idea of a wave algorithm designed to find the shortest paths in crisp graphs [15]: 1°. To the vertex xs we assign a fuzzy reachable time set in the form of a pair τ˜s = {}. Here 1 is the reachability degree of xs itself in cyclic time (0,0). We assign τ˜j = {}. In other words, the vertex x5 is reachable from the vertex x1 in 2 takt (shortest time) with degree 0.8 and reachable with the largest degree 0.9 in 5 takt.

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4 Strong Connectivity of PFG By analogy with strong connectivity of crisp graph [1] consider the concept of strong connectivity PFG. Definition 6. PFG is strongly connected if there is some path between any two vertices. ˜ = (V , U˜ t , T ) be PFG. Let’s construct a crisp graph G = (V , U ) as follows: Let G if for some time t the degree μt (xi , xj ) = 0, then the graph G has a directed edge (xi , xj ) ∈ U . The following property is true: ˜ to be strongly connected, it is necessary and sufficient that crisp Property. For PFG G graph G is also strongly connected. ˜ shown in Fig. 1, the crisp graph G = (V , U ) is shown in Fig. 3: Example 5. For PFG G

Fig. 3. Crisp graph G.

To find the strong connectivity of crisp graph G, we use the Malgrange method [1]. To do this, we take an arbitrary vertex x i ∈ V and find for it a transitive closure Γ (xi ) − and an inverse transitive closure Γ (xi ). If their intersection coincides with the vertex set V , then crisp graph G is strongly connected. − For crisp graph G shown in Fig. 3 we have: Γ (x1 ) ∩ Γ (x1 ) = V . Whence it follows ˜ is also strongly connected. that PFG G







5 Conclusions The paper introduced the concept of a periodic fuzzy graph, which can be effectively used for modeling of complex systems where the objects have fuzzy relationships that may change over discrete time periods and have the property of cyclicity. The notion of a fuzzy reachable set between vertices and the notion of a strong connection PFG

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were considered. A algorithm for determining a fuzzy reachable set was proposed. The relationship between PFG strong connectivity and crisp graph strong connectivity was considered. In the future, it is planned to investigate issues related to determining the degree of strong connectivity of PFG and finding the reachability time between any two vertices, at which PFG will have a given degree of reachability. Acknowledgments. The research was funded by the Russian Science Foundation Project No. 23–21-00206, https://rscf.ru/en/project/23-21-00206/ implemented by the Southern Federal University.

References 1. Kaufmann, A.: Introduction a la theorie des sous-ensemles flous. Masson, Paris (1977) 2. Christofides, N.: Graph theory. An algorithmic approach. Academic press, London (1976) 3. Ferreira, A.: Building a reference combinatorial model for manets. Netw. IEEE 18(5), 24–29 (2004) 4. Flocchini, P., Mans, B., Santoro, N.: On the exploration of time-varying networks. Theoret. Comput. Sci. 469, 53–68 (2013) 5. Othon, M.: An introduction to temporal graphs: an algorithmic perspective. Internet Math. 12, 239–280 (2016) 6. Kostakos, V.: Temporal graphs. In: Proceedings of Physica A: Statistical Mechanics and its Applications, vol. 6, no. 388, pp. 1007–1023. Elsevier (2008) 7. Bramsen, P.: Doing Time: Inducing Temporal Graphs. Technical report, Massachusetts Institute of Technology (2006) 8. Dittmann, F., Bobda, C.: Temporal graph placement on mesh-based coarse grain reconfigurable systems using the spectral method. from specification to embedded systems application, vol. 184, pp. 301–310. Springer (2005) 9. Bozhenyuk, A., Belyakov, S., Knyazeva, M.: Modeling objects and processes in GIS by fuzzy temporal graphs. Stud. Fuzziness Soft Comput. 393, 277–286 (2020) 10. Knyazeva, M., Bozhenyuk, A., Kaymak, U.: Fuzzy temporal graphs and sequence modelling in scheduling problem. Commun. Comput. Inf. Sci. 1239, 539–550 (2020) 11. He, Z., Ma, Z., Li, Z., Giua, A.: Parametric transformation of timed weighted marked graphs: applications in optimal resource allocation. IEEE/CAA J. Autom. Sin. 8(1), 179–188 (2021) 12. He, Z., Li, Z.W., Giua, A.: Cycle time optimization of deterministic timed weighted marked graphs by transformation. IEEE Trans. Control Syst. Technol. 25(4), 1318–1330 (2017) 13. Tellache, N.E.H., Boudhar, M.: Flow shop scheduling problem with conflict graphs. Ann. Oper. Res. 261(1–2), 339–363 (2017). https://doi.org/10.1007/s10479-017-2560-x 14. Chen, Y., Goebel, R., Lin, G., Su, B., Zhang, A.: Open-shop scheduling for unit jobs under precedence constraints. Theor. Comput. Sci. 803, 144–151 (2020) 15. Lee, C.Y.: An algorithm for path connections and its applications. IRE Trans. Electr. Comput. EC-10(2), 364—365 (1961)

Intuitionistic Fuzzy Sets

An Automatic Rating System Based on Review Sentiments and Intuitionistic Fuzzy Sets Mustafa Ünver(B) Department of Industrial Engineering, Gebze Technical University, 41400 Gebze/Kocaeli, Turkey [email protected]

Abstract. Natural language processing (NLP) and text mining methodologies are beneficial tools for utilizing from large scale service review datasets. Sentiment analysis (SA) is among them and it is used to label texts according to their polarity degrees. In this paper, a novel review scoring approach is introduced based on sentiment polarity scores and intuitionistic fuzzy sets. The approach is then implemented on a review dataset which is retrieved from several websites for a certain hotel service. The proposed approach presents promising results for more reliable product/service evaluation relatively to crisp rating scores. Keywords: NLP · Text Mining · Sentiment Analysis · Intuitionistic Fuzzy Sets

1 Introduction The Natural language processing (NLP) is a collection of computational techniques that automatically analyze the human language in order to gain beneficial insights inside the textual datasets. The textual preprocessing, textual representations, classification or clustering the textual data, sentiment analysis, topic modelling etc. are included in NLP techniques. The NLP methodologies have been frequently used by the researchers in marketing, social science, economics etc. [1]. The sentiment analysis (SA) is a computational study that aims to find the sentiments of the people on the texts as their opinion, attitude or judgments according to polarity of the text and conversion of the polarity degrees into “positive”, “negative” or “neutral” sentiments. SA can be implemented on the text in aspect level, sentence level and document levels. There are both machine learning based SA approaches and lexicon based SA approaches in the literature. All the SA approaches are elaborated in detail in [2]. The tremendous sales volumes in recent years in e-commerce websites (for both products and services) increase the importance of online marketing. By that context online shoppers always consider other customers review about the service or products before they make the shopping decisions. But e-commerce web sites carry immense amount of information in post-sale reviews and it makes very difficult to consider every review for a shopping/buying decision of a potential customer [3].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 177–184, 2023. https://doi.org/10.1007/978-3-031-39774-5_22

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NLP tools are very efficient for extracting valuable information too fast to compare with human cognition, and able to present useful insights for potential customers and product service developers in seconds. Intuitionistic fuzzy set (IFS) theory is developed in order to present a soft cognition for the belonging of an item into a set. The belonging is defined as in three aspects: Membership degree (μ), Non-Membership degree (V) and a hesitancy margin (π ). The sum of all these three components is equal to 1 [4]. The main objective of that study is that developing an automatic product/service star rating system based on only customer reviews. By means of new scoring system, more accurate and coherent ratings are expected for both potential customers and product/service system developers. Thus, a novel automatic scoring approach is introduced based on sentence and document based SA which is integrated with an IFS framework. Then the proposed approach is implemented on a review dataset which is retrieved from several websites such as tripadvisor.com, expedia.com, hotels.com etc. and includes post-sale reviews and customer star ratings. The next parts of the study are composed of the following topics: Sect. 2 contains literature survey, Sect. 3 introduces proposed scoring approach and its implementation on a service system review dataset and Sect. 4 points out conclusion and future research issues.

2 Literature Review The related literature for service/product scoring by using NLP and SA is surveyed. The related studies comprise different approaches for scoring for social media, hotels, movies, politics or e-commerce reviews etc. In most of the studies which are surveyed, star rating values are predicted by using a classification techniques or statistical analysis. Most of the predictions are crisp classification approaches. In [5], fuzzification of word polarity sentiment scores is analyzed by using two different lexicons and fuzzy cardinality rules. Another study of sentiment analysis on reviews can be found as [6]. In that work, sentiment analysis tools of some web-services are used on several datasets and classification accuracy of sentiments are given in order to present their efficiency on SA. Rating values are used to present classification accuracy. In [7], star ratings and SA based ratings are statistically analyzed on doctor reviews, hotel reviews and amazon product reviews. A similar work can be found as [8]. In that study, SA based scores and customer ratings are used for several classification methods and the results are statistically compared. In [9], SA is conducted on mobile application reviews after topic modeling is applied on these datasets as a feature based SA analysis. SA results are found as much more accurate in capturing customer sentiments rather than star ratings [10] can be given another example of aspect based SA on restaurant reviews. A predictive approach is used for star ratings by using SA in [11] for Yelp review dataset by means of a logit model. A linear regression model is built up for the prediction of star ratings from reviews sentiments polarity for mobile apps in [12]. Another interesting implementation of sentiment analysis on product ratings is done in [13]. In this work, an aspect level of sentiment analysis is considered for the rankings of alternative cell phones and their features to support consumers’ purchase decisions using the intuitionistic fuzzy

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scoring function and the “vertical projection distance” method. As can be deducted from the papers surveyed, SA output can be used from the IFS viewpoint, in order to create an automatic star rating system.

3 The Proposed Scoring Approach and the Application The proposed approach for automatic product/service star rating system based on SA is illustrated in Fig. 1. It includes four stages: data preprocessing, sentiment analysis, intuitionistic fuzzy set fitting and rating score computation.

Pre-Processing

Senment Analysis

• Data Cleaning • Word / Sentence Tokenizaon

• Vader Senment Analyzer • Senment Intensity Analyzer

IFS Fing

Rang Score Computaon

• Posive Senments • Negave Senments • Neutral Senments

• Individual Review Rang • Final Rang Score

Fig. 1. The proposed approach for automatic service rating

In data preparation stage, the review dataset is cleaned and some NLP techniques are applied on it to make it ready for the SA stage. In SA stage, “Sentiment Intensity Analyzer” of “vader sentiment” - a python library, is used to extract the polarity degrees of each sentence in reviews and total polarity of the review text. The vader sentiment analyzers returns polarity degrees in [−1, 1] range as float values. By using total polarity score of the reviews, star values are reassigned. [−1, 1] range is divided into five equal parts. [−1, −0.6) is considered as one star, [−0.6, −0.2) is considered as two stars, [−0.2, 0.2) is considered as three stars, [0.2, 0.6) is considered as four stars, [0.6, 1] is considered as five stars. After extracting consumers’ sentiments polarity on his/her review, each sentence is classified as “positive”, “negative” or “neutral”. The classification of sentences is done by using the logic as in Eq. 1. ⎧ ⎫ polarity degree > 0.1 ⎨ Positive ⎬ The Sentiment = Neutral −0.1 < polarity degree < 0.1 (1) ⎩ ⎭ Negative polarity degree < −0.1 IFS is introduced in [4]. An item may belong to a set with three components: Membership Degree (μ), Non-Membership Degree (V) and a Hesitancy Margin(π). The relation between these components are given in Eq. 2. μ + V + π = 1 where 0 ≤ μ ≤ 1; 0 ≤ V ≤ 1; 0 ≤ π ≤ 1

(2)

After sentiment determinations, number of sentiments in each review is used as IFS viewpoint. For positive review polarity scores, rate of positive sentences is considered

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as membership degree of its total sentiment of the review. Similarly, its negativity rate is considered as non-membership degree and neutrality sentence rate is considered as hesitancy degree. For negative review polarity scores, rate of negative sentences is considered as membership degree of its total sentiment of the review, its positivity rate is considered as non-membership degree and neutrality sentence rate is considered as hesitancy margin. As for the neutral polarity reviews; IFS components are computed as in Eq. 3. ⎫ ⎧ ⎧ ⎫ ⎪ Number of neutral sentences ⎪ ⎪ ⎬ ⎨μ⎬ ⎪ ⎨ Total number of sentences |Nr. of Pos. sent. − Nr. of Neg. sent.| (3) IFS V = ⎪ ⎩ ⎭ ⎪ Total number of sentences ⎪ ⎪ ⎭ ⎩ π 1−μ−V For instance, Review_1332 is “The front desk service was very good. The location and proximity to tourist attractions was great. The beds were very uncomfortable. We did have a problem with our room being cleaned one day. It affected our ability to come back to the room and rest prior to dinner. However, when this was brought up to the front desk they immediately addressed… More”. That review has three positives, three negatives and one neutral sentiments in total seven sentences. Total polarity of this review is found as 0.7351 by SA and considered as 5 stars, and the real star rating value of this review is three by the customer. Since it has a positive overall sentiment, 3/7 is considered as membership degree, 3/7 is considered as non-membership degree and 1/7 is considered as hesitancy margin. As a result, review score of Review_1332 is going to be an intuitionistic fuzzy number as in Eq. 4. 

3 3 1 RatingIF (1331) = {Polarity Rating, (μ, V , π )} = 5, , , (4) 7 7 7 After defining a new star value from IFS viewpoint of each review, n intuitionistic fuzzy numbers are obtained. Than a fuzzy aggregating operator is used to find the final score of the service. Generalized intuitionistic fuzzy weighted average (GIFWA) is used to determine the overall star rating of the hotel service. The GIFWA is introduced in [14]. By using GIFWA, n intuitionistic fuzzy numbers can be aggregated. In Eq. 5 and Eq. 6; sum operator (⊕) in IFS [4] and GIFWA operator formulas [14] are given respectively. In Eq. 6.; α represents a intuitionistic fuzzy value, w represents weight of each intuitionistic fuzzy value, λ represents cut value for the IFS components. α1 ⊕ α1 = (μ1 + μ1 − μ1 μ2 , V1 V2 )

(5) 1

GIFWA(α1, α2 , . . . , αn ) = (w1 α1λ ⊕ w2 α2λ ⊕ . . . ⊕ wn αnλ ) λ

(6)

3.1 The Review Dataset The review dataset is crawled from several websites such as tripadvisor.com, expedia.com, hotels.com etc. and includes post-sale reviews and customer star ratings for

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a specific hotel. The dataset includes 10.000 review text from different customer in English. The mean star value is 3.98 and standard deviation is 1.18 for star ratings. Thus the service is approximately scored as four stars overall. There are 596 “one star”, 660 “two stars”, 1459 “three stars”, 2900 “four stars” and 4385 “five stars” ratings along with their review text. Some descriptive statistics of the dataset are given in Fig. 2, Fig. 3, and Fig. 4 and a word cloud is provided in Fig. 5.

Fig. 2. Words/Review Statistics

Fig. 3. Sentences/Review Statistics

3.2 The Implementation of the Proposed Approach and the Output All the implantations are done in Python 3.9.by using “VaderSentiment”, “nltk” and “matplotlib” libraries. A sample view of the output is given in Table 1. The output includes individual evaluations for all reviews in the dataset.

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Fig. 4. Words/Sentences Statistics

Fig. 5. The word cloud of the dataset. The most frequent words are illustrated in bigger font sizes.

In GIFWA operator, all the weights are considered as equal and chosen as 1/total number of reviews = 1/10000 and λ is set as 1. By applying GIFWA, overall score is found as 3.2 stars with 0.91 membership degree, 0.06 non-membership degree and 0.03 hesitancy margin. The proposed approach automatically presented star ratings by using customer sentiments. When the output is compared with user defined star ratings, the proposed approach presented more realistic star rating then the user defined rating average. The proposed approach determined the service quality as around three star, as moderate quality, with reliable intuitionistic fuzzy degrees. However, in user defined star evaluation system, the same service had been rated as 4 stars as a good overall quality and presents no reliability level.

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Table 1. A sample from the output of the proposed approach Review ID

Overall Polarity

# of Positive Sent

# of Neutral Sent

# of Negative Sent

User’s Rating

Polarity Rating

IFS Components

4

0

0

1

2

2

3

1331

0.7351

3

1

3

3

5

( 13 , 13 , 13 ) ( 37 , 37 , 17 )

9097

–0.93

1

2

8

2

1

8 , 1 , 2 ) ( 11 11 11

4 Conclusion and Future Research In this study, a novel review scoring approach is introduced based on sentiment polarity scores and intuitionistic fuzzy sets. The approach is then tested on a customer review dataset which is crawled from several websites for a certain hotel service. The proposed approach automatizes the service rating score once a customer declares his/her comments for the service. The proposed approach presents promising results for more reliable product/service evaluation relatively to crisp rating scores, defined by the service users. As a future research challenge, enhancing the proposed approach is planned so as to make it capable of producing scores for all aspect of the service by utilizing topic modelling approaches. Another interesting future research issue can be the implantation of SA within a spherical fuzzy set or neutrosophic set framework. Acknowledgement. The author acknowledges for the support of Scientific and Technological Research Council of Turkey (TÜB˙ITAK) on Program 2224-B.

References 1. Kang, Y., Cai,Z., Tan, C., Huang, Q., Liu, H.: Natural language processing (NLP) in management research: a literature review, J. Manag. Anal. 7(2), 139–172 (2020) 2. Medhat, W., Hassan, A., Korashy, H.: Sentiment analysis algorithms and applications: a survey. Ain Shams Eng. J. 5(4), 1093–1113 (2014) 3. Zhang, Z., Varadarajan B.: Utility scoring of product reviews. In: Proceedings of the 15th ACM international conference on Information and knowledge management (CIKM ‘06). Association for Computing Machinery, New-York, USA, pp. 51–57 (2006) 4. Atanassov, K.: Intuitionistic fuzzy sets. Int. J. Bioautom. 20, 1–6 (2016) 5. Vashishtha, S., Susan, S.: Fuzzy interpretation of word polarity scores for unsupervised sentiment analysis. In: 11th International Conference on Computing, Communication and Networking Technologies (ICCCNT), Kharagpur, India, pp. 1–6 (2020) 6. Serrano-Guerrero, J., Olivas, J.A., Romero, F.P., Herrera-Viedma, E.: Sentiment analysis: a review and comparative analysis of web services. Inf. Sci. 311, 18–38 (2015) 7. Lak, P., Turetken, O.: Star ratings versus sentiment analysis − a comparison of explicit and implicit measures of opinions. In: 47th Hawaii International Conference on System Sciences, Waikoloa, HI, USA, pp. 796–805 (2014)

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8. Hogenboom, A., Boon, F., Frasincar, F.A.: Statistical approach to star rating classification of sentiment. In: Management Intelligent Systems. Advances in Intelligent Systems and Computing, vol. 171, Springer, Berlin, Heidelberg, (2012). https://doi.org/10.1007/978-3-64230864-2_24 9. Luiz,W., et al.: A feature-oriented sentiment rating for mobile app reviews. In: Proceedings of the World Wide Web Conference (WWW 2018). International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, CHE, pp. 1909–1918 (2018) 10. Mahadevan, A., Arock, M.: Integrated topic modeling and sentiment analysis: a review rating prediction approach for recommender systems. Turk. J. Electr. Eng. Comput. Sci. 28(1) (2020) 11. Qiu, J., Liu, C., Li, Y., Lin, Z.: Leveraging sentiment analysis at the aspects level to predict ratings of reviews. Inf. Sci. 451–452, 295–309 (2018) 12. Monett, D., Stolte, H.: Predicting star ratings based on annotated reviews of mobile apps. In: Federated Conference on Computer Science and Information Systems (FedCSIS), Gdansk, Poland, pp. 421–428 (2016) 13. Yang, Z., Xiong, G., Cao, Z., Li, Y., Huang, L.: A decision method for online purchases considering dynamic information preference based on sentiment orientation classification and discrete DIFWA operators. IEEE Access 7, 77008–77026 (2019) 14. Zhao, H., Xu, Z., Ni, M., Liu, S.: Generalized aggregation operators for intuitionistic fuzzy sets. Int. J. Intell. Syst. 25, 1–30 (2010)

Circular Intuitionistic Fuzzy Analysis of Variance on the Factor Season of Apple Sales Velichka Traneva(B)

and Stoyan Tranev

“Prof. Asen Zlatarov” University, “Prof. Yakimov” Blvd, Bourgas 8000, Bulgaria [email protected], [email protected] http://www.btu.bg Abstract. The circular intuitionistic fuzzy sets developed by Atanassov are a more versatile tool for defining economic fuzziness than the intuitionistic fuzzy ones. ANOVA is a technique for determining how a certain factor affects the event or process under research. This study extends the intuitionistic fuzzy analysis of variance (IFANOVA) to a circular IFANOVA (C-IFANOVA), where all of its parameters are circular intuitionistic fuzzy numbers. Pessimistic, optimistic, and averaging possibilities are presented to the decision-maker for a final decision. To check the created approach of C-IFANOVA and demonstrate its efficacy, a comparison analysis is conducted using a traditional variation analysis on the sales of the Apple firm for 2019-2022, broken down by the factor “season”. Keywords: C-IFANOVA · Circular Intuitionistic Fuzzy Sets Matrix · Variation Analysis

1

· Index

Introduction

Fisher invented the ANOVA method for identifying statistical differences between group means [11]. Due to the current uncertain economic climate, some of the data used in this research may be ambiguous. The classical apparatus of ANOVA cannot analyze unclear numbers. This necessitates updating ANOVA to be applied to this type of data. In the following, we give a brief overview of various scientific works on fuzzy (FANOVA) and intuitionistic fuzzy ANOVA (IFANOVA). Zadeh suggests fuzzy sets in [33]. Based on fuzzy random variables [25], FANOVA is implemented in [19]. In [17], a method for ANOVA with ambiguous data is shown in which the moment correction is used to eliminate the ambiguity’s influence on the sum of squares. According to [12], a bootstrap proposal to FANOVA has been made. According to [9], FANOVA is created using variance confidence intervals. In [15,23], a one-way triangular FANOVA is provided. Using Zadeh’s extension concept, a one-way FANOVA is suggested in [20]. Work on Sects. 2 and 3 is supported by the A. Zlatarov University through project Ref. No. NIX-482/2023 “Application of Intuitionistic Fuzzy Logic in Decision Making”. Work on Sects. 1, 4, and 5 is supported by the project Ref. No. NIX-486/2023 “Modeling Management Decisions with New Analysis Tools in a Fuzzy Environment”. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 185–193, 2023. https://doi.org/10.1007/978-3-031-39774-5_23

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In [13], a least squares-based FANOVA strategy is devised. In a publication [22], a novel method for FANOVA is illustrated over typical fuzzy numbers, extending the traditional ANOVA. The paper [16] examines the IFANOVA approach for determining student homogeneity. In the study [1], the one-way IFANOVA method is created by converting intuitionistic fuzzy data into precise ones. To determine the influence of the factor over IF data that appeared in an uncertain environment, an index-matrix interpretation of the conventional ANOVA in the form of the IFANOVA is developed in [30,31] by applying the index-matrix view (IM, [3]) and IF logic [4]. In the study [29], one-way IFANOVA is carried out using the software tool that was built. The circular intuitionistic fuzzy sets (CIFSs), one of the most recent extensions of intuitionistic fuzzy sets (IFSs, [2]), provide a more efficient toolkit for ANOVA in an uncertain environment. By expanding the intuitionistic fuzzy ANOVA (IFANOVA, [30]), an index-matrix approach to the ANOVA is proposed in this study, in which all of its parameters are circular intuitionistic fuzzy (C-IF) numbers. Let’s use C-IFANOVA to indicate this IFANOVA variant. The decision-maker is presented with three options for a final decision: pessimistic, optimistic, and averaged. ANOVA and C-IFANOVA are used to analyze the sales of the Apple corporation for the years 2019 through 2022 to test the created approach and demonstrate its efficacy. The work’s scientific contribution stems from the specified C-IFANOVA and its application of the stated C-IFANOVA method to Apple’s sales by a season factor. The remaining article’s sections are as follows: Basic explanations from the theories of IMs and IF logic are provided in Sect. 2. The ANOVA is described in Sect. 3 along with an analysis of how it was applied to Apple sales based on a season factor. The C-IFANOVA is introduced in Sect. 4. The new method is used to analyze the circular IF data on Apple sales to demonstrate statistically how sales are affected by the year season. The conclusion and potential directions for future research are suggested in Sect. 5.

2 2.1

Fundamental Definitions of the IMs and C-IF Logic Concepts Operations on Circular Intuitionistic Fuzzy Pairs (C-IFPs) and Relations

C-IFSs are established in 2020 as an extension of the IFSs and have a different from IFS by including a circle with radius r of the number consisting of membership degree and non-membership degree [6]. The C-IFP has the following form: a(p), b(p); r = μ(p), ν(p); r, where a(p), b(p) ∈ [0, 1] and a(p) + b(p) ≤ 1 are used to√ evaluate a proposition p [6,8]. The circle’s radius around a(p), b(p) is r ∈ [0, 2]. Two C-IFPs x = a, b; r1  and y = c, d; r2 , shall be used. Let us define an operation ∗ ∈ {min, max}. The further operations are suggested in [6,8,32]:

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x ∨1∗ y = max(a, c), min(b, d); ∗(r1, r2 ); x ∧1∗ y = min(a, c), max(b, d); ∗(r1, r2 ); x ∧2∗ y = x + y = a + c − a.c, b.d; ∗(r1, r2 ); b+d x ∨2∗ y = x.y = a.c, b + d − b.d; ∗(r1, r2 ); x@∗ y =  a+c 2 , 2 ; ∗(r1, r2 ); x −∗ y = max(0, a − c), min(1, b + d, 1 − a + c); ∗(r1, r2 ); ⎧ min(1, a/c), min(max(0, 1 − a/c), ⎪ ⎨ ⎪ x :∗ y = max(0, (b − d)/(1 − d))); ∗(r1, r2 )if c  0 &d  1 ⎪ ⎪ 0, 1; ∗(r1, r2 ) otherwise ⎩ The definition of Szmidt and Kacprzyk’s IF Hamming distance [26], is expanded for the C-IFSs in [8]. We have proposed the following relation for comparison of C-IFPs [32], extending the definition for IFPs measure R = √ 0.5(1 + πx )distance(1, 0; 2, x that may be used for ranking the alternatives. c c ≤ Rcir x ≥R cir c y iff Rcir a,b ;r1  c,d ;r2 

(1)

√ where the distance from the ideal positive alternative 1, 0; 2 to x is √   | 2 − r| cir c R a,b;r  = 0.25(2 − a − b) +1−a . √ 2 2.2

Description, Usage, and Relations to Circular if IMs (C-IFIMs)

In 1987, according to [3], the theory of index matrices (IMs) was developed. Assume that the set of induces I is fixed. The definition of two-dimensional C-IFIM A = [K, L, {μki ,l j , νki ,l j ; rki ,l j }] with index sets K and L (K, L ⊂ I) is the following, which is similar to that of IFIM [5]: l1 ... lj k1 μk1,l1 , νk1,l1 ; rk1,l1  . . . μk1,l j , νk1,l j ; rk1,l j  A= . .. .. .. .. . . . k m μkm,l1 , νkm,l1 ; rkm,l1  . . . μkm,l j , νkm,l j ; rkm,l j 

... ln . . . μk1,ln , νk1,ln ; rk1,ln  .. .. . .

. . . μkm,ln , νkm,ln ; rkm,ln 

The definition of a 3-D C-IFIM extends the 2-D C-IFIM definition and is comparable to the one in [5]. There are various operations, relations, and operators over C-IFIMs A = [K, L, {μki ,l j , νki ,l j ; rki ,l j }] and B = [P, Q, {ρ pr ,qs , σpr ,qs ; δki ,l j }] defined in [5,32]. Certain of them are: Reduction [5]: An IM A’s operation (k, ⊥)-reduction is defined as follows: A(k,⊥) = [K − {k}, L, {ctu ,vw }], where ctu ,vw = aki ,l j (tu = ki ∈ K − {k}, vw = l j ∈ L). Projection [5]: Let M ⊆ K and N ⊆ L. Then, pr M, N A = [M, N, {bki ,l j }], where for each ki ∈ M and each l j ∈ N, bki ,l j = aki ,l j .   Substitution [5]: pk ; ⊥ A = (K − {k}) ∪ {p}, L, {ak,l } Addition-(◦1, ◦2, ∗): A ⊕(◦1,◦2,∗) B = [K ∪ P, L ∪ Q, {φtu ,vw , ψtu ,vw ; ηtu ,vw }], where ◦1, ◦2  ∈ {max, min, min, max,  average, average} and ∗ ∈ {max, min}. φtu ,vw , ψtu ,vw ; ηtu ,vw  = ◦1 (μki ,l j , ρ pr ,qs ), ◦2 (νki ,l j , σpr ,qs ); ∗(rtu ,vw , δtu ,vw ).

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Termwise Subtraction-(max,min): A −(max, min,∗) B = A ⊕(max, min,∗) ¬B. Termwise Multiplication: A⊗(◦1,◦2,∗) B = [K ∩P, L ∩Q, {φtu ,vw , ψtu ,vw ; ηtu ,vw }], where φtu ,vw , ψtu ,vw ; ηtu ,vw  = ◦1 (μki ,l j , ρ pr ,qs ), ◦2 (νki ,l j , σpr ,qs ); ∗(rtu ,vw , δtu ,vw ). Aggregation Operations. The aggregation operations #q, (q ≤ i ≤ 3) from [28] are expanded in [32] for scaling C-IFPs x = a, b; r1  and y = c, d; r2 : x#1, ∗y = min(a, c), max(b, d); ∗(r1, r2 ); x#2, ∗y = average(a, c), average(b, d); ∗(r1, r2 ); x#3, ∗y = max(a, c), min(b, d); ∗(r1, r2 ) and ∗ ∈ {min, max}. Let the fixed index be k0  K. The aggregation operation by the dimension K over 3-D C-IFIM A is defined as follows [5,28]: hg ∈ H αK, #q ,∗ (A, k0 ) =

k0

l1

...

ln

m

m

i=1

i=1

#q, ∗ μki ,l1,hg , νki ,l1,hg ; rki ,l1,hg  . . . #q, ∗ μki ,ln,hg , νki ,ln,hg ; rki ,l1,hg 

,

where 1 ≤ q ≤

3. Aggregate Global Internal Operation [27]: AGIO ⊕(#q ,∗) (A) . If q = 1, q = 2 or q = 3, we will get a pessimistic, averaged, or optimistic situation, respectively. Internal Subtraction of IMs’ Components [27]:

 IO−(max, min,∗) ( ki, l j , A , pr , qs, B) = [K, L, {γtu ,vw , δtu ,vw }].

3

Apple’s Dependence on the “season” as a Factor in Sales

The one-way ANOVA [11] is used in this section on the Apple company’s sales for the 2019-2022 period by the factor “season” [34]. The ANOVA’s primary steps are: Let yki ,l j (i = 1, 2, ..., m and j = i1, i2, ...iI (1 ≤ iI ≤ n)) represent the data from the factor’s ki −th level and the l j −th observation. Assume there are N observations. The formulas for the mean sums of squares for error (MSE), treatment (MSC), and the mean sum of squares (MST) are provided in [21]. If F=

MSE 1 MSC 1 ≥ F(α,m−1, N −m) or = ≤ = F(1−α, N −m,m−1), MSE F MSC F(α,m−1, N −m)

(2)

where F(α,m−1, N −m) is α−quantile of F−distribution, then the factor influences the values of the studied quantity on significance level α [10,11]. Before doing an Anova on the data, we must use the Kolmogorov-Smirnov test with a significance threshold of 5% to determine whether the data distribution is normal [10]. Winter, spring, summer, and autumn are the four levels of the season factor under study. Table 1 at a significance level of 5% shows the outcomes of an ANOVA by “season” factor done by the SPSS [18] for the sales of the Apple corporation for 2019-2022:

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Table 1. ANOVA to Apple sales by the “season” factor Source

SS

df MS

Between groups Within groups Total

4316.93 3

F

p-value F crit

1438.98 5.69 0.012

3037

12 253.11

7354

15

3.49

The results of the ANOVA show that for the four years, the “season” factor determined the measure of Apple sales. The Fig. 1 compares Apple sales by season from 2019 to 2022: With the application of ANOVA, the following findings can be drawn: – According to a one-way ANOVA, there is a significant seasonal difference in Apple’s sales at the selected 95% confidence level, with the company’s sales being at their highest in the autumn. – The lowest season for Apple sales is spring, followed by summer and winter. The springtime sales strategy for Apple management should be to boost sales.

4

One-Way Circular Intuitionistic Fuzzy ANOVA on Apple Sales by the “Season” Factor

We have suggested one-way IFANOVA in articles [30,31], combining the ANOVA with the benefits of intuitionistic fuzzy logic [2] and IMs [3] theories. Here, IFANOVA will be made more comprehensive so that it may be used with CIFPs. To represent C-IFANOVA, pseudocode, and mathematical language will be employed.

Fig. 1. Apple’s sales based on the “season” parameter for 2019-2022.

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Step 1. An evaluation IFIM EV[K, L, E, {ev ki ,l j ,ds }] is created, where K =

{k1, . . . , ki, . . . , k m }, L = l1, . . . , l j , . . . , ln , E = {d1, . . . , ds, . . . , dD } and the element {ev ki ,l j ,ds } = μki ,l j ,ds , νki ,l j ,ds  (for 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ s ≤ D) is the estimate of the ds -th expert for the ki -th level of the studied factor for the l j -th value of ( j = 1, . . . , n). Due to changes in some uncontrolled factors, the expert is unsure about the evaluation according to a particular criterion, and his evaluations take the shape of IFPs. To determine the averaged IF value of the ki -th factor level against the l j -th value in a present moment h f  E, the αE -th aggregation operation is used to obtain the IM PI[K, L, h f , {pi ki ,l j ,h f }] hf

...

l1 D

ln D

k1 #2 μk1,l1,ds , νk1,l1,ds  . . . #2 μk1,ln,ds , νk1,ln,ds  | h f  E. = .. .. .. . . . . s=1 . . s=1 D

D

s=1

s=1

k m #2 μkm,l1,ds , νkm,l1,ds  . . . #2 μkm,ln,ds , νkm,ln,ds 

Afterward, we receive a C-IFIM Y [K, L, {yki ,l j }], whose components are the measured values by the various levels of the factor under study, where K = y y y {k1, k2, . . . , k m }, L = {l1, l2, . . . , ln, Sr i, Sr } . yki ,l j = μki ,l j , νki ,l j ; rki ,l j for 1 ≤ i ≤ m, 1 ≤ j ≤ n, are created by converting the IFPs pi ki ,l j ,h f into C-IFPs by { for j = 1 y pi y pi to n, i = 1 to m μki ,l j = μki ,l j ,h f ; νki ,l j = νki ,l j ,h f and 

y pi pi rki ,l j = max | (μev − μki ,l j ,h f )2 + (νkevi ,l j ,ds − νki ,l j ,h f )2 | . {yki ,Sr i , yki ,Sr } are ki ,l j ,ds 1 ≤s ≤D

defined as C-IFPs. In the papers [6,14,32], a comparable method for creating C-IFPs is established. For empty cells of IM Y , we use the notation “⊥”. Then, an IM S = [K, L, {ski ,l j }] is generated with the form, such that S = Y i.e. (ski ,l j = yki ,l j , ∀ki ∈ K, ∀l j ∈ L). The IMs are then created as follows: S1 [K, L/{Sr 1, Sr }] = prK, L/{Sr 1,Sr } S, S2 [K, {Sr 1 }] = αL, #2,∗ (S1, Sr1 ). We get a new form of IM S1 by using the operation “addition” as shown below: S := S ⊕(◦1,◦2,∗) S2, go to Step 2. Step 2. The sample’s mean is determined by S3 [k0, {Sr 1 }] = αK, #1,∗ (S2, k0 )(Sr 1  L/{Sr 1, Sr }).   1 for i = 1 to m {S := S ⊕(◦1,◦2,∗) kk0i ; Sr Sr S3 }. We subtract the mean of each row of the matrix S corresponding to a certain factor level from each element of the matrix S: for i = 1 to m 

 

for j = 1 to n {S4 = IO−(max, min,∗) ki, l j , prK, {L/{Sr,Sr 1 } S , ki, Sr 1, prK, {Sr 1 }S }. MSE =

  1 AGIO ⊕(◦1,◦2,∗) S4 ⊗(◦1,◦2,∗) S4 . Go to Step 3. N − km

Step 3. The mean sums of squares MSC are computed using the following techniques:

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– S5 [K, {Sr 1 }] = prK, {Sr 1 } S4 and S6 [K,{Sr }] = prK, {Sr }S4 ;

 f Di f – Let Di f  K ∪ L then S7 [K, Di f ] = ⊥; Di Sr 1 S5 ⊕−,∗ ⊥; Sr S6 ;   – MSC = km1−1 AGIO ⊕(◦1,◦2,∗) S7 ⊗(◦1,◦2,∗) S7 ; Then proceed to Step 4.

Step 4. The main statistic of the C-IFANOVA estimator using Pietraszek’s approach ([24],2016) FC−I F(1−α, N −km,m−1;0) value is determined. Hypothesis H0 is denied on significance level α if MSE 1 = ≤ FC−I F (1 − α, N − k m, k m − 1; 0), F MSC otherwise H0 is accepted on level α else H0 is accepted on level α. The Algorithm is Complete. The next application clarifies the proposed CIFANOVA in this section to ascertain whether the levels of the “season” factor have a sizable impact on Apple’s sales numbers from 2019 to 2022. The problem is: The values of the sales for the years 2019 to 2022 {l1, ..., l4 } must be evaluated by a team of experts {d1, d2, d3 } using 4 levels of the season factor. The solution algorithm is: Step 1. 3-D assessment IFIM EV[K, L, E, {ev ki ,l j ,ds }] is formed. {ev ki ,c j ,ds } (for 1 ≤ i ≤ 4, 1 ≤ j ≤ 4, 1 ≤ s ≤ 3) is the estimate of the ds -th expert for the ki -th level at the l j -th value of sales value. Finally, using Apple sales by the “season” factor for 2019-2022, we derive C-IFIM Y [K, L/{Sr1, Sr }] without the last two columns: Winter Spring Summer Autumn

l1 0.107, 0, 89; 0.06 0.05, 0.94; 0.03 0.18, 0.81; 0.03 0.56, 0.44; 0.04

l2 0.11, 0.88; 0.05 0.12, 0.87; 0.04 0.19, 0.8; 0.05 0.81, 0.18; 0.04

l3 0.53, 0.47; 0.04 0.41, 0.58; 0.03 0.44, 0.55; 0.04 0.89, 0.1; 0.04

l4 0.63, 0.36; 0.05 0.43, 0.56; 0.05 0.53, 0.46; 0.03 0.01, 0.99; 0.03

Let’s use C-IFANOVA in the case of a positive scenario, therefore ◦1, ◦2, ∗ = max, min, min. Pietraszek’s method is used to construct the C-IF estimator of the C-IFANOVA key statistic F. [24]. The traditional core statistic F(0, 95; 12; 3) = 0, 27 is modified in the C-IF criterion FC−I F (0, 95; 12; 3) = 0, 97; 0; 0. Because of this, 0.144, 0.01; 0.03 =

1 ≤ FC−I F (0, 95; 12; 3) = 0.97, 0; 0. F

The “season” has an impact on Apple’s sales. The effects of the “season” factor are seen in the C-IFANOVA and ANOVA results for Apple sales.

5

Conclusion

In this study, we modified IFANOVA [30] to C-IFANOVA so that it could be used with C-IFPs. The approach can be easily extended to be used on other types

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of fuzzy data. To assess the substantial impact of the levels of the “season” factor on Apple’s sales values for 2019-2022, an application of the recommended approach explains the developed IMs interpretation of ANOVA under the form of the C-IFANOVA. Pessimistic, optimistic, and average scenarios are put out to the decision-maker for consideration before making a final decision. In the future, we will enhance the new C-IFANOVA to incorporate elliptic IF data and software for its implementation [7].

References 1. Anuradha, D., Kalpanapriya, D.: Intuitionistic fuzzy ANOVA and its application in medical diagnosis. Res. J. Phar. Technol. 11(2), 653–656 (2018) 2. Atanassov, K.: Intuitionistic fuzzy sets. In: Proceedings of VII ITKR’s Session, Sofia (1983) (in Bulgarian) 3. Atanassov, K.: Generalized index matrices. Comptes rendus de l’Academie Bulgare des Sci. 40(11), 15–18 (1987) 4. Atanassov, K.: On intuitionistic fuzzy sets theory. STUDFUZZ, vol. 283. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29127-2 5. Atanassov, K.T.: Index Matrices: Towards an Augmented Matrix Calculus. SCI, vol. 573. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10945-9 6. Atanassov, K.: Circular intuitionistic fuzzy sets. J. Intell. Fuzzy Syst. 39(5), 5981– 5986 (2020) 7. Atanassov, K.: Elliptic intuitionistic fuzzy sets. C. R. Acad. Bulg. Sci. 74(65), 812–819 (2021) 8. Atanassov, K., Marinov, E.: Four distances for circular intuitionistic fuzzy sets. Mathematics 9(10), 11–21 (2021). https://doi.org/10.3390/math9101121 9. Buckley, J.J.: Fuzzy probability and statistics. Springer, Berlin Heidelberg (2006). https://doi.org/10.1007/3-540-33190-5 10. Doane, D., Seward, L.: Applied statistics in business and economics. McGraw-Hill Education, New York, USA (2016) 11. Fisher, R.: Statistical Methods for Research Workers, London (1925) 12. Gil, M.A., Montenegro, M., Gonz´ alez-Rodr´ıguez, G., Colubi, A., Casals, M.R.: Bootstrap approach to the multi-sample test of means with imprecise data. Comput. Statist. Data Anal. 51, 148–162 (2006) 13. Jiryaei, A., Parchami, A., Mashinchi, M.: One-way ANOVA and least squares method based on fuzzy random variables. Turk. J. Fuzzy Syst. 4(1), 18–33 (2013) 14. Kahraman, C., Alkan, N.: Circular intuitionistic fuzzy TOPSIS method with vague membership functions: supplier selection application context. NIFS 27(1), 24–52 (2021) 15. Kalpanapriya, D., Pandian, P.: Fuzzy hypothesis testing of ANOVA model with fuzzy data. Int. J. Mod. Eng. Res. 2(4), 2951–2956 (2012) 16. Kalpanapriya, D., Unnissa, M.: Intuitionistic fuzzy ANOVA and its application using different techniques. In: Madhu, V., Manimaran, A., Easwaramoorthy, D., Kalpanapriya, D., Mubashir Unnissa, M. (eds.) Advances in Algebra and Analysis. Trends in Mathematics, pp. 457–468. Birkh¨ auser, Cham (2017) 17. Konishi, M., Okuda, T., Asai, K.: Analysis of variance based on fuzzy interval data using moment correction method. Int. J. Innov. Comput. Inf. Control 2(1), 83–99 (2006)

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18. Marques de S´ a, J.: Applied statistics using SPSS, STATISTICA, MATLAB and R. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71972-4 19. Montenegro, M., Gonzalez-Rodriguez, G., Gil, M.A., Colubi, A., Casals, M.R.: Introduction to ANOVA with fuzzy random variables. In: Lopez-Diaz, M., Gil, M.A., Grzegorzewski, P., Hryniewicz, O., Lawry, J. (eds.) Soft Methodology and random information systems, vol. 26, pp. 487–494. Springer, Berlin (2004). https:// doi.org/10.1007/978-3-540-44465-7 60 20. Nourbakhsh, M.R., Parchami, A., Mashinchi, M.: Analysis of variance based on fuzzy observations. Int. J. Syst. Sci. 44(4), 714–726 (2013) 21. Ostertagov´ a, E., Ostertag, O.: Methodology and application of one-way ANOVA. Am. J. Mech. Eng. 1(7), 256–261 (2013) 22. Parchami, A., Nourbakhsh, M., Mashinchi, M.: Analysis of variance in uncertain environments. Complex Intell. Syst. 3(3), 189-196 (2017) 23. Parchami, A., Mashinchi, M., Kahraman, C.: A case study on vehicle battery manufacturing using fuzzy analysis of variance. In: Kahraman, C., Cevik Onar, S., Oztaysi, B., Sari, I.U., Cebi, S., Tolga, A.C. (eds.) INFUS 2020. AISC, vol. 1197, pp. 916–923. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-51156-2 106 24. Pietraszek, J., Kolomycki, M., Szczotok, A., Dwornicka, R.: The fuzzy approach to assessment of ANOVA results. In: Nguyen, N.-T., Manolopoulos, Y., Iliadis, L., Trawi´ nski, B. (eds.) ICCCI 2016. LNCS (LNAI), vol. 9875, pp. 260–268. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45243-2 24 25. Puri, M., Ralescu, D.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986) 26. Szmidt, E., Kacprzyk, J.: Amount of information and its reliability in the ranking of Atanassov intuitionistic fuzzy alternatives. In: Rakus-Andersson, E., Yager, R.R., Ichalkaranje, N., Jain, L.C. (eds.) Recent Advances in Decision Making. Studies in Computational Intelligence, vol. 222, pp. 7–19. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02187-9 2 27. Traneva, V.: Internal operations over 3-dimensional extended index matrices. Proceed. Jangjeon Math. Soc. 18(4), 547–569 (2015) 28. Traneva, V., Tranev, S., Stoenchev, M., Atanassov, K.: Scaled aggregation operations over 2- and 3-dimensional IMs. Soft. Comput. 22(15), 5115–5120 (2018) 29. Traneva, V., Mavrov, D., Tranev, S.: Software utility of one-way intuitionistic fuzzy ANOVA. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) Intelligent and Fuzzy Systems. INFUS 2022. Lecture Notes in Networks and Systems, vol. 504, pp. 681–689. Springer, Cham (2022). https://doi.org/10. 1007/978-3-031-09173-5 79 30. Traneva, V., Tranev, S.: Intuitionistic fuzzy analysis of variance of movie ticket sales. In: Kahraman, C., Cevik Onar, S., Oztaysi, B., Sari, I.U., Cebi, S., Tolga, A.C. (eds.) INFUS 2020. AISC, vol. 1197, pp. 363–371. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-51156-2 43 31. Traneva, V., Tranev, S.: Digital interpretation of movie sales revenue through intuitionistic fuzzy analysis of variance. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds) Intelligent and Fuzzy Systems. INFUS 2022. Lecture Notes in Networks and Systems, vol. 504, pp. 581–588. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5 67 32. Traneva, V., Petrov, P., Tranev, S.: Circular IF Knapsack problem. Lecture Notes in Networks and Systems (2023) (in press) 33. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 34. Apple Homepage. https://www.apple.com/bg/. Accessed 4 Mar 2023

Petrol Station Franchisor Selection Through Circular Intuitionistic Fuzzy Multicriteria Analysis Velichka Traneva(B)

and Stoyan Tranev

“Prof. Asen Zlatarov” University, “Prof. Yakimov” Blvd, 8000 Burgas, Bulgaria [email protected], [email protected] http://www.btu.bg Abstract. In a rapidly unstable business situation, the development of an ideal multi-criteria franchisor selection method is essential for effective business operations. Atanassov introduced the theories of index matrices and circular intuitionistic fuzzy sets in 1987 and 2020, respectively, which are appropriate theoretical tools for simulating a multi-criteria system for choosing a successful franchisor. For choosing a gas station franchisor, the research suggests an index-matrix interpretation of a circular intuitionistic fuzzy decision-making model (C-IFFr), in which the proposed algorithm takes into account the ratings of the experts as well as the priorities of the evaluation criteria. The evaluation criteria values and their relative importance are introduced as circular intuitionistic fuzzy numbers. Three scenarios - optimistic, average, and pessimistic - are suggested to help the entrepreneurial firm to make the best choice. Using the suggested C-IFFr analysis to select a franchisor for a chain of gas stations was successful. Keywords: Circular Intuitionistic Fuzzy Sets Matrix

1

· Franchisor · Index

Introduction and Literature Review

Franchising is a successful business approach for expanding into new markets. Making the best choice for the franchise business is crucial for an entrepreneur seeking a franchisor. The problem of choosing the best franchisor based on the system of criteria is a type of multi-criteria decision-making problem (MCDM). The shifting evaluations of the criteria are caused by the economic environment’s dynamics. In this regard, it is necessary to extend the optimal selection algorithms so that they can be used in an uncertain environment. It is appropriate to use intuitionistic fuzzy sets (IFSs, [1]), the earliest extension of fuzzy set theory [22], to describe the fuzziness in the selection criterion for the best Work on Sects. 2 and 3 is supported by the A. Zlatarov University through project Ref. No. NIX-482/2023 “Application of Intuitionistic Fuzzy Logic in Decision Making”. Work on Sects. 1, 4, and 5 is supported by the project Ref. No. NIX-486/2023 “Modeling Management Decisions with New Analysis Tools in a Fuzzy Environment”. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 194–203, 2023. https://doi.org/10.1007/978-3-031-39774-5_24

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franchisee. Using the ideas of index matrices (IMs, [2]) and intuitionistic fuzzy sets (IFSs), a novel technique IFIMFr for selecting the best franchisee candidate is proposed in the work [18]. IFSs theory has some extensions, with the circular IFSs (C-IFSs) being the most recent. By utilizing a radius around the degrees of membership and non-membership, C-IFSs can manage imprecision and ambiguity [5]. A novel current worth analysis, based on interval-valued IF and C-IF sets, is evolved in [10]. A C-IFSs method of evaluation using deviations from the mean solution was proposed in [12]. In [11], a brand-new C-IF MCDM technique is developed. The formulation of new C-IF frameworks for TOPSIS and VICOR is found in [14,15]. The C-IF AHP and VIKOR are used to form an integrated MCDA strategy that is proposed in [16]. In our study, we use C-IF logic [5] and index matrices (IMs, [2]) theory to create a circular IF decisionmaking model (C-IFFr) for the best franchisor selection. In this model, experts and their ratings determine the criteria values in the form of C-IF numbers. The creation of a C-IFFr technique for choosing the most qualified franchisee candidates and its use in choosing a chain franchisor for gas stations are the key contributions of the paper. The originality of this study consists in introducing an algorithm for choosing a franchise chain based on the theories of IMs and C-IF logic. The remainder of this work is structured as follows: Preliminaries for IM ideas and C-IFSs are provided in Sect. 2. Section 3 formulates a specific C-IF problem for choosing a franchisor and offers an IM innovative strategy for solving it. The actual C-IFFr problem for choosing a franchisor for chain gas stations is resolved in Sect. 4. Section 5 contains recommendations for the future.

2

Preliminaries of IMs and C-Intuitionistic Fuzzy Pairs

In this section, the preliminaries of C-IF pairs and IMs are given with definitions. 2.1

Some Operations and Relations on Circular Intuitionistic Fuzzy Pairs (C-IFPs)

A C-IFP has the following form: a(p), b(p); r = μ(p), ν(p); r, where a(p), b(p) ∈ [0, 1] and a(p) + b(p) ≤ 1 are used to evaluate a proposition p [5,6]. The √ circle’s radius around a(p), b(p) is r ∈ [0, 2]. Two C-IFPs x = a, b; r1  and y = c, d; r2 , will be used. Let us define an operation ∗ ∈ {min, max}. The further operations are introduced in [5,6,14,19]: x ∨1∗ y = max(a, c), min(b, d); ∗(r1 , r2 ); x ∧1∗ y = min(a, c), max(b, d); ∗(r1 , r2 ); x ∧2∗ y = x + y = a + c − a.c, b.d; ∗(r1 , r2 ); x ∨2∗ y = x.y = a.c, b + d − b.d; ∗(r1 , r2 ).

The definition of Szmidt and Kacprzyk’s IF Hamming distance [17] is expanded for the C-IFSs in [6]. We have proposed the following relation for comparison of C-IFPs [19], √ extending the definition for IFPs measure R = 0.5(1 + πx )distance(1, 0; 2, x that may be used for ranking the alternatives. circ circ ≤ Rc,d;r x ≥Rcirc y iff Ra,b;r 1 2

where the distance from the ideal positive alternative 1, 0;   √2−r| +1−a . 0.25(2 − a − b) | √ 2

(1) √

circ 2 to x is Ra,b;r =

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Three-Dimensional C-Intuitionistic Fuzzy Index Matrix (3-D C-IFIM)

Let be given a fixed set of indices I. Using the definition of 3-D IFIM from [3,20], let us we define a 3-D C-IFIM [K, L, H, {μki ,lj ,hg , νki ,lj ,hg ; rki ,lj ,hg }] as follows: hg ∈ H l1 ... ln k1 μk1 ,l1 ,hg , νk1 ,l1 ,hg ; rk1 ,l1 ,hg  . . . μk1 ,ln ,hg , νk1 ,ln ,hg ; rk1 ,ln ,hg  , .. .. .. . ... . . km μkm ,l1 ,hg , νkm ,l1 ,hg ; rkm ,l1 ,hg  . . . μkm ,ln ,hg , νkm ,ln ,hg ; rkm ,ln ,hg 

(2)

where (K, L, H ⊂ I) and its elements are C-IFPs. There are various operations, relations, and operators over C-IFIMs A = [K, L, H, {μki ,lj ,hg , νki ,lj ,hg ; r1ki ,lj ,hg }] and B = [P, Q, R{ρpr ,qs ,te , σpr ,qs ,te ; r2pr ,qs ,te }] [3,19,20]. Certain of them are: Addition-(◦1 , ◦2 , ∗): A ⊕(◦1 ,◦2 ,∗) B = [K ∪ P, L ∪ Q, H ∪ R, {φtu ,vw ,xy , ψtu ,vw ,xy ; ηtu ,vw ,xy }], where ◦1 , ◦2  ∈ {max, min, min, max,  average, average} and ∗ ∈ {max, min}. φtu ,vw ,xy , ψtu ,vw ,xy ; ηtu ,vw ,xy  = ◦1 (μki ,lj ,xy , ρpr ,qs ,xy ), ◦2 (νki ,lj ,xy , σpr ,qs ,xy ); ∗(r1tu ,vw ,xy , r2tu ,vw ,xy ).

Multiplication: A (◦1 ,◦2 ,∗) B = [K ∪ (P − L), Q ∪ (L − P ), H ∪ R, {φtu ,vw ,xy , ψtu ,vw ,xy ; ηtu ,vw ,xy }], where φtu ,vw ,xy , ψtu ,vw ,xy  is defined in [3] and ηtu ,vw ,xy = ∗(r1tu ,vw ,xy , r2tu ,vw ,xy ). Aggregation Operation by One Dimension [21]: The aggregation operations #q , (q ≤ i ≤ 3) from [21] are expanded in [19] for scaling C-IFPs x = a, b; r1  and y = c, d; r2 : x#1 , ∗y = min(a, c), max(b, d); ∗(r1 , r2 ); x#2 , ∗y = average(a, c), average(b, d); ∗(r1 , r2 ); x#3 , ∗y = max(a, c), min(b, d); ∗(r1 , r2 ) and ∗ ∈ {min, max}. / K. The aggregation operation by a dimension K Let the fixed index be k0 ∈ over 3-D C-IFIM A is defined as follows [3,21]: hg ∈ H αK,#q ,∗ (A, k0 ) =

3

k0

l1

...

ln

m

m

i=1

i=1

#q , ∗ μki ,l1 ,hg , νki ,l1 ,hg ; rki ,l1 ,hg  . . . #q , ∗ μki ,ln ,hg , νki ,ln ,hg ; rki ,l1 ,hg 

.

A Model for the Selection of a Franchisor Using a Circular Intuitionistic Fuzzy Index-Matrix

The algorithm for solving a form of C-IF franchisor selection problem will be created in this section, extending the IFFr approach [18]. Let us consider

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the following problem: An entrepreneurial company wants to select a suitable franchisor of a gas station chain and has developed an evaluation system with criteria {c1 , . . . , cj , . . . , cn } (for j = 1, ..., n) for businesses offering a franchise. It is necessary for the experts {d1 , . . . , ds , . . . , dD } to assess the chains of gas stations ({k1 , . . . , ki , . . . , km }) based on criteria. The ratings of the experts {r1 , . . . , rs , . . . , rD } are defined according to their involvement in the evaluation of the franchise procedures {r1 , . . . , rs , . . . , rD }, and they are provided to the experts in the form of C-IFPs δs , s ; rads (1 ≤ s ≤ D). The assessments of the experts are made, and they are under the form of IF data ev ki ,cj ,ds (for 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ s ≤ D). The final estimates of franchise gas station chains are formed in the form of C-IFPs f iki ,cj ,ds (for 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ s ≤ D), taking into account the C-IF priorities pk cj ,ve of the criteria cj (for j = 1, ..., n) for the entrepreneur. The optimal goal is to determine which franchise chain of gas stations is the most qualified. The steps in the solution process, which we refer to as C-IFIMFr, are as follows: Step 1. An IFIM EV [K, C, E, {ev ki ,cj ,ds }], K = {k1 , k2 , . . . , km }, C = {c1 , c2 , . . . , cn } and E = {d1 , d2 , . . . , dD } is constructed. Due to the uncertainty of valuation experts in the dynamic economic environment, the elements {ev ki ,cj ,ds } = μki ,cj ,ds , νki ,cj ,ds  (for 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ s ≤ D) of the IM EV are the IF valuations of the ds -th expert for the ki -th candidate by the cj -th criterion. Next, we go on to Step 2. Step 2. Each expert’s score coefficient rs = δs , s ; rads , (s ∈ E) should be specified by a C-IFP, whose components can be understood as indicating how competent or incompetent they are. The IM EV ∗ [K, C, E, {ev ∗ ki ,cj ,ds }] = r1 prK,C,d1 EV ⊕(◦1 ,◦2 ) r2 prK,C,d2 EV . . . ⊕(◦1 ,◦2 ) rD prK,C,dD EV. EV := EV ∗ (evki ,lj ,ds = evk∗i ,lj ,ds , ∀ki ∈ K, ∀lj ∈ L, ∀ds ∈ E) is constructed. An application of the second αE -th aggregation operation results in the following calculation of the averaged IF assessment of the ki -th franchisor on the cj -th criterion / E moment. in a moment hf ∈ ⎧ ⎫ cj hf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ k1 #2 μk1 ,cj ,ds , νk1 ,cj ,ds  ⎬ | c P I = αE,#2 (EV, hf ) = ∈ C . Next, we go on to j .. ⎪ ... s=1 ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ km #2 μkm ,cj ,ds , νkm ,cj ,ds  ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

Step 3.

s=1

Step 3. The franchise business candidate’s evaluation system will be optimized at this stage. To improve the franchisee rating system, we suggest using the inter-criteria analysis (ICrA, [7,8]) to get rid of slower or more expensive criteria that have been discovered to closely correlate with other criteria

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under intuitionistic fuzzy settings. Let α, β be an IFP. The criteria Ck and Cl are in (α, β)-positive consonance, if μCk ,Cl > α and νCk ,Cl < β; (α, β)negative consonance, if μCk ,Cl < β and νCk ,Cl > α; (α, β)-dissonance, otherwise. The ICrA algorithm is used to search the P I matrix for the criteria that are consonant. Using the IM reduction operation over P I, more costly, slower, or more complex criteria are removed from the evaluation franchise system. We calculate the transposed IM of P I T = [K, C, hf , piT ki ,cj ,hf ]. Then, C-IFIM Y [K, C, hf , {yki ,cj ,hf }] is created whose components are the rated franchise gas station chains by criteria. yki ,cj ,hf = μyki ,cj ,hf , νkyi ,cj ,hf ; rkyi ,cj ,hf  for 1 ≤ i ≤ m, 1 ≤ j ≤ n, are created by converting the IFPs piT ki ,cj ,hf into C T y piT IFPs by { for j = 1 to n, i = 1 to m μyki ,cj ,hf = μpi ki ,cj ,hf ; νki ,cj ,hf = νki ,cj ,hf

piT piT ev 2 2 . In the and rkyi ,cj ,hf = max | (μev ki ,cj ,ds − μki ,cj ,hf ) + (νki ,cj ,ds − νki ,cj ,hf ) | 1≤s≤D

papers [5,14], a comparable method for creating C-IFPs is established. Next, we go on to Step 4. Step 4. At this stage, a 3-D C-IFIM P K is created, and the coefficients used in the following operation determine the weighting of each evaluation criterion for the applicant ve for a gas station franchise: ve hf c1 pk c1 ,ve ,hf .. .. . . P K[C, ve , hf {pk cj ,ve ,hf }] = , cj pk cj ,ve ,hf .. .. . . cn pk cn ,ve ,hf where C = {c1 , c2 , . . . , cn } . The complete estimations of the ki -th gas station franchise are calculated into the evaluation IFIM F I[K, ve , hf {bki ,ve ,hf }] = Y (◦1 ,◦2 ,∗) P K (for 1 ≤ i ≤ m) for the entrepreneur ve . Go to Step 5. Step 5. At this stage, the franchise candidate ve selects the chain of gas stations that is the most advantageous based on an aggregation operation αK,#q (F I, k0 ) using pessimistic, averaging, or optimistic scenarios: ve αK,#q (F I, k0 ) =

m

k0 #q μki ,ve , νki ,ve 

,

(3)

i=1

/ K, 1 ≤ q ≤ 3. Go to Step 6. where k0 ∈ Step 6. This step obtains the new rating coefficients of the experts. Let the expert ds (s = 1, ..., D) has participated in γs evaluation procedures for the

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selection of a franchisee, based on which his score rs = δs , s , φs  is determined, then after his participation in the current procedure, his new score will be changed by [4]: ⎧ δγ+1 γ   γ+1 , γ+1 ; ∗(φs , φs ), if the expert has assessed correctly ⎪ ⎨     δγ γ δs , s ; φs  =  γ+1 , γ+1 ; ∗(φs , φs ), if the expert has not given any estimation ⎪  ⎩ δγ γ+1  γ+1 , γ+1 ; ∗(φs , φs ), if the expert has assessed incorrectly

(4) The algorithm is complete. If the operations ◦1 , ◦2  = min, max are applied, the pessimistic scenario has been utilized. The operation “∗ = max” is used in cases of more ambiguity, otherwise “∗ = min .” The proposed C-IFIMFr algorithm has the complexity O(Dm2 n2 ), based on the complexity of ICrA [9]).

4

Using C-IFFr to Solve the Challenge of Choosing a Franchise Chain Gas Station

C-IFFr approach from Sect. 3 can be successfully used in determining the most appropriate franchisor of a gas station chain selection. The following problem is formulated: A business venture wants to effectively franchise a chain of petrol stations. It creates a system of criteria for assessing potential franchisors ki (for 1 ≤ i ≤ 4) for this goal using the expertise of experts d1 , d2 , and d3 . The four groups of criteria that make up the franchisee selection system are as follows: C1 - choosing a well-known, respectable brand with a sizable market share requires consideration of its reputation and market share; C2 - location: pick a franchise with a prime spot that can attract customers and be easily seen and reached; C3 - operating and start-up costs, business model: to calculate initial and ongoing operating costs, such as franchise fees, royalties, and marketing expenses; to assess the franchise business model’s potential for profitability, such as fuel and retail margins, volume discounts, and inventory control systems; and to determine whether a franchise is right for you; C4 - evaluation of the franchisor’s degree of training and support, including continuous marketing and technical support. According to the criteria of the multi-criteria system, franchise gas station chains were assessed by experts. Their assessments are IF data ev ki ,cj ,ds (for 1 ≤ i ≤ 4, 1 ≤ j ≤ 4, 1 ≤ s ≤ 3). The final estimates of franchise gas station chains are formed in the form of C-IFPs f iki ,cj ,ds (for 1 ≤ i ≤ 4, 1 ≤ j ≤ 4, 1 ≤ s ≤ 3), taking into account the C-IF priorities pk cj ,ve of the criteria cj (for j = 1, ..., 4) for the entrepreneur. The goal of the problem is to determine which chain of gas stations is the most suitable franchisor. Solution of the Problem: Step 1. At this stage, the 3-D expert assessment IFIM EV [K, C, E, {eski ,cj ,ds }] is created with the expert’s estimations in the ds-th criterion for the kith franchisor (for 1 ≤ i ≤ 4, 1 ≤ j ≤ 4, 1 ≤ s ≤ 3), and its form is:

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⎧ ⎪ ⎪ d1 c1 c2 c3 c4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k 0.25, 0.35 0.15, 0.55 0.55, 0.25 0.15, 0.55 ⎪ 1 ⎪ ⎨

d2

c1

c2

c3

c4

k1 0.35, 0.45 0.05, 0.75 0.65, 0.15 0.25, 0.55

k2 0.05, 0.65 0.35, 0.45 0.35, 0.55 0.35, 0.45 , k2 0.15, 0.85 0.25, 0.55 0.55, 0.25 0.55, 0.15 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k3 0.35, 0.25 0.05, 0.75 0.15, 0.45 0.55, 0.25 k3 0.25, 0.45 0.25, 0.65 0.05, 0.75 0.35, 0.45 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k4 0.05, 0.75 0.15, 0.75 0.15, 0.75 0.35, 0.55

d3 k1 k2 k3 k4

k4 0.10, 0.65 0.20, 0.35 0.15, 0.65 0.25, 0.35

⎫ c1 c2 c3 c4 ⎪ ⎪ ⎪ 0.05, 0.75 0.15, 0.75 0.35, 0.45 0.35, 0.45⎪ ⎬ 0.05, 0.85 0.25, 0.65 0.15, 0.65 0.45, 0.25 ⎪ 0.25, 0.55 0.15, 0.75 0.25, 0.65 0.35, 0.55⎪ ⎪ ⎪ ⎭ 0.05, 0.85 0.25, 0.55 0.05, 0.75 0.25, 0.65

Step 2. The experts’ rating indices are as follows: {r1 , r2 , r3 } = {0.75, 0.15, 0.65, 0.15, 0.85, 0.15}. The following operations are used to create the IM EV ∗ [K, C, E, {ev ∗ }] : EV ∗ = r1 prK,C,d1 EV ⊕(◦1 ,◦2 ) r2 prK,C,d2 EV ⊕(◦1 ,◦2 ) r3 prK,C,d3 EV ; EV := EV ∗ (5) The averaged aggregate procedure P I[K, hf , C] = αE,#2 (EV, hf )(EV, hf ) is then used to determine the averaged IF evaluation of the ki -th franchisor on the cj -th criterion in the present. Step 3. At this step, we applied the ICrA with α = 0.80 and β = 0.10 over P I. The result is that there are no criteria that depend on consonants. The findings are shown as an IM in μ - ν view result matrix (see Fig. 1) after using the software that implements ICrA [13].

Fig. 1. The IFPs provide the InterCriteria correlations.

C-IFIM Y [K, C, hf , {yki ,cj ,hf }] is created whose components are the rated franchise gas station chains by criteria. hf k1 Y = k2 k3 k4

c1 c2 c3 c4 0.16, 0.59; 0.04 0.09, 0.73; 0.02 0.39, 0.39; 0.03 0.19, 0.60; 0.03 0.06, 0.82; 0.02 0.21, 0.62; 0.02 0.26, 0.56; 0.02 0.34, 0.39; 0.02 0.21, 0.5; 0.01 0.11, 0.67; 0.02 0.11, 0.67; 0.01 0.31, 0.5; 0.01 0.05, 0.79; 0.01 0.15, 0.62; 0.01 0.09, 0.79; 0.01 0.21, 0.59; 0.02

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Step 4. This stage involves creating an IM P K from the firm ve using the weight coefficients of the assessment criterion following their precedence in an optimistic scenario: hf c1 P K[C, ve , hf , {pk cj ,ve ,hf }] = c2 c3 c4

ve 0.85, 0.15; 0.02 0.75, 0.15; 0.02 0.55, 0.25; 0.02 0.75, 0.15; 0.01

hf k1 and F I = Y (◦1 ,◦2 ,min) P K = k2 k3 k4

ve 0.624, 0.096; 0.02 0.638, 0.090; 0.02 0.644, 0.097; 0.02 0.489, 0.220; 0.01

(6)

(7)

Step 5. According to the optimistic aggregation operation αK,#3 ,min (F I, k0 ), k3 is the best franchisor for the gas station chain in Bulgaria, with a maximum acceptance degree of 0.644 and a minimum rejection degree of 0.097. The decision-makers will select the candidate k4 with the minimum degree of membership 0.489 and the greatest degree of non-membership 0.22 in a gloomy scenario if the future is unclear and the decision-making environment is unpredictable.

Step 6. At the last step, we assume that the experts’ assessments are correct from the point of view of C-intuitionistic fuzzy logic [4] and their new rating coefficients are equal to {0.77, 0.14; 0.02, 0.68, 0.14; 0.02, 0.86, 0.14; 0.02}.

5

Conclusion

In the study, based on the theories of IFSs and IMs, we created a novel CIFIMFr business model for the most efficient selection of franchisors by C-IF assessments from independent experts which takes into account the ratings of the experts as well as the priorities of the evaluation criteria. The suggested method is illustrated using a case study for selecting a franchisor for a gas station chain. The developed C-IFFr approach can be applied to both circular numbers and crisp ones. There are no limitations on its application, and it is easily adaptable to different forms of data that are found in fuzzy contexts. In the future, a software program will be developed by C-IFFr to automate the selection of franchisors, with the evaluation criteria taking the form of circular intuitionistic fuzzy values.

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References 1. Atanassov, K.: Intuitionistic fuzzy sets. In: Proceedings of VII ITKR’s Session, Sofia, (1983). (Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84) 2. Atanassov, K.: Generalized index matrices. Comptes rendus de l’Academie Bulgare des Sciences 40(11), 15–18 (1987) 3. Atanassov, K.: Index Matrices: Towards an Augmented Matrix Calculus. Studies in Computational Intelligence, vol. 573. Springer, Cham (2014). https://doi.org/ 10.1007/978-3-319-10945-9 4. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. STUDFUZZ, vol. 283. Springer, Cham (2012). https://doi.org/10.1007/978-3-642-29127-2 5. Atanassov, K.: Circular intuitionistic fuzzy sets. J. Intell. Fuzzy Syst. 39(5), 5981– 5986 (2020) 6. Atanassov, K., Marinov, E.: Four distances for circular intuitionistic fuzzy sets. Mathematics 9(10), 11–21 (2021). https://doi.org/10.3390/math9101121 7. Atanassov, K., Mavrov, D., Atanassova, V.: Intercriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues IFSs Generalized Nets 11, 1–8 (2014) 8. Atanassov, K., Szmidt, E., Kacprzyk, J., Atanassova, V.: An approach to a constructive simplification of multiagent multicriteria decision making problems via ICrA. Comptes rendus de lAcademie bulgare des Sciences 70(8), 1147–1156 (2017) 9. Atanassova, V., Roeva, O.: Computational complexity and influence of numerical precision on the results of intercriteria analysis in the decision making process. Notes Intuitionistic Fuzzy Sets 24(3), 53–63 (2018) 10. Bolt¨ urk, E., Kahraman, C.: Interval-valued and circular intuitionistic fuzzy present worth analyses. Informatica 33(4), 693–711 (2022) 11. C ¸ akır, E., Ta¸s, M.A., Ulukan, Z.: Circular intuitionistic fuzzy sets in multi criteria decision making. In: Aliev, R.A., Kacprzyk, J., Pedrycz, W., Jamshidi, M., Babanli, M., Sadikoglu, F.M. (eds.) 11th International Conference on Theory and Application of Soft Computing, Computing with Words and Perceptions and Artificial Intelligence, ICSCCW 2021. LNNS, vol. 362, pp. 34-42. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-92127-9 9 12. Chen, T.-Y.: A circular intuitionistic fuzzy evaluation method based on distances from the average solution to support multiple criteria intelligent decisions involving uncertainty. Eng. Appl. Artif. Intell. 117, 105499 (2023) 13. Ikonomov, N., Vassilev, P., Roeva, O.: ICrAData - software for InterCriteria analysis. Int. J. Bioautomation 22(1), 1–10 (2018). https://doi.org/10.7546/ijba.2018. 22.1.1-10 14. Kahraman, C., Alkan, N.: Circular intuitionistic fuzzy TOPSIS method with vague membership functions: supplier selection application context. NIFS 27(1), 24–52 (2021) 15. Kahraman, C., Otay, I.: Extension of VIKOR method using circular intuitionistic fuzzy sets. In: Kahraman, C., et al. (eds.) Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation, INFUS 2021. LNNS, vol. 308, pp. 48–57. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-85577-2 6 ˙ Kahraman, C.: A novel circular intuitionistic fuzzy AHP & VIKOR 16. Otay, I, methodology: an application to a multi-expert supplier evaluation problem. Pamukkale Univ. J. Eng. Sci. 28(1), 194–207 (2021)

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17. Szmidt, E., Kacprzyk, J.: Amount of information and its reliability in the ranking of Atanassov intuitionistic fuzzy alternatives. In: Rakus-Andersson, E., Yager, R.R., Ichalkaranje, N., Jain, L.C. (eds.) Recent Advances in Decision Making, SCI, vol. 222, pp. 7–19. Springer, Heidelberg (2009). https://doi.org/10.1007/9783-642-02187-9 2 18. Traneva, V., Tranev, S.: Intuitionistic fuzzy model for Franchisee selection. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., et al. (eds.) Intelligent and Fuzzy Systems. INFUS 2022. LNNS, vol. 504, pp. 632–640. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5 73 19. Traneva, V., Petrov, P., Tranev, S.: Circular IF Knapsack Problem. Lecture Notes in Networks and Systems (2023, in press) 20. Traneva, V., Tranev, S.: Index Matrices as a Tool for Managerial Decision Making. Publishing House of the Union of Scientists, Sofia (2017). (in Bulgarian) 21. Traneva, V., Tranev, S., Stoenchev, M., Atanassov, K.: Scaled aggregation operations over 2- and 3-dimensional IMs. Soft Comput. 22(15), 5115–5120 (2018) 22. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

On Characterizations of Watson Crick Intuitionistic Fuzzy Automata N. Jansirani1

, N. Vijayaraghavan2(B)

, and V. R. Dare3

1 Queen Mary’s College, Chennai 600004, India 2 KCG College of Technology, Chennai 600097, India

[email protected] 3 Madras Christian College, Chennai 600059, India

Abstract. The Watson Crick Intuitionistic Fuzzy Automata are created by combining the principles of Watson Crick Automata and Intuitionistic Fuzzy Automata. The Watson Crick Intuitionistic Fuzzy Automata are presented in Automata Theory to effectively manage the idea of impreciseness. Watson Crick Intuitionistic Fuzzy Automata offer a wide range of applications in real-world imprecise circumstances. Few of the characteristics of the Watson Crick Intuitionistic Fuzzy Automata, such as intuitionistic successor property, intuitionistic exchange property, intuitionistic retrievable property and intuitionistic Fuzzy Switchboard machine are investigated in detail in this research work. Keywords: Characterization · Intuitionistic · Watson Crick

1 Introduction Classical automata theory is incapable of dealing with system uncertainty. The notion of fuzzy finite automata was suggested to cope with system uncertainty. Fuzzy automata can be utilized in a variety of applications, including defect detection, pattern matching, measuring the fuzziness of strings, natural language description, neural network and many more. As an extension of fuzzy finite state machines, intuitionistic fuzzy finite state machines were presented using the notion of intuitionistic fuzzy sets. The Watson Crick Fuzzy Automata and Watson Crick Intuitionistic Fuzzy Automata (WKIFA) were developed to effectively handle the idea of impreciseness in Automata theory. They were developed in order to study the ideas of Fuzziness and Intuitionistic Fuzziness in Watson Crick Automata Theory. The second section reviews the literature and reminds the reader of the necessary preliminary steps. The third portion examines some Watson Crick Intuitionistic Fuzzy Automata’s characterizations. The conclusion and suggested next steps are provided in the final section along with the application part.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 204–211, 2023. https://doi.org/10.1007/978-3-031-39774-5_25

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2 Literature Review and Preliminaries 2.1 Literature Review One example of a mathematical model abstracting biological features for computing is the Watson-Crick automaton. It was introduced by Rosenberg in 1997. They operate on double stranded sequences as finite automata with two reading heads. The fact that characters on equivalent places from the two input strands are associated by a complementarity relation akin to the Watson-Crick complementarity of DNA nucleotides is one of the major characteristics of these automata. Read-only heads controlled by a shared state individually scan each of the input’s two strands from left to right. A number of Watson-Crick automata variations and limitations have been introduced and studied over time. In several disciplines, Zadeh’s notion of fuzzy sets has been extremely successful. Intuitionistic fuzzy sets developed by Atanassov have been determined to be among the most effective higher order fuzzy sets for addressing vagueness. As an extension of fuzzy finite state machines, Jun developed intuitionistic fuzzy finite state machines and thoroughly studied their characterizations using the idea of intuitionistic fuzzy sets. To effectively handle the idea of uncertainty in automata theory and study their associated languages, the current authors developed the idea of WKIFA. 2.2 Preliminaries The Definitions which are required to discuss the characterizations of WKIFA are discussed in [2–4].

3 Characterizations of WKIFA Here we discuss the Characterizations of WKIFA. Definition 3.1 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA.  letp,  the imme  q ∈ Q. Then qis called  a a1 ∈ diate intuitionistic successor of p, if ∃ such that μA p, 1 , q > 0 a2 a2      a1 , q < 1 and q is called the intuitionistic successor of p if ∃ and γA p, a2    ∗         x1  x1 x ∗ ∗ ∈ and such that μA p, , q > 0 and γA p, 1 , q < 1. x2 ∗ x2 x2 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA and letp ∈ Q, the set of all intuitionistic successors of p is denoted as WKIS(p) Proposition 3.1 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA. Let us assume that p, q, r ∈ Q, then we have the following (1) q ∈ WKIS(q)

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(2) p ∈ WKIS(q)and r ∈ WKIS(p), then r ∈ WKIS(q) Proof:         λ λ ∗ (1) Since μA q, , q = 1 > 0 and γA q, , q = 0 < 1, then it is clear λ λ that q is a intuitionistic successor of q, this implies q ∈ WKIS(q). (2) Given p ∈ WKIS(q)   and  r ∗ ∈ WKIS(p), then by the  definition   3.1, we y  x x1 , 1 ∈ and such that μA ∗ q, 1 , p > 0 and have ∃ x2 y2 ∗ x2             x1 y1 y ∗ ∗ ∗ , p < 1. And μA p, , r > 0 and γA p, 1 , r < 1. γA q, x2 y2 y2 Therefore, we have              x y x y μA ∗ q, 1 1 , r = μA ∗ q, 1 , s ∩ μA ∗ s, 1 , r x2 y2 x2 y2 ∗

s∈Q

        x y ≥ μA ∗ q, 1 , p ∩ μA ∗ p, 1 , r > 0 x2 y2              x y x y Also, γA ∗ q, 1 1 , r = s∈Q γA ∗ q, 1 , s ∪ γA ∗ s, 1 , r x2 y2 x2 y2         x1 y ∗ ∗ , p ∪ γA p, 1 , r < 1. Therefore r ∈ WKIS(q). ≤ γA q, x2 y2 Proposition 3.2 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA. Let us assume that A, B ⊂ Q, then we have the following (I) (II) (III) (IV)

If A ⊂ B, WKIS(A) ⊂ WKIS(B). A ⊂ WKIS(A) WKIS(A ∪ B) = WKIS(A) ∪ WKIS(B) WKIS(A ∩ B) ⊂ WKIS(A) ∩ WKIS(B)

Proof The proofs of I, II, III, IV are straight forward. Definition 3.2 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA. letp, q ∈ Qand A ⊆ Q. We say that M satisfies intuitionistic exchange property if whenever p ∈ WKIS(A ∪ {q}) and p ∈ / WKIS(A), then q ∈ WKIS(A ∪ {p}). Proposition 3.3 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA, then the following statements are equivalent.

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(i) M satisfies intuitionistic exchange property. (ii) for every p, q ∈ Q, p ∈ WKIS(q) ⇔ q ∈ WKIS(p) Proof The proof consists of two parts. (i)⇒ (ii): Assume that M satisfies intuitionistic exchange property. Let p, q ∈ Q be such that p ∈ WKIS(q)= WKIS(∅ ∪ q). Also it is to note that p ∈ / WKIS(∅), which means q ∈ WKIS(∅ ∪ p)= WKIS(p). Similarly the other part can be established by tracing the above steps. (ii)⇒ (i): suppose that we have for every p, q ∈ Q, p ∈ WKIS(q) ⇔ q ∈ WKIS(p). / Also let A ⊆ Q. Then by the Definition 3.2, we know that p ∈ WKIS(A ∪ {q}) and p ∈ WKIS(A), then q ∈ WKIS(A ∪ {p}), which means p ∈ WKIS(q), Hence q ∈ WKIS(p) ⊆ WKIS(A ∪ {p}). Therefore M satisfies intuitionistic exchange property. Definition 3.3 Suppose that M = (Q, , ρ, A, I, F)is a WKIFA.  ∗  Then M is said to be intuitionis y1 ∈ and there exists s ∈ Q, such that tic retrievable if for all q ∈ Q, y2 ∗            ∗ y y x  such μA ∗ q, 1 , s > 0 and γA ∗ q, 1 , s < 1⇒ there exists 1 ∈ y2 y2 x2 ∗         x x that μA ∗ s, 1 , q > 0 and γA ∗ s, 1 , q < 1. x2 x2 Definition 3.4 Suppose that M = (Q, , ρ, A, I , F) is aWKIFA. q and  Let∗p,q, r ∈ Q Then  r are  said  to  y y1 ∈ such that μA ∗ p, 1 , q > be intuitionistic p-related if there exists y2 ∗ y            2 y y y 0, μA ∗ p, 1 , r > 0, γA ∗ p, 1 , q < 1 and γA ∗ p, 1 , r < 1. We y2 y2 y2 also say that q and r are intuitionistic p-twins if q and r are intuitionistic p-related and WKIS(q) = WKIS(r). Proposition 3.4 Suppose that M = (Q, , ρ, A, I , F) is a WKIFA. Then M is Intuitionistic retrievable if and only if, for all p, q, r ∈ Q, if q and r are said to be intuitionistic p-related then q and r are intuitionistic p-twins. Proof The proof is a direct consequence of the Definitions 3.3 and 3.4. Definition 3.5 Suppose = (Q, , is a WKIFA.   thatM   ρ,A, I ,F)    M  iscalled switching   if and  only if a1 a1 a1 a1 , q = μA q, , p γA p, , q = γA q, , p and for μA p, a2 a a2 a2    2  a1 in all p, q in Q and . a2 

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If M is both switching and commutative [2] then M is called Watson Crick Intuitionistic Fuzzy switchboard Machine (WKIFSM). Proposition 3.5

  x Suppose that M = (Q, , ρ, A, I , F) is a WKIFSM, then ∀p, q ∈ Q, ∀ 1 ∈ x2  ∗              ∗ q, x1 , p ∗ p, x1 , q ∗ q, x1 , p (i) μ = μ and (ii) γ = A A A ∗ x2 x2 x2     x γA ∗ p, 1 , q x2  ∗ . We prove the result by induction on |x1 | = |x2 | = n. ∗         λ x x1 = = If n = 0, that is if , then we have μA ∗ q, 1 , p x2 x2 λ             λ λ x and have μA ∗ q, ,p = μA ∗ p, ,q = μA ∗ p, 1 , q x2 λ λ                 x λ λ x γA ∗ q, 1 , p = γA ∗ q, , p = γA ∗ p, , q = γA ∗ p, 1 , q . x2 x2 λ λ Hence the result is true for n = 0.    ∗  z with |z1 | = |z2 | = n − 1, n > 0. Assume that the result is true for all 1 ∈ z2 ∗             z1 z1 z ∗ ∗ ∗ ,p = μA p, , q and γA q, 1 , p = That is μA q, z2 z2 z2     z γA ∗ p, 1 , q z2     ∗         x1 z1 a1 a1 x ∈ ∈ be such that = , Let . 1 a2 x2 ∗ x2 z2 a2               x z a ∗ q, z1 , r = μA ∗ q, 1 1 , p = ∩ then μA ∗ q, 1 , p μ A r∈Q x z2 a2 z2    2 a μA r, 1 , p a2          z a = r∈Q μA ∗ r, 1 , q ∩ μA p, 1 , r z2 a2          z a = r∈Q μA ∗ r, 1 , q ∩ μA ∗ p, 1 , r z2 a2          a1 z ∗ ∗ , r ∩ μA r, 1 , q = r∈Q μA p, a z2   2      z a a z 1 1 1 1 ∗ ∗ , q = μA p, ,p = μA p, a z z2 a2   2 2  x = μA ∗ p, 1 , q x2 Proof

Let p, q ∈ Q,



x1 x2





∈

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        x z a (ii)γA ∗ q, 1 , p =γA ∗ q, 1 1 , p x z a 2     2 2    z a = r∈Q γA ∗ q, 1 , r ∪ γA r, 1 , p z2 a2          z1 a ∗ , q ∪ γA p, 1 , r = r∈Q γA r, z a   2   2    z a 1 , q ∪ γA ∗ p, 1 , r = r∈Q γA ∗ r, z2 a2          a z = r∈Q γA ∗ p, 1 , r ∪ γA ∗ r, 1 , q a2 z2         a z z a = γA ∗ p, 1 1 , q = γA ∗ p, 1 1 , p a z z2 a2   2 2  x = γA ∗ p, 1 , q . Hence the proof. x2 Proposition 3.6

    x1 y1 Suppose that M = (Q, , ρ, A, I , F) is a WKIFSM, then μA q, ,p = x y            2 2 y x x y y x μA ∗ q, 1 1 , p and γA ∗ q, 1 1 , p = γA ∗ q, 1 1 , p for all p,q in y2 x2 x2 y2 y2 x2      ∗ y  x1 , 1 ∈ Q and x2 y2 ∗ ∗

    ∗ y  x1 , 1 ∈ x2 y2 ∗ | = |y2 |= n. We prove this theorem on |y1    by induction      λ x λ λ x1 y1 x x y = Let n = 0, then = 1 = , and hence 1 1 = 1 y2 x2 y2 x2 λ x2 λ λ x2   y1 x1 = y2 x2 Therefore                 x1 y1 x1 λ x1 λ x1 ∗ ∗ ∗ ∗ , p =μA q, , p =μA q, ,p μA q, , p =μA q, x y x2 λ x2 λx  2 2          2 y x x y x λ and γA ∗ q, 1 1 , p = γA ∗ q, 1 = μA ∗ q, 1 1 , p ,p = y2 x2 x2 y2 x2 λ             x λ x1 y x , p = γA ∗ q, 1 1 , p . Hence the basis is γA ∗ q, 1 , p = γA ∗ q, x2 y2 x2 λ x2 true.    ∗ z  Assume that the result is true for all 1 ∈ with |z1 | = |z2 | = n − 1, n > 0. z2 ∗

Proof

(i) Let p,q be in Q and



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            x z x z x z That is μA ∗ q, 1 1 , p = μA ∗ p, 1 1 , q and γA ∗ q, 1 1 , p = x z x z x2 z2     2 2     2 2  ∗    x z a1   y1 y ∈ be such that = γA ∗ p, 1 1 , q . Let . 1 ∈ x2 z2 a2 y2 ∗ y2    z1 a1 , then z2 a2         x1 y1 x1 z1 a1 ∗ ∗ , p = μA q, ,p μA q, x2 y2 x2 z2 a2          x z a = r∈Q μA ∗ q, 1 1 , r ∩ μA r, 1 , p x2 z2 a2          z x a = r∈Q μA ∗ q, 1 1 , r ∩ μA r, 1 , p z2 x2 a2          z1 x1 a ∗ , q ∩ μA p, 1 , r = r∈Q μA r, z x a2   2 2       x a1 z 1 1 ∗ , r ∩ μA r, ,q = r∈Q μA p, a2 z2 x2     a z x = μA ∗ q, 1 1 1 , p a2 z2 x2          a1 z1 x ∗ ∗ , r ∩ μA r, 1 , q = r∈Q μA p, a z x   2 2    2   a z x = r∈Q μA ∗ p, 1 1 , r ∩ μA ∗ r, 1 , q z2 a2 x2         z a x z a x = μA ∗ p, 1 1 1 , q =μA ∗ q, 1 1 1 , p z a x z2 a2 x2   2 2 2  y x = μA ∗ q, 1 1 , p y2 x2         x1 y1 x1 z1 a1 ∗ ∗ , p =γA q, ,p (ii)γA q, x y2 x z a 2     2 2 2   x z a = r∈Q γA ∗ q, 1 1 , r ∪ γA r, 1 , p x2 z2 a2          z x a = r∈Q γA ∗ q, 1 1 , r ∪ γA r, 1 , p z2 x2 a2          z x a = r∈Q γA ∗ r, 1 1 , q ∪ γA p, 1 , r z2 x2 a2          a1 z1 x1 ∗ , r ∪ γA r, ,q = r∈Q γA p, a2 z2 x2     a z x = γA ∗ q, 1 1 1 , p a2 z2 x2          a z x = r∈Q γA ∗ p, 1 1 , r ∪ γA ∗ r, 1 , q a2 z2 x2

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        ∗ p, z1 a1 , r ∪ γ ∗ r, x1 , q γ A r∈Q A z2 a2 x2         z a x z a x = γA ∗ p, 1 1 1 , q =γA ∗ q, 1 1 1 , p z a x z2 a2 x2   2 2 2  x y = γA ∗ q, 1 1 , p . y2 x2 Therefore the result is true for every n. Hence the proof.

=



4 Application Using the characterizations of Watson Crick Intuitionistic Fuzzy Automaton, Pattern Recognition process can be effectively done. The Watson Crick Intuitionistic Fuzzy Automaton can handle the diversity of patterns by establishing imprecise models.

5 Conclusion and Future Scope This research paper examined in depth a few of the Characteristic properties of WKIFA, including intuitionistic successor property, intuitionistic exchange property, intuitionistic retrievable property, and intuitionistic Fuzzy Switchboard machine. Future scope intends to study Watson Crick Intuitionistic Fuzzy Grammars.

References 1. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 2. Jansirani, N., Vijayaraghavan, N., Dare, V.R.: Watson crick intuitionistic fuzzy automata. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) Intelligent and Fuzzy Systems: Digital Acceleration and the New Normal-Proceedings of the INFUS 2022 Conference, vol. 1, pp. 606–615. Cham: Springer International Publishing (2022). https://doi. org/10.1007/978-3-031-09173-5_70 3. Jun, Y.B.: Intuitionistic fuzzy finite state machines. J. Appl. Math. Comput. 17(1–2), 109–120 (2005) 4. Jun, Y.B.: Intuitionistic fuzzy finite switchboard state machines. J. Appl. Math. Comput. 20(1– 2), 315–325 (2006) 5. Srivastava, A.K., Tiwari, S.P.: Intuitionistic fuzzy automata and associated fuzzy topologies. In: 2007 International Conference on Computing: Theory and Applications (ICCTA’07), pp. 267– 271. IEEE (2007)

An Intelligent Data Analysis Approach for Women with Menopausal Genitourinary Syndrome with Intuitionistic Fuzzy Logic Pavel Dobrev1,2 and Evdokia Sotirova1,3(B) 1 Faculty of Public Health and Health Care, Prof. Assen Zlatarov University,

Burgas 8010, Bulgaria [email protected], [email protected] 2 Oncology Complex Center - Burgas, 86 Demokratsiya Blvd, Burgas 8000, Bulgaria 3 Laboratory of Intelligent Systems, Prof. Asen Zlatarov University, Burgas 8010, Bulgaria

Abstract. Genitourinary syndrome of menopause is a set of symptoms that develop after and around menopause, due to low estrogen levels and hypo- and atrophic changes in the genitourinary tract of women. The study analyzed data on 67 patients with genitourinary syndrome of menopause after surgical and medical (hormonal and chemotherapy) castration aged 30 to 45 years. 35 patients had a surgical castration and 32 patients are with drug therapy (chemo and/or hormone therapy). For the analysis an InterCriteria Analysis method is used. It is based on two mathematical theories: the theory of intuitionistic fuzzy sets, and the theory of index matrices. The investigated data contains information about different parameters: Age, Type of castration, Main disease, Therapy carried out, Vaginal PH, Vaginal cleanliness, Amount of lactobacilli, Subjective symptoms related to genitourinary syndrome, Vulvovaginal symptoms, etc. By applying the ICA method, the dependencies between the criteria (clinical results) by which the different patients are evaluated were investigated. The correlations between each pair of criteria are determined in the form of intuitionistic fuzzy pairs of values in the interval [0; 1]. This supports the decision-making process in patients with oncological disease (cervical cancer, mammary carcinoma, ovarian carcinoma) after surgical and medical castration regarding vaginal cleanliness, lactobacilli count, vaginal pH and subjective symptoms of vaginal and urinary component of menopausal genitourinary syndrome. As a result of the study, conclusions were made about the predictors of the genitourinary syndrome. Keywords: InterCriteria Analysis · Intuitionistic Fuzzy Logic · Genitourinary Syndrome

1 Introduction Genitourinary syndrome of menopause (GSM) is a hypoestrogenic condition of the external genitalia with urological and sexual manifestations [8] that affects more than 50% of postmenopausal women [15]. Because of sexual difficulty and the sensitivity © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 212–219, 2023. https://doi.org/10.1007/978-3-031-39774-5_26

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of discussing symptoms, GSM is largely underdiagnosed [10, 13]. Early detection and individually tailored pharmacologic (estrogen therapy, selective estrogen receptor modulator, synthetic steroids, etc.) and/or non-pharmacologic (laser therapies, moisturizers and lubricants, homeopathic remedies, lifestyle) treatment is paramount and important not only to improve the quality of life, but also to prevent exacerbation of symptoms in women with such a condition. GSM is most often diagnosed when the patient has dyspareunia (secondary to vaginal dryness). Common signs and symptoms, in order of prevalence and degree of atrophy, include vaginal dryness (in 75% of postmenopausal women), dyspareunia (38%), and vaginal itching and pain (15%) [14, 23]. When the vulvovaginal epithelium is insufficiently moistened, ulcers and fissures can develop during intercourse, causing dyspareunia. Vaginismus, or painful spasm of the vaginal muscles, can also occur as a physiological response when there is anxiety (anxiety) from anticipated sexual pain. Sexual manifestations (signs) are a continuation of those of the external genitalia. To manage and treat genitourinary syndrome in postmenopausal patients after surgical and medical castration, it is essential to analyze the subjective symptoms associated with the vaginal and urinary components of genitourinary syndrome and the frequency of their occurrence. The aim of this study was to investigate the relationship between parameters associated with oncological disease of GSM patients after surgical and medical castration (type of oncological disease: cervical cancer, mammary carcinoma, ovarian carcinoma; type of castration: surgical, medical; amount of lactobacilli; vaginal cleanliness; vaginal pH and subjective symptomatology of vaginal and urinary components of GSM). The ICA method was used to determine the correlations between the observed parameters by which each patient was assessed, [3]. ICA has shown very good results for analyzing a medical and health care data [9, 12, 21, 22], data on patients with oncological diseases [17], Malignant Neoplasms of the Digestive Organs [16, 19], Malignant Melanoma [18]. It can be applied successfully for other malignant data [20]. The paper is organized as follows: First, an introduction to the area under study is given in Sect. 1. The problem and its significance are presented. Section 2 includes a description of the dataset. 67 patients were studied, 35 with surgical castration and 32 with medical castration. The ICA method chosen to analyze the data, is presented. Section 3 presents the obtained correlations between the parameters observed for patients in the form of intuitionistic fuzzy pairs. In Sect. 4 some conclusions of the study are drawn. Section 5 lists the bibliography used for the study.

2 Dataset and Research Method 2.1 Description of Dataset The present study analyzed data on women with genitourinary syndrome of menopause who underwent surgical and medical castration. A questionnaire was developed to assess subjective symptoms, with which 67 patients (35 with surgical castration and 32 with medicinal castration) were surveyed [6, 7].

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Based on the analyzed parameters, the patients were divided as follows: – 3 age groups: “30–35” (10 patients); “36–40” (20 patients) and “41–46” (20 patients). – The age ranged from 30 to 45 years, and the mean age was 40.26 ± 3.91 years. – 3 main diseases: cervical cancer (24 patients), ovarian cancer (10 patients) and breast cancer (32 patients). – 2 types of castration: surgical castration (34 patients), drug therapy (32 patients). – 4 types of therapy conducted: surgery and radiotherapy (19 patients); breast surgery and drug therapy (32 patients); surgery and chemotherapy (8 patients); surgical intervention (7 patients). – 3 types of Vaginal PH: acidic pH (12 patients); alkaline pH (26 patients); highly alkaline pH (28 patients). – 4 types of Vaginal cleanliness: 1st degree (3 patients); II degree (22 patients); III degree (28 patients); IV degree (13 patients). – 3 types of Amount of lactobacilli: normal amount (14 patients); reduced amount (29 patients); missing amount (23 patients). 52.2% of the analyzed patients are with surgical castration, and 47.8% are with drug therapy (chemo and/or hormone therapy). Distribution of patients by main disease and age groups is shown on Fig. 1. 17

18 16

14

14 12

10

10 8 6

5

6

6 4

5

4 2

0

0

Cervical cancer patients with surgical castration

Ovarian cancer patients Breast cancer patients with with surgical castration medical castration

30-35

36-40

41-46

Fig. 1. Distribution of patients by main disease and age groups.

2.2 Introduction to the ICA Method For determination the correlations between the analyzed parameters for each patient the ICA approach is used [3]. It uses the Intuitionistic Fuzzy Sets (IFSs) [2], Index Matrices (IMs) [1], and Intuitionistic fuzzy pairs (IFPs) [4]. This gives a great possibility to determine the correlation between individual object parameters. The ICA method is applied on an indexed matrix on the columns of which are the analyzed objects (e.g., GSM patients) and on the rows are the criteria by which the

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objects are evaluated. After applying the ICA approach to the dataset, a new indexed matrix is obtained, along the rows and columns of which are the criteria. Its elements are intuitionistic fuzzy pairs of values between 0 and 1, [5] and represent the computed correlations for each pair of criteria. If the criteria are dependent on each other, they are in a consonance, and if they are independent of each other, they are in a dissonance. The scale for interpretation the obtained correlation between analyzed criteria is given on Fig. 2, [4].

Fig. 2. A scale for the degree of consonance and dissonance between observed parameters.

3 Implementation Data testing using the ICA method is carried out using specially developed software that is freely available online at: http://intercriteria.net/software [11]. The output data matrix on which the ICA method is applied is with 9 rows (for the criteria for evaluating the subjective symptomatology of patients) and 68 columns (for patients with GSM). After applying the ICA method, a new index matrix with 9 rows and 9 columns was obtained. Its elements present the correlations between any two criteria in the form of IFPs. The considered criteria are the following: C1: Age; C2: A type of castration; C3: Underlying disease; C4: Conducted therapy; C5: Vaginal PH; C6: Vaginal cleanliness; C7: Amount of lactobacilli; C8: Subjective symptomatology associated with GSM; C9: Vulvo-vaginal symptoms. After testing with the ICA simulator, two indexed matrices containing membership part (Table 1) and non-membership part (Table 2) of the IFPs that give the correlations of each pair of criteria were obtained.

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One IFP is in Positive consonance: “Type of castration” – “Main disease” with evaluation 0,875; 0,004. This mean that the type of castration directly depends on the main disease. The other 35 IFPs are in weak dissonance, dissonance or strong dissonance. This means, that the criteria are absolutely independent, i.e. the criteria are well chosen. Table 1. Result matrix with membership parts of the IFPs.

Table 2. Result matrix with non-membership parts of the IFPs.

The visualization of the obtained IFPs in the Intuitionistic fuzzy triangle is presented on Fig. 3.

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Fig. 3. Visualization of the obtained IFPs in the Intuitionistic fuzzy triangle.

4 Conclusions This paper examines data collected through a survey method among 67 female patients with genitourinary syndrome of menopause after surgical and medical (hormonal and chemotherapy) castration aged 30 to 45 years. A multicriteria decision-making approach was used to analyze the data, looking for existing dependencies in the set of criteria by comparing the values of the patients’ test results on these criteria. The selected data show an adequate determination of the investigated parameters. When these criteria are met, adequate results can be obtained without the excess information that occurs with highly correlated data. When conducting the research, one of the two parameters, which are connected in a strict consonance, can be removed. This could be the subject of a new study. The high coefficient of consonance obtained in the research done shows a strong relationship between the underlying disease and the type of castration, which shows the connection between these two parameters and could influence the doctor in making a decision-making process in patients with oncological disease (cervical cancer, mammary carcinoma, ovarian carcinoma) after surgical and medical castration regarding vaginal cleanliness, lactobacilli count, vaginal pH and subjective symptoms of vaginal and urinary component of menopausal genitourinary syndrome. In future work, the authors will apply the ICA approach to data collected after conducting a GSM treatment. To collect this data, a questionnaire with 14 questions related to the assessment of treatment outcomes (criteria) was developed. Finding correlations between the analyzed criteria evaluating the results of the conducted topical estrogen/dehydroepiandrosterone therapy and systemic estrogen therapy will support the treatment process. The use of an intelligent approach to predict the relief of GSM symptoms will assist in improving the quality of life of patients.

References 1. Atanassov, K. Index Matrices Towards an Augmented Matrix Calculus. Studies in Computational Intelligence Series, Vol. 573, Springer, Cham (2014) https://doi.org/10.1007/978-3319-10945-9 2. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012)

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3. Atanassov, K., Mavrov, D., Atanassova, V.: InterCriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic Fuzzy sets. Issues in Intuitionistic Fuzzy Sets and Generalized Nets 11, 1–8 (2014) 4. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. Notes on Intuitionistic Fuzzy Sets 19(3), 1–13 (2013) 5. Atanassov, K, Atanassova, V., Gluhchev, G.: InterCriteria analysis: ideas and problems, Notes on Intuitionistic Fuzzy Sets 21(1), 81–88 (2015) 6. Dobrev, P., Yordanov, A., Strashilov, St.: Synchronous primary cervical carcinoma and ovarian fibroma: challenge in surgery. Gazzetta Medica Italiana-Archivio per le Scienze Mediche 179(5), 375–7 (2020) 7. Dobrev, P., Strashilov, St., Yordanov, A.: Metastasis of malignant melanoma in ovarium simulating primary ovarian cancer: a case report. Gazetta Medica Italiana – Archivio per le Scienze Medicine 180, 867–869 (2021) 8. Gandhi, J., et al.: Genitourinary syndrome of menopause: an overview of clinical manifestations, pathophysiology, etiology, evaluation, and management. Am. J. Obstet. Gynecol. 215(6), 704–711 (2016) 9. Jekova, I., Vassilev, P., Stoyanov, T., Pencheva, T.: InterCriteria analysis: application for ECG data analysis. Mathematics 9(8), 854 (2021) 10. Hutchinson-Colas, J., Segal, S.: Genitourinary syndrome of menopause and the use of laser therapy. Maturitas 82, 342–345 (2015) 11. InterCriteria Analysis Portal, Software, https://intercriteria.net/software/ 12. Krumova, S., Todinova, S., Mavrov, D., Marinov, P., Atanassova, V., Atanassov, K., Taneva, S. G.: Intercriteria analysis of calorimetric data of blood serum proteome. Biochimica et Biophysica Acta (BBA)-General Subjects 1861(2), 409–417 (2017) 13. Labrie, F., Archer, D.F., Koltun, W., et al.: Efficacy of intravaginal dehydroepiandrosterone (DHEA) on moderate to severe dyspareunia and vaginal dryness, symptoms of vulvovaginal atrophy, and of the genitourinary syndrome of menopause. Menopause 23, 243–256 (2016) 14. North American Menopause Society, The role of local vaginal estrogen for treatment of vaginal atrophy in postmenopausal women: 2007 position statement of the North American Menopause Society. Menopause 14(quiz 370–1), 355–369 (2007) 15. Palacios, S., Castelo-Branco, C., Currie, H., et al.: Update on management of genitourinary syndrome of menopause: a practical guide. Maturitas 82, 308–313 (2015) 16. Sotirova, E., Bozova, G., Bozov, H., Sotirov, S., Vasilev, V.: Application of the InterCriteria Analysis Method to a data of Malignant neoplasms of the digestive organs for the Burgas region for 2014–2018, 2019 Big Data, Knowledge and Control Systems Engineering (BdKCSE), Sofia, Bulgaria, 1–6 (2019) 17. Sotirova, E., Petrova, Y., Bozov, H.: InterCriteria Analysis of oncological data of the patients for the city of Burgas. Notes on Intuitionistic Fuzzy Sets 25(2), 96–103 (2019) 18. Sotirova E., Bozova G., Bozov H., Sotirov S., Vasilev V. Application of the InterCriteria Analysis Method to a Data of Malignant Melanoma Disease for the Burgas Region for 2014– 2018. In: Atanassov K.T. et al. (eds) Advances and New Developments in Fuzzy Logic and Technology. Advances in Intelligent Systems and Computing, vol 1308. 166–174 (2021) 19. Sotirov S., Bozova G., Vasilev V., Krawczak M. Clustering of InterCriteria Analysis Data Using a Malignant Neoplasms of the Digestive Organs Data. In: Atanassov K.T. et al. (eds) Advances and New Developments in Fuzzy Logic and Technology. Advances in Intelligent Systems and Computing, vol 1308 193–201 (2021)

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20. Stoyanov, V., Petkov, D., Bozdukova, P.: Pott’s puffy tumor: a case report - Trakia journal of sciences, ISSN: 1313–3551, 18, Suppl. 1, 93–96 (2020) 21. Todinova, S., et al.: Blood plasma thermograms dataset analysisby means of intercriteria and correlation analyses for the case of colorectal cancer. Int. J. Bioautomation 20(1), 115 (2016) 22. Zaharieva, B., Doukovska, L., Ribagin, S., Michalíková, A., Radeva, I.: Intercriteria analysis of Behterev’s kinesitherapy program. Notes Intuitionistic Fuzzy Sets 23(3), 69–80 (2017) 23. Wines N.,Willsteed E. Menopause and the skin. Australas J Dermatol. 42 (quiz 159): 149–158 (2001)

Measuring Happiness: Evaluation of Elementary School Students’ Perception of Happiness Assessed by Intuitionistic Fuzzy Logic Gergana Avramova-Todorova

and Veselina Bureva(B)

“Prof. Dr. Assen Zlatarov” University, “Yakimov” Blvd., 8010 Burgas, Bulgaria [email protected], [email protected]

Abstract. The paper explores happiness as an emotional construct, part of people’s emotional intelligence. The motivation for studying the phenomenon is the statement, supported by a number of empirical studies, that the feeling of happiness is a sought-after state to which a person strives hard. Conceptually important is the question of how to define happiness as an emotional state and whether it can be validly measured. An experimental study was conducted with elementary school students regarding the possibility of awareness of this emotion. Various factors that can be predictors of happiness and their relationship in assessing the degree of satisfaction experienced by the surveyed respondents are discussed. The obtained results are processed by making use of intuitionistic fuzzy set (IFS) techniques. The information obtained is a starting point for future studies on the possibilities of achieving a sense of happiness and its objective evaluation. Keywords: Happiness Measurement · Intuitionistic Fuzzy Logic · Elementary Students

1 Introduction The pursuit of happiness as key to the meaning of human existence still remains one of the main human goals [5]. Happiness has been and continues to be the subject of a wide range of research. However, despite diverse studies, the question of its measurement in children of early school age remains controversial. This provides the focus of the present research, which aims to explore the possibilities of a new, proprietary metric to refine the measurement of happiness in children. The concept of happiness has many formulations, some of which even describe it broadly as a positive psychological state [6]. In the most general sense, they are united around the idea that it consists in a preponderance of positive emotions and a feeling of general well-being. Well-being can provide a useful point of view on how people are doing [8]. Well-being reflects not only healthy functioning and happiness [9], but also serves an evaluative function in the self-determination process [10]. When we discuss the subjective feeling of happiness, we should keep in mind that it can be considered in emotional and cognitive terms. The emotional or affective aspect of happiness refers to what people feel at a certain moment. The cognitive aspect of happiness refers to what they think about their experiences [12]. These two types of sense of well-being can coexist, but not compulsory. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 220–226, 2023. https://doi.org/10.1007/978-3-031-39774-5_27

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In this article, the focus is on the emotional aspect of happiness, based on the understanding that in children of early school age, experiential affect is the leading one, and the cognitive component is subordinate to feelings. Furthermore, at this stage of childhood development, it is difficult to predict cognitive well-being, which refers more to a whole life attitude and includes a greater number of predictors that are not applicable at this age - for example, satisfaction with social status, career, family status and others. There are many mental constructs that are considered in relation to happiness. One of the factors found to be related to well-being is life values. Each person has his own scale of values, which are graded by importance. We can define one part of them as personal, and the rest as social - that is, important for society as a whole. Despite the variety of characteristics of values, we dwell here on Schwartz’s conceptual framework. His model contains 10 types of values that a person experiences in the process of pursuing various goals in everyday life, such as: security, conformity, tradition, benevolence, universalism, self-directedness, stimulation, hedonism, achievement and power [11]. The table below presents a comparison between the model of Schwartz and the corresponding values in our proposed model (Table 1). Table 1. Comparison between the model of Schwartz and Questionnaire model of values Schwartz’s model of values

Questionnaire model of values

self-directedness

the choice to be yourselfs

benevolence

kind gestures

hedonism

to look at life on the bright side

universalism

wonderment about the world

stimulation

curiosity

The paper has the following structure. Section 1 presents a brief introduction on the definitions of happiness measurement. Section 2 presents the methods used for the investigation and study design. In Sect. 3 the ICA applications are discussed. Section 4 presents conclusion and future research remarks.

2 Methods and Study Design The questionnaire was created as a tool to assess the current state of happiness. It is suitable for children of primary school age. It is constructed on the basis of individual life values, set as symbolic themes, which are supposed to be related to the feeling of happiness. The values that make up the scales of the questionnaire are: the choice to be yourself, kind gestures, looking at life from the “good” side, curiosity about the world, inquisitiveness. Three statements are assigned to each of them, which examine the attitude to happiness, giving a numerical expression of the degree of the experienced emotion. The subject should rate the degree of happiness using a 5-point Likert-type scale: a) never: 1 point; b) rarely: 2 points; c) sometimes: 3 points; d) often: 4 points; e) all the time: 5 points. The total score of the methodology varies from 15 to 75 and forms the momentary “happiness quotient” of the researched respondent.

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The questionnaire items are: I have a good opinion of myself; I feel confident; I know what my strengths are; I like to socialize with friends; I care about my friends; I like to please my friends; I feel cheerful; I am positive about the future; I think of good things; I am interested in new things; I like being an explorer; I feel good when something surprises me; I like to learn new things; I get excited about unknown things; I like challenges. The research was conducted with a group of 83 respondents - students aged 8–9 years. They were randomly selected from the second grades of a general secondary school in the city of Burgas, Bulgaria. Participation is completely voluntary, with prior parental consent obtained and the experiment coordinated with the school administration.

3 Results and Discussion In the current investigation the results of the happiness measurements to elementary school students are analyzed using InterCriteria Analysis [3]. ICA is a tool for determining possible relationships between the data. It is based on the theories of intuitionistic fuzzy sets [2, 4] and index matrices [1]. ICA compare the relations between the objects/criteria and present intuitionistic fuzzy pairs as result. The investigation is made in two parts: in the first case ICA is applied to determine the dependencies between happiness measurements for the elementary school students; in the second case ICA is used to investigate the behavior between the groups of indicators. The investigation is applied over the data for 5 groups of indicators and 84 happiness measurements of the elementary school students. The ICrAData software is used for the intuitionistic fuzzy comparison between the indicators and elementary school students [7]. 3.1 An Application of ICA for Determining Relationships Between the Elementary School Student’s Happiness Indicators The results of ICA application for elementary school students are presented in two index matrices by 84 × 84 containing degrees of membership and degrees of non-membership. The outcomes of ICA application are presented directly. The results are interpreted in the intuitionistic fuzzy triangle. It is obvious that the elementary school students have different level of happiness according to the indicators. The ICA application over the happiness measurement data give us the following outcomes in the table below, where elementary school students (ESS) are given with their number from the overall list with their results (Table 2). The outcome is: 3 pairs of elementary school students are in weak positive consonance; 2 pairs of elementary school students are in positive consonance and 1 pair of elementary school students is in strong positive consonance. The degrees of happiness between the selected elementary school students are very similar (Fig. 1). The degrees of happiness between all other elementary school students are independent. The results present 3 group of elementary school students that have different relation between their happiness degrees.

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Fig. 1. Results of ICA for determining student’s happiness relations averaged by groups of indicators Table 2. Outcome of the personal “happiness quotient” of the researched respondents The elementary school students (ESS) with opposite degrees of happiness: • ESS15–ESS37, 0.00, 1.00; • ESS02–ESS25, 0.00, 0.90; • ESS01–ESS28, 0.10, 0.90;

• ESS02–ESS27, 0.00, 0.80; • ESS02–ESS65, 0.10, 0.80; • ESS02–ESS04, 0.20, 0.80;

The elementary school students (ESS) with very similar degrees of happiness are: • ESS03–ESS23, 1.00, 0.00; • ESS04–ESS59, 0.90, 0.00; • ESS01–ESS16, 0.80, 0.00;

• ESS01–ESS12, 0.80, 0.10; • ESS02–ESS41, 0.90, 0.10; • ESS04–ESS08, 0.80, 0.20

3.2 An Application of ICA for Determining the Accuracy of the Selected Indicators The results of ICA application over the group of indicators are presented in two index matrices representing the degrees of membership and degrees of non-membership in Table 3 and Table 4. Table 3. Degrees of membership for groups of indicators μ

Indicator 1

Indicator 2

Indicator 3

Indicator 4

Indicator 5

Indicator 1

1.00

0.48

0.53

0.45

0.51

Indicator 2

0.48

1.00

0.48

0.47

0.50

Indicator 3

0.53

0.48

1.00

0.55

0.58

Indicator 4

0.45

0.47

0.55

1.00

0.60

Indicator 5

0.51

0.50

0.58

0.60

1.00

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G. Avramova-Todorova and V. Bureva Table 4. Degrees of non-membership for groups of indicators

ν

Indicator 1

Indicator 1 Indicator 2

Indicator 2

Indicator 3

Indicator 4

Indicator 5

0.00

0.24

0.25

0.31

0.27

0.24

0.00

0.25

0.24

0.22

Indicator 3

0.25

0.25

0.00

0.23

0.22

Indicator 4

0.31

0.24

0.23

0.00

0.18

Indicator 5

0.27

0.22

0.22

0.18

0.00

The received outcomes determine the information that the groups of indicators are correctly selected. 2 pairs of groups of indicators are in dissonance and 8 pairs of groups of indicators are in strong dissonance (Table 5). All groups of indicators are independent: they are suitable for measuring happiness of the elementary school students. Table 5. Relationships between the groups of indicators Type of correlations

Number of pairs of indicators

strong positive consonance [0,95; 1,00]

–

positive consonance [0,85; 0,95)

–

weak positive consonance [0,75; 0,85)

–

weak dissonance [0,67; 0,75)

–

dissonance [0,57; 0,67)

2

strong dissonance [0,43; 0,57)

8

dissonance [0,33; 0,43)

–

weak dissonance [0,25; 0,33)

–

weak negative consonance [0,15;0,25)

–

negative consonance [0,05;0,15)

–

strong negative consonance [0,00;0,05)

–

The results are visualized in the intuitionistic fuzzy triangle (Fig. 2). Obviously, all indicators are in the area of dissonance. The indicators are correctly selected for the elementary school students’ happiness measurement.

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Fig. 2. Results of ICA for criteria relations determining

4 Conclusion The derived data analysis shows a very good intercriteria correlation between the indicators of happiness in the proposed tool for its measurement. This leads us to think that they are correctly selected and applied in the questionnaire for the evaluation of happiness quotient for children in primary school age. The collected and analyzed data can be the basis for further research on the validity of the proposed instrument, as well as for additional analysis of a possible relationship between happiness and its predisposition with personality traits, such as extraversion and emotional stability. Acknowledgments. The authors are thankful for the support provided by the European Regional Development Fund and the Operational Program “Science and Education for Smart Growth” under contract UNITe No. BG05M2OP001-1.001-0004-C01 (2018–2023).

References 1. Atanassov, K.: Index Matrices Towards an Augmented Matrix Calculus. SCI, vol. 573, Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10945-9 2. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012). https://doi.org/ 10.1007/978-3-642-29127-2 3. Atanassov, K., Mavrov, D., Atanassova, V.: InterCriteria decision making: a new approach for multicriteria decision making, based on index matrices and intuitionistic fuzzy sets. Issues Intuitionistic Fuzzy Sets General. Nets 11, 1–8 (2014) 4. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. Notes Intuitionistic Fuzzy Sets 19(3), 1–13 (2013) 5. Del Río, N. Holder, M.D.: Happiness in children. Measurement, correlates and enhancement of positive subjective well-being. Appl. Res. Qual. Life 8, 407–408 (2013) 6. Ingelström, M., van der Deijl, W.: Can happiness measures be calibrated? Synthese 199(3–4), 5719–5746 (2021). https://doi.org/10.1007/s11229-021-03043-5

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7. Ikonomov, N., Vassilev, P., Roeva, O.: ICrAData – software for InterCriteria analysis. Int. J. Bioautomation 22(1), 1–10 (2018) 8. Wong, P.T.: Positive psychology 2.0: towards a balanced interactive model of the good life. Can. Psychol. 52, 69–81 (2011) 9. Ryan, R.M., Huta, V.: Wellness as health functioning or wellness as happiness: the importance of eudemonic thinking. J. Posit. Psychol. 4, 202–204 (2009) 10. Ryan, R.M., Huta, V., Deci, E.L.: Living well: a self-determination theory perspective on eudaimonia. J. Happiness Stud. 9, 139–170 (2008) 11. Schwartz, S.H.: Universals in the content and structure of values: theoretical advances and empirical tests in 20 countries. In: Zanna, M.P. (ed.) Advances in Experimental Social Psychology, vol. 25, pp. 1–65. Academic Press, New York (1992) 12. Weinstein, J.: Be happier now (2023)

Selecting an Employer: Evaluation of University Students’ Perception About Business Companies Assessed by Intuitionistic Fuzzy Logic Milen Todorov and Veselina Bureva(B) “Prof. Dr. Assen Zlatarov” University, “Prof. Yakimov” Blvd., Burgas 8010, Bulgaria [email protected], [email protected]

Abstract. Employer attractiveness could be associated with benefits that a potential job seeker sees in working for a particular company. Inducement concerning work may be distinctive for each generation, and would require adaptation in management practices. It was found that different generations tend to prioritize different elements in the workplace. Generally, these elements can be related to the employer attractiveness recognized as instrumental (e.g. salary package, flexible schedule, location, etc.) and symbolic e.g. business innovation degree, culture, prestige, etc.) attributes. Identifying the influence of those attributes can contribute companies to set employer branding and internal marketing strategies. The paper shed light on significant factors related to both attributes which attract students from largest national university in south-east Bulgaria “Prof. Dr. Asen Zlatarov” to employers. Data was gathered as a result of self-completion electronic questionnaire. The obtained results are processed by making use of intuitionistic fuzzy set (IFS) technique. Intuitionistic Fuzzy estimations are defined to investigate the student’s preferences according to the business companies. The received results are analyzed and future directions for employer’s behavior are determined. It was found that some attributes, for example interpersonal relations and organizational culture are considered by the students to be more important than others. Considering the dynamic of labor market these findings are expected to be valuable for all business companies in order to keep their internal marketing strategy up to date. Keywords: Intuitionistic Fuzzy Sets · Modelling · Employer attractiveness

1 Introduction From marketing point of view, the term brand is usually related to a product, but many things can also be branded, including business companies or organizations. In broader definition “employer brand” could be related to functional, economic, and psychological benefits [7]. The employer branding alternatively called internal marketing is accepted as significant market factor [11]. The role of internal marketing has significant contribution on employees’ motivation and satisfaction [8]. By making use of internal marketing the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 227–233, 2023. https://doi.org/10.1007/978-3-031-39774-5_28

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job positions can be managed in a similar manner to products, and company should use marketing techniques to offer jobs that meet both the employees’ and the firm’s needs. Another brand element - brand attractiveness [4] could be defined as set of factors. Five factors have been considered as most important: social value, economic value, interest value, development value and application value. Other authors have identified that employer brand attractiveness could be related to personal characteristics [1, 5, 14]. Employer branding could be represented by three-step process: (i), organizations create a value proposition, (ii) market this proposition to the external customers, and (iii) application of internal marketing of the employer brand in order to create a workforce committed and loyal to the organization employees’ [10]. A specific group that is of special interest as young talents for companies are university students. The key aspect that can be attributed to this group is that they normally have limited or no job experience. On a national (or even regional) level it will be valuable to survey the university student’s attitudes toward employers. The obtained information could be used successfully in collaborative activities between academic career centers and human resources departments. The aim of this work is to identify the employee benefits that students from largest University in south-east Bulgaria - “Prof. Dr. Asen Zlatarov”, Burgas relate to companies, for which they will work. The obtained results are expected to be valuable for employers in order to adapt their job proposal taking into account the opinion of their future employees. The paper has the following structure. Section 1 presents a brief introduction on the process of selecting an employer and the evaluation of university students’ perception about business companies assessed. Section 2 contains the methodology and the evaluation of the student’s preferences according to the process of selecting an employer. In Sect. 3 the received results of the investigation are discussed. Section 4 presents conclusion and future remarks.

2 Materials and Methods 2.1 Methodology For data collection, an electronic questionnaire has been sent to the students with active institutional profile. The questionnaire contained 33 questions (Table 1) related to economic value, interest value, social value, development value and application. The return rate was about 35% (clicked/responded by data from e-mail campaign), with a final number of 92 responses. The survey was attended by 36% (33) of students (1st , 2nd and 3rd year of education) and 64% (59) of graduates (4th year of education and masters). 77% of women and 23% of men participated in the study. The structure of the respondents does not differ from the structure of the University’s students, who are mostly female. 2.2 Data Processing by Intuitionistic Fuzzy Sets The concept of intuitionistic fuzzy sets is proposed by Krassimir Atanassov in 1983. The intuitionistic fuzzy set is an extension of fuzzy set [2]. The function μA (x) defines

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the degree of membership of an element x to set A, evaluated in the interval [0; 1]. The function ν A (x) defines the degree of non-membership of an element x to set A, evaluated in the interval [0; 1]. Thereafter the degree of uncertainty can be defined as π A (x) = 1 − μA (x) − ν A (x). The intuitionistic fuzzy set has the following form A = {x, μA (x), νA (x) | x ∈ E}. The Intuitionistic Fuzzy Pair (IFP) is an object with the form a, b, where a, b ∈ [0, 1] and the condition a + b ≤ 1 is satisfied. Its components (a and b) are interpreted as degrees of membership and non-membership. IFP is used as an estimation of some object or process [3, 6, 9, 12, 13]. Comparison between elements of any two IFSs is performed using pairwise comparisons between their respective elements’ degrees of membership and non-membership to both sets. The input dataset for intuitionistic fuzzy evaluation contains 33 questions containing the numerical answers of 92 respondents (students). The students answer using the numbers from 1 to 7 to assign its preferences of the employers. The numbers from the students answers are distributed using the following scale: • • • •

Rather positive answers (p) – the numbers 6 and 7; Rather uncertain answers (a) – the values 5 and 4; Rather negative answers (n) – the numbers 3, 2, 1. The set of all answers (s) – the set of all values from 1 to 7.

The intuitionistic fuzzy evaluations are presented. The degree of membership and the degree of non-membership are calculated using the following formulas: μ=

n p and ν = s s

The degree of uncertainty has the following form: π=

a s

3 Results and Discussion The degree of membership and the degree of non-membership for all questions and answers are written in the form of intuitionistic fuzzy pairs in Table 1. The degrees of memberships from Table 1 are visualized in the Fig. 1. The numbers from 1 to 33 represent the number of the question in the survey. The obtained results and could be interpreted as follows: 1) Eight questions (6, 8, 12, 14, 19, 22, 24, and 25) has values below 0.60. The lowest value of 0.37 is obtained for question # 12 addressing the praise. It could be assumed that praise without connection with specific achievement is not considered as important benefit in daily work process. Next, offering additional household services covered by the employer (#24), is assessed as minor benefit. Here, the largest value in the group is given to question #6 – which concerns the possibility for remote work / flexible working hours.

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Table 1. Intuitionistic fuzzy evaluations represented as intuitionistic fuzzy pairs for each question #

Question text

Intuistuinistic Fuzzy Estimation of the Students Answers

1

Opportunity for career growth

0.77, 0.03

2

Good interpersonal relationships

0.83, 0.04

3

Organizational culture

0.83, 0.01

4

Nature of work

0.74, 0.03

5

Opportunity to generate new ideas

0.73, 0.05

6

Possibility for remote work / flexible working hours

0.54, 0.10

7

Innovations and technologies implemented in the work

0.63, 0.02

8

The company’s portfolio (products and services)

0.51, 0.07

9

Additional benefits Recognition and appreciation for a job well done

0.68, 0.07

10

Respect for both professional and significant 0.61, 0.03 personal issues and events

11

Providing benefits

0.67, 0.03

12

Praise

0.37, 0.13

13

Providing amenities in the workplace

0.72, 0.01

14

Providing a fun work atmosphere

0.52, 0.11

15

Reduction of stress factors

0.76, 0.02

16

Employee support in the necessary moment of personal crises

0.71, 0.07

17

Recognition and appreciation of well-done projects

0.76, 0.01

18

Special bonuses for successfully obtained professional certificates and diplomas

0.77, 0.02

19

Providing benefits and privileges for the employee’s family

0.41, 0.21

20

Providing additional medical care

0.62, 0.09

21

Providing training and personal professional 0.71, 0.00 development programs

22

Supporting and promoting work and personal life

0.52, 0.08

23

Develop and offer flexible working hours

0.64, 0.07 (continued)

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Table 1. (continued) #

Question text

Intuistuinistic Fuzzy Estimation of the Students Answers

24

Offering additional household services

0.39, 0.08

25

Understanding the individual needs of the employee and complying with them

0.52, 0.08

26

Listening to the employee and showing interest in his ideas

0.72, 0.03

27

Granting of additional leave

0.61, 0.11

28

Evaluate new ideas and encourage the acceptance of calculated risk

0.66, 0.02

29

Support for individual initiatives

0.62, 0.08

30

Encouraging creativity

0.77, 0.03

31

Creating an atmosphere of trust

0.80, 0.02

32

Encourage the development of employee talent

0.73, 0.00

33

Hiring the right people from the beginning

0.86, 0.00

Fig. 1. Quantitative evaluation of the student’s perception about business companies

2) Sixteen questions (1, 2, 3, 4, 5, 13, 15, 16, 17, 18, 21, 26, 30, 31, 32 and 33) which correspond to 48.5% of the total number of questions has values above 0.70. Among questions one should pointed out those with high values. Creating an atmosphere of trust (question #31) is found to be assessed with value of 0.80. Good interpersonal relationships and Organizational culture (#2 and #3) correspond to value of 0.83 for both. The question with highest value of 0.86 is related to one of the most important factors in the companies – “Hiring the right people from the beginning”.

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4 Conclusions The dynamic changes in labor market requires evaluation of many contributing factors in order to support professional career of young talents. In this study the perception of university students about business companies as preferred employer is investigated. The received results are analyzed and future directions for employer’s behavior are determined. It was found that some attributes, for example the rule “Hiring the right people from the beginning” and “Interpersonal relations” and “Organizational culture” are considered by the students to be most important than others. Alternatively, attributes such as “Praise” and “Additional household services covered by the employer” was assessed with low importance. These findings are expected to be valuable for internal marketing strategy of companies in order to set the focus on attributes that corresponds to preferences of recently university graduates. The received results are analyzed and future directions for employer’s behavior are determined. Acknowledgments. The authors are thankful for the support provided by the European Regional Development Fund and the Operational Program “Science and Education for Smart Growth” under contract UNITe No. BG05M2OP001-1.001-0004-C01 (2018–2023).

References 1. Angelova, G.: Creativity in axiety – techniques to deal with anxiety in children. Knowl. Int. J. 43(2), 371–376 (2020) 2. Atanassov, K.: Generalized Nets and Intuitionistic Fuziness in Data Mining, Professor Marin Drinov Publishing House of Bulgarian Academy of Sciences (2020) 3. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. In: 17th International Conference on IFSs, Sofia, 1–2 November 2013, Notes on Intuitionistic Fuzzy Sets, vol. 19, no. 3, pp. 1–13 (2013) 4. Berthon, P., Ewing, M., Hah, L.L.: Captivating company: dimensions of at-tractiveness in employer branding. Int. J. Advert. 24(2), 151–172 (2005) 5. Edwards, M.R.: An integrative review of employer branding and OB theory. Pers. Rev. 39(1), 5–23 (2010) 6. Kim, T., Sotirova, E., Shannon, A., Atanassova, V., Atanassov, K., Jang, L.: Interval valued intuitionistic fuzzy evaluations for analysis of a student’s knowledge in university e-learning courses. Int. J. Fuzzy Logic Intell. Syst. 18(3), 190–195 (2018) 7. Kotler, P.: Marketing Management: Analysis, Planning, Implementation and Control, 8th edn. Prentice-Hall Inc., Englewood Cliffs (1994) 8. Love, L.F., Singh, P.: Workplace branding: leveraging human resources management practices for competitive advantage through “best employer” surveys. J. Bus. Psychol. 26, 175–181 (2011) 9. Madera, Q., Castillo, O., García-Valdez, M., Mancilla, A., Sotirova, E., Sotirov, S.: A method for optimizing a bidding strategy for online advertising through the use of fuzzy intuitionistic systems. In: ICIFSTA’2016, 20–22 April 2016, Beni Mellal, Morocco, Notes on Intuitionistic Fuzzy Sets, vol. 22, no. 2, pp. 99–107 (2016) 10. Moroko, L., Uncles, M.: Characteristics of successful employer brands. J. Brand Manag. 16, 160–175 (2008)

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11. Mosley, R.: Customer experience, organisational culture and the employer brand. J. Brand Manag. 15, 123–134 (2007) 12. Sotirova, E., Shannon, A., Kim, T., Krawczak, M., Melo-Pinto, P., Rieˇcan, B.: Intuitionistic fuzzy evaluations for the analysis of a student’s knowledge in university e-learning courses. In: Hadjiski, M., Atanassov, K. (eds.) Intuitionistic Fuzziness and Other Intelligent Theories and Their Applications. SCI, vol. 757, pp. 95–100 (2018). https://doi.org/10.1007/978-3-31978931-6_6 13. Sotirova, E., Petkov, T., Krawczak, M.: Generalized net modelling of the intuitionistic fuzzy evaluation of the quality assurance in universities. In: Kacprzyk, J., Szmidt, E., Zadro˙zny, S., Atanassov, K.T., Krawczak, M. (eds.) IWIFSGN/EUSFLAT -2017. AISC, vol. 643, pp. 341– 347. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66827-7_31 14. Wilden, R., Gudergan, S., Lings, I.: Employer branding: strategic implications for staff recruitment. J. Mark. Manag. 26(1–2), 56–73 (2010)

A Novel Intuitionistic Fuzzy Grey Model for Forecasting Electricity Consumption Bahadır Yörür1(B)

, Nihal Erginel2

2 , and Sevil Sentürk ¸

1 Kütahya Dumlupınar University, Kütahya 43100, Turkey

[email protected] 2 Eski¸sehir Technical University, Eski¸sehir 26520, Turkey

Abstract. The fact that electrical energy, which is a secondary energy type, is closely related to the country’s economy and social life and that it is difficult and costly to store reveals the problem of consumption forecasting. It is possible to express electricity consumption values as a single base value. Still, the values can include some uncertainty due to showing up and down deviations within the period under consideration. Therefore, it is reasonable to express consumption values with triangular fuzzy numbers. Thus, the values correspond to fuzzy numbers. For this reason, electricity consumption values can be said with membership and non-membership values and can make an estimation. This study proposes a triangular grey GM(1,1) model based on intuitionistic fuzzy sets called TIFGM(1,1) for long-term electricity consumption estimation. The theoretical background integrating the one-variable triangular grey model with intuitionistic fuzzy numbers was presented. Then a long-term consumption forecasting study was carried out for Turkey. For the evaluation of the proposed method, the results are compared with the triangular fuzzy number grey model (TFGM(1,1)) and standard GM(1,1) results. Keywords: Intuitionistic Fuzzy Sets · Grey Model · Electricity Consumption · Forecasting

1 Introduction Electricity consumption estimation is essential for electricity demand management, and many studies have been carried out using different methods to reveal an accurate consumption model. Among current methods used for this purpose, grey prediction models, which require less data than other methods, have been increasing. In this way, the desired estimation results can be achieved, and the risk of uncertainty caused by extensive sample data can be avoided [1]. The grey method was first proposed by Deng [2]. Compared to other methods, the grey model has advantages in working with incomplete information and small samples [3]. Recently, many researchers have extended the structure of the univariate first-order standard GM(1,1) model in different ways to improve prediction accuracy. Among these are studies in which a variable is considered in triangular fuzzy numbers including its © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 234–242, 2023. https://doi.org/10.1007/978-3-031-39774-5_29

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preference and range of variation [4]. In this study, the triangular fuzzy grey model used earlier is expanded with intuitionistic fuzzy numbers. The triangular intuitionistic fuzzy grey model (TIFGM(1,1)) is proposed for the first time in the literature. The remainder of the study is structured as follows. The fuzzy number literature in electricity consumption estimation is given in the second part. The third and fourth chapters present the theoretical background of the triangular fuzzy grey model and the proposed triangular intuitionistic fuzzy grey model, respectively. While the results of the application and analysis are given in the fifth section, the results are presented in general in the last section.

2 Literature Review As in many areas, grey estimation models have been used frequently in electricity consumption estimation. In parallel with these studies, fuzzy approaches are integrated into the methods. These studies performed the estimation process by converting the intervalvalued time series to the real number series [5–7]. However, this conversion method does not allow GM(1,1) to be applied to the interval value sequence. To overcome this problem, Zeng et al. proposed a GM(1,1) method developed to estimate a triangular fuzzy number sequence called TFGM(1,1). In the proposed method, there is no need to convert the sequence to a series of real numbers [8]. The triangle fuzzy grey model (TFGM (1,1)) present in the literature is very useful in giving the data’s maximum, minimum and average values. Zor and Çebi applied the TFGM (1,1) method to estimate the demand of a hospital [9]. In another study, the parameter optimization in the TFGM (1,1) model was made using by Moth Flame algorithm. Then the method was applied to hourly electric consumption data [10]. Electricity consumption values may vary during the period under consideration, whether in the short or long term. Therefore, electricity consumption values that can be characterized as a ternary interval number or a triangular fuzzy number can be expressed intuitionistic as a fuzzy number with membership and non-membership. Intuitionistic fuzzy numbers are beneficial in providing a flexible model for elaborating the uncertainty involved in real-life problems. The intuitionistic fuzzy set was introduced by Atanassov in 1983 as an extension of the fuzzy set [11]. This study uses the GM(1,1) method under fuzzy environment to estimate long-term electricity consumption. For this purpose, the intuitionistic fuzzy set was combined with TFGM(1,1) to increase the prediction accuracy. A new model called the triangular intuitionistic fuzzy grey model TIFGM(1,1) was developed and applied for the first time electricity consumption estimation. In this way, although there are few studies on combining intuitionistic fuzzy numbers with the grey system model, there is no study in which intuitionistic fuzzy numbers are integrated with grey estimation [12]. With the proposed TIFGM(1,1) model, GM (1,1) and triangular fuzzy grey model TFGM(1,1) was applied to the same data and the accuracy of the estimation methods was compared.

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3 Triangular Fuzzy Grey Model – TFGM(1,1) The model whose theoretical background is given below belongs to a model presented by integrating triangular fuzzy number with GM(1,1) [8]. X˜ (0) stands for triangular fuzzy number time series data.   (1) X˜ (0) = X˜ (0) (1), X˜ (0) (2), . . . , X˜ (0) (n)   (0) (0) (0) Equation (1) can be expressed explicitly as X˜ (0) (k) = XL (k), XM (k), XR (k) . “k” indicates the order of the data, and “0” indicates the original data. The steps of the TFGM (1,1) model are as follows. Step 1. The first order accumulating generating operation (AGO) is applied to the original triangular fuzzy sequential series X˜ (0) . Thus, X˜ (0) is transformed into a monotonically increasing series and expressed as in Eq. (2).   (2) X˜ (1) = X˜ (1) (1), X˜ (1) (2), . . . , X˜ (1) (n) X˜ (1) (k) is obtained by adding the previous original number as clearly seen in Eq. (3). (3) Step 2. The grey differential equation of the TFGM(1,1) model is obtained as in (4). ˜ k = 2, 3, . . . , n X˜ (0) (k) + aZ˜ (1) (k) = b,

(4)

Step 3. Equation (5) is obtained by the least squares method.          −1 −1 −1 aL a a = AT A AT YL , M = BT B BT YM , R = C T C C T YR (5) bL bM bR Each expression in Eq. (5) is clearly given in Eq. (6). In other expressions, it will be obtained as given in Eq. 6. ⎤ ⎡ ⎡ ⎤ (1) −Z L (2) − 1 −X L (2) ⎢ −X (3) ⎥ ⎢ ⎥ (1) ⎥ ⎢ ⎢ −Z L (3) − 1 ⎥ L ⎥ ⎢ ⎢ ⎥ (6) A=⎢ . ⎥ ⎥, YL = ⎢ . ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ ⎣ ⎦ . . (1) −X L (n) −Z L (n) − 1 The Z˜ value in Eq. (4) and (6) is expressed as in Eq. (7), and the λ value is generally taken as 0.5 as a net number.  ∼ ∼ Z˜ (1) (k) =λ X˜ (1) (k − 1) + 1− λ X˜ (1) (k), k = 2, 3, . . . , n (7)   (1) (1) (1) It is clearly expressed in the form of .Z˜ (1) (k) = ZL (k), ZM (k), ZR (k) .

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The improvement parameter “a” in Eq. (4) is the exact number and is obtained from the weighted average of aL , aM ve aR . (Eq. (9)). b˜ is the grey parameter and the triangular fuzzy number. It is expressed as b˜ = [bL , bM , bR ]. a = αaL + βaM + γ aR

(8)

In the weights used in Eq. (11), β must be equal to or greater than the other coefficients and the sum of the three coefficients must be “1” as shown in Eq. (9). α + β + γ = 1, β ≥ αveβ ≥ γ

(9)

Step 4. The reverse accumulating generating operation is applied. The time sensitive series are respectively; obtained by applying Eqs. (10), (11) and (12). (0)



X˜

(1) = X˜ (0) (1)

Equation (13) is taken as the first condition.   (1) ˜ b e−a(k−1) + X˜ (k) = X˜ (1) (1) − a 

(0)



X˜



(k) = X˜

(1)

(1)

(10)

b˜ , k = 2, 3, . . . , n a

(11)



(k) − X˜

(k − 1), k = 2, 3, . . . , n

(12)

4 Triangular Intuitionistic Fuzzy Grey Model – TIFGM(1,1) The intuitionistic fuzzy set (IFS) concept is characterized by three main degrees, including the degree of hesitation, where the sum of the degree of membership and degree of non-membership is 1 or less than 1 [13]. An IFS A on the U discourse universe is defined as an object of the form [11]:

In this expression, μA and ϑA represent the membership and non-membership degrees of element x in the set A, respectively. 0 ≤ μA (x) + ϑA (x) ≤ 1 The proposed TIFGM model extends the triangular fuzzy grey model with intuitionistic fuzzy numbers. The similarities and differences between the proposed TIFGM(1,1) model with the TFGM(1,1) model are presented below. The first step of the proposed method is applied in the same way as in the triangular fuzzy grey model approach. In this way, cumulative series are obtained. In the ˜ grey parameter calculations TFGM(1,1) model, “a” improvement parameter and “b” are made separately for the left, middle and right value series (Step 3). In the proposed TIFGM(1,1) model, all three values belonging to the same period are included in the

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model as triangular fuzzy numbers at the same time, together with the membership and ˜ parameters obtained non-membership values. The calculation method of the “a” and “b” by the least squares method is given in Eq. 13. A and Y matrices are clearly given in Eqs. 14 and 15.    −1 aL aM aR = AT A AT Y (13) bL bM bR ⎡ ⎤ (1) (1) (1) (−Z L (2)−Z M (2)−Z R (2); μ, ϑ) − 1 ⎢ ⎥ (1) (1) (1) ⎢ (−Z L (3)−Z M (3)−Z R (3); μ, ϑ) − 1 ⎥ ⎢ ⎥ (14) A=⎢ ⎥ . ⎢ ⎥ ⎣ ⎦ . (1) (1) (1) (−Z L (n)−Z M (n)−Z R (n); μ, ϑ) − 1 ⎡ ⎤ (−X L (2)−X M (2)−X R (2); μ, ϑ) ⎢ (−X (3)−X (3)−X (3); μ, ϑ) ⎥ ⎢ ⎥ M R L ⎢ ⎥ Y =⎢ (15) . ⎥ ⎢ ⎥ ⎣ ⎦ . (−X L (n)−X M (n)−X R (n); μ, ϑ) While performing operations in Eq. 13, triangular intuitionistic arithmetic operations are used. In the final results obtained, the parameters of membership and nonmembership values are also available. Therefore, the defuzzification process is applied to the parameters. By weighting the coefficients aL , aM veaR obtained through Eq. 13, the crisp “a” parameter is obtained as given in Eq. 16. Finally, the predictive values are obtained using Eqs. 17 and 1, similar to the triangular fuzzy grey model. a = αaL + βaM + γ aR (1)





X˜

˜ (1)

(k) = X (0)



X˜

 b˜ −a(k−1) + e (1) − a

(1)



(k) = X˜

(1)

b˜ , k = 2, 3, . . . , n a

(16)

(17)



(k) − X˜

(k − 1), k = 2, 3, . . . , n

(18)

5 Applications This section applies the TIFGM(1,1) model is for long-term electricity consumption estimation. To evaluate the estimation results of the proposed model, standard GM(1,1) and TFGM(1,1) models were also applied for the same data. Absolute percentage error (APE) and Mean Absolute Percentage Error (MAPE) criteria, the most commonly used in the literature, were used to measure the accuracy of the above three methods. The formulations of the evaluation criteria used are given in Eqs. 19 and 20.  (0)    X (k) − X (0) (k) × 100% (19) APE = X (0) (k) 

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MAPE =

n

k=1

X (0) (k)

× 100%

(20)

In this study, an integrated model is presented in this study by extending the triangular fuzzy grey model to intuitionistic fuzzy numbers. The electricity consumption data used in the application are given in Table 2 [16]. Table 1 shows low, high and base (medium) electricity consumption values between 2013–2021. The last five years will be estimated using the first four periods. Each method used in the study is univariate model and is based on its sequence of system characteristics, independent of influencing factors [12]. In applying the standard GM(1,1) model, the median value for each year is used. The triangular fuzzy number shows the system characteristic’s lower, middle and upper limit points [12]. The basic principle of the other two methods is that the system property has the highest probability of getting the middle boundary point. The central boundary point is also known as the preference value, and the likelihood of the system property getting values from the central boundary point to both sides decreases gradually. The values in Table 1 represent the triangular fuzzy number’s left, middle and right values. As mentioned in theory given in the second chapter, the development parameter ˜ is used as a of the model “a” is used as a crisp number and the grey parameter “b” triangular fuzzy number. Table 1. Turkey net electricity consumption values (Gwh). Year

Low

Mean

Right

2013

192689,25

198045,18

206178,45

2014

196369,23

207375,08

218479,13

2015

203560,88

217312,25

231058,21

2016

213945,41

231203,75

249935,32

2017

226726,19

249022,65

272855,79

2018

230428,32

258232,18

286104,71

2019

227689,13

257273,13

282539,46

2020

240036,67

262702,13

288457,20

2021

265378,14

286691,46

307525,36

˜ is treated as the intuitionistic fuzzy number in the intuitionThe grey parameter “b” istic fuzzy grey model. In this way, real-life uncertainty is included in more models in calculating the background value of the grey parameter. In the use of the intuitionistic fuzzy grey model, membership and non-membership values for the first four years of data used as the training set were determined as in Table 2 to give more weight to the last period. In both models, the horizontal smoothing coefficient λ = (0,5;0,5;0,5), and the weights used in the calculation of the improvement parameter “a” are taken as α =

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0,25, β = 0,50, γ = 0,25. In addition, Eq. 21 is used for the defuzzification process [17]. df =

aiL + aiM + aiU aL + aiM + aU i + i 3 τ

(21)

Table 2. Membership and non-membership values for the first four years. Year

(μA ; ϑA )

2013

(0,6; 04)

2014

(0,7; 0,3)

2015

(0,8; 0,2)

2016

(0,9; 0,1)

The results of the three models are given in Table 3. The MAPE value of all three models is below 10%, and the model prediction grades are excellent. For model evaluation, the predictive effect of the model is more important than the degree of fit because the predictive effect better reflects the stability and reliability of the model’s predictive ability. When Table 3 is examined, the MAPE values of the GM(1,1), TFGM(1,1) and TIFGM(1,1) models were respectively 4.63, 4.78 and 3.94. It is seen that the proposed TIFGM(1,1) model gives more successful results than other models. Table 3. Estimated values of electricity consumption according to different models. GM(1,1)

TFGM(1,1)

TIFGM(1,1)

Year Actual value Forecasting APE (%) Forecasting APE (%) Forecasting APE (%) value value value 2017 249022,65

243589,32

2,18

243989,86

2,02

243035,92

2,40

2018 258232,18

257275,55

0,37

254792,93

1,33

255770,04

0,95

2019 257273,13

271730,76

5,62

274376,87

6,64

269129,30

4,60

2020 262702,13

286998,13

9,25

285785,86

8,78

283208,47

7,80

2021 286691,46

303123,32

5,73

301468,57

5,15

298024,18

3,95

MAPE (%)

4,63

4,78

3,94

The estimated values of electricity consumption for the next five years, made with the TIFGM(1,1) model, are presented in Table 4.

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Table 4. Electricity consumption forecast values (Gwh). 2022

2023

2024

2025

2026

313614,96

330021,34

347286,013

365453,86

384572,14

6 Conclusion This study introduced a new model named TIFGM to realize long-term electricity consumption. Triangular intuitionistic fuzzy numbers are used to model the uncertainty in the real-life problem. For this, we defined membership and non-membership values for the periods determined for the training set. Unlike the TFGM(1,1) model previously introduced by Zeng [11], we defined the model with intuitionistic fuzzy numbers. We allowed the low, medium and high values of the same period to be entered into the analysis at the same time. Although the error values of the other two methods used for comparison were satisfactory, the error values of the proposed TIFGM(1,1) model gave better results. In future studies, variables such as population, imports or exports, which are thought to directly effect on electricity consumption, can be included in the model. Parameter optimization can be achieved from a different perspective, or the model can be repeated with neutrosophic numbers.

References 1. Wang, M., Wang, W., Wu, L.: Application of a new grey multivariate forecasting model in the forecasting of energy consumption in 7 regions of China. Energy 243, 123024 (2022) 2. Deng, J.: Introduction to grey system theory. J. Grey Syst. 1, 1–24 (1989) 3. Pu, B., Nan, F., Zhu, N., Yuan, Y., Xie, W.: UFNGBM(1,1): a novel unbiased fractional grey Bernoulli model with Whale Optimization Algorithm and its application to electricity consumption forecasting in China. Energy Rep. 7, 7405–7426 (2021) 4. Zeng, X., Shu, L., Yan, S., Shi, Y., He, F.: A novel multivariate grey model for forecasting the sequence of ternary interval numbers. Appl. Math. Model. 69, 273–286 (2019) 5. Zeng, B., Liu, S.-F., Meng, W.: Prediction model of discrete grey number based on kernels and areas. Control Decis. 26(9), 1421–1424 (2011) 6. Dong, C.X.: Grey interval model of the temperature rise prediction of massive concrete in construction process. Appl. Mech. Mater. 357, 631–634 (2013) 7. Zeng, X., Shu, L., Jiang, J.: Fuzzy time series forecasting based on grey model and Markov chain. Int. J. Appl. Math. 46(4), 464–472 (2016) 8. Zeng, X.-Y., Shu, L., Huang, G.-M., Jiang, J.: Triangular fuzzy series forecasting based on grey model and neural network. Appl. Math. Model. 40, 1717–1727 (2016) 9. Zor, C., Cebi, F.: Demand prediction in health sector using fuzzy grey forecasting. J. Enterp. Inf. Manag. 31(6), 937–949 (2018) 10. Bilgiç, C.D., Bilgiç, B., Ferhan, Ç.: Fuzzy grey forecasting model optimized by moth-flame optimization algorithm for short time electricity consumption. J. Intell. Fuzzy Syst. 42(1), 129–138 (2022) 11. Atanassov, K.: Intuitionistic fuzzy set. Fuzzy Sets Syst. 20, 87–96 (1986) 12. Govindan, K., Ramalingam, S., Broumi, S.: Traffic volume prediction using fuzzy greyMarkov model. Neural Comput. Appl. 33, 12905–12920 (2021)

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Generalized Net Model with Intuitionistic Fuzzy Estimations of the Humanoid Robot Behavior During Navigation Tasks Simeon Ribagin1,2(B) , Sotir Sotirov2 , and Krassimir Atanassov1,2 1 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia,

Bulgaria [email protected], [email protected] 2 University “Prof. D-R Asen Zlatarov”, Burgas, Bulgaria [email protected]

Abstract. In the present study we propose an approach to model the humanoid service robot behavior during a specific navigation task based on two mathematical formalisms, the Intuitionistic Fuzzy Sets (IFSs) and the Generalized Nets (GNs). Moreover, we include some possibilities for the possible ways for evaluating the overall performance of the robot. To do this we can apply estimations of the intuitionistic fuzzy sets. The estimations, which signify the degree of successfulness (μ) and non-successfulness (ν) of the current navigation task, are represented by ordered pairs of real numbers from the set [0, 1]. In addition, we study the degree of uncertainty π = 1 − μ − ν that appears in those cases when the navigation task is not fully performed by some reason, either due to the environment or the robot itself. The intuitionistic fuzzy estimations in the proposed model give information for the possible reasons of the task failure, the accuracy of the performance of the robot and the efficiency of the robot when performing navigation tasks. Keywords: Humanoid robot · Intuitionistic fuzzy estimation · Generalized nets · Generalized net model

1 Introduction Navigation is one of the most important functionalities of the humanoid mobile robots to move around in their surroundings. This allows the robot to perform various tasks and to interact with the indoor or outdoor environment. The main objective of the robot navigation is to arrive at a goal position seamlessly and without human interaction during the execution of the specific navigational task. The mobile robot is supposed to be aware of obstacles and move freely in different working scenarios. Depending on the design, purpose of the robot and the assignment, different strategies for navigation and locomotion have been proposed [8]. While performing the task of navigation, most of the humanoid robots are equipped with many intelligent autonomous sensing peripherals or sensors. Sensors are required to model the environment and localize its position, control © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 243–251, 2023. https://doi.org/10.1007/978-3-031-39774-5_30

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the motion, detect obstacles, and avoid obstacles by using navigational techniques. Safe path planning in order to avoid static or moving obstacles, from the start to the end of the navigation task is the most important function of any navigational technique. Therefore, the proper selection of the navigational technique is the most important step in the path planning of a robot when working in a simple and complex environment [7]. Recently the so called “vision-based navigation method” has gained popularity and it can be categorized as vision-based navigation for indoor robots and vision-based navigation for outdoor robots. The vision-based indoor navigation fall into three broad group methods: map-based navigation method, map-building based navigation method and maples navigation method [5]. Vision based navigation methods relies mainly on the sensory information. Sensors are the key to making a robot perceived environment a reality and sensing the environment is the first step when a robot performs a specific task. The proper function, the level of accuracy of the sensors, as well as the analyzed sensory information is one of the most important factors for reaching the full potential of the humanoid robot navigation and especially those build for public services. Simultaneous localization and mapping (SLAM) is also a reliable navigation technique. SLAM simultaneously creates an internal map and localize the robot during the environment exploration [6] For the purpose of the present paper and following [9] we will briefly describe the main navigation functionalities and sensor specifications of the UBTECH’s humanoid Cruzr robot [10]. Cruzr is one of the most advanced fully-programmable HSRs that provides a new generation of developers with service applications for a variety of industries and research fields. The robot has anthropomorphic features and it is able to function in complex environments thanks to his depth perception camera, Lidar and IR sensors that enable him to navigate autonomously. However, there are a number of restrictions related to the sensors itself. For example, the signal from the LIDAR is ineffective against glass, stainless steel and pure black materials. The RGBD sensor signal is ineffective against black light-absorbing materials. The infrared sensor is ineffective against complex surfaces etc. When navigating, the distance from the reference objects should be less than 16m, because of the radar sight distance itself. It is known that the success rate of the mapping process is higher, when the total size of the map is within 1.000 m2 . Never the less, hazardous objects like tables, chairs, glass walls, steps and moving objects including humans further complicate the process of navigation. In a view of this it is important to develop models or strategies that include and take into account the degree of uncertainty during the mapping, the navigation process and the execution of the specific tasks. In order to autonomously navigate, the robot needs an accurate static map. The intuitionistic fuzzy estimations in the proposed model give information for the possible reasons of the task failure, the accuracy of the performance of the robot and the efficiency of the robot when performing navigation tasks.

2 Materials and Methods The paper presents an approach to model a humanoid service robot functionalities and the information of the embedded sensors, based on the apparatus of the Generalized Nets (GNs) [1, 2] and the Intuitionistic Fuzzy Sets (IFs) [3, 4]. GNs represent a significant extension and generalization of the concept of Petri nets, as well as of other Petri nets

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extensions and modifications. They contain places having priorities and capacities, transitions having special matrices (called index matrices) with predicates that determine the direction of the tokens from input to output places of the transition, and tokens that enter the GN with initial characteristic and that, after each transfer form an input to an output place, obtain a new characteristic. On the other hand, the IFSs are extensions of Zadeh’s fuzzy sets. In them, two degrees – of membership and non-membership exist and they determine a third degree – of uncertainty (indeterminacy). The difference between fuzzy sets and intuitionistic fuzzy sets (IFSs) is in the presence of a second function ν A (x) in addition to the membership function μA (x) that defines the non-membership of the element x to the set A, where μA (x) ∈ [0, 1] ν A (x) ∈ [0, 1], under the condition: μA (x) + ν A (x) ∈ [0, 1]. The IFS itself is formally denoted by: A = {x, μA (x), ν A (x) | x ∈ E}. The presented approach assumes deterministic evaluation, while, in practice, some inaccuracy is always present. Here we will construct a reduced. GN-model of the humanoid robot behavior during a navigation task, based on the information and the feedback from the embedded navigational sensors, the generated two dimensional map of space by the robot and the developed mapping algorithm (Fig. 1).

Fig. 1. Cruzr robot mapping algorithm

In addition to constructing a model, here we will define an assessment of the degree of successfulness (μ) and non-successfulness (ν) of the current navigation task and the degree of uncertainty (π = 1 − μ − ν) which arises after the mapping process. Initially when still no information has been obtained from the current navigation task, all estimations are given initial values of 0, 0 this is the starting position of the robot (see Fig. 2). On the next step,   the robot  starts to execute the navigation route, the  when estimation obtains values of 1n , 0 , 2n , 0 , 3n , 0 , ... Etc., until the robot reaches the end of the current navigation task with evaluation of 1, 0. If the robot reaches an unexpected obstacle or other hazardous object blocking its path after s steps, we obtain the following evaluation ns , 0 . In general, there are several scenarios for the robot. First the robot can wait until the object is removed, second the robot can choose a different route, third the robot can return to its starting position and fourth the robot can stop the execution of the current navigation task on the spot. Here, we will describe and give evaluations of the robot performance only for the th first case. When the robot  an obstacle on the s step, it stops and the estimation  reaches s 1 and obtains values of n+1 , n+1 . If it waits r time-moments, the robot waits until the

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. Then the robot  s+1 r continues the rest of the navigation route with estimation of n+r . If there are no , n+r other obstacles during the  current task, the robot reaches its final destination with the n r . If there are other obstacles during the navigation route for p , n+r estimation of n+r   r+p n time-moment, the final evaluation is n+r+p . , n+r+p obstacle is removed and we obtain the following evaluation

s r n+r , n+r



Fig. 2. The 2-dimentional map scanned by the robot and the navigation task route: a) navigation route without an obstacles and b) navigation route with an obstacle

3 GN-Model of the Humanoid Robot Behavior During Navigation Tasks The proposed GN-model (see Fig. 3) has five transitions and 22 places with the following meaning: transition Z 1 represents the control module of the robot, transition Z2 represents the data storage of the robot, transition Z 3 represents the LIDAR sensor of the robot, transition Z4 represents the RGBD sensor of the robot, while transition Z5 represents the robot’s editing module. The proposed GN-model contains six types of tokens: α, β, σ, γ, τ and ϕ. The five transitions of the GN-model exhibit the so called “special places”, where a token stays and collects information about the specific part of the robot, and these are represented as follows: in place l5 , token α permanently stays and collects information from the robot’s embedded sensors and control module, in place l8 , token β permanently stays to collect information regarding the data storage, in place l11 , token γ permanently stays to collect information regarding the robots’ LIDAR sensor, in place l15 , token τ permanently stays to collect information regarding the RGBD sensor, in place l 22 , token ϕ permanently stays to collect information regarding the robot’s editing module. During the GN-model functioning, some of the tokens can split, thus generating new tokens that circulate around the net and are respectively being assigned new token

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characteristics. At certain time moments, these tokens would merge with some of the tokens α, β, σ , γ , τ and ϕ, while at the initial moment of the generalized net’s functioning they enter the following places with the following initial characteristics: • token α  enters the net through place l1 with characteristic: “the robot network connection is turned on”, • token σ 2 enters the net through place l8 , with characteristic: “environment signal for the LIDAR sensor”,

Fig. 3. The GN-model of humanoid robot behavior during a navigation task

• token σ 3 enters the net through place l12 with characteristic: “environment signal for the RGBD sensor”, • token σ 4 enters the net through place l16 with characteristic: “new navigation task”. The transition Z 1 of the GN-model has the following form: Z1 = {l1 , l5 , l9 , l14 , l20 }, {l2 , l3 , l4 , l5 }, r1 , where:

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r1

l2 fаlse W5,2 fаlse fаlse fаlse

l1 l5 l9 l14 l20

l3 fаlse W5,3 fаlse fаlse fаlse

l4 fаlse W5,4 fаlse fаlse fаlse

l5 truе truе truе truе truе

and the predicates in the index matrix are defined as follows: • W 5,2 = “the scanner application is uploaded on the robot” ∧ “the data from the sensors is transferring correctly” ∧ “all previous mapping processes has been canceled”; • W 5,3 = ¬W 5,2 ; • W 5,4 = “the intuitionistic fuzzy estimation for the current navigation task is evaluated”. The tokens from all input places of transition Z 1 transfer to place l 5 where they merge with token α that obtains the above mentioned characteristic. Conversely, token α splits to four tokens – the same token α that resides permanently in place l5 , and tokens α 1 , α 2 and α 3 , whose further movement in the net is determined by the values of the predicates. When predicate W 5,2 holds true, token α 1 enters place l2 where it is assigned the characteristic: “select the scanning route and start the mapping process”. Since W 5,3 is the negated predicate W 5,2 , at any other moment, the token α 2 enters place l3 where it obtains the characteristic: “check the download process and the robot settings, cancel the current mapping process”. Finally, when the predicate W 5,4 becomes true, token α 3 enters place l 4 where it is assigned the characteristic: “current intuitionistic fuzzy estimation of the performed navigation task”. The next transition Z 2 of the IFGN-model has the following form: Z2 = {l2 , l4 , l7 }, {l6 , l7 }, r2 , where:

r2

l2 l4 l7

l6 l7 fаlse truе fаlse truе W7,6 truе

and W 7,6 = “the information from the map scanning process is successfully stored in the data storage”. The tokens from all input places of transition Z 2 transfer to place l8 where they merge with token β that obtains the above mentioned characteristic. On the contrary, token β splits into two tokens – the same token β that permanently resides in place l8 , and token β 1 in in place l 6 . When the above formulated predicate W 7,6 becomes true, token β 1 is assigned with the characteristic: “start the editing process for the new navigation task”.

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The transition Z 3 of the IFGN-model has the following form: Z3 = {l8 , l11 }, {l16 , l17 , l18 }, r3 , where:

r3

l8 l11

l9 l10 l11 fаlse fаlse truе W11,9 W11,10 truе

and the predicates in the index matrix are defined as follows: • W 11,9 = “the signal from the LIDAR sensor is transferred correctly”, • W 11, 10 = ¬W 11,9 . The tokens from all input places of transition Z 3 transfer to place l 11 and merge there with token γ that obtains the above mentioned characteristic. Conversely, token γ splits to three tokens – the same token γ that permanently occupies place l11 and tokens γ 1 and token γ 2 . When the predicate W 11,9 holds true, token γ 1 enters place l 9 and there it obtains a characteristic: “current signal for the environment from the LIDAR sensor”. Conversely, when its negation, predicate W 11,10 is valid, token γ 2 enters place l10 and it is assigned there the characteristic: “check the LIDAR sensor or the environment for hazardous objects”. The transition Z 4 of the IFGN-model has the following form: Z4 = {l12 , l15 }, {l13 , l14 , l15 }, r4 , where:

r4

l13 l14 l15 l12 fаlse fаlse truе l15 W15,13 W15,14 truе

and the predicates above have the following sense: • W 15,13 = “the signal from the RGBD sensor is transferred correctly”, • W 15, 14 = ¬W 15,13 . The tokens from all input places of transition Z 4 move to place l15 where they merge with token τ thus obtaining the above mentioned characteristic. Token τ itself splits to three tokens – the same token τ that permanently occupies place l15 and tokens τ  and τ  . When the predicate W 15,13 holds true, token τ  enters place l13 where is is assigned the characteristic: “current signal for the environment from the RGBD sensor”. Alternatively, when its negated predicate W 15,14 holds true, the other token τ  enters place l 14 where

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it obtains a characteristic: “check the RGBD sensor or the environment for hazardous objects”. The last transition Z 5 of the presented IFGN-model is described as follows: Z5 = {l20 , l25 , l28 }, {l26 , l27 , l28 }, r5 , where:

r5

l6 l16 l21 l22

l17 l18 l19 fаlse fаlse fаlse fаlse fаlse fаlse fаlse fаlse fаlse W22,17 W22,18 W22,19

l20 l21 fаlse fаlse fаlse fаlse fаlse fаlse truе W22,21

l22 truе truе truе truе

and the predicates in the index matrix have the following descriptions: • • • • •

W 22,17 = “the robot has executed the current navigation task”, W 22,18 = “the robot has not executed the current navigation task”, W 22,19 = “the intuitionistic fuzzy estimation of the current navigation task is evaluated”, W 22,21 = “the points of interest and the current behavior of the robot for each point of interest is set”.

The tokens from all input places of transition Z 5 enter place l22 where they merge with token ϕ thus obtaining the above mentioned characteristic. Conversely, token ϕ splits to six tokens – the same token ϕ that resides permanently in place l22 , and tokens ϕ 1 , ϕ 2 ϕ 3 , ϕ 4 and ϕ 5 . When the predicate W 22,17 holds true, token ϕ 1 enters place l 17 where it is assigned the characteristic: “the current successful navigation task”. When the predicate W 22,18 holds true, token ϕ 2 transfers to place l 18 and obtains there the characteristic: “the current non-successful navigation task”. In case of validity of predicate W 22,19 , token ϕ 3 enters place l19 where it obtains the characteristic: “the intuitionistic fuzzy estimation of the current navigation task”, while token ϕ 4 moves to place l20 with the respective characteristic for the data storage. Finally, when the predicate W 22,21 holds true, token ϕ 5 enters place l 21 where its characteristic becomes “start the navigation”.

4 Conclusions The developed GM model describes a navigation behavior of the humanoid robot Cruzr with intuitionistic fuzzy estimations. The applied intuitionistic fuzzy estimations in the model describe the degree of successfulness of the performed navigation task, based on the scanning and the editing process. Our model will permit the development of a more detailed and complex model allowing optimization and improvement of the robot navigation when performing a specific task in different environment.

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Acknowledgements. This work is partially supported by the grant: BG05M20P001-1.002-0011 “Centre of competence MIRACle - Mechatronics, Innovation, Robotics, Automation, Clean technologies”.

References 1. Atanassov, K.: Generalized Nets. World Scientific, Singapore (1991) 2. Atanassov, K.: On Generalized Nets Theory. Prof. M. Drinov Academic Publishing House, Sofia (2007) 3. Atanassov, K.: Generalized Nets and Intuitionistic Fuzziness in Data Mining. Prof. M. Drinov Academic Publishing House, Sofia (2020) 4. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Studies in Fuzziness and Soft Computing, vol. 283. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29127-2 5. Desouza, G.N., Kak, A.C.: Vision for mobile robot navigation: a survey. IEEE Trans. on Pattern Anal. Mach. Intell. 24(2), 237–267 (2002) 6. Durrant-Whyte, H., Bailey, T.: Simultaneous localisation and mapping (SLAM): part I the essential algorithms. Robot. Autom. Mag. 13 (2006) 7. Patle, B.K., Ganesh Babu, L., Anish Pandey, D.R.K., Parhi, A.J.: A review: on path planning strategies for navigation of mobile robot. Defense Technol. 15(4), 582–606 (2019) 8. Lobo, J., Marques, L., Dias, J., Nunes, U., Almeida, A.T.d.: Sensors for mobile robot navigation, Lecture Notes in Control and Information Sciences,50–81 (1998) 9. Ribagin, S., Sotirov, S., Sotirova, E., Hristozov, I., Atanassov, K.: Intuitionistic fuzzy generalized net model of the humanoid service robot functionalities. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) INFUS 2022. LNNS, vol. 504, pp. 529–536. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5_62 10. UBTECH Homepage, http://www.ubtrobot.com/Cruzr/index.aspx

Intuitionistic Fuzzy Generalized Net Model of a Human-Robot Interaction Simeon Ribagin1,2(B) , Sotir Sotirov2 , and Evdokia Sotirova2 1 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia,

Bulgaria [email protected] 2 University “Prof. D-R Asen Zlatarov”, Burgas, Bulgaria {ssotirov,ssotirova}@btu.bg

Abstract. In the present paper we propose an approach to model the humanoid service robot interaction with a human, based on two mathematical formalisms, the Intuitionistic Fuzzy Sets (IFSs) and the Generalized Nets (GNs). Moreover, in the present work we are using one of the extensions of the ordinary GNs, the so called, Intuitionistic Fuzzy GNs of the first type (IFGN1). The input data for the proposed model is namely from the embedded sensors and the peripherals of the robot which give the possibility of multi-modal interaction. The IFGN1model allows the development of a more detailed and complex model for optimization and improvement of the human-robot interaction in industrial, service, co-manipulation, medical and healthcare applications. Keywords: Human-robot interaction · Intuitionistic fuzzy estimation · Generalized nets · Intuitionistic fuzzy generalized net of first type

1 Introduction Nowadays the human-robot interaction (HRI) is a very extensive and diverse research and design activity. One of the fundamental goals of studying the HRI is the development of the principles and algorithms for robot systems that make them capable of direct, safe and effective interaction with humans [6]. This will permit the application of service humanoid robots almost in every aspect of industrial, social and medical areas. Because of the high tech sensors and peripherals available in most of the robots, a bidirectional communication channels can be established, thus allowing the human and a robot to work together. When we consider human-robot interaction we cannot but mention the humanoid robots and namely the humanoid service robots. Humanoid service robots are unique type of robots that possess human-like details not only in their physical appearance, but also and a human-like interactions and human social skills [1], characterized by kinematics, recreated from the human locomotor system and in a way by the possibility of reproducing the cognitive skills of humans. The focus of these robots lies on the performance of semi-automatic and fully automatic services. One very distinctive future is the high degree of interaction with their changeable environment in which they © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 252–258, 2023. https://doi.org/10.1007/978-3-031-39774-5_31

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work autonomously and communicate with humans. The interaction with humans takes place, in work areas, leisure time or the health care industry etc. However humanoid service robots are still far from the physical and mental performance of a human, even if developments in some functional areas are promising [10]. Dealing with uncertainty and real-time perception are some of the most enduring challenges in HRI. For HRI, the perceptual challenges are particularly complex, because of the need to perceive, understand, and react to human activity in real-time. For the purpose of the present study and following [8], we will briefly describe the main interaction functionalities and sensor specifications of one of the most advanced fully-programmable HSR currently on the market, namely the UBTECH’s humanoid Cruzr robot [9]. In general, Cruzr uses several communication modes making the communication fluid and very similar to human communication. The robot can speak apprehensible several different languages, thanks to the integrated two-channel stereo speakers. Cruzr is equipped with a 13MP camera and a software that can detect facial images in milliseconds it has an open source platform that also allows users to identify emotions, gender, age, nationality, etc. The embedded 11.6inch touchscreen tablet allows visualization of any type of information (text, images, videos…) - all of which makes human-robot communication smoother [11]. Cruzr can be programmed to act differently and do things based on a specific event or series of events. The information and interaction with the surrounding environment is possible thanks to the embedded: cliff sensor, RGBD sensor, e-Skin sensors, infrared and sonar sensor, IMU sensor, LIDAR sensor, geomagnetic sensor, temperature and humidity sensors. All of these high tech futures, makes this robot a unique piece of AI and engineering. However, in an unexpected scenario or in complex dynamic environment some limitations related to the communication sensors, may lead to a significant reduction in the efficiency of the robot. The present paper presents an approach to model the human-robot interaction of a humanoid service robot based on IFGN1s. We discuss some possible ways for evaluating the multimodal synergies and coordination of the robot when performing a task related to the interaction with a person. In addition, we study the degree of uncertainty of the IFPs: π = 1 − μ − v that appears when the human-robot interaction is corrupted by some reason, either due to the environment or the information and the feedback from the embedded sensors thanks to which the communication takes place. The input data for the proposed model is namely from the depth-perception camera, the RGBD sensor, the microphone of the robot and the touchscreen display of the robot.

2 Materials and Methods The Generalized Nets (GNs) [2, 3] were introduced as extension of the Petri nets and the other Petri net modifications and extensions. One of the first extensions of the GNs were Intuitionistic Fuzzy GNs of first type (IFGN1s) [4, 5]. The tokens of the GNs enter the net with initial characteristics and during their transfers through the net, they obtain next characteristics. Moreover, the transitions of the GNs have special matrices with predicates that determine the direction of the tokens transfers from the transition inputs to their outputs. In the IFGN1s, these predicates are evaluated by Intuitionistic Fuzzy Pairs (IFPs) that have the form , where μ, ν, μ + ν \in [0, 1] and they represent the degrees of the validity and of the non-validity of the predicate.

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All transition condition predicates W i,j have degrees of validity (μ(W i,j )) and nonvalidity (ν(W i,j )), so the truth-value of W i,j is μi, j , ν i,j . For each of these predicates we can determine two thresholds – lower ( t l ∈ [0,1]) and higher ( t h ∈ [0,1]), so that t l + t h ≤ 1. The token from place li will go to place l j if the following conditions hold:     μ Wi,j > tl and v Wi,j < 1 − th

3 The IFGN1-Model of a Human-Robot Interaction The proposed IFGN1-model (Fig. 1) has 6 transitions, 24 places and it contains seven types of tokens: ε, α, β, σ, γ, τ and ϕ. We assume, that initially the robot is powered up and has an established Wi-Fi connection. The six transitions have the following meaning: Z 1 represents the camera module of the robot; Z2 – the microphone module of the robot; Z 3 - the main module of the robot (control module); Z4 – the speaker module of the robot; Z5 – the display module of the robot; Z 6 - data storage (cloud server). The 6 transitions of the GN model have a so called “special place”, where a token stays and collects information about the specific part of the robot, represents as follows: In place l4 , token α collects information about the current state of the robot’s camera. In place l 8 , token β collects information about in the current state of the robot’s microphone. In place l 15 , token σ collects information from the main module of the robot. In place l 18 , token γ collects information about the current state of the robot’s speaker. In place l 21 , token τ collects information about the current state of the robot’s touchscreen display. In place l 24 , token ϕ collects information about the cloud server. During the IFGN1-model functioning, some of the tokens can split, thus generating new tokens that circulate around the net and are respectively being assigned new token characteristics. At certain time moments, these tokens would merge with some of the tokens α, β, σ, γ, τ and ϕ, while at the initial moment of the generalized net’s functioning they enter in places l1 , l 5 and l 9 with the following initial characteristics: token ε1 with a characteristic: “images from the camera “, token ε2 with a characteristic: “signals from the microphone “, token ε3 with a characteristic: “current interaction task (voice interaction, face and speech recognition etc.) “ We will give a formal description only for the transition Z1 and the transition Z2 of the IFGN1-model. All of the transitions and predicates can be described formally in a similar way and the threshold values can be chosen accordingly to the specific values obtained from the communication peripherals of the robot, the environment and the person interacting with the robot. The transition Z 1 of the IFGN1-model has the following form: Z1 = {l1 , l4 , l10 }, {l2 , l3 , l4 }, r1  where:

Intuitionistic Fuzzy Generalized Net Model

r1 =

l2

l3

255

l4

l1 l4

fаlsе W4,2

fаlsе truе W4,3 W4,4

l10

fаlsе

fаlsе trие

• W 4,2 = “the camera of the robot is in active interaction mode and the robot recognized the current face image” • W 4,3 = “the camera of the robot is in active interaction mode and the robot detects a face image” • W 4,4 = “the robot detects an image but doesn’t recognized it” The predicates of the transition Z 1 can be estimated by their intuitionistic fuzzy degrees of the facial image recognition from the facial image database.

Fig. 1. The IFGN1-model of the human-robot interaction

We must mention that the predicate P is valid if for f (P) = a, b holds a >

1 2

> b.

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Let

    Icu − Itol I − Icu f W4,2 = , , I I     I − Icu Icu − Itol f W4,3 = , , I I W4,4 = ¬W4,2 ∧ ¬W4,3.

where I, I cu and I tol , are the set of all face images in the face image database, the current face image and the tolerance of the camera. Token α 1 obtains a characteristic: “current face image is recognized” in place l2, “check and upgrade the face image database” in place l2, “wait until the face image is recognized” in place l4. The transition Z 2 of the IFGN1-model has the following form: Z2 = {l5 , l8 , l14 }, {l6 , l7 , l8 }, r2  where:

r2 =

l5

l6 false

l7 l8 false true

l8 l14

W8,6 false

W8,7 W8,8 false true

• W 8,6 = “the robot’s microphone is in active interaction mode and the robot recognized the current speech sound signal” • W 8,7 = “the robot’s microphone is in active interaction mode and the robot detects a speech sound signal” • W 8 8 = “ the robot detects a sound signal but doesn’t recognized it” Token β 1 obtains a characteristic: “current speech is recognized” in place l6, “check and upgrade the speech database” in place l7, “wait until the speech is recognized (standby mode)” in place l8. Below for brevity for the rest of the transitions, we’ll mention only the token characteristics and the transition condition predicates. The predicates of the transition Z3 are as follows: W 15,10 = “the SDK for face tracking, face detection, and facial recognition is integrated”, W 15,11 = “the voice commands are integrated”, W 15, 12 = “the current human-robot interaction has been performed”, W 15,13 = “the touchscreen display of the robot is in active interaction mode” and W 15,14 = “the SDK for the speech (speech recognition, speech synthesis, and command issuing functions) interaction is integrated”. The intuitionistic fuzzy estimations of these predicates can be obtained from the accuracy and non-accuracy of the quality of the transferred data. Depending on the degree of validity of these predicates, tokens σ 1 , σ 2 ,

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σ 3 , σ 4 and token σ 5 will obtain a characteristic: “start the face image initialization,” in place l 10 , “ the new voice command” in place l11 , “protocol of the result of the current human-robot interaction for the data storage” in place l12 , “current message on the display of the robot” in place l13 ““ and “start the speech initialization” in place l14 . The predicates of the transition Z4 are as follows: W 18,16 = “the speaker of the robot is in active interaction mode” and W 18, 17 = “the speaker is off”. The intuitionistic fuzzy estimations of these predicates can be obtained from the correctness and non- correctness of the sound signal due to the surrounding environment or the speech service command set. Depending on the degree of validity of these predicates, token γ 1 will obtain a characteristic: “speech reply from the robot for the current interaction task” in place l16 and γ 2 “check the robot speaker” in place l17 . The predicates of the transition Z5 are as follows: W 21,19 = “the touchscreen display of the robot is in active interaction mode” and W 21, 20 = “the display of the robot is off”. The intuitionistic fuzzy estimations of these predicates can be obtained from the level of the touchscreen display sensitivity and the predefined display messages. Depending on the degree of validity of these predicates, token τ 1 will obtain a characteristic: “the display message for the current interaction task” in place l20 and τ 2 “check the robot display” in place l19 . The predicates of the transition Z6 are as follows: W 24,22 = “there is an established network connection between the robot and the server” and W 24, 23 = “the robot cannot connect to the network server”. The intuitionistic fuzzy estimations of these predicates can be obtained from the accuracy and non-accuracy of the transferred data due to the quality of the internet connection. Depending on the degree of validity of these predicates, token ϕ 1 will obtain a characteristic: “check the network quality” in place l22 and ϕ 2 “select a human-robot interaction task from the data storage” in place l23 .

4 Conclusions The developed IFGN1 model describes the human-robot interaction of a humanoid service robot Cruzr. The applied intuitionistic fzzy estimations in the model give the possibility to predict and prevent failure of the human-robot interaction, when the robot is executing a specific service or social tasks. The proposed IFGN1 model can be complicated and detailed thus allowing optimization and improvement of the human-robot interaction in industrial, service, co-manipulation, medical and healthcare applications for example. Acknowledgments. This work is partially supported by the grant: BG05M20P001–1.002–0011 “Centre of Competence MIRACle - Mechatronics, Innovation, Robotics, Automation, Clean technologies”.

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References 1. Alotaibi, S.B., Manimurugan S.: Humanoid robotic system for social interaction using deep imitation learning in a smart city environment. Front. Sustain. Cities 4, 2624–9634 (2022) 2. Atanassov, K.: Generalized Nets. World Scientific, Singapore (1991) 3. Atanassov, K.: On Generalized Nets Theory. Prof. M. Drinov Academic Publishing House, Sofia (2007) 4. Atanassov, K.: Generalized Nets and Intuitionistic Fuzziness in Data Mining. Prof. M. Drinov Academic Publishing House, Sofia (2020) 5. Atanassov, K.T.: On Intuitionistic Fuzzy Sets Theory. Springer Berlin Heidelberg, Berlin, Heidelberg (2012) 6. Feil-Seifer, D., Matari´c, M.J.: Human robot interaction. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 4643–4659. Springer New York, New York, NY (2009). https://doi.org/10.1007/978-0-387-30440-3_274 7. Rautiainen, S., et al.: Multimodal interface for human–robot collaboration. Machines 10(10), 957 (2022). https://doi.org/10.3390/machines10100957 8. Ribagin, S., Sotirov, S., Sotirova, E., Hristozov, I., Atanassov, K.: Intuitionistic fuzzy generalized net model of the humanoid service robot functionalities. In: Cengiz Kahraman, A., Tolga, C., Onar, S.C., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) Intelligent and Fuzzy Systems: Digital Acceleration and The New Normal - Proceedings of the INFUS 2022 Conference, Volume 1, pp. 529–536. Springer International Publishing, Cham (2022). https://doi.org/10. 1007/978-3-031-09173-5_62 9. UBTECH Homepage. http://www.ubtrobot.com/Cruzr/index.aspx 10. https://robotik.thws.de/en/thws-robotics/topic/robot-categories/ 11. https://kinemarobotics.eu/

Intuitionistic Fuzzy Evaluations of Garbage Sorting Using a Robotic Arm Petar Petrov1,2

, Veselina Bureva1(B)

, and Krassimir Atanassov1,3

1 Laboratory of Intelligent Systems, “Prof. Dr. Assen Zlatarov” University, “Prof. Yakimov”

Blvd., Burgas 8010, Bulgaria [email protected] 2 Vocational School of Electrical Engineering and Electronics “Konstantin Fotinov”, Burgas, Bulgaria 3 Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str. Block 105, Sofia 1113, Bulgaria [email protected]

Abstract. Garbage sorting is an important part of the recycling process. Recycling is vital for green and circular economics and can be improved drastically by using modern technologies such as robotics and AI. This paper demonstrates the calculation of Intuitionistic Fuzzy Evaluations (IFEs) for a batch of garbage that contains mixed materials, such as paper, plastic, metals, and other types of waste. A robotic arm equipped with a sensor analyzes each item of garbage and determines whether it is recyclable. This process is modeled and illustrated using DOBOT Magician and a conveyor belt. The different types of garbage are represented by cubes of different colors on the conveyor belt. Based on the color sensor data, an IFE is given for the batch of items on their recyclability. Keywords: Garbage sorting · Clean technologies · Robotics · DOBOT Magician · Intuitionistic Fuzzy Evaluations

1 Introduction In the current research the process of garbage sorting using a robotic arm is discussed. The realization is performed using two DOBOT Magicians and a Mini Conveyor Belt. The idea is to perform garbage sorting with intuitionistic fuzzy estimation. In the current study garbage sorting using intuitionistic fuzzy evaluations is presented for the first time. The aim is to present the degrees of successful sorting. The first DOBOT Magician robot picks a given garbage item using a suction pump and drops it onto the mini conveyor belt. Thereafter, the item color is scanned using a color sensor of the second DOBOT Magician robot. The results received after the run of the conveyor belt in the DOBOT Studio console are evaluated. The area of sorting is represented by intuitionistic fuzzy interpretation of “Red-Green-Blue (RGB) color system”. In order to achieve the intuitionistic fuzzy evaluations, a transformation process using the coordinates of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 259–264, 2023. https://doi.org/10.1007/978-3-031-39774-5_32

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colors and transformation formulas is performed. Research works of garbage sorting are discussed in [5, 6]. Examples of intuitionistic fuzzy estimations in different fields of science are already published [7–10]. The paper has the following structure. Section 1 presents an introduction of the research work. Section 2 describes the notation of intuitionistic fuzzy sets and the interpretation of “Red-Green-Blue (RGB) color system”. Section 3 introduces a constructed conceptual prototype of the garbage sorter using DOBOT Magician. Section 4 describes an example of Python implementation for garbage sorting and calculating IFEs. Section 5 presents discussion of data mining potential of garbage data, conclusion and future remarks.

2 Brief Remarks on Intuitionistic Fuzzy Sets (IFS) Intuitionistic Fuzzy Sets (IFS) theory is presented in [1–4]. Let E denotes the universe and the subset A will be given. Thereafter, the intuitionistic fuzzy set has the following form A∗ = {x, μA (x), vA (x)|x ∈ E} where 0 ≤ μA (x) + vA (x) ≤ 1; μA : E → [0, 1] and vA : E → [0, 1] are called degree of membership and degree of non-membership of element x ∈ E. The degree of uncertainty is defined as: π(x) = 1 − μA (x) − vA (x). Each element x has the geometrical interpretation shown on Fig. 1.

Fig. 1. A geometrical interpretation of element x.

In the paper, the intuitionistic fuzzy evaluations are constructed to estimate the recyclability of the garbage batch. The realization is performed using two DOBOT Magicians and a Mini Conveyor Belt. The garbage is represented by cubes in different colors similar to red, green, blue and yellow. The first DOBOT Magician recognizes the square and pick it on the Mini Conveyor Belt. The second DOBOT Magician recognizes the square and distributes it according to the selected groups of garbage. The intuitionistic fuzzy interpretation of “Red-Green-Blue (RGB) color system” is used for garbage sorting intuitionistic fuzzy evaluations calculation. The IF-geometrical interpretation is presented by an equilateral triangle with vertices assigned to the three basic colors “Red”, “Green” [2].   and √ “Blue” √ The basic colors “Red”, “Green”, “Blue” have the coordinates 0, 0, 2 3 3 , 0 , and 33 , 1, respectively. The equilateral triangle which interprets the

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“Red-Green-Blue (RGB) color system is presented in Fig. 2. It is another geometrical interpretation of element x.

Fig. 2. The equilateral triangle interpreting the “Red-Green-Blue (RGB) color system

The scope of the DOBOT Magician is represented by the equilateral triangle interpreting the “Red-Green-Blue (RGB color system). Afterwards, the results from the recognition are transformed into the rectangular intuitionistic fuzzy triangle using the transformation function F that has the form:  √  x 3−y u, v = F(x, y) = ,y . 2 Then, the three vertices of the equilateral triangle are transformed in the rectangular triangle in Fig. 1. They have the following form: F(0, 0) = 0, 0,   √ 2 3 , 0 = 1, 0, 3 √  3 F , 1 = 0, 1. 3

F

3 Conceptual Prototype of Garbage Sorting Installation Using DOBOT Magician As proof of concept for the following research we set up an installation consisting of two DOBOT Magicians and a Mini Conveyor Belt (Fig. 3). The garbage items are represented as color cubes of varying colors. The first DOBOT Magician robot is responsible for picking a given garbage item using a suction pump. After the item is picked it is dropped onto the mini conveyor belt. Then the item color is scanned using a color sensor which is controlled by the second DOBOT Magician robot. The color is checked in a dataset which contains information for possible color values and their recyclability. After the

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color is determined the garbage item is either taken by the robot and separated if it is potentially recyclable or it is let to drop out of the conveyor belt if the color detection shows that the item is not recyclable. The case when an item is indefinite is also covered. If the robot could not determine the recyclability or the color, then the item is let to drop out of the conveyor belt.

Fig. 3. Dobot Magicians and Mini Conveyor Belt

In this example, recyclable item colors are considered to be red, green and blue while yellow is non-recyclable. In case of poor results of the color sensor the item’s recyclability is decided to be indefinite. Then for a batch of nine items we can determine its recyclability. Please note that the data used in this prototype is simulated and does not reflect actual recyclability of materials. The prototype could be upgraded by using a camera with computer vision in order to detect materials as well and not just colors. By using such means the practical application of the prototype would be more accurate.

4 Calculating Intuitionistic Fuzzy Evaluations for a Batch of Garbage Items After sorting through the conveyor belt a calculation of the Intuitionistic Fuzzy Evaluations (IFEs) is performed. The received values for RGB colors from Dobot Magicians are in the form (a, b, c) where a represents the degree of red color participation, b – degree of green color participation and c- degree of blue color participation. This part is presented in Table 1. After that, the following intuitionistic fuzzy evaluations are calculated:   b a , a+b+c a+b+c where the third component is c a+b+c

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For example, for result (137, 207, 101) we have (0.31, 0.47, 0.22).  √    √ x 3−y 0.31 3 − 0.47 u, v = F(x, y) = ,y = , 0.47 = 0.03, 0.47. 2 2 The results received after the run of the conveyor belt in the DOBOT Studio console are displayed in Table 1. The values of RGB colors for each object are determined. The received color name is written. Finally, the item is classified as recyclable, nonrecyclable and indefinite. Table 1. Results received after the run of the conveyor belt in the DOBOT Studio console RGB (Red, Green, Blue)

Color Name

Recyclable

(137, 207, 101)

Green

Yes

(104, 174, 85)

Green

Yes

(178, 223, 138)

Green

Indefinite

(70, 132, 49)

Green

Yes

(157, 200, 100)

Green

Yes

(218, 57, 57)

Red

Yes

(255, 99, 71)

Red

Yes

(233, 79, 55)

Red

Indefinite

(210, 77, 87)

Red

Yes

(72, 61, 139)

Blue

Yes

(25, 25, 112)

Blue

Indefinite

(0, 191, 255)

Blue

Yes

(65, 105, 225)

Blue

Yes

(255, 196, 0)

Yellow

No

(238, 130, 238)

Purple

No

(255, 215, 0)

Golden

No

(255, 140, 105)

Orange

No

(128, 0, 128)

Purple

No

(218, 112, 214)

Purple

No

(255, 228, 196)

Peach

No

(255, 99, 71)

Orange

Indefinite

(255, 165, 0)

Orange

Indefinite

(128, 128, 0)

Yellow

Indefinite

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5 Conclusion In the context of IFE, it is possible that given city areas have better membership values than others. These could be areas where people have a higher culture of recycling and green thinking. On the other hand, places with bigger recyclability non-membership values of their garbage could point out to places where more actions towards people are required so that they start recycling more. Based on that data a map could be constructed. The map visualizes with different colors given areas based on their IFEs for the garbage batches which come from them. Please note that the data in this paper is simulated. A potentially useful aspect of the data mining could be to determine the most commonly used non-recyclable materials in a given area. Using this data, different types of events and initiatives could be organized so that people start using more eco-friendly and recyclable materials in the targeted areas. Another useful potential of this kind of evaluations could be in decision making and forecasting for the garbage picking process. Acknowledgments. The authors are thankful for the support provided by the Operational Program “Science and Education for Smart Growth” under contract BG05M20P001–1.002–0011 “Centre of competence MIRACle - Mechatronics, Innovation, Robotics, Automation, Clean technologies”.

References 1. Atanassov, K.: Generalized Nets and Intuitionistic Fuzziness in Data Mining. Professor Marin Drinov Academic Publishing House (2020) 2. Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999) 3. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012) 4. Atanassov, K., Szmidt, E., Kacprzyk, J.: On intuitionistic fuzzy pairs. Notes Intuitionistic Fuzzy Sets 19(3), 1–13 (2013) 5. Fu, H., Xu, D., Wu, J.: Robotic arm intelligent grasping system for garbage recycling. In: 2021 China Automation Congress (CAC), Beijing, China, pp. 6821–6826 (2021). https://doi.org/ 10.1109/CAC53003.2021.9728547 6. Gupta, S., Kruthik, H.M., Hegde, C., Agrawal, S., Prashanth, B.: Gar-Bot: garbage collecting and segregating robot. ICMAI 2021, J. Phys.: Conf. Ser. 1950, 012023 (2021). IOP Publishing. https://doi.org/10.1088/1742-6596/1950/1/012023 7. Kim, T., Sotirova, E., Shannon, A., Atanassova, V., Atanassov, K., Jang, L.: Interval valued intuitionistic fuzzy evaluations for analysis of a student’s knowledge in university e-learning courses. Int. J. Fuzzy Logic Intell. Syst. 18(3), 190–195 (2018) 8. Madera, Q., Castillo, O., García-Valdez, M., Mancilla, A., Sotirova, E., S. Sotirov: A method for optimizing a bidding strategy for online advertising through the use of fuzzy intuitionistic systems. In: ICIFSTA 2016, 20–22 April 2016, Beni Mellal, Morocco, Notes on Intuitionistic Fuzzy Sets, Vol. 22, No. 2, pp. 99–107 (2016) 9. Sotirova, E., Shannon, A., Kim, T., Krawczak, M., Melo-Pinto, P., Rieˇcan, B.: Intuitionistic fuzzy evaluations for the analysis of a student’s knowledge in university e-learning courses. In: Hadjiski, M., Atanassov, K. (eds) Intuitionistic Fuzziness and Other Intelligent Theories and Their Applications. Studies in Computational Intelligence. vol 757. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-78931-6_6 10. Sotirova, E., Petkov, T., Krawczak, M.: Generalized net modelling of the intuitionistic fuzzy evaluation of the quality assurance in universities, advances in fuzzy logic and technology 2017. Adv. Intell. Syst. Comput. 643, 341–347 (2017)

Picture Fuzzy Sets

Picture Fuzzy Internal Rate of Return Analysis Elif Haktanır(B) Department of Industrial Engineering, Bahcesehir University, 34349 Istanbul, Turkey [email protected]

Abstract. The internal rate of return (IRR) is the interest rate that sets the net present value of all future cash flow from a project to zero. The most common use of the IRR is when an organization uses it to consider investing in a new project or to increase investment in an already ongoing project. For example, when considering the situation of a company that chooses to establish a new facility or expand the operation of an existing facility, information about the profitability of the project can be obtained by calculating the IRR and a decision to accept or reject the project can be made. Although there are many applications of the IRR method under the classical approach in the literature, its application under fuzzy environment is quite limited. In fact, there is no implementation of it using extensions of ordinary fuzzy sets. Picture fuzzy sets, one of the most up-to-date and accepted ordinary fuzzy sets extensions, were used in this study, in the method that was developed to deal with the uncertainties in cash flow. The developed method has been applied in the field of a technology company’s decision to purchase a new equipment. Keywords: Picture Fuzzy Sets · Internal Rate of Return Analysis · Trial-and-Error Method · Interpolation

1 Introduction Internal rate of return (IRR) is a very important financial concept used in evaluating different investment options according to their profitability. The IRR also expresses the extent to which the investment will create added value. The IRR can also be defined as the discount rate that equates the net present worth (NPW) of a cash flow to zero. The IRR is easy to calculate if z < the annual net cash flows from the investment are constant. However, the periodic net cash flows from investments are often different. In this case, the IRR is found by using the trial-and-error method and the interpolation method together. In the calculation, there are two discount rates that give a negative and positive NPW result. The IRR is determined by interpolating between these two values. In the selections made among alternative investments, priority is given to the project with the highest IRR. If the investment will be financed entirely using debt, the IRR represents the highest interest rate that can be paid. In making an investment decision, the IRR is compared with the rate of return that the investor expects from the investment. Although it varies according to the risk of the project, the expectation of the investor, etc., the minimum value of the profitability © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 267–274, 2023. https://doi.org/10.1007/978-3-031-39774-5_33

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rate expected from the investment is accepted as the cost of capital. As a result of the comparison, it is decided to accept or reject the project. If the IRR calculated for any project is higher than the expected profitability from the investment, the project is accepted, otherwise the project is rejected. Figure 1 presents the acceptance or rejection decision based on the relationship between NPW and IRR.

Fig. 1. Acceptance or rejection decision based on NPW and IRR.

In the literature the IRR analysis is generally handled under deterministic environment which does not consider possible changes in the market. For example, a change in the first cost or the salvage value could significantly change the acceptance or rejection decision of a project. Fuzzy sets provide us an important opportunity to deal with uncertainty, especially when past data is not available. Fuzzy logic is developed by Zadeh [1] in 1965 as an efficient tool to present values with membership and non-membership degrees to present the complexity and uncertainty more realistically in the assessments of the decision analyses. After its introduction, ordinary fuzzy sets have been extended by many researchers. One of the latest and mostly accepted extensions of the ordinary fuzzy sets is picture fuzzy sets (PFSs). PFSs are proposed by Cuong [2] in 2014 as a direct extension of the ordinary fuzzy sets and the intuitionistic fuzzy sets. PFSs are presented with three parameters: degree of positive membership, degree of neutral membership, and degree of negative membership. PFSs can be used when human opinions include more answers as yes, refusal, neutral, and no. In this study, for the first time, IRR analysis is handled under PF environment and original equations are developed. The method is illustrated with an application of technology manufacturer of six-axis, electric servo-driven robots. The remaining sections of the study are given as follows. Section 2 includes a literature review on IRR. Section 3 presents the preliminaries of PFSs. Section 4 presents the proposed PF-IRR analysis method. Section 5 includes an application of the proposed approach on an example. Section 6 presents the conclusions and recommendations for future study.

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2 Literature Review on IRR Analysis A detailed literature search on IRR was conducted in the Scopus database. A total of 4,754 publications were reached when the titles, abstracts, and keywords including the term “internal rate of return” were examined. Figure 2 shows the publication numbers per year. The first study on IRR was published in 1935 and especially after 2000s, the studies in this field gained a momentum and attracted more authors. 45.6% of the IRR studies were collected in the first three subject areas which are Energy, Engineering, and Environmental Science. While 15.1% of the studies were done in the United States, 74.9% are article type of documents.

Fig. 2. IRR publications per year.

Although IRR has been frequently handled in the literature with classical approach, the number of studies integrating IRR analysis with fuzzy sets is very limited. Some of the most recent fuzzy IRR publications are as follows. López-marín et al. [3] studied the IRR of four biodegradable raffias by using fuzzy logic-ordered weighted averages to aggregate the information. Hašková and Fiala [4] compared the conventional approach and the fuzzy approach of IRR estimation of subsidized projects. Deveci et al. [5] determined the degree of importance of criteria including IRR, affecting site selection of solar photovoltaic projects under fuzzy environment. Jin et al. [6] proved that the sensitivity of the IRR of project investment from high to low is unit price of electricity sold, electricity sold and operation cost under fuzzy environment of a power investment project. Morozov et al. [7] developed of mathematical model to assess the investment risks of startup projects in the IT industry based on fuzzy set theory and took the main financial parameters of the project as NPW and IRR. Vatani et al. [8] calculated the profitability of the health, safety and environment-management system implementation by defining the values of the financial process flow in terms of fuzzy numbers and using the IRR. Serrano-Gomez and Muñoz-Hernandez [9] transformed the experts judgments into triangular fuzzy values to assess the influence of risks on NPW and IRR. Banda [10] developed a fuzzy techno-financial methodology based on five capital budgeting

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techniques including NPW and modified IRR. Lavrynenko et al. [11] evaluated the effectiveness and risks of investment projects based on fuzzy logic for the formalization of uncertainty by considering some input variables like NPW and IRR. Pullteap et al. [12] calculated some financial data’s NPW and IRR to determine the investment feasibility level by using fuzzy logic. To the best knowledge of the authors, there is no study in the literature handling the uncertainty in IRR with any extensions of ordinary fuzzy sets.

3 Preliminaries: Picture Fuzzy Sets Some definitions and operations about PFSs are given below [2]. Definition 1. A PFS on a A˜ P of the universe of discourse U is given by Eq. (1).      A˜ P = u, μA˜ P (u), ηA˜ P (u), νA˜ P (u) u ∈ U

(1)

where μA˜ P (u) : U → [0, 1], ηA˜ P (u) : U → [0, 1], νA˜ P (u) : U → [0, 1]

(2)

0 ≤ μA˜ P (u) + ηA˜ P (u) + νA˜ P (u) ≤ 1 ∀u ∈ U

(3)

and

For each u, μA˜ p (u), ηA˜ P (u), and νA˜ P (u) are called the degree of positive membership, ˜ the degree of neutral membership, and the degree of   negative membership of u to AP , respectively. ρ = 1 − μA˜ P (u) + ηA˜ P (u) + νA˜ P (u) is called as the degree of refusal membership of u in the PFS A˜ P . Definition 2. Some basic operations of PFSs are given in Eqs. (4–7) [2].   ˜ P ⊕ B˜ P = µ ˜ + µ ˜ − µ ˜ µ ˜ , η ˜ η ˜ , ν ˜ ν ˜ A AP BP AP BP AP BP AP BP   ˜ P ⊗ B˜ P = µ ˜ µ ˜ , η ˜ + η ˜ − η ˜ η ˜ , ν ˜ + ν ˜ − ν ˜ ν ˜ A AP BP AP BP AP BP AP BP AP B P λ · A˜ P =

 λ  λ λ , ηA˜ , νA˜ for λ > 0 1 − 1 − μA˜ P P

P

   λ  λ

 A˜ λP = μλA˜ , 1 − 1 − ηA˜ P , 1 − 1 − νA˜ P for λ > 0 P

(4) (5) (6) (7)

  Definition 3. Let A˜ P = μA˜ P , ηA˜ P , νA˜ P be a picture fuzzy number (PFN). Then, the relative magnitude index (RMI) for a PF number is calculated as follows.   (8) RMIA˜ P = (1/2) × 2 × μA˜ P − ηA˜ P /2 − νA˜ P

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4 Picture Fuzzy Internal Rate of Return Analysis In this section PF-IRR analysis method will be presented in steps. Step 1: First, the number of the real positive i∗ s (IRRs) should be determined. To do this, the net cash flow (NCF) rule of signs method will be applied. The number of real positive i∗ s for a project is always less than or equal to the number of sign changes in the NCF where zero cash flow is ignored. In a NCF a change from either positive to negative or negative to positive is counted as one sign change. If the rule of NCF signs indicates more than one possible positive i∗ value, then it should be proceeded to the accumulated cash-flow sign test (ACF) to eliminate some possibility of multiple rates of return. Step 2: The ACF is the sum of the NCFs up to and including a given life. If the ACF starts with a negative value and changes sign only once, then we can say there is a unique positive i∗ . Step 3: If a unique positive i∗ value is found in Step 2, it is continued with the trial-and-error method with an educated guess at the value of i∗ to compute the NPW of NCFs and check whether it is positive, negative, or zero. For instance, if NPW is found as negative, since we search for a value of i∗ that makes NPW equal to zero, the NPW of the cash flow should be increased. Thus, the tried i∗ value should be lowered, and the process should be repeated until the NPW is approximately equal to zero. Step 4: The point where the NPW is bounded by a positive and a negative value, the linear interpolation approach is applied to approximate i∗ by using Eq. (9). IRR = ia +

NPWa (ib − ia ) (NPWa − NPWb )

(9)

where ia = lower interest rate ib = higher interest rate NPWa = NPW using the lower interest rate NPWb = NPW using the higher interest rate

5 PF-IRR Application on a Technology Manufacturer Company A technology company that manufactures six-axis, electric servo-driven robots, predicts the cash flows given in Table 1 for a new machine. The predictions represent the consensus of experts team including top managers of the company. Each parameter is represented by a cash value and the experts’ trust to these values. The company wants to determine the number of possible IRR values and all i∗ values between 0 and 100%. Also, since the company’s MARR is known to be 2%, the company wants to know if this investment is justifiable. First, the NCF is calculated as given in Table 2 by using Eq. (4). Since there are four sign changes in the NCF, there may be up to four i∗ values. In the next step accumulated NCF should be calculated as in Table 3 to eliminate some possibility of IRRs.

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E. Haktanır Table 1. PF parameter estimations of the given investment.

Year

Expense, $

Savings, $

0

(−33,000; 0.8, 0.1, 0.1)

(0; 1.0, 0.0, 0.0)

1

(−15,000; 0.85, 0.5, 0.1)

(18,000; 0.95, 0.0, 0.5)

2

(−40,000; 0.90, 0.0, 0.1)

(38,000; 0.9, 0.0, 0.1)

3

(−20,000; 0.9, 0.5, 0.5)

(55,000; 0.85, 0.0, 0.15)

4

(−13,000; 0.75, 0.1, 0.1)

(12,000; 0.8, 0.1, 0.1)

Table 2. Net cash flow. Year

Expense, $

Savings, $

NCF, $

0

(−33,000; 0.8, 0.1, 0.1)

(0; 1.0, 0.0, 0.0)

(−33,000; 1.0, 0.0, 0.0)

1

(−15,000; 0.85, 0.5, 0.1)

(18,000; 0.95, 0.0, 0.5)

(+3,000; 0.99, 0.0, 0.05)

2

(−40,000; 0.90, 0.0, 0.1)

(38,000; 0.9, 0.0, 0.1)

(−2,000; 0.99, 0.0, 0.01)

3

(−20,000; 0.9, 0.5, 0.5)

(55,000; 0.85, 0.0, 0.15)

(+35,000; 0.99, 0.0, 0.08)

4

(−13,000; 0.75, 0.1, 0.1)

(12,000; 0.8, 0.1, 0.1)

(−1,000; 0.95, 0.01, 0.01)

Table 3. Cumulative net cash flow. Year

NCF, $

Cumulative NCF, $

0

(−33,000; 1.0, 0.0, 0.0)

(−33,000; 1.0, 0.0, 0.0)

1

(+3,000; 0.99, 0.0, 0.05)

(−30,000; 1.0, 0.0, 0.0)

2

(−2,000; 0.99, 0.0, 0.01)

(−32,000; 1.0, 0.0, 0.0)

3

(+35,000; 0.99, 0.0, 0.08)

(3,000; 1.0, 0.0, 0.0)

4

(−1,000; 0.95, 0.01, 0.01)

(2,000; 1.0, 0.0, 0.0)

Cumulative NCF starts with a negative value and occurs only one sign change which indicates one possible positive i∗ value. So, it can be proceeded to the trial-and-error method for the PW equation given below after the defuzzification of the values in cumulative NCF by using Eq. (8).    0 = −33, 000+3, 000 P/F, i∗ , 1 − 2, 000 P/F, i∗ , 2 + 35, 000 P/F, i∗ , 3  −1, 000 P/F, i∗ , 4 For i∗ = 2, the NPW equals to $75.5 and for i∗ = 3, the NPW equals to $-832.5. Next, the interpolation formula given in Eq. (9) should be applied. IRR = 2% +

75.5(3% − 2%) = 2.08% 75.5 − (−832.5)

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Since i∗ is calculated as 2.08%, and the company’s MARR is known to be 2%, the company can accept the investment. Figure 3 presents the IRR’s sensitivity with the change of the first cost when it is increased or decreased by 1,000$ each time. In each situation the cash flow passed the accumulated NCF sign test.

Fig. 3. IRR’s change depending on different first cost values.

When the first cost is decreased up to −36,000$, the IRR becomes a negative value as −0.98% and when it is increased up to −5,000$, the IRR becomes greater than 100% (104.89%). Negative IRR occurs when the aggregate amount of NCF caused by an investment is less than the amount of the first cost. In such situation, a negative return on the investment will be experienced. Thus, if a negative IRR is calculated the investment should not be made.

6 Conclusions IRR analysis is an economic analysis method, just like net present worth analysis and annual worth analysis, that is frequently used to measure the profitability of an investment, or to evaluate more than one alternative and choose the most profitable one. Although IRR analysis has been discussed in many studies in the literature, there are very few studies integrated with fuzzy set theory. The IRR study, especially considering fuzzy sets extensions, is quite limited. However, studies that consider real-life uncertainties will provide more accurate results in the use of the IRR method. PFSs, one of the most up-to-date and widely accepted fuzzy sets, was used in this study, which handles future cash forecasts as probabilistic, not deterministic. Thanks to PFSs, parameters such as positive membership, the degree of neutral membership, and the degree of negative membership related to the predictions made by decision-making experts for the future were also taken into account.

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For the future research, it is recommended to extend the fuzzy IRR methods by using some of the other recent extensions of ordinary fuzzy sets such as Pythagorean fuzzy sets or spherical fuzzy sets. Also, by using PF aggregation operators, multiple experts’ evaluations can be aggregated.

References 1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 2. Cuong, B.C.: Picture fuzzy sets. J. Comput. Sci. Cybern. 30(4), 409–420 (2014) 3. López-marín, J., Gálvez, A., Del Amor, F.M., Piñero, M.C., Brotons-martínez, J.M.: The cost-benefits and risks of using raffia made of biodegradable polymers: the case of pepper and tomato production in greenhouses. Horticulturae 8(2), 133 (2022) 4. Hašková, S., Fiala, P.: Internal rate of return estimation of subsidised projects: conventional approach versus fuzzy approach. Comput. Econ. (2022). https://doi.org/10.1007/s10614-02210299-7 5. Deveci, M., Cali, U., Pamucar, D.: Evaluation of criteria for site selection of solar photovoltaic (PV) projects using fuzzy logarithmic additive estimation of weight coefficients. Energy Rep. 7, 8805–8824 (2021) 6. Jin, X., Liu, Q., Long, H.: Impact of cost–benefit analysis on financial benefit evaluation of investment projects under back propagation neural network. J. Comput. Appl. Math. 384, 113172 (2021) 7. Morozov, V., Tsesliv, O., Kolomiiets, A., Kolomiiets, S.: Construction of a mathematical model for analyzing the effectiveness of IT startups. CEUR Workshop Proc. 2851, 283–292 (2021) 8. Vatani, J., Gholipour, M., Poorhaji, Z., Khalighdost, F., Torshizi, Y.F.: Comparison of economic evaluation methods of classic NPV1, MIRR2, AIRR3, IRR4 in performance evaluation of health, safety and environment-management system (HSE-MS). Iran Occup. Health 17(1), 35 (2020) 9. Serrano-Gomez, L., Muñoz-Hernandez, J.I.: Risk influence analysis assessing the profitability of large photovoltaic plant construction projects. Sustainability (Switzerland) 12(21), 1–16 (2020) 10. Banda, W.: A fuzzy techno-financial methodology for selecting an optimal mining method. Nat. Resour. Res. 29(5), 3047–3067 (2020) 11. Lavrynenko, S., Kondratenko, G., Sidenko, I., Kondratenko, Y.: fuzzy logic approach for evaluating the effectiveness of investment projects. Int. Sci. Tech. Conf. Comput. Sci. Inf. Technol. 2, 297–300 (2020) 12. Pullteap, S., Samartkit, P., Kheovichai, K., Seat, H.C.: A software development for investment analysis of LED lighting production project using fuzzy logic technique. Sci. Eng. Health Stud. 14(2), 83–100 (2020)

LR-Type Spherical Fuzzy Numbers and Their Usage in MCDM Problems Cengiz Kahraman(B) , Sezi Cevik Onar, and Basar Öztaysi Department of Industrial Engineering, Istanbul Technical University, 34367 Istanbul, Turkey [email protected]

Abstract. Spherical fuzzy sets (SFS) are recent extension of ordinary fuzzy sets, which are composed of three parameters whose squared sum is at most one. SFS are defined by discrete values of these three parameters and provide a larger assignment volume than picture fuzzy sets. We propose a non-linear continuous membership, non-membership, and/or hesitancy function for spherical fuzzy sets based on LR-type nonlinear fuzzy numbers. We illustrate their usage in multiple criteria supplier selection problems. The paper is concluded by limitations and suggestions for further research. Keywords: spherical fuzzy sets · LR-fuzzy numbers · MCDM · supplier selection

1 Introduction After the introduction of intuitionistic fuzzy sets, new extensions of fuzzy gained a significant acceleration (Oztaysi et al., 2017). Spherical fuzzy sets have been developed by Kahraman and Kutlu Gündo˘gdu (2018) as an extension of picture fuzzy sets, providing a larger domain volume than picture fuzzy sets. Triangular prism of picture fuzzy sets becomes one-eighth of the unit sphere in spherical fuzzy sets. The squared sum of membership, non-membership and hesitancy degrees can be at most equal to 1.0 in spherical fuzzy sets. Spherical fuzzy sets have been used in many decision making papers such as spherical fuzzy outranking techniques (Akram et al., 2023a, 2023b; Zahid et al., 2022), spherical fuzzy VIKOR (Akram et al., 2021; Kutlu Gundogdu and Kahraman, 2019), spherical fuzzy TOPSIS (Akram et al., 2021), spherical fuzzy regret (Cevik Onar et al., 2020), spherical distance and similarity measures (Donyatalab et al., 2021, 2022), spherical fuzzy CRITIC (Kahraman et al., 2021, 2022), spherical Fuzzy EXPROM (Kahraman et al., 2021), spherical fuzzy TOPSIS (Kutlu Gundogdu and Kahraman, 2019a, 2019b; Kutlu Gundogdu et al., 2021; Karasan and Kahraman, 2019), spherical fuzzy AHP (Kutlu Gundogdu and Kahraman, 2019; Oztaysi et al., 2020a, 2020b), spherical fuzzy QFD (Kutlu Gundogdu and Kahraman, 2020; Haktanır and Kahraman, 2019), spherical fuzzy CODAS (Kutlu Gundogdu and Kahraman, 2019), spherical fuzzy WASPAS (Kutlu Gundogdu and Kahraman, 2019), spherical fuzzy optimization (Kutlu Gundogdu et al., 2021), spherical similarity measures (Kutlu Gundogdu et al., 2020), analysis of usability © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 275–283, 2023. https://doi.org/10.1007/978-3-031-39774-5_34

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(Kutlu Gundogdu et al., 2020), spherical fuzzy REGIME (Oztaysi et al., 2021), spherical fuzzy AHP-VIKOR (Oztaysi et al., 2020a, 2020b), fuzzy PROMETHEE (Senvar et al., 2014). There is not yet any work proposing LR-type spherical fuzzy numbers in the literature. Spherical fuzzy numbers are defined as discrete numbers composed of three parameters whose squared sum is limited to 1.0. SFNs can be defined by continuous membership functions as LR-type fuzzy numbers. Then non-linear SF LR-type fuzzy numbers can be used in the new extensions of MCDM methods such as LR-type spherical fuzzy TOPSIS method, LR-type spherical fuzzy AHP, or LR-type spherical fuzzy VIKOR. This paper developed continuous spherical fuzzy numbers and use it to define LR-type spherical fuzzy numbers. Even the complexity of the LR-type SF numbers are larger than the single valued SF numbers, the obtained results are more correct and realistic. The rest of the paper is organized as follows. Section 2 presents LR-type fuzzy numbers. Section 3 gives spherical fuzzy numbers. Section 4 introduces LR-type spherical fuzzy numbers. Section 5 applies the proposed LR-type spherical fuzzy numbers to MCDM problems. Section 6 concludes the paper.

2 LR-Type Fuzzy Numbers (LR-Type FNs) The main idea of LR-type fuzzy numbers is to divide the nonlinear membership function into a left side nonlinear function and a right side nonlinear function. An LR-type fuzzy number is represented by (m, α, β)LR where α is the left support and is β the right support and m is the center value of the LR-type fuzzy number.

Fig. 1. A trapezoidal LR-type FN

Fig. 2. A triangular LR-type FN

The left and right side functions in Fig. 1 are given by Eqs. (1–3): μ(x) = 1, m1 ≤ x ≤ m2  m1 − x , x ≤ m1 μL (x) = L(x) = f α   x − m2 , x ≥ m2 μR (x) = R(x) = f β

(1)



The left and right side functions in Fig. 2 are given by Eqs. (4–5):   m−x L(x) = f ,x ≤ m α

(2) (3)

(4)

LR-Type Spherical Fuzzy Numbers and Their Usage in MCDM Problems

 R(x) = f

 x−m ,x ≥ m β

277

(5)

3 Spherical Fuzzy Numbers (SFNs) Spherical fuzzy sets are one of the latest extensions of picture fuzzy sets and Pythagorean fuzzy sets. Spherical fuzzy sets (SFSs) present a larger domain for the expert to assign membership degrees. Each decision-maker can determine his/her hesitancy (indeterminacy) information as a separate parameter under the spherical fuzzy environment. Definition 1. A Single-valued spherical fuzzy set (SVSFSs) of the universe X is given by (Kutlu Gündoˇgdu and Kahraman, 2019):    (6) A˜ s = x, μA˜ s (x), ϑA˜ s (x), πA˜ s (x)x ∈ X } where μA˜ s (u), ϑA˜ s (u), πA˜ s (u) : U → [0, 1] are the degrees of membership, nonmembership, and indeterminacy of x to A˜ S , respectively, and 0 ≤ μ2A˜ (x) + ϑA2˜ (x) + πA2˜ (x) ≤ 1 s

 Then,

s

(7)

s

  1 − μ2˜ (x) + ϑ 2˜ (x) + π 2˜ (x) is defined as the refusal degree of x in X . As

As

As

Definition 2. Assume that A˜ s and B˜ s be any two spherical fuzzy sets. So the basic operations of SFSs can be defined as follows (Kutlu Gündoˇgdu and Kahraman, 2019):

 μ2˜ + μ2˜ − μ2˜ μ2˜ , ϑA˜ s ϑB˜ s , (1 − μ2˜ )π 2˜ + (1 − μ2˜ )π 2˜ − π 2˜ π 2˜ Bs B s As As As Bs A s Bs A s Bs

 A˜ s ⊗ B˜ s = μA˜ s μB˜ s , ϑ 2˜ + ϑ 2˜ − ϑ 2˜ ϑ 2˜ , (1 − ϑ 2˜ )π 2˜ + (1 − ϑ 2˜ )π 2˜ − π 2˜ π 2˜

A˜ s ⊕ B˜ s =

Bs

As



As Bs

Bs

As

As



Bs

As Bs

(8) (9)

,k > 0

(10)

   ˜Aks = μk , 1 − (1 − ϑ 2 )k , (1 − ϑ 2 )k − (1 − ϑ 2 − π 2 )k , k > 0 ˜ ˜ ˜ ˜ A˜

(11)

k A˜ s =

k 1 − (1 − μ2˜ ) ,ϑAk˜ , As s

s

As

k (1 − μ2˜ ) As

− (1 − μ2˜ As

As

As

k − π 2˜ ) As

As

Definition 3. For the SFS A˜ s = (μAs , ϑAs , πAs ) and B˜ s = (μBs , ϑBs , πBs ), the followings are valid, provided that k, k1 and k2 ≥ 0 (Kutlu Gündoˇgdu and Kahraman, 2019): I. A˜ s ⊕ B˜ s = B˜ s ⊕ A˜ s

(12)

II. A˜ s ⊗ B˜ s = B˜ s ⊗ A˜ s

(13)

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III. k(A˜ s ⊕ B˜ s ) = k A˜ s ⊕ k B˜ ss

(14)

IV. k1 A˜ s ⊕ k2 A˜ s = (k1 + k2 )A˜ s

(15)

k V. (A˜ s ⊗ B˜ s ) = A˜ ks ⊗ B˜ sk

(16)

VI. A˜ ks 1 ⊗ A˜ ks 2 = A˜ ks 1 +k2

(17)

Definition 4. Suppose that A˜ s and B˜ s be two Spherical fuzzy numbers. Then to compare these SFS, use the score (SC) and accuracy (AC) functions which are defined as follows (Kutlu Gündoˇgdu and Kahraman, 2019):

 πA˜ 2 πA˜ 2 (18) SC A˜ s = (μA˜ s − s ) − (ϑA˜ s − s ) 2 2

 (19) AC A˜ s = μ2A˜ + ϑA2˜ + πA2˜ s

s

s

After the calculation of the results of score and accuracy functions, the comparison rules are as follows: 

 If SC A˜ s > SC B˜ s , then A˜ s > B˜ s ;







 If SC A˜ s = SC B˜ s and AC A˜ s > AC B˜ s , then A˜ s > B˜ s ;







 If SC A˜ s = SC B˜ s , AC A˜ s = AC B˜ s , then A˜ s > B˜ s ;







 If SC A˜ s = SC B˜ s , AC A˜ s = AC B˜ s , then A˜ s = B˜ s ; Definition 5. Spherical Fuzzy Weighted  Arithmetic Mean (SFWAM ) with respect to, w = (w1 , w2 , . . . , wn ); wi ∈ [0, 1]; ni=1 wi = 1, is specified as (Kutlu Gündoˇgdu and Kahraman, 2019):

 SFWAM w A˜ S1 , A˜ S2 , . . . , A˜ Sn = w1 A˜ S1 + w2 A˜ S2 + · · · + wn A˜ Sn ⎧ ⎫   n n n n ⎨ ⎬ wi       w w 1 − μ2As , = 1 − ϑAwsi ,  (1 − μ2As ) i − (1 − μ2As − πA2s ) i ⎩ ⎭ i=1

i=1

i=1

i=1

(20) Definition 6. Spherical Fuzzy Weighted  Geometric Mean (SFWGM ) with respect to, w = (w1 , w2 , . . . , wn ); wi ∈ [0, 1]; ni=1 wi = 1, is proofed as (Kutlu Gündoˇgdu and Kahraman, 2019):

 i ˜ wi ˜ wi SFWGM w A˜ S1 , A˜ S2 , . . . , A˜ Sn = A˜ w S1 + AS2 + · · · + ASn ⎧ ⎫    n n n n ⎨ ⎬ wi       i  2 2 − π 2 )wi  (1 − ϑ 2 )wi − 1 − ϑ = μw , 1 − , (1 − ϑ As As As As As ⎩ ⎭ i=1

i=1

i=1

i=1

(21)

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Definition 7. Let x ∈ X be a universe set and A˜ and B˜ be two sample spherical fuzzy ˜ B) ˜ is given by Eq. (22) (Ashraf et al., sets. Then normalized Euclidean distance Ed SFS (A, 2023).   n  1   ˜ ˜ Ed SFS (A, B) =  (μA (xi ) − μB (xi ))2 + (ϑA (xi ) − ϑB (xi ))2 + (πA (xi ) − πB (xi ))2 n

(22)

i=1

4 LR-Type Spherical Fuzzy Numbers (LR-Type SFNs) A triangular LR-type spherical fuzzy number satisfying Eq. (7) can be represented by Fig. 3.   (23) xμ = mμ , αμ , βμ LR xπ = (mπ , απ , βπ )RL

(24)

xϑ = (mϑ , αϑ , βϑ )RL

(25)

The left and right side functions of the membership functions in Fig. 3 are given by Eqs. (26–31): Then a triangular LR-type spherical fuzzy number A˜ s,LR can be given as in Eq. (32):   mμ − x Lμ (x) = f , x ≤ mμ (26) αμ  x − mμ Rμ (x) = f (27) , x ≥ mμ βμ   x − mϑ , x ≤ mϑ (28) Lϑ (x) = f βϑ   mϑ − x , x ≥ mϑ Rϑ (x) = f (29) αϑ   x − mπ , x ≤ mπ (30) Lπ (x) = f βπ   mπ − x , x ≥ mπ Rπ (x) = f (31) απ   !  (32) A˜ s,LR = x, mμ , αμ , βμ LR , (mϑ , αϑ , βϑ )RL , (mπ , απ , βπ )RL x ∈ X } Or  ! A˜ s,LR = x, Lμ (x), Rμ (x) , Lϑ (x), Rϑ (x) , Lπ (x), Rπ (x) x ∈ X }

(33)

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Fig. 3. LR-type spherical fuzzy number

5 Multi-criteria Decision Making with LR-Type SFNs Lets consider the fuzzy decision matrix given in Eq. (34) whose linguistic terms are represented by LR-type spherical fuzzy numbers with m alternatives and n criteria. We developed the linguistic scale given in Table 1 for this purpose. The numbers in this table are symmetrical but any LR type spherical fuzzy numbers can be used based on the given linguistic terms. For instance, if you are between low (L) and middle (M), you can assign a non-symmetrical LR-type spherical fuzzy number such as (0.45., 0.2, 0.1)LR , (0.55, 0.1, 0.1)RL , (0.35, 0.1, 0.2)RL . Any spherical fuzzy LR type number (SFLRN) can be defuzzified to a crisp number by Eq. (35). Then Eq. (29) can be transformed to Eq. (36). Then, using a multi-criteria decision making method such as TOPSIS, ELECTRE, VIKOR, or EDAS, the problem can be solved by crisp values. ⎡

˜ LR−Z = D

 ⎤  mμ , αμ , βμ LR , (mϑ , αϑ , βϑ )RL , mμ , αμ , βμ LR , (mϑ , αϑ , βϑ )RL , · · · ⎢ ⎥ (mπ , απ , βπ )RL (mπ , απ , βπ )RL ⎢ 11 1m ⎥ ⎢ ⎥ ⎢ ⎥ . . .. .. .. ⎢ ⎥ . ⎢ ⎥     ⎢  ⎥ ⎣ mμ , αμ , βμ LR , (mϑ , αϑ , βϑ )RL , ⎦ mμ , αμ , βμ LR , (mϑ , αϑ , βϑ )RL , ··· (mπ , απ , βπ )RL (mπ , απ , βπ )RL 

n1

(34)

nm

   Def (SFLRFN ) = Def mμ , αμ , βμ LR , (mϑ , αϑ , βϑ )RL , (mπ , απ , βπ )RL = 2mμ − α μ + βμ − (2mϑ + αϑ − βϑ ) − (2mπ + απ − βπ ) (35) ⎤ (Def (SFLRN ))11 · · · (Def (SFLRN ))1m ⎥ ⎢ .. .. .. =⎣ ⎦ . . . ⎡

˜ LR−Z D

(Def (SFLRN ))n1 · · · (Def (SFLRN ))nm

(36)

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Table 1. Linguistic scale for LR-type spherical FNs Linguistic terms for restriction

Corresponding LR-type SF numbers

Absolutely Low, AL

(0.2, 0.1, 0.1)LR , (0.8, 0.1, 0.1)RL , (0.1, 0.1, 0.1)RL

Very Low, VL

(0.3, 0.1, 0.1)LR , (0.7, 0.1, 0.1)RL , (0.2, 0.1, 0.1)RL

Low, L

(0.4, 0.1, 0.1)LR , (0.6, 0.1, 0.1)RL , (0.3, 0.1, 0.1)RL

Middle, M

(0.5, 0.1, 0.1)LR , (0.5, 0.1, 0.1)RL , (0.4, 0.1, 0.1)RL

High, H

(0.6, 0.1, 0.1)LR , (0.4, 0.1, 0.1)RL , (0.3, 0.1, 0.1)RL

Very High, VH

(0.7, 0.1, 0.1)LR , (0.3, 0.1, 0.1)RL , (0.2, 0.1, 0.1)RL

Absolutely High, AH

(0.8, 0.1, 0.1)LR , (0.2, 0.1, 0.1)RL , (0.1, 0.1, 0.1)RL

6 Conclusions Spherical fuzzy sets provided a larger domain volume than picture fuzzy sets for the assignment of membership parameters. However, spherical fuzzy sets are defined by discrete values of these parameters whereas continuous functions must be used for continuous fuzzy numbers. This has been settled by spherical fuzzy LR type numbers in this paper. SFLRNs can be used in multiple criteria decision making to obtain more realistic results than the discrete fuzzy numbers based decision making methods. Since the page limitations, we could not present a decision making application based on SFLRNs in this paper. For further research, we suggest some MCDM methods in industrial engineering to be extended by SFLRNs such as SFLRN based TOPSIS method or SFLRN based AHP method (Kahraman, 2006; Kahraman et al., 2006).

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Analyzing Customer Requirements Based on Text Mining via Spherical Fuzzy QFD Sezen Ayber1

, Nihal Erginel2(B) , Mustafa Ünver3 and Ahmet Aydın4

, Gökhan Göksel4

,

1 Institute of Graduate Programs, Eski¸sehir Technical University, Eski¸sehir, Turkey 2 Industrial Engineering Department, Eski¸sehir Technical University, Eski¸sehir, Turkey

[email protected]

3 Industrial Engineering Department, Gebze Technical University, Kocaeli, Turkey 4 Computer Engineering Department, Eski¸sehir Technical University, Eski¸sehir, Turkey

Abstract. Customer requirements (CRs) and their interpretation of the product is a crucial stage for a company’s existence in a competitive environment. QFD provides a structured methodology to define and rank the product technical requirements (TRs) related to CRs. Because the importance levels of CRs and the relationship between CRs and TRs have ambiguous information, it is profitable to use fuzzy numbers while implementing the QFD process. Nowadays, online reviews of users are an important source for collecting customer requirements. Text mining methods are helpful for extracting meaningful expressions on product attributes. This study proposed a methodology that integrates text mining methods for collecting CRs and spherical fuzzy numbers with QFD for ranking TRs. By the new proposed method, the CRs on smartwatches are extracted through Latent Dirichlet Allocation (LDA) method with specific suggestion words that is the first proposed in this study. Then, spherical fuzzy QFD is implemented to acquire weights of TRs. The smartwatch case is the first applied with text mining and QFD with spherical fuzzy numbers. Keywords: Customer Requirements · Text mining · Spherical Fuzzy Numbers · Quality Function Deployment · Latent Dirichlet Allocation

1 Introduction Quality Function Deployment (Quality Function Deployment, QFD) is a four-house systematic approach that transforms customer requests and expectations into technical requirements and then production parameters. The House of Quality, on the other hand, is the first house that transforms customer requests and expectations, which is the first step of this approach, into the technical requirements of the product. QFD is a systematic and customer-oriented method used to transform customer needs into engineering specifications [1]. Traditional methods for collecting customer needs and requirements such as questionnaires, interviews, and telephone surveys are costly, cumbersome and time consuming [2]. Nowadays the number of internet users prefer e-commerce sites to select and buy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 284–292, 2023. https://doi.org/10.1007/978-3-031-39774-5_35

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the products. Also, they can express purchase experiences, perceptions, opinions, complaints, satisfactions and feedback about purchased products through online e-commerce sites such as Amazon, eBay, Alibaba or online social media platforms such as twitter. The customer reviews on online e-commerce platforms are trustworthy because customers who purchased products only access them [3]. The analysis of customer reviews is not only useful for customers to make more reasonable purchasing decisions but also useful for companies to improve technical requirements of products, develop better products and R&D strategies. In this study, a methodology is proposed for ranking technical requirements of a product via House of Quality (HoQ) that is the first house of QFD, based on customer reviews that text mining techniques applied. The aim of the proposed approach is to find the “recommendation sentences” of online customer reviews who have the experiences on the product, and use them for expressing the customer requirements and needs in HoQ. As a case study, the smartwatch is selected. The smartwatch has become a popular wearable product that was first released in the early 2000s [4]. The rest of the paper is organized as follows. Section 2 provides a literature review about QFD and text mining. Section 3 describes proposed methodology and Sect. 4 presents a case of smartwatch to show validation of the proposed methodology. Section 5 gives the conclusions.

2 Literature Review In the literature, there are many studies on the use of the QFD method in product design and development. Vonderembse and Raghunathan [5] have shown that it is possible to reduce cost and time and increase customer satisfaction by using QFD with the articles in the literature they examined. Tan, Xie and Shen [6] presented a case study using QFD and the Kano Model for an NPD process and emphasized that the use of QFD provides competitive advantage. There are many studies in the literature on the analysis of QFD with fuzzy sets. Ertay et al. [7] combined QFD with the Analytical Hierarchy Process (AHP) approach in a fuzzy environment and described the relationships between customer needs and technical requirements with fuzzy numbers. Erginel [8] proposed a fuzzy failure QFD matrix to prioritize product defects, and set up a multi-objective decision model to select technical requirements and presented its solution. Haktanır and Kahraman [9] proposed the interval valued Pythagorean fuzzy QFD method and applied it to the development of solar photovoltaic technology. Wang et al. [10] applied the fuzzy QFD for a vehicle design in China and proposed a framework. Gündo˘gdu and Kahraman [11] performed the spherical fuzzy SF_QFD analysis to include linguistic variables in the model in the QFD to determine customer requirements’ important degree and their relationships with design requirements. Finger and Lima-Junior [12] proposed a model using QFD with hesitant fuzzy sets to make supplier development programs. There are few articles in the literature that use text mining in the QFD approach. Ozdagoglu et al. [13] used QFD and topic modeling to analyze the comments of a restaurant made by the customer in the digital environment and to reveal the real customer needs. Asadabadi et al. [14] suggested a method by using NLP, text mining and QFD.

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They used online product reviews to obtain VoC. Ming and Zhang [14] analyzed customer comments obtained from e-commerce sites using LDA and Apriori algorithms and combining heuristic fuzzy numbers with Kano, AHP and KFY methods. Text mining methods aim to find valuable information and hidden patterns from the given textual dataset. A considerable amount of free format or natural language textual data is the subject of the text mining methods [16]. A typical text mining application begins with a set of sequential text preprocessing techniques. According to a specific problem, a number of those techniques can be applied to the text. Each technique transforms the given text into a different form. Text preprocessing techniques include breaking the text into sentences, tokenization, normalization, stop-word removal, stemming, filtering, etc. [17]. The main goal of those techniques is to contribute to the efficiency and effectiveness of text mining applications.

3 Proposed Methodology Stage 1: Collecting Customer Reviews Collecting whole customer reviews on products is easy from e-commerce platforms such as Amazon, eBay, Alibaba or social media platforms like twitter, YouTube, etc. Stage 2: Selecting Recommendation Sentences English grammar has a lot of recommendation words. People use these words for giving their thinking or suggestions on product developments to get friendlier and more functional of the product. The sentences that include these words reflect their experiences on the product. Customers write their recommendation sentences to the reviews of the e-commerce sites by using “can, could, may, might, shall, should, will, would” modals. These words and phrases are strong indicators of negative sentiment, dissatisfaction, or issues of development of product features, and the text mining methods are inevitable tools to extract recommendation sentences from a large amount of textual data by using those words. For example, if a customer gives reviews as “I wish the battery could last for 36 h without a charge” which clearly reflects the thinking and recommendation to improve the product. Stage 3: Preprocessing of Recommendation Sentences Removing smileys, unrelated texts, regularization of user typing mistakes are some cleaning operations while adding missing values or texts into dataset such as translation from foreign languages are called imputation. After applying cleaning operations, preprocessing of review sentences related to the customer expectation and requirements are performed by standard operations such as tokenization, filtering stop words, normalization of text, stemming. Stage 4: Topic Modeling Topic modeling is an unsupervised tool in machine learning which assigns a set of texts to some certain number of topics that are determined by the users. In other words, the aim of topic modeling is to discover hidden semantic relations inside any text data. Topics are

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composed of some set of words that have some similar co-occurrences or frequencies. [18] Topic modeling process generally begins with retrieval of texts (i.e. documents) from different sources. After documents retrieval some preprocessing operations are applied on raw text in order to prepare data for topic modeling. There are several topic modeling methodologies in the literature such as Latent Semantic Analysis (LSA), Latent Dirichlet Allocation (LDA), Random Projection (RP), Non-negative Matrix Factorization (NMF) and Principal Component Analysis (PCA). [18]. Latent Dirichlet Allocation (LDA) is a topic modeling tool which produces the topics as a finite mixture of probabilities by utilizing Dirichlet distribution which is a multivariate generalization of the beta distribution. As an output of LDA, each document can be represented by topic probabilities which add up to 1. [19]. LDA is preferred as a topic modeling methodology. In LDA, each topic is word distribution and each document is a topic distribution. Number of topics is determined by the users as an input parameter for LDA. Dirichlet distribution is used in LDA and it is a continuous multivariate probability distribution which has an alpha vector as a parameter which is a positive real number. It is known as multivariate beta distribution (MBD) and generally denoted as Dir (α). Here alpha is also known as concentration parameter. Dirichlet distribution is the basis of LDA and creates a k dimensional simplex. LDA output can be validated by using two different metrics: Perplexity and Coherence scores. Coherence refers to how similar N words in a topic, Briefly, the coherence score measures how similar these words are to each other. The higher the coherence, the better the topic modeling [20]. Perplexity is a statistical measure of how well a probability model predicts a sample. Lower perplexity values are desirable for any LDA outputs. Stage 5: Extracting Customer Requirements LDA gives topics and related topic representation by a pile of words along with belonging probabilities of each word to topics. Then, most representative sentences (MRS) for each topic are found by using mean (M _BSP i ) and standard deviation of probabilities of belonging sentences (S). MRS’s for each topic are extracted by starting from the sentence with highest probability (max(M _BP i )) up to the sentence with the probability higher than max(M _BP i )-(1)*Si . . As a result of the MRS determination process, the number of MRS is variant for each topic. Finally, we extracted most representative sentences (MRS) for each topic and number of sentences which are allocated into each topic in order to use them for QFD analysis. Stage 6: Weighted the TRs by Using HoQ with Spherical Fuzzy Numbers After extracting customer requirements, importance degrees of customer requirements are calculated by their frequencies. Number of sentences for positive cosine values is determined and normalized values are used. Then, by using online product description data, engineering characteristics are defined. The correlation matrix among ECs is constructed. “ +” symbol demonstrates a positive correlation, “-” symbol demonstrates a negative correlation and blank shows no correlation. After that, relationships between each CRs and TRs are determined using linguistic terms. Spherical fuzzy numbers (Eq. 1) corresponding to linguistic terms are found and relationship matrix between CRs and

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TRs is constructed. A spherical fuzzy number A in X is defined by Gündo˘gdu and Kahraman [11];   A = < x, μA˜ (x), vA˜ (x), πA˜ (x)) > : x ∈ X , μA˜ (x), vA˜ (x), πA˜ (x) ∈ [0, 1] 0 ≤ μ2A˜ (x) + vA2˜ (x) + πA2˜ (x) ≤ 1

(1)

Spherical fuzzy numbers can be used while determining relationship degrees between CRs and TRs [11]. Importance degrees of each TRs are calculated by multiplying importance degrees of CRs and relationship degrees between CRs and TRs. Then results are converted into crisp values Relative importance degrees of each TR are calculated by dividing importance levels of each DR to the cumulative degree of importance. Technical requirements are ranked, and according to these ranks, actions are determined.

4 Application of Proposed Method on Smart Watches Application is implemented on smart watches. Apple I-Watch 7 and its customer reviews are extracting Amazon e-commercial site on December 14, 2021. 2140 reviews and 9020 sentences are crawled from Amazon. According to the steps of the proposed algorithm, firstly 1035 recommendation sentences are extracted by using recognized modal auxiliary verbs. After collecting the reviews, we split the reviews into sentences. Then, those sentences are filtered by modal auxiliary verbs to get recommendation sentences from all reviews. Then, preprocessing operations are applied to these sentences. LDA is implemented on these determined recommendation sentences for topic modeling. In this study, all the LDA implementations are done on Python 3.9 by using Scikit-learn library. For validation and corpus preparation, we used Gensim library, some preprocessing implementations are done by using nltk library and matplotlib is used for graphical illustrations. Firstly, we tried to determine the number of topics in the dataset by using the elbow method by choosing k from 2 to 40 and we measured both coherence and perplexity scores for each k setting. As a result of elbow representation, we decided to set k as 16, since less improvements in both scores are observed as the k (number of topics) increases. We decided to set alpha as 0.3 and eta as 0.2. Each topic has 10 most frequent words. Table 1 shows LDA output for Topic 0 and 1. Then, for all topics defined with LDA, we counted the number of sentences which are assigned to that topic with a probability which is greater than 0.1. Since LDA gives sentences belonging probabilities to a specific topic with a probability which is greater than 0.1, we listed these belonging sentence probabilities (BP i ). Then, most representative sentences (MRS) for each topic are found by using mean (M _BP i ) and standard deviation of probabilities of belonging sentences (Si ). Finally, HoQ is conducted with CRs and TRs as given in Table 2, and by using spherical fuzzy relationships between CRs and TRs as given in Fig. 1. The most significant TRs are; present applications with 28.02, microprocessor with 21.40, MEMS sensor with 13.08 and microbiosensors with 11.85.

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Table 1. Topic 0 and 1 and their representations provided by LDA output as an example. Topic ID Topic Representation by Words along with their belonging probabilities to topics Topic 0

(‘0.026*“turn” + 0.014*“work” + 0.012*“track” + 0.009*“find” + 0.009*“time” + 0.008*“workout” + 0.008*“count” + 0.007*“o2” + 0.007*“easili” + 0.007*“way”’)

Topic 1

(‘0.033*“face” + 0.032*“thing” + 0.028*“see” + 0.027*“chang” + 0.013*“keep” + 0.011*“new” + 0.010*“good” + 0.010*“differ” + 0.010*“activ” + 0.009*“detect”’)

Table 2. Customer requirements and technical requirements for smart watches Customer requirements

Technical requirements

CR1

Fast response time of the watch (Switch speed between interfaces)

TR1

CR2

The ability to make emergency calls and fall TR2 detection

Present applications

CR3

Ease of use, friendly user interface

TR3

Micro biosensors

CR4

Applications work properly

TR4

Wireless network

CR5

Long lasting battery life

TR5

Global Position System

CR6

The measurement of blood oxygen

TR6

Microprocessor

CR7

The washing hands timer

TR7

Speaker and microphone

CR8

Measurements of exercises (calorie count, count of steps, different workout metrics)

TR8

Weight

CR9

Sleep tracking

TR9

Display size

CR10

Customizability about apps

TR10

Screen resolution

CR11

Measurement of ECG (heart rate)

TR11

Wristband material

CR12

Accurate data about exercises

TR12

Inner material

CR13

Tracking emails without smart phone

TR13

Battery capacity

CR14

Use of non-irritating watch bands

TR14

RAM

CR15

Reminders about exercise and moving

TR15

Memory size

CR16

Automatically sync with other devices

CR17

Compatible with all internet service providers

CR18

Low price

CR19

Alerts for weather, calendar and appointments

CR20

Durability

MEMS sensor

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+ + + +

+ +

+

+ +

+

+ + +

Customer requirements

CR1

Importance Evaluations

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0,221

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+ +

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0,021

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0,020

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0,012

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0,011

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0,009

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0,002

Absolute importance Relative absolute importance (%)

+ +

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+ HOWS

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AMR HR AMR

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1.26

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VHR 11.85

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0.25

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0.22

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1.91

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Fig. 1. Linguistic HoQ and relative absolute importance degrees of technical requirements

5 Conclusion Reaching many customer needs and requirements is easy as a digital from e-commerce sites. Therefore, analyzing customer voice for development of product attributes is possible by using HoQ. The linguistic expression of relationships between CRs and TRs can be modeled for more accuracy under fuzzy environments. In this study, CRs are extracted from Amazon customer reviews for smart watches, sentences with “recommendation modals” are filtered and LDA is used for classifying words and finding topics. Topics representation words are obtained and sentences are extracted for each topic. Then customer wants are determined from selected sentences. HoQ is obtained with TRs by using spherical fuzzy numbers. Finally, TRs are weighted. Thus, product development according to customer voice is provided by analyzing and improvement of the significant TRs. The contribution of this study is to consider “recommendation models” for improving product attribute and the second is to use text mining methods for evaluating

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huge customer reviews. Main findings of that study are “fast response time of the watch (Switch speed between interfaces)” and “the ability to make emergency calls and fall detection” customer needs are the most important for the smartwatch and “MEMS sensor”, present applications”, “micro biosensors” and “wireless network” have the highest technical importance due to their relative absolute importance in QFD. As for the future research issue, competitors’ product comparatives are considered. Another interesting future research can be integrating other extended fuzzy sets into the proposed approach.

References 1. Sullivan, L.P.: Quality function deployment. Qual. Prog. 19(6), 39–50 (1986) 2. Tezuka, T., Tanaka, K.: Landmark extraction: a web mining approach. In: Cohn, A.G., Mark, D.M. (eds.) COSIT 2005. LNCS, vol. 3693, pp. 379–396. Springer, Heidelberg (2005). https:// doi.org/10.1007/11556114_24 3. Trappey, A.J.C., Trappey, C.V., Fan, C.Y., Ian, J.Y.L.: Consumer driven product technology function deployment using social media and patent mining. Adv. Eng. Inform. 36, 120–129 (2018) 4. Wang, Y.H., Lee, C.H., Trappey, A.J.: Service design blueprint approach incorporating TRIZ and service QFD for a meal ordering system: a case study. Comput. Ind. Eng. 107, 388–400 (2017) 5. Vonderembse, M.A., Raghunathan, T.S.: Quality function deployment’s impact on product development. Int. J. Qual. Sci. 2(4), 253–271 (1997) 6. Tan, K.C., Xie, M., Shen, X.X.: Development of innovative products using Kano’s model and QFD. Int. J. Innovation Manag. 3(3), 271–286 (1999) 7. Ertay, T., Büyüközkan, G., Kahraman, C., Ruan, D.: Quality function deployment implementation based on analytic network process with linguistic data: an application in automotive industry. J. Intell. Fuzzy Syst. 16(3), 221–232 (2005) 8. Erginel, N.: Construction of a fuzzy QFD failure matrix using a fuzzy multiple-objective decision model. J. Eng. Des. 6, 677–692 (2010) 9. Haktanır, E., ve Kahraman, C.: A novel interval-valued Pythagorean fuzzy QFD method and its application to solar photovoltaic technology development. Comput. Ind. Eng. 132, 361–372 (2019) 10. Wang, H., Fang, Z.G., Wang, D.A., Liu, S.F.: An integrated fuzzy QFD and grey decisionmaking approach for supply chain collaborative quality design of large complex products. Comput. Ind. Eng. 140, 106212 (2020) 11. Gündo˘gdu, F.K., Kahraman, C.: A novel spherical fuzzy QFD method and its application to the linear delta robot technology development. Eng. Appl. Artif. Intell. 87, 103348 (2020) 12. Finger, G.S.W., Lima-Junior, F.R.: A hesitant fuzzy linguistic QFD approach for formulating sustainable supplier development programs. Int. J. Prod. Econ. 247, 108428 (2022) 13. Ozdagoglu, G., Kapucugil ˙Ikiz, A., Çelik, A.F.: Topic modelling-based decision framework for analysing digital voice of the customer. Total Qual. Manag. Bus. Excell. 29, 1545–1562 (2018) 14. Asadabadi, M.R., Saberi, M., Sadghiani, N.S., Zwikael, O., Chang, E.: Enhancing the analysis of online product reviews to support product improvement: integrating text mining with quality function deployment. J. Enterp. Inf. 36(1), 275–302 (2022) 15. Ming, L., Zhang, J.: Integrating Kano Model, AHP, and QFD methods for new product development based on text mining, intuitionistic fuzzy sets, and customers satisfaction. Math. Probl. Eng. Manag. 2349716, 1741–0398 (2018)

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16. Dumais, S.: Using SVMs for text categorization, microsoft research. IEEE Intell. Syst. Mag. 13, 18–28 (1998) 17. Bhujade, V., Jhanwe, N.J.: Knowledge discovery in text mining technique using association rules extraction. In: International Conference on Computational Intelligence and Communication Systems, pp. 498–502 (2011) 18. Albalawi, R., Yeap, T.H., Benyoucef, M.: Using topic modeling methods for short-text data: a comparative analysis. Front. Artif. Intell. 3, 42 (2020) 19. Blei, D.M., Ng, A.Y., Jordan, M.I.: Latent dirichlet allocation. J. Mach. Learn. Res. 3, 993– 1022 (2003) 20. Stevens, K., Kegelmeyer, P., Andrzejewski, D., Buttler, D.: Exploring topic coherence over many models and many topics. In: Proceedings of the 2012 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning, pp. 952–961 (2012, July)

A Method Based on Picture Fuzzy Graph Coloring for Determining Traffic Signal Phasing on an Intersection Isnaini Rosyida1,2(B) , Ch. Rini Indrati3 , and Sunny Joseph Kalayathankal4 1

Post-Doc Fellow, Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia 2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Semarang, Semarang, Indonesia [email protected] 3 Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia [email protected] 4 Jyothi Engineering College, Cheruthuruthy Thrissur, Kerala, India [email protected]

Abstract. Traffic jam in an intersection is a problem that often happens during a peak time in a city. Hence, regulation of traffic flows in an intersection is an important part to handle the problems. There are many methods that offer traffic flow arrangement in an intersection, including the fuzzy graph theory method. The research aims to propose the concept of PFG’s coloring based on strong and weak adjacencies between vertices and to construct an algorithm based on the coloring of PFG for handling traffic signal phasing at an intersection. The proposed steps include: presenting traffic movements as vertices and all conflicts between movements as edges, transforming traffic flows on all movements into a picture fuzzy vertex set, presenting traffic flows on conflicting movements in a picture fuzzy edge set, constructing the PFG model, determining the minimum number of phases and traffic signal phasing using the PFG’s coloring. The proposed method is the novelty of this research. Further, we evaluate the algorithm through a case study at an intersection in Semarang City, Indonesia, namely Kariadi intersection. The result showed that the minimum number of phases is 3 and the traffic signal phasing obtained from the proposed algorithm is safe since there are no traffic flows from merging conflicts that move simultaneously at the same phase. Keywords: Coloring · Picture fuzzy graph Traffic signal phasing · Intersection

1

· Chromatic number ·

Introduction

Traffic signal phasing and phase timings are important parts to handle traffic jam in an intersection. Many papers discussed mathematics tool to handle traffic light c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 293–303, 2023. https://doi.org/10.1007/978-3-031-39774-5_36

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problems including compatible graph [5]. Further, Adeniji et al. proposed static method using Webster’s formula to determine the total cycle length of signal and allocation of green-time in a cycle [1]. Further, they offered an adaptive method that used sensors in conjunction with control unit to calculate the cycle length and apportion time for green and red lights based on the length of queue and arrival rate on each phase at an intersection. Wang et al. used Wardrop’s formula to calculate passing ability of interweave section [20]. In Kasatkina and Vavilova [11], a mathematical model based on a stochastic method is offered for improving traffic flows in an urban setting. All of papers discussed above were done in a crisp environment. However, many real-world problems contain indeterminate phenomena including traffic light problems. For example, we do not know the real condition of traffic flows at an intersection. Therefore, some researchers handle indeterminate phenomena in traffic flows using fuzzy mathematical tools, such as fuzzy graph theory [9,15], intuitionistic fuzzy graph (IFG) theory ([13,14]), and fuzzy logic [12]. In fuzzy graph coloring and IFG coloring models for traffic light problems, two types of conflicts in an intersection, i.e., crossing and merging conflicts, have not been accommodated [13,15]. Since it’s possible for two merging flows to move in the same phase at once, the traffic signal phasing that both theories produce is occasionally unsafe. In 2021, we proposed a picture fuzzy graph (PFG) coloring based on α-cut for α ∈ [0, 1]. However, there was a problem with its computation since we should use different α to determine the α-cut chromatic number. The aim of this research is to construct an algorithm based on picture fuzzy graph (PFG) coloring for handling traffic signal phasing at an intersection. At the first step, we propose the coloring of PFG based on strong and weak adjacencies between vertices. In the next step, the algorithm is implemented in determining the minimum number of phases and traffic signal phasing at an intersection and this is the novelty of this research. Traffic movements are presented as vertices and all conflicts between movements are presented as edges. We distinguish between crossing conflict and merging conflict to model traffic flows at an intersection. We prove that all edges that come from crossing and merging conflicts connect strongly adjacent vertices. We evaluate the algorithm through a case study at an intersection in Semarang City, Indonesia, namely Kariadi intersection. The structure of the paper is as follows: Sect. 1 gives an introduction. In Sect. 2, some basic concepts are reviewed. Further, an algorithm for determining traffic phase scheduling is given in Sect. 3. Section 4 discusses a case study and the conclusion is presented in Sect. 5. For the sake of simplicity, the following acronyms are used in this paper: picture fuzzy set (PFS), picture fuzzy graph (PFG), positive membership (PosM), neutral membership (NeuM), and negative membership (NegM).

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Literature Review

At the beginning of this section, we look at a few theories related to the traffic light issue. A “phase” is the portion of a cycle where multiple streams of traffic are permitted to use the route at once. The study of “traffic flow” involves tracking how individual drivers and vehicles move between two points and interact with one another. The number of vehicles going through a specific stretch of road in a specific amount of time is known as the traffic flow. The number of vehicles per hour, or passenger car units (pcu) per hour, is used to measure the traffic flow. In this paper, we use pcu per hour with the conversion factors as follows: 1 for motor cycle (MC), 1.3 for light vehicle (LV) such as “passenger cars or pick-up trucks”, and 0.2 for heavy vehicle (HV) such as “two-axle trucks or buses” [17]. A PFS was first introduced by Cuong and Kreinovich [6]. Further, it has been developed by many researchers including Memis [10], also Dutta and Ganju [7]. Based on the concept of PFS, basic theories of PFG were proposed by several researchers such as Al-hawary et al. [2], Amanathulla and Pal [3], and Zuo et al. [22]. Definition 1. Given a set A˜ = {(v, μA˜ (v), ηA˜ (v), νA˜ (v)) : v ∈ X}. We call A˜ as a PFS on X if it has a PosM degree μA˜ (v) ∈ [0, 1], a NeuM degree ηA˜ (v) ∈ [0, 1], and a NegM degree νA˜ (v) ∈ [0, 1] of element v in the PFS A˜ that satisfy 0 ≤ μA˜ (v) + ηA˜ (v) + νA˜ (v) ≤ 1. Further, πA˜ (v) = 1 − (μA˜ (v) + ηA˜ (v) + νA˜ (v)) is called a refusal membership degree of v in A˜ [6]. Definition 2. Let V be a universal set that consists of vertices. The graph ˜ = (V˜ , E) ˜ is called a PFG if V˜ = {(v, μ ˜ (v), η ˜ (v), ν ˜ (v))} is a picG V V V ˜ = {(uv, μ ˜ (uv), η ˜ (uv), ν ˜ (uv))} is a picture fuzzy vertex set on V and E E E E ture fuzzy edge set on E ⊆ V × V such that μE˜ (uv) ≤ min{μV˜ (u), μV˜ (v)}, ηE˜ (uv) ≤ min{ηV˜ (u), ηV˜ (v)}, νE˜ (uv) ≤ max{νV˜ (u), νV˜ (v)}, and 0 ≤ μE˜ (uv) + ηE˜ (uv) + νE˜ (uv) ≤ 1 for each uv ∈ E. The notations μV˜ (v), ηV˜ (v), νV˜ (v) mean PosM, NeuM, and NegM degrees of v in V, respectively. Further, the degrees μE˜ (uv), ηE˜ (uv), νE˜ (uv) stand for PosM, NeuM, and NegM degrees of adjacency between u and v. [2]. ˜ = (V˜ , E) ˜ be a PFG on the universal set V . A graph Definition 3. Let G ∗ ∗ ∗ ˜ if μ ˜ (x) > 0, η ˜ (x) > 0, G (V , E ) is mentioned as an underlying graph of G V V ∗ and νV˜ (x) > 0 for each x ∈ V [2].

3

Coloring of Picture Fuzzy Graphs

In this part, we propose the concept of coloring of PFGs based on strong and weak adjacencies between vertices. We first initiate the notion of strong and weak adjacencies in PFGs in this paper.

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˜ = (V˜ , E) ˜ be a PFG wherein V˜ is a PFS on V . We call Definition 4. Let G u, v ∈ V as strongly adjacent vertices if 1 1 min{μV˜ (u), μV˜ (v)} ≤ μE˜ (uv); min{ηV˜ (u), ηV˜ (v)} ≤ ηE˜ (uv); 2 2 1 max{νV˜ (u), νV˜ (v)} ≤ νE˜ (uv). 2 Otherwise, u and v are called weakly adjacent vertices.

(1)

When ηE˜ (uv) = 0 for each uv ∈ E ⊆ V × V , Definition 4 becomes a strong adjacency between vertices in an IFG as in [14]. When νE˜ (uv) = 0 also happens for each uv ∈ E, it is transformed into a strong adjacency in a fuzzy graph as introduced in [8]. ˜ = (V˜ , E) ˜ where V˜ is a Definition 5. Let us consider the PFG G ˜ = PFS on V with V˜ = {(x, μ1 (x), η1 (x), ν1 (x))|x ∈ V }. Further, E {(xy, μ2 (xy), η2 (xy), ν2 (xy))|xy ∈ E} is a PFS on E ⊆ V × V. The family Γ = {γ1 , γ2 , . . . , γk } of picture fuzzy subsets of V˜ where γi = {(x, μγi (x), ηγi (x), νγi (x))} ˜ if for 1 ≤ i ≤ k is called a k-vertex coloring of G k 1. i=1 γi = V˜ , i.e., max{μγi (x)} = μ1 (x), max{ηγi (x)} = η1 (x), min{νγi (x)} = ν1 (x)

(2)

2. γi ∩ γj = ∅pf s , ∀i = j, i.e., min{μγi (x), μγj (x)} = 0, min{ηγi (x), ηγj (x)} = 0, max{νγi (x), νγj (x)} = 1, (3) 3. Every pair of strongly adjacent vertices xy with x, y ∈ V meets the conditions: min{μγi (x), μγi (y)} = 0, min{ηγi (x), ηγi (y)} = 0, max{νγi (x), νγi (y)} = 1, (4) for 1 ≤ i, j ≤ k. ˜ denoted by χ(G), ˜ is the minimum value k for The chromatic number of G, ˜ has k-vertex coloring. which G Definition 5 becomes k-vertex coloring based on strong and weak adjacencies between vertices in an IFG if ηE˜ (vi vj ) = 0 for each vi vj ∈ E ⊆ V × V [14]. Additionally, it becomes k-vertex coloring in a fuzzy graph when νE˜ (vi vj ) is equal to zero for each vi vj ∈ E and i, j ∈ {1, 2, · · · , n} [8]. We give an illustration of Definition 5 in Example 1.

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Fig. 1. The PFG for Example 1

˜ = (V˜ , E) ˜ in Fig. 1 where V = {A, B, C, D, E} Example 1. Given a PFG G ˜ and V = {(A, 0.1, 0.3, 0.3), (B, 0.2, 0.1, 0.6), (C, 0.2, 0.2, 0.1), (D, 0.2, 0.3, 0.5), (E, 0.2, 0.2, 0.3)}. The pairs of vertices: AB, BC, CD, DE, and AD are strongly adjacent vertices. Meanwhile, BE and AD are weakly adjacent vertices. Therefore, we can construct a family of picture fuzzy subsets of V˜ : Γ = {γ1 , γ2 , γ3 } where γ1 = {(B, 0.2, 0.1, 0.6), (E, 0.2, 0.2, 0.3)}, γ2 = {(A, 0.1, 0.3, 0.3), (D, 0.2, 0.3, 0.5)}, and γ3 = {(C, 0.2, 0.2, 0.1)}. 3 The family Γ satisfies the criteria in Definition 5, i.e., i=1 γi = V˜ , γi ∩ γj = ∅pf s , ∀i = j, and for every pair of strongly adjacent vertices xy with x, y ∈ V : min{μγi (x), μγi (y)} = 0, min{ηγi (x), ηγi (y)} = 0, max{νγi (x), νγi (y)} = 1 for ˜ = 3. 1 ≤ i ≤ 3. Hence, the chromatic number χ(G)

4

An Algorithm Based on Picture Fuzzy Graph Coloring to Determine Traffic Signal Phasing at an Intersection

In this section, we discuss an algorithm (Algorithm 1) for determining traffic signal phasing based on PFG coloring. A traffic movement in an intersection is represented by a vertex in V and two vertices are connected by an edge if there is a conflict between the two movements. We distinguish the conflicts into crossing conflict and merging conflict. The crossing conflict is a collision that occurs when two separate directions of traffic try to cross paths at one spot. Whereas, the merging conflict is a conflict happened when vehicles from multiple lanes or directions merge into a single lane traveling in a single direction [19].

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Algorithm 1. An algorithm to implement the PFG coloring in determining traffic signal phasing Input: – – –

Elements in an intersection which consist of Traffic movement data: V = {vi }, i = 1, 2, · · · , n. Traffic flow data: Fv = {fvi } Crossing conflict data: Ec = {ep }, p = 1, 2, · · · , m, where ep = vi vj , i = j ∈ {1, · · · , n}. – Merging conflict data: Em = {eq }, q = 1, 2, · · · , m . ˜ Output: Chromatic number: χ(G). 1: 2: 3: 4: 5: 6: 7:

8: 9: 10:

11: 12: 13:

Steps Cluster the traffic flow data. Determine the interval for each cluster. Calculate the PosM degrees of traffic flow data {μ1 (fvi )} for i = 1, 2, . . . , n. Obtain the picture fuzzy vertex set (PFVS) V˜ = {(vi , μ1 (fvi ), η1 (fvi ), ν1 (fvi ))} for i = 1, 2, . . . , n where η1 (fvi ) = 0 and ν1 (fvi ) = 1 − μ1 (fvi ) for each vi ∈ V. = vi vj : Determine the volumes on crossing conflicts of er f (er ) = min{fvi , fvj } = fmin for i = j,r ∈ {1, 2, . . . , m}, and i, j ∈ {1, 2, . . . , n}. Calculate the PosM degree of each traffic flow on crossing conflict: μ2 (er ) = μ2 (f (er )) = μ1 (fmin ). Obtain the picture fuzzy edge set (PFES) of traffic flows on crossing conflicts: ˜c = {(ei , μ2 (ei ), η2 (ei ), ν2 (ei ))} where η2 (ei ) = 0, ν2 (ei ) = 1 − μ2 (ei ) for each E ei ∈ Ec . = vp vq : Determine the volume on each merging conflict es  for p = q, s ∈ {1, 2, . . . , m }, and p, q ∈ {1, 2, . . . , n}. f  (es ) = min{fv p , fv q } = fmin Calculate the PosM degree of each traffic flow on merging conflict:  ) for each es ∈ Em . μ2 (es ) = μ2 (f  (es )) = μ1 (fmin Obtain the picture fuzzy edge set (PFES) of traffic flows on merging conflicts: ˜m = {(ei , μ2 (ei ), η2 (ei ), ν2 (ei ))} where η2 (ei ) = 0, ν2 (ei ) = 1 − μ2 (ei ) for i = E 1, 2, · · · , m . ˜ = (V˜ , E) ˜ of traffic flows in the intersection. Draw the PFG model G ˜ Find the minimum number of phases using the chromatic number of PFG G. Determine the phase scheduling, i.e. elements of family Γ as in Definition 5.

We verify the property related to crossing and merging conflicts in Theorem 1. ˜ = (V˜ , E) ˜ with V˜ is a PFS on V = {v1 , v2 , . . . , vn }. Theorem 1. Given PFG G ˜ All edges in E which are obtained from crossing conflict or merging conflict connect strongly adjacent vertices. Proof. Any vertex in V represents traffic movement at an intersection and two vertices in V are connected by an edge if there is a conflict between the two movements. The degree of any vertex v ∈ V is (μ1 (v), η1 (v), ν1 (v)) where μ1 (v) = μ1 (fv ), fv stands for traffic flow on movement v, ν1 (v) = 1 − μ1 (v), and η1 (v) = 0. Let e = vi vj be any edge in a set of crossing conflict or merging conflict E with the degree (μ2 (e), η2 (e), ν2 (e)). The PosM degree represents the degree of crowdness of traffic flow on conflicting movements e, i.e. μ2 (e) = μ1 (fe ) where fe = min{fvi , fvj }. Meanwhile, we stipulate the NeuM

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η2 (e) = 0, and the NegM ν2 (e) = 1 − μ2 (e). Since f (e) = min{fvi , fvj }, we get 1 1 2 min{μ1 (fvi ), μ1 (fvj )} ≤ μ2 (e). Further, 2 min{η1 (fvi ), η1 (fvj )} = 0 = η1 (e) 1 and 2 max{ν1 (fvi ), ν1 (fvj )} ≤ ν2 (e) are always satisfied. Hence, vi and vj are strongly adjacent vertices.

5

Case Study

We take a case study at the Kariadi intersection in Semarang City, Indonesia to evaluate the algorithm. The sketch of the intersection is depicted in Fig. 2. The symbols W = West (Kaligarang Street), N = North (Dr Sutomo Street), and S = South (Dr Sutomo Street).

Fig. 2. The sketch of the Kariadi intersection in Semarang City, Indonesia.

The inputs used in the case study are the following: 1. The set of traffic movements (vertices), i.e., V = {N S, N W, SN, SW, W N, W S}. Two vertices are connected by an edge if the two movements are crossing conflict or merging conflict. 2. The traffic flow data are presented in Table 1 which were taken from a survey in August, 2021 [18]. 3. The sets of crossing and merging conflicts which are provided in Table 3.

Table 1. Traffic movement and traffic flow data in Kariadi intersection (source: [18]) No

1

Movements

WN SN

Traffic flows 248 Clusters

2 383

3

4

5

6

NS

WS

SW

NW

199

607

440

391

Low Medium Low High Medium Medium

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In Step 1, we cluster the traffic flow data into low, medium, and high. In Step 2, we use the range Y = minvol-2:0.1:maxvol+2 and the intervals [0 0 minvol-2 minvol-2+r] for low, [minvol-2+r-20 R maxvol+2-r+20] for medium, and [maxvol+2-r maxvol+2 maxvol+10 maxvol+10] for high traffic flow in the triangular and trapezoidal membership functions as presented in Fig. 3, where . r = round((maxvol − minvol + 4)/3) and R = (maxvol+minvol) 2

Fig. 3. Membership functions for determining PosM degree of traffic flow data in Kariadi intersection.

In Steps 3–4, we calculate the PosM degree of each vertex and obtain the picture fuzzy vertex set (PFVS) in Table 2. Table 2. The PFVS of traffic flows in Kariadi intersection Traffic Movement Traffic flow Degree WN SN NS

248 383 199

Traffic Movement Traffic flow Degree

(0.6277 0 0.3723) WS (0.7753 0 0.2247) SW (0.9854 0 0.0146) NW

607 440 391

(0.9854 0 0.0146) (0.5843 0 0.4157) (0.8652 0 0.1348)

In Steps 5–10, we calculate the degrees of edges (crossing and merging conflicts) and they are presented as PFES in Table 3. ˜ = (V˜ , E) ˜ of traffic flows in the Further, Step 11 provides the PFG model G Kariadi intersection as depicted in Fig. 4.

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Table 3. The PFES of crossing and merging conflicts in Kariadi intersection Edges

Degrees

Crossing conflicts: (WS,NW) (0.8652 0 0.1348) (WS,SN) (0.7753 0 0.2247) (SN,NW) (0.7753 0 0.2247)

Edges

Degrees

Merging conflicts: (NS,WS) (0.9854 0 0.0146) (NW,SW) (0.8652 0 0.1348) (SN,WN) (0.6277 0 0.3723)

Fig. 4. The PFG model of traffic flows in the Kariadi intersection.

In Step 12, we verify the chromatic number of PFG model in Fig. 4 using the PFG’s coloring and we could arrange traffic flows in the Kariadi intersection in three phases. Finally, Step 13 provides the traffic signal phasing for traffic flows in the Kariadi intersection that is indicated by elements of family Γ in Algorithm 1. We observe that there are no merging flows moving in the same phase simultaneously in each pattern in the following signal phasing: 1. Pattern 1: SN, NS; NW, WN; SW, WS 2. Pattern 2: NW ,NS; SW, SN; WS, WN The performance of Algorithm 1 is evaluated using python.

6

Conclusion

We have constructed the notion of PFG’s coloring based on strong and weak ajacencies between vertices and have proposed a method based on coloring of picture fuzzy graph (PFG) for solving traffic signal phasing at an intersection. Traffic movements are represented as vertices and two traffic movements are connected by an edge if there is a conflict between traffic flows from both movements. Further, an edge that came from crossing or merging conflicts connected two strongly adjacent vertices. It means that the traffic flows from crossing or merging conflicts could not move at the same phase.

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We took a case study at an intersection in Semarang City, Indonesia, namely Kariadi intersection to evaluate the method. The result indicated that the minimum number of phases needed at the intersection was 3 and there were two patterns for traffic signal phasing. Since there were no merging flows that moved simultaneously at the intersection, the proposed method gave a traffic signal phasing that was safer than the signal phasing obtained from the fuzzy graph coloring method. In further research, we can improve the algorithm by combining it with realtime traffic data collection.

References 1. Adeniji, S.T., Ohize, H.O., Dauda, U.S.: Development of a mathematical model for traffic light control. In: 1st International Conference on Multidisciplinary Engineering and Applied Science (ICMEAS), Abuja, Nigeria, pp. 1–6 (2021) 2. Al-hawary, T., Mahmood, T., Jan, N., Ullah, K.: On intuitionistic fuzzy graphs and some operations on picture fuzzy graphs. Italian J. Pure Appl. Math. 32, 1–16 (2018) 3. Amanathulla, S., Pal, M.: An Introduction to picture fuzzy graph and its application to select best routes in an airlines network. In: Broumi, S. (eds.) Handbook of Research on Advances and Applications of Fuzzy Sets and Logic, pp. 385–411. IGI Global (2022) 4. Atanassov, K.T.: On intuitionistic fuzzy graphs and intuitionistic fuzzy relations. In: Proceedings of the VI IFSA World Congress, pp. 551–554 (1995) 5. Baruah, A.K., Baruah, N.: Signal groups of compatible graph in traffic control problems. Int. J. Adv. Networking Appl. 4(1), 1473 (2012) 6. Cuong, B.C., Kreinovich, V.: Picture fuzzy sets - a new concept for computational intelligence problems. In: Proceedings of the 2013 3rd World Congress on Information, Communication, and Technology, WICT, pp. 1–6 (2013) 7. Dutta, P., Ganju, S.: Some aspects of picture fuzzy set. Trans. A. Razmadze Math. Inst. 172, 164–175 (2018) 8. Eslahchi, C., Onagh, B.N.: Vertex-strength of fuzzy graphs. Int. J. Math. Math. Sci. 2006(043614), 1–9 (2006) 9. Mahapatra, T., Ghorai, G., Pal, M.: Fuzzy fractional coloring of fuzzy graph with its application. J. Ambient. Intell. Humaniz. Comput. 11, 5771–5784 (2020) 10. Memis, S.: A Study on picture fuzzy sets. In: Proceedings of 7th IFS Contemporary Mathematics and Engineering Conference, Turkey, pp. 125–132 (2021) 11. Kasatkina, E.V., Vavilova, D.D.: Mathematical modeling and optimization of traffic flows. J. Phys. Conf. Ser. 2134, 012002 (2021) 12. Koukol, M., Zaj´ıˇckov´ a, L., Marek, L., Tuˇcek, P.: Fuzzy logic in traffic engineering: a review on signal control. Math. Probl. Eng. 2015, 1–14, Article ID 979160 (2015) 13. Prasanna, A. Rifayathali, M., Ismail Mohideen, S.: Strong intuitionistic fuzzy graph coloring. Int. J. Latest Eng. Res. Appl. 02(08), 163–169 (2017) 14. Rifayathali, M.A., Prasanna, A., Mohideen, S.I.: Intuitionistic fuzzy graph coloring. Int. J. Res. Analytical Rev. 5(3), 734–742 (2018) 15. Rosyida, I., Nurhaida, Narendra, A.: Widodo: Matlab algorithm for traffic light assignment using fuzzy graph, fuzzy chromatic number, and fuzzy inference system, MethodsX 7, 101–136, (2020)

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16. Rosyida, I., Suryono, S.: Coloring of picture fuzzy graphs through their cuts and its computation. Int. J. Adv. Intell. Inform. 7(1), 63–75 (2021) 17. Sumadji, S., Asmoro, D., Sastrosoegito, S.: Indonesian Highway Capacity Manual: Urban and Semi Urban Traffic-Facilities. Ministry of Public Works, Directorate General of Highways (1993) 18. Suprayogi, A., Rosyida, I., Wiyanti, D.T., Safaatullah, M.F.: Implementation of mamdani fuzzy implication in predicting traffic volume and duration of green lights on an intersection. J. Phys. Conf. Ser. 2106(012020), 1–16 (2021) 19. Thompson, M.J., Kwon, O.H., Park, M.J.: The application of axiomatic design theory and conflict technique for the design of intersections: Part 1. In: Proceedings of The Fifth International Conference on Axiomatic Design, pp. 121–127 (2009) 20. Wang, Zhijiang, Wang, Kaili, Zhang, Huancheng: Application of mathematical model in road traffic control at circular intersection. In: Zhu, Rongbo, Zhang, Yanchun, Liu, Baoxiang, Liu, Chunfeng (eds.) ICICA 2010. CCIS, vol. 106, pp. 436–443. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-163395 58 21. Xiao, W., Dey, A., Son, L.H.: A study on regular picture fuzzy graph with applications in communication networks. J. Intell. Fuzzy Syst. 39(3), 3633–3645 (2020) 22. Zuo, Z., Pal, A., Dey, A.: New concepts of picture fuzzy graphs with application. Mathematics 7(470), 1–18 (2019)

Type-2 Fuzzy Sets

Interval Type-2 Fuzzy CODAS: An Application in Flight Selection Problem Erdem Akın1 and Ba¸sar Öztay¸si2(B) 1 Technology Istanbul, Istanbul, Turkey 2 Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey

[email protected]

Abstract. When travelers try to make a travel plan they face a decision problem about selecting their flights, in the literature this problem is defined as flight selection problem (FSP). The problem occurs since there are numerous companies and flight options and the options makes it hard to select the best alternative. The factors which are generally involved in FSP are cost of the flight, the duration of the flight, the number of connected flights, the departure and arrival times, and the convenience of the airports. Flight selection problem can be considered as a multicriteria decision making problem since it involves various criteria. COmbinative Distance-based ASsessment (CODAS) is a multi-criteria decision making method which is based on distance of alternatives to ideal solutions. Interval Type-2 Fuzzy Sets (IT2FS) are extensions of fuzzy sets which can handle more uncertainty in decision problems. In this study, we propose IT2F-CODAS method which incorporates IT2FS and CODAS method and apply the method to solve flight selection problem. Keywords: Interval Type-2 Fuzzy Sets · CODAS · Flight Selection

1 Introduction The flight selection problem (FSP) raises when travelers try to choose the best flight out of various options for a given trip. Since travel sector provides various companies with different flight options, travelers face with large number of options. Travellers may consider different criteria for the selection such as cost, duration, number of connections, departure and arrival times, and convenience of airports. From a travelers perspective, the solution of FSP can significantly affect the overall experience of a trip since finding the best flight can enhance the overall satisfaction of the trip, save time and money and be a more comfortable travel experience. Solving FSP requires considering various factors and making trade-offs between them to find the flight that best meets the traveler’s needs and preferences. FSP can be handled as a multi-criteria-decision-making problem (MCDM) focuses on evaluating and choosing between multiple options, where each option is evaluated based on multiple criteria. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 307–317, 2023. https://doi.org/10.1007/978-3-031-39774-5_37

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Fuzzy sets, proposed by Zadeh [1], are important for decision making problems because they provide a way to model and represent uncertain and imprecise information, which is often present in real-world decision-making problems. Fuzzy sets provide the mathematical foundations for representing and operating with imprecision and vagueness in decision making problems. An interval type-2 fuzzy set (IT2FS) is an extension of ordinary fuzzy sets which handle uncertainty in a better way [8]. In ordinary fuzzy sets, the membership of an element in a set is represented by a single membership value between 0 and 1, where 0 indicates that the element is definitely not a member of the set and 1 indicates that the element is definitely a member of the set. However, the membership of an element to a set is also represented by a fuzzy set. Combinative distance-based assessment (CODAS) [2] is a multi-criteria decision making (MCDM) method that uses distance measures to assess the similarity or dissimilarity between alternatives. The method is based on the idea that the best alternative is the one that is closest to an ideal solution in a multi-dimensional space defined by the criteria. In this study, we propose a novel method, interval type-2 fuzzy CODAS. We also apply the proposed method to a real-world flight selection problem. The remainder of this paper is structured as follows: the preliminaries on Interval Type-2 fuzzy sets is given in Sect. 2. The methodology is introduced in Sect. 3, and an application on FSP is given in Sect. 4. Finally, in Sect. 5 the conclusions are provided.

2 Interval Type-2 Fuzzy Sets (IT2FS) In this section, definitions regarding IT2FS are clarified [3]. A T2FS  A˜ defined in X can be represented as in Eq. 1 [8]:     A˜˜ = (x, u), μ ˜˜ (x, u) ∀x ∈ X , ∀u ∈ Tx ⊆ [0, 1], 0 ≤ μ ˜˜ (x, u) ≤ 1 , A

A

(1)

where μ ˜˜ represents the membership function and Tx represents [0,1]. The T2FS A˜˜ also A can be represented as follows [3]:  (2) A˜˜ = ∫x∈X ∫u∈Tx μA˜˜ (x, u) (x, u), where Tx ⊆ [0, 1] and ∫ ∫ denote union over all admissible x and u. A special case of T2FS are IT2FS [4]. An IT2FS A˜˜ can be represented as follows [3]: A˜˜ = ∫x∈X ∫u∈Tx 1 (x, u),

(3)

where Tx ⊆ [0, 1]. For interval type-2 fuzzy set, we can assume there are two membership functions, the upper and lower membership functions which are ordinary fuzzy membership functions. A trapezoidal interval type-2 fuzzy set is illustrated as             U U U U L L L L ˜L ˜ U , ai1 ai1 , ai2 , ai3 , ai4 ; H1 A˜ U , ai2 , ai3 , ai4 ; H1 A˜ Li , H2 A˜ Li A˜˜ i = A˜ U i ; Ai = i , H2 Ai

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U U U U L L L L ˜L where A˜ U i and Ai are type-1 fuzzy sets,ai1 , ai2 , ai3 , ai4 , ai1 , ai2 , ai3 and ai4 are the A˜ i , Hj A˜ U references points of the IT2FS  i ; denotes the membersip value of the element   U aj(j+1) A˜ Li denotes the in the upper trapezoidal membership function (A˜ U ), 1 ≤ 2, H j i L in the lower trapezoidal membership function membership value of the element aj(j+1)         L U U A˜ , 1 ≤ j ≤ 2, H1 A˜ ∈ [0, 1], H2 A˜ ∈ [0, 1], H1 A˜ L ∈ [0, 1], H2 A˜ L ∈ [0, 1] i

i

i

i

i

and 1 ≤ i ≤ n [5]. When two trapezoidal IT2FSs are to be summed, the addition operation given in Eq. 4 can be used [5]. A˜˜ 1 ⊕ A˜˜ 2 =



U U U U U U U a11 + a21 , a12 + a22 , a13 + a23 , a14            U L L L L L ˜U ˜U ˜U , a11 + a24 ; min H1 A˜ U + a21 , a12 + a22 , a13 1 ; H1 A2 , min H2 A1 ; H2 A2           L L L + a23 , a14 + a24 ; min H1 A˜ L1 ; H1 A˜ L2 , min H2 A˜ L1 ; H2 A˜ L2

(4)

When a trapezoidal IT2FS is subtracted from an other trapezoidal IT2FS subtraction function given in Eq. (5) can be used [5]. A˜˜ 1  A˜˜ 2 =



U U U U U U U a11 − a24 , a12 − a23 , a13 − a22 , a14           U L L L L L ˜U ˜U ˜U , a11 − a21 ; min H1 A˜ U − a24 , a12 − a23 , a13 1 ; H1 A2 , min H2 A1 ; H2 A2           L L L − a22 , a14 − a21 ; min H1 A˜ L1 ; H1 A˜ L2 , min H2 A˜ L1 ; H2 A˜ L2



(5)

For multiplying two IT2FSs Eq. (6) is used. A˜˜ 1 ⊗ A˜˜ 2 ∼ =



U U U U U U U a11 × a21 , a12 × a22 , a13 × a23 , a14           U L L L L L ˜U ˜U ˜U , a11 × a24 ; min H1 A˜ U × a21 , a12 × a22 , a13 1 ; H1 A2 , min H2 A1 ; H2 A2           L L L × a23 , a14 × a24 ; min H1 A˜ L1 ; H1 A˜ L2 , min H2 A˜ L1 ; H2 A˜ L2



(6)

Aritmetic operations between a crisp number and IT2FS is given in the following:       U U U U L L ˜U kA˜˜ 1 = k × a11 , k × a12 , k × a13 , k × a14 ; H1 A˜ U 1 , H2 A1 , k × a11 , k × a12 , k     L L × a13 , k × a14 ; H1 A˜ L1 , H2 A˜ L1

    1 1 A˜˜ 1 U 1 U 1 U 1 U L 1 L 1 , H , × a12 , × a13 , × a14 ; H1 A˜ U , × a12 , = × a11 × a11 A˜ U , 2 1 1 k k k k k k k k      L 1 L , × a14 ; H1 A˜ L1 , H2 A˜ L1 × a13 k

(7) (8)

Defuzzification of IT2FS is an important research area. For a given trapezoidal IT2FS, defuzzification can be done by using Eq. 9 [6]

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DTraT =

(uU −lU +(βU .m1U −lU )+(αU .m2U −lU )) 4

+ lU ]+[ (uL −lL +(βL .m1L −l4 L )+(αL .m2L −lL )) + lL 2

(9)

The rank-based distance function between trapezoidal IT2FS are calculated as in Eq. 10 [7]:    1     H1 A˜ U λ a2L 1 L U ˜ ˜ 2H1 A1 H1 A1       −a1L − a2U + a1U − a4L − a3L − a2L + a1L H1 A˜ L1 a4U − a3U − a4L + a3L

    ˜˜ 1˜˜ = 1 − aL − λ aL − aU + aU − aL − Rd A, 4 1 1 4 4

(10)         ˜U ˜L ˜L where H1 A˜ U 1 = H2 A1 and H1 A1 = H2 A1 . The above mentioned ranking-distance function has a parameter λ, which can be viewed as a reflection of the decision maker’s attitude. If the decision maker is optimistic, λ is set to 0, if neutral, λ is set to 0.5, and if pessimistic, λ is set to 1. The distance between the two trapezoidal interval type-2 fuzzy sets are calculated as in the following.          (11) d A˜˜ 1 , A˜˜ 2 = Rd A˜˜ 1 , 1˜˜ − Rd A˜˜ 2 , 1˜˜ 

3 Methodology: Interval Type-2 CODAS CODAS is a MCDM method based on distances between the alternatives [2]. In this study we propose IT2F-CODAS which uses IT2FS with in the decision making process. Step 1. Decision makers express their judgement according to the scale given in Table 1 and fill the decision matrix (DM). Step 2. IT2FS DM (X), is constructed by transforming linguistic evaluations to IT2FS, as shown in the following: X˜˜ = x˜˜ ij

n×m

⎡

x˜˜ 11 ⎢ .. =⎣ . x˜˜ n1

⎤ · · · x˜˜ 1m . . .. ⎥ . . ⎦ · · · x˜˜ nm

(12)

where x˜˜ ij represents the fuzzy judgements value of ith alternative on jth criterion. Step 3. Normalized DM is obtained by using Eq. (13) to normalize the judgement values. ⎧ ˜ x˜ ij ⎪ ⎪ ⎨ max x˜˜ ij ifj ∈ Tb i (13) n˜˜ ij = min x˜˜ ij ⎪ ⎪ ⎩ i ifj ∈ T c ˜ x˜ ij

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Table 1. Linguistic variables and the corresponding TIT2FSs [9] Usage

Linguistic variables

Trapezoidal interval type-2 fuzzy scales

For weighting criteria

Very Low (VL)

(0,0,0,0.1;1,1) (0,0,0,0.05;0.9,0.9)

Low (L)

(0,0.1,0.15,0.3;1,1) (0.05,0.1,0.15,0.2;0.9,0.9)

Medium Low (ML)

(0.1,0.3,0.35,0.5;1,1) (0.2,0.3,0.35,0.4;0.9,0.9)

Medium (M)

(0.3,0.5,0.55,0.7;1,1) (0.4,0.5,0.55,0.6;0.9,0.9)

Medium High (MH)

(0.5,0.7,0.75,0.9;1,1) (0.6,0.7,0.75,0.8;0.9,0.9)

High (H)

(0.7,0.85,0.9,1;1,1) (0.8,0.85,0.9,0.95;0.9,0.9)

Very High (VH)

(0.9,1,1,1;1,1) (0.95,1,1,1;0.9,0.9)

For rating alternatives Very Poor (VP)

(0,0,0,1;1,1) (0,0,0,0.5;0.9,0.9)

Poor (P)

(0,1,1.5,3;1,1) (0.5,1,1.5,2;0.9,0.9)

Medium Poor (MP)

(1,3,3.5,5;1,1) (2,3,3.5,4;0.9,0.9)

Fair (F)

(3,5,5.5,7;1,1) (4,5,5.5,6;0.9,0.9)

Medium Good (MG) (5,7,7.5,9;1,1) (6,7,7.5,8;0.9,0.9) Good (G)

(7,8.5,9,10;1,1) (8,8.5,9,9.5;0.9,0.9)

Very Good (VG)

(9,10,10,10;1,1) (9.5,10,10,10;0.9,0.9)

where Tb shows the benefit criteria and Tc shows the cost criteria. Step 4. The Weighted Normalized Decision Matrix is Calculated by Using Eq. (14). r˜˜ij = w˜˜ i n˜˜ ij where the weight of jth criterion is shown by w˜˜ i . Step 5. The negative-ideal solution is obtained as follows:   ns = nsj 1×m nsj = mini  r˜ ij

(14)

(15) (16)

Step 6. Distances of the alternatives from the negative-ideal solution is obtained using Eq. (17) and Eq. (18).  2 m  Ei = r˜˜ij − nsj (17) j=1

 m   Ti = r˜˜ij − nsj  j=1

(18)

Step 7. The relative assessment matrix is obtained, by using Eqs. (19–20). Ra = [hik ]n×n

(19)

hik = (Ei − Ek ) + (ψ(Ei − Ek ) × (Ti − Tk ))

(20)

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E. Akın and B. Öztay¸si

the threshold function, ψ, shows to identify the likeness of the Euclidean distances of two alternatives, and the threshold function is defined as in the following:  1 if |x| ≥ τ ψ(x) = (21) 0 if |x| < τ The function uses a threshold parameter, τ, which can be selected by the decision maker. Studies in the literature recommend to assign a value between 0.01 and 0.05 for τ. In cases where the difference between the Euclidean distances of two alternatives is smaller than τ, the function compares those two alternatives using the Manhattan distance instead. In academic literature, the value of τ is commonly set to 0.02 for the purpose of the calculations. Step 8. The assessment score values for the alternative are calculated using Eq. (22). n hik (22) Hi = k=1

Step 9. Sort the alternatives in order of decreasing assessment scores (Hi). The alternative with the highest assessment score is considered the optimal choice among the available alternatives.

4 Flight Selection Problem: An Application In order to show the validity of the proposed method an application in flight selection problem is provided. The Flight Selection problem refers to the problem of finding the optimal flight schedule for a given set of flights. After a brief literature review [10–13] the criteria are reviewed with sector experts and the criteria in Table 2 are selected for the decision model. For numerical application, five alternative flights are selected from Tokyo to New York. The attributes of the flight options are given in Table 3. Three decision makers evaluate the criteria weights and rate alternatives with respect to each criterion. Table 4 represents the linguistic evaluations of the criteria. Table 5 represents the linguistic evaluations of the flight alternatives by each-decision maker.

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313

Table 2. Definitions of flight selection criteria Criteria Name

Definition

C1

Price

Price of the flight alternative

C2

Total Elapsed Time Total elapsed time of the whole flight solution including waiting time in stops and on air

C3

Service Quality

Service quality of the airline serves for the alternative in terms of in-flight entertainment, food & beverages, aircraft type and comfort, staff attitude, etc

C4

Change Policy

“Change policy” in the pricing rules whether it is allowed to change flights or not and if flight change is allowed with or without penalty

C5

Refund Policy

“Refund policy” in the pricing rules whether it is allowed to refund or not and if refund is allowed with or without penalty

Table 3. Attributes of the flight options Alternatives

Attributes

F1

Total elapsed time 13 h 5 min (Non-stop), price 2497 USD, does not allow refunds, allows changes with penalty

F2

Total elapsed time 20 h 30 min (1 Stop), price 1522 USD, does not allow refunds, allows changes without penalty

F3

Total elapsed time 27 h 20 min (1 Stop), price 1495 USD, allows refunds with penalty, allows changes with penalty

F4

Total elapsed time 25 h 10 min (1 Stop), price 3359 USD, allows refunds with penalty, allows changes with penalty

F5

Total elapsed time 26 h 25 min (1 Stop), price 1685 USD, allows refunds with penalty, allows changes with penalty

Table 4. Decision Makers’ weight values of the criteria Criteria

Decision Makers Dm1

Dm2

Dm3

C1

L

MH

VH

C2

VH

MH

L

C3

H

H

L

C4

H

M

L

C5

L

MH

H

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E. Akın and B. Öztay¸si Table 5. The rating values of the flight alternatives by each decision-maker

Decision Makers DM1

DM2

DM3

Alternatives

Criteria C1

C2

C3

C4

C5

F1

F

VG

MG

F

P

F2

G

F

MP

G

P

F3

G

P

G

F

F

F4

P

MP

G

F

F

F5

G

MP

G

F

F

F1

P

G

MG

F

VP

F2

F

MG

P

VG

VP

F3

F

F

VG

F

MG

F4

VP

F

G

F

MG

F5

F

F

VG

F

MG

F1

VP

VG

VG

MG

VP

F2

P

G

G

VG

VP

F3

P

MG

VG

MG

MP

F4

VP

MG

VG

MG

MP

F5

P

MG

VG

MG

MP

Following the steps of proposed method, the weighted normalized weights of the criteria and rating values of the alternatives are calculated using Eqs. (12–14). The negative-ideal solution is obtained by using Eqs. (15–16). Table 6 represents the results of the weighted normalized TIT2FSs and the negative-ideal solution. Euclidean and Manhattan distances of alternatives are calculated using Eqs. (17–18). The relative assessment matrix is obtained by using Eqs. (19–20). Finally, the assessment score of each alternative is calculated by using Eq. (22). The results are represented in Table 7. According to assessment scores the flight options are ranked as follows F5 > F3 > F4 > F2 > F1 . The results show that F5 is the best alternative. In order to show the validity of the proposed methodology, the results are compared with Interval Type-2 Fuzzy TOPSIS (IT2F-TOPSIS) and Interval Type-2 Fuzzy VIKOR (IT2F-VIKOR). The ranking results of the comparison is shown in Table 8. The results show that the proposed methodology provides similar results with compared methods.

Criteria

((0.23,0.54,0.7,1.47;1,1), (0.38,0.54,0.7,0.93;0.9,0.9))

((0,0.04,0.07,0.37;1,1), (0.02,0.04,0.07,0.16;0.9,0.9))

((0.23,0.54,0.7,1.47;1,1), (0.38,0.54,0.7,0.93;0.9,0.9))

F3

F4

F5

((0,0.04,0.07,0.37;1,1), (0.02,0.04,0.07,0.16;0.9,0.9))

((0.23,0.54,0.7,1.47;1,1), (0.38,0.54,0.7,0.93;0.9,0.9))

F2

ns

((0.07,0.23,0.31,0.81;1,1), (0.14,0.23,0.31,0.45;0.9,0.9))

C1

F1

Alt

((0.12,0.27,0.32,0.56;1,1), (0.19,0.27,0.32,0.4;0.9,0.9))

((0.14,0.31,0.37,0.62;1,1), (0.22,0.31,0.37,0.44;0.9,0.9))

((0.14,0.31,0.37,0.62;1,1), (0.22,0.31,0.37,0.44;0.9,0.9))

((0.12,0.27,0.32,0.56;1,1), (0.19,0.27,0.32,0.4;0.9,0.9))

((0.23,0.42,0.49,0.76;1,1), (0.33,0.42,0.49,0.58;0.9,0.9))

((0.39,0.59,0.64,0.88;1,1), (0.49,0.59,0.64,0.73;0.9,0.9))

C2

((0.12,0.26,0.32,0.55;1,1), (0.2,0.26,0.32,0.4;0.9,0.9))

((0.39,0.59,0.66,0.92;1,1), (0.5,0.59,0.66,0.76;0.9,0.9))

((0.36,0.56,0.64,0.92;1,1), (0.48,0.56,0.64,0.75;0.9,0.9))

((0.39,0.59,0.66,0.92;1,1), (0.5,0.59,0.66,0.76;0.9,0.9))

((0.12,0.26,0.32,0.55;1,1), (0.2,0.26,0.32,0.4;0.9,0.9))

((0.3,0.5,0.57,0.86;1,1), (0.4,0.5,0.57,0.67;0.9,0.9))

C3

((0.12,0.28,0.35,0.61;1,1), (0.2,0.28,0.35,0.43;0.9,0.9))

((0.12,0.28,0.35,0.61;1,1), (0.2,0.28,0.35,0.43;0.9,0.9))

((0.12,0.28,0.35,0.61;1,1), (0.2,0.28,0.35,0.43;0.9,0.9))

((0.12,0.28,0.35,0.61;1,1), (0.2,0.28,0.35,0.43;0.9,0.9))

((0.28,0.48,0.54,0.8;1,1), (0.38,0.48,0.54,0.64;0.9,0.9))

((0.12,0.28,0.35,0.61;1,1), (0.2,0.28,0.35,0.43;0.9,0.9))

C4

((0,0.03,0.06,0.41;1,1), (0.01,0.03,0.06,0.16;0.9,0.9))

((0.17,0.5,0.66,1.71;1,1), (0.32,0.5,0.66,0.98;0.9,0.9))

((0.17,0.5,0.66,1.71;1,1), (0.32,0.5,0.66,0.98;0.9,0.9))

((0.17,0.5,0.66,1.71;1,1), (0.32,0.5,0.66,0.98;0.9,0.9))

((0,0.03,0.06,0.41;1,1), (0.01,0.03,0.06,0.16;0.9,0.9))

((0,0.03,0.06,0.41;1,1), (0.01,0.03,0.06,0.16;0.9,0.9))

C5

Table 6. The weighted normalized rating values of the flight alternatives and the negative-ideal solution

Interval Type-2 Fuzzy CODAS 315

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Table 7. Euclidean and Manhattan distances of alternatives from the negative-ideal solution and assessment scores Alternatives

Ei

Ti

Hi

F1

0.547

0.945

−4.642

F2

0.852

1.206

−1.816

F3

1.202

1.990

3.911

F4

0.889

1.224

−1.542

F5

1.204

2.047

4.089

Table 8. Ranking results of comparison between the proposed Interval Type-2 Fuzzy CODAS, IT2F-TOPSIS and IT2F-VIKOR Alternatives

ProposedCODAS

IT2F-TOPSIS

IT2F-VIKOR

F1

5

5

4

F2

4

3

3

F3

2

2

2

F4

3

4

5

F5

1

1

1

5 Conclusion In this paper we propose Interval Type-2 Fuzzy CODAS method and apply it on a sample flight selection problem. In the numerical example, five criteria are selected and the methodology is applied to five alternatives by the evaluations of three decision makers. In order to show the validity of the proposed method, the results of the proposed methodology are compared with IT2F-VIKOR and IT2F-TOPSIS The results show that the proposed method provides similar results with other techniques. For the further studies, the results can also be compared with the results of crisp MCDM methods and other fuzzy MCDM methods.

References 1. Zadeh, L.A.: Fuzzy sets, Inform. Control 8(3), 338–353 (1965) 2. Ghorabaee, M.K., Zavadskas, E.K., Turskis, Z., Antucheviciene, J.: A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Econom. Comput. Econom. Cybernet. Stud. Res. 50(3), 25–44 (2016) 3. Mendel, J.M., John, R.I., Liu, F.L.: Interval type-2 fuzzy logical systems made simple. IEEE Trans. Fuzzy Syst. 14(6), 808–821 (2006) 4. Buckley, J.J.: Fuzzy hierarchical analysis. Fuzzy Sets Syst. 17, 233–247 (1985) 5. Chen, S.-M., Lee, L.-W.: Fuzzy multiple attributes group decision-making based on the interval type-2 TOPSIS method. Expert Syst. Appl. 37, 2790–2798 (2010)

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6. Kahraman, C., Öztay¸si, B., Sarı, ˙I.U., Turano˘glu, E.: Fuzzy analytic hierarchy process with interval type-2 fuzzy sets. Knowl.-Based Syst. 59, 48–57 (2014) 7. Qin, J., Liu, X., Pedrycz, W.: An extended VIKOR method based on prospect theory for multiple attribute decision making under interval type-2 fuzzy environment. Knowl.-Based Syst. 86, 116–130 (2015) 8. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning– I. Inform. Sci. 8(3), 199–249 (1975) 9. Ghorabaee, M.K., Amiri, M., Zavadskas, E.K., Turskis, Z., Anucheviciene, J.: A new multicriteria model based on interval type-2 fuzzy sets and EDAS method for supplier evaluation and order allocation with environmental considerations. Comput. Ind. Eng. 112, 156–174 (2017) 10. Chang, Y.H., Yeh, C.H.: Evaluating airline competitiveness using multiattribute decision making. Omega 29(5), 405–415 (2001) 11. Park, J.W., Robertson, R., Wu, C.L.: The effect of airline service quality on passengers’ behavioural intentions: a Korean case study. J. Air Transp. Manag. 10, 435–439 (2004) 12. Park, J.W.: The effect of frequent flyer programs: A case study of the Korean airline industry. J. Air Transp. Manag. 16, 287–288 (2010) 13. Liao, C.-N.: A fuzzy approach to business travel airline selection using an integrated AHPTOPSISs-MSGP methodology. Int. J. Inf. Technol. Decis. Mak. 12(01), 119–137 (2013)

Stabilization of a D.C Motor Controller Using an Interval Type-2 Fuzzy Logic System Designed with the Bee Colony Optimization Algorithm Leticia Amador-Angulo(B) and Oscar Castillo Tijuana Institute of Technology, Tijuana, Mexico [email protected], [email protected]

Abstract. In this paper a Bee Colony Optimization algorithm (BCO) to stabilization of a D.C Motor control is analyzed. The focus of the proposal algorithm is found the distribution of the optimal parameters in the Membership Functions (MFs) of an Interval Type-2 Fuzzy Logic System (IT2FLS). The final experimentation can highlight by implementing the stabilization in an IT2FLS through of the BCO algorithm excellent results are obtained in the evaluation of the Fuzzy Logic Controller (FLC). A comparative analysis with Type-1 FLS and others relevant algorithms is presented to demonstrated and verified that IT2FLS-BCO found minimal errors. Keywords: Bio-inspired algorithm · Interval Type-2 Fuzzy Logic System · D.C. motor · Uncertainty · Speed

1 Introduction In recent years, the problems have increased their complexity, that is why meta-heuristics algorithm have been combined with techniques, such as; fuzzy logic systems for stabilization in control for the complex problems. The BCO algorithm is an example of these bio-inspired algorithms, some problems studied with BCO are; Lee et al. in [1] this algorithm is used on support vector regression, Ziyadullaev et al. in [2] use this algorithm to develop a traditional transport system, Bala et al. in [3] use this algorithm to FCM-based image segmentation, and Zhao et al. in [4] study the BCO algorithm applied in the hybridization strategy to solve problems related with sensor networks. The main contribution is highlighting the main contribution that the hybridization the interval type-2 FLS and BCO algorithm contribute to obtain good results to optimize the speed in the FLC problem, the real problem is simulated with a IT2FLC to find the smallest error in the simulation. The organization in each section is presented below. Section 2 shows some relevant Related Works. Section 3 outlines the BCO algorithm. Section 4 shows the real problem. Section 5 outlines the design presented of the IT2FLS. Section 6 shows the results and a comparative analysis with others bio-inspired algorithms, and, Sect. 7 shows some important conclusions and some recommendations to improve this paper. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 318–325, 2023. https://doi.org/10.1007/978-3-031-39774-5_38

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2 Related Works The problem analyzed is identified as “DC Speed Motor Controller”, this problem is a real problem that several authors have studied, for example; in [5] a review on brush dc motor control techniques by Bokam et al., in [6] an optimization of PID controller parameters for this problem is used with two algorithms proposal by Pandey, in [7] a proposal with this problem using NARXnet based FOPID controller is studied by Munagala et al., in [8] a design of an affordable fault-tolerant control system of the brushless for this problem is presented by Yang et al., in [9] an automation of camel race by controlling this problem is implemented by Chandramma et al., and in [10] a design of a fuzzy logic PID controller with this problem is presented by Rahman et al., The bio-inspired algorithm studied in this paper, the BCO algorithm has been studied for different types of problems for some mention, in [11] a BCO is used to find important values in two parameters through an IT3FLS, in [12, 13] are implemented to solve problem of traffic control, in [14] is applied to stabilize benchmark problems in fuzzy controller for interval type-2 FS, and in [15] is used with the hybridization of type-2 fuzzy to solve problems of motor vehicle crash involvement.

3 Bee Colony Optimization Algorithm Teodorovi´c Dušan introduce the first idea for this algorithm. The methodology in the BCO algorithm is based in explores collective intelligence through honeybees with the goal to collect nectar [16]. Two phases present: backward pass and forward pass. During all the process in the BCO, a bee can express three roles; follower bee, scout bee and bee [17]. Equations 1–4 express the dynamics of BCO algorithm; α  1 β · dij =  α  1 β  · dij j∈Ai,n ρij,n 

Pij,n

ρij,n

Di = K · Pf i =

Pf i

(2)

Pf colony

1 , Li = TourLength LI

Pf colony =

1

N Bee 

NBee

i=1

(1)

Pf i

(3)

(4)

Table 1 describes the main parameters for the BCO algorithm, and the equation related with this parameter. An important mechanism is the waggle dance that a bee presents to alert the discovery of a good solution found [18], and Fig. 1 illustrates the step to step of the BCO algorithm.

320

L. Amador-Angulo and O. Castillo Table 1. Description in the function in main parameters for the BCO algorithm

Parameter

Function

Equation

k

Probability

(1)

i

Actual node

j

Following node

Nk

i

ρ ij β

All nodes in a neighborhood Rating value exploration

d ij

Heuristic distance

α

Best solution

K

Waggle dance

Pf i

Probability score

(3)

Pf colony

Average or the probability in all colony

(4)

(2)

Fig. 1. Detail step to step in the BCO algorithm.

4 Fuzzy Logic Controller Problem The real problem called DC speed motor consist in moving starting from an initial state at a speed based in 40 rad/s, Fig. 2 shows the behavior with the base model, and Fig. 3 illustrates the model in the FLC.

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Fig. 2. Trajectory in the speed response with the base model.

Fig. 3. Model of control for the real problem.

5 Proposed Design of the IT2FLS 5.1 State of the Art for the IT2FLS Zadeh creates the first idea of the FLS in 1965 [19, 20]. An extension of the T1FLS is presented by Mendel et. al in 2002 called IT2FLS [21], and Mamdani in 1974 propose a case of Fuzzy Controller to implementation of the FLS [22]. Figure 4 shows the graphic description of an Interval Type-2 FLC.

Fig. 4. Graphic description of an IT2FLC. ∼

An IT2FS A, expressed by µ ∼ (x) and µ ∼ (x) is defined by the lower and upper MFs _A

A

of μ (x). Where x ∈ X. Equation (5) describes the definition IT2FS [23]. A

∼

A= {((x, u), 1)|∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]}

(5)

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5.2 Proposed Design of the IT2FLS The proposal in the IT2FLS is designed with a Mamdani type of system, the Fig. 5 (a) shows the distribution of the MFs (triangular and trapezoidal), the number de input and output and the names for each linguistic variable. Figure 5(b) illustrates the number of rules in the IT2FLS.

Fig. 5. a) Design of the proposed IT2FLS, and b) Proposed Fuzzy Rules for the IT2FLS.

In the BCO algorithm an individual indicates a possible solution in the execution, in this case a bee, for the real problem the vector solution is created for the number the parameters that represent the sum of the MFs values in the IT2FLS, a trapezoidal FMs requires of 8 values, and a triangular MFs requires of 6 values, therefore a total of 90 values in the MFs are found by this algorithm.

6 Results in the Experimentation The settings in the relevant parameters of this algorithm as the following: population (N) of 50 values, Follower Bee of 25 values, α of 0.5, β of 2.5, and iterations of 30. The metric used in the function fitness is the RMSE that represent the Root Mean Square Error and is detailed by Eq. (6).  2 1 N Xt − X t ε= (6) t=1 N

Others important metrics used in fuzzy controller are presented by Eqs. (7 – 11). ∞ ISE =

e2 (t)dt

(7)

|e (t)|dt

(8)

0

∞ IAE = 0

Stabilization of a D.C Motor Controller

323

∞ ITSE =

e2 (t)tdt

(9)

|e (t)|tdt

(10)

0

∞ ITAE = 0 n

2 1  MSE = Y i − Yi n

(11)

i=1

The experimentation was developed with a total of 30 executions. The average (AVG) is presented in Table 2 for minimum values found by BCO algorithm. Table 2. Final Errors for the BCO algorithm Performance Indexes

Best

Worst

AVG

σ

ITAE

9.79E+00

1.70E+02

6.69E+01

4.37E+01

ITSE

5.69E+01

4.55E+03

1.16E+03

1.20E+03

IAE

3.55E+00

3.11E+02

5.26E+01

1.17E+02

ISE

1.89E+01

747E+02

4.07E+02

4.10E+02

MSE

1.95E–07

8.90E+01

5.53E+00

1.67E+01

Table 2 shows the average MSE that BCO algorithm to find is of 1.95E-07 indicates a stabilization in the speed of the IT2FLC. Table 3 shows a comparative with other algorithms with the goal in highlight the good results with the proposal method, such as; BCO with Type-1 FLS, Chicken Search Optimization (CSO), Fuzzy Harmony Search (FHS) and Fuzzy Differential Evolutional (FDE). Table 3. Comparison between the BCO, FHS and FDE algorithms Performance Indexes

IT2FLS-BCO

T1FLS-BCO

CSO [24]

FHS [25]

FDE [25]

Best

3.69E-04

3.69E-02

1.38E-02

2.36E-01

2.73E-01

Worst

7.87E+00

7.87E+00

9.17E+00

7.00E-01

6.06E-01

AVG

1.23E+00

1.04E+00

5.18E-01

4.52E-01

4.35E-01

Table 3 presents the, BCO with T1FLS with a value of 3.69E-02, CSO with a RMSE value of 1.38E-02, FHS with a 2.36E-01 and FDE with a 2.73E-01, comparing results the RMSE to BCO with IT2FLS is of 3.69E-04 value. The best results is with the proposal method regarding BCO T1FLS, CSO, FHS and FDE algorithm. Figure 6(a) shows the convergence for this algorithm, and Fig. 6(b) shows the speed response in the real problem.

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Fig. 6. A) Best Convergence on the results in the proposal, and b) Speed response in the real problem with the BCO algorithm.

7 Conclusions The real problems are analyzed and stabilized with greater precision when is implemented an IT2FLS, this is because the uncertainty is better evaluated. The proposal method shows a minimal error of 3.69E-04 (See Table 4). Another important point to highlight is that BCO with IT2FLS shows premature low-error convergence (See Fig. 6(a)). The comparative with others meta-heuristics shows that BCO with IT2FLS maintains a stabilization in the DC speed motor (See Fig. 6(b)). A strategy to improve this research is to add perturbation or disturbance in the FLC with the main objective of exploiting in greater depth the outputs of this algorithm. Other idea is to increase the extension of the Fuzzy Sets (FS) with the Generalized Type- 2 FLS and to be able to analyze in more detail the levels of the uncertainty in the real problem.

References 1. Lee, Z.J., Luo, X.: Predicting rainfall-induced landslide using bee colony algorithm based on support vector regression. In: Recent Advances in Computer Science and Communications (Formerly: Recent Patents on Computer Science), vol. 16, no. 1, pp. 33–37 (2023) 2. Ziyadullaev, D., Muhamediyeva, D., Ziyaeva, S., Xoliyorov, U., Kayumov, K., Ismailov, O.: Development of a traditional transport system based on the bee colony algorithm. In: E3S Web of Conferences, vol. 365, p. 01017, EDP Sciences (2023) 3. Bala, A., Sharma, A.K.: FCM-based image segmentation using bio-inspired optimization techniques: a comprehensive study. Specialusis Ugdymas 1(44), 514–529 (2023) 4. Zhao, C., Yao, Y., Zhang, N., Chen, F., Wang, T., Wang, Y.: Hybrid scheduling strategy of multiple mobile charging vehicles in wireless rechargeable sensor networks. Peer-to-Peer Networking Appl. 16, 1–17 (2023) 5. Bokam Divakar, D.R., Divya, G., Yugandhar, S., Sindhusha, D.: A review on brushless Dc motor control techniques. J. Pharmaceutical Negative Results, 6821–6828 (2023) 6. Pandey, S.: Optimization of PID controller parameters for speed control of DC motor using firefly and fminsearch algorithms. Available at SSRN 4378784 (2023) 7. Munagala, V.K., Jatoth, R.K.: A novel approach for controlling DC motor speed using NARXnet based FOPID controller. Evol. Syst. 14(1), 101–116 (2023)

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8. Yang, L., Qu, C., Jia, B., Qu, S.: The design of an affordable fault-tolerant control system of the brushless DC motor for an active waist exoskeleton. Neural Comput. Appl. 35(3), 2027–2037 (2023) 9. Chandramma, P., Prakash, P., Nandankar, H.R., Kathir, I., Singh, P.: Automation of camel race by controlling DC motor speed using Blynk application through IoT. In: AIP Conference Proceedings, vol. 2690, no. 1, p. 020047. AIP Publishing LLC (2023) 10. Ab Rahman, N.N., Yahya, N.M., Sabari, N.U.M.: Design of a fuzzy logic proportional integral derivative controller of direct current motor speed control. IAES Int. J. Robot. Autom. 12(1), p. 98 (2023) 11. Amador-Angulo, L., Castillo, O., Melin, P., Castro, J.R.: Interval Type-3 fuzzy adaptation of the bee colony optimization algorithm for optimal fuzzy control of an autonomous mobile robot. Micromachines 13(9), p. 1490 (2022) 12. Jovanovi´c, A., Stevanovi´c, A., Dobrota, N., Teodorovi´c, D.: Ecology based network traffic control: a bee colony optimization approach. Eng. Appl. Artif. Intell. 115, 105262 (2022) 13. Jovanovi´c, A., Teodorovi´c, D.: Fixed-time traffic control at superstreet intersections by bee colony optimization. Transp. Res. Rec. 2676(4), 228–241 (2022) 14. Amador-Angulo, L., Castillo, O.: Stabilization of a Fuzzy Controller Using an Interval Type2 Fuzzy System designed with the bee colony optimization algorithm. In: Intelligent and Fuzzy Systems: Digital Acceleration and The New Normal-Proceedings of the INFUS 2022 Conference, vol. 2, pp. 713–721. Springer, Cham (2022) ˇ ˇ cevi´c, S., Trifunovi´c, A., Dobrodolac, M.: A bee 15. Cubrani´ c-Dobrodolac, M., Švadlenka, L., Ciˇ colony optimization (BCO) and type-2 fuzzy approach to measuring the impact of speed perception on motor vehicle crash involvement. Soft. Comput. 26(9), 4463–4486 (2021). https://doi.org/10.1007/s00500-021-06516-4 16. Teodorovi´c, D., Davidovi´c, T., M. Šelmi´c, and M. Nikoli´c, “Bee Colony Optimization and its Applications”. Handbook of AI-based Metaheuristics, pp. 301–322, 2021 17. J. C. Biesmeijer, and T. D. Seeley, “The use of waggle dance information by honey bees throughout their foraging careers, Behavioral Ecology and Sociobiology”, vol. 59, no. 1, pp. 133–142, 2005 18. Dyler, F.C.: The biology of the dance language. Annu. Rev. Entomol. 47, 917–949 (2002) 19. Zadeh, L.A.: The concept of a Linguistic Variable and its Application to Approximate Reasoning. Part II, Information Sciences 8, 301–357 (1975) 20. Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1(1), 3–28 (1978) 21. Mendel, J.M., John, R.i.B.: Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst. 10(2), 117–127 (2002) 22. Mamdani, E.H.: Application of fuzzy algorithms for control of simple dynamic plant. In Proceedings of the Institution of Electrical Engineers 121(12), 1585–1588 (1974) 23. Karnik, N.N., Mende, J., Liang, Q.: Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 7(6), 643–658 (1999) 24. L. Amador-Angulo, L. and O. Castillo, Stabilization of a DC motor speed controller using Type-1 fuzzy logic systems designed with the chicken search optimization algorithm. In: Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation: Proceedings of the INFUS 2021 Conference, held August 24–26. vol. 1, pp. 492–499. Springer International Publishing (2022). Doi: https://doi.org/10.1007/978-3-030-85626-7_58 25. Castillo, O., et al.: A high-speed interval type 2 fuzzy system approach for dynamic parameter adaptation in metaheuristics. Eng. Appl. Artif. Intell. 85, 666–680 (2019)

Requirement Prioritization by Using Type-2 Fuzzy TOPSIS Basar Oztaysi(B)

, Sezi Cevik Onar , and Cengiz Kahraman

Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey [email protected]

Abstract. Effective results in business analysis projects depend on the prioritization of requirements. The literature provides several techniques for the prioritization process, it can be challenging due to the involvement of various stakeholders and conflicting criteria. To address this, multi-criteria decision making methods can be used to empower the process. Interval Type-2 Fuzzy Sets is a type of fuzzy set where the membership function is represented as an interval rather than a single value. This allows for a more comprehensive representation of uncertainty and ambiguity in the data being modeled. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision-making method used to determine the best alternative among a set of options based on multiple criteria. In this study, we propose using the Interval Type-2 Fuzzy TOPSIS method for requirements prioritization. The process involved three experts, five criteria, and ten requirements. Keywords: Interval Type-2 Fuzzy Sets · Multi-criteria decision making · TOPSIS

1 Introduction Business analysis is the process of identifying business needs and finding solutions to meet those needs. It is a critical function that helps organizations improve their performance and achieve their goals by identifying opportunities for growth and optimization. A business analyst is typically responsible for conducting a range of activities, such as gathering and analyzing data, defining business requirements, identifying problems and opportunities, and recommending solutions to address them. The key objective of business analysis is to ensure that the organization is focused on delivering value to its customers and stakeholders by improving business processes, products, and services. By working closely with stakeholders and subject matter experts, business analysts can identify areas for improvement, recommend solutions, and help to implement changes that deliver measurable business benefits. Requirements prioritization is the process of identifying and ranking the importance of requirements for a particular project or product. It involves analyzing and categorizing requirements based on various and conflicting criteria. Requirements prioritization can © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 326–334, 2023. https://doi.org/10.1007/978-3-031-39774-5_39

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be handled as a multi criteria decision making problem since various criteria are involved and different stakeholder evaluations are important for the final decision. Zadeh proposed Type-2 fuzzy sets in 1975 as a response to the criticism that ordinary fuzzy sets did not consider uncertainty in their membership functions. Unlike ordinary fuzzy sets, Type-2 fuzzy sets use fuzzy numbers to represent the degree of membership of an element to a set. Additionally, the membership function itself is also allowed to be fuzzy in Type-2 fuzzy sets, which makes them a more flexible representation of uncertainty. TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution), developed by Hwang and Yoon (1981), is a multi-criteria decision-making method that is used to identify the best alternative among a set of options based on multiple criteria and ideal solutions. TOPSIS has been extended by using fuzzy sets in order to involve the uncertainty and vagueness in decision makers’ evaluations (Chen and Lee 2010). In this study, a decision model is proposed for requirements prioritization problem and it is solved by using Type-2 Fuzzy TOPSIS methodology. The results are compared with other TOPSIS methods to show the validity of the proposed method. The rest of the paper is organized as follows. In section three requirements prioritization is explained in detail. In Sect. 3, Type-2 Fuzzy TOPSIS is summarized. In Sect. 4 a case study is given and solved by using Type-2 Fuzzy TOPSIS. Finally, in Sect. 5, conclusions are given.

2 Literature Review In the context of business analysis, a requirement refers to a specific feature, function, or capability that a product, system, or software must possess in order to meet the needs of its stakeholders (Bukhsh et al. 2020). Effective requirement gathering and prioritization is critical to the success of a business analysis project, as it helps ensure that the final product or system meets the needs of its users and stakeholders. Requirements prioritization is the process of identifying and ranking the importance of requirements for a particular project or product (Berander and Andrews 2005) It involves analyzing and categorizing requirements based on their relative value, urgency, and impact on project success. The purpose of requirements prioritization is to ensure that the most critical and high-value requirements are addressed first, and that resources are allocated efficiently (Achimugu et al. 2014) By prioritizing requirements, stakeholders can focus on delivering the features and functionalities that are most important to customers, users, and the overall success of the project. Ultimately, the outcome of requirements prioritization is a clear understanding of which requirements are most important and should receive the highest priority during project planning and execution. This can help to ensure that resources are focused on delivering the most critical and valuable features and functionalities, leading to a successful project outcome (Ahmad et al. 2017). Requirements prioritization problem has been investigated in the literature as a multicriteria decision making problem. Analytic Hierarchy Process has been widely used in the literature (Danesh and Ahmad 2009; Perini et al. 2009; Dabbagh et al. 2016; Singh et al. 2018) also TOPSIS method has been employed in some studies (Kukreja, 2023). Interval Type-2 Fuzzy sets have been employed to empower decision making processes such as strategic decision selection (Cevik Onar et al. 2022), warehouse location

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selection (Ucal Sari et al. 2013), information systems selection (Oztaysi 2015), capability analysis (Parchami et al. 2017) renewable energy alternatives selection (Oztaysi and Kahraman 2017) evaluation of creative thinking tools (Cevik et al. 2022).

3 Interval Type-2 Fuzzy TOPSIS Type-2 fuzzy sets introduced by Zadeh (1975) are fuzzy sets whose membership grades themselves are fuzzy. They are usually preferred when it is difficult to determine an exact membership function for a fuzzy set. ∼

A type 2 fuzzy set A˜ in the universe of discourse X can be represented by a type 2 membership function µ∼ , shown as follows (Zadeh 1975): A˜

∼

A˜ = {((x, u), µ∼ (x, u))|∀x ∈ X, ∀u ∈ J x ⊆ [0, 1], 0 ≤ µ∼ (x, u) ≤ 1} A˜

A˜

(1)

where Jx denotes an interval [0,1].

∼    U , aU ; H A ˜U , A triangular interval type 2 fuzzy set is illustrated as A˜ i = ailU , aim ir i    L L L L L U U U U L ail , aim , air ; H A˜ i where A˜ i ve A˜ i are type-1 fuzzy sets, ail , aim , air , ailL , aim ∼   denotes and airL are the references points of the interval type 2 fuzzy set A˜ i , H A˜ U i

the membership value of the element aiU in the upper triangular membership function   ˜ L denotes the membership value of the element aL in the lower triangular A˜ U i , H Ai i     L U L ˜ ˜ ˜ membership function Ai , H Ai ∈ [0, 1], H Ai ∈ [0, 1], , and 1 ≤ i ≤ 2 (Mendel, 2006). Interval Type-2 fuzzy TOPSIS steps are given as follows (Chen and Lee 2010): ∼

∼

˜ ) are constructed. The Step 1. The decision matrix Y˜ and weighting matrix (W linguistic variables used for evaluations are given in Table 1. ⎡∼ ∼ ⎤ ˜f 11 · · · f˜ 1n ⎥ ∼ ⎢ ⎢ ⎥ (2) Y˜ = ⎢ ... . . . ... ⎥ ⎣∼ ⎦ ∼ f˜ m1 · · · f˜ mn ∼

where f˜ ij is an interval type-2 fuzzy set which denotes the evaluation of jth alternative based on ith criteria, 1 ≤ i ≤ m, 1 ≤ j ≤ n, m denotes the number of criteria, n represents the number of alternatives. 

∼ ∼ ∼ ∼ ˜ = w˜ 1 , w˜ 2 , (3) W . . . w˜ m

∼

where w˜ i is an interval type-2 fuzzy set which denotes the importance of ith criterion.

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Table 1. Interval Type-2 Fuzzy scales Linguistic scale for alternative evaluations

Linguistic scale for criteria evaluations

VL

((0,0,0,0.1;1,1), (0,0,0,0.05;0.9,0.9))

VP

((0,0,0,1;1,1), (0,0,0,0.5;0.9,0.9))

L

((0,0.1,0.15,0.3;1,1), (0.05,0.1,0.15,0.2;0.9,0.9))

P

((0,1,1.5,3;1,1), (0.5,1,1.5,2;0.9,0.9))

ML

((0.1,0.3,0.35,0.5;1,1), (0.2,0.3,0.35,0.4;0.9,0.9))

MP

((1,3,3.5,5;1,1), (2,3,3.5,4;0.9,0.9))

M

((0.3,0.5,0.55,0.7;1,1), (0.4,0.5,0.55,0.6;0.9,0.9))

F

((3,5,5.5,7;1,1), (4,5,5.5,6;0.9,0.9))

MH

((0.5,0.7,0.75,0.9;1,1), (0.6,0.7,0.75,0.8;0.9,0.9))

MG

((5,7,7.5,9;1,1), (6,7,7.5,8;0.9,0.9))

H

((0.7,0.85,0.9,1;1,1), (0.8,0.85,0.9,0.95;0.9,0.9))

G

((7,8.5,9,10;1,1), (8,8.5,9,9.5;0.9,0.9))

VH

((0.9,1,1,1;1,1), (0.95,1,1,1;0.9,0.9))

VG

((9,10,10,10;1,1), (9.5,10,10,10;0.9,0.9))

Step 2. In case of multiple decision makers, the decision matrices and weight matrices are collected from each decision maker and aggregated by using geometric means operation.



∼

∼

∼U

Let a˜ be a interval type-2 fuzzy set, a˜ i =

∼L

L , aL , aL , H ˜i ai1 1 a i2 i3

∼L

, H2 a˜ i

U , aU , aU , H ai1 ˜i 1 a i2 i3 ∼

∼U

, H2 a˜ i ∼

, ∼

, , geometric mean ( r˜ i ) of m values ( a˜ 1 , a˜ 2 ,

∼

. . . a˜ m ) is calculated by using Eq. (4) ∼ r˜ i

1/m

∼ ∼ ∼ = a˜ 1 ⊗ a˜ 2 ⊗ . . . ⊗ a˜ m

(4)

where the multiplication operation in Eq. (4) is made by using Eq. (3) and mth root of an interval type-2 fuzzy set is calculated as in Eq. (5). 

   m

∼

a˜ =

m

 m

a1U ,

a1L ,

 m

m

a2U ,

a2L ,

 m

m

∼U

a3U , H1 a˜

a3L ,

∼L

H1 a˜

∼U

, H2 a˜

∼L

, H2 a˜

,

(5)

Step 3. The aggregated decision matrix is normalized by dividing all aij parameters in the aggregated decision matrix to the highest aij value in the linguistic scale. In this extension of fuzzy TOPSIS, normalization is not required since all the evaluations are

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done by using the same scale and all the values in the decision matrix are between 0 and 1. ∼ Step 4. Weighted decision matrix Y˜ w is calculated. ⎡∼ ⎢ v˜ 11 ∼ ⎢ ˜ Y w = ⎢ ... ⎣∼ v˜ m1 ∼

∼

··· .. .

∼ v˜ 1n

⎤

⎥ .. ⎥ . ⎥ ⎦ ∼ · · · v˜ mn

(6)

∼

where v˜ ij = w˜ i ⊗ f˜ ij , 1 ≤ i ≤ m, 1 ≤ j ≤ n. Step 5. The positive and negative ideal solutions are calculated. In order to identify the positive and negative ideal solutions, Ranking values of the interval type-2 fuzzy sets are calculated and the ranking weighted decision matrix Yw∗ is constructed.  ∼  ∗ (7) Yw = Rank v˜ ij m×n ∼

Ranking value of an interval type-2 fuzzy set v˜ ij is calculated as follows Eq. (4) (Chen and Lee 2010) ∼       + M1 A˜ Li + M2 A˜ U Rank A˜ i = M1 A˜ U i i       1     + M3 A˜ Li − + S1 A˜ Li S1 A˜ U + M2 A˜ Li + M3 A˜ U i i 4             U L U ˜ ˜ ˜ + S2 Ai + S2 Ai + S3 Ai + S3 A˜ Li + S4 A˜ U + S4 A˜ Li i         + H1 A˜ Li + H2 A˜ U + H2 A˜ Li + H1 A˜ U i i

(8)

  + and the negative-ideal solution The positive ideal solution x+ = v1+ , v2+ , . . . vm   − , are obtained by using Eq. (9). x− = v1− , v2− , . . . vm ⎫ ⎫ ⎧ ⎧     ∼ ∼ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎬ ⎨ max Rank( v˜ ij ) , if fi ∈ F1 ⎪ ⎨ min Rank( v˜ ij ) , if fi ∈ F1 ⎪ 1≤j≤n 1≤j≤n + −    , andv vi = = ∼ ∼ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ min Rank( v˜ ij ) , if fi ∈ F2 ⎪ ⎩ max Rank( v˜ ij ) , if fi ∈ F2 ⎪ 1≤j≤n

1≤j≤n

(9) where F1 denotes the set of benefit attributes, F2 denotes the set of cost attributes and 1 ≤ i ≤ m.   Step 6. The distance d ID xj between each alternative xj and the ideal solutions are calculated by using Eq. (10):  ∼  2 m    ID Rank v˜ ij − viID d xj = (10) i=1

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Step 7. The relative degree of closeness C(vi ) of vi with respect to the ideal solutions is calculated by using Eq. (11): C(vi ) =

d − (vi ) d + (vi ) + d − (vi )

(11)

Step 8. The values of C(vi ) are sorted in a descending sequence, where 1 ≤ i ≤ n. The larger the value of C(vi ), the higher the preference of the alternative vi . Perform a sensitivity analysis to check the robustness of the solutions.

4 Application A towel manufacturing company wants to establish a web site for their new brand. At the beginning of the project main requirements are determined. In order to provide effective results, the requirements should be prioritized. The requirements are as follows: 1. Listing of products, 2. Detail pages for products with features (color, size, variation, etc.), 3. Adding products to the basket, 4. Free shipping campaign over a specific basket amount, 5. Use of discount coupons specific to the product and/or order, 6. Follow-up of the order preparation process, 7. Order-specific cargo tracking, 8. Payment by credit card, 9. Pay at the door, 10 Ability to create an order-specific support request. Three experts are involved into the prioritization process and they identify five criteria for prioritization namely; benefits, penalty costs, production costs, risk, and related requirements. The experts are asked to evaluate the criteria and the alternatives by using the linguistic scale given in Table 1. In Table 2 the criteria weights determined by each expert are represented. Table 2. Experts’ criteria evaluations. E-1

E-2

E-3

C1

H

H

H

C2

H

MH

MH

C3

H

VH

VH

C4

VH

VH

VH

C5

MH

H

VH

Next, the experts evaluate the requirements with respect to each criteria using the linguistic scale given in Table 1. Experts’ evaluations for each alternative with respect to each criterion are given in Table 3. The steps of Type-2 Fuzzy TOPSIS is applied using the given inputs. Due to page limitations only some of the tables can be provided in this paper (Table 4). Using the rank value, the positive ideal and negative ideal solutions are obtained. Then distance of each requirement to ideal solutions are calculated. Finally, degree of closeness value of each requirement is obtained. The values are given in Table 5.

332

B. Oztaysi et al. Table 3. Expert evaluations for the requirements C1

C2

C3

C4

C5

E-1 E-2 E-3 E-1 E-2 E-3 E-1 E-2 E-3 E-1 E-2 E-3 E-1 E-2 E-3 R1

MG G

F

R2

G

MG MG G

MG VG G

R3

MG G

F

MG G

MG G

VG G

VG G

VG MG G

MG

R4

MP F

P

F

F

MP P

MG G

MG F

F

F

R5

F

P

G

F

MP

R6 R7

G

G

G

F

G

VG VG VG VG G

P

VG MG G

VG VG VG VG MG MG MG

MG F

MG MG MG VP

P

MP P

F

MP F

MP VG G

VP

MP MP F

MG F

P

F

P

R8

F

MP MP P

MP F

G

G

MG VG VG VG F

R9

F

MP F

P

P

F

MG MG MG MG G

R10 F

F

F

F

MP G

MG

VP

VG G

MG G

F

VG MG MG MG

VG VG VG G

VG MP F

MG

VG VG VG MG MG F

MG F

F

P

P

P

P

Table 4. Rank values for the weighted aggregated decision matrix. C1

C2

C3

C4

C5

R1

8.135

8.262

9.393

9.393

8.807

R2

8.872

7.731

9.157

9.641

8.468

R3

8.135

7.731

8.921

9.393

8.807

R4

5.766

6.438

4.814

8.269

7.157

R5

7.397

7.466

4.051

8.582

6.72

R6

4.963

6.438

5.891

9.393

8.468

R7

5.113

6.78

5.193

9.641

8.533

R8

6.153

5.43

8.385

9.641

7.594

R9

6.568

4.77

8.921

9.641

8.031

R10

6.983

7.47

8.085

6.008

4.621

Using the degree of closeness values the requirements are ranked. The rank values show the prioritization of the requirements. According to these results; detail pages have the highest priority,listing of products and adding products to basket is the following requirements according to the prioritization. Discount coupons and free shipping campaign takes place at the end of the prioritization list.

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Table 5. Distance to ideal solutions, Degree of Closeness and Rank values. Dist. to PIS

Dist. to NIS

Degree of Closeness

Rank

R1

0.605

79.785

0.992

2

R2

0.453

78.126

0.994

1

R3

1.111

71.531

0.985

3

R4

38.547

15.560

0.288

10

R5

36.820

24.236

0.397

9

R6

31.048

32.432

0.511

6

R7

34.041

33.884

0.499

7

R8

17.894

42.673

0.705

5

R9

18.344

51.112

0.736

4

R10

36.642

27.625

0.430

8

5 Conclusion Prioritization of requirements are vital for effective results in business analysis projects. The literature provides a couple of techniques for the prioritization process however since the prioritization process include various stakeholders and various conflicting criteria, the process can be empowered by multi-criteria decision making methods. In this study, we propose using Interval Type-2 Fuzzy TOPSIS method for requirements prioritization process. Three experts, five criteria and ten requirements are involved into the process. The experts were asked to evaluate the criteria ad alternatives individually. At the end of the process the results were presented to the experts and it was seen that the results were satisfying. In the future studies, the decision model can be improved by using methods like Analytic Hierarchy Process to obtain the weight of the criteria. Also other fuzzy multicriteria decision making methods can be used with novel extensions of fuzzy sets and the results can be compared with the result of this study.

References Bukhsh, F.A., Bukhsh, Z.A., Daneva, M.: A systematic literature review on requirement prioritization techniques and their empirical evaluation. Comput. Stand. Interfaces 69, 103389 (2020) Berander, P., Andrews, A.: Requirements prioritization, Engineering and managing software requirements, pp. 69–94. Springer (2005) Achimugu, P., Selamat, A., Ibrahim, R.: A systematic literature review of software requirements prioritization research. Inf. Softw. Technol. 56(6), 568–585 (2014) Ahmad, K.S., Ahmad, N., Tahir, H., Khan, S.: Fuzzy_MoSCoW: a fuzzy based MoSCoW method for the prioritization of software requirements. In: 2017 (ICICICT), Kerala, India, pp. 433–437 (2017)

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Danesh, A.S., Ahmad, R.: Study of prioritization techniques using students as subjects. In: International Conference on Information Management and Engineering, 2009. ICIME’09. IEEE, pp. 390–394 ((2009)) Perini, A., Ricca, F., Susi, A.: Tool-supported requirements prioritization: comparing the AHP and CBRank methods. Inf. Softw. Technol. 51(6), 1021–1032 (2009) Dabbagh, M., Lee, S.P., Parizi, R.M.: (2016) Functional and non-functional requirements prioritization: empirical evaluation of IPA, AHP-based, and HAM-based approaches. Soft Comput. 20, 4497–4520 (2016) Singh, Y.V., Kumar, B., Chand, S., Kumar, J.: A comparative analysis and proposing ‘ANN fuzzy AHP model for requirements prioritization 10(4), 55–65 (2018) Kukreja, N.: Decision theoretic requirements prioritization A two-step approach for sliding towards value realization. In: 35th International Conference on Software Engineering (ICSE), San Francisco, CA, USA, 2013, pp. 1465–1467 (2013) Cevik, O.S., Oztaysi, B., Kahraman, C.: A comparison of fuzzy extensions of TOPSIS method: evaluating creative thinking tool alternatives. J. Multiple-Valued Logic Soft Comput. 39(5–6), 401–443 (2022) Ucal sari, I., Oztaysi, B., Kahraman, C.: Fuzzy Analytic Hierarchy Process Using Type-2 Fuzzy Sets: An Application to Warehouse Location Selection, Multicriteria Decision Aid and Artificial Intelligence, pp. 285–308 (2013) Oztaysi, B.: A group decision making approach using interval type-2 fuzzy AHP for enterprise information systems project selection. J. Multiple-Valued Logic Soft Comput. 24 (2015) Oztaysi, B., Kahraman, C.: Evaluation of renewable energy alternatives using hesitant fuzzy TOPSIS and interval type-2 fuzzy AHP. In: I. Management Association (Ed.), Renewable and Alternative Energy: Concepts, Methodologies, Tools, and Applications, pp. 1378–1412 (2017) Parchami, A., Onar, S.Ç., Öztay¸si, B., Kahraman, C.: Process capability analysis using interval type-2 fuzzy sets. Int. J. Comput. Intell. Syst. 10(1), 72 (2017) Chen, S.-M., Lee, L.-W.: Fuzzy multiple attributes group decision-making based on the interval type-2 TOPSIS method. Expert Syst. Appl. 37(4), 2790–2798 (2010) Hwang, C.L., Yoon, K.: Multiple Attribute Decision Making: Methods and Applications. SpringerVerlag, New York, A State-of-the-Art Survey (1981) Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning—1. Inform. Sci. 8, 199–249 (1975)

Fuzzy Z-Numbers

Picture Fuzzy Z-AHP: Application to Panel Selection of Solar Energy Nurdan Tüysüz1,2(B)

and Cengiz Kahraman1

1 Department of Industrial Engineering, Istanbul Technical University, 34367, Besiktas Istanbul,

Turkey [email protected] 2 Department of Industrial Engineering, Istanbul Gelisim University, 34310, Avcilar Istanbul, Turkey

Abstract. In this study, picture fuzzy sets are expanded using Z-fuzzy numbers that can model reliability information in addition to the restrictive judgments of experts. Then, the Analytic Hierarchy Process (AHP) method, which is one of the most commonly used multi-criteria decision making methods, is extended to the Picture Fuzzy Z-AHP method to create a more inclusive and reliable pairwise comparison structure for decision makers. An illustrative example for the solar energy panel selection problem is presented to demonstrate the practicality of the proposed method. Finally, a comparative analysis is performed to demonstrate the effects of reliability information on the results, which also justifies the importance of considering reliability information. Keywords: Picture Fuzzy Sets · MCDM · Z-fuzzy numbers · AHP

1 Introduction Fuzzy sets extensions like fermatean fuzzy sets, hesitant fuzzy sets, spherical fuzzy sets, orthopair fuzzy sets and picture fuzzy sets aim to model uncertainty within their descriptive properties and membership functions. Picture fuzzy sets [1] use membership, indeterminacy and non-membership degrees to define an element to a set. Z-fuzzy numbers [2] provide an opportunity to symbolize the fuzzy sets in a 2-tuple number that represents the reliability function in addition to the restriction function. This study integrates picture fuzzy sets (PFSs) together with Z-fuzzy numbers. It also aims at better representing the uncertainty modeling ability of PFSs and thus reaching more comprehensive and reliable results than the current. In the literature, only one paper [3] uses picture fuzzy Z-linguistic sets in defining experts’ words, different from our study. AHP method [4] constructs a significant decision making system to define the criteria weights based on pairwise comparison principle. In order to represent linguistic terms in more reliable decision structure, many extensions of fuzzy AHP method have been incorporated by Z-fuzzy numbers such as ordinary Z-fuzzy AHP [5], intuitionistic Zfuzzy AHP [6], spherical Z-fuzzy AHP [7] and interval type 2 fuzzy Z-AHP [8]. However, best of our knowledge, AHP method have not yet been extended to picture fuzzy Z-AHP © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 337–345, 2023. https://doi.org/10.1007/978-3-031-39774-5_40

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method. Main objective of this study is to extend AHP method to picture fuzzy Z-AHP method to capture the full judicial capacity of experts, which can be an effective solution proposal for the real life applications. The remaining paper is constructed as follows. Section 2 includes a literature review on PFSs and Z-fuzzy AHP methods. Section 3 gives the basics of Z-fuzzy numbers and PFSs. Section 4 gives the steps of proposed picture fuzzy Z-AHP method. Section 5 presents an illustrative example on panel selection of solar energy. Finally, conclusions and further research recommendations are given.

2 Literature Review Literature review is conducted in two parts: picture fuzzy MCDM methods and Z-fuzzy AHP methods in the literature. The picture fuzzy MCDM studies between the years 2020–2023 because of the page limitation are summarized as follows. Gündo˘gdu et al. [9] developed a hybrid decision making approach by integrating picture fuzzy AHP method and linear assignment model. They applied it to public transport development problem. Haktanır and Kahraman [10] proposed a picture fuzzy CRITIC®IME methodology and selected most preferable wearable health technology. Senapati and Chen [11] established a decision making environment under picture fuzzy WASPAS method. They showed the utility of the method by an application of air-conditioning system selection. The Z-fuzzy AHP method is considered to represent the pairwise comparison of the experts more trustworthy because it considers the reliability degrees of the judgments obtained from the experts. There are few Z-fuzzy AHP methods on various areas in the literature: evaluation of universities [12], solar PV power plant location selection [13], performance evaluation of law offices [14], selection of underground coal gasification location [15], prioritization of social sustainable development factors [5], conceptual design evaluation [16], ranking of public services’ digitalization [17], supplier selection [7], hydrogen storage technology selection [6], covid-19 risk evaluation of occupations [8].

3 Preliminaries 3.1 Z-Fuzzy Numbers   ˜ R˜ which are introduced by Zadeh [2] represent the reliability Z-fuzzy numbers Z A, ˜ A simple triangular Z-fuzzy ˜ in addition to the restriction function (A). function (R) number is given in Fig. 1. 3.2 Picture Fuzzy Sets Definition 1. A PFS on a A˜ P of the universe of discourse X is given by Eq. (1) [1]:    (1) A˜ P = x, μA˜ P (x), ηA˜ P (x), νA˜ P (x)|x ∈ X }

Picture Fuzzy Z-AHP

339

Fig. 1. A simple triangular Z-fuzzy numbers

where μA˜ P (x), ηA˜ P (x), νA˜ P (x) : X → [0, 1] are the degrees of membership, neutral membership (indeterminacy), and non-membership of x to A˜ P , respectively, and 0 ≤ μA˜ P (x) + ηA˜ P (x) + νA˜ P (x) ≤ 1

(2)

  Then, 1 − μA˜ P (x) + ηA˜ P (x) + νA˜ P (x) is defined as the refusal degree of x in X Definition 2. Let A˜ P = (μA˜ P , ηA˜ P , νA˜ P ) and B˜ P = (μB˜ P , ηB˜ P , νB˜ P ) be any two PFSs. The basic arithmetic operations of PFSs are given by Eqs. (3–6) [18, 19].   (3) A˜ P ⊕ B˜ P = μA˜ P + μB˜ P − μA˜ P μB˜ P , ηA˜ P ηB˜ P , νA˜ P νB˜ P   A˜ s ⊗ B˜ s = μA˜ P μB˜ P , ηA˜ P + ηB˜ P − ηA˜ P ηB˜ P , νA˜ P + νB˜ P − νA˜ P νB˜ P

(4)

  k  k A˜ P = 1 − 1 − μA˜ P , ηAk˜ , νAk˜ ; k > 0

(5)

 k k    ˜AkP = μk , 1 − 1 − η ˜ ;k > 0 AP , 1 − 1 − νA˜ P A˜

(6)

P

P

P

Definition 3. Let A˜ P = (μA˜ P , ηA˜ P , νA˜ P ) be a PFS. In order to obtain crisp value for PFS, a modified defuzzification formula is given by Eq. (7) [20, 21]. Def (A˜ P ) = μA˜ P +

ηA˜ P 2

−δ∗(

where δ is the net membership coefficient, 0 ≤ δ ≤ 1.

νA˜ P 2

)

(7)

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4 Proposed Method In this section, we present the proposed Picture Fuzzy Z-AHP (PF Z-AHP) method. Step 1. Determine the main-criteria (j = 1,2,…,m), sub-criteria and alternative (i = 1,2,…,n) sets of the decision problem by creating a hierarchical structure. Step 2. Define PF Z restriction and reliability scales for pairwise comparisons. Table 1. PF Z restriction and reliability scales for pairwise comparisons Linguistic Terms for Restriction

Linguistic Terms for Reliability

(μ, η, ν)

Absolutely High Importance (AHI)

Absolutely High Reliable (AHR)

(0.80, 0.05, 0.00)

Very High Importance (VHI)

Very High Reliable (VHR)

(0.70, 0.10, 0.10)

High Importance (HI)

High Reliable (HR)

(0.60, 0.15, 0.20)

Slightly High Importance (SHI)

Slightly High Reliable (SHR)

(0.50, 0.20, 0.30)

Equally Importance (EI)

Medium Reliable (MR)

(0.40, 0.20, 0.40)

Slightly Low Importance (SLI)

Slightly Low Reliable (SLR)

(0.30, 0.20, 0.50)

Low Importance (LI)

Low Reliable (LR)

(0.20, 0.15, 0.60)

Very Low Importance (VLI)

Very Low Reliable (VLR)

(0.10, 0.10, 0.70)

Absolutely Low Importance (ALI)

Absolutely Low Reliable (ALR)

(0.00, 0.05, 0.80)

Step 3. Apply a questionnaire using Table 1 to collect pairwise comparisons from experts. In this study, three experts compromise on the pairwise evaluations. This compromised pairwise comparison matrix is given in Eqs. (8–9).  

˜ R˜ ˜Z = A, (8) ⎡



˜ R˜ A, 11 ⎢ ⎢ A, ˜ ˜ R ⎢ 21 ⎢ .. ⎢ ⎢  . Z˜ = ⎢ ⎢ A, ⎢ ˜ R˜ i1 ⎢ ⎢ .. ⎢ ⎣ .  ˜ R˜ A, n1

  ˜ R˜ A,  12 ˜ R˜ A, 22

..  . ˜ R˜ A, ..  . ˜ R˜ A,

i2

n2

ij n×n

  ⎤ ˜ R˜ ... A,  1n ⎥ ⎥ ˜ R˜ ... A, ⎥ 2n ⎥ .. ⎥ .. ⎥ .    .  ⎥i = 1, 2, . . . , n, j = 1, 2, . . . , n ⎥ ˜ R˜ ˜ R˜ A, A, ⎥ ij in ⎥ ⎥ .. .. ⎥ .  . ⎦ ˜ R˜ ... A,

(9)

nn

Step 4. Calculate consistency ratio (CR) based on Saaty’s classical consistency check procedure for pairwise comparison matrices using the corresponding crisp values of linguistic terms for restriction functions. Corresponding crisp values of picture fuzzy restriction and reliability values can be obtained by Eqs. (10–11), respectively. ⎧ ⎨ 11 ∗ μA˜P + 4 ∗ ηA˜ P − 11 ∗ νA˜ P , for AHI , VHI , HI , SHI , EI .   Def A˜ = (10) 1   ⎩  11∗ν +4∗η −11∗μ , for SLI , LI , VLI , ALI . A˜ P

A˜ P

A˜ P

Picture Fuzzy Z-AHP

341

Def R˜ = 0.95 − 0.5 ∗ νR˜ P for AHR, VHR, HR, SHR, MR, SLR, LR, VLR, ALR.

(11)

Step 5. Defuzzify PF reliability functions (R˜ ij ) using Eq. (11). Then, multiply each restriction value by square root of defuzzified PF reliability values using Eq. (12). Thus, PF Z values are transformed to PF values.

  Z = cij n×n

 ⎡  ˜ 11 × def R˜ 11 A ⎢ ⎢   ⎢ ⎢ A˜ 21 × def R˜ 12 ⎢ ⎢ ⎢ .. ⎢ . ⎢   =⎢ ⎢ ˜ ⎢ Ai1 × def R˜ 1i ⎢ ⎢ .. ⎢ ⎢ . ⎢   ⎣ ˜An1 × def R˜ 1n

   A˜ 12 × def R˜ 12    A˜ 22 × def R˜ 22



⎤  def R˜ 1n ⎥  ⎥  ⎥ ... A˜ 2n × def R˜ 2n ⎥ ⎥ ⎥ ⎥ .. .. .. ⎥ . . .      ⎥    ⎥ ⎥ A˜ i2 × def R˜ 2i A˜ ij × def R˜ ij A˜ in × def R˜ in ⎥ ⎥ ⎥ .. .. ⎥ .. ⎥ . . .    ⎥   ⎦ ... A˜ n2 × def R˜ 2n A˜ nn × def R˜ nn ...

A˜ 1n ×

(12) where def R˜ ij represents the defuzzified   reliability value.  vector using Picture Fuzzy Weighted Arithmetic Step 6. Define the PF mean PFM Mean (PFWAM ) or Picture Fuzzy Weighted Geometric Mean  (PFWGM ) operator given by Eqs. (14–15) [19], respectively. At the end of this step, cij n×n matrix is transformed   to mij n×1 matrix given by Eq. (13). ⎤ m ˜ 11 ⎢m ˜ 21 ⎥ ⎥  =⎢ PFM ⎢ . ⎥ . ⎣ . ⎦ ⎡

m ˜ i1

(13) n×1

where m ˜ ij = (μm˜ ij , ηm˜ ij , νm˜ ij ) and, ⎧ ⎫ n  n  n  ⎨ wj  wj  wj ⎬  PFWAM (m ˜ i1 ) = 1 − 1 − μm˜ ij , ηm˜ ij , νm˜ ij ⎩ ⎭ j=1

j=1

(14)

j=1

⎧ ⎫ n  n  n  ⎨ wj wj wj ⎬   μm˜ ij , 1 − 1 − ηm˜ ij , 1 − 1 − νm˜ ij PFWGM (m ˜ i1 ) = (15) ⎩ ⎭ j=1

j=1

j=1

Step 7. Apply Steps (3–6) for all PF Z pairwise comparison matrices of experts. Global weights are calculated using PF comparison matrices by multiplying the local alternative weights and the criteria weights. The calculated PF weighted average is defuzzified by using Eq. (7) in order to compute the crisp alternative scores (δ = 0.1).

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5 Application An illustrative example on panel selection of solar energy is presented to demonstrate applicability of proposed PF Z-AHP method. Step 1. We determine the alternatives and the criteria. There are 5 criteria (C1: Mechanical, C2: Electrical, C3: Economic, C4: Environmental, C5: Satisfaction) and 3 alternative solar panels (A1, A2, A3) in the problem. Step 2–3. Three experts use Table 1 for evaluations and they compromise on the pairwise comparisons. The obtained comparison matrix for the compromised evaluations of criteria is given in Table 2 whereas the other compromised comparison matrices are given in Table 3. Table 2. Compromised PF Z pairwise comparisons for criteria C1

C2

C3

C4

C5

C1

(EI, AHR)

(SHI, SLR)

(SLI, HR)

(AHI, VHR)

(SHI, SHR)

C2

(SLI, SLR)

(EI, AHR)

(VLI, LR)

(SHI, SHR)

(SLI, VLR)

C3

(SHI, HR)

(VHI, LR)

(EI, AHR)

(VHI, SLR)

(SHI, SHR)

C4

(ALI, VHR)

(SLI, SHR)

(VLI, SLR)

(EI, AHR)

(SLI, LR)

C5

(SLI, SHR)

(SHI, VLR)

(SLI, SHR)

(SHI, LR)

(EI, AHR)

CR = 0.028

Table 3. Compromised PF Z pairwise comparisons for alternatives with respect to criteria C1 A1 A1 (EI, AHR) A2 (ALI, SHR) A3 (SLI, HR) CR=0.078 C3 A1 A1 (EI, AHR) A2 (VHI, HR) A3 (HI, SHR) CR=0.063

C5 A1 A1 (EI, AHR) A2 (VHI, SLR) A3 (SHI, SHR) CR=0.007

A2 (AHI, SHR) (EI, AHR) (VHI, LR) A2 (VLI, HR) (EI, AHR) (SLI, VLR)

A3 (SHI, HR) (VLI, LR) (EI, AHR)

C2 A1 A1 (EI, AHR) A2 (SHI, SLR) A3 (LI, HR) CR=0.063 A3 C4 A1 (LI, SHR) A1 (EI, AHR) (SHI, VLR) A2 (LI, LR) (EI, AHR) A3 (SLI, HR) CR=0.037

A2

A3

(VLI, SLR) (EI, AHR) (SLI, LR)

(SLI, SHR) (SHI, LR) (EI, AHR)

A2 (SLI, SLR) (EI, AHR) (VLI, SHR)

A3 (HI, HR) (VHI, SHR) (EI, AHR)

A2 (HI, LR) (EI, AHR) (SHI, SHR)

A3 (SHI, HR) (SLI, SHR) (EI, AHR)

Step 4. The consistency of each pairwise comparison matrix is calculated and is given at the bottom of Tables 2–3. Since all the evaluations are in the consistency limits, we continue to Step 5.

Picture Fuzzy Z-AHP

343

Step 5. We transformed PF Z pairwise comparison matrices to PF pairwise comparison matrices using Eqs. (11–12). Table 4 gives the PF pairwise comparisons for criteria. Only one of the obtained PF pairwise comparison matrices is given for C1 in Table 5 because of the page limitation. Table 4. PF pairwise comparisons for criteria C1

C2

C3

C4

C5

C1

(0.39, 0.19, 0.39)

(0.42, 0.17, 0.25)

(0.28, 0.18, 0.46)

(0.76, 0.05, 0.00)

(0.45, 0.18, 0.27)

C2

(0.25, 0.17, 0.42)

(0.39, 0.19, 0.39)

(0.08, 0.08, 0.56)

(0.45, 0.18, 0.27)

(0.23, 0.15, 0.39)

C3

(0.46, 0.18, 0.28)

(0.56, 0.08, 0.08)

(0.39, 0.19, 0.39)

(0.59, 0.08, 0.08)

(0.45, 0.18, 0.27)

C4

(0.00, 0.05, 0.76)

(0.27, 0.18, 0.45)

(0.08, 0.08, 0.59)

(0.39, 0.19, 0.39)

(0.24, 0.16, 0.40)

C5

(0.27, 0.18, 0.45)

(0.39, 0.15, 0.23)

(0.27, 0.18, 0.45)

(0.40, 0.16, 0.24)

(0.39, 0.19, 0.39)

Table 5. PF pairwise comparisons for alternatives with respect to C1 C1

A1

A2

A3

A1

(0.39, 0.19, 0.39)

(0.72, 0.04, 0.00)

(0.46, 0.18, 0.28)

A2

(0.00, 0.04, 0.72)

(0.39, 0.19, 0.39)

(0.08, 0.08, 0.56)

A3

(0.28, 0.18, 0.46)

(0.56, 0.08, 0.08)

(0.39, 0.19, 0.39)

 vectors for each pairwise comparison matrix using Eqs. Step 6. We calculate PFM  vectors are given in Table 6. Thus, criteria weights (13) and (15). The obtained PFM and local alternative weights are defined as PF numbers.  vectors of alternatives with respect to criteria Table 6. PFM C1 (0.43, 0.03, 0.10) A1 (0.51, 0.02, 0.08) A2 (0.00, 0.02, 0.34) A3 (0.40, 0.03, 0.13) C4 (0.02, 0.02, 0.31) A1 (0.45, 0.03, 0.09) A2 (0.26, 0.03, 0.19) A3 (0.37, 0.03, 0.15)

C2 (0.24, 0.03, 0.18) A1 (0.38, 0.03, 0.12) A2 (0.47, 0.02, 0.08) A3 (0.19, 0.02, 0.29) C5 (0.34, 0.03, 0.14) A1 (0.21, 0.03, 0.23) A2 (0.46, 0.02, 0.07) A3 (0.35, 0.03, 0.13)

C3 (0.48, 0.02, 0.06) A1 (0.19, 0.02, 0.29) A2 (0.46, 0.02, 0.07) A3 (0.37, 0.03, 0.11)

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Step 7. Using the PF criteria weights and PF alternative weights in Step 6, we calculate the weighted average of the alternatives. By Eq. (7), we defuzzify PF alternative scores and obtain the crisp scores and normalized crisp weights as in Table 7. Table 7. Final weights and ranking of alternatives PF weights

Crisp scores

Crisp weights

Ranks

A1

(0.407, 0.000, 0.002)

0.407

0.323

3

A2

(0.422, 0.000, 0.001)

0.422

0.335

2

A3

(0.432, 0.000, 0.002)

0.432

0.342

1

According to Table 7, it is found that the most preferred solar energy panel is A3 with the weight of 0.342. The overall ranking is A3 > A2 > A1. To understand whether the reliability information is important and whether it is necessary, we compared these results with the picture fuzzy AHP method (PF AHP), which does not use reliability information. Results of PF AHP method are given in Table 8. Table 8. Results of PF AHP method PF weights

Crisp scores

Crisp weights

Ranks

A1

(0.477, 0.000, 0.006)

0.477

0.315

3

A2

(0.519, 0.000, 0.004)

0.519

0.343

1

A3

(0.518, 0.000, 0.004)

0.518

0.342

2

6 Conclusion This study proposes a new picture Z-fuzzy number together with the combination of PFSs and Z-fuzzy numbers. Then, the PF Z-AHP method, which can offer a reliable pairwise comparisons based structure to experts, is suggested. The practical utility of proposed method is demonstrated with an illustrative example. Finally, PF AHP method is applied with the same data except reliability degrees of judgments. The obtained results demonstrate that reliability information can change the results and it should be considered in decision processes. In the future, new types of Z-fuzzy numbers can be developed by using other fuzzy set extensions such as interval-valued Pythagorean Z-numbers.

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References 1. Cuong, B.C.: Picture fuzzy sets. J. Comput. Sci. Cybern. 30(4), 409 (2014) 2. Zadeh, L.A.: A note on Z-fuzzy numbers. Inf. Sci. 181(14), 2923–2932 (2011) 3. Chen, L., Wang, Y., Yang, D.: Picture fuzzy Z-linguistic set and its application in multiple attribute group decision-making. J. Intell. Fuzzy Syst. 181, 1–15 (2022) 4. Saaty, T.L.: The Analytic Hierarchy Process. Mcgrawhill, New York (1980) 5. Tüysüz, N., Kahraman, C.: Evaluating social sustainable development factors using multiexperts Z-fuzzy AHP. J. Intell. Fuzzy Syst. 39(5), 6181–6192 (2020) 6. Haktanır, E., Kahraman, C.: Hydrogen storage technology selection using a novel intuitionistic Z-Ahp & Z-Topsis Methodology. Available at SSRN 4175200 (2022) 7. Alkan, N., Kahraman, C.: Fuzzy analytic hierarchy process using spherical Z-Numbers: supplier selection application. In: Intelligent and Fuzzy Systems: Digital Acceleration and The New Normal-Proceedings of the INFUS 2022 Conference, vol. 1, pp. 702–713. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5_81 8. Sari, I.U., Tüysüz, N.: COVID-19 Risk Assessment of Occupations Using Interval Type 2 Fuzzy Z-AHP & Topsis Methodology. J. Multiple-Valued Logic Soft Comput. 38 (2022) 9. Gündo˘gdu, F.K., Duleba, S., Moslem, S., Aydın, S.: Evaluating public transport service quality using picture fuzzy analytic hierarchy process and linear assignment model. Appl. Soft Comput. 100, 106920 (2021) 10. Haktanır, E., Kahraman, C.: A novel picture fuzzy CRITIC & REGIME methodology: wearable health technology application. Eng. Appl. Artif. Intell. 113, 104942 (2022) 11. Senapati, T., Chen, G.: Picture fuzzy WASPAS technique and its application in multi-criteria decision-making. Soft. Comput. 26(9), 4413–4421 (2022). https://doi.org/10.1007/s00500022-06835-0 12. Azadeh, A., Saberi, M., Atashbar, N. Z., Chang, E., Pazhoheshfar, P.: Z-AHP: A Z-number extension of fuzzy analytical hierarchy process. In: 2013 7th IEEE International Conference on Digital Ecosystems and Technologies (DEST), pp. 141–147. IEEE (2013) 13. Kahraman, C., Otay, I.: Solar PV power plant location selection using a Z-fuzzy number based AHP. Int. J. Analytic Hierarchy Process 10(3) (2018) 14. Kahraman, C., Oztaysi, B., Onar, S.C.: Performance comparisons of law offices and optimum allocation of debt files using Z-Fuzzy AHP. In: 11th Conference of the European Society for Fuzzy Logic and Technology, pp. 446–451. Atlantis Press (2019) 15. Rafiee, R., Azarfar, A., Najafi, M., Jalali, S.M.E.: Z-number-based selection of suitable underground coal gasification site considering information reliability. Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, pp. 1–21 (2020) 16. Liu, Q., Chen, J., Wang, W., Qin, Q.: Conceptual design evaluation considering confidence based on Z-AHP-TOPSIS method. Appl. Sci. 11(16), 7400 (2021) 17. Sergi, D., Ucal Sari, I.: Prioritization of public services for digitalization using fuzzy Z-AHP and fuzzy Z-WASPAS. Compl. Intell. Syst. 7(2), 841–856 (2021) 18. Wei, G.: Picture 2-tuple linguistic Bonferroni mean operators and their application to multiple attribute decision making. Int. J. Fuzzy Syst. 19, 997–1010 (2017) 19. Wei, G., Alsaadi, F.E., Hayat, T., Alsaedi, A.: Picture 2-tuple linguistic aggregation operators in multiple attribute decision making. Soft. Comput. 22(3), 989–1002 (2016). https://doi.org/ 10.1007/s00500-016-2403-8 20. Son, L.H., Van Viet, P., Van Hai, P.: Picture inference system: a new fuzzy inference system on picture fuzzy set. Appl. Intell. 46(3), 652–669 (2016). https://doi.org/10.1007/s10489-0160856-1 21. Xu, X.G., Shi, H., Xu, D.H., Liu, H.C.: Picture fuzzy Petri nets for knowledge representation and acquisition in considering conflicting opinions. Appl. Sci. 9(5), 983 (2019)

LR-Type Z Fuzzy Numbers and Their Usage in MCDM Problems Cengiz Kahraman(B) , Sezi Cevik Onar, and Basar Öztaysi Department of Industrial Engineering, Istanbul Technical University, 34367 Istanbul, Turkey [email protected]

Abstract. Z fuzzy numbers are given by a restriction function and a reliability function to better define an ordinary fuzzy number. In this paper we propose LR fuzzy numbers in the definition of z-fuzzy numbers. Besides, the new extensions of ordinary fuzzy sets are used in the development of LR-type Z fuzzy numbers such as intuitionistic fuzzy LR type Z-numbers. The proposed new numbers are planned to be employed in multi-criteria decision making problems such as LR type Z-fuzzy TOPSIS or intuitionistic fuzzy LR type VIKOR methods. The paper is concluded by limitations and future research suggestions. Keywords: z-fuzzy numbers · LR-fuzzy numbers · MCDM

1 Introduction LR-type fuzzy numbers are the continuous numbers having non-linear memberships functions defined by a left side function and a right side function of a triangular or trapezoidal shaped function. The simplified LR-type fuzzy numbers are the fuzzy numbers having linear membership functions called triangular and trapezoidal fuzzy numbers. Z-fuzzy numbers were introduced by Zadeh (2011) to indicate the reliability of a fuzzy judgment. A restriction function and a reliability function together define a Z-fuzzy number. Integration of an LR-type fuzzy number and a Z-fuzzy number and their usage in Multi-Citeria Decision Making (MCDM) is the contribution of this paper. There is not yet any work integrating LR-type fuzzy numbers with Z-fuzzy numbers in the literature, to the best knowledge of the authors. Many MCDM methods have been extended to their fuzzy versions by using Z-fuzzy numbers: Z-fuzzy MULTIMOORA (Peng et al., 2021), Z-fuzzy DEMATEL (Jiang et al., 2020; Karasan and Kahraman, 2019), Z-fuzzy CODAS (Yıldız and Kahraman, 2020), Z-fuzzy ANP-TODIM (Liu et al., 2020), Z-fuzzy TOPSIS (Forghani et al., 2018; Gardashova, 2018), Z-fuzzy VIKOR (Das et al., 2020; Shen and Wang, 2018), Z-fuzzy AHP (Tuysuz and Kahraman, 2020, 2023; Kahraman and Otay, 2018; Yıldız and Kahraman, 2018; Kahraman et al., 2019; Alkan and Kahraman, 2022; Azadeh et al., 2013), Z-fuzzy DEA (Azadeh and Kokabi, 2016; Mohtashami and Ghiasvand, 2020). Among other applications, we can mention Z-fuzzy QFD (Haktanır and Kahraman, 2022; Haktanir and Kahraman, 2019), Z-fuzzy hypothesis testing (Haktanır and Kahraman, 2019), intuitionistic Z-fuzzy numbers (Uçal and Kahraman, 2020), and renewable energy selection (Chatterjee and Kar, 2018). The rest of the paper is organized © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 346–353, 2023. https://doi.org/10.1007/978-3-031-39774-5_41

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as follows. Section 2 presents LR-type fuzzy numbers. Section 3 gives the preliminaries of Z-fuzzy numbers. Section 4 introduces LR-type Z fuzzy numbers. Section 5 applies the proposed LR-type Z-fuzzy numbers to MADM problems. Section 6 concludes the paper.

2 LR-Type Fuzzy Numbers (LR-Type FNs) The main idea of LR-type fuzzy numbers is to divide the nonlinear membership function into a left side nonlinear function and a right side nonlinear function. An LR-type fuzzy number is represented by (m, α, β)LR where α is the left support and is β the right support and m is the center value of the LR-type fuzzy number. Figures 1 and 2 show a trapezoidal LR-type fuzzy number and a triangular LR-type fuzzy number, respectively.

Fig. 1. A trapezoidal LR-type fuzzy number

Fig. 2. A triangular LR-type fuzzy number

The left and right side functions in Fig. 1 are given by Eqs. (1–3): μ(x) = 1,

m1 ≤ x ≤ m2   m1 − x μL (x) = L(x) = f , x ≤ m1 α   x − m2 μR (x) = R(x) = f , x ≥ m2 β The left and right side functions in Fig. 2 are given by Eqs. (4–5):   m−x L(x) = f , x≤m α   x−m R(x) = f , x≥m β

(1) (2) (3)

(4) (5)

3 Z Fuzzy Numbers (Z-FNs) Zadeh (2011) introduced the Z-fuzzy numbers to the literature, which is an ordered pair of ˜ B) ˜ as given in Fig. 3. The first component A˜ is a restriction function fuzzy numbers, Z(A, whereas the second component B˜ is a measure of reliability of the restriction function. The figure on the left is  on the right is the part   the part of restriction, and the figure of reliability. Let A˜ = x, μA˜ (x)|μ(x) ∈ [0, 1] and R˜ = x, μR˜ (x)|μ(x) ∈ [0, 1] , μA˜ (x) is a trapezoidal membership function, μR˜ (x) is a triangular membership function.

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˜ B) ˜ Fig. 3. A simple Z-fuzzy number, Z(A,

Definition 1. Let a fuzzy set A be defined on a universe X may be given as: A˜ = ˜ The mem{x, µA˜ (x)|xX } where µA˜ : X → [0, 1] is the membership function A. ˜ The Fuzzy bership value µA˜ (x) describes the degree of belongingness of x ∈ X in A. Expectation of a fuzzy set is denoted as in Eq. (6):  (6) EA (x) = xμA (x)dx Definition 2: Converting a Z-fuzzy number to a Regular Fuzzy Number (1) The second part (reliability) is converted into a crisp number by Eq. (7).  xμ ˜ (x)dx α=  R μR˜ (x)dx

(7)

Alternatively, the defuzzification equation (a1 + 2 ∗ a2 + 2 ∗ a3 + a4 )/6 for symmetrical trapezoidal fuzzy numbers and (a1 + 2 ∗ a2 + a3 )/4 for symmetrical triangular fuzzy numbers can be used. (2) Convert the Z fuzzy number (weighted restriction) to ordinary fuzzy number. The ordinary fuzzy set can be denoted as in Eq. (8) and by Fig. 4.    ˜Z = x, μ ˜ (x)|μ ˜ (x) = μ ˜ √x , μ(x) ∈ [0, 1] (8) Z Z A α

Fig. 4. Ordinary fuzzy number converted from Z-fuzzy number

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4 LR-Type Z Fuzzy Numbers (LR-Type Z-FNs) An LR-type Z –fuzzy number is presented in Fig. 5. The trapezoidal LR restriction membership function is given by Eqs. (9–11). The left and right side functions of triangular LR reliability function in Fig. 3 are given by Eqs. (12–13). Defuzzified reliability membership function is obtained by Eq. (14). And the defuzzification of the restriction function after the reliability is integrated with the defuzzification is given by Eq. (15). When the LR type fuzzy number is a symmetrical number, Eq. (15) gives the center point of the fuzzy number.   1 m1 − x = (9) μRs,L (x) = L(x) = f 2 , x ≤ m1

m1−x α 1+ α μRs (x) = 1,  μRs,U (x) = R(x) = f

x − m2 β 

μRl,L (x) = L(x) = f  μRl,R (x) = R(x) = f

∫m x=m−α



m−x α x−m β

∫m x=m−α x Def μRl (x) =

m1 ≤ x ≤ m2 =

1+

 =  =

1+ 1+

1 2 dx 1+( m−x α ) 1+(

a) LR Type restriction function

1



1 x−m2 β

)

2 , x ≥ m2

(11)

1

m−x 2 , x ≤ m

(12)

1

(13)

α

x−m β

2 , x ≥ m

m+β

+ ∫x=m x m+β

m−x 2 α

(10)

dx + ∫x=m

1

2 dx 1+ x−m β

1

2 dx 1+ x−m β

b) LR Type reliability function

Fig. 5. An LR-Type Z-fuzzy Number

(14)

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5 Multi-criteria Decision Making with LR-Type Z-FNs Let’s consider the fuzzy decision matrix given in Eq. (16) whose linguistic terms are represented by LR-type fuzzy numbers with m alternatives and n criteria. We developed the linguistic scale given in Table 1 for this purpose. The numbers in this table are symmetrical but any LR type fuzzy number can be used based on the given linguistic terms. For instance, if you are between low reliability and middle reliability, you can assign a non-symmetrical LR-type fuzzy number such as (4.5, 1, 1.5). Using Eq. (16), LR-Z FNs can be transformed to an ordinary trapezoidal LR fuzzy number as illustrated in Fig. 6. These ordinary LR fuzzy numbers can be used in Eq. (16) and thus Eq. (16) is obtained: Alternatively, using Eqs. (14) and (15), Eq. (16) can be transformed to Eq. (17): Then, using a multi-criteria decision making method such as TOPSIS, ELECTRE, VIKOR, or EDAS, the problem can be solved by a crisp MCDM method.   m+β  m 1 1  x=m−α x 1+( m−x )2 dx+ x=m x 1+ x−m 2 dx  m1 α  β 1  m+β dx+ 1 1 x=m1−α x 1+( m−x )2   m dx+

2 dx 2 x=m−α x=m α x−m 1+( m−x 1+ β α )   m+β  m 1 1 x dx+ x

2 dx  x=m−α 1+( m−x )2 x=m  m2+β 1+ x−m α  β 1  m+β dx 1 1 x=m2 x x−m 2   m

2 dx 2 dx+ x=m x=m−α 1+ β 1+( m−x 1+ x−m α ) β  Def μRl&Rs (x) =  m+β  m 1 1  x=m−α x 1+( m−x )2 dx+ x=m x 1+ x−m 2 dx  m1 α  β 1  m+β dx+ 1 1 x=m1−α 1+( m−x )2   m dx+

2 dx 2 x=m−α x=m α x−m 1+( m−x 1+ α ) β   m+β  m 1 1 x dx+ x

2 dx  x=m−α 1+( m−x )2 x=m  m2+β 1+ x−m α  β 1  m+β dx

2   m 1 1 x=m2

2 dx 2 dx+ x=m x=m−α 1+ x−m β 1+( m−x 1+ x−m α ) β (15) A trapezoidal intuitionistic fuzzy (TraIF) LR type fuzzy number can be illustrated as given in Fig. 7. Then an intuitionistic fuzzy LR type Z-number can be defined as given in Fig. 8 with a TraIF LR type-restriction function and a Triangular intuitionistic fuzzy (TriIF) LR-type reliability function. ⎡

    ⎤    m1 Def μRl (x), m2 Def μRl (x), m1 Def μRl (x), m2 Def μRl (x),     ··· ⎢ ⎥ ⎢ ⎥ α Def μRl (x), β Def μRl (x) α Def μRl (x), β Def μRl (x) LR,11 LR,1m ⎥ ⎢ ⎢ ⎥ . . .. ⎥ ˜ LR−Z = ⎢ .. .. D ⎢ ⎥ .    ⎢  ⎥    ⎢ m Def μ (x), m Def μ (x), ⎥ m m Def μ Def μ (x), (x), 1 2 1 2 ⎣ ⎦ Rl Rl Rl Rl ··· α Def μRl (x), β Def μRl (x) α Def μRl (x), β Def μRl (x) LR,n1

⎡ ˜ LR−Z D

⎢ =⎣

LR,nm

⎤

(Def μRl&Rs (x))11 · · · (Def μRl&Rs (x))1m ⎥ .. .. .. ⎦ . . . (Def μRl&Rs (x))n1 · · · (Def μRl&Rs (x))nm

(16) (17)

LR-Type Z Fuzzy Numbers and Their Usage in MCDM Problems Table 1. Linguistic scale for LR-type Z-FNs Linguistic terms for restriction

Linguistic terms for reliability

Corresponding LR-type fuzzy restriction number

Corresponding LR-type fuzzy reliability number

Absolutely Low, AL

Absolutely Low Reliability, ALR

(2, 2, 2)LR

(0.2, 0.2, 0.2)LR

Very Low, VL

Very Low Relaibility, VLR

(3, 2, 2)LR

(0.3, 0.2, 0.2)LR

Low, L

Low Relaibility, LR

(4, 2, 2)LR

(0.4, 0.2, 0.2)LR

Middle, M

Middle reliability, MR

(5, 2, 2)LR

(0.5, 0.2, 0.2)LR

High, H

High Reliability, HR (6, 2, 2)LR

(0.6, 0.2, 0.2)LR

Very High, VH

Very High Reliability, VHR

(7, 2, 2)LR

(0.7, 0.2, 0.2)LR

Absolutely High, AH

Absolutely High Reliability, AHR

(8, 2, 2)LR

(0.8, 0.2, 0.2)LR

Fig. 6. Ordinary trapezoidal LR type FN

Fig. 7. A trapezoidal intuitionistic fuzzy LR type fuzzy number

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Fig. 8. Intuitionistic fuzzy LR- type Z-number

6 Conclusions Linear membership functions such as trapezoidal or triangular fuzzy membership functions are used to simplify the fuzzy operations. The membership functions of real problems are nonlinear functions such as Gaussian or exponential or Gamma functions. LR-type nonlinear fuzzy numbers have been integrated with Z-fuzzy numbers and thus the reliability to the restriction function could be considered by nonlinear functions better modeling the real events. Through the developed equations and the scale, multi-criteria decision making problems can be solved closer to the real solution. However, one limitation is the complexity of the proposed Z-fuzzy number, which makes the model harder but more realistic. For further research, we suggest the developed LR-type fuzzy Z numbers to be applied in MCDM methods such as TOPSIS, CODAS, COPRAS, WASPAS, MOORA, and PROMETHEE (Senvar et al. 2014). Besides, the other extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets or picture fuzzy sets or spherical fuzzy sets can be integrated with Z-fuzzy numbers.

References Alkan, N., Kahraman, C.: Fuzzy analytic hierarchy process using spherical Z-numbers: supplier selection application. In: Proceedings of INFUS 2022 Conference, 19–21 July 2022, Izmir, Turkey, vol. 1, pp. 702–713 (2022) Azadeh, A., Kokabi, R.: Z-number DEA: a new possibilistic DEA in the context of Z-numbers. Adv. Eng. Inform. 30(3), 604–617 (2016) Azadeh, A., Saberi, M., Atashbar, N.Z., Chang, E., Pazhoheshfar, P.: Z-AHP: a Z-number extension of fuzzy analytical hierarchy process. In: 2013 7th IEEE International Conference on Digital Ecosystems and Technologies (DEST), pp. 141–147. IEEE (2013) Chatterjee, K., Kar, S.: A multi-criteria decision making for renewable energy selection using Z-numbers in uncertain environment. Technol. Econ. Dev. Econ. 24(2), 739–764 (2018) Das, S., Dhalmahapatra, K., Maiti, J.: Z-number integrated weighted VIKOR technique for hazard prioritization and its application in virtual prototype based EOT crane operations. Appl. Soft Comput. 94, 106419 (2020) Forghani, A., Sadjadi, S.J., Moghadam, B.F.: A supplier selection model in pharmaceutical supply chain using PCA, Z-TOPSIS and MILP: a case study. PLoS ONE 13(8), e0201604 (2018) Gardashova, L.A.: Z-number based TOPSIS method in multi-criteria decision making. In: Aliev, R.A., Kacprzyk, J., Pedrycz, W., Jamshidi, Mo., Sadikoglu, F.M. (eds.) ICAFS 2018. AISC, vol. 896, pp. 42–50. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-04164-9_10

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Haktanır, E., Kahraman, C.: New Product Design Using Chebyshev’s Inequality Based IntervalValued Intuitionistic Z-Fuzzy QFD Method, accepted, INFORMATICA (2022) Haktanır, E., Kahraman, C.: Z-fuzzy hypothesis testing in statistical decision making. J. Intell. Fuzzy Syst. 37(5), 6545–6555 (2019) Haktanır, E., Kahraman, C.: A novel interval-valued Pythagorean fuzzy QFD method and its application to solar photovoltaic technology development. Comput. Ind. Eng. 132, 361–372 (2019) Jiang, S., Shi, H., Lin, W., Liu, H.C.: A large group linguistic Z-DEMATEL approach for identifying key performance indicators in hospital performance management. Appl. Soft Comput. 86, 105900 (2020) Kahraman, C., Otay, I.: Solar PV power plant location selection using a Z-fuzzy number based AHP. Int. J. Anal. Hierarchy Process 10(3) (2018) Kahraman, C., Oztaysi, B., Cevik Onar, S.: Performance comparisons of law offices and optimum allocation of debt files using Z-fuzzy AHP, EUSFLAT 2019, Chech Republic, Prague, 9–13 September 2019 (2019) Kara¸san, A., Kahraman, C.: A novel intuitionistic fuzzy DEMATEL – ANP – TOPSIS integrated methodology for freight village location selection. J. Intell. Fuzzy Syst. 36(2), 1335–1352 (2019) Liu, Y.H., Peng, H.M., Wang, T.L., Wang, X.K., Wang, J.Q.: Supplier selection in the nuclear power industry with an integrated ANP-TODIM method under Z-number circumstances. Symmetry 12(8), 1357 (2020) Mohtashami, A., Ghiasvand, B.M.: Z-ERM DEA integrated approach for evaluation of banks & financial institutes in stock exchange. Expert Syst. Appl. 147, 113218 (2020) Peng, H.G., Wang, X.K., & Wang, J.Q.: New MULTIMOORA and pairwise evaluation-based MCDM methods for hotel selection based on the projection measure of Z-numbers. Int. J. Fuzzy Syst. 1–20 (2021) Senvar, O., Tuzkaya, G., Kahraman, C.: Multi criteria supplier selection using fuzzy PROMETHEE method. In: Kahraman, C., Öztay¸si, B. (eds.) Supply Chain Management Under Fuzziness. SFSC, vol. 313, pp. 21–34. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-64253939-8_2 Shen, K., Wang, J.: Z-VIKOR method based on a new comprehensive weighted distance measure of Z-number and its application. IEEE Trans. Fuzzy Syst. 26(6), 3232–3245 (2018). https:// doi.org/10.1109/TFUZZ.2018.2816581 Tuysuz, N., Kahraman, C.: A novel Z-fuzzy AHP&EDAS methodology and its application to wind turbine selection. Infomatica (2023) Tuysuz, N., Kahraman, C.: Evaluating social sustainable development factors using multi-experts Z-fuzzy AHP. J. Intell. Fuzzy Syst. 39(5), 6181–6192 (2020) Ucal Sari, I., Kahraman, C.: Intuitionistic fuzzy Z-numbers. In: International Conference on Intelligent and Fuzzy Systems (INFUS 2020), 21–23 July 2020, Istanbul Turkey, pp. 1316–1324 (2020) Yildiz, N., Kahraman, C.: Evaluation of social sustainable development factors using buckley’s fuzzy AHP based on Z-numbers. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds.) INFUS 2019. AISC, vol. 1029, pp. 770–778. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-23756-1_92 Yildiz, N., Kahraman, C.: CODAS method using Z-fuzzy numbers. J. Intell. Fuzzy Syst. 38(2), 1649–1662 (2020) Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181(14), 2923–2932 (2011)

The Use of Z-numbers to Assess the Level of Motivation of Employees, Taking into Account Non-formalized Motivational Factors Alekperov Ramiz Balashirin1,2(B) 1 Department of Computer Engineering, Odlar Yurdu University, Baku AZ1072, Azerbaijan

[email protected] 2 Azerbaijan State Oil and Industry University, Baku AZ1010, Azerbaijan

Abstract. This work is devoted to the development of an approach to assessing the level of motivation of the company’s employees, taking into account nonformalized motivational factors. The approach is based on the application of the theory of fuzzy sets, which allows us to mathematically formulate difficult-toformalize factors affecting the process of assessing the level of employee motivation. To assess the level of motivation of the company’s employees, it is proposed to use V. Vroom’s theory of expectation in combination with the use of Z-numbers. Based on what is in the concept in. In general, all factors are evaluated with a certain probability, in this regard, they can be represented as Z–numbers. It is proved that each of the combinations of valence and the probability of the influence of valence on employee motivation corresponds to the concept of Z-numbers from the theory of fuzzy sets. This approach is also effective because the Z–numbers take into account the peculiarities of human reasoning, which is based on the experience and intuition of the company’s employees and allow for a more effective assessment of the features of the events being held. Basically, the idea is that the assessment of the level of motivation is carried out taking into account the deviation of the level of motivation desired by employees from the actual current level prevailing in the enterprise. The results obtained using this approach can be useful to managers and HR departments of companies in the formation of personnel policy, the development of a system for attracting, motivating and increasing the level of commitment of personnel. Keywords: Personnel management · Personnel motivation · Level of personnel motivation · Fuzzy set theory · Z-numbers

1 Introduction Motivation and remuneration of personnel in enterprises are key issues in the personnel management process. The motivational process requires a well-founded remuneration management system that includes policies, processes and practices for rewarding employees and takes into account the contribution, skills and competencies of personnel to the work of the enterprise. In this regard, more attention is paid to the issues of motivation, due to the fact that personnel, along with monetary interest, also show interest in the intangible component. The original version of this chapter was revised: The reference [1] has been corrected. The correction to this chapter is available at https://doi.org/10.1007/978-3-031-39774-5_90 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023, corrected publication 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 354–362, 2023. https://doi.org/10.1007/978-3-031-39774-5_42

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1.1 Brief Review of the Literature The currently existing theories of motivation can be conditionally divided into three groups in accordance with the interpretation of the phenomenon under study [1–4]. The initial theories are based on the historical experience of people’s behavior and the use of simple incentives of coercion and encouragement (material and moral) in relation to them. This group includes: Theory “X” (F. Taylor). The theory of “U” (D. McGregor). The theory of “Z” (V. Ouchi). Meaningful theories are based on the definition of internal needs that motivate people to take certain actions. This group includes: The theory of the hierarchy of needs (A. Maslow). The theory of existence, connection and growth – ERG-needs (K. Alderfer). The theory of acquired needs (D. McClelland). The theory of two factors (F. Herzberg). Procedural theories – they are based on models of human behavior, their perception and cognition. This group includes: The theory of justice (S. Adams). Porter-Lawler motivation model (L. Porter, E. Lawler). The theory of expectations (V. Vroom). Existing methods can also be divided into heuristic and mathematical methods, which can be attributed to: – Questionnaire: One of the most common methods for assessing motivation is a questionnaire. This can be done through a questionnaire containing questions about what motivates the employee, job satisfaction, and expectations from the career and work environment. These include the following methods: interview, observation, performance evaluation and testing. – Statistical Methods. There are also mathematical methods for assessing the level of employee motivation, which are based on statistical methods. Some of them include: regression analysis method, variance analysis method, cluster analysis method. Each of these methods has its own advantages and limitations, and the choice of method will depend on the specific situation and goals of motivation assessment. In general, mathematical methods can help in the systematization and analysis of data on the level of employee motivation, but their use may be limited by the availability of data and the need for special training in their use. 1.2 Actuality of the Problem An analysis of existing methods shows that in most cases it is necessary to rely on a combination of heuristic and mathematical methods. However, it is also necessary to take into account the expected probability of satisfaction from the motivational activities carried out. Our analysis revealed that Vroom’s expectancy theory [2] combines intuitions and probabilistic factors in the expectation that the motivation of employees will give the desired result and they will successfully achieve their goals only if they have an active need. That is, employees with a certain probability are sure that the efforts made and the actions taken will help to get what they want. On the other hand, the process of questioning employees is influenced by factors of uncertainty, incompleteness and insufficient confidence (certain probability) in the answers. As well as the responses of employees to the account of satisfaction with the results of the level of motivation are mostly verbal and based on the intuition of experts.

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In this regard, we propose an approach for assessing the level of motivation of employees of an enterprise, which is based on a combination of W. Vroom’s expectation theory, with fuzzy set theory [5], in particular fuzzy Z-numbers [6–8, 10, 11]. The work is structured as follows. The introduction substantiates the expediency of using Z-numbers to assess the levels of motivation. Section 2 describes basic concepts such as fuzzy sets, membership function, fuzzy numbers, Z-numbers, operations on Znumbers, distances between Z-numbers. Section 3 describes an approach for assessing the level of motivation of employees of an enterprise using Z-numbers. Some examples are given, according to which it is possible to more clearly present the formulas for calculating the level of motivation. Closing comments are included in the “Conclusion” section.

2 Basic Concepts and Definitions Definition 1. The Vroom model is built around such concepts as valence (or value), significance (or instrumentality) and expectancy, so the theory is usually called VIE (Valence, Instmmentality, Expectancy) theory. The concept of V. Vroom is based on the fact that the motivation of employees is an important factor in achieving high efficiency and success of the organization. The effectiveness of the labor activity of the staff depends on the degree of motivation, which can be determined through a formula that includes valence (V), respectively: s - money (salary), r - career, m - good attitude of the administration, l - authority among colleagues and k - preservation workplace and the parameter P—the probability of the influence of these motivational factors on the efficiency of the labor activity of the personnel. Valence mainly depends on the work experience of the individual and the degree to which his basic needs are met. It should be noted that valencies by nature are difficult to formalize factors. The essence of the concept of V. Vroom lies in the following provisions: the degree of motivation of the personnel for the high efficiency of their labor activity (E) can be calculated by the formula: E = Vs Ps + Vr Pr + Vm Pm + Vl Pl + Vk Pk

(1)

Based on the fact that in the concept of V. Vroom all factors are simultaneously evaluated with a certain probability, in this regard, they can be represented as Z-numbers. In other words, each of these VP combinations corresponds to the concept of Z-numbers [9]. According to the proposed formula V. Vroom can be described as follows: E = ws Zs + wr Zr + wm Zm + wl Zl + wk Zk ,

(2)

where (W = {ws , wr , wm , wi , wk }) - weight coefficients determining the importance of each factor in the overall procedure for assessing motivation. Definition 2. A fuzzy set [Zadeh,Aliyev, Alekperov] A is defined on a universe X may be given as: A = {x, A (x)| x ∈ X}

(3)

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where µA : X → [0; 1] is the membership function A. The membership value µA (x) describes the degree of belongingness of x ∈ X in A. Definition 3. Fuzzy numbers can be represented using trapezoidal and triangular membership functions [5, 9]. Triangular fuzzy numbers can be described by the following triple (x1 , x2 , x3 ), where the membership can be determined as the following equation.

µ(x) =

⎧ ⎪ ⎪ ⎪ 0 0 < x ≤ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x−x1 , x1 ≤ x ≤ x2 x2 −x1

⎪ x3 −x ⎪ ⎪ ⎪ x3 −x2 , x2 ≤ x ≤ x3 ⎪ ⎪ ⎪ ⎪ ⎩ 0, x3 ≤ x

(4)

Definition 4. Fuzzy Z–number. A Z-number [6] (continuous/discrete) is an ordered ˜ B) ˜ of fuzzy numbers (Fig. 1). The first component - the part A˜ expressed by a pair Z = (A, continuous/discrete fuzzy number is a restriction on the values that an indefinite variable A˜ can take on the real number axis R. The part B˜ expressed by a continuous/discrete ˜ fuzzy number is a measure of confidence or certainty of B.

Fig. 1. Graphical representation of a fuzzy Z-number

Example: X equals A with B confidence, or Salary = (high, medium confidence), Management attitude = (good, high confidence). If we express these values in the form of trapezoidal and triangular membership functions, then these Z-numbers can be represented as follows: Z salary = (4.5, 4.8, 4.9, 5.0) (0.6, 0.65, 0.75, 0.8) or Z administartion = (3, 4, 5) (0.7, 0.8,0.9). Definition 5. The product of a scalar value and a Z-number. The product of a scalar quantity k and a Z-number - Znew = k · Z, gde k  R, is determined by the formula: Znew = (k A, B)

(5)

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Definition 6. Operations on Z-numbers. Basic operations with Z-numbers are described in [7, 8]. They are based on operations performed with fuzzy numbers (NF) given by trapezoidal and triangular membership functions (PC). The results of operations on Z-numbers are also represented as Z-numbers. For example Let’s say we want to add two Z-numbers: Z1 = (A1 = {4.5, 4.8, 4.9, 5.0}, B1 = {0.6, 0.65, 0.75, 0.8}) i Z2 = (A2 = {2.0, 3.5, 4.0, 5.0}, B2 = {(0.05, 0.25, 0.35, 0.55}). Then using the formulas described in [3] the sum of these Z-numbers will be equal to: Z1+ Z2 = Z3 = (A3 = {6.0, 8.3, 8.9, 10.0}, B3 = {(0.04, 0.21, 0.30, 0.47}). Definition 7. Distance between Z-numbers. According to the method proposed in [8], the distance between fuzzy Z-numbers Z1 and Z2 , parts of which are given by trapezoidal fuzzy numbers A1 = (a11 ,a12 ,a13 ,a14 ), B1 = (b11 ,b12 ,b13 ,b14 ), A2 = (a21 ,a22 ,a23 ,a24 ), B2 = (b21 ,b22 ,b23 ,b24 ), is calculated using the following formula: ⎞ ⎛ 4 4   b1j − b2j ⎠ (6) D(Z1 , Z2 ) = 0.5 · ⎝ |a1i − a2i | + i=1

j=1

3 An Approach for Assessing the Level of Motivation of Employees of an Enterprise Using Z-Numbers To determine the overall level of motivation in an enterprise using the Vroom method using an additive formula using Z-numbers, you must perform the following steps: 1. Definition of motivation factors. Let’s say you are a department head in a company and you want to analyze the motivation of your employees. You have selected 5 factors that you think can affect employee motivation: salary (S), professional development opportunities (R), freedom to choose working methods (M), recognition of contribution to work (L) and good relations with colleagues (K). It is necessary to draw up a table that indicates five motivation factors (salary, opportunities for professional development, freedom to choose working methods, recognition of contribution to work and good relations with colleagues) and their weighting coefficients (W = {ws , wr , wm , wi , wk }), defining the importance of each factor in the overall motivation. Weighting coefficients can be determined based on a survey of employees who can predict the degree of influence of each factor on motivation. There are many methods for normalizing the responses of respondents, with the help of which it is possible to bring the weight coefficients into the interval [0..1]. 2. Questioning of employees in order to determine the desired level of motivation that satisfies the requirements of the majority of employees. Ask each employee how they assess the extent to which the above factors influence the level of motivation. Answers must be specified as Z-numbers. For example, Table 1 shows exemplary employee responses in terms of how confident they are that each of the above factors will affect motivation, where the first component of the Z-number of each of the factors

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(S, R, M, L, K) is represented as a trapezoid fuzzy number and can get the following values: A = {NI = “not important”, LI = “low importance”, I = “important”, VI = “very important”} ili A = {DS = “dissatisfied”, LS = “low satisfaction”, AS = “average satisfaction”, GS = “good satisfaction”, VS = “very satisfied”}. The second component of Z-numbers is also represented as a trapezoidal fuzzy number and the following values are obtained: B = { LC = “low confidence”, WC = “weak confidence”, GC = “good confidence”, HC = “high confidence”}.

Table 1. Experts’ opinions on the influence of factors on the level of motivation. Factors

№

Weight coefficients (W)

Employee 1 (ZI∗1 ,

Employee 2 (ZI∗2 ,

Employee 3 (ZI∗3 ,

Employee 4 (ZI∗4 ,

Employee 5 (ZI∗5 ,

i = 1..5)

i = 1..5)

i = 1..5)

i = 1..5)

i = 1..5)

S

1

w1

(I,HC)

(I,HC)

(VI,HC)

(I,HC)

(I,HC)

R

2

w2

(I,HC)

(VI, GC)

(I,HC)

(I,HC)

(I,HC)

L

3

w3

(I, GC)

(I,HC)

(I,HC)

(I,HC)

(I,HC)

M

4

w4

(VI,HC)

(VI,HC)

(VI, GC)

(VI,HC)

(VI, GC)

K

5

w5

(I,HC)

(VI,HC)

(I,HC)

(VI, GC)

(VI, GC)

∗1 Edesired

∗2 Edesired

∗3 Edesired

∗4 Edesired

∗5 Edesired

Formula Evaluation 2 ∗ Edesired

∗i 3. Find the desired solution (level of motivation) that satisfies all employees - Edesired . According to Vroom’s formula, the level of motivation is determined by the answers of each employee. To do this, it is necessary for each employee to multiply the assessment of the level of motivation for each factor, presented in the form of a Znumber, by the appropriate weighting factor and add the results obtained for all five factors. Thus, the amount of motivation for each employee is calculated as follows:

E*i desired =

5 5  

wj Z*i i , i = 1..5, j = 1..5

(7)

i=1 j=1

where i is the number of employees, j is the number of motivation factors. According to definition 5, the average value is found - the desired value of the level of motivation, satisfied by all:

E ∗i 1 ∗ ∗i ∗ A∗i = desired = , B (8) Edesired desired desired i = 1..5 n n 4. Questioning of employees in order to determine the real level of motivation. At this stage, a survey is conducted among employees in order to determine how satisfied they are with the current level of motivation in the enterprise. For example,

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the following table (Table 2) shows the approximate answers of employees to what extent they assess the current state of the level of motivation and how confident they are in their answers. Table 2. Employee Opinions. Assessment of the current level of motivation. Factors

№

Weight coefficients (W)

Employee 1 ZIr1

Employee 2 ZIr2

Employee 3 ZIr3

Employee 4 ZIr4

Employee 5 ZIr15

S

1

w1

(AS, GC)

(AS, GC)

(VS,HC)

(VS, HC)

(VS, GC)

R

2

w2

(AS,HC)

(VS, GC)

(VS,HC)

(VS,HC)

(AS,HC)

L

3

w3

(LS,HC)

(VS,HC)

(AS,HC)

(LS,HC)

(VS,HC)

M

4

w4

(VS, HC)

(LS,HC)

(VS,HC)

(GS,HC)

(LS,HC)

K

5

w5

(VS, GC)

(VS,GC)

(GS, GC)

(LS, GC)

(VS,HC)

1 Ereally

2 Ereally

3 Ereally

4 Ereally

5 Ereally

Formula 2 score

i Further, for each employee, the assessment of the real level of motivation Ereally ,i

= 1..5, according to formula 2, is calculated as follows: i Ereally =

5  5 

wj Zrii , i = 1..5, j = 1..5

(9)

i=1 j=1 ∗ 5. The distance between the desired solution Edesired and the assessments of each i employee of the real level of motivation Ereally The distance between them is calculated according to definition 7. 6. Clustering. Classes can be defined like this:   i ∗ i Edesired ≤ , Ereally Classes = {Class 1: “high level difference” if 0 < Dreally   i ∗ i Edesired ≤ 0.5”; 0.3”; Class 2: “medium level difference” if 0.3 < Dreally , Ereally   i Class 3: “close levels” if 0.5 < Dreally E∗ ≤ 0.7”; Class 4: “very close , Ei   desired really i ∗ i levels” if 0.7 < Dreally Edesired , Ereally ≤ 1.0”}. Employee responses are classified according to the following formula:   i ∗ i   Dreally Edesired , Ereally i ∗ i  ∗  , Dreally Edesired = , Ereally (10) i Edesired , E0∗ Dreally

where E0∗ is the assessment of the absence or the lowest level of motivation, which is  

i ∗ i Edesired , Ereally also determined similarly to the desired level of motivation.. Dreally   i ∗ Edesired and Dreally , E0∗ is determined based on formula 6. This formula allows you to determine the difference in opinion employees, which are defined in the interval [0..1].

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7. Ranking of classes by the number of employees whose opinions correspond to the classes defined in step 6. For example, suppose the following values are obtained after the corresponding calculations: The number of employees whose opinions correspond to the classes: Class1 = 0; Class2 = 1; Class3 = 3; Class4 = 0. Determination of the degree of deviation from the desired level of motivation. To determine the degree of deviation from the desired level of motivation, it is necessary, after ranking, to take the class with the largest number of employees and compare the result with the desired level of motivation. Let’s say after ranking we got that Class3 has the largest number of employees. Then we can say that the desired level of motivation and the actual level of motivation are close. What can we say about the fact that a large number of employees are approximately 70% satisfied with the level of motivation carried out in the enterprise.

4 Conclusion Thus, the use of Z-numbers, when assessing the level of motivation based on the Vroom theory, allows you to determine the overall level of motivation in the enterprise, taking into account the measure of confidence or certainty, which is expressed as a function of the probability of the influence of these motivation factors on the effectiveness of the work of the staff. But unlike Vroom’s theory of multiplying the valency values of motivation factors by the probability of their influence, their combinations are used in the form of Z-numbers, which more naturally takes into account the peculiarities of human reasoning. Instead of the usual multiplication of valency and probability used in Vroom’s theory, operations are performed according to operations performed on Znumbers. This approach is based on the fact that, taking into account the peculiarities of human reasoning, based on the experience and intuition of the employees of the enterprise, to assess the features of ongoing activities. This approach can be used by the personnel department of enterprises, as well as marketing centers and marketers. However, for the full use of this method by personnel departments, it is necessary to conduct additional research on the creation of the necessary program modules and the selection and formalization of motivational factors.

References 1. Lambovska, M.: A fuzzy logic model for evaluating the motivation for high-quality publications: evidence from a Bulgarian University. Manag. J. Contemp. Manag. Issues 27(2), 87–108 (2022). https://doi.org/10.30924/mjcmi.27.2.6 2. Vroom, V.H., Deci, E.L.: Management and motivation/V.H. Vroom, E. L. Deci. - Harmondsworth (Mx); Baltimore (Md): Penguin Books, 399 p. (1970) 3. Wigfield, A.: Expectancy-value theory of achievement motivation: a development perspective. Educ. Psychol. 6(1), 49–78 (1994) 4. Herzberg, F.: One more time: how do you motivate employees? Harvard Bus. Rev. 81(1), 87–96 (2003) 5. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 6. Zadeh, L.A.: Note on Z-numbers. Inf. Sci. 181(14), 2923–2932 (2011)

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7. Aliev, R.A., Huseynov, O.H., Aliyev, R.R., Alizadeh, A.V.: The arithmetic of Z-numbers. Theory and Applications Singapore, p. 316. World Scientific (2015) 8. Aliev, R.A., Pedrycz, W., Huseynov, O.H., Eyupoglu, S.Z.: Approximate reasoning on a basis of Z-number-valued if–then rules. IEEE Trans. Fuzzy Syst. 25(6), 1589–1600 (2017) 9. Balashirin, A.R.: The use of fuzzy numbers for the rational choice of the structure of the distribution channel of goods. In: Aliev, R.A., Kacprzyk, J., Pedrycz, W., Jamshidi, M., Babanli, M.B., Sadikoglu, F. (eds.) ICAFS 2022. LNNS, vol. 610, pp. 626–633. Springer, Cham (2023) 10. Alekperov, R.: Solution of a multi-criteria problem of choosing the location of retail outlet using z-numbers. IOSR J. Comput. Eng. 23, 43–47 (2021). https://doi.org/10.9790/0661-230 4024347 11. Alekperov, R.: Using fuzzy Z - numbers when processing flexible queries. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) INFUS 2022. LNNS, vol. 504, pp. 435–444. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5_52

Literature Review

A Literature Review on Fuzzy ELECTRE Methods Beril Akkaya1,2(B)

and Cengiz Kahraman1

1

2

Department of Industrial Engineering, Istanbul Technical University, Macka, Besiktas, 34367 Istanbul, Turkey {akkayab20,kahramanc}@itu.edu.tr, [email protected] Department of Industrial Engineering, Middle East Technical University, Cankaya, Ankara 06800, Turkey

Abstract. Decision-making is one of the crucial processes in all operations. As the concept comprises many complexities and impreciseness, there is a need for systematic approaches. This need drives many multicriteria decision-making (MCDM) methodologies in the literature. The outranking methods are one of the classifications of these methodologies, which utilizes binary comparisons to rank the alternatives. Elimination and Choice Translating Reality (ELECTRE) is one of the most useful outranking MCDM approaches. Since most decision-making processes require linguistic decision terms and the fuzzy sets are generally accepted as great expressers of linguistic terms, researchers have offered various fuzzy extensions of MCDM methods and ELECTRE methods. This study aims to provide a comprehensive literature review regarding the fuzzy extensions of the ELECTRE methods. Keywords: Multi-criteria decision-making (MCDM) · Outranking methods · ELECTRE · Fuzzy sets · Extensions · Literature review

1

Introduction

MCDM indicates the decision-making process while dealing with complex situations comprising multiple conflicted criteria. MCDM methods are systematic quantitative instruments that increase decision-makers’ capability to handle complex objective and subjective criteria [18]. They serve as a practical framework, improving decision accuracy while reducing cost and time [6]. The classical MCDM method can be classified as hierarchical approaches, multi-objective mathematical programming, and outranking methods. The outranking methods evaluate alternatives based on outranking relations. The two main procedures of the methods are pairwise comparisons of the criteria to build outranking relations and utilization of these outranking relations to get the final decision [6]. There are various outranking methods such as ELECTRE methods, PROMETHEE methods, ORESTE, MAPPAC, and TACTIC [18]. Among these outranking methods, ELECTRE is the first introduced one and still the most powerful in representation [21]. It allows decision-makers to c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 365–375, 2023. https://doi.org/10.1007/978-3-031-39774-5_43

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take the most advantageous action among others with maximum advantage and minimum conflict while considering various criteria [6]. There is more than one type of ELECTRE method in the literature designated for different decision problems. While ELECTRE I best suits selection problems, ELECTRE II, III, and IV are designed for ranking problems, and ELECTRE TRI for assignment problems [32].

Fig. 1. Subject areas of the ELECTRE methods publications.

Fig. 2. Frequencies of the ELECTRE methods publications over years.

As the method is practical for more than one type of problem, the methods of the ELECTRE family are the most widely used outranking methods [12]. There are many studies in the literature that covers various subject areas. Some

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examples of recent studies are risk evaluation [2], prioritization of public transport systems [25], quality assessment [41], optimization of water supply systems [38], and sustainable supplier selection [30]. Figure 1 shows the distribution of subject areas of publications on the ELECTRE methods, while Fig. 2 shows the frequencies over the years. The purpose of this research is to present a comprehensive literature review on the fuzzy extensions of the ELECTRE methods. The rest of the paper is organized as follows. First, a detailed literature review on fuzzy extensions of ELECTRE methods is given in section two. In section three, we comment on how the future of ELECTRE methods can be shaped. Finally, section four concludes the paper with suggestions that may be a guide for the future directions.

2

Fuzzy ELECTRE Methods

For most decision-making approaches, using crisp numbers for evaluation can be considered inefficient, especially when using linguistic assessment terms, as human thoughts can be imprecise. Zadeh [49] introduced the fuzzy set theory to the literature in 1965 to deal with such problems, and today there are more than 20 extensions of fuzzy sets [24]. According to the fuzzy set theory, an element can partially belong to a fuzzy set, while membership of an element is binary for crisp sets [49]. Thus, fuzzy sets provide a broader range of frames than classical sets, and they are convenient for projecting problems more realistically [8]. In the literature, there can be found many studies that combine different types of fuzzy sets with decision-making approaches. As ELECTRE is one of the methods that make use of linguistic evaluation terms, various fuzzy extensions of the method have been studied. 2.1

Ordinary Fuzzy ELECTRE Methods

The ordinary fuzzy ELLECTRE method is developed for dealing with the ambiguity and imprecision of decision-making, helping decision-makers during the assignment of weights to the criteria. As a result, DMs can deal with a broader range of perspectives on weights rather than assigning crisp numbers [8]. There exist numerous studies on extensions of ELECTRE based on ordinary fuzzy sets. For example, Sevkli [52] proposed the fuzzy ELECTRE I method and its application to a supplier selection problem. In addition, Hatami-Marbini and Tavana [22] extended the ELECTRE I to trapezoidal fuzzy sets as a method for group decision-making. Mohamadghasemi et al. [35] developed a triangular fuzzy ELECTRE III group decision-making method integrated with the fuzzy weighted average approach. Noori et al. [38] proposed triangular fuzzy ELECTRE III to solve an optimization problem of water supply selection. 2.2

Type-2 Fuzzy ELECTRE Methods

Type-2 fuzzy sets differ from type-1 fuzzy sets in terms of their membership functions. The membership functions of type-2 fuzzy sets, namely the secondary

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membership function, are also fuzzy. This fact increases the fuzziness of the sets, giving more space for vagueness [12]. Interval type-2 fuzzy sets (IT2FS) are the type-2 fuzzy sets whose secondary membership function equals 1 [12]. As the computational process of the IT2FS is more straightforward, there are extensions of the ELECTRE methods based on this type of type-2 fuzzy sets in the literature. Chen [12] introduced an ELECTRE-based outranking group decisionmaking method based on IT2FS. She developed signed distances in IT2FS and compared pairs of alternatives based on their signed distances. Finally, she applied the proposed method to a supplier selection problem and proved the method’s effectiveness by a comparative analysis. C ¸ elik et al. [11] extended the ELECTRE to IT2FS as a multi-criteria decision-making method. They apply the proposed method to a green logistic provider selection problem. Another extension is proposed by Zhong and Yao [50] as a multi-criteria group-decision methodology based on ELECTRE I and IT2FS. An application to a supplier selection problem is also provided. Selvaraj and Jeon [42] extended the ELECTRE III method to trapezoidal IT2FS. They used their method to evaluate OECD countries based on innovational capabilities. Ayyıldız et al. [9] used the trapezoidal IT2FS ELECTRE method with AHP (analytical hierarchy process) for individual credit ranking evaluation. They developed trapezoidal IT2FS AHP to evaluate criteria, then ranked the experts using trapezoidal IT2FS ELECTRE. 2.3

Intuitionistic Fuzzy ELECTRE Methods

Intuitionistic fuzzy sets (IFS) were first introduced by Atanassov in 1986, and some of the operations of the sets are defined [7]. The intuitionistic fuzzy ELECTRE (IF ELECTRE) is first developed by Ming-Che and Chen [45]. They defined three types of concordance and discordance sets based on the accuracy function, score function, and intuitionistic index and developed concordance and discordance matrices using these sets. They introduced the procedures of the method in detail and illustrated two hypothetical examples. A new extension of the IF ELECTRE method is proposed by Vahdani et al. [43] as a multi-criteria group decision-making method. They defined a novel intuitionistic discordance index derived from fuzzy distance measure in their methodology. They applied the introduced method to the selection problem of flexible manufacturing systems for a company. They also provide a comparative analysis. Another IF ELECTRE group decision-making method is introduced by Devi and Yadav [16] and applied to a plant location selection problem. Further, Xu and Shen [47] proposed an IF ELECTRE by developing a novel IF entropy measure. They provide an application example on supplier selection. Ebrahimnejad et al. [17] extended the IF ELECTRE by introducing novel indexes. Apply method to evaluation of outsourcing providers. There are many other studies to extend the methods on areas such as supplier risk assessment [20], site selection for offshore wind powers [46], job selection [13], carbon emission strategy selection [37], and evaluation of investments in green flight activities [5].

A Literature Review on Fuzzy ELECTRE Methods

2.4

369

Pythagorean Fuzzy ELECTRE Methods

Pythagorean fuzzy sets (PFS) were proposed by Yager [48]. He defined PFS, developed operations within the sets, and compared the novel sets with intuitionistic fuzzy sets. He claimed that the PFS are more efficient in representing human thinking as their membership functions can represent a broader range of space. Akram et al. [1] presented a new ELECTRE method based on PFS, called Pythagorean fuzzy ELECTRE (PF ELECTRE). They mentioned that their motivation to extend the ELECTRE method on PFS arose from the fact that the representation space on PFS is greater than IFS and their more extensive capability of compensating uncertainty in decision-making problems. In the study, they developed a novel method, clarified the procedures, and provided two illustrative examples of health safety and environmental management. Other studies are extending ELECTRE methods on secondary products of PFS, such as interval-valued Pythagorean fuzzy sets [14] and hesitant Pythagorean fuzzy sets [2,3]. Furthermore, there are different application studies of the methods in different areas. Some examples are financial decision-making [14], risk evaluation [2], and selection of an e-waste disposal system [40]. 2.5

Picture Fuzzy ELECTRE Methods

Cuong [15] introduced Picture fuzzy sets (PiFS) in 2014. He described PiFS as direct extensions of the fuzzy and intuitionistic fuzzy sets with three parameters. The PiFS were first used with the ELECTRE method by Liang et al. [29], not as a proposed PiFS ELECTRE method, but as an integration of evaluation based on distance from average solution (EDAS) with the ELECTRE method. They extended this integrated method to PiFS as the sets contribute decision-making process by increasing the capability of representing uncertainty, as the sets are composed of three parameters. They applied the proposed integrated method to evaluate cleaner production levels. Further, Liang et al. [28] introduced an integrated TODIM and ELECTRE method to evaluate the performance of Green Mine in China. Kumar et al. [19] extended PiFS sets by introducing a new picture fuzzy entropy measure and developed an integrated TODIM and ELECTRE method based on this novel extension. They also introduce an application for a supplier selection problem. There are only five publications referring PiFS ELECTRE methods from 2018 to 2023. 2.6

Spherical Fuzzy ELECTRE Methods

Kutlu G¨ undo˘ gdu and Kahraman [27] defined spherical fuzzy sets (SFS) and basic arithmetic operations in addition to the defuzzification operation. The spherical fuzzy ELECTRE method was proposed by Menek¸se and Camg¨oz Akda˘g [33] in 2022. They propose two different approaches to extend the ELECTRE method on spherical fuzzy sets. In the first approach, they provided spherical fuzzy strong, weak, and moderate concordance and discordance sets, then

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compared concordance and discordance indexes of alternatives by the proposed formulation based on these sets. In the second approach, on the other hand, they used single types of concordance and discordance sets to compare alternatives. In their study, they also proposed an interval-valued spherical fuzzy ELECTRE method. They also illustrated the novel methods by applying the procedures on internal audit planning of an organization. Furthermore, Menek¸se and Camg¨oz Akda˘g [34] proposed an integrated AHP ELECTRE method based on spherical fuzzy sets. As a first step, they used the AHP method to determine the weights of the criteria with linguistic terms that correspond to spherical fuzzy numbers. After alternatives were evaluated by decision-makers, they obtained a spherical fuzzy decision matrix. Then they applied the ELECTRE method to rank the alternatives based on obtained spherical fuzzy decision matrix. Finally, they apply their model to a higher education institution’s IT management assessment. Akram et al. [4] introduced another two approaches as integrated spherical fuzzy AHP ELECTRE method, focusing mainly on group decision-making processes. They briefly mentioned that the proposed SF ELECTRE method by Menek¸se and Camg¨oz Akda˘g [33] needs to be revised for group decision-making. In the study, they used AHP to decide criteria weights; then, they proposed SF ELECTRE I and SF ELECTRE II procedures by extending the used structure of spherical fuzzy sets on the ELECTRE method to handle the group decisionmaking process. Besides, they applied these two novel methods separately to a real case of the public transportation problem in Turkey. 2.7

Neutrosophic Fuzzy ELECTRE Methods

Ji et al. [23] introduced an integrated MABAC (multi-attribute border approximation area comparison) ELECTRE method that relies on neutrosophic fuzzy sets. They suggested that MABAC is based on the assumption that criteria are complementary; however, the integration of two methods allows the consideration of non-complementary criteria in their outsourcing provider selection problem. First, they used the MABAC method to obtain weighted decision and difference matrixes in a neutrosophic fuzzy decision environment. Then, after the steps of MABAC, they introduce the ELECTRE method to obtain neutrosophic fuzzy concordance and discordance matrices. Finally, they rank the alternatives according to alternatives based on these matrices. Ma et al. [31] proposed an interval neutrosophic fuzzy ELECTRE III method. They developed the novel method to evaluate cloud services based on their trustworthiness to recommend to small and medium-sized enterprises. 2.8

Q-Rung Orthopair Fuzzy ELECTRE Methods

Pınar and Boran [39] developed the q-rung orthopair fuzzy (q-ROF) ELECTRE method. They mentioned that q-ROF sets to increase the flexibility of the decision-making process in comparison to other fuzzy sets. They provide a new distance measure with q-ROF sets and integrate it into their novel ELECTRE

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method. Furthermore, they formulated strong, moderate, and weak concordance and discordance matrices to get the aggregate dominance matrix. They further suggest that if the ELECTRE can not eliminate all the alternatives, the qROF TOPSIS (technique for order preference by similarity to an ideal solution) method, also introduced in the same study, can be integrated to overcome the problem. The study aims to develop new methods for supplier selection problems, and they applied their novel method to a case of a selecting supplier for a construction company. Bao and Shi [10] proposed a new linguistic q-ROF ELECTRE method to solve multi-attribute decision-making problems. First, they developed a new distance measure based on q-ROF numbers to evaluate the weights of attributes. Then apply q-ROF ELECTRE to the robot selection problem of an energy company. Narayanamoorthy et al. [36] introduce an interval-valued q-ROF ELECTRE III as a multi-attribute group decision-making method. They describe the procedure in detail and present the method as an efficient approach to solving solid waste management problems during an emergency, with the example of the COVID-19 pandemic. 2.9

T-Spherical Fuzzy ELECTRE Methods

Only one study on T-spherical fuzzy (T-SF) sets with ELECTRE methods exists in the literature. Wang and Chen [44] proposed the T-SF ELECTRE method and claimed that this set is differentiated from other fuzzy sets in terms of the representation capability of uncertainty. They argued some recently developed T-SF distance measures and score functions for their effectiveness in the sphere of the ELECTRE method. They proposed the T-SF ELECTRE I method with the use of concordance, discordance, and prioritization Boolean matrices, and they evolved the method to a T-SF ELECTRE II with two novel T-SF Minkowski distance measures. Finally, they presented an application case for the investment evaluation of a company and a comparative analysis to show their method’s validity. 2.10

Fermatean Fuzzy ELECTRE Methods

Fermatean fuzzy ELECTRE is first introduced by Zhou et al. [51]. They developed a new distance measure for fermatean fuzzy numbers relying on JensenShannon divergence and three novel outranking relationship measures. Then, they introduced strong, medium, and weak concordance and discordance matrices based on the fermatean fuzzy sets. Next, they apply their method to a site selection problem considering a local type of mobile field hospital during the COVID-19 pandemic in Wuhan. Lastly, they proved their method’s superiority over the fermatean fuzzy TOPSIS and MABAC methods in terms of maintaining vivid and optimal ranking orders. Further, Kiri¸s¸ci et al. [26] developed another fermatean fuzz ELECTRE I method to solve group decision-making problems. Based on the score and accuracy functions approach, they defined fermatean fuzzy strong, medium, and weak

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concordance and discordance sets. Finally, they obtained the dominant alternative by using their method on the biomedical material selection problem.

3

Future of ELECTRE Methods

Fig. 3 demonstrates the frequency of studies on crisp and fuzzy ELECTRE methods over the years. These graph clearly shows that the researches on ELECTRE methods have an increasing trend. Therefore, it is reasonable to interfere that both crisp and fuzzy methods will be studied widely in the future. Besides, according to recent studies, the fuzzy extensions of ELECTRE methods have expanded their portion in the general studies on ELECTRE methods. Thence, further studies on ELECTRE methods are likely to exploit extensions of fuzzy sets more.

Fig. 3. Frequencies of the ELECTRE methods publications over years.

4

Conclusion

The ELECTRE method family is one of the most utilized outranking MCDM methods. There exist many extensions of the methods in the literature, yet, future researches will reveal new ones. Furthermore, the studies on fuzzy sets have expanded steadily, and new extensions of the fuzzy sets are very likely to be explored. Hence, we may also expect the new fuzzy extensions of ELECTRE methods in the future, as the fuzzy sets thoroughly represent linguistic decision terms. Future studies may consider extending fuzzy ELECTRE methods by utilizing the secondary products of the existing fuzzy sets as well as the prospective ones to broaden the scope of linguistic expression representation.

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References 1. Akram, M., Ilyas, F., Garg, H.: Multi-criteria group decision making based on ELECTRE i method in Pythagorean fuzzy information. Soft. Comput. 24(5), 3425–3453 (2019). https://doi.org/10.1007/s00500-019-04105-0 2. Akram, M., Luqman, A., Alcantud, J.C.: An integrated ELECTRE-i approach for risk evaluation with hesitant Pythagorean fuzzy information. Exp. Syst. Appl. 200, 116945 (2022) 3. Akram, M., Luqman, A., Kahraman, C.: Hesitant Pythagorean fuzzy ELECTREii method for multi-criteria decision-making problems. Appl. Soft Comput. 108, 107479 (2021) 4. Akram, M., Zahid, K., Kahraman, C.: Integrated outranking techniques based on spherical fuzzy information for the digitalization of transportation system. Appl. Soft Comput. 134, 109992 (2023) 5. Aksoy, T., Y¨ uksel, S., Dincer, H., Hacioglu, U., Maialeh, R.: Complex fuzzy assessment of green flight activity investments for sustainable aviation industry. IEEE Access PP, 1 (2022) 6. Asghari, F., Amidian, A.A., Muhammadi, J., Rabiee, H.: A fuzzy ELECTRE approach for evaluating mobile payment business models, pp. 351 – 355 (2010) 7. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 8. Aytac, E., IS ¸ IK, A.T., Kundakci, N.: Fuzzy ELECTRE i method for evaluating catering firm alternatives. Ege Acad. Rev. 11(5), 125–134 (2011) 9. Ayyildiz, E., Taskin Gumus, A., Erkan, M.: Individual credit ranking by an integrated interval type-2 trapezoidal fuzzy Electre methodology. Soft. Comput. 24(21), 16149–16163 (2020). https://doi.org/10.1007/s00500-020-04929-1 10. Bao, H., Shi, X.: Robot selection using an integrated MAGDM model based on ELECTRE method and linguistic q-rung Orthopair fuzzy information. Math. Prob. Eng. 2022, 1–13 (2022) 11. Celik, E., Taskin, A., Erdo˘ gan, M.: A new extension of the ELECTRE method based upon interval type-2 fuzzy sets for green logistic service providers evaluation. J. Testing Eval. 44, 20140046 (09 2016) 12. Chen, T.Y.: An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets. Inf. Sci. 263, 1-21 (2014) 13. Chen, T.Y.: An IVIF-ELECTRE outranking method for multiple criteria decisionmaking with interval-valued intuitionistic fuzzy sets. Technol. Econ. Develop. Econ. 22, 1–37 (2015) 14. Chen, T.Y.: An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its application to financial decision making. Appl. Soft Comput. 71, 460–487 (2018) 15. Cuong, B.C., Kreinovich, V.: Picture fuzzy sets. J. Comput. Sci. Cybern. 30(4), 409–420 (2014) 16. Devi, K., Yadav, S.: A multicriteria intuitionistic fuzzy group decision making for plant location selection with ELECTRE method. Int. J. Adv. Manufact. Technol. 66, 1219–1229 (2012). https://doi.org/10.1007/s00170-012-4400-0 17. Ebrahimnejad, S., Hashemi, H., Mousavi, S., Vahdani, B.: A new interval-valued intuitionistic fuzzy model to group decision making for the selection of outsourcing providers. Econ. Comput. Econ. Cybern. Stud. Res./Acad. Econ. Stud. 49, 1–22 (2015) 18. Fahmi, A., Kahraman, C., Bilen, .: ELECTRE i method using hesitant linguistic term sets: an application to supplier selection. Int. J. Comput. Intell. Syst. 9, 153–167 (2016)

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19. Goel, S., Arya, V., Kumar, S., Dahiya, A.: A new picture fuzzy entropy and its application based on combined picture fuzzy methodology with partial weight information. Int. J. Fuzzy Syst. 24, 3208–3225 (2022) 20. Govindan, K., Jepsen, M.: Supplier risk assessment based on trapezoidal intuitionistic fuzzy numbers and ELECTRE TRI-C: a case illustration involving service suppliers. J. Oper. Res. Soc. 67, 339–376 (2015) 21. Govindan, K., Jepsen, M.B.: ELECTRE: a comprehensive literature review on methodologies and applications. Eur. J. Oper. Res. 250(1), 1–29 (2016) 22. Hatami-Marbini, A., Tavana, M.: An extension of the ELECTRE i method for group decision-making under a fuzzy environment. Omega 39, 373–386 (2010) 23. Ji, P., Zhang, H.Y., Wang, J.: Selecting an outsourcing provider based on the combined MABAC-ELECTRE method using single-valued neutrosophic linguistic sets. Comput. Indust. Eng. 120, 429–441 (2018) ¨ 24. Kahraman, C., Oztay¸ si, B., Otay, I., C ¸ evik, S.: Extensions of ordinary fuzzy sets: a comparative literature review, pp. 1655–1665 (2021) ¨ ¨ 25. Kalifa, M., Ozdemir, A., Ozkan, A., Banar, M.: Application of multi-criteria decision analysis including sustainable indicators for prioritization of public transport system. Integr. Environ. Assess. Manage. 18, 25–38 (2021) 26. Kiri¸sci, M., Demir, I., Simsek, N.: Fermatean fuzzy ELECTRE multi-criteria group decision-making and most suitable biomedical material selection. Artif. Intell. Med. 127, 102278 (2022) 27. Kutlu G¨ undo˘ gdu, F., Kahraman, C.: Spherical fuzzy sets and spherical fuzzy topsis method. J. Intell. Fuzzy Syst. 36, 1–16 (2018) 28. Liang, W., Bing, D., Zhao, G., Wu, H.: Performance evaluation of green mine using a combined multi-criteria decision making method with picture fuzzy information. IEEE Access PP, 1 (2019) 29. Liang, W., Zhao, G., Luo, S.Z.: An integrated EDAS-ELECTRE method with picture fuzzy information for cleaner production evaluation in gold mines. IEEE Access PP, 1 (2018) 30. Lu, H., Jiang, S., Song, W., Ming, X.: A rough multi-criteria decision-making approach for sustainable supplier selection under vague environment. Sustainability 10, 2622 (2018) 31. Ma, H., Zhuoxuan, H., Zhang, X., Zhang, H., Wang, J.: Cloud service recommendation for small and medium-sized enterprises: a context-aware group decision making approach. J. Intell. Fuzzy Syst. 42, 1–21 (2021) 32. Marzouk, M.M.: ELECTRE III model for value engineering applications. Autom. Constr. 20(5), 596–600 (2011) 33. Menek¸se, A., Camgoz-Akdag, H.: Internal audit planning using spherical fuzzy ELECTRE. Appl. Soft Comput. 114, 108155 (2021) 34. Menek¸se, A., Camgoz-Akdag, H.: Information technology governance evaluation using spherical fuzzy AHP ELECTRE, pp. 757–765 (2022) 35. Mohamadghasemi, A., Hadi-Vencheh, A., Lotfi, F., Khalilzadeh, M.: An integrated group FWA-EELECTRE III approach based on interval type-2 fuzzy sets for solving the MCDM problems using limit distance mean. Complex Intell. Syst. 6, 130 (2020) 36. Narayanamoorthy, S., et al.: Assessment of the solid waste disposal method during COVID-19 period using the ELECTRE III method in an interval-valued q-rung orthopair fuzzy approach. Comput. Model. Eng. Sci. 131, 1229–1261 (2022) 37. Niu, X., Y¨ uksel, S., Dincer, H.: Emission strategy selection for the circular economy-based production investments with the enhanced decision support system. Energy 274, 127446 (2023)

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38. Noori, A., Bonakdari, H., Salimi, A., Gharabaghi, B.: A group multi-criteria decision-making method for water supply choice optimization. Socio-Econ. Plann. Sci. 77, 101066 (2020) 39. Pinar, A., Boran, F.E.: A q-rung orthopair fuzzy multi-criteria group decision making method for supplier selection based on a novel distance measure. Int. J. Mach. Learn. Cybern. 11(8), 1749–1780 (2020). https://doi.org/10.1007/s13042020-01070-1 40. Ramya, L., Narayanamoorthy, S., Thangaraj, M., Samayan, K., Kang, D.: An extension of the hesitant pythagorean fuzzy ELECTRE III: techniques for disposing of e-waste without any harm. Appl. Nanosci. 13, 1939–1957 (2022) 41. Rocha, A., Costa, A., Figueira, J., Ferreira, D., Marques, R.: Quality assessment of the portuguese public hospitals: a multiple criteria approach. Omega 105, 102505 (2021) 42. Selvaraj, G., Jeon, J.: Assessment of national innovation capabilities of OECD countries using trapezoidal interval type-2 fuzzy ELECTRE III method. Data Technol. Appl. 55, 0154 (2020) 43. Vahdani, B., Mousavi, S., Tavakkoli-Moghaddam, R., Hashemi, H.: A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment. Appl. Math. Modell. 37, 1781-1799 (2013) 44. Wang, J.C., Chen, T.Y.: A T-spherical fuzzy ELECTRE approach for multiple criteria assessment problem from a comparative perspective of score functions. J. Intell. Fuzzy Syst. 41, 1–20 (2021) 45. Wu, M.C., Chen, T.Y.: The ELECTRE multicriteria analysis approach based on Atanassov’s intuitionistic fuzzy sets. Expert Syst. Appl.- ESWA 38, 12318–12327 (2011) 46. Wu, Y., Zhang, J., Yuan, J., Geng, S., Zhang, H.: Study of decision framework of offshore wind power station site selection based on ELECTRE-III under intuitionistic fuzzy environment: a case of china. Energy Convers. Manage. 113, 66–81 (2016) 47. Xu, J., Shen, F.: A new outranking choice method for group decision making under Atanassov’s interval-valued intuitionistic fuzzy environment. Knowl.-Based Syst. 70, 177–188 (2014) 48. Yager, R.: Pythagorean fuzzy subsets, pp. 57–61 (2013) 49. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 50. Zhong, L., Yao, L.: An ELECTRE I-based multi-criteria group decision making method with interval type-2 fuzzy numbers and its application to supplier selection. Appl. Soft Comput. 57, 556–576 (2017) 51. Zhou, L.P., Wan, S.P., Dong, J.Y.: A Fermatean fuzzy ELECTRE method for multi-criteria group decision-making. Informatica 33, 1–44 (2021) 52. S ¸ evkli, M.: An application of the fuzzy ELECTRE method for supplier selection. Int. J. Product. Res. 48, 3393–3405 (2010)

Evolution of Fuzzy Sets: A Comprehensive Literature Review Murat Gülbay1(B)

and Cengiz Kahraman2

1 University of Gaziantep, Gaziantep, Turkey

[email protected]

2 Istanbul Technical University, Istanbul, Turkey

[email protected]

Abstract. Introduced by Zadeh (1965), ordinary fuzzy sets theory (OFS) has been widely used in many fields to handle uncertainty. Zadeh represented ordinary fuzzy sets with singletons with some elements whose degrees of membership of each element to the set is defined as [0,1] and non-membership of the elements represented by the complement of the membership degree to 1. The complementary feature of ordinary fuzzy sets was criticized by various researchers on the basis of lining up with the necessity that membership degree of an element should be also fuzzy and redundancy for the complementary feature of OFS. Based on these two criticisms, ordinary fuzzy sets have been successfully extended by many researchers to describe membership functions with more details using up to three parameters. This study deals with the evolution of fuzzy sets and their attraction in the literature based on the document statistics. Keywords: Fuzzy sets · Ordinary fuzzy sets · Extension of ordinary fuzzy sets

1 Introduction Ordinary fuzzy sets (OFS) introduced by Zadeh (1965) are represented with singletons with some elements and their degrees of membership to the set [1]. The non-membership of this element is the complement of the membership degree to 1. This complementary feature of ordinary fuzzy sets was criticized by various researchers. According to these researchers, the membership degree of an element should be also fuzzy and there should be no necessity for the complementary feature in OFS. Thus, by adding the fuzziness to membership degrees and/or by removing the complementary feature, OFS have been extended to several new extensions to describe membership functions with more details. In this paper evolution of extensions of OFS are reviewed and some statistics on them are illustrated.

2 Extensions of Ordinary Fuzzy Sets Having introduced by Zadeh, OFS have been extended by many researchers aiming to explain the membership of an element to a fuzzy set with more parameters and in more details. In a chronological order, extensions of ordinary fuzzy sets with their basics and statistics are presented in the following subsections. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 376–388, 2023. https://doi.org/10.1007/978-3-031-39774-5_44

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2.1 Interval-valued Fuzzy Sets (IVFS) Interval-valued fuzzy sets were independently introduced by Zadeh (1975) [2], GrattanGuiness (1976) [3], Sambuc (1976) [4] and Jahn (1975) [5]. Different from the Type-1 Fuzzy Sets, interval-valued fuzzy sets also let the membership functions be considered as fuzzy. Document statistics are illustrated in Fig. 1. Between 1981 and 2022, 2358 documents on IVFS were published. Documents per year for IVFS shows that IVFS has gained importance after 2004 (Fig. 1-a), reached to 238 documents in 2022. Among 139 sources, as can be seen from Fig. 1-b the most of the documents on IVFS were published by Journal of Intelligent and Fuzzy Systems (146), Information Sciences (93) and Fuzzy Sets and Systems (78). IVFS were studied by 159 authors between 1981 and 2022. By analyzing documents per authors of IVFS (Fig. 1-c), the top 5 authors (documents) were listed as Bustince, H. (57), Chen, S.M. (54), Mousavi, S.M. (39), Garg, H. (38) and Xu, Z. (35). IVFS has been covered by 24 subject areas and as shown in Fig. 1-d, the most 3 subject areas were seen as Computer Science, Mathematics and Engineering with 1720, 1303 and 904 documents, respectively.

(a) Documents by year

(b) Documents by sources

(c) Documents by authors

(d) Documents by subject areas

Fig. 1. Statistics for documents on IVFS

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2.2 Type-2 Fuzzy Sets (T2FS) Based on the criticism that membership functions should also be fuzzy, Zadeh (1975) introduced the concept of T2FS whose membership grades are themselves type-1 fuzy sets [2]. Type1 fuzzy systems deals with a fixed membership function, while in T2FS membership function is expressed as fluctuating. Document statistics between 1987 and 2022 for T2FS are illustrated in Fig. 2. During this period, 744 documents were published by 159 authors on 99 sources. As can be seen from Fig. 2-a, the topic of T2FS have been increasingly attracted by the researchers between 2006 and 2019, and a decrease was seen in the last three years. The most 3 sources published documents on T2FS were found as IEEE Int. Conf. On Fuzzy Systems (74), IEEE Transactions on Fuzzy Systems (52) and Information Sciences (34) (Fig. 2-b). As illustrated in Fig. 2-c, among 159 authors authors with top 5 documents were noted as Mendel, J.M.(67), Tahayori, H. (26), Chen, S.M. (24), Mo, H.(23) and Konar, A. (22). Figure 2-d shows that T2FS were mostly focused on the subject areas of Computer Science (613 documents, 40.4%), Mathematics (404 documents, 26.6%) and Engineering (296 documents, 19.5%).

(a) Documents by year

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Fig. 2. Statistics for documents on T2FS

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2.3 Intuitionistic Fuzzy Sets (IFS) Developed by Atanassov (1986), studies on IFS have gained importance and attracted by many researchers in many fields. The concept of IFS is based on the membership and nonmembership functions. Leta set X be fixed. An fuzzy set A˜ i in X is an  intuitionistic  i ˜ object having the form A = x, μA˜ i (x), vA˜ i (x) : xX , where μA˜ i (x) : X → [0, 1] and vA˜ i (x) : X → [0, 1] define the degree of membership and degree of non-membership respectively, of the xX to the set A˜ i , which is a subset of X, for every element xX , 0 ≤ μA˜ i (x) + vA˜ i (x) ≤ 1 [6]. Based on the documents published between 1986 to 2022, documents statistics are illustrated in Fig. 3. IFS has been studied in 6221 documents. As can be seen from Fig. 3-a, documents about IFS per year dramatically increased after 2003 and reached to 611 documents in 2022. Figure 3-b shows the leading sources (documents) are Journal of Intelligent and Fuzzy Systems (361), Advances in Intelligent Systems and Computing (179) and Information Sciences (151). IFS were studied by 159 authors between 1986 and 2022. By analyzing documents per authors of IFS (Fig. 3-c) the top 5 authors (documents) are ordered as Xu, Z. (158), Atanassov, K. (148), Garg, H. (91), Szmidt, E.(82) and Kacprzyk, J. (80). IFS has been covered by 25 subject areas and as shown in Fig. 3-d, the most 3 subject areas were seen as Computer Science, Mathematics and Engineering with 4161, 3177 and 2450 documents, respectively. 2.4 Neutrosophic Sets (NS) NS, an extension of OFS, were developed by Smarandache in 1998, considering 3 parameters as the truth, indeterminacy and falsity membership grades in order to tackle uncertainty. Let U be a universe of discourse. Neutrosophic set A˜ in U is an object    ˜ having the form A = u, TA˜ (u), IA˜ (u), FA˜ (u) |u ∈ U , where TA˜ (u) is the truthmembership function, IA˜ (u) is the indeterminacy-membership function, and FA˜ (u) is the falsity membership function and for every element uU , 0 ≤ TA˜ (u)+IA˜ (u)+FA˜ (u) ≤ 3 [7]. During the period of 2006 and 2022, 274 documents were found in the literature and as can be seen from Fig. 4-a studies on NS are concentrated after 2017. Documents of NS by sources are illustrated in Fig. 4-b where the leading sources (documents) are found as Neutrosophic Sets and Systems (41), Journal of Intelligent and Fuzzy Systems (23), Symmetry (18). Between 2006 and 2022, 160 authors published documents on NS. The top 3 authors (documents) are found to be Smarandache, F. (25), Broumi, S. (11), Fahmi, A. (9), Mohammed, F.M. (9) (Fig. 4-c). NS studies have been applied to 20 subject areas and as illustrated in Fig. 4-d, the most 3 subject areas were seen as Mathematics (30.8%), Computer Science (28.3%) and Engineering (16.5%) with 167, 154 and 90 documents, respectively.

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(a) Documents by year

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Fig. 3. Document statistics for IFS

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Fig. 4. Statistics for documents on NS

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2.5 Hesitant Fuzzy Sets (HFS) Torra and Narukawa (2009) defined hesitant fuzzy sets in terms of a function that returns a set of membership values for each element in the domain. Let M = {μ1 , . . . , μN } be a set of N membership functions. Then the HFS associated with M, that is hM , is defined as hM (x) = ∪μ∈M {μ(x)}. These fuzzy sets force the membership degree of an element to be possible values between zero and one [8]. 2090 HFS studies in 148 sources were found in the literature between 2009 and 2022. Documents statistics for HFS are illustrated in Fig. 5.

(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

Fig. 5. Statistics for documents on HFS

HFS has been studied by 159 authors where the top 5 authors (documents) were found as Xu, Z. (220), Liao, H. (71), Wei, G. (50), Farhadinia, B. (34) and Meng, F. (32). 2.6 Pythagorean Fuzzy Sets (PyFS) PyFS are proposed by Yager & Abbassov (2013) [9] and Yager (2014) [10] by two parameters, membership and nonmembership functions. PyFS was proposed for the cases where the sum of the membership and nonmembership function is greater than one, that is the condition for IFS does not holds good. It is based on the sum of the squares of the membership and nonmembership value does not exceed 1 as 0 ≤ (μP (x))2 +(vP (x))2 ≤ 1 [9]. Document statistics are shown in Fig. 6., 862 documents on PyFS were found in the literature. As shown in Fig. 6-a, PyFS documents were started to increase after 2016, reached to its greatest value of 230 documents in 2022. Among 148 sources, the most

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(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

Fig. 6. Statistics for documents on PyFS

3 sources published documents on PyFS (Fig. 3-b) was found as International Journal of Intelligent Systems with 91 documents, Journal of Intelligent and Fuzzy Systems with 87 documents and IEEE Access with 31 documents. PyFS has been studied by 160 authors from 110 affiliations in 57 countries. The top 5 authors (documents) were found (Fig. 3-c) as Abdullah, S. (46), Akram, M. (41), Garg, H. (37), Wei, G. (35) and Chen, T.Y. (30). The most studied 3 subject areas were found (Fig. 3-d) to be Computer Science, Mathematics and Engineering. 2.7 Picture Fuzzy Sets (PFS) PFS were developed by Coung & Kreinovich (2013). PFS is more suitable for the cases when human opinions involving more answers of type such that yes, abstain, ˜ ˜ no,  and refusal. A PFS on a AS of the universe of discourse U is given by AS = u, (μA˜ S (u), vA˜ S (u), πA˜ S (u))|u ∈ U  where μA˜ S (u) : U → [0, 1], vA˜ S (u) : U → [0, 1], πA˜ S (u) : U → [0, 1] and 0 ≤ μA˜ S (u) + vA˜ S (u) + πA˜ S (u) ≤ 1 for every u ∈ U [11]. Figure 7 shows document statistics for PFS. Between 2014 and 2022, 348 documents were found and as shown in Fig. 7-a, studies on PFS has been sharply increasing reaching to 114 documents in 2022. The most 3 sources (Fig. 7-b) were found as Journal of Intelligent and Fuzzy Systems (27), Symmetry (19) and Soft Computing (14). Among 160 authors of PFS, the top 5 authors (documents) were listed (Fig. 7-c) as Abdullah, S. (17), Wei, G. (13), Akram, M. (12), Ashraf, S. (12), Pal, M. (12), Son, L.H. (12),

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Mahmood, T. (10), Qiyas, M. (10) and Thong, P.H. (9) (Fig. 7-c). With respect to the subject areas PFS has applied to 19 subject areas and the top 3 subject areas were found (Fig. 7-d) as Computer Science (234 documents, 33.5%, Mathematics (189 documents, 27.1%) and Engineering (117 documents, 16.8%).

(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

Fig. 7. Statistics for documents on PFS

2.8 Q-Rung Orthopair Fuzzy Sets (q-ROFS) Yager (2016) proposed a general class of these sets called q-rung orthopair fuzzy sets in which the sum of the qth power of the support against is bonded by one. They noted that as q increases, the space of acceptable orthopairs also increases and thus gives the ˜ user more freedom in expressing their belief about membership  grade. A q-ROFS O is ˜ an abject having the form O = u, (μO˜ (u), vO˜ (u)|u ∈ U  , where μO˜ (u) → [0, 1], q q vO˜ (u) → [0, 1] and 0 ≤ μ ˜ (u) + v ˜ (u) ≤ 1, and where μ and v are membership and O O nonmembership degrees, respectively [12]. Documents statistics for q-ROFS are demonstrated in Fig. 8. Between 2018 and 2022, increasing year by year (Fig. 8-a), a total of 348 documents were published on q-ROFS. The sources that most published documents on q-ROFS (Fig. 8-b) were seen as International Journal of Intelligent Systems with 57 documents, Journal of Intelligent and Fuzzy Systemswith 38 documents and IEEE Access with 22 documents. Figure 8-c

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shows the top 10 authors with their document numbers. The top 3 authors (documents) were seen as Mahmood, T. (35), Ali, Z. (30) and Liu, P. (25).

(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

Fig. 8. Statistics for documents on q-ROFS

As shown in Fig. 8-d, q-ROFS studies were mostly focused on the subject areas of Computer Science (279 documents, 35.2%), Mathematics (217 documents, 27.4%) and Engineering(143 documents, 18.0%). 2.9 Spherical Fuzzy Sets (SFS) SFS were proposed by Kahraman & Kutlu Gündo˘gdu (2018). SFS are based on the degree of membership, nonmembership and hesitancy. A SFS on a A˜ S of the universe of  discourse U is given by A˜ S = u, (μA˜ S (u), vA˜ S (u), πA˜ S (u))|u ∈ U  where μA˜ S (u) : U → [0, 1], vA˜ S (u) : U → [0, 1], πA˜ S (u) : U → [0, 1] and 0 ≤ (μA˜ S (u))2 + (v A˜ S (u))2 + (πA˜ S (u))2 ≤ 1 for every u ∈ U [13]. Document statistics for SFS are demonstrated in Fig. 9. From 2019 to 2022, 233 documents were published. As can be seen from Fig. 9-a, documents by year for SFS have been increasing year by year, reaching to 107 documents in 2022. The leading source for SFS studies was found as Journal of Intelligent and Fuzzy Systems with 23 documents (Fig. 9-b). On SFS, 160 authors from 28 countries published documents. Among 160 authors of SFS, the top 5 authors (documents) were listed as Kahraman, C.

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(39), Kutlu Gündo˘gdu, F. (18), Abdullah, S. (17), Ashraf, S. (15) and Onar, S.C. (12) (Fig. 9-c). SFS have been applied to 19 subject areas and top 3 subject areas were found as (Fig. 9-d) Computer Science (169 documents, 32.4%), Mathematics (132 documents, 25.3%) and Engineering (110 documents, 21.1%) (Fig. 11-d).

(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

Fig. 9. Statistics for documents on SFS

2.10 T-Spherical Fuzzy Sets (T-SFS) T-SFS are proposed by Mahmood et al. (2018) as a generalization of SFS using any positive power (n) instead of 2 and it is given as 0 ≤ (μA˜ S (u))n +(v A˜ S (u))n +(πA˜ S (u))n ≤ 1 [14]. Document statistics for T-SFS are illustrated in Fig. 10. Between 2018 and 2022, 77 documents on T-SFS were found in an increasing numbers year by year (Fig. 10-a). Among 34 sources with their documents hosted, as can be seen from Fig. 10-b, the top 3 sources were found as Symmetry (10), Journal of Intelligent and Fuzzy Systems (8) and IEEE Access (6). Among 160 authors who published documents on T-SFS, the top 5 authors (Fig. 10-c) with respect to the documents published were listed as Ullah, K. (25), Mahmood, T. (24), Ali, Z. (12), Garg, H. (12) and Jan, N. (9). Figure 11-d shows that T-SFS documents were mostly based on the Computer Science (56 documents,29.3%), Mathematics (51 documents, 26.7%), Engineering (35documents, 18.3%).

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(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

Fig. 10. Statistics for documents on T-SFS

2.11 Fermatean Fuzzy Sets (FFS) FFS proposed by Senapati and Yager (2019) can handle uncertain information more easily in the process of decision making. It is defined as follows. Let X be a universe of discourse. A Fermatean fuzzy set F in X is an object having the form F = {x, αF (x), βF (x) : xX }, where αF (x) : X → [0, 1] and βF (x) : X → [0, 1], including the condition 0 ≤ (αF (x))3 + (βF (x))3 ≤ 1 for all x ∈ X, where αF (x) and , βF (x) denote the degree of membership and the degree of nonmembership of the element x in the set F [15]. Document statistics for FFS are shown in Fig. 11. 94 studies on FFS were found in the literature. Documents per year as illustrated in Fig. 11-a shows that FFS has gained attraction after 2021. FFS studies were published by 57 sources and as can be seen in Fig. 11-b, sources (documents) with respect to their documents published, the most 3 sources are found as International Journal of Intelligent Systems (6), Journal of Intelligent and Fuzzy Systems (6), Journal of Ambient Intelligence and Humanized Computing(5) and Mathematical Problems in Engineering (4). FFS has been studied by 160 authors. The top 5 authors (documents) are found as (Fig. 11-c) Mishra, A.R. (10), Akram, M.(9), Rani, P. (9), Senapati, T. (7), Pamucar, D. (5), Shahzadi, G. (5), Garg, H. (4), and Yager, R.R. (4). With respect to the subject areas (documents) FFS studies (Fig. 11-d) are mainly focused on Computer Science (70), Mathematics (52), Engineering (40).

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(a) Documents by year

(b) Documents by sources

(c) Documents by Authors

(d) Documents by subject areas

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Fig. 11. Statistics for documents on FFS

3 Conclusions Several extensions of OFS are proposed to describe membership functions with more details using more parameters. IFS, PyFS, q-ROFS, and FFS are based on two parameters while NS, PFS, SFS and T-SFS are defined with three parameters. For the cases, where the sum conditions are not satisfied, new extensions are proposed by using sum of the powers of the parameters, i.e., PyFS, q-ROFS, SFS, T-SFS and FFS. As an overall comparison of the statistics of the extensions of OFS, documents by year for 11 extensions are illustrated in Fig. 14. As can be seen from Fig. 12, IFS can be regarded as the most attracted extension while PyFS can be referred as the extension that mostly gained importance in the last 5 years. Although their newly proposed years, SFS, PFS and q-ROFS are found to be gaining more attraction in the following years. In order to take their life in the literature into consideration, that may enable a better comparison, average documents per year through their life are presented in Fig. 13. As shown in Fig. 13, the first 5 attractive extension were found as IFS, HFS, PyFS, q-ROFS and SFS. Analysis also showed that documents on the extensions of OFS are mostly belonged to the subject areas of Computer Science, Mathematics and Engineering showing that theoretical and practical studies are well integrated in the literature.

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Fig. 12. Documents by year for extensions of OFS Fig. 13. Average documents by year for extensions of OFS

References 1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 2. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning– 1. Inf. Sci. 8, 199–249 (1975) 3. Grattan-Guiness, I.: Fuzzy membership mapped onto interval and many-valued quantities. Math. Log. Q. 22(1), 149–160 (1976) 4. Sambuc, R.: Fonction -flous. In: Application a l’aide au Diagnostic en Pathologie Thyroidienne. University of Marseille (1976) 5. Jahn, K.: Intervall-wertige Mengen. Math. Nachr. 68, 115–132 (1975) 6. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst 20, 87–96 (1986) 7. Smarandache, F.A.: Unifying field in logics. In: Neutrosophy: Neutrosophic Probability, Set and Logic. American Research Press, Rehoboth (1998) 8. Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: IEEE International Conference on Fuzzy Systems, Jeju Island, Korea (2009) 9. Yager, R.R., Abbassov, A.M.: Pythagorean membership grades, complex numbers and decision making. Int. J. Intell. Syst. 28, 436–452 (2013) 10. Yager, R.R.: Pythagorean membership grades in multi-criteria decision-making. IEEE Trans Fuzzy Syst 22(4), 958–965 (2014) 11. Cuong, B., Kreinovich, V.: Picture fuzzy sets - a new concept for computational intelligence problems. In: 3rd World Congress on Information and Communication Technologies, WICT (2013) 12. Yager, R.R.: Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 25, 1222–1230 (2016) 13. Kahraman, C., Kutlu Gündo˘gdu, F.: From 1D to 3D membership: spherical fuzzy sets. In: BOS/SOR 2018 Conference, Var¸sova/POLONYA (2018) 14. Mahmood, T., Ullah, K., Khan, Q., Jan, N.: An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Comput. Appl. 31(11), 7041–7053 (2018). https://doi.org/10.1007/s00521-018-3521-2 15. Senapati, T., Yager, R.R.: Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making. Informatica 30, 391–412 (2019)

Fuzzy Multicriteria Decision Making in Earthquake Supply Chain Management: A Literature Review Sezi Cevik Onar(B) , Cengiz Kahraman, and Basar Oztaysi Industrial Engineering Department, Istanbul Technical University, Macka, Istanbul, Turkey [email protected]

Abstract. The earthquake supply chain management is crucial for minimizing causalities and increasing the effectiveness of relief logistics systems. It requires effective and efficient actions to maintain the support and restoration efforts. Under these conditions, the capacity to make knowledgeable decisions becomes essential to minimize the harmful consequences of earthquakes. Fuzzy multiple-criteria decision-making can be considered as a forceful method for solving the challenges and ambiguities in managing earthquake supply chains. This review article tries to present the current research on fuzzy multiple-criteria decision-making in earthquake supply chain management. Scopus database is utilized for analyzing both earthquake supply chain management and earthquake multi-criteria decision-making literature. Keywords: Fuzzy Multicriteria Decision Making · Literature Review · Supply Chain Management

1 Introduction Earthquakes continue to significantly impact the global population, resulting in the loss of thousands of lives, displacement of millions, and extensive damage to structures, infrastructure, and the environment. For example, the 2023 Turkey-Syria earthquake led to a confirmed death toll of 59,259 people, with 50,783 in Turkey and 8,476 in Syria. Such disasters occur in various regions around the world, from Asia to the Americas. While certain areas are more prone to earthquakes, the severity and impacts depend on factors such as geological conditions, seismic vulnerability, earthquake characteristics, the affected region, development level, and preparedness. Effective planning and decision-making are essential to minimize losses and disruptions caused by earthquakes, throughout all disaster management stages, including mitigation, preparedness, response, and recovery. In recent years, disaster management literature has focused on supply chain management. Apart from pre-earthquake preparedness, an effective response strategy can significantly reduce the damage caused by earthquakes. However, earthquake evacuations require different approaches compared to short-notice disasters like hurricanes and floods, which allow 24–72 h of lead time for evacuations [1]. The uncertainties surrounding earthquake © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 389–395, 2023. https://doi.org/10.1007/978-3-031-39774-5_45

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evacuations, such as the location and number of evacuees, route unavailability due to road damage and debris, and other factors, demand careful planning and decision-making. Earthquakes have widespread consequences, often resulting in severe injuries requiring emergency care and long-term rehabilitation, and potentially triggering secondary disasters due to aftershocks. This paper addresses the challenges associated with earthquake supply chain management and summarizes studies on decision-making processes for such problems. Fuzzy multicriteria decision-making approaches have proven successful in addressing these complex and uncertain issues. Previous studies have shown that earthquakes have lasting effects on the environment, society, and economy, and supply chain management can help minimize these negative impacts. Effective supply chain management enables the rapid and efficient distribution of relief materials and ensures the continuity of critical infrastructure and services necessary for rescue and relief operations. However, the complexity and unpredictability of earthquake supply chain management present significant decision-making challenges [2–5]. Earthquake supply chain management includes various stages, such as demand forecasting, procurement, inventory management, transportation, and distribution. Each stage presents unique challenges, like accurate demand forecasting, maintaining adequate inventory levels, selecting optimal transportation routes, and ensuring timely and efficient distribution of relief materials. Fuzzy multicriteria decision-making methods have emerged as a potential tool for handling these complexities and uncertainties. In this paper we aim to provide a literature review on earthquake supply chain management and fuzzy multicriteria decision-making models in earthquake problems especially in earthquake supply chain management problems. The paper is organized as follows: first we provide an introduction, in the second section we provide a literature review on challenges in earthquake supply chain management; and in the third section we provide a literature review of fuzzy multicriteria decision-making in earthquake problems and earthquake supply chain management problems. By summarizing existing research on fuzzy multicriteria decision-making in this area, we hope to offer insights for researchers, practitioners, and decision-makers to effectively manage earthquake-related supply chain challenges.

2 Earthquake Supply Chain Management To gain deeper insights into the current state of research concerning earthquake supply chain management, an exhaustive literature review was carried out using the Scopus database. The search was performed using the keywords “Earthquake Supply Chain Management,” and it was limited to journal articles while excluding conference papers. The keywords needed to be present in the titles, abstracts, or keyword fields. A total of 191 articles were found to be relevant to the search criteria. A network analysis was conducted on the identified articles and their associated keywords using VOSviewer. The strength of the association between the keywords was utilized to create a network visualization with a minimum cluster size of three. Figure 1 displays the results of the network analysis. The network analysis visualization offers a clear summary of the most frequently occurring keywords in the earthquake supply chain management research. The size of each node in the network signifies the frequency of the corresponding keyword in the articles, while the

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thickness of the lines connecting the nodes represents the association strength between the keywords. The clusters based on network analysis have formed clusters that are related with several keywords such as “humanitarian logistics,” “business continuity,” “stochastic programming,” “scenario analysis,” and “operations management.“ These clusters indicate that research in earthquake supply chain management encompasses a broad range of topics, including risk management, decision-making processes, logistics and transportation, and disaster preparedness and recovery.

Fig. 1. Network Analysis of the Earthquake Supply Chain Management Literature Review Kewords

Overall, the literature review and network analysis highlight the importance of earthquake supply chain management as a critical aspect of disaster management and recovery efforts. The broad range of topics and research areas associated with earthquake supply chain management emphasizes the complexity and multi-faceted nature of the field. The results of the analysis also provide a foundation for further research and development in the field, as well as potential areas for collaboration and knowledge sharing among researchers and practitioners. Upon analysis of these terms, it has been found that the studies can be clustered based on their focus areas, such as studies in humanitarian logistics, disaster management, risk management, and mathematical models. The keywords within each cluster are presented in Table 1. The literature review on earthquake supply chain management has revealed the complexity of this critical area of study. Numerous factors come into play when managing the supply chain in the aftermath of an earthquake, and these factors have been

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Table 1. Earthquake Supply Chain Management Literature Review Keywords and Their Clusters Cluster 1

Cluster 4

automotive industry

business continuity

business continuity

computer simulation

earthquake

customer satisfaction

tsunami earthquakes

decision making

global supply chain

disaster response

Great east Japan earthquake

emergency management

industrial management

health care

industry losses

health risks

manufacture

life cycle profitability

resilience

risk mitigation

restoration

scenario analysis

risk assessment

simulation model

risk management

supply chain disruption system

risk perception

dynamics

seismic response

vulnerability

seismology supply chain

vulnerability assessment

supply chain risk management

Cluster 5

supply chains

commerce disaster

Cluster 2

disaster planning earthquake

disaster prevention

emergency health service

disaster relief earthquake

epidemic

disaster emergency logistics

health care delivery

emergency services

human

facility location

natural disaster

humanitarian aid

organization and mana relief work

humanitarian logistics

resource allocation

humanitarian supply chains

Cluster 6

integer programming inventory

chains

location logistics optimization

competition

pre-positioning

data mining

stochastic models

disaster relief operation:

stochastic programming

economics (continued)

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Table 1. (continued) stochastic systems supply chain network uncertainty

humanitarian supply chains

uncertainty analysis warehouses

inventory control

Cluster 3

inventory management

automobile industry carbon emission

management science

China disaster management

planning probability

earthquake event economic impact

public policy

humanitarian relief input-output analysis

supply chain management technology

Japan natural disasters numerical model operations management Peru Sichuan earthquake storms sustainability sustainable development transportation system tsunami Wenchuan earthquake

approached from a variety of different perspectives in academic literature. From humanitarian logistics to disaster management, from risk management to mathematical modeling, researchers have explored various aspects of earthquake supply chain management. Despite this diversity of approaches, however, one common theme emerges: the need for a multi-criteria decision-making process. When we consider the problem’s complexity and the numerous factors to be addressed, a single perspective is not sufficient, rather, a more comprehensive and adaptable approach that considers the distinct aspects of each earthquake and the requirements is mandatory. Fuzzy multi-criteria decision-making can have the potential for dealing with this complex and uncertain problem.

3 Fuzzy Multicriteria Decision Making in Earthquake and Supply Chain Management In this section, we performed a search on the Scopus database for revealing the usage of fuzzy decision-making in earthquake-related issues and we found out that a total of 169 studies focusing on this topic. Figure 2 shows the number of studies that have used fuzzy multi-criteria decision-making in relation to earthquake issues. The figure shows an increasing trend in studies using this approach in recent years. This shows that additional research and development in this area can increase the effectiveness of earthquake supply chain management and improve disaster response and recovery efforts. Among these studies risk assessment and mitigation has an important role, Song et al. [6] developed a disaster prevention and mitigation index assessment system for green buildings using the fuzzy analytic hierarchy process. The system includes four primary

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20 15 10 5 2021

2019

2017

2015

2013

2011

2009

2007

2005

2003

2001

1999

1997

1995

1993

0

Number of earthquake studies that utilizes fuzzy multicriteria decision making Fig. 2. Number of studies that have used fuzzy multi-criteria decision-making in relation to earthquake issues.

aspects: structural safety, disaster prevention and mitigation design, facility settings, and resource utilization. Yet, in earthquake supply chain management the usage is very limited, only a few studies utilized fuzzy multi-criteria decision-making methods in earthquake supply chain management. Supply chain management is a complex area [7] that needs to be considered from different perspectives, thus, we think that this is a significant gap in literature.

4 Conclusion In summary, the literature review on fuzzy multi-criteria decision-making method usage in earthquake supply chain management has showed the multi-criteria characteristics of this important field of study. A single perspective or considering only one aspect for earthquake supply chain management may not be appropriate, thus, a more comprehensive and adaptable approach is necessary. Fuzzy logic and multi-criteria decision-making process is very valuable in many fields [8–11] and it can be considered as a valuable method for dealing with the challenges and uncertainties associated with earthquake supply chain management. Although Scopus database search shows that the application of FMCDM is limited in earthquake supply chain management, in the future studies addressing this approach will be very beneficial.

References 1. Çoban, B., Scaparra, M.P., O’Hanley, J.: Use of OR in earthquake operations management: a review of the literature and roadmap for future research. Int. J. Disaster Risk Reduct. 65, 102539 (2021) 2. Parwanto, N.B., Oyama, T.: A statistical analysis and comparison of historical earthquake and tsunami disasters in Japan and Indonesia. Int. J. Disaster Risk Reduct. 7, 122–141 (2014). https://doi.org/10.1016/j.ijdrr.2013.10.003 3. Rahman, N., Ansary, M.A., Islam, I.: GIS based mapping of vulnerability to earthquake and fire hazard in Dhaka city, Bangladesh. Int. J. Disaster Risk Reduct. 13, 291–300 (2015). https://doi.org/10.1016/j.ijdrr.2015.07.003

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4. Amini Hosseini, K. Hosseini, M., Izadkhah, Y.O., Mansouri, B., Shaw, T.: Main challenges on community-based approaches in earthquake risk reduction: case study of Tehran, Iran. Int. J. Disaster Risk Reduct. 8, 114–124 (2014). https://doi.org/10.1016/j.ijdrr.2014.03.001 5. Kappos, A., Sextos, A., Stefanidou, S., Mylonakis, G., Pitsiava, M., Sergiadis, G.: Seismic risk of inter-urban transportation networks. Procedia Econ. Financ. 18, 263–270 (2014) 6. Song, S., Che, J., Yuan, X.: Disaster prevention and mitigation index assessment of green buildings based on the fuzzy analytic hierarchy process. Sustainability (Switzerland) 14(19), art. no. 12284 (2022) 7. Cevik Onar, S., Aktas, E., Ilker Topcu, Y., Doran, D.: An analysis of supply chain related graduate programmes in Europe. Supply Chain Manag. Int. J. 18(4), 398–412 (2013) 8. Kahraman, C., Çevik Onar, S., Öztay¸si, B.: Engineering economic analyses using intuitionistic and hesitant fuzzy sets. J. Intell. Fuzzy Syst. 29(3), 1151–1168 (2015) 9. Kahraman, C., Öztay¸si, B., Çevik Onar, S.: A multicriteria supplier selection model using hesitant fuzzy linguistic term sets. In: Proceedings of the 11th International on Decision Making and Soft Computing (2014) 10. Kahraman, C., Onar, S.C., Cebi, S., Oztaysi, B.: Extension of information axiom from ordinary to intuitionistic fuzzy sets: an application to search algorithm selection. Comput. Ind. Eng. 105, 348–361 (2017) 11. Kahraman, C., Parchami, A., Cevik Onar, S., Oztaysi, B.: Process capability analysis using intuitionistic fuzzy sets. J. Intell. Fuzzy Syst. 32(3), 1659–1671 (2017)

Artificial Intelligence and Household Energy-Saving Policies: A Literature Review Samrand Toufani1 , Irem Ucal Sari2(B) , and Gizem Intepe3 1 School of Business, Western Sydney University, Parramatta, Australia 2 Industrial Engineering Department, Istanbul Technical University, Macka, Istanbul, Turkey

[email protected] 3 School of Computer, Data, and Mathematical Sciences, Western Sydney University,

Rydalmere, Australia

Abstract. The increasing need for energy in the world brings the necessity for efficient use of resources. Energy efficiency can be examined under three headings; efficiency in energy production, distribution, and consumption. In particular, the prevention of losses in energy consumption is directly related to the habits of the users. It can be predicted that researching the current level of energyconsuming activities, determining the activities that can increase energy saving, and taking and implementing the necessary measures in these areas will greatly reduce the energy need. This study examines studies focusing on the impact of consumer habits on energy saving. Namely, energy consumption habits, energy control mechanisms, and building energy efficiency applications, which have an important dimension in end-user energy consumption, are examined in order to develop energy-saving policies for the end user. The study also examines artificial intelligence applications in energy-saving literature. Keywords: Household energy consumption · Energy Saving Policies · Artificial Intelligence

1 Introduction Energy is one of the most important resources of our age. It has been observed more clearly how important it is for countries to provide the energy they need in sustainable ways, especially after natural disasters and wars in recent years. Countries are in a position to produce much more energy than consumed, especially due to losses in energy production and efficiency rates. In order to use limited resources more efficiently, energy efficiency has become a very important scientific field of study. In addition to the energy loss experienced in production and distribution, it can be observed that the habits of the end users and the level of energy efficiency awareness will directly affect the energy need. This study aims to examine the studies carried out in this field from the past to the present, to help decision-makers to develop policies for the energy efficiency of end users.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 396–403, 2023. https://doi.org/10.1007/978-3-031-39774-5_46

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The paper is structured as follows: In Sect. 2, a visual summary of studies on household energy saving is presented. Sect. 3 provides a summary of major publications in the literature. In Sect. 4, policies that can be created for households to increase energy savings are exemplified.

2 Literature Review In this study, the Scopus database is used for the investigation. 2519 publications on household energy-saving, 209 of which included artificial intelligence, are examined. Figure 1 shows the distribution of the publications on household energy-saving by country. The largest proportion of research has been done in China, followed by the United States, Japan, and the United Kingdom.

Fig. 1. Publications on household energy-saving by country

In Fig. 2, studies involving “artificial intelligence”, which is among these studies, are presented. When Fig. 1 and Fig. 2 are examined together, it can be seen that the density of the studies is similar according to the countries.

Fig. 2. Publications on artificial intelligence and household energy-saving by country

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The distribution chart of the studies by year is given in Fig. 3. Not surprisingly, the topic is increasing in popularity, both number of the studies on household energy-saving and the use of artificial intelligence in these articles are increasing.

1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022

350 300 250 200 150 100 50 0

Household energy saving

AI in household energy saving

Fig. 3. Publications by year

Figure 4 shows that the subject area of the studies is concentrated in the fields of engineering, energy, environmental science, and computer science (Fig. 4).

AI ENGINEERING ENERGY ENVIRONMENTAL… COMPUTER… SOCIAL SCIENCES ECONOMICS,… MATHEMATICS BUSINESS,… PHYSICS AND… MATERIALS… AGRICULTURAL… EARTH AND… CHEMICAL… MEDICINE CHEMISTRY DECISION… PSYCHOLOGY MULTIDISCIPLIN… BIOCHEMISTRY,… ARTS AND… NURSING IMMUNOLOGY… HEALTH… PHARMACOLOGY,… VETERINARY NEUROSCIENCE

1400 1200 1000 800 600 400 200 0

Fig. 4. Publications by subject area

Total

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3 Household Energy-Saving When the literature on energy losses and energy saving was reviewed, it was observed that the studies focused on examining the energy consumption behaviors of households and building energy efficiency. Addressing the energy-saving behavior of households, Chen et al. [3] examine occupant behavior in buildings in terms of space use, interactions, and behavioral efficiency. In this study, it was concluded that real space use and interactions with the building are the main factors that determine the building energy consumption. Behavioral efficiency has been identified as an effective and economical method compared to replacement technologies, however, it has been emphasized that the classification, quantification, and validation of behavioral inputs need to be updated. For example, in the study, window opening behavior was rarely taken into account when calculating the energy impact, as it is rare in most centrally air-conditioned buildings. Another study examined the attitudes and sustainable behaviors of the residents of an industrial city in Russia focusing on environmental problems, socio-demographic factors, mental health and subjective well-being, and physical health status of households by using a survey. It was concluded that the overwhelming majority of the inhabitants of the city (more than 80% of the participants) were extremely dissatisfied with the state of the environment and 70% believed that they could contribute to improving the situation. In the same study, it is stated that indecisive behaviors toward energy saving depend on psychological factors and are associated with “emotion-sensation-behavior”. [10]. Another study aimed to model end-user behaviors and employed the Pareto approach to highlight energy losses from minor events that were created by home users in Australia, and a two-stage simulation model was proposed to examine electricity consumption behaviors in residences. In the aforementioned study, practical solutions are foreseen to cope with the variability of energy losses, and new demand management strategies are proposed [16]. Liu et al. [12] focused on energy-saving potential prediction models that play an important role in developing a retrofit scheme. In retrofit measures, reliable estimation and quantification of energy savings are important as it is often used to guide policy decision-makers. The article reviews current approaches to predict energy-saving impacts for large-scale building retrofitting, including data-driven, physics-based, and hybrid approaches, as well as key factors. The review focuses on addressing key issues that are not considered in current models for estimating the energy-saving impact of large-scale building renovation. Due to the difficulty of applying existing user behavior modeling methods and the inability to obtain a holistic map from survey data on various behavior models, He et al. [7] proposed a holistic survey and simulation-based framework for estimating the energy-saving potential of user behavior. Based on the survey results, they identified seven typical models of user behavior and represented different levels of user-conscious behavior with four behavioral styles (base, wasteful, moderate, and strict) according to modeling difficulty. Based on a case study by a nationwide survey in Singapore, it was seen that there would be significant energy savings potential when user behavior was improved. It has been found that building energy consumption can be reduced by up to 21% with advanced behavior improvement, while it can be reduced by up to 9.5% with moderate behavior improvement [7]. Alsalemia et al. [1] aim to create an accurate profile especially for household energy users to encourage energy-saving

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behaviors. The article presents an innovative method for accurately understanding household energy use behaviors through a classification system based on short energy-related events called micro-moments. Micro-moments are classified according to a given set of criteria given household appliance. The proposed classification system offers a new contribution to household energy profiling to take energy efficiency to the next level. When household energy consumption is analyzed, integrated systems for building energy efficiency and energy control policies have a great effect on energy saving. Especially with the developments in intelligent systems, the energy efficiency of buildings has increased significantly. The fact that many different machine learning tools can be used in algorithms developed for energy saving is a problem in determining the methods to be used in modeling. Lee et al. [11] proposed a hierarchical workflow to guide users in choosing learning, optimization, and control tools to save energy. The workflow, developed based on the data from existing studies in the literature, provides qualitative and quantitative energy-saving effects for applications in various fields. It is stated that the proposed workflow in the study can justify 35% energy cost savings in the building, resulting in energy savings of 25% in heating, ventilation, and air conditioning equipment, 50% in artificial lighting systems, and up to 70% in information transfer and communication power. Summarizing the optimization algorithms and various control strategies with their benefits and drawbacks in achieving energy reduction, Hossain et al. [9] examined the effect of the use of both active and passive solutions on building energy efficiency in net zero energy buildings. They also discussed the relationship of different energy management systems with goals such as user comfort, energy policy, data privacy, and security; and outlined the future trends and issues for developing an effective building energy management system for the realization of the United Nations Sustainable Development Goals. In the study, the importance of the data security problem that can be created by IoT-based building management systems, especially in multi-store buildings, is mentioned and it is concluded that more importance should be given to the optimization of energy storage systems in order to prevent the costs of interruptions in renewable energies. It was emphasized that for comfort optimization, optimization should be made by taking into account all indoor air comfort index parameters such as thermal, visual, acoustic, and air quality characteristics, and passive design solutions should be used more to increase energy efficiency. Spudys et al. [14] aimed to bring a new approach to improving existing building energy control practices by adapting building information models and related data. The study demonstrates the potential of the evaluation process using building information models for the component energy audit procedure, focusing on developing new tools in this area for greater adoption of digitalization practices in the conduct of energy audits. The main output of the study is the analysis of the Industrial Base Class (IFC) schema-building information model data structure and relationships so that the information obtained can be used to digitize energy audit procedures. It also includes the digitization of the building energy consumption assessment, which is affected by the building envelope properties, and the economic evaluations for possible building envelope optimization scenarios. In the literature research, it has been seen that studies have been carried out to reveal the effect of building renovation on energy saving. Uddin et al. [15], one of the many studies examining the energy-saving effects of building renovation, found that the interior

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layout of the building can affect the presence and movement of building occupants, which in turn may lead them to a specific activity, eg energy-saving behavior occurring in a specific location within an enclosed space. By integrating Agent-Based Modeling, System Dynamics, and Building Information Modeling in their work, they provide a comprehensive modeling framework for investigating the impact of interior layouts on the energy-saving behavior of building occupants. As a result, it has been shown that adjusting the interior layout can increase the energy performance of the building by 14.9%. In another study examining the renovation of old urban buildings in Beijing in terms of energy savings, improvements such as increasing the thermal insulation of the buildings, modernizing the air-conditioning systems, making the lighting system more efficient, and using renewable energy sources were made in the selected buildings. In the study, it was observed that the thermal insulation performance of the building facade is an important factor affecting the energy consumption of old urban buildings and the energy saving rate of old urban buildings can reach 65.53% after renovation [17]. Elsharkawy and Rutherford [5] investigated how user consumption habits are positively and negatively affected by user awareness and behavior. They examined home energy use and performance before and after a comprehensive renovation in 150 households in England. The results showed that while the renovation significantly improved home conditions and reduced energy consumption, homes failed to achieve the estimated annual savings in energy bills. This is due to users’ established habits for energy use, preferences for higher comfort levels, insufficient awareness of energy consumption, and the lack of sufficient information to help users better manage their home energy use after renovation. When building energy efficiency applications are examined, it is observed that artificial lighting is a popular research area due to its place in total energy consumption and its easy management. Shankar et al. [13] proposed a smart lighting system that collects daylight with a translucent photovoltaic module integrated into the building, which can produce clean energy and allows less daylight to be used, in order to increase daylight benefit without affecting the comfort of the household. In the study, the cost analysis of the proposed lighting system was made and it was stated that it was more suitable than the existing electricity network. Another factor that directly affects building energy efficiency is the central hot water systems. Bocian et al., [2] state that the share of hot water systems in the total energy consumption of buildings, especially in zero-energy buildings, is increasing, and focuses on the problem of excessive heat losses in the circulation lines of the installation in the residences fed from the central heating network. Studies examining the effects of government incentives and behavioral interventions were also found in the literature review. Hong et al. [8] investigated the impact and ways of influence of government incentive measures and psychological factors on the energysaving behavior of households. In the study, where data were collected by the survey method, it was observed that the energy-saving attitude and environmental responsibility of the households had a significant and positive effect on the energy-saving behavior, while the consumption values had no effect. In the study, it was concluded that government incentive policies played an important role in promoting energy-saving behavior, and had a significant positive regulatory effect on attitude-based energy-saving behavior. But unlike the findings of other studies in the literature, these policies had a significant

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negative regulatory effect on environmental responsibility-based energy-saving behavior. Du and Pan [4] aimed to examine from a systematic perspective how behavioral interventions affect behaviors related to air cooling in buildings. Using the theory of social practice, the study created a theoretical framework that shows the dynamics of work done in three linked stages: “material”, “behavior” and “perception”. Accordingly, the behavior change process was simulated using building energy, user behavior, and social psychology sub-models. The simulation results show that behavioral interventions can help reduce air conditioning electricity use by 15–18%. Han and Cudjoe [6] investigated the effects of households’ knowledge about energy issues, degree of anxiety, perceived energy-saving control, and sense of responsibility on energy-saving behaviors in underdeveloped countries. In the study, the survey data were tested with multiple regression method and it was shown that knowledge about energy issues, degree of anxiety, perceived energy-saving control and sense of responsibility positively and significantly related to energy-saving behaviors. Of the four determinants, it was concluded that the degree of anxiety had the most significant impact on the energy-saving behavior of urban residents.

4 Conclusion With the help of the literature review examined, the sub-headings that the energy efficiency policy for households should include are summarized below: Planning of New Housing Projects in Harmony with Energy-Efficient Smart Systems: Especially considering the importance of the cost factor, the construction and implementation of building design projects that make use of passive and active energy systems at the highest level will significantly reduce the end-user energy needs. The systems that should be considered during the design phase should provide a high level of energy efficiency as active and passive energy systems. Increasing the Awareness of End-Consumer Energy Efficiency: It should be aimed to increase energy awareness with public spots. Awareness should be raised about how many resources a user needs for the energy consumed by concrete examples and the labor-time technology spent to make this energy usable. It is also necessary to analyze the differentiation caused by factors such as the effect of the end consumer behavior habits on energy consumption, cultural structure, and geographical structure, which are also frequently discussed in the literature research. Based on the energy consumption behavior habits map to be revealed regionally, sub-policies should be formed to reshape the habits that disrupt energy efficiency specific to groups.

References 1. Alsalemi, A., et al.: Endorsing domestic energy saving behavior using micro-moment classification. Appl. Energy 250, 1302–1311 (2019) 2. Bocian, M., Siuta-Olcha, A., Cholewa, T.: On the circulation heat losses in domestic hot water systems in residential buildings. Energy Sustain. Dev. 71, 406–418 (2022)

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3. Chen, S., Zhang, G., Xia, X., Chen, Y., Setunge, S., Shi, L.: The impacts of occupant behavior on building energy consumption: a review. Sustain. Energy Technol. Assess. 45, 101212 (2021) 4. Du, J., Pan, W.: Evaluating energy saving behavioral interventions through the lens of social practice theory: a case study in Hong Kong. Energy and Buildings 251, 111353 (2021) 5. Elsharkawy, H., Rutherford, P.: Energy-efficient retrofit of social housing in the UK: lessons learned from a Community Energy Saving Programme (CESP) in Nottingham. Energy Build. 172, 295–306 (2018) 6. Han, M.S., Cudjoe, D.: Determinants of energy-saving behavior of urban residents: evidence from Myanmar. Energy Policy 140, 111405 (2020) 7. He, Z., Hong, T., Chou, S.K.: A framework for estimating the energy-saving potential of occupant behaviour improvement. Appl. Energy 287, 116591 (2021) 8. Hong, J., She, Y., Wang, S., Dora, M.: Impact of psychological factors on energy-saving behavior: moderating role of government subsidy policy. J. Clean. Prod. 232, 154–162 (2019) 9. Hossain, J., et al.: A review on optimal energy management in commercial buildings. Energies 16(4), 1609 (2023) 10. Krupnova, T.G., Rakova, O.V., Shefer, E.A., Semenenko, D.P., Saifullin, A.F.: Domestic energy-saving behavior index as sustainability indicator: are Russians ready for sacrifices to protect the environment? Environ. Sustain. Indicators 16, 100209 (2022) 11. Lee, D.S., Chen, Y.T., Chao, S.L.: Universal workflow of artificial intelligence for energy saving. Energy Rep. 8, 1602–1633 (2022) 12. Liu, S., et al.: Energy-saving potential prediction models for large-scale building: a state-ofthe-art review. Renew. Sustain. Energy Rev. 156, 111992 (2022) 13. Shankar, A., Vijayakumar, K., Babu, B.C.: Energy saving potential through artificial lighting system in PV integrated smart buildings. J. Build. Eng. 43, 103080 (2021) 14. Spudys, P., Jurelionis, A., Fokaides, P.: Conducting smart energy audits of buildings with the use of building information modelling. Energy Build. 285, 112884 (2023) 15. Uddin, M.N., Chi, H.L., Wei, H.H., Lee, M., Ni, M.: Influence of interior layouts on occupant energy-saving behaviour in buildings: an integrated approach using agent-based modelling, system dynamics and building information modelling. Renew. Sustain. Energy Rev. 161, 112382 (2022) 16. Zaghwan, A., Gunawan, I.: Resolving energy losses caused by end-users in electrical grid systems. Designs 5(1), 23 (2021) 17. Zhang, X., Nie, S., He, M., Wang, J.: Energy-saving renovation of old urban buildings: a case study of Beijing. Case Stud. Thermal Eng. 28, 101632 (2021)

Optimization

Optimization and Intellectualization of Adaptive Control of Investment Projects of Multi-agent Network Industrial Complexes with Fuzzy Data Andrey Shorikov1

and Elena Butsenko2(B)

1 Institute of Economics of the Urals Branch, Russian Academy of Sciences, Ekaterinburg,

Russia 2 Ural State University of Economics, Ekaterinburg, Russia

[email protected]

Abstract. The paper considers the network structure of the industrial complex, which models the processes of investment projecting by the relevant agents (economic entities). It is assumed that investment projects are being implemented in the multi-agent network industrial complex with the corresponding general indicators of their effectiveness. In this regard, the development of a model for optimizing the adaptive control of investment projects of multi-agent network industrial complexes is based on procedures that use the presence of information and control links between agents. Such procedures involve the formation of a set of acceptable positions of project management processes, as well as the implementation of feedback in the form of appropriate reactions of agents’ control actions to possible changes in situations during the implementation of investment projecting processes. To solve the problem under consideration, the paper proposes a methodology for optimizing the adaptive control of investment projects of network industrial complexes, which allows developing numerical algorithms for creating intelligent management decision support systems. The practice of applying the proposed methodology is illustrated by the example of an investment project implemented by a network complex that unites food industry, catering and trade enterprises. Keywords: multi-agent network industrial complex · investment project management · adaptive control optimization · economic and mathematical modeling · network planning and management methods

1 Introduction The formation and development of the network structure of the regional industrial complex ensures its competitiveness, stability and adaptability. This circumstance makes it necessary to study the system of self-adjustment and self-organization of processes and relationships in the network structure of the industrial complex [2]. Due to the fact that the participants of the network industrial complex enter into long-term partnerships, it is relevant to develop strategies for their effective partnership [1]. In addition, it is necessary to update the procedures for the formation and development of the network structure © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 407–414, 2023. https://doi.org/10.1007/978-3-031-39774-5_47

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of industrial complexes [1]. For the food industry, this issue is most acute, because. The digital transformations of this industry become apparent almost immediately and entail changes in other industries. The relevance of this problem, the growing need for its solution at the theoretical, methodological and practical levels determined the choice of the topic of this study. The scientific novelty of the research results consists in solving an important scientific problem – the development of new methodological techniques, ways to optimize adaptive control and practical recommendations that reveal the features and prospects for the formation and development of network structures of regional industrial complexes aimed at producing food and providing services related to their consumption. The article contains such sections as introduction, literature review, methods and models, research results, conclusion, acknowledgments and references.

2 Literature Review The problem of integration interaction of economic agents is reflected in the works of such authors as V. Bautin, B. Garrett, S. Dokholyan, R. Coase, B. Petrosyants, N. Williams, M. Tugan-Baranovsky, etc. However, in the works of these authors, insufficient attention is paid to the theoretical substantiation of the features and prospects for the development of network models of a regional industrial complex, such as the food industry. Russian and foreign authors have attempted graphic, verbal, imitative and economic-mathematical modeling of backbone links in the structure of the industrial complex [3–5]. However, until now, not enough attention has been paid to the problem of forming a system of optimal self-tuning (adaptive) control and self-organization of intelligent connections in the network structure of an industrial complex.

3 Methods and Models Adaptive control optimization of investment projecting is presented in detail in [6]. The main principles of this method are as follows. Network economic and mathematical modeling is carried out for a specific investment project, based on the values of its parameters, a critical path is formed that determines the a priori minimum (optimal) project implementation time, the calendar schedule for the implementation of all investment projecting processes described by the corresponding work-operations, as well as the set of admissible positions (states) of investment projecting processes corresponding to a specific control period determined by the event (node) of the critical path, and the set of all admissible strategies for adaptive control of investment projecting processes corresponding to a fixed position. In accordance with the rules for the formation of a strategy for optimal adaptive project management [6], at each step of implementing the management of investment projecting processes, which is determined by the initial event of the corresponding critical path, the quality of the actual implementation of the workoperations included in it is assessed based on the prescribed duration of their execution and data current schedule. Then, the deviation of the planned values of the parameters from their actual implementations is established, and if there is a deviation, then the model parameters are adjusted in order to optimize it in time, which is necessary for

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the implementation of the investment project as a whole. Next, a new critical path and a new schedule for the next period of time are formed, and such a recurrent procedure for optimizing the adaptive control of investment projecting processes is repeated until the implementation of the investment project as a whole.

4 Research Results Implementation of optimization of adaptive control of investment projecting of network industrial complexes is considered on the example of an investment project carried out by a regional group of enterprises in three industries – food industry, public catering and trade. Table 1. Initial data of the network investment project. Work No

The content of the work

The industry and location of the business entity that is part of the project network

Duration of work, weeks

Previous works

R1 (0)

Project concept development

Food industry, public catering, trade, Ekaterinburg

1 (0) = 2

–

R2 (0)

Search for regional network partners

Food industry, trade, 2 (0) = 4 Ekaterinburg, N.Tagil, Kamensk-Uralsky, Irbit

R3 (0)

Project franchise development

Food industry, public catering, trade, Ekaterinburg

R4 (0)

R1 (0)

3 (0) = 2

R1 (0), R2 (0)

Determining the Public catering, range of products that Ekaterinburg the dealer should purchase

4 (0) = 1

R1 (0), R2 (0), R3 (0)

R5 (0)

Determining terms of the pricing policy

Trade, Ekaterinburg

5 (0) = 1

R1 (0), R2 (0), R3 (0), R4 (0)

R6 (0)

Coordination of transportation conditions to the municipality for the organization of sales, timely notification of customers and delivery of the order

Trade, Ekaterinburg, Nizhny Tagil, Kamensk-Uralsky, Irbit

6 (0) = 1

R1 (0), R2 (0), R3 (0)

(continued)

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Work No

The content of the work

The industry and location of the business entity that is part of the project network

Duration of work, weeks

Previous works

R7 (0)

Advertising and Public catering, trade, 7 (0) = 3 promotion of products Ekaterinburg

R1 (0), R2 (0), R3 (0)

R8 (0)

Product certification

Public catering, Ekaterinburg

8 (0) = 4

R1 (0), R2 (0), R3 (0)

R9 (0)

Conclusion of contracts

Food industry, public catering, trade, Ekaterinburg

9 (0) = 2

R1 (0), R2 (0), R3 (0)

R10 (0)

Sales launch

Trade, Ekaterinburg, N.Tagil, Kamensk-Uralsky, Irbit

10 (0) = 1

R9 (0)

The investment projecting of the considered regional network industrial complex contains several hundred works-operations. Table 1 presents the initial data in the form of a description of the aggregated operations of a network investment project, their code, the industry of the partner company, the duration of the work and previous works. The following conditions-restrictions for the implementation of the project have been formed: a specific range of products has been drawn up, which the dealer must purchase (about 2 thousand units of finished products), this is enough to start; the terms of the pricing policy were formed and a clause was added stating that there can only be one representative in each region; dealers undertake transportation of products to the point of sale, their duties also include informing customers on time and delivering the order; the organization of advertising and promotion of products is undertaken by the founding company. Several companies participate in the project, the first is a manufacturer of meat raw materials, the second is engaged in the development of dishes and their preparation, the third sells finished products to the public. In Ekaterinburg, they are developing new dishes and the company has everything you need up to an autoclave. In Nizhny Tagil, all technological maps are created, dishes are worked out, and then the recipe is sent to the technologist in Ekaterinburg. All the company’s dishes are prepared in Ekaterinburg. Freight companies (SDEK, Business lines) are used to deliver products to customers. Let us consider the application of the methodology for optimizing the adaptive control of the regional network industrial complex to this project. In accordance with the methodology outlined in [6], at the first stage, the auxiliary parameters τ := 0, s: = 0 and based on the initial data on the investment projecting processes under consideration, as well as the rules of network economic and mathematical modeling, form the corresponding network model for the implementation of (e) investment projecting processes WM(e) τ (R(τ )) = WM0 (R(0)) ∈WM0 (R(0)), where

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WM0 (R(0)) – the set of all admissible network models corresponding to the initial period of time, in the form of a network graph, which is shown in Fig. 1 and corresponds to the initial array of work-operations R(τ ) = {R1 (τ ), R2 (τ ), . . . , Rnτ (τ )} == Rτ = R(0) = {R1 (0), R2 (0), . . . , R10 (0)} = R0 , as well as the initial array of duration values for the execution of work-operations (τ ) = {1 (τ ), 2 (τ ), . . . , nτ (τ )} = τ == (0) = {1 (0), 2 (0), . . . , 10 (0)} = 0 . At the second stage, based on the initial data, including – the initial array of work-operation R(τ ) = {R1 (τ ), R2 (τ ), . . . , Rnτ (τ )} = Rτ = R(0) = {R1 (0), R2 (0), . . . , R10 (0)} = R0 , the corresponding array of values for the duration of  the execution of work-operations (τ ) = 1 (τ ), 2 (τ ), . . . , nτ (τ ) = τ = (0) = {1 (0), 2 (0), . . . , 10 (0)} = 0 , as well as the generated network model for the implementation of investment projecting processes, it is necessary to optimize the network model in terms of the time parameter – to find the critical path, the critical time and form an appropriate calendar schedule for the implementation of investment projecting processes in general, that is, to solve the scheduling task. R6(0)

4 5 F2(0) δ2(0)=0 4 2 R1(0) 1(0)=2

1 4

4 F1(0) δ1(0)=0

R2(0) 2(0)=4

R3(0) 3(0)=2 4 3

4 8

6(0)=1

F6(0)

R7(0) F3(0) F4(0) δ3(0)=0 δ4(0)=0 R4(0)

R5(0)

74

4(0)=1

R9(0) 9(0)=2

6 4

δ6(0)=0

7(0)=3

10 4

5(0)=1

R10(0) 10(0)=1

F5(0) R8(0)

δ5(0)=0 94

8(0)=4

Fig. 1. Network model of the investment project of the regional network industrial complex (τ = 0).

For this network model, there are three critical paths, which are shown in Fig. 2 and are highlighted in gray with a thick line, with the duration of each of them being 9 weeks. To perform further actions, any of these critical paths is selected, and specifically, we select a path that consists of a set of workAs a result, operations: {R2 (τ ), R8(τ ), R10 (τ )} = {R2 (0), R8 (0), R10 (0)}.   the criti(cr) (cr) (cr) (cr) cal path R(cr.) (τ ) == R1 (τ ;τ1 ), R1 (τ1 ;τ2 ), . . . , R (cr) τn(cr) −1 ;τn(cr) = Rτ = τ τ n τ   (κp.) (cr) (cr) (cr) (cr) R1 (0; 4), R2 (4; 8), R3 (8; 9) = R(cr) (0) = R0 is formed, where nτ = 3. The (κp.)

(κp.)

duration of the implementation of the formed critical path Rτ = R0 determines (e) (e) the critical time Tτ = T0 = 9, i.e. the optimal (minimum) time required to perform all the work-operations that form the entire set of measures for the implementation of investment projecting processes as a whole. Next, an integer array of time periods is formed to implement the optimization of adaptive control of the processes of the regional

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network industrial complex Tτ = {τk }

(κp.)

k∈0, nτ

= T0 = {τk }k∈0, 4 = {τ0 , τ1 , . . . , τ4 } =

{0; 4; 8; 9}, corresponding to the events {1; 3; 9; 10} of the critical path Rτ(cr) = R0(cr) . For the generated network model, based on the found critical path and the corresponding array of the duration of the execution of work-operations, the task of scheduling is solved – the formation of a calendar schedule, that is, a description of the acceptable timing for the execution of all work-operations. The abscissa shows the duration of the project work-operations in weeks; along the y-axis – ordered work-operations; rectangles with black color indicate non-critical work-operations; rectangles with gray color indicate critical work-operations; dashed rectangles indicate free time reserves for the execution of work-operations.

Fig. 2. Critical paths of the network model of the project (τ = 0).

At the third stage, for the generated network model, the optimization of adaptive control of the processes of the project under consideration is implemented, which is carried out on the basis of the method of network formalization and optimization of adaptive control of investment projecting processes, described in [6]. There was a delay of 1 week during the operation R8 (τ ) – “Certification of products”, and the network model of the project will take the form shown in Fig. 3.  For a period of time τ1 ∈ Tτ , we will call the set p(τ1 ) = τ1 , R(τ1 |Rτ , Tτ ) τ1 position for the project management process under consideration, where the set of workR(τ1 |Rτ , Tτ ) is determined by the following relationship: R(τ1 |Rτ , Tτ ) operations   = R1 (τ1 ), R2 (τ1 ), . . . , Rnτ1 (τ1 ) ; nτ1 ≤ nτ ; ∀k ∈ 1, nτ1 : Rk (τ1 ) – work-operation, of work-operations R(τ ), that is Rk (τ1 ) ∈ Rτ =  which is included in the array R1 (τ ), R2 (τ ), . . . , Rnτ (τ ) and which was implemented in the period of time t1 : (t1 ∈ Tτ ) ∧ (t1 < τ1 ).   Let be P(τ1 ) = {pk (τ1 )}k∈1, rτ = τ1 , Rk (τ1 |Rτ , Tτ ) k∈1, r – finite set of all τ1 1 admissible τ1 -positions for the investment project management process under consideration (rτ1 ∈ N). For τ1 ∈ Tτ an admissible strategy of adaptive project management Ua as a mapping Ua : P(τ1 ) → 2R(τ ) , which assigns to each admissible τ1 -position

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Fig. 3. Critical path in the network model of the project, taking into account the delay in the execution of work R8 (τ ).

  ˆ 1 ) ∈ 2R(τ ) , namely: p(τ1 ) = τ1 , R(τ1 |Rτ , Tτ ) ∈ P(τ1 ) a set of work-operations R(τ ˆ 1 ), Ua (p(τ1 )) = R(τ where

ˆ 1) R(τ

=

  Rˆ 1 (τ1 ), Rˆ 2 (τ1 ), . . . , Rˆ nτ1 (τ1 ) ,

  ∀k ∈ 1, nτ1 : Rˆ k (τ1 ) ∈ Rτ \R(τ1 |Rτ , Tτ ) ; here and below, for any set M, the symbol 2M will denote the set of all its subsets. Denote by symbol Ua∗ the set of all admissible strategies Ua for adaptive project management. When implementing the process of optimizing the adaptive control of the considered investment project, the formation of optimal posi (e) tions p(e) (τ1 ) = τ1 , R (τ1 |Rτ , Tτ ) ∈ P(τ1 ), non-optimal positions p(τ1 ) =      τ1 , R(τ1 |Rτ , Tτ ) ∈ P(τ1 )\ p(e) (τ1 ) and the implementation of the optimal strat(e) egy Ua ∈ Ua∗ are carried out in accordance with the method of network formalization and optimization of adaptive control of investment projecting by the way  (e) (e) (e) of forming elements:pa (4) = p (τ1 = 4) = τ1 , R (τ1 |Rτ , Tτ ) ∈ P(τ1 ),   (e) (e) (e) (e) (e) R (τ1 = 4|Rτ , Tτ ) = {R1 (τ1 ), R2 (τ1 )} = {R1 (0), R2 (0)}, Ua pa (4) = {R3 (0), R4 (0), . . . , R10 (0)}, τ := τ1 = 4; Tτ = T4 = {τk }k∈0, 2 = {τ0 , τ1 , τ2 }     (e) (e) |R = {4; 8; 9}, pa (8) = p(τ1 = 8)  = τ1 , R(τ1 τ , Tτ ) ∈ P(τ1 )\p (τ1 = 8) , R(τ1 = 8|Rτ , Tτ ) = R1 (τ1 ), R2 (τ1 ), . . . , R6 (τ1 ), R˜ 8 (τ1 ), R9 (τ1 ) =    {R1 (0), R2 (0), . . . , R6 (0), R˜ 8 (0), R9 (0) , Ua(e) (p(τ1 = 8)) = R7 (0), R8 (0), R10 (0) , = T8 = {τk }k∈0, 2 = {τ0 , τ1 , τ2 } = {8; 9; 10}, pa(e) (9)   (e) (e) = p(e) (τ1 = 9) = τ1 , R (τ1 |Rτ , Tτ ) ∈ P(τ1 ), R (τ1 = 9|Rτ , Tτ ) =    (e) (e) (e) (e)

R1 (τ1 ), R2 (τ1 ), . . . , R9 (τ1 ) = {R1 (0), R2 (0), . . . , R9 (0)}, Ua p(e) (τ1 = 9) = {R10 (0)}, τ := τ1 = 9; Tτ = T9 = {τk }k∈0, 1 = {τ0 , τ1 } = {9; 10},   (e) (e) (e) pa (10) = p(e) (τ1 = 10) = τ1 , R (τ1 |Rτ , Tτ ) ∈ P(τ1 ), R (τ1 = 10|Rτ , Tτ ) τ

:= τ1 = 8; Tτ

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 (e) (e) (e) = {R1 (0), R2 (0), . . . , R10 (0)} = R0 , R1 (τ1 ), R2 (τ1 ), . . . , R10 (τ1 )

 (e) (e) (e) Ua p(e) (τ1 = 10) = ∅; τ1 = τn(κp.) = Tτ = T9 = 10. τ At the fourth stage, the output results of the adaptive project management   (e) optimization process are formed: Ra = R(0) = R1 (0), R2 (0), . . . , Rn0 (0) = = R0 = {R1 (0), R2 (0), . . . , R10 (0)} – the initial set of work-operations  of  (e) (e) (e) the project; WMa = WM0 (R0 ) – optimal network model; pa (tk ) ==   (e) (e) (e) (e) pa (4), pa (8), pa (9), pa (10) – a set of tk -positions k ∈ 1, 4 corresponding to the implementation of the strategy of optimal adaptive control of investment project (e) (e) (e) (e) Ua ∈ Ua∗ ; Ta = Tτ = T9 = 10 – the optimal time for the implementation of a regional network project. Note that if the proposed methodology is not applied, then in   the time period τ = 8 works R8 (0) (R8 (0) = R˜ 8 (0) ∪ R8 (0)) and R10 (0) will be carried out in accordance with the initial schedule at the same time, which is unacceptable from the conditions of its implementation and will lead to the failure of the project as a whole.

=



5 Conclusion The article considers the problem of optimizing the adaptive control of investment project of multi-agent network industrial complexes. Its solution was carried out on the basis of network economic and mathematical modeling, taking into account feedback. Constant monitoring of the implementation of investment project is consistently used to form and implement an optimal adaptive control strategy. A practical experiment on the implementation of the proposed approach was carried out for a specific investment project in the food industry. Acknowledgments. The work was carried out with financial support by the Russian Science Foundation (Project No. 22-28-01868 “Development of an agent-based model of the network industrial complex in the context of digital transformation”).

References 1. Akberdina, V., Romanova, O.: Regional aspects of industrial development: a review of approaches to the formation of priorities and regulatory mechanisms. Region. Econ. 17(3), 714–736 (2021). https://doi.org/10.17059/ekon.reg.2021-3-1 2. Agarkov, A., Golov, R.: Projecting and Formation of Innovative Industrial Clusters, 2nd edn. Publishing and Trade Corporation “Dashkov and K”, Moscow (2021) 3. Kleiner, G.: System Economics: Development Steps. Foreword by Academician Makarov, Publishing House “Scientific Library”, Moscow (2021) 4. Sirotkina, N., Shan, Ya.: On the dominant approaches to the management of regional economic systems. Region Syst. Econ. Manag. 1(40), 24–32 (2018) 5. Stukalo, O.: Network interaction of subjects of the regional economy (on the example of the food sector of the regional economy). (Ed. by, Sirotkina, N.). Publ. Voronezh State Pedagogical University, Voronezh (2016) 6. Shorikov, A., Butsenko, E.: Optimization of adaptive network economic and mathematical modeling of business planning process control. Econ. Math. Methods 3(57), 110–125 (2021). https://doi.org/10.31857/S042473880016413-3

Fuzzy Model for Multi-objective Airport Gate Assignment Problem Mert Paldrak1(B)

, Melis Tan Tacoglu2

, and Mustafa Arslan Örnek1

1 Industrial Engineering Department, Ya¸sar University, Bornova, ˙Izmir, Turkey

{mert.paldrak,arslan.ornek}@yasar.edu.tr

2 Logistics Management Department, Ya¸sar University, Bornova, ˙Izmir, Turkey

[email protected]

Abstract. To address the increasing demand for air transportation, the management, allocation, and efficient utilization of limited airport resources, such as bridge-equipped gates, are of paramount importance to improve the efficiency of the air transportation system. Bridge-equipped gates are scarce and immobile resources have a significant impact on airport management, airlines, and passenger convenience when utilized properly. Hence, the gate assignment problem is an important problem involving multiple stakeholders with conflicting objectives. This study proposes a fuzzy model to tackle two objectives: maximization of overall utility of flight-gate assignments and maximization of the robustness of the assignment schedule simultaneously. Fuzzy variables are employed in order to represent the uncertainty of idle times between two consecutive flights served by the same bridge-equipped gate, and their membership degrees express their effect on assignment robustness. An adjustment function is applied to combine these two objective functions into one. To handle this NP-hard problem in a reasonable amount of computational time, a constructive heuristic algorithm is employed. The performance of the proposed fuzzy model is evaluated with the help of various test in instances of different sizes Two fuzzy distribution functions are tested, and their comparison is provided. The simulation results demonstrate the applicability and effectiveness of the fuzzy model in addressing multi-objective airport gate assignment problem. Keywords: Multi-objective · Airport Gate Assignment Problem · Fuzzy Logic and Optimization · Constructive Heuristics

1 Introduction To meet the demand for air transportation, methodologies for management and allocation of scarce resources in hub-and-spoke airports are rendered important to increase the efficiency of the overall air transportation system. Among these resources, bridge-equipped gates hold significant importance as limited and fixed assets that directly affect airport operations, airline efficiency and passenger conveniences. The Airport Gate Assignment Problem (AGAP) arises as a critical decision-making problem with multiple stakeholders to be handled in daily basis. The chief aim of the problem is to generate an assignment schedule where flights are appropriately assigned to bridge-equipped gates while © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 415–425, 2023. https://doi.org/10.1007/978-3-031-39774-5_48

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maintaining safety and security, minimizing costs incurred due to gate assignment and increasing the satisfaction of airlines and passengers. In recent years, research has recognized the importance of incorporating airline preferences into the gate assignment problem by considering the utility of gates. Airlines generally have specific preferences regarding gate assignments, which is based on such factors as proximity to airline lounges, airport entrance and exit, operational conveniences, code-sharing requirements, or connecting flight considerations. These objectives are used to capture the unique requirements and priorities of each airline, allowing for more tailored gate assignments. By considering airline preferences, gate assignment schedule can be optimized to enhance operational efficiency, improve customer experience, and foster better partnerships between airport management and airlines. Traditionally, the gate assignment problem focused more on the maximization of metrics like minimizing delays, reducing congestion, and efficient resource usage. Nonetheless, the importance of robustness in gate assignments is inevitable and increasingly recognized by authors. Robustness refers to ability of an assignment plan to handle disruptions and uncertainty, such as flight delays or equipment failures, without any major disruptions to airport operations. Ensuring robustness in gate assignments is vital for maintaining operational stability, minimizing assignment disruptions, and increasing the reliability of the airport transportation system. Assignments sensitive to problemrelated uncertainties can provoke costly reassignments, delays or cancellations, and inconvenience for airlines and passengers. In this study, the airport gate assignment problem is formulated as a multi-objective Binary Integer Programming (BIP) where conflicting two objective functions are taken into account. It focuses on providing a compromise solution for two conflicting objectives by converting multi-objective functions into a single objective one using fuzzy membership functions.

2 Literature Review The gate assignment problem is a real-life problem decision-making problem that is related to multiple stakeholders, namely airport management, airlines, and passengers. Hence, there is no surprise that this problem is formulated as a multi-objective problem while respecting some hard and soft constraints. In recent years, numerous mathematical models and analytical tools have been applied to this problem to solve it. Traditionally, there are two widely studied optimization objectives, namely minimization of total walking distance of passengers to enhance customer satisfaction (Yan & Huo (2001) [1], Xu and Bailey (2001) [2], Mangoubi & Mathaisel (1985) [3], Ding et al. (2005) [4], Bolat (1999) [5]). This objective makes the problem NP-hard proven by Obata (1980) [6] because it is a special version of a quadratic assignment problem. Another important aspect of the problem is related to the consideration of the preferences of airlines over bridge-equipped gates. A flight-to-gate assignment considers the expectation of airlines and their exclusive-use, preferential-use, or common-use agreements (Ornek et al. (2022) [7]). Even though most airports are equipped with more infrastructure resources such as several runways and multiple terminals, they are not sufficient because of participation of new airlines, and an increase in the number of arriving flights at the airport (Ding et al. (2004) [8]).

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Flight delays are common, and static flight-gate assignments are vulnerable to unforeseeable changes in flight schedules. Consequently, enhancing the robustness of flightgate assignment schedules to handle dynamic changes is another objective of the problem. In the gate assignment literature, there exist numerous models proposed to distribute gate idle times evenly among bridge-equipped gates and minimize the dispersion of idle time periods (Bolat (1999) [5], Bolat (2001) [9]). Besides these approaches, stochastic programming models have also been developed to address robust gate assignment by considering gate conflict probabilities (Lim & Wang (2005) [10], Paldrak & Örnek (2022) [11]). In the context of increasing the robustness of a schedule, one approach widely used is to insert a fixed buffer time between two consecutive flights assigned to the same gate (Yan & Huo (2001) [1]). Nonetheless, determination of an appropriate buffer time length is challenging due to the varying flight densities in the schedule. To address this issue, flexible buffer times are designed, considering specific arrival and departure times of flights (Yan et al. (2002) [12]). This allows for a more tailored approach to buffer time allocation. In existing literature, robustness is often expressed as absolute constraints, which establish a rigid relationship between flights and bridge-equipped gates (Wei & Liu (2009) [13]). However, this approach is considered inefficient in capturing the complexity of the problem. In this study, a fuzzy model is proposed to handle two main conflicting objectives. To represent the degree of compatibility or match between flights and bridgeequipped gates, fuzzy membership functions are employed. This fuzzy model allows for a more flexible and nuanced representation of the assignment problem, considering the inherent uncertainties and variations in flight schedules. In the upcoming section, the problem will be described in detail, providing a comprehensive understanding of the airport gate assignment problem and its various aspects. Section 4 presents the formulation of the multi-objective Mixed Integer Programming (MIP) model for the problem. Additionally, a fuzzy model for proposed model, utilizing two fuzzy distribution functions to define the membership values. Section 5 introduces a problem-specific heuristic algorithm, coupled with constraint satisfaction technique. This algorithm is specifically designed to handle large instances of the airport gate assignment problem, providing effective and efficient solutions. In Sect. 6, the study presents computational results and interpretation of comparisons analyses.

3 Problem Definition The gate assignment problem involves the allocation of bridge-equipped gates to a set of arriving and departing flights within a specified operational period. In this study, we focus on one of the busiest airport airports in Turkey, which has multiple aprons and bridge-equipped gates. Upon arrival, each flight is assigned to a gate until it is ready for its next flight. When all gates are occupied, new arrival flights are assigned to remote stands. This problem includes two types of flights: regular flights and night-stand flights. Night-stand flights, due to their longer occupational time, are not preferred to be assigned to bridge-equipped gates to efficient source utilization. The gates in the airport have varying sizes, and certain gates are reserved for emergency purposes only. Thus, not all flights are eligible to be assigned to every bridge-equipped gate, considering

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the compatibility of aircraft size with gate capacity. Additionally, airport management emphasizes apron safety aims to reduce traffic congestion and flight delays caused by ground operations. To actualize this, airlines serviced by same ground-handling firm are assigned to adjacent gates. Upon determination of the objectives of this study, three chief stakeholders along with the robustness of the static schedule are taken into consideration. The objectives include maximization of overall utility of flight-to-gate assignments based on airline preferences for specific gates and minimization of the total flight conflict probability assigned to bridge-equipped gates. In this study, the term “utility” refers to the appropriateness of a flight being assigned to a bridge-equipped gate, which measures the overall effectiveness of the static gate assignment. On the other hand, the robustness of the static assignment is assessed by analyzing the flight conflict probability, which considers only the scheduled arrival and departure times of flights.

4 Multi-objective Mathematical Model for AGAP The proposed multi-objective airport gate assignment model is given in detail as follows: Sets

Index s: index of periods i,j: index of flights k,r: index of parking areas c: index of ground handling service companies y: index of night-stand flights m: index of bridge-equipped parking areas to which night-stand planes cannot be assigned d: index of parking areas that are already occupied

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Parameters

a j : Scheduled arrival period of flight j , a j ∈ S g j : Scheduled departure period of flight j , g j ∈ S . f j : Ground service company of flight j , f j ∈ C tk : Earliest available period of parking area

W jk :

Utility of assigning flight j to parking area k,

wk : Utility of parking area k

M: A big number which is considered as an upper bound for flights b : Buffer time E (p(i, j)): Expected conflict probability of flight i and j, Decision Variables

Mathematical Model Maximize Z1 =



Wjk ∗ wk ∗ Xjk

(1a)

j∈U k∈N

Minimize Z2 =





(E(p(i, j)) ∗ Yijk

(1b)

k∈N i,j∈U |i3600

11728

0.336

0.0472

6.3 Summary of Results Based on tables demonstrated in the previous section, it is easy to realize that problemspecific heuristic algorithm outperforms in terms of computational time.

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Table 2. Comparison of MOGAP and fuzzy AGAP with parabolic fuzzy distribution function # of Flights

80

B&B for the First Objective Function

B&B for the Second Objective Function

Heuristic Model for Fuzzy Airport Gate Assignment Problem

Objective

Time (s)

Objective

Time (s)

1st Obj.

0.27

0.02157

3600

7010

2nd Obj.

Time (s)

6902

0.0117

>0.02

90

8320

245.2

0.1630

3600

7810

0.1288

0.02

105

8832

204

0.2784

3600

8347

0.1728

0.03

120

10333

348.58

0.3688

3600

9647

0.2612

0.04

140

11992

3600

0.4963

>3600

11584

0.375

0.0502

Taking the first objective into account, namely total flight-to-gate utility, solutions obtained through proposed heuristic methodology approximates the best solution found using the exact solution technique, namely B&B. Furthermore, although B&B seems to have outperformed the proposed heuristic algorithm with respect to solution quality, the robustness of the solutions obtained through problem-specific heuristic is better. It is noteworthy that B&B algorithm fails to provide optimal solutions for the instance with 140 flights in 3600 s. Considering second objective, namely total conflict probability, the exact algorithm is unable to find good solutions in a short computational time. The proposed heuristic algorithm outperforms B&B in terms of solution quality. On the other hand, solutions obtained using heuristic algorithm are more reasonable with respect to both objective functions. Comparing the performance of proposed fuzzy distributions functions, it is noteworthy to highlight that there is not a significant difference between solutions when problem size is small, in other words airport density is low. However, when problem complexity increases due to increasing number of flights, the robustness of the solutions obtained using proposed heuristic algorithm with parabolic fuzzy distribution function is better because parabolic fuzzy distribution function is more sensitive to small changes in idle times.

7 Conclusion and Future Directions In this study, we formulate the AGAP as a multi-objective optimization problem aiming at maximization of total flight-to-gate assignment utility and minimization of total flight conflict probability. Because these two objective functions are naturally conflicting, there is no algorithm that can find an optimal solution in terms of one objective function without degrading the other. Inspired by [13], fuzzy membership of idle times is used in order to have a plausible trade-off between these two conflict objective functions. In view of NP-hardness of the problem, a heuristic algorithm is proposed to solve fuzzy AGAP model in a reasonable computational time. To test the performance of this heuristic algorithm, two real-life and three test instances with different size are employed.

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From constructing test results, it is concluded that the proposed heuristic algorithm is a promising technique to capture good solutions for each objective function in a reasonable amount of computational time, especially for AGAP with big sizes. As a future direction, the proposed fuzzy model can be modified by adding different objective functions. Since heuristic algorithm is only a constructive type, local search strategies can be applied to improve the solutions obtained using this algorithm.

References 1. Yan, S., Huo, C.M.: Optimization of multiple objective gate assignments. Transport. Res. Part A Policy Pract. 35(5), 413–432 (2001) 2. Xu, J., Bailey, G.: The airport gate assignment problem: mathematical model and a tabu search algorithm. In: Proceedings of the 34th Annual Hawaii International Conference on System Sciences, 10-p. IEEE (2001) 3. Mangoubi, R.S., Mathaisel, D.F.: Optimizing gate assignments at airport terminals. Transp. Sci. 19(2), 173–188 (1985) 4. Ding, H., Lim, A., Rodrigues, B., Zhu, Y.: The over-constrained airport gate assignment problem. Comput. Oper. Res. 32(7), 1867–1880 (2005) 5. Bolat, A.: Assigning arriving flights at an airport to the available gates. J. Oper. Res. Soc. 50(1), 23–34 (1999) 6. Obata, T.: Quadratic Assignment Problem: Evaluation of Exact and Heuristic Algorithms (1980) 7. Ornek, M.A., Ozturk, C., Sugut, I.: Integer and constraint programming model formulations for flight-gate assignment problem. Oper. Res. Int. J. 22(1), 135–169 (2020). https://doi.org/ 10.1007/s12351-020-00563-9 8. Ding, H., Lim, A., Rodrigues, B., Zhu, Y.: New heuristics for over-constrained flight to gate assignments. J. Oper. Res. Soc. 55(7), 760–768 (2004) 9. Bolat, A.: Models and a genetic algorithm for static aircraft-gate assignment problem. J. Oper. Res. Soc. 52(10), 1107–1120 (2001) 10. Lim, A., Wang, F.: Robust airport gate assignment. In: 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAI’05), 8–p. IEEE (2005) 11. Paldrak, M., Örnek, M.A.: A GRASP algorithm for multi-objective airport gate assignment problem. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) INFUS 2022. LNNS, vol. 505, pp. 548–557. Springer, Cham (2022). https://doi.org/10.1007/ 978-3-031-09176-6_63 12. Yan, S., Shieh, C.Y., Chen, M.: A simulation framework for evaluating airport gate assignments. Transport. Res. Part A Policy Pract. 36(10), 885–898 (2002) 13. Wei, D.X., Liu, C.Y.: Fuzzy model and optimization for airport gate assignment problem. In: 2009 IEEE International Conference on Intelligent Computing and Intelligent Systems, vol. 2, pp. 828–832. IEEE (2009)

Two-Stage Transportation Model for Distributing Relief Aids to the Affected Regions in an Emergency Response Under Uncertainty Jency Leona Edward and K. Palanivel(B) Department of Mathematics - School of Advanced Sciences, Vellore Institute of Technology (VIT), Vellore, Tamil Nadu 632014, India [email protected], [email protected]

Abstract. Natural disasters have become increasingly frequent, complex, and lengthy, causing devastating effects on people and their belongings. Relief materials aim to save lives, reduce vulnerability, and distribute aid quickly. Despite challenges in distribution, transportation models can reduce costs, time, and other factors. This study proposes a conventional transportation model with two stages to distribute relief aid quickly and efficiently to victims in uncertain scenarios. The model minimizes shipping costs and distributes aid to the least and most affected areas. Fuzzy triangular membership functions are used to redistribute surplus relief items. The model’s effectiveness is demonstrated through numerical illustrations and comparative analysis using conventional and traditional methods. The results are obtained using MATLAB, and the model provides new insights through sensitivity analysis. The proposed transportation model is efficient and effective in distributing relief aid during natural disasters. Keywords: Triangular fuzzy numbers · two-stage fuzzy transportation problem · ∝ -cut interval · VAM · LCM · Russell’s method · Heuristic method-1 and Modi method · Sensitivity analysis · optimal solution

1 Introduction Hitchcock initially tackled transportation issues, which are a subset of linear programming challenges that aim to minimize transportation expenses as products move from multiple sources to different destinations. In reality, uncertainties regarding product availability, demand, and transport costs can make decision-making challenging. This is where the concept of fuzzy transportation problems comes in, and Zadeh’s fuzzy set theory can be used to make decisions in a vague environment. The FTP considers uncertain transportation costs, supply, and demand, and a Two-Stage FTP was proposed to address this. Additionally, Pandian and Natarajan suggested a zero-point method to solve the FTP [13]. Additionally, Ritha and Vinotha presented two stage FTP that minimizes costs while adhering to multiple constraints, and fuzzy geometric programming © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 426–439, 2023. https://doi.org/10.1007/978-3-031-39774-5_49

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can be used to find the best solution to a two-stage multi-objective FTP with trapezoidal supplies and demands [16]. Muruganandam and Srinivasan described a two-stage cost-minimizing FTP with multiple solutions to a multi-objective, fuzzy two-stage transportation problem [10] in 2016. Samanta, Mondal, and Das outlined a multi-objective solid transportation problem (MOSTP) [17] in 2015, with cost and travel time as the two important criteria. They suggested an appropriate discount policy (AUD) for transportation costs, which depends on the number of transportation units, as the shipment cost, supply quantity, demand, etc. are unpredictable in their model. In 2015, Pramanik, Jana, Mondal, and Maiti proposed two mathematical models for two-stage supply chain networks in a Gaussian fuzzy type 2 environment with inaccurate capacitated fixed charge problems [15]. The transportation process involves two stages, from a manufacturing center to potential distribution centers (DCs), and then from DCs to business centers with specific requirements. In 2010, Mohideen and Kumar discussed fuzzy transportation problems using trapezoidal fuzzy numbers and introduced a new algorithm called the fuzzy zero-point approach to quickly find a fuzzy optimal solution to a TP [9]. Liu and Kao highlighted the importance of the transportation model in logistics and supply chain management [7] in 2004, and this work proposes a method to estimate the fuzzy objective value of the FTP, where the supply and demand quantities as well as the cost coefficients are all fuzzy numbers. In 2012, Poonam, Abbas, and Gupta proposed a ranking method with an alpha-optimal solution for handling transportation problems with fuzzy demand and supply, both represented as trapezoidal fuzzy numbers [14]. Narayanamoorthy and Kalyani presented a new method for obtaining the initial basic feasible solution for an FTP [11] in 2015. Dinagar and Palanivel introduced the FTP using membership functions [3] in 2009, and researchers have developed various algorithms and ranking functions to convert fuzzy data into crisp data to address the FTP. Kaliyaperumal and Das proposed a new fuzzy optimization model [4] in 2022, while Khalifa, Kumar, and Alharbi examined a multi-objective fractional STP with fuzzy cost coefficients, supply quantities, demand levels, and/or conveyances to minimize costs [6]. The concept of fuzzy efficiency is introduced to extend the crisp efficient solution, and Basirzadeh proposed a strategy for using it to rank fuzzy numbers in solving various problems [1] in 2011. In a fuzzy transportation problem, there may be crisp or fuzzy quantities, and a new approach is proposed that can solve any type of transportation problem, regardless of whether the objective function is maximized or minimized. Pandian and Natarajan introduced the fuzzy zero-point method for an FTP in 2010, while Chandran and Kandaswamy proposed a method for resolving a complex transportation problem with unknown demand, supply and transportation costs [2] in 2016. Kaur and Kumar used generalized trapezoidal fuzzy numbers to represent product availability, demand, and transportation costs in their proposed method [5] in 2011. Narayanamoorthy, Saranya, and Maheswari proposed a new approach for the initial basic feasible solution to a fuzzy transportation problem, named the fuzzy Russell’s method [12], in 2013, where fuzzy numbers can be of various types, such as triangular, trapezoidal, normal, or abnormal. Louveaux described that logistics for disaster relief is a critical aspect of disaster management, involving the distribution of appropriate supplies to individuals in the right quantities and at right times and locations [8] in 1993. The procedures involve two steps: the preparation phase, where the locations of distribution points and suppliers

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are determined, and the response phase, where decisions about the best suppliers and aid transportation are made. The mathematical challenge of distributing humanitarian supplies is described in the study, which involves reallocating surplus relief supplies from areas that have recovered to those still in need (Fig. 1).

Fig. 1. Schematic representation of the area affected by the catastrophe

2 Fuzzy Concepts Here, we define some of the fundamental definitions necessary for triangular fuzzy numbers. Definition (Fuzzy set): The characteristic function μA (x) of a crisp set A ⊆ X assigns a value of either 0 or 1 to each member in X. This function can be generalized to a function μA (x) such that the value assigned to the element of the universal set X fall within a specified range i.e. μA : X → [0,1]. The assigned value indicates the membership grade of the element in set A. The function μA (x) is called the membership function and the set A = {(x, μA (x)): x ∈ A and μA (x) ∈ [0, 1]} is called a fuzzy set. Definition (Characteristics of a Fuzzy Set): A fuzzy set A, defined on the set of real numbers R is said to be a fuzzy number if its membership function μA : R → [0,1] has the following characteristics. • A is normal. It means that there exists an x ∈ R such that μA (x) = 1. • A is convex. It means that for every x1 , x2 ∈ R, μ ∼ (λ x1 +(1-λ)x2 ) ≥ min {μA (x1 ), A μA (x2 )}, λ ∈ [0,1] • μA is upper semi-continuous. • Support (A) is bounded in R.

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Definition (Fuzzy Number): A fuzzy number A˜ is said to be a non-negative fuzzy number if and μA˜ (x) = 0, ∀ x < 0. Definition (Triangular Fuzzy Number): A fuzzy number A in R is said to be a triangular fuzzy number if its membership function μA : R → [0,1] has the following characteristics. ⎧ x−a 1 ⎪ ⎪ a −a , a1 ≤ x ≤ a2 ⎪ ⎨ 2 1 1, x = a2 μ ∼ (x) = a3 −x A ⎪ ⎪ a3 −a2 , a2 ≤ x ≤ a3 ⎪ ⎩ 0, otherwise a(3)

It is denoted by A˜ = (a(1) , a(2) , a(3) ) Where a(1) is core (A), a(2) is left width and is right width.

Definition 2.5 (Ranking technique of Triangular Fuzzy number): Several approaches for the ranking of fuzzy numbers have been proposed. An efficient approach for comparing the fuzzy numbers is by the use of a ranking function based on their graded means, That is, for every A= (a(1) , a(2) , a(3) ) ∈ F(R), the ranking function : F(R) → R by graded mean is defined as (A)=( a1 +4a62 +a3 ). For any two triangular fuzzy   numbers A = a(1) , a(2) , a(3) and B = (b(1) , b(2) , b(3) ), in F(R), we have the following comparison. • • • •

A B If and only if (A) > (B) A ≈ B If and only if (A) >= (B) A − B If and only if (A) − (B) = 0

A triangular fuzzy number A = (a(1) , a(2) , a(3) ) in F(R) is said to be positive if (A) > 0 and denoted by A > 0. Also if (A) > 0, then A > 0 and if (A) = 0, then A ≈0. If (A) = (B), then the triangular numbers A and B are said to be equivalent and is denoted by A ≈ B.

3 Theoretical Development The catastrophe-prone zone is divided into two phases, with phase I involving the minimum fuzzy demand bq of a particular commodity at destination q and the fuzzy availability ap of the same product at origin p. The Two-stage Fuzzy Cost Minimization Transportation Problem (FCMTP) aims to distribute the minimum demand of destinations in phase I, while the excess quantity p ap − bq left over after phase I is completed is supplied to the destinations in phase II from the origin. The mathematical description of the phase I problem is provided. min [c1 (x)] = min [(cpq (xpq ))]

x∈oi1

x∈oi2

(1)

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where the set oi1 is given by 

z

oi1 =

xpq ≤ ap p = 1, 2, ..y p=1 xpq = bq q = 1, 2, ..z

yq=1

to a feasible solution x = (xpq ) of the phase-I problem, xpq ≥ 0, ∀ (p, q) corresponding   the set oi2 = {x = xpq } of feasible solutions of the phase-II problem is given by



xpq ≥ 0 ∀ (p, q) Where a p is the remaining quantity at the pth origin so the phase-I     is a p , ie., a p =ap − q xpq . It is clearly explained as, p a p = p ap − q bq . Thus the phase-II problem would be mathematically stated as:

min [c2 (x)] = min [max(cpq (xpq ))]

x∈oi2

x∈oi2

|x|

(2)

The objective is to determine the feasible solution x = (xpq ) for the phase-I problem, which results in the minimum sum of transportation costs for phase-II. This can be defined as the Two-stage Fuzzy Cost Minimization Transportation Problem (FCMTP). min [c1 (x) + min c2 (x)]

x∈oi2

x∈oi2

(3)

3.1 Algorithm Step 1: Construct the FTP, where supply and demand takes the triangular fuzzy number. Step 2: Calculate the α-cut interval for supply and demand, Where ∝ ∈ (0,1]. Step 3: Calculate any integer which lies any integer which lies in the γ-cut interval for the supply and demand, which is considered as the α− optimal parameters. The integers  y for ap and bq should be chosen in such a way that p=1 ap = zq=1 bq . Step 4: Divide FTP into two-stages based on α − optimal parameters, where each stage is a balanced one. Step 5: Apply the conventional traditional methods namely VAM, LCM, Russell’s, Heuristic-1 and MODI method to solve the FTP in two stages to obtain the optimal objective value namely C1 and C2 . Step 6: Finally calculate the optimal objective value for the FCMTP namely Minimize Z = C1 + C2 3.2 Numerical Example This manuscript presents a novel approach to providing disaster relief materials to victims in catastrophe management. The disaster-prone zone is divided into two phases: the

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less affected and most affected areas, with the aim of minimizing transportation costs. During the first stage, relief goods are transported to heavily flooded areas such as Thiruvallur, Chennai, and Kancheepuram, using trucks, lorries, and special vehicles. In the second stage, surplus relief materials from other destinations are supplied to areas like T Nagar, Ashok Nagar, and Teynampet, which are still in need of relief assistance. Due to road obstructions, relief materials are transported via dumpers, ropes, and boats. This approach ensures that relief supplies are distributed in a timely and cost-effective manner to the affected regions, particularly those that are most in need of assistance (Figs. 2, 3 and Table 1).

Fig. 2. Graphical representation of Phase I illustrated in a chart

Fig. 3. Graphical representation of Phase II illustrated in a chart

Table 1. The fuzzified Transportation Problem and its related values D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

(9, 10, 11)

Oi2

14

21

27

33

30

(14, 15, 16)

Oi3

15

23

29

17

26

(19, 20, 21)

Oi4

13

18

24

32

28

(24, 25, 26)

Fuzzy Demand

(5, 6, 7)

(13, 14, 15)

(10, 11, 12)

(20, 21, 22)

(17, 18, 19)

Let ∝ = 0.5 a1 ∈ [9.5, 10.5], a2 ∈ [14.5, 15.5], a3 ∈ [19.5, 20.5], a4 ∈ [24.5, 25.5]. b1 ∈ [5.5, 6.5], b2 ∈ [13.5, 14.5], b3 ∈ [10.5, 11.5], b4 ∈ [20.5, 21.5], b5 ∈ [17.5, 18.5]. The α-optimal parameters are (Tables 2, 3, 4 and 5) a1 = 10, a2 = 15, a3 = 20, a4 = 25; b1 = 6, b2 = 14, b3 = 11, b4 = 21, b5 = 18. (4)

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D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

10

Oi2

14

21

27

33

30

15

Oi3

15

23

29

17

26

20

Oi4

13

18

24

32

28

25

6

14

11

21

18

Fuzzy Demand

Let n = 2, Where Supply and Demand is distributed as per the requirement in Phase-1and Phase-2 respectively. 1st phase We take a1 = 3, a2 = 6, a3 = 8, a4 = 7; b1 = 2, b2 = 4, b3 = 3, b4 = 7, b5 = 8. Table 3. The fuzzified capacity and demand values used in Phase I computation D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

3

Oi2

14

21

27

33

30

6

Oi3

15

23

29

17

26

8

Oi4

13

18

24

32

28

7

2

4

3

7

8

Fuzzy Demand

C1 = 502. 2nd phase We take a1 = 7, a2 = 9, a3 = 12, a4 = 18; b1 = 4, b2 = 10, b3 = 8, b4 = 14, b5 = 10. Table 4. The fuzzified capacity and demand values used in Phase II computation D1 Oi1

7

Oi2

14(4)

D2

D3

D4

D5

12

16(7)

19

25

7

27

33(2)

30(3)

9

21

Fuzzy Capacity

Oi3

15

23

29

17(12)

26

12

Oi4

13

18(10)

24(1)

32

28(7)

18

Fuzzy Demand

4

10

8

14

10

C2 = 928.

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Table 5. The optimal table for the second phase D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

7

Oi2

14

21

27

33

30

9

Oi3

15

23

29

17

26

12

Oi4

13

18

24

32

28

18

4

10

8

14

10

Fuzzy Demand

Therefore the optimal objective value for the given FTP is Minimum Z = C1 +C2 = Rs. 1430.

4 Results and Discussion This section discusses supply and demand using the ∝ – optimal parameters from Eq. (4). An exemplary case is shown in Tables 6 and 7. The comparison of the optimal outcomes obtained is presented in the following Table 8 along with a graphical representation in Fig. 4. Additionally, the computational result is provided in the case, when the supply and demand takes the unbalanced ∝ − optimal parameters as given in Tables 9 and 10, the ideal objective value obtained here is lower than the objective value provided in the Eq. (5). Table 6. When the supply and demand take the α-optimal parameters as in Eq. (4) D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

10

Oi2

14

21

27

33

30

15

Oi3

15

23

29

17

26

20

Oi4

13

18

24

32

28

25

6

14

11

21

18

Fuzzy Demand

The optimal objective value obtained in this case when the supply and demand take the α-optimal parameters as per in Eq. (4) is Min Z = Rs. 1430

(5)

Thus FTP must be divided into 2-phases in such a way that each stage must be a balanced one, to get the exact optimal objective value namely Rs. 1430, which coincide with the Eq. (5). When the supply and demand, take the α-optimal parameters in such a way that

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J. L. Edward and K. Palanivel Table 7. After applying the VAM, the optimal table is D1

D2

D3

D4

D5

Fuzzy Capacity

12

16(10)

19

25

10

30(8)

15

Oi1

7

Oi2

14(5)

21

27

33(2)

Oi3

15(1)

23

29

17(19)

26

20

Oi4

13

18(14)

24(1)

32

28(10)

25

Fuzzy Demand

6

14

11

21

18

Table 8. Optimal objective value obtained by the conventional traditional methods Phases

VAM

LCM

Russell’s

Heuristic method-1

VAM and MODI method

Phase-1

502

502

502

502

502

Phase-2

928

928

928

928

928

Fig. 4. Chart showing the graphical representation of values obtained using traditional methods

y



= zq=1 bq (ie.,) a1 = 9, a2 = 16, a3 = 19, a4 = 24; b1 = 5, b2 = 13, b3 = 10, b4 = 22, b5 = 19. The optimal result obtained here is Minimize Z = 1414. The optimal objective value obtained here is less than the Eq. (5). p=1 a p

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Table 9. Represents the supply and demand are unbalanced D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

9

Oi2

14

21

27

33

30

16

Oi3

15

23

29

17

26

19

Oi4

13

18

24

32

28

24

5

13

10

22

19

Fuzzy Demand

Table 10. After applying VAM method, the optimal solution is D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16(9)

19

25

Oi2

14(6)

21

27

33(3)

30(8)

16

Oi3

15

23

29

17(19)

26

19

13

18(13)

24(1)

32

28(10)

24

Oi4 Odummy

0

0

0

0

0(1)

Fuzzy Demand

5

13

10

22

19

9

1

5 Sensitivity Analysis A sensitivity analysis is a useful tool for examining the impact of changes in input parameters on the output of a model. In this context, a sensitivity analysis is conducted on the input and output data from Tables 8, 9, 10, and 11, using the VAM method. By varying the output data, this analysis allows us to better understand how changes in the inputs affect the model’s output. This approach is particularly useful in the context of disaster management, where small changes in input parameters can have significant effects on the overall outcome. Therefore, conducting a sensitivity analysis can help identify key factors that contribute to successful relief operations and ensure that resources are allocated effectively and efficiently (Figs. 5, 6 and 7). Here C1 = 500. Here C2 = 475. Therefore the optimal objective value for the given FTP for Tables 12, 13 and 14 is (Table 15). Minimum Z = C1 +C2 = Rs. 975.

436

J. L. Edward and K. Palanivel Table 11. Represents the supply and demand for 1st phase D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

4

Oi2

14

21

27

33

30

8

Oi3

15

23

29

17

26

7

Oi4

13

18

24

32

28

5

2

4

3

7

8

Fuzzy Demand

Table 12. Optimal result for 1st phase D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16(3)

19

25(1)

4

Oi2

14(2)

27

33

30(6)

8

29

17(7)

26

7 5

Oi3

21

15

23

Oi4

13

18(4)

24

32

28(1)

Fuzzy Demand

2

4

3

7

8

Table 13. Represents the supply and demand for 2nd phase D1

D2

D3

D4

D5

Fuzzy Capacity

Oi1

7

12

16

19

25

3

Oi2

14

21

27

33

30

6

Oi3

15

23

29

17

26

8

Oi4

13

18

24

32

28

7

4

3

2

8

7

Fuzzy Demand

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437

Table 14. Applying VAM method, the optimal result for 2nd phase D1 Oi1

7

Oi2

14(4)

D2

D3

12

16(2)

21

D4

D5

Fuzzy Capacity

19

25(1)

3

27

33

30(2)

6

Oi3

15

23

29

17(8)

26

8

Oi4

13

18(3)

24

32

28(4)

7

Fuzzy Demand

4

3

2

8

7

Table 15. The optimal objective value as determined by standard ways Phases

VAM

LCM

Russell’s

Heuristic method-1

VAM and MODI method

Phase-1

500

500

500

500

500

Phase-2

475

475

475

475

475

Fig. 5. Input representations used for the first phase of the study presented in a table

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D1 7 14 15 13 Fuzzy Capacity 3687

8 6 4 2 0

D5 25 30 26 28

D2 12 21 23 18

D3 16 27 29 24 D4 19 33 17 32

Fig. 6. Input representations used for the second Phase of the study presented in a table

Fig. 7. Graphical representation of sensitivity analysis for obtained values using traditional methods

6 Conclusion Transportation models optimize the transportation of goods while minimizing costs by considering factors like transportation mode availability, vehicle capacity, and distance. Fuzzy transportation problems use fuzzy numbers to represent supply, demand, and costs, allowing for imprecision and uncertainty. Solving these problems can be challenging, and conventional methods may not be suitable. Techniques like fuzzy linear programming, dynamic programming, and heuristic algorithms have been developed. The manuscript discusses a two-stage fuzzy transportation problem of distributing relief materials to disaster-stricken areas. The first phase meets minimum demand, and the

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second phase supplies surplus materials to areas in need. The objective is to minimize fuzzy transportation costs in both phases, ensuring efficient delivery of relief materials.

References 1. Basirzadeh, H.: An approach for solving fuzzy transportation problem. Appl. Math. Sci. 5(32), 1549–1566 (2011) 2. Chandran, S., Kandaswamy, G.: A fuzzy approach to transport optimization problem. Optim. Eng. 17(4), 965–980 (2012). https://doi.org/10.1007/s11081-012-9202-6 3. Dinagar, D.S., Palanivel, K.: The transportation problem in fuzzy environment. Int. J. Algorithms Comput. Math. 2(3), 65–71 (2009) 4. Kaliyaperumal, P., Das, A.: A mathematical model for nonlinear optimization which attempts membership functions to address the uncertainties. Mathematics 10(10), 1743 (2022) 5. Kaur, A., Kumar, A.: A new method for solving fuzzy transportation problems using ranking function. Appl. Math. Model. 35(12), 5652–5661 (2011) 6. Khalifa, H.A.E.-W., Kumar, P., Alharbi, M.G.: On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment. J. Intell. Syst. 30(1), 620–635 (2021) 7. Liu, S.-T., Kao, C.: Solving fuzzy transportation problems based on extension principle. Eur. J. Oper. Res. 153(3), 661–674 (2004) 8. Louveaux, F.: Stochastic Location Analysis: Location Science 1, 127–154. Location Science (1993) 9. Mohideen, S.I., Kumar, P.S.: A comparative study on transportation problem in fuzzy environment. Int. J. Math. Res. 2(1), 151–158 (2010) 10. Muruganandam, S., Srinivasan, R.: Optimal solution for multi-objective two stage fuzzy transportation problem. Asian J. Res. Soc. Sci. Human. 6(5), 744–752 (2016) 11. Narayanamoorthy, S., Kalyani, S.: Finding the initial basic feasible solution of a fuzzy transportation problem by a new method. Int. J. Pure Appl. Math. 101(5), 687–692 (2015) 12. Narayanamoorthy, S., Saranya, S., Maheswari, S.: A method for solving fuzzy transportation problem (FTP) using fuzzy Russell’s method. Int. J. Intell. Syst. Appl. 5(2), 71 (2013) 13. Pandian, P., Natarajan, G.: A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4(2), 79–90 (2010) 14. Poonam, S., Abbas, S., Gupta, V.: Fuzzy transportation problem of trapezoidal numbers with cut and ranking technique. Int. J. Fuzzy Math. Syst. 2(3), 263–267 (2012) 15. Pramanik, S., Jana, D.K., Mondal, S.K., Maiti, M.: A fixed-charge transportation problem in two-stage supply chain network in Gaussian type-2 fuzzy environments. Inf. Sci. 325, 190–214 (2015) 16. Ritha, W., Vinotha, J.M.: Multi-objective two stage fuzzy transportation problem (2009) 17. Samanta, S., Mondal, S.K., Das, B.: A multi-objective solid transportation problem with discount and two-level fuzzy programming technique. Int. J. Oper. Res. 24(4), 423–440 (2015)

A Fuzzy Based Optimization Model for Nonlinear Programming with Lagrangian Multiplier Conditions K. Palanivel1(B)

and Selcuk Cebi2

1 Department of Mathematics – School of Advanced Sciences, Vellore Institute of Technology

(VIT), Vellore, Tamil Nadu 632014, India [email protected], [email protected] 2 Department of Industrial Engineering, Yildiz Technical University, Istanbul, Turkey [email protected]

Abstract. A fuzzy based mathematical model on Lagrangian multiplier conditions has been proposed to address the Non-linear Programming (NLP) with equality constraints. Furthermore, the model demonstrates how multivariable optimization issues can be solved using membership functions. This model is excellent for problem-solving because there is no need to explicitly solve the conditions and utilize them to eliminate additional variables. Then the sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors of the bordered Hessian matrix of second derivatives of the Lagrangian expression. It also demonstrates how the Lagrange multiplier method can be used for proving the Jacobian matrix. Additionally, the model can be considered in three stages: that is, mathematical formulation, computational procedures, and numerical illustration with comparative analysis. Likewise, the model illustrates the considered problem using two distinct approaches, namely membership functions (MF) and robust ranking index. Finally, the comparison analysis provides detailed results and discussion that justify the optimal outcome in order to address the vagueness of certain NLPPs. Keywords: Nonlinear optimization · Fuzzy nonlinear programming problem · Lagrange’s multiplier conditions with fuzziness · Trapezoidal fuzzy membership functions.

1 Introduction A linear relation between most of the model parameters seems to be very accurately adaptable to linear algebraic mostly resulting in complicated issues. Because of the relationship in nonlinearity, there is a significant difference between local and global results. It further implies that any local optimum solution must be evaluated for optimality over the whole feasible region, rather than just at the extreme points, which is possible in an LPP. This furthermore suggests that simplex-type algorithms are unsuitable for solving NLPPs. Moreover, NP often describes higher-level problems than LP. As a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 440–453, 2023. https://doi.org/10.1007/978-3-031-39774-5_50

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result, this scenario is always complex and tough, even when part of the constraints are linear and the cost function is nonlinear. For example, the feasible set may or may not be convex, and the optimum solution may be located within the feasible set, on its boundary, or at its vertex. The scientific programming challenge, for the most part, oversees the optimal use or distribution of constrained resource to achieve the ideal goal. Because of the ambiguous, emotive nature of the problematic definition or has a specific layout, the fuzzy NLPP is useful in dealing with concerns. An objective function must improve in this situation while operating within specified limits. [1, 2] established fuzzy theory and fuzzy rule-based decision-making, and in decision challenges, the right choice is employed to achieve an optimal result [3]. In most real-life circumstances, the verdicts are vague, and preliminary attempts at the choices are required to build a suitable model or cases to be analyzed. Similarly, we include fuzziness in our models of such scenarios to offer methods for dealing with fuzzy data [4]. An optimum solution in linear programming is one that meets both the constraints and the goal function. Accordingly, such a problem has an objective function, as well as variables, which include constraints and coefficients all are described in the form of fuzzy [4–6]. If the objective or limitations are nonlinear, at that point, we think of it as a nonlinear programming problem. In this model to tackle such an optimization problem, a fuzzy mathematical model has been proposed to address the NLP with equality constraints in terms of fuzziness using Lagrange’s multiplier condition to find an optimum solution. In addition, the model suggests how multivariable optimization problems have been demonstrated using boarded Hessian matrix. However, this model represents the mathematical formulation and is followed by the computational procedure with the numerical illustration. The procedure employs trapezoidal fuzzy MFs and its mathematical calculations to describe the illustrated numerical results of NLP. It has offered a fuzzy based optimization model the NLP, it handles the vagueness and also clarified its optimum results in terms of MFs [6–8]. In addition, the proposed model was carried out numerically in two different cases: the first case was conferred with fuzziness, while the second case was presented with a robust ranking. Finally, the investigation of the optimal solution in the preceding two situations indicates the newness and costeffectiveness of a fuzzy model, clearing the ambiguity and providing significantly more optimum results.

2 Review of Literature The following research works illustrate the literature on fuzzy NLP: Tang and Wang [9] proposed a nonsymmetrical model for NLPPs with fuzzy penalty coefficients. It also tried to develop a structured model to deal with the problem in order to demonstrate the available resources and constraints using fuzzy, which is a sort of nonlinear membership function. Tang et al. [10] used an approach to the genetic algorithm for a penalty function and gradient search to develop a hybrid genetic algorithm with a genetic disorder in along with the weighted gradient for a certain kind of NLP. Fung et al. [11] extended the hybrid genetic algorithm and concentrated on critical techniques for extending to NLPPs with the two kinds of constraints. Sarimveis and Nikolakopoulos [12] offered a strategy for constrained penalty weight optimization issues focused on the LUDE algorithm. Syau and Lee [13] discussed fuzzy convex optimization techniques and also gave numerous

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instances of multiobjective programming. Chen [14] presented a mathematical model for creating a cost-based queueing decision system by membership functions using Yager’s ranking index approach. Qin et al. [15] proposed an interval parameter nonlinear system for controlling stream water quality in a fuzzy situation, in which fuzzy programming and interval techniques are coupled in a single structure to illustrate the fuzziness of nonlinear restrictions on both the sides. Kassem [16] created an approach to estimate the consistency of optimal findings for multiobjective NLPPs. Ravi Sankar et al. [17] presented a novel method for optimizing the nonlinear objective function that employs a genetic algorithm with coefficients and constraints under fuzziness. Abd-El-Wahed et al. [18] suggested a hybrid methodology that combines particle swarm optimization and genetic algorithms, it associates two heuristic optimization techniques. Jameel and Sadeghi [19] distinguished the crisp and fuzzy NLPs, it demonstrating a more accurate outcomes. Further, the scheduling problem was initially modeled as a stochastic optimization problem, specifically NLP, and then converted as a multiobjective optimization issue that provides optimal scheduling based on multiple models [20]. Bi-level preferable operation issues have been proposed through a methodology, using the Kuhn-Tucker approach for estimating durable characteristics [21]. The fuzzy Lagrangian technique has been defined which allows vector machines in the framework of digital information to be obtainable for biological data interpretation [22]. However, the optimization frameworks that deal with renewable energy, automation, risk control, manufacturing inventory [23], and the production workflow [24] are effectively addressed. Those exact formulations are vague and inaccurate, which can be investigated in terms of fuzziness [25]. A number of studies offered multiple approaches for handling nonlinear issues such as scheduling and allocation problems in supply chain management, quadratic programming, and several other problems are investigated in a fuzzy situation; it begins with the most appropriate method to take care of all of them to meet the stack holder’s needs cost-effectively [26, 27].

3 Preliminaries In this section, we will look over certain important fundamental notions and perspectives on fuzzy arithmetic [6, 7], subsequently it continues to address a couple of necessary definitions: Definition [7]: A trapezoidal fuzzy number M can be represented as M = [m1 , m2 , m3 , m4 ] with the ⎧ following MF μM (x) is illustrated below. 1 ⎪ mx−m , m1 ≤ x ≤ m2 ⎨ 2 −m1 μM (x) = 1, m2 ≤ x ≤ m3 , ⎪ ⎩ x−m4 , m ≤ x ≤ m 3 4 m3 −m4 Definition [7] (α cut): Assumed a fuzzy set M in X and any real number α in [0, 1], then the α -cut of M, denoted by α M is the crisp set α M = {x ∈ X : μM (x) ≥ α}. For illustration, let M be a fuzzy set whose membership function is given as above μM (x). To find the α-cut of M, where α ∈ [0, 1], let us set the reference functions of M to each

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left and right. i.e., α =

x(1) − m1 x(2) − m4 &α = m2 − m1 m3 − m4

Expressing x to α, where x(1) = (m2 − m1 )α + m1 andx(2) = m4 + (m3 − m4 )α = [x(1) , which provides the α – cut of M is α M (2) x ] = [(m2 − m1 )α + m1 , m4 + (m3 − m4 )α]. Definition [7] (Robust ranking index): The robust ranking index satisfies compensation, homogeneity, and additive properties, and produces results that are controlled by human perception. If M is a fuzzy number, the robust ranking 1   index is measured as R(M ) = 0 (0.5) ∗ [MαL , Mαu ]d α, Where [MαL , Mαu = [(m2 − m1 )α + m1 , m4 + (m3 − m4 )α] is the α cut of the fuzzy number M. Here the robust ranking index R(M) offers the numerical significance of fuzzy M.

4 The Fuzzy Model for Nonlinear Programming The attention to fuzzy optimization difficulties in NLP studies is primarily limited, particularly fuzzy quadratic programming, it has received little attention. Aside from that, there are numerous types of fuzzy nonlinear difficulties in various real-world issues, most notably in complex industrial systems. In the discipline of NLP, the attention to fuzzy optimization challenges is often limited. Although there is little concern in using NLP to address the issues about vagueness. In addition from that, many different types of fuzzy NLPs arise in various real-world challenges, particularly in complex systems of production. Conventional approaches cannot represent and tackle it. Even so, research on modeling and improving methodologies for NLP programming in a fuzzy framework is not only important in the notion of fuzzy optimization, but it is also vital and important in the application to the challenges. Accordingly, the fuzzy optimization model has been formulated in three stages: namely problem definition, computational process, and numerical illustration ending with a comparative analysis. 4.1 Formulation of the NLPP with Equality Constraints in Terms of Fuzziness [8] The

difficulty a fuzzy vector has been well-defined in fuzzy  of finding 

NLPP.  (k) (k) (k) (k) , for all k = 1, 2, 3, 4., where xi , i = 1, 2, 3, x1 , x2 , · · · , xn ··· , n &for all k = 1, 2, 3, 4 is a trapezoidal fuzzy membership function, which optimizes (maximize/minimize) the objective function Z, which is a real-valued function of ‘n’ fuzzy variables defined by

 

 (k) (k) (1) Z (k) = f x1 , x2 , · · · , xn(k) , for all k = 1, 2, 3, 4.

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Under the constraints,

 

 

 (k) (k) (k) gj x1 , x2 , · · · , xn(k) = bj ; for all k = 1, 2, 3, 4 & j = 1, 2, · · · , m. where g j s are ‘m’ real-valued function of ‘n’ fuzzy variables and bj s are ‘m’ fuzzy constants, and

 (k) xi ≥ 0, i = 1, 2, 3, · · · , n & for all k = 1, 2, 3, 4. Moreover, as stated before, can be restated as.  

the problem   (k) (k) (k) Maximize Z (k) = f x1 , x2 , · · · , xn , for all k = 1, 2, 3, 4. Under the constraints (2) where g j s are ‘m’ real-valued function of ‘n’ fuzzy variables and bj s are ‘m’ fuzzy constants, and

 (k) xi ≥ 0, i = 1, 2, 3, · · · , n & for all k = 1, 2, 3, 4. (3) The fuzzy vector that satisfies conditions (2) and (3) is a feasible solution to the fuzzy NLP.  

 (k) (k) X (k) = x1 , x2 , · · · , xn(k) , for all k = 1, 2, 3, 4.

4.2 Computational Procedure This section provides the detailed computational procedure of the model for the algorithm of the Lagrangian multiplier necessary conditions followed by the multivariate optimization with Jacobian matrices. 4.3 Algorithm of Lagrangian Multiplier Conditions Using the Lagrange’s multiplier conditions, the following step by step procedure will help to find an optimum solution to a fuzzified NLPP.

 (k) Step 1: Gj (x(k) ) = gj x(k) − bj , for all k = 1, 2, 3, 4 & j = 1, 2, · · · , m.

 (k) and m Step 2: Form the Lagrangean function with the objective function f xi 

(k) (k) 

  (k) (k) (k) , i = 1, 2, 3, · · · , n & constraints are L x , λ − m = f xi j=1 λj gj x for all k = 1, 2, 3, 4., where λ(k) is called Lagrangean multipliers.

 (k) are given by the The necessary conditions for maximum or minimum of f xi existence of extreme point: ∂L ∂xi(k)

= 0, i = 1, 2, 3, · · · , n & for all k = 1, 2, 3, 4.

A Fuzzy Based Optimization Model for Nonlinear Programming

∂L (k)

∂λi

445

= 0, i = 1, 2, 3, · · · , n & for all k = 1, 2, 3, 4.

Step 3: Solve the equations in step 2 to obtain the extreme points for the Lagrangean function. Step 4: Determine the extreme points obtained from the maximum or minimum points using the sufficiency conditions. 4.4 Multivariable Optimization A multivariable optimization is a scalar function of a several variables. In this section we solve the fuzzy nonlinear programming problem with equality constraint using Lagrange multipliers method with the help of Jacobian matrix. 4.4.1 Jacobian Matrix Now, the problem becomes:

 

  (k) (k) x1 , x2 , · · · , xn(k) , for all k = 1, 2, 3, 4. Maximize Z (k) = f Subject to  the

constraints 

 (k) (k) (k) = 0, j = 1, 2, 3, · · · , m & for all k = 1, 2, 3, 4, gj x1 , x2 , · · · , xn exists, it is contained among the solutions to system ∂L ∂xi(k) ∂L

= 0, i = 1, 2, 3, · · · , n & for all k = 1, 2, 3, 4.

= 0, i = 1, 2, 3, · · · , n & for all k = 1, 2, 3, 4. ∂λi(k)



 (k) (k) and gj xi ; j = 1, 2, 3, · · · , m & for all k = 1, 2, 3, 4 all have Provided f xi continuous first order partial derivatives and the m × n Jacobian matrix, ∂g J = (k)j , i = 1, 2, 3, · · · , n, j = 1, 2, 3, · · · , m & for all k = 1, 2, 3, 4, has rank ∂xi

matX = X ∗ . The method of Lagrange multipliers is equivalent to using the constraint equations to eliminate certain of the x variables from the objective function and then solving an unconstrained maximization problem in the remaining x variables.

5 Numerical Illustration This section addresses two cases of numerical illustrations that can simplify the models for solving the problem of fuzzy NLP using trapezoidal membership functions and their mathematical calculations [6, 7, 25]. The fuzzy model explains the procedure which employs the membership function approach in case 1 and, the same problem was investigated with the help of the robust ranking approach in case 2. Let us consider a nonlinear programming problem using the Lagrange multipliers method and the Jacobian matrix: the fuzzified form of the considered NLPP can be stated as below:





 (k) (k) (k) Maximize − [−1, 0, 2, 3] x1 − [−1, 0, 2, 3] x2 − [−1, 0, 2, 3] x3 ,

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for all k = 1, 2, 3, 4. Subject to the constraints,

2

 (k) (k) + [−1, 0, 2, 3] x2 − [1, 2, 4, 5] = [−2, −1, 1, 2], [−1, 0, 2, 3] x1 for all k = 1, 2, 3, 4.

   (k) (k) (k) x1 , x2 , x3 ≥ 0, for all k = 1, 2, 3, 4.

(4)

5.1 Case (I): NLP with Fuzzy Membership Functions The above NLPP has been optimized with fuzziness by employing the Lagrange’s multiplier conditions, as discussed earlier. The Lagrangian function is then

Now the fuzzified form of the necessary conditions for the Lagrange multiplier method for maximizing the above NLPP is,

 (k) (k) (k) −[−1, 0, 2, 3] − λ1 [0, 1, 3, 4] x2 − λ2 [−1, 0, 2, 3] = 0, for all k = 1, 2, 3, 4. (5) (k)

(k)

−[−1, 0, 2, 3] − λ1 [−1, 0, 2, 3] − λ2 [1, 2, 4, 5] = 0, for all k = 1, 2, 3, 4. (6) (k)

−[−1, 0, 2, 3] − λ2 [0, 1, 3, 4] = 0, for all k = 1, 2, 3, 4.

(7)

2

 (k) (k) + [−1, 0, 2, 3] x2 − [1, 2, 4, 5] = 0, for all k = 1, 2, 3, 4. [−1, 0, 2, 3] x1 (8)





 (k) (k) (k) −[−1, 0, 2, 3] x1 + [1, 2, 4, 5] x2 + [0, 1, 3, 4] x3 − [5, 6, 8, 9] = 0,

A Fuzzy Based Optimization Model for Nonlinear Programming

for all k = 1, 2, 3, 4.

447

(9)

The relevant computations for obtaining the FMF for the above Lagrangian necessary conditions are as follows-for the Eq. (5), ⎧ x+1 for − 1 ≤ x ≤ 0 ⎪ ⎪ ⎪ ⎨ 1 for 0 ≤ x ≤ 2 , = μA˜ (x) ⎪ −x + 3 for 2 ≤ x ≤ 3 ⎪ ⎪ ⎩ 0 otherwise ⎧  ⎪ ± x1 λ1 + 4 for 0 ≤ x ≤ x1 λ1 ⎪ ⎪ ⎪ ⎨ 1 for x1 λ1 ≤ x ≤ 3x1 λ1 μB(·)  ˜ D ˜ C(·) ˜ (x) = ⎪ ± x1 λ1 + 4 for 3x1 λ1 ≤ x ≤ 4x1 λ1 ⎪ ⎪ ⎪ ⎩ 0 for otherwise ⎧  ⎪ ⎪ −λ + 3 ± λ22 − 2λ2 + 1 2 ⎪ ⎪ ⎪ for − λ2 ≤ x ≤ 0 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ 1 for 0 ≤ x ≤ 2λ2 μE(·) (x) =  ˜ F˜ ⎪ ⎪ λ2 + 5 ± λ22 − 2λ2 + 1 ⎪ ⎪ ⎪ ⎪ for 2λ2 ≤ x ≤ 3λ2 ⎪ ⎪ 2 ⎪ ⎪ ⎩ 0 otherwise Therefore,  ⎧ ⎪ −2x + λ2 − 5 ± λ22 − 2λ2 + 2x1 λ1 + 9 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ + 1 ≤ x ≤ −x1 λ1 for λ ⎪ 2 ⎪ ⎪ ⎪ ⎨ 1 for − x1 λ1 ≤ x ≤ −3x1 λ1 − 2λ2 − 2 μ−A−  ˜ B(·) ˜ C(·) ˜ D− ˜ E(·) ˜ F˜ (x) = ⎪ ⎪ ⎪ 2x − λ2 − 11 ± λ22 − 2λ2 + 2x1 λ1 + 9 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ for − 3x1 λ1 − 2λ2 − 2 ≤ x ≤ −4x1 λ1 − 3λ2 − 3 ⎪ ⎪ ⎩ 0 otherwise

Similarly for the Eq. (6), (7), (8) and (9) we get,  ⎧ ⎪ −2x + λ1 + λ2 − 5 ± λ21 + λ22 − 2λ1 − 6λ2 + 10 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ for λ1 − λ2 + 1 ≤ x ≤ −2λ2 ⎪ ⎪ ⎪ ⎪ ⎨ 1 for − 2λ2 ≤ x ≤ −2λ1 − 4λ2 − 2 μ−A− (x) =  ˜ B(·) ˜ C− ˜ D(·) ˜ E˜ ⎪ 2 2 ⎪ ⎪ ⎪ 2x − λ1 − λ2 − 11 ± λ1 + λ2 − 2λ1 − 6λ2 + 10 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ − 4λ for − 2λ 1 2 − 2 ≤ x ≤ −3λ1 − 5λ2 − 3 ⎪ ⎪ ⎩ 0 otherwise

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μ−A− ˜ B(·) ˜ C ˜ (x) =

⎧ −x + λ2 − 3 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨

for

1 ≤ x ≤ −λ2

for − λ2 ≤ x ≤ −3λ2 − 2

 2x − λ2 − 12 ± λ22 − 4λ2 + 4 ⎪ ⎪ ⎪ for − 3λ2 − 2 ≤ x ≤ −4λ2 − 3 ⎪ ⎪ 2 ⎪ ⎪ ⎩ 0 otherwise

 ⎧ ⎪ x + 2x − 5 ± 2x12 + x22 − 2x2 + 9 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ 2 ⎪ for x1 + x2 + 1 ≤ x ≤ 2 ⎪ ⎪ ⎪ ⎪ ⎨ 1 for 2 ≤ x ≤ −2x12 − 2x2 + 4 μ−A(·)  ˜ B(·) ˜ C− ˜ D(·) ˜ E+ ˜ F˜ (x) = ⎪ ⎪ ⎪ −x2 − 2x + 5 ± 2x12 + x22 − 2x2 + 9 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ for − 2x ⎪ 1 − 2x2 + 4 ≤ x ≤ −3x1 − 3x2 + 5 ⎪ ⎩ 0 otherwise  ⎧ ⎪ x + x + x − 2x + 4 ± x12 + x22 − 2x1 − 6x2 + 10 1 2 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ − x − 5 ≤ x ≤ −2x2 − x3 − 6 for x ⎪ 1 2 ⎪ ⎪ ⎪ ⎨ 1 for − 2x2 − x3 − 6 ≤ x ≤ −2x1 − 4x2 − 3x3 − 8 μ−A(·)  ˜ B− ˜ C(·) ˜ D− ˜ E(·) ˜ F− ˜ G ˜ (x) = ⎪ ⎪ ⎪ −x1 − x2 − x3 + 2x − 36 ± x12 + x22 − 2x1 − 6x2 + 10 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ − 4x − 3x3 − 8 ≤ x ≤ −3x1 − 5x2 − 4x3 − 9 for − 2x 1 2 ⎪ ⎪ ⎩ 0 otherwise

Fuzzified solution of the above five equations, 

−[−1, 0, 2, 3] − [0, 1, 3, 4][x1 − 2, x1 − 1, x1 + 1, x1 + 2] [λ1 − 2, λ1 − 1, λ1 + 1, λ1 + 2] − [λ2 − 2, λ2 − 1, λ2 + 1, λ2 + 2]

=0

(10)

⇒ [λ2 + 1, −x1 λ1 , −3x1 λ1 − 2λ2 − 2, −4x1 λ1 − 3λ2 − 3] = 0

 −[−1, 0, 2, 3] − [λ1 − 2, λ1 − 1, λ1 + 1, λ1 + 2] =0 −[1, 2, 4, 5][λ2 − 2, λ2 − 1, λ2 + 1, λ2 + 2]

(11)

⇒ [λ1 − λ2 + 1, −2λ2 , −2λ1 − 4λ2 − 2, −3λ1 − 5λ2 − 3] −[−1, 0, 2, 3] − [0, 1, 3, 4][λ2 − 2, λ2 − 1, λ2 + 1, λ2 + 2] = 0 ⇒ [1, −λ2 , −2 − 3λ2 , −3 − 4λ2 ]   [x1 − 2, x1 − 1, x1 + 1, x1 + 2] − =0 +[x2 − 2, x2 − 1, x2 + 1, x2 + 2] − [1, 2, 4, 5] ⇒

[x12

+ x2 + 1, 2,

−2x12

− 2x2 + 4,

−3x12

− 3x2 + 5]

(12)

(13)

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 +[1, 2, 4, 5] + [x2 − 2, x2 − 1, x2 + 1, x2 + 2] − =0 +[0, 1, 3, 4] + [x3 − 2, x3 − 1, x3 + 1, x3 + 2] − [5, 6, 8, 9] ⇒ [x1 − x2 + 5, −2x2 − x3 + 6, −2x1 − 4x2 − 3x3 + 8, −3x1 − 5x2 − 4x3 + 9] (14) Successively solving the Eqs. (10), (11), (12), (13), and (14) we get the values of x1 , x2 , x3 and λ1 , λ2 as x1 , x2 , x3 and λ1 , λ2 as [x1(1) , x1(2) , x1(3) , x1(4) ] = [−3.5, −2, 1, 2.5] [x2(1) , x2(2) , x2(3) , x2(4) ] = [0.25, 1.5, 4, 5.25] [x3(1) , x3(2) , x3(3) , x3(4) ] = [−4.625, −2.5, 1.75, 3.875] and (2) (3) (4) [λ(1) 1 , λ1 , λ1 , λ1 ] = [−2.5, −1, 2, 3.5] (2) (3) (4) [λ(1) 2 , λ2 , λ2 , λ2 ] = −[−0.5, 0, 1, 1.5]

Hence the optimal solution is, Z˜ = −[−7.875, −3, 6.75, 11.625]. Using the results to find Z ∗ , Since the first partial derivatives of f (x1 , x2 , x3 ), g1 (x1 , x2 , x3 ), and g2 (x1 , x2 , x3 ) are all continuous, and  ∂g ∂g ∂g    1 1 1 2x1 1 0 ∂x1 ∂x2 ∂x3 = J = ∂g 2 ∂g2 ∂g2 1 32 ∂x ∂x ∂x 1

2

3

J is of rank 2 everywhere, either [x1(1) , x1(2) , x1(3) , x1(4) ] = [−3.5, −2, (1) (2) (3) (4) (1) (2) (3) 1, 2.5], [x2 , x2 , x2 , x2 ] = [0.25, 1.5, 4, 5.25], , [x3 , x3 , x3 , (4) x3 ] = [−4.625, −2.5, 1.75, 3.875] is the optimal solution to Eq. (4) or no optimal solution exists. Checking feasible points in the region around ([−3.5, −2, 1, 2.5], [0.25, 1.5, 4, 5.25], [−4.625, −2.5, 1.75, 3.875]), we find that this point is indeed the location of a (global) maximum for Eq. (4). Therefore, it is also the location of a global minimum for the original equation is Z ∗ = [−7.875, −3, 6.75, 11.625]. 5.2 Case (ii): NLPP with the Approach of Robust Ranking Let’s address the aforementioned NLPP using a robust ranking approach [7]. Furthermore, the ranking indices of R[−1, 0, 2, 3] and its fuzzy membership function are as follows. ⎧ x + 1, −1 ≤ x ≤ 0 ⎪ ⎪ ⎨ 1, 0 ≤ x ≤ 2 μR[−1, 0, 2, 3] (x) = ⎪ −x + 3, 2 ≤ x ≤ 3 ⎪ ⎩ 0, otherwise The functions of α will be used to describe the confidence intervals for each degree α and the trapezoidal shapes in the following manner.

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(α)

Here α = x1 + 1 & α = −x2 + 3. Therefore,

Similarly, for every other fuzzy number, the ranking index is computed and 1   outlined as below: R(M) = R[1, 2, 4, 5] = (0.5) ∗ Mα,L Mα,U d α = 3, 0

1

1   R(M) = R[0, 1, 3, 4] = (0.5) ∗ Mα,L Mα,U d α = 2, R(M) = R[5, 6, 8, 9] = (0.5) ∗ 0

1    L U Mα, Mα, d α = 7, R(M) = R[−2, −1, 1, 2] = (0.5) ∗ Mα,L Mα,U d α = 0.

0

0

Using the above framework, the fuzzy NLPP can be converted to the conventional crisp problem, it has stated below: Maximize Z = −x1 − x2 − x3 . Subject to the constraints: x1 2 + x2 − 3 = 0, x1 + 3x2 + 2x3 − 7 = 0andx1 , x2 , x3 ≥ 0. Further, employ the existing conventional approach to the NLPP by using the Lagrange’s multiplier method and obtained the optimum results for the above is x1 = −0.5, x2 = 2.75, x3 = −0.375, λ1 = 0.5, λ2 = −0.5 & Max Z = −1.875.

5.3 Comparative Analysis with Results and Discussion The table (Table 1) below compares the optimum outcomes obtained from the conventional approach for the fuzzy NLPP preferred in the numerical illustration above for both FMF and robust ranking approaches. The table findings show that similar outcomes are obtained irrespective of whether existing or fuzzy membership or ranking approaches are employed. It demonstrates the novelty of the proposed model, also how to address the vagueness, hence the decision-maker may utilize this particular kind of model to clear the ambiguity of any applicable situation in order to get the best optimum outcomes. From the results stated above, it has been suggested to apply any of the approaches presented based on decision-maker’s situations under uncertainty [7]. By utilizing the suggested framework, the optimum outcomes for the fuzzy NLPP is [−7.88, −3, 6.75, 11.68], which could be a novel attempt to eliminate ambiguity. It will be persistently larger than −7.88 whereas fewer than 11.68, with the most likely outcome being elsewhere between −3 and 6.75. Figure 1 depicts the cost variations that have a high possibility of occurring. Furthermore,

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Table 1. The optimal results between existing and fuzzy based approaches Conventional approach results x1

x2

x3

Max Z

−0.5

2.75

−0.375

−1.875

Conventional approach results in terms of fuzziness (i = 1, 2, 3, 4)





 (k) (k) (k) x1 = x2 = x3 = [−4.6, [0.25, 1.5, 4.5, 5.25] −2.5, 1.7, 3.9] [−3.5, −2, 1, 2.5] Robust ranking approach results



 (k) (k) x1 = x2 = [0.25, 1.5, 4.5, 5.25] = [−3.5, −2, 1, 2.5] R[0.25, 1.5, 4.5, 5.25] =

R[−3.5, −2, 1, 2.5] = 0.5

= 2.75

Z [−7.88, −3, 6.75, 11.68]

 (k) x3

Z = [−7.88, −3, 6.75, 11.68] = [−4.6, −2.5, 1.7, 3.9] = = R[−7.88, −3, 6.75, 11.68] R[−4.6, −2.5, 1.7, 3.9] = 1.875 = −0.375

Fig. 1. Fuzzy optimal results of trapezoidal FMF

the generated fuzzy optimum outcomes xij may be physically interpreted. Let x be the value of the overall fuzzified NLPP values, and subsequently μmax (x) is the proportion of favorability in the DM, where ⎧ x+7.88 ⎪ for − 7.88 ≤ x ≤ −3 ⎪ ⎪ ⎨ −3+7.88 1 for − 3 ≤ x ≤ 6.75 μmax (X ) = x−11.68 ⎪ for 6.75 ≤ x ≤ 11.68 ⎪ 6.75−11.68 ⎪ ⎩0 otherwise

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6 Conclusion The fuzzy based NLPP using Lagrange multiplier conditions has been discussed with numerical illustration, through this the aim of a proposed model is justified with obtained optimal outcomes. Additionally, the formulated model has presented and clarifies how to deal vagueness in NLPP with appropriate illustrations, in which the considered problem attained the optimal outcomes for both the way of membership functions as well as the robust ranking index. In a fuzzy scenario, the membership function plays an important part in the formulation of a model. Further the model provides an efficient approach to deal the NLPP, also it incorporates multiobjective issues by utilizing Jacobian matrices. Besides, the ideal outcome was determined by the way of fuzziness and it was compared with existing approaches and explained in the result and discussion. The concept is also explained by employing a depiction of trapezoidal membership functions. Likewise, the comparative study could be a novel technique for dealing with NLPPs under uncertainties. The model attempts to overcome the uncertainties and personal perspectives of decision-makers, and it could be useful in addressing decision-making difficulties particularly in NLPPs. This similar framework can be suggested for further research on fuzzy optimization in different applications and also multidimensional frameworks as well as other forms of nonlinear optimization problems.

References 1. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 2. Bellman, R., Zadeh, L.: Decision making in a fuzzy environment. Manag. Sci. 17(4), 141–164 (1970) 3. Zimmermann, H.J.: Fuzzy Sets, Decision Making, and Expert Systems, vol. 10. Springer, Dordrecht (2010). https://doi.org/10.1007/978-94-009-3249-4 4. Vasant, P.: Nagarajan, R.: Yaacob, S: Fuzzy linear programming: a modern tool for decision making. Comput. Intell. Model. Predict. 383–401 (2005) 5. Kheirfam, B.: Hasani, F.: Sensitivity analysis for fuzzy linear programming problems with fuzzy variables. Adv. Model. Optim. 12, 257–272 (2010) 6. Palanivel, K.: Contributions to the study on some optimization techniques in fuzzy membership functions. Bharathidasan University, Trichy, Tamil Nadu, India (2013) 7. Palanivel, K.: Fuzzy commercial traveler problem of trapezoidal membership functions within the sort of alpha optimum solution using ranking technique. Afr. Mat. 27, 263–277 (2016) 8. Saranya, R.: Palanivel, K.: Fuzzy nonlinear programming problem for inequality constraints with alpha optimal solution in terms of trapezoidal membership functions. Int. J. Pure Appl. Math. 119, 53–63 (2018) 9. Tang, J., Wang, D.: A nonsymmetric model for fuzzy nonlinear programming problems with penalty coefficients. Comput. Oper. Res. 24, 717–725 (1997) 10. Tang, J., Wang, D., Ip, A., Fung, R.: A hybrid genetic algorithm for a type of nonlinear programming problem. Comput. Math. Appl. 36, 11–22 (1998) 11. Fung, R.Y., Tang, J., Wang, D.: Extension of a hybrid genetic algorithm for nonlinear programming problems with equality and inequality constraints. Comput. Oper. Res. 29, 261–274 (2002) 12. Sarimveis, H., Nikolakopoulos, A.: A line up evolutionary algorithm for solving nonlinear constrained optimization problems. Comput. Oper. Res. 32, 1499–1514 (2005)

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13. Syau, Y.-R.: Stanley Lee, E.: Fuzzy convexity and multiobjective convex optimization problems. Comput. Math. Appl. 52, 351–362 (2006) 14. Chen, S.-P.: Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method. Eur. J. Oper. Res. 177, 445–457 (2007) 15. Qin, X.S., Huang, G.H., Zeng, G.M., Chakma, A.: Huang, Y.F.: An interval-parameter fuzzy nonlinear optimization model for stream water quality management under uncertainty. Eur. J. Oper. Res. 180, 1331–1357 (2007) 16. Kassem, M.A.E.-H.: Stability achievement scalarization function for multiobjective nonlinear programming problems. Appl. Math. Model. 32, 1044–1055 (2008) 17. Shankar, N.R., Rao, G.A.,Latha, J.M., Sireesha, V.: Solving a fuzzy nonlinear optimization problem by genetic algorithm. Int. J. Contemp. Math. Sci. 5, 791–803 (2010) 18. Abd-El-Wahed, W.F.: Mousa, A.A.: El-Shorbagy, M.A.: Integrating particle swarm optimization with genetic algorithms for solving nonlinear optimization problems. J. Comput. Applied Math. 235, 1446–1453 (2011) 19. Jameel, A.F., Sadeghi, A.: Solving nonlinear programming problem in fuzzy environment. Int. J. Contemp. Math. Sci. 7, 159–170 (2012) 20. Ahmadini, A. A.H., Varshney, R., Mradula, A.I.: On multivariate-multiobjective stratified sampling design under probabilistic environment: a fuzzy programming technique. J. King Saud Univ. Sci. 33, 101448 (2021) 21. Khan, M.F.,Modibbo, U.M., Ahmad, N., Ali, I.: Nonlinear optimization in bi-level selective maintenance allocation problem. J. King Saud Univ. Sci. 34, 101933 (2022) 22. Gupta, D., Borah, P., Sharma, U.M., Prasad, M.: Data-driven mechanism based on fuzzy Lagrangian twin parametric-margin support vector machine for biomedical data analysis. Neural Comput. Appl. (2021) 23. Lin, L., Lee, H.-M.: Fuzzy nonlinear programming for production inventory based on statistical data. J. Adv. Comput. Intell. Intell. Inform. 20, 5–12 (2016) 24. Lu, T., Liu, S.-T.: Fuzzy nonlinear programming approach to the evaluation of manufacturing processes. Eng. Appl. Artif. Intell. 72, 183–189 (2018) 25. Saberi Najafi, H., Edalatpanah, S.A., Dutta, H.: A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Alexandria Eng. J. (2016) 26. Silva, R.C., Cruz, C., Verdegay, J.L.: Fuzzy costs in quadratic programming problems. Fuzzy Optim. Decis. Mak. 12, 231–248 (2013) 27. Gani, A.N., Saleem, R.A.: Solving Fuzzy Sequential Quadratic Programming Algorithm for Fuzzy Non-Linear Programming (2018)

Biometric Identification of Hand by Particle Swarm Optimization (PSO) Algorithm Hatem Ghodbane1 , Hichem Amar2 , Monir Amir3 , Badreddine Babes2(B) and Noureddine Hamouda2

,

1 University Mohamed Khider of Biskra, Biskra, Algeria 2 Research Center in Industrial Technologies (CRTI) Algiers, Chéraga, Algeria

[email protected] 3 Electronic Department, University Mouloud Mammeri of Tizi-Ouzou, Tizi-Ouzou, Algeria

Abstract. Today, biometrics is an expanding field of research, several identification and verification systems are now being developed, but their performance remains insufficient in the face of increased security needs. Hand biometric recognition has been successfully developed for authentication or biometric identification. In most cases, the use of a single biometric modality reduces the reliability of these systems, which has prompted us to combine several modalities. In this article, we propose a multi-biometric fusion approach for the identification of the individual. Indeed, we use two types of biometrics: the palm print and the shape of the hand. The Extraction of the texture was made by the technique of the transformed into (DCT). Then, the identification process fuses the geometric characteristics of the fingers and the texture of the analyzed palm with the PSO. The simulation results show the basic good of this approach with a recognition rate of more than 85% for a population of 230 individuals. Keywords: Particle Swarm Optimization (PSO) · Transformed into discrete cosines · multimodal fusion · Identification · Palm imprint · Geometric fingers

1 Introduction The field of automatic ratification has been growing for some years. Given that its obvious imposition as the technology of the future in the field of safety, also of ease of use. That’s why a lot of the research is looking at that forecast. Biometrics are playing an increasingly important role in authentication and identification systems. Biometric recognition processes allow recognition of individuals based on physical or behavioral characteristics. Various technologies have been developed such as fingerprints [1], iris [2], face [3], signature [4], hand [5] this last method relies on a study of the shape of the hand and the texture of the palm. It has many advantages over other technologies. The characteristics of the hand are more numerous than those of the fingerprints are and can be determined with low-resolution images. In addition, this system is well accepted by users and the hand leaves few traces unlike a system based on the fingerprint. Several studies have already shown that the combination of different biometric modalities improves the performance of systems based on a single modality. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 454–461, 2023. https://doi.org/10.1007/978-3-031-39774-5_51

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In this work we are concerned with the modeling and representation of these two modalities. It is a question of extracting, from the images of palmprint and the images of geometry of the hand, information relating to the identity, and to estimate with the latter a model to merge the modalities, which allow us identification. We are particularly interested in the optimization by swarm of particles, it is estimated from the model a better identification to have a solid and robust system. Optimization methods based on artificial intelligence have been widely launched by researchers to ensure optimal planning of electrical networks, we mention for example: Genetic Algorithms (GA), Particle Swarm Optimization (PSO), Ant Colonies Optimization (ACO), and neural networks (NN). The goal of our work is to maximize the rate of multimodal fusion identification. Using the PSO method to maximize the identification rate and make it efficient.

2 Hand Biometrics 2.1 Palmprint (The Palm Print) The man’s palm is defined as the inner part of the hand from the wrist to the root of the fingers. Printing is an impression made when the palm is pressed against any surface. A palm print thus illustrates the physical properties of the skin model such as lines, dots, minutes and texture. The identification palm print can be considered as the ability to uniquely identify a person among others, by an appropriate algorithm using the characteristics of palm prints. The characteristics of palmar prints are mainly developed during life processes, due to biological phenomena, the growth of fetuses in the uterus. Even a minute change in these inherent phenomena, the process of complete life changes and therefore, the structure of two different palm trees should be never again the same. The main characteristics of interest of the palm are as follows: 1. 2. 3. 4.

Characteristics of minute palm papillary crest, The main lines, which are the most, go straight on the palm lines, The lines are thinner and more irregular, compared to lines principle said of wrinkles, Reference points, which are the endpoints of the main lines [6].

2.2 The Joints The joints of the phalanx bones of the hand man generate distinct patterns on the surface texture of the doigtback, also known as the back of the hand. In particular, the formation of structures of the image of the flexion of the finger joints is quite unique and allows this surface a distinctive character of biometric identification. Figure 4 identifies three finger knuckles, each of which can potentially be used for personal identification purposes. The Geometry of the Hand The shape anatomy of the hand counts on the give geometry, length, width of fingers, and duration of the hand in different dimensions. The biometric geometry of the hand is not considered suitable for personal identification for the large population of the scale user as the geometry of the hand traits are not very distinctive.

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The Veins of the Hand The veins are hidden under the skin, and are usually invisible to the naked eye and other visual inspections. The pattern of blood veins is unique to each person, even in identical twins. Veins are the internal structure responsible for performing blood from one part of the body to the other. The veins that are present in the fingers, palm and back of the hand surface are of particular interest to biometric identification. There are mainly two types of veins found on the back of the hand, namely cephalic and basilica. Basil veins are the group of veins attached to the surface of the hand. The Skin The skin is a protective barrier that contains nerve receptors for a variety of sensations, regulates temperature, allows the passage of sweat and sebaceous oils, and houses hair and nails. The skin of papillary ridges is differentiated from thin skin not only by the presence of raised papillary ridges but also by the epidermis which is much thicker and structurally more complex, by increasing sensory abilities, by the absence of hair, and by the absence of sebaceous glands. The skin throughout the body consists of three basic layers: the hypodermis, the dermis and the epidermis.

3 The PSO Algorithm The problem of optimizing biometric identification consists in choosing solutions to minimize the false rejection rate as well as the false acceptance rate, so it is a complex optimization problem. Complex problem-solving methods can be grouped into three classes: deterministic methods, metaheuristic methods, hybrid methods. Among the metaheuristic methods is the particle swarm optimization method (OEP or PSO), which is a metaheuristic optimization method, invented by Russel Eberhart (electrical engineer) and James Kennedy (socio-psychologist) in 1995. This algorithm is inspired by the origin of the living world. It is based in particular on a model developed by Craig Reynolds in the late 1980s, allowing to simulate the movement of a group of birds. Another source of inspiration, claimed by the authors, is sociopsychology. 3.1 Definition and Algorithm Particle swarm optimization is an evolutionary technique that uses “a population” of candidate solutions to develop an optimal solution to the problem. The optimum, the basic principle is to define a set of particles (the swarm) by initial positions and speeds, then to make it evolve according to a reasoning-decision process at each time step, which, informally, is the following, from a particle perspective: 1. 2. 3. 4.

I remember my best position ever found (best position reached); I asks each of my neighbors their best position reached; I consider only the best of them; I combine three tendencies: my propensity to follow my own path (“voluntaries” attitude), my “conservative” tendency (to return to my best position) and my “following” tendency (to go to my best neighbor) to set my new speed/direction of travel; 5. I make this move [7].

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3.2 Operating Principle At the start of the algorithm, a swarm is randomly distributed in the search space, each particle also having a random speed. Then, at each time step: • A physical component: Each particle is able to evaluate the quality of its position and to keep in memory its best performance. • A social component: Each particle is able to interrogate a number of its congeners and to obtain from each of them its own best performance. • A cognitive component: With each step of time, each particle chooses the best of the best performances. The first point is easily understood, but the other two require some clarification. The informants are defined once and for all as follows (Fig. 1), the size three information group of particle 1 is composed of particles 1, 2 and 7.

1

2

7

3

6 4 5 Fig. 1. The virtual circle for a swarm of seven particles [7].

3.3 Characteristics of PSO The PSO has some interesting properties, which make it a good tool for many optimization problems [8, 9] (Fig. 2).

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Towards its best performance New position Towards the best performance of the swarm Current position

Towards the reachable point with its current speed

Fig. 2. Principle diagram of the displacement of a particle [10].

4 Proposed Method The proposed system is composed of two subsystems, the first is based on the verification of palm impressions, while the second is based on the verification of the shape of the human hand, each of them follows a well-defined process as in Fig. 3.

Fig. 3. Learning and Testing Flowchart.

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4.1 Feature Extraction Module This step is one of the most important, as it will condition the rest of the treatments. First, we will give a frequency representation with the Discrete Cosine Transformation (DCT). The DCT is a mathematical function that allows to change the field of representation of a signal. Moreover, this is his mathematical equation. −1 N −1 N   (2y + 1)jπ 1 (2x + 1)iπ ) cos( ) (1) DCT (i.j) = √ C(t)C(f ) plxel(x.y) cos( 2N 2N 2 x=0 y=0

1 C(x) = √ if x worth 0 and if x ≥ 0 2

(2)

5 Results and Discussion The simulations are carried out on the two modalities separately then on their fusion, we built two types of databases: Base1, Base2. 5.1 The Geometry of the Hand From several variations on the number of vectors in the learning base, we have obtained the following results (Table 1): Table 1. Geometry results. Vp

140

180

220

260

300

340

380

400

TR (%)

42.17

41.73

43.26

43.47

43.47

43.47

43.91

43.69

We find that the results obtained after the test phase are not really satisfactory either with or without an increase in the number of vectors it comes back to the pre-treatment phase where we obtained a recognition rate TR = 43,91 with a number of own learning vectors Vpro = 380. 5.2 Palm Print The results obtained at the palm level are quite satisfactory because we have segmented images in the database which facilitates the extraction of their characteristics where we obtained a recognition rate TR = 80.43 with a number of own vectors V p = 45 (Table 2).

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Vp

10

20

30

40

50

60

70

80

90

100

TR (%)

55.21

72.82

77.39

79.56

80.43

80

80

79.56

78.26

78.04

Table 3. Palmprint results. Method

Geometry of the hand

Palm print

Fusion by PSO

TR (%)

43.26

80.43

83.04

5.3 Merge Results The application of fusion is to take the best result for each modality, the recognition rate of each modality is considered a local solution pbest the output of our algorithm is a global solution gbest . The results are described in the Table 3. As we said previously the recognition rate at the geometry level is low, despite this disappointment the fusion by PSO gave us an acceptable result which is TR = 83,04, each PSO program execution gets a different value than the previous one, after six execution we got these results (Table 4): Table 4. Different PSO executions. Ex (n)

1

2

3

4

5

6

7

TR (%)

82.04

81.73

83.04

78.04

82.17

83.47

83.27

6 Conclusion In this work, we presented a system of identification based on two modalities hand geometry and palm imprint and specifically the fusion of these two modalities. The results demonstrate that the palmar identification offers interesting identification rates (80.43%). Nevertheless, it has the following disadvantages: requires significant amounts of learning data, identification by hand geometry offers us important identification rates (43.91%). Although both rates of identifications are acceptable but the identification of individuals requires a higher rate of identification. Because of these results, he guided us to use multimodal fusion.

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References 1. Iancu, I., Constantinescu, N., Colhon, M.: Fingerprints identification using a fuzzy logic system. Int. J. Comput. Commun. Control 5(4), 525–531 (2010) 2. De Mira, J., Neto, H.V., Neves, E.B., Schneider, F.K.: Biometric-oriented iris identification based on mathematical morphology. J. Sig. Process. Syst. 80, 181–195 (2015) 3. Osadchy, M., Pinkas, B., Jarrous, A., Moskovich, B.: Scifi - a system for secure face identification. In: 2010 IEEE Symposium on Security and Privacy, pp. 239–254. IEEE, May 2010 4. Impedovo, D., Pirlo, G., Russo, M.: Recent advances in offline signature identification. In: 2014 14th International Conference on Frontiers in Handwriting Recognition, pp. 639–642. IEEE, September 2014 5. Demongin, L.: Identification guide to birds in the hand. Brit Birds 109, 553–555 (2016) 6. Zayed, H.L., Hamid, H.M.A., Kamal, Y.M., et al.: A comprehensive survey on finger vein biometric. J. Adv. Inf. Technol. 14(2) (2023) 7. Ross, A., Jain, A.K.: Biometrics, overview. In: Li, S.Z., Jain, A.K. (eds.) Encyclopedia of Biometrics. Springer, Boston (2015). https://doi.org/10.1007/978-1-4899-7488-4 8. Ross, A., Poh., N.: Multibiometric systems: overview, case studies and open issues. In: Tistarelli, M., Li, S.Z., Chellappa, R. (eds.) Handbook of Remote Biometrics. Advances in Pattern Recognition, pp. 273–292. Springer, London (2009). (Ross, A., Jain, A.K.: Information fusion in biometrics. Pattern Recogn. 24(13), 2115–2125 (2003)). https://doi.org/10. 1007/978-1-84882-385-3_11 9. Clerc, M., Siarry, P.: Une nouvelle métaheuristique pour l’optimisation difficile : la méthode des essaims particulaires. In: J3eA, vol. 3, p. 007. EDP Sciences (2004) 10. Hamouda, N., Babes, B., Kahla, S., Boutaghane, A., Beddar, A., Aissa, O.: ANFIS controller design using PSO algorithm for MPPT of solar PV system powered brushless DC motor based wire feeder unit. In: International Conference on Electrical Engineering 11. Kahla, S., Bechouat, M., Amieur, T., Sedraoui, M., Babes, B., Hamouda, N.: Maximum power extraction framework using robust fractional-order feedback linearization control and GM-CPSO for PMSG-based WECS. Wind Eng. 45(4), 1040–1054 (2020) 12. Hamouda, N., Babes, B., Kahla, S., Hamouda, C., Boutaghane, A.: Particle swarm optimization of fuzzy fractional PDµ +I controller of a PMDC motor for reliable operation of wire-feeder units of GMAW welding machine. Przeglad Elektrotechniczny. 96(12), 40–46 (2020) 13. Babes, B., Albalawi, F., Hamouda, N., Kahla, S., Ghoneim, S.S.M.: Fractional-fuzzy PID control approach of photovoltaic-wire feeder system (PV-WFS): simulation and HIL-based experimental investigation. IEEE Access 9, 159933–159954 (2021)

Optimization of Alkaline Zinc Plating Process in a Company Using Taguchi Model Based on Fuzzy Logic Furkan Atik(B)

and Ahmet Sarucan

Konya Technical University, Konya 42100, Turkey [email protected], [email protected]

Abstract. In order to effectively prevent corrosion and ensure a long service life, the galvanizing (alkaline zinc plating) process is a frequently used process in the industrial field. The galvanizing process, which has a wide range of applications, especially in the automotive industry, the construction sector and the white goods industry, offers various benefits, especially the production of corrosion-resistant and long-lasting products to reduce costs. As a process with a large number of parameters, Galvanized coating can be made more efficient for industrial enterprises. In order to optimize the process, Taguchi method and fuzzy logic were used in the study. 8 factors and 4 performance characteristics that are effective in the process were tried to be improved with 27 experiments determined according to the Taguchi method. In order to find the optimal levels of the parameters, the signal-to-noise (S/N) ratio was determined and the best combination of factors was determined. In order to test the effectiveness and efficiency of this approach, verification experiments were conducted and the results were presented. Keywords: Alkaline zinc coating · Fuzzy Logic · Taguchi

1 Introduction Alkaline zinc plating is a metal coating process that is performed using alkaline substances. This coating takes advantage of the corrosion resistance properties of zinc. Alkaline zinc plating plays an important role in protecting various metal parts, especially those used in the industrial and aviation sectors. It increases the resistance of metal surfaces to environmental factors and provides long-lasting protection. Additionally, the alkaline zinc plating process gives metal parts an aesthetic appearance. A high-volume product made from processed automaton steel, with dimensions longer than 90 mm and a diameter less than 10 mm, used in the automotive spare parts industry, was found to have been subjected to a large number of faulty galvanization coatings. As a result of faulty coatings, it was observed that time, chemical, labor, and other costs increased, but more importantly, a decrease was observed in coating thickness and other quality factors. The low quality significantly reduced the corrosion resistance. In this study, the Taguchi method and fuzzy logic were used together to optimize four quality parameters. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 462–470, 2023. https://doi.org/10.1007/978-3-031-39774-5_52

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The Taguchi experimental design method was used to reduce the number of experiments required in the study and thereby minimize factors such as time and cost. Fuzzy logic was used to address uncertainties such as chemicals and time flow during the coating process. Through this study, facilities engaged in alkaline zinc plating can generate costeffective solutions to the problems they encounter during the coating process. The article highlights the industrial relevance of optimizing the alkaline zinc plating process, which is commonly utilized in various industries. It emphasizes the importance of improving the quality and efficiency of the process to meet industry demands. By showcasing the effectiveness and feasibility of the optimization methods employed, the article serves as a practical resource for industrial practitioners. The paper is organized as follows. Section 2 Literature review, Sect. 3 explains Taguchi method and fuzzy logic, Sect. 4 explains the application of the Taguchi method and fuzzy logic to improve the quality of a problematic steel product, Sect. 5 results and suggestions.

2 Literature Review In their study conducted in 2010, Gologlu and Mızrak used Fuzzy Logic (FL) and Taguchi experimental design to enable customers to determine their own product based on their needs [1]. In 2016, George and Kyatanavar optimized a process in the sugar industry to reduce energy consumption and improve sugar quality using the Taguchi method and fuzzy logic [2]. Rajabloo et al., in 2014, applied the Taguchi method and a fuzzy logic model to minimize the band-gap energy in cobalt sulfide nanostructures with the CoxSy formula [3]. Basar and Kahraman, in 2018, conducted a study comparing and predicting the surface hardness in ball burnishing of aluminum alloy using the Taguchi technique, fuzzy logic, and regression models [4]. Wang et al., proposed a fuzzy logic-based Taguchi method in 2015 to find an optimal fast charging pattern for Li-ion batteries, maximizing charging cost-effectiveness [5].

3 Taguchi Method and Fuzzy Logic 3.1 Taguchi Method Quality has been defined by many people as “ensuring tolerances”, “zero errors” or “customer satisfaction”, but these definitions have not revealed a methodology decoupling the relationship between quality and cost. However, Taguchi’s understanding of quality has enabled decoupling the relationship between quality and cost. Taguchi’s quality system involves the quality concept and quality cost through all phases of a products life cycle that begins with product planning [6]. The Taguchi method is a low-cost and high-efficiency high-quality engineering method that can accurately predict the results by testing a special combination based on orthogonal sequence and statistical analysis [7]. The Taguchi method is widely used in the industrial sector with the ability to determine the most effective parameters according to the experimental results [8]. The steps in applying the Taguchi method [9], which simplifies analysis, reduces the time spent on analysis and finds a good solution, used to evaluate the effect of some parameters on a

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specific variable, can be listed as identifying factors, defining levels, choosing a factorial design, assigning factors and designing interactions, running simulations, analyzing data with Taguchi, optimizing results [10]. The Taguchi method depends on orthogonal sequences, in which response variables are measured for different input variables, which include decision variables and uncontrollable noise factors. A uniformly distributed orthogonal array has the advantages of comparability and reliable representation for experimental schemes [11]. In classical experimental design techniques, the number of experiments must increase when the number of process parameters increases, but in the Taguchi method, the number of experiments is reduced by using orthogonal experimental design, signal-to-noise ratio (S/N) and analysis of variance (ANOVA). According to the Taguchi method, the best decision variables that reduce the effect of noise factors should be determined [12]. The signal-to-noise (S/N) ratio proposed by Taguchi is useful for finding the optimal levels of parameters [13]. The Taguchi method converts objective function values into a signalto-noise (S/N) ratio, allowing the performance characteristics of control factor levels to be measured against these factors [8]. The S/N ratio used to measure the performance characteristics extracted from the desired values is determined according to three basic performance characteristics, such as smaller-better, larger-better and nominal-best. 3.2 Fuzzy Logic The concept of fuzzy logic, which can be defined as a strict mathematical order established for the expression of uncertainties and working with uncertainties, coincides with people’s ability to think with imprecise statements [15]. In classical logic, in inter-set relations, an element belongs to a set, it does not belong Decently, or it belongs equally to both sets. In fuzzy logic, on the other hand, cluster elements can belong to clusters to different degrees [14]. In contrast to classical logic, which starts from the assumption that every proposition is either true or false, fuzzy logic has defined a third proposition, which is defined as Decoherence between two propositions. Fuzzy logic systems are applied for process simulation or control and consist of four components, including the fuzzifier, fuzzy rule base, inference engine, and defuzzifier [16]. The integration of fuzzy logic with the Taguchi method can be used for optimization of multiple performance features, since the optimization of multiple performance features can be converted to the optimization of a single performance index via fuzzy logic [17]. The process of mapping from a crisp input universe of discussion into the fuzzy interval (0, 1) that depicts the membership of fuzzy input variables is defined as fuzzification [18].

4 Application of Taguchi Method and Fuzzy Logic In this study, an optimization study was carried out for the alkali zinc coating plant of a company that has been producing spare parts in the automotive industry. Zinc coatings, in addition to providing good corrosion protection and mechanical properties to the material, allow welding of the material and painting of the material. The alkali zinc coating process consists of many stages, and the flow of the processing is given in the Fig. 1. The way to use the Taguchi method in this process is as shown in the Fig. 2.

Optimization of Alkaline Zinc Plating Process Hot degreasing

Acidic degreasing

Blue passivation

Electrolyte degreasing

Alkaline zinc bath

465

Nitric rinse

Lacquering

Fig.1. The flow of the processing

In the company, the alkali zinc coating process is applied for a special production gland with deep holes. However, as a result of the process, it was found that the inside of the glands was not completely coated with alkaline zinc and this situation was determined as the problem of the study. As a result of the alkali zinc coating process studied, considering the quality standards, the fact that the inside of the gland is coated with a thickness of 10–20 µ (IL- interior lining), shining (SH), smoothness (SM), white color stability (ST) has been determined as four important performance criteria. 3 Levels have been determined for 8 factors (voltage- V, duration-T, amount of zinc-Zn, NaOH, carrier chemical-CH, conditioner-C, polisher-PO, purifier-P) that are effective in the alkaline zinc coating process and can be controlled (Table 1). Table 1. Factors and levels Factor

V

T

Zn

NaOH

CH

C

PO

P

Unit

V

min

gr/l

gr/l

%

%

%

%

5

2

1

1

1st level

9

45

12

120

2nd level

10

60

13.30

130

9

4

3

2

3rd level

11

75

15

140

13

6

5

3

The method was applied according to the L27 orthogonal sequence. The 27 experimental designs proposed by the Taguchi method are given in the Table 2. The experimental results obtained in the Taguchi method were converted into Signal-to-Noise ratio and “the larger the better” formula was used. The fact that the inside of the gland is coated with a thickness of 10–20 µ and the most effective factor in the color stability criterion is voltage, while polisher was found to be bright and zinc was found to be smooth. According to the ANOVA test result, the best combination of factors was found to be 11 V, 60 min, 15 g/lt Zn, 130 g/lt NaOH, 13% chemical activator, 4% conditioner, 5% polisher and 1% purifier. Taking into account the results of the Taguchi experiment, a set of rules for determining input variables and output variables was created and structured with the Mamdanitype fuzzy inference system. The established fuzzy model is given in the Fig. 3. Signalto-noise ratios have been used for fuzzy sets. An example of membership functions and working rules in a fuzzy logic model is shown. This function and the rules work separately for each input (Fig. 4).

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Fig. 2. The flow chart of Taguchi method Table 2. Taguchi experiment design Row

V

T

Zn

NaOH

CH

C

PO

P

1

9

45

12.0

120

5

2

1

1

2

9

45

12.0

120

9

4

3

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3

9

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13.5

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13.5

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3

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13.5

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5

1

7

9

75

15.0

140

5

2

1

3

8

9

75

15.0

140

9

4

3

1

9

9

75

15.0

140

13

6

5

2

10

10

45

13.5

140

5

4

5

1

11

10

45

13.5

140

9

6

1

2

12

10

45

13.5

140

13

2

3

3

13

10

60

15.0

120

5

4

5

2

14

10

60

15.0

120

9

6

1

3

15

10

60

15.0

120

13

2

3

1

16

10

75

12.0

130

5

4

5

3

(continued)

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Table 2. (continued) Row

V

T

Zn

NaOH

CH

C

PO

P

17

10

75

12.0

18

10

75

12.0

130

9

130

13

6

1

1

2

3

19

11

45

15.0

130

2

5

6

3

1

20

11

21

11

45

15.0

45

15.0

130

9

2

5

2

130

13

4

1

3

22 23

11

60

11

60

12.0

140

5

6

3

2

12.0

140

9

2

5

3

24

11

25

11

60

12.0

140

13

4

1

1

75

13.5

120

5

6

3

3

26 27

11

75

13.5

120

9

2

5

1

11

75

13.5

120

13

4

1

2

Fig. 3. The Fuzzy Model

Fig. 4. MATLAB image of rules running in a fuzzy logic rule system

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5 Conclusions and Future Study When one of the surfaces (Fig. 5a) is examined as an example of the fuzzy logic system, it is seen that there is a correlation between the brightness of the inner coating and the thickness of the inner coating, and the quality increases with the increase of both values. Similar inferences can be made for other surface indicators (Fig. 5 b, c, d). A comparison of the data obtained as a result of the verification experiment and the change in performance characteristics obtained as a result of the use of the factor combination used by the company before the experiment in the process is given in the Table 3. The change in the inner coating, which is the most important criterion for the success of the process, shows the success of the model. Although the amount of deviation increased in all quality factors such as interior lining (IL), shining (SH), smoothness (SM), white color stability (ST), the average quality amount increased. In addition, as a result of fuzzy logic, it is seen that the quality factors increase in direct proportion to each other on the surfaces of these quality factors. In future studies, it can be suggested that the results obtained can be compared with the results of this study by examining the problem parameter, multi-target optimization, new performance measurement development, and further customizing the problem or methods such as grey relational analysis. Table 3. Change in performance characteristics IL First Sit

SH

SM

ST

Mean

Std. Dev

Mean

Std. Dev

Mean

Std. Dev

Mean

Std. Dev

60

10

70

10

73,33

5,77

66,66

5,77

Last Sit

86,66

5,7

83,3

5,77

86,66

5,77

80

10

Difference

26,66

−4,22

13,3

−4,22

13,33

0

13,33

4,22

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Fig. 5. Example of a fuzzy logic system characteristic surface

References ˙ 1. Gologlu, C., Mızrak, C.: Bulanik Mantik Ve Taguchi Yakla¸simli Mü¸steri Istemli Ürün Belirleme. Gazi Üniv. Müh. Mim. Fak. Der. 25(1), 9–19 (2010) 2. George, S., Kyatanavar, D.: Optimization of multiple effect evaporator using fuzzy logic integrated with Taguchi technique. In: International Conference on Electrical, Electronics, and Optimization Techniques, pp. 1415–1419 (2016) 3. Rajabloo, T., Ghafarinazari, A., Faraji, L., Mozafari, M.: Taguchi based fuzzy logic optimization of multiple quality characteristics of cobalt disulfide nanostructures. J. Alloy. Compd. 607, 61–66 (2014) 4. Basar, G., Kahraman, F.: Prediction of surface hardness in a burnishing process using Taguchi method, fuzzy logic model and regression analysis. Sigma J. Eng. Nat. Sci. 36(4), 1283–1295 (2018) 5. Wang, S., Chen, Y., Liu, Y., Huang, y.: A fast-charging pattern search for li-ion batteries with fuzzy-logic-based Taguchi method. In: 10th Conference on Industrial Electronics and Applications, pp. 855–859 (2015) 6. Taguchi, G.: Elsayed, A., Hsiang, T.: Quality Engineering in Production Systems, McGraw Hill Inc, New York (1989) 7. Guo, B., Xu, J., Zheng, S., Wang, K., Chang, J.: Mathematical problems in engineering, pp. 1–13 (2022) 8. Akıncıo˘glu, S.: Facta Universitatis, series: mechanical engineering 20(2), 237–253 (2022) 9. Stojanovic, N., Ghazaly, N., Grujic, I., Doric, J.: Determination of noise caused by ventilated brake disc with respect to the rib shape and material properties using Taguchi method. Trans. FAMENA. 6(4), 19–30 (2022) 10. Aabid, A., Ibrahim, Y.E., Hrairi, M., Ali, J.S.M.: Optimization of structural damage repair with single and double-sided composite patches through the finite element analysis and Taguchi method. Materials 16, 1581 (2023)

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11. Shi, Z., et al.: Optimizing the transient performance of thermoelectric generator with PCM by Taguchi method. Energies 16, 805 (2023) 12. Balhasan, S., Alnahhal, M., Towler, B., Salah, B., Ruzayqat, M., Tabash, M.: I. Energies (19961073). 15(15), 5424-5424 (2022). 19p 13. Neag, E., Stupar, Z., Varaticeanu, C., Senila, M., Roman, C.: Optimization of lipid extraction from spirulina spp. by ultrasound application and mechanical stirring using the Taguchi method of experimental design. Molecules 27, 6794 (2022) 14. Terzi, Ü.: Taguchi Yöntemi Ve Bulanık Mantık Kullanılarak Çok Yanıtlı Kalite Karakteristiklerinin E¸szamanlı En ˙Iyilenmesi (2004) 15. Salimiasl, A., Rafighi, M.: Titre¸sim ve Kesme Kuvveti Esaslı Takım A¸sınmasının Bulanık Mantıkla ˙Izlenmesi ve Tahmini. Politeknik Dergisi 20(1), 111–120 (2017) 16. Tsai, T., Liukkonenb, M.: robust parameter design for the micro-BGA stencil printing process using a fuzzy logic-based Taguchi method. Appl. Soft Comput. 48, 124–136 (2016) 17. Sánchez, R., Blas, M., Fernández, M., Nolasco, S.: Applying a Taguchi-based fuzzy logic approach to optimize hydrothermal pretreatment of canola seeds using multi-response performance index. OCL 28, 8 (2021) 18. Raja, P., Malayalamurthi, R., Sakthivel, M.: Experimental investigation of cryogenically treated HSS tool in turning on AISI1045 using fuzzy logic – Taguchi approach bulletin of the polish academy of sciences technical sciences. Bull. Polish Acad. Sci. Tech. Sci. 67(4), 687–696 (2019)

Study of Multiverse Optimizer Variations with Chaos Theory and Fuzzy Logic Over Benchmark Optimization Lucio Amézquita , Oscar Castillo(B) , Jose Soria , and Prometeo Cortes-Antonio Tijuana Institute of Technology, Tijuana, México {lucio.amezquita19,prometeo.cortes}@tectijuana.edu.mx, [email protected], [email protected]

Abstract. In this work, we are presenting multiple variations of the Multiverse Optimizer algorithm incorporating the use of Fuzzy inference systems and chaotic maps (FCMVO) with the purpose of analyzing their performance in a benchmark with some mathematical functions. We implement some of the most used chaotic maps over literature for metaheuristics in order to replace multiple parameters of the original algorithm, and change the original behavior, and by the other hand, we use Fuzzy Logic to adapt dynamically some of the parameters. By adding both Chaos theory and Fuzzy Logic, the resulting algorithm can have the best of both, resulting in a competitive variant. By using Chaos theory, we saw that not all the chaotic maps behave a good behavior in MVO, then we select a set of six variants that we called Elitist FCMVO. In the study, we do a brief comparison between the resulting variants with the benchmark functions, where each variant changes a set of three selected chaotic maps. The main objective is to define the best set of chaotic maps that FCMVO can used, and it could be used in other cases of study. Keywords: multiverse optimizer · fuzzy logic · benchmark · chaotic maps · chaos theory · Elitist · FCMVO · variants · mathematical functions · fuzzy inference systems

1 Introduction Each time we try to solve a problem, it starts with an analysis to comprehend truly the origin of it, keeping focus on the context where it is being presented; then, we start thinking of multiple solutions to achieve the main goal, which is, to find a solution. Sometimes, we are inspired by how in nature are solved similar problems, constructing a process that resembles on those behaviors presented by animals, plants or even weather conditions. When a solution to the problem is chosen, it is not always the best one that could have been done in that moment, perhaps because the lack of time or options presented to solve that particular problem. As technology advances, there are some particular areas, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 471–479, 2023. https://doi.org/10.1007/978-3-031-39774-5_53

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like is the case of computational intelligence techniques, that can aid in the search of multiple solutions for the problem in process. Some of these computational intelligence techniques [1] are inspired by natures behaviors, or even in some man-made behaviors; they are known as metaheuristics, and they are used to find multiple solutions to obtain the best result over an expected goal, this by applying his respective optimization algorithm. Even, for optimization algorithms can be challenging to obtain the best possible solution to a problem; to overcome this, in the literature, some of these metaheuristics use other techniques to improve the generation of solutions by including improved randomization, like the use of levy flights [2]. In previous works, we have used the Multiverse Optimizer (MVO) algorithm, to achieve some of the best solutions on benchmark optimization and control cases. We adapted the algorithm to include the use of Fuzzy Logic to dynamically adapt [3] some parameters, and over this new study, we have included the use of Chaos Theory to improve further the algorithm, by selecting some of the best performing chaotic maps and observe the behavior of them. In this paper, we are organizing it as follows: Sect. 2 describes optimization algorithm and chaotic maps in literature, Sect. 3 presents the methodology proposed for the MVO algorithm, Sect. 4 presents some of the results obtained with the methodology and the last section has the conclusions achieved in this work.

2 Optimization Algorithms and Chaotic Maps The optimization algorithms are a part of computational intelligence techniques used to find optimal solutions [4] for the problem that are solving, and their behavior depends on different inspirations in nature, or even on synthetic ones, that are being interpreted by mathematic models, and then, into computer code in different programming languages. The use of metaheuristics in scientific research has been adopted for many years, so, new algorithms are being inspired not only in nature, for this reason, many of these algorithms can solve some specific set of problems. To improve further, some algorithms use other techniques to improve, and such is the case of levy flights. Given the nature of Chaos theory, is used to generate numbers that cannot be obtained by other randomization methods, and depend on mathematic models called chaotic maps. Because of the behavior of chaotic maps, they can be used in different cases [5] to improve optimization algorithms, avoiding local stagnation and helping convergence in the algorithm; there are several maps over literature, but only some sets are commonly used to boost the performance of metaheuristics. Chaos has gained popularity as an improvement for metaheuristics, such as Particle Swarm Optimization (PSO) [6], Krill herd algorithm (KH) [7], Biogeography-based optimization algorithm (BBO) [8], to mention some; where chaos in being embedded in some of the multiple phases of the algorithm, like the formation of initial population in several cases of metaheuristics [9]. Even, in some algorithms, because of the nature of chaos [10], they have principles in the formation of fractals in the formation of chaotic maps, of which, other methods have incorporated to improve further the algorithm in search of solutions [11].

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3 Fuzzy Multiverse Optimizer and Chaos Theory The MVO algorithm has proven, in previous works [12–16] to be competitive against other metaheuristics. It is one of the multiple algorithms proposed by Mirjalili et al. [17], and has its behavior inspired in the concepts of black holes, white holes and worm holes, that can be found in cosmology. Since his appearance in 2015, other scientists have made some improvements of the algorithm [18] by implementing neural networks, use of levy flights, or other randomization techniques [19–22]. One of the improvements we made to the algorithm corresponds to the use of Fuzzy Logic, where we dynamically adapt two main concepts of the algorithm, corresponding to Travel Distance Rate (TDR) and Worm hole Existence Probability (WEP) presented in (1) and (2). Here, WP represents WEP, TR is TDR, amax and amin are the maximum and minimum values of WEP, l and L are the actual and maximum iterations, and p is the exploitation accuracy. A Mamdani fuzzy inference system is implemented for these equations and can be observed in Fig. 1 and 2 for WEP and TDR respectively.   amax − amin (1) WP = amin + l L TR = 1 −

l 1/p L1/p

(2)

input Lightyears Low

High

1

Degree of membership

Degree of membership

1

output WEP

Medium

0.8 0.6 0.4 0.2 0

Low

Medium

High

0.8 0.6 0.4 0.2 0

0

0.2

0.4 0.6 Lighyears

0.8

1

0

0.2

0.4 0.6 WEP

0.8

1

Fig. 1. Input and Output of fuzzy inference system for WEP.

This variation of the algorithm is called Fuzzy Multiverse Optimizer (FMVO) and improves mainly because WEP and TDR are used for exploration and exploitation on the algorithm, adjusting dynamically over iterations as we can observe from our works [14] on the algorithm. For Chaos theory, we mainly use chaotic maps in the equations of worm hole, white j hole and black hole, presented in (3) and (4), where xi,t is the j th parameter of the i th solution, Ui,t is the i th solution, NI (Ui,t ) is the normalized inflation rate, r1 , r2 , r3 , r4 j j are random numbers in [0, 1], xk,t is the j th parameter of the k th solution; xbest,t is the j

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output TDR High

1

Degree of membership

Degree of membership

1

Low

0.8 0.6 0.4 0.2 0

Low Medium

High

0.8 0.6 0.4 0.2 0

0

0.2

0.4 0.6 Lighyears

0.8

1

0

0.2

0.4 0.6 TDR

0.8

1

Fig. 2. Input and Output of fuzzy inference system for TDR.

th parameter of the best solution so far, lbj and ubj are the lower and upper limits of the variable. ⎧ j  j j j ⎪ ⎨ xbest,t + TR ub − lb r4 + lb r3 < 0.5yr2 < WP j j xi,t+1 = xbest,t (3) − TR ubj − lbj r4 + lbj r3 ≥ 0.5yr2 < WP ⎪ ⎩ j xi,t r2 ≥ WP

j xk,t r1 < NI (Ui,t ) j xi,t+1 = (4) j xi,t r1 ≥ NI (Ui,t ) The chaotic maps are used in the r1 , r2 , r4 parameters of the equations from a set of 10 maps, where after some tests, we selected the best performing maps in each parameter, this selection can be observed in Table 1, with the result of 6 new variants. These variants use fuzzy logic and chaotic maps, so we called it Fuzzy-Chaotic Multiverse Optimizer (FCMVO); and the resulting variants are referred as Elitist FCMVO. Table 1. Elitist Chaotic Variants of MVO. Variant name

r1

r2

r4

RT1

Gauss

Singer

Sinusoidal

RT2

Tent

RT3 RT4

Circle Iterative

Sinusoidal

RT5

Tent

RT6

Circle

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4 Testing and Results For the testing of the elitist variants of FCMVO, we are using 13 benchmark mathematical functions [17] of the original MVO, performing 30 tests with 5, 50 and 100 dimensions and 500 iterations; using Z-test to compare our results between the original algorithm, the chaotic variant and the Fuzzy variant with a 95% of confidence. As we observed in Table 1, we selected only some of the 10 chaotic maps, this is because they were the best performing ones in independent tests for each parameter. Because of the length if this paper, we are only presenting a summary of the best results of these variants. In Table 2 can be observed a comparison with only Chaotic Multiverse Optimizer (CMVO) variants of the algorithm, with some of the six combinations, as for Table 3, we are comparing between all the CMVO variants in function 1. Table 2. Test results in 5 dimensions comparing MVO with CMVO variants. MVO

CMVO RT3

CMVO RT6

Function

Average

SD

Average

SD

z

Average

SD

z

F1

1.22 × 10–3

8.04 × 10−4

5.79 × 10−4

4.07 × 10−4

−3.87

4.94 × 10−4

3.64 × 10−4

− 4.48

F2

5.84 × 10−3

2.46 × 10−3

2.22 × 100

4.67 × 100

2.60

2.25 × 100

5.36 × 100

2.30

F3

2.40 × 10−3

1.75 × 10−3

1.64 × 10−3

1.28 × 10−3

−1.92

1.19 × 10−3

9.10 × 10−4

−3.36

F4

2.14 × 10−2

7.75 × 10−3

1.55 × 10−2

5.96 × 10−3

−3.35

1.23 × 10−2

5.52 × 10−3

−5.28

F5

3.97 × 100

5.84 × 100

2.18 × 102

4.65 × 102

2.52

2.67 × 102

5.58 × 102

2.59

F6

1.12 × 10−3

8.48 × 10−4

6.94 × 10−4

5.75 × 10−4

−2.28

5.37 × 10−4

4.33 × 10−4

−3.35

F7

8.82 × 10−4

6.00 × 10−4

1.16 × 10−3

8.84 × 10−4

1.41

9.33 × 10−4

6.84 × 10−4

0.31

F8

1.70 × 103

1.85 × 102

1.43 × 103

2.13 × 102

−5.30

1.40 × 103

3.33 × 102

−4.36

F9

8.63 × 10–1

8.15 × 10–1

3.65 × 100

1.98 × 100

7.13

2.82 × 100

2.42 × 100

4.19

F10

1.91 × 10–2

6.16 × 10–3

1.55 × 100

3.01 × 100

2.78

7.22 × 10–1

1.06 × 100

3.64

F11

1.07 × 10–1

7.08 × 10–2

1.90 × 10–1

1.24 × 10–1

3.20

2.21 × 10–1

1.33 × 10–1

4.14 (continued)

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CMVO RT3

CMVO RT6

Function

Average

SD

Average

SD

Average

SD

F12

1.82 × 10–4

2.52 × 10–4

8.11 × 10–1

1.05 × 100

z 4.22

8.04 × 10–1

1.06 × 100

z 4.15

F13

6.49 × 10–4

2.00 × 10–3

9.03 × 10–3

8.17 × 10–3

5.46

8.84 × 10–3

1.03 × 10–2

4.27

Table 3. Comparison of z-value between MVO and CMVO for F1 in 5 dimensions. F1

MVO

MVO RT1

MVO RT2

MVO RT3

MVO RT4

MVO RT5

MVO RT6

Average 1.22E−03 1.82E−03 1.29E−03 5.79E−04 1.94E−03 1.04E−03 4.94E−04 MVO

–

1.9825

0.2601

−3.8707

1.5577

−0.7588

−4.4773

MVO RT1

−1.9825

–

−1.4810

−4.4906

0.2452

−2.4148

−4.8327

MVO RT2

−0.2601

1.4810

–

−2.8402

1.3007

−0.8346

−3.2075

MVO RT3

3.8707

4.4906

2.8402

–

3.0329

2.3094

−0.8494

MVO RT4

−1.5577

−0.2452

−1.3007

−2.8402

–

−1.8873

−3.2298

MVO RT5

0.7588

2.4148

0.8346

−2.8402

1.8873

–

−2.7752

MVO RT6

4.4773

4.8327

3.2075

−2.8402

3.2298

2.7752

–

In the comparison of Table 2 and 3, we observed some cases where the CMVO variant with the selected chaotic maps, can perform better than the original algorithm, as is the case of RT3 and RT6 variants. The results obtained in Table 4 present the new FCMVO variant in one of the mentioned combinations of chaotic maps. For the last results we can observe one of the best performing variants of the FCMVO algorithm, which consists of the Gauss, Singer and Circle map, surpassing the original MVO algorithm and the chaotic variant in some of the functions, proving that the combination of fuzzy logic and chaos theory in the MVO algorithm improves the results.

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Table 4. Test results in 50 dimensions comparing FCMVO with MVO and CMVO variants Function

FCMVO RT3

MVO

CMVO RT3

Average

SD

Average

SD

z

Average

SD

z

F1

1.71 × 100

4.53 × 10–1

1.04 × 101

2.12 × 100

−21.98

3.85 × 101

1.13 × 101

−17.84

F2

5.30 × 1015

2.86 × 1016

4.29 × 102

1.40 × 103

1.02

1.68 × 1012

8.00 × 1012

1.02

F3

1.39 × 104

3.15 × 103

5.87 × 103

1.42 × 103

12.72

1.65 × 104

3.09 × 103

−3.24

F4

6.65 × 101

6.70 × 100

1.66 × 101

6.53 × 100

29.19

6.80 × 101

9.43 × 100

−0.69

F5

2.49 × 103

3.32 × 103

6.64 × 102

6.96 × 102

2.94

5.54 × 103

8.51 × 103

−1.83

F6

1.84 × 100

4.32 × 10–1

1.06 × 101

2.71 × 100

−17.42

3.70 × 101

7.93 × 100

−24.23

F7

2.91 × 10–1

7.87 × 10–2

1.19 × 10–1

4.03 × 10–2

10.65

2.62 × 10–1

6.61 × 10–2

1.57

F8

1.16 × 104

1.09 × 103

1.25 × 104

8.00 × 102

−3.54

1.17 × 104

8.58 × 102

−0.47

F9

3.53 × 102

3.37 × 101

2.54 × 102

4.94 × 101

9.06

2.62 × 102

2.28 × 101

12.17

F10

1.68 × 101

2.65 × 100

3.49 × 100

3.08 × 100

17.99

5.85 × 100

8.58 × 10–1

21.61

F11

8.57 × 10–1

6.76 × 10–2

1.09 × 100

1.82 × 10–2

−18.22

1.34 × 100

8.60 × 10–2

−23.96

F12

2.96 × 101

1.32 × 101

6.57 × 100

2.64 × 100

9.39

3.58 × 101

1.58 × 101

−1.64

F13

8.24 × 101

2.74 × 101

9.08 × 100

1.34 × 101

13.15

9.09 × 101

3.00 × 101

−1.15

5 Conclusions In our experimentation over MVO algorithm and the variants CMVO and FCMVO, we could achieve some improvement over the original algorithm to perform competitively on benchmark mathematical functions. Our research included the results achieved from previous works on fuzzy logic and, some of them from implementation of chaotic maps. Over the results obtained, we observed that certain chaotic maps could improve the original algorithm, and with the combination of the best maps, some of them could perform better than the others, such as the case of CMVO RT3 and RT6 variants, that used Gauss-Singer-Circle and Gauss-Iterative-Circle maps respectively. From this analysis, we can conclude that Gauss and Circle maps in r1 and r4 parameters are the most influencing chaotic maps for the MVO algorithm.

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After implementing the best chaotic maps, we moved forward with the FCMVO variant, testing with the same chaotic variants, and observing that fuzzy logic can improve further the chaotic variant of the algorithm. As a main conclusion, the FCMVO variant with the elitist chaotic MVO variant can perform competitively on benchmark mathematical functions, opening a path to test on some application cases, such as fuzzy controllers and, also by using type-2 fuzzy logic on the fuzzy inference system [23]. For future research, we can advance our experimentation with some applications such as fuzzy logic controller design, in which, the original MVO algorithm [14] performed competitively over other metaheuristics, which can be inherited by the FCMVO. This research also opened a path to test with other implementations to improve further the MVO algorithm, and close the gap of some of the problems of the original algorithm, such as local optima stagnation.

References 1. Lagunes, M.L., Castillo, O., Valdez, F., Soria, J.: Multi-metaheuristic competitive model for optimization of fuzzy controllers. Algorithms 12, 90 (2019). https://doi.org/10.3390/A12 050090 2. Guerrero-Luis, M., Valdez, F., Castillo, O.: A review on the cuckoo search algorithm. In: Castillo, O., Melin, P. (eds.) Fuzzy logic hybrid extensions of neural and optimization algorithms: theory and applications. SCI, vol. 940, pp. 113–124. Springer, Cham (2021). https:// doi.org/10.1007/978-3-030-68776-2_7 3. Ochoa, P., Castillo, O., Soria, J.: Optimization of fuzzy controller design using a differential evolution algorithm with dynamic parameter adaptation based on Type-1 and Interval Type2 fuzzy systems. Soft Comput. 24, 193–214 (2020). https://doi.org/10.1007/S00500-01904156-3/TABLES/14 4. Bernal, E., Lagunes, M.L., Castillo, O., Soria, J., Valdez, F.: Optimization of Type-2 fuzzy logic controller design using the GSO and FA algorithms. Int. J. Fuzzy Syst. 23, 42–57 (2021). https://doi.org/10.1007/S40815-020-00976-W/TABLES/18 5. Atan, Ö., Kutlu, F., Castillo, O.: Intuitionistic fuzzy sliding controller for uncertain hyperchaotic synchronization. Int. J. Fuzzy Syst. 22, 1430–1443 (2020). https://doi.org/10.1007/ S40815-020-00878-X/TABLES/2 6. Sánchez, D., Melin, P., Castillo, O.: Comparison of particle swarm optimization variants with fuzzy dynamic parameter adaptation for modular granular neural networks for human recognition. J. Intell. Fuzzy Syst. 38, 3229–3252 (2020). https://doi.org/10.3233/JIFS-191198 7. Kondepogu, V., Bhattacharyya, B.: A novel sparse multipath channel estimation model in OFDM system using improved Krill Herd-deep neural network. J. Ambient Intell. Humaniz. Comput. 14, 2567–2583 (2023). https://doi.org/10.1007/S12652-022-04503-7/FIGURES/13 8. Zhu, M., Xu, Z., Zang, Z., Dong, X.: Design of FOPID controller for pneumatic control valve based on improved BBO algorithm. Sensors 22, 6706 (2022). https://doi.org/10.3390/S22 176706 9. Tang, R., Fong, S., Dey, N., Tang, R., Fong, S., Dey, N.: Metaheuristics and chaos theory. Chaos Theory. (2018). https://doi.org/10.5772/INTECHOPEN.72103 10. Anter, A.M., Gupta, D., Castillo, O.: A novel parameter estimation in dynamic model via fuzzy swarm intelligence and chaos theory for faults in wastewater treatment plant. Soft Comput. 24, 111–129 (2020). https://doi.org/10.1007/S00500-019-04225-7/FIGURES/9 11. Talatahari, S., Azizi, M.: Chaos game optimization: a novel metaheuristic algorithm. Artif. Intell. Rev. 54, 917–1004 (2021). https://doi.org/10.1007/S10462-020-09867-W/TABLES/25

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12. Amézquita, L., Castillo, O., Soria, J., Cortes-Antonio, P.: Optimal design of fuzzy controllers using the multiverse optimizer. In: Abraham, A., Hanne, T., Castillo, O., Gandhi, N., Nogueira Rios, T., Hong, T.-P. (eds.) HIS 2020. AISC, vol. 1375, pp. 289–298. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-73050-5_29 13. Amézquita, L., Castillo, O., Cortes-Antonio, P.: Fuzzy-chaotic variant of the multiverse optimizer algorithm in benchmark function optimization. In: In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) INFUS 2022. LNNS, vol. 504, pp. 53–63. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5_8 14. Amézquita, L., Castillo, O., Soria, J., Cortes-Antonio, P.: A fuzzy variant of the multi-verse optimizer for optimal design of fuzzy controllers. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds.) INFUS 2021. LNNS, vol. 307, pp. 537–545. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-85626-7_63 15. Amézquita, L., Castillo, O., Soria, J., Cortes-Antonio, P.: Optimization of membership function parameters for fuzzy controllers in cruise control problem using the multi-verse optimizer. In: Castillo, O., Melin, P. (eds.) Fuzzy Logic Hybrid Extensions of Neural and Optimization Algorithms: Theory and Applications. SCI, vol. 940, pp. 15–40. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-68776-2_2 16. Amézquita, L., Castillo, O., Soria, J., Cortes-Antonio, P.: A novel study of the multi-verse optimizer and its applications on multiple areas of computer science. In: Melin, P., Castillo, O., Kacprzyk, J. (eds.) Recent Advances of Hybrid Intelligent Systems Based on Soft Computing. SCI, vol. 915, pp. 133–144. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-587 28-4_7 17. Mirjalili, S., Mirjalili, S.M., Hatamlou, A.: Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput. Appl. 27(2), 495–513 (2015). https://doi.org/10. 1007/s00521-015-1870-7 18. Mirjalili, S., Jangir, P., Mirjalili, S.Z., Saremi, S., Trivedi, I.N.: Optimization of problems with multiple objectives using the multi-verse optimization algorithm. Knowledge-Based Syst. 134, 50–71 (2017). https://doi.org/10.1016/j.knosys.2017.07.018 19. Sayed, G.I., Darwish, A., Hassanien, A.E.: A new chaotic multi-verse optimization algorithm for solving engineering optimization problems. J. Exp. Theor. Artif. Intell. 30, 293–317 (2018). https://doi.org/10.1080/0952813X.2018.1430858 20. Elfattah, M.A., Hassanien, A.E., Abuelenin, S., Bhattacharyya, S.: Multi-verse optimization clustering algorithm for binarization of handwritten documents. In: Bhattacharyya, S., Mukherjee, A., Bhaumik, H., Das, S., Yoshida, K. (eds.) Recent Trends in Signal and Image Processing. AISC, vol. 727, pp. 165–175. Springer, Singapore (2019). https://doi.org/10.1007/ 978-981-10-8863-6_17 21. Hu, C., Li, Z., Zhou, T., Zhu, A., Xu, C.: A multi-verse optimizer with levy flights for numerical optimization and its application in test scheduling for network-on-chip. PLoS ONE 11, 1–22 (2016). https://doi.org/10.1371/journal.pone.0167341 22. Sayed, G.I., Darwish, A., Hassanien, A.E.: Quantum multiverse optimization algorithm for optimization problems. Neural Comput. Appl. 31(7), 2763–2780 (2017). https://doi.org/10. 1007/s00521-017-3228-9 23. Castillo, O., Amador-Angulo, L.: A generalized type-2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design. Inf. Sci. (Ny) 460–461, 476–496 (2018). https://doi.org/10.1016/J.INS.2017.10.032

Fuzzy Goal Programming Model for Sequencing Multi-model Assembly Line with Sequence Dependent Setup Times in Garment Industry Elvin Sarı(B) , Mert Paldrak , Tunahan Kuzu, Devin Duran, Yaren Can, Sude Dila Ceylan, Mustafa Arslan Örnek , and Ba¸sak Erol Industrial Engineering Department, Ya¸sar University, Bornova, ˙Izmir, Turkey [email protected], {mert.paldrak,arslan.ornek}@yasar.edu.tr

Abstract. In today’s competitive market, the pressure on organizations to find ways to create value for customers and meet their requirements becomes stronger. In this manner, clothing manufacturers focus on the production of various products with low stock to minimize their costs. In the garment industry, assembly lines are commonly used production systems whose balance is the main concern of production managers. Meeting customer demand on-time is of outsized importance for the reputation of a clothing manufacturer and several objective functions must be considered simultaneously. In this study, two objective functions, namely minimization of setup times and minimization of lateness are handled to increase the efficiency of the assembly line and convenience to customers. A real-life problem in the garment industry is defined and formulated as a Fuzzy Goal Programming model since the aspiration levels of each objective are not certainly known. Two different Weighted Fuzzy Goal Programming Models are proposed, and the proposed mathematical models are tackled with the help of ILOG IBM CPLEX OPTIMIZATION STUDIO version 20.1 and the solutions are interpreted from a decision-maker perspective. Keywords: Garment Industry · Multi-Model Assembly Line · Sequence Dependent Setup times · Fuzzy Logic · Fuzzy Goal Programming

1 Introduction The garment industry is of outsized importance for global economy that consists of the production, design and marketing of clothing and textiles. The industry has a rich history spanning even centuries and has been evolving. The sector includes a wide range of manufacturing processes, technologies, and materials to meet the demand requirements of customers changing by time. This industry plays a pivotal role in the global trades, and it also employs millions of people worldwide. On the other hand, garment industry is an ever-changing and dynamic field, with new trends and changing customer preferences emerging perpetually. Hence, garment manufacturers need to stay up to date with the latest industry trends to stay competitive in this challenging market. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 480–490, 2023. https://doi.org/10.1007/978-3-031-39774-5_54

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There exist numerous processes in garment production, yet the sewing process is a critical aspect because it uses various tools and techniques to join fabric pieces together. Although the sewing process is performed by computerized sewing machines with the advent of recent technologies developments, manual techniques are still employed in most garment companies. Since garment industry is notorious for its labor intensiveness, efficient utilization of machines and workers is the issue of line managers. The maximization of the efficiency of a line relies on the assignment of tasks as equally as possible to balance the line. Consequently, line balancing problem emerges in garment industry to increase efficiency of both workers and machines. Line balancing is employed in order to optimize the production process by minimizing idle time and ensuring maximum efficiency of the machines within the production line. By balancing the workload across different workstations, line balancing helps reduce bottlenecks, increase productivity, and ameliorate the overall efficiency of the production process. Line balancing in garment industry is especially important since the manufacturing process involves multiple stages such as cutting, stitching, finishing, and packaging. In the context of a sewing line, line balancing refers to the process of distributing sewing work across different workstations to minimize idle time and maximize efficiency. To balance a sewing line, the production process must be analyzed to identify the tasks involved to obtain the final product. The tasks that can be performed in parallel are grouped into workstations, and the workload is distributed evenly among these workstations. In this study, we deal with the problem of sewing line balancing problem in a garment company where efficiency of the sewing line is related to completion time of each product. Due to the variety of products, there is a set-up time for preparing machine or equipment between two consecutive tasks, which has a huge impact on the overall efficiency and productivity of the sewing line. Therefore, we formulate a mathematical model where we aim to minimize total set-up time of the sewing line and total lateness. The remaining sections of the study are as follows. In Sect. 3, a multi-objective Fuzzy Goal Programming model for our problem is formulated and two different weighted fuzzy goal programming models are presented. In Sect. 4, our numerical examples, and computational studies along with results are given for three instances with different levels. Lastly, In Sect. 5, our findings are summarized, and future work has been prospected.

2 Literature Review The garment industry is a complex and tremendously competitive market that necessitates garment manufacturers to be efficient and innovative in their operations to make profit. In this section, a literature review on the garment industry and sewing line is conducted to provide a general overview of the key issues and challenges that garment manufacturers and companies face regularly considering goal programming methods. In the literature, [2] introduced Goal Programming as a practical technique for addressing problems related to multi-objective programming. Goal programming is a commonly used technique for addressing multi-objective optimization problems, as it simplifies complex trade-offs between multiple objectives by transforming them into a

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standard single-objective programming problem. To accomplish this, deficiency variables that are non-negative in value are introduced to represent the degree to which goals or other soft constraints that are not necessarily required to be strictly enforced are not met [5]. A Goal Programming model aims to satisfy all goals to the greatest extent possible by minimizing a weighted sum of the deficiency variables [5]. The use of the fuzzy approach offers a different means of modeling imprecision and uncertainty that is more effective than traditional methods, particularly in cases where no historical data is available [1]. The implementation of fuzzy set theory has proven to be highly effective in addressing various scheduling issues, particularly in situations where decision-making and intuition are critical factors [5]. For example, customer demand, processing times, production due dates or job precedence relations [3]. In the literature, [4] examined a scheduling problem related to a single machine, where in due dates were modeled using fuzzy set theory. Conventional goal programming assumes that the aspiration levels of objectives are precisely known, but in practice, determining these values with certainty is often challenging. To address this issue, fuzzy set theory can be employed to manage a goal with an imprecise aspiration level. Fuzzy goal programming has been employed to address multi-objective transportation problems. The assumption made by traditional goal programming methods is that the decision-maker can pinpoint an exact ambition level for each of the objectives. The aspiration levels are not known for sure in most realworld challenges, though. Fuzzy goal programming may be used in these circumstances [6].

3 Mathematical Modeling In this study, a real-life production management problem in garment industry is described and modelled using operational research methodologies where the objectives are minimization of setup times and minimization of total lateness of products. These objectives are vital to the company since the management focuses on increasing customer satisfaction level while maximizing profit. These objectives are conflicting, and we formulated a multi-objective MIP model and converted the model into two Fuzzy Goal Programming Models. The assumptions made while creating this mathematical model are as follows: • Initially, all models are assumed to be ready to go into the production line. • Setup times between two consecutive models are known in advance. • No pre-emption between models is allowed. In other words, a model is not allowed to be interrupted in the middle of execution until its process terminates. • The due date of each model is determined according to the customer orders. Mathematical Model. Indices: i, k ∈ I = {0, . . . , n} –index of models produced on the line. j, a ∈ J = {0, . . . , n} –index of position (sequence) of a model on the line.

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Parameters: Pti = Processing time of model i. DDi = Due date of model i. STik = Setup time between model i and model k. Decision Variables: CTj = Completion time of a model in the position j. Xij = 1; if model i is processed in position j, otherwise 0. Yij = 1; if model i is processed before its due date in position j, otherwise 0. Zjik = 1; if model k is processed immediately after model i in position j, otherwise 0. Lj = Lateness of a model in position j. Ej = Earliness of a model in position j. 3.1 Fuzzy Goal Programming Model In this study, to solve MOLP, two different Weighted Fuzzy Goal Programming formulations inspired by [6] are developed. Goal Programming technique is a well-known approach to handling multi-objective problems since it helps reduce a complex multiobjective problem to a standard single objective programming using weights for each objective. In most cases, there does not generally exist an optimal solution that simultaneously optimize all objectives of a multi-objective problem. Hence, decision makers seek suitable compromise solutions for such problems. Although there are numerous techniques to deal with multi-objective problems, goal programming is most efficient one. Conventional goal programming models consider that decision maker(s) is able to determine a precise aspiration level for each of the objective functions where the model aims to minimize the deviation from goals. Nonetheless, in real-life problems, decision maker cannot know the aspiration levels of each objective with certainty. In such cases, fuzzy goal programming can be employed. In this paper, it is thought that the aspiration level of each of the objective functions is not known precisely. Therefore, the following model is to be optimized. Objective Functions: OPT min

n  n n  

Zjik ∗STik Z1

(1)

j=0 i=0 k=0 n 

Lj Z2

(2)

Xij = 1, ∀i

(3)

Xij = 1, ∀j

(4)

X00 = 1

(5)

min

j=0

Constraints: n  j=0 n  i=0

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Zjik = 1, ∀j; j = 0(j = 1, . . .)

(6)

i=0 k=0

Xij−1 + Xkj − Zjik ≤ 1, ∀j, ∀i and ∀k; k = i; j = 0(j = 1, . . . ., n)

(7)

Z0ik = 0, ∀i, ∀k

(8)

CTj =

j n   a=1 i=0

Pti∗ xia +

n n  

Zjik ∗STik , ∀j

CTj − DDi ∗Xij + M*Yij ≥ 0, ∀jand ∀i n 

(9)

i=1 k=i

(10)

DDi ∗Yij − CTj ≤ Ej , ∀j

(11)

DDi ∗Xij − CTj + Lj ≥ 0, ∀j

(12)

i=0 n  i=0

Lj ≥ 0, ∀j

(13)

Ej ≥ 0, ∀j

(14)

Xij , Yij , Zjik binary, ∀j, i, k, a

(15)

Our objectives are to minimize total setup times (1) and total lateness (2) on the assembly line while changing the model type. Constraints (3) and (4) ensure that only one model can be assigned to a position and only one position to a model. Since the last position of the previous model type is the first position of the next model, the assignment of model 0 to the 0th position is provided by constraint (5). For all positions except the starting position, the process of only one model immediately after a model is provided by the equation in constraint (6). As seen in constraint (7), if two models are processed one after the other, the first model should be processed right after the other model. In the equation in Constraint (8), a model at position 0 indicates that any model cannot be found in its previous position. Constraint (9) calculates the completion time of the model at each position. Constraint (10), (11), and (12) determine earliness and lateness of a position according to the assigned model. As seen in constraint (13) and (14) the lateness and earliness are integer variables. Constraint (15) gives the domain of decision variables. In this model, the symbol  is elaborated as “essentially less than” and each one of two fuzzy goals is represented by using a fuzzy set defined over the feasible set with the fuzzy membership function. Applying piecewise linear membership functions to express fuzzy goals and using weighted goal programming technique, the following model is achieved.

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3.2 Fuzzy Weighted Goal Programming Model-1 Objective Function: 2 

min

pl  lR

(16)

Zjik ∗STik − p1 ≤ Z1

(17)

Lj − p2 ≤ Z2

(18)

wl ∗

l=1

Constraints: n  n n   j=0 i=0 k=0 n  j=0

pl ≤ 1, l = 1, 2  lR

(19)

pl ≥ 0, l = 1, 2

(20)

Constraints Between (3) and (15) In this model, wl represents of the weight of the lth fuzzy goal which are normalized such that the sum of wl s is equal to 1. Moreover, the slack variable of the constraint pl th lR ≤ 1 equals to the degree of membership function for the l fuzzy model. 3.3 Fuzzy Weighted Goal Programming Model-2 To propose the new model, instead of using deviational variables pl , l = 1, 2 in Fuzzy Goal Programming Model-1, it is suggested to use deviational function r(1 − wl ). Consequently, to solve our fuzzy multi-objective model, the following fuzzy goal programming-oriented model is stated as below. Objective Function: min Z = r

(21)

Constraints: n  n n  

Zjik ∗STik − r*(1 − w1 ) ≤ Z1

(22)

Lj − r*(1 − w2 ) ≤ Z2

(23)

j=0 i=0 k=0 n  j=0

r*(1 − wl ) ≤  lR, l = 1, 2

(24)

r≥0

(25)

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Constraints Between (3) and (15) Where Zl is considered as optimal solution of the lth objective function. And  lR is determined by max Zn − Zm n = 1, .., l n = m It is noteworthy that in this previous model, values of the deviational function are consistent with the priority of the objectives. In other words, larger value of wk will result in a smaller value of deviation function. By means of this property, the objective function with higher priority will be closer to its aspiration level.

4 Numerical Examples and Computational Results The data set for the problem is collected after face-to-face meetings with sewing line production managers. However, due to the confidential agreements, we are not allowed to use these data as they are. Consequently, we had to generate our data with the help of MATLAB R2016.We employed uniform distribution with relevant parameters to obtain processing time and due date of each job and set-up time between two jobs. Our instances are classified as small, medium and large in terms of number of jobs. Since proposed model is NP-hard, it is impossible to solve the problem in a reasonable computational time, especially when its size is quite large. Taking this fact into considerations, the number of jobs is taken as 5 for a small instance, 8 for a medium instance and 10 for a large instance. Fuzzy weighted goal programming model for each instance is solved using IBM ILOG CPLEX OPTIMIZATION STUDIO version 20.1 on a personal computer. The results that are obtained by solving both models for different weights are demonstrated in Tables 1, 2, 3, 4, 5, 6 and 7. Table 1. Optimal Solution Values of Each Instance for each Objective Function INSTANCE

OBJECTIVE

Setup Time

Lateness

Z1

Z2

n=5

Setup Time

84

3527

84

1066

Lateness

154

1066

n=8

Setup Time

98

8745

98

6415

Lateness

169

6415

n = 10

Setup Time

75

14822

75

11552

Lateness

192

11522

As could be seen in above tables, it can be seen that deviational variables are not consistent with the priority of objectives in Weighted Fuzzy Goal Programming-1. For example, considering medium instance with n = 8 (Table 4), when w1 = 0.3 and w2 = 0.7, the deviational variables are p1 = 37 and p2 = 66, respectively. It means that the first objective function which is less preferred has the value which is closer to

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Table 2. Optimal Objective Values Obtained from Fuzzy Model-1 for n = 5 n=5 w1

w2

1R

2R

p1

p2

w1*(p1/1R) + w2*(p2/2R)

0,1

0,9

70

2461

52

22

0,0823

0,2

0,8

70

2461

52

22

0,1557

0,3

0,7

70

2461

52

22

0,2291

0,4

0,6

70

2461

21

607

0,2680

0,5

0,5

70

2461

21

607

0,2733

0,6

0,4

70

2461

1

1624

0,2725

0,7

0,3

70

2461

0

1683

0,2052

0,8

0,2

70

2461

0

1683

0,1368

0,9

0,1

70

2461

0

1683

0,0684

Table 3. Optimal Objective Values Obtained from Fuzzy Model-2 for n = 5 n=5 w1

w2

1R

2R

r

r*(1-w1)

r*(1-w2)

0,1

0,9

70

2461

73,333

65,9997

7,3333

0,2

0,8

70

2461

82,5

66

16,5

0,3

0,7

70

2461

74,286

52,0002

22,2858

0,4

0,6

70

2461

86,667

52,0002

34,6668

0,5

0,5

70

2461

104

52

52

0,6

0,4

70

2461

130

52

78

0,7

0,3

70

2461

173,333

51,9999

121,3331

0,8

0,2

70

2461

245

49

196

0,9

0,1

70

2461

490

49

441

its aspiration level. However, consistency exists in Table 5 which consists of the results obtained by Weighted Fuzzy Goal Programming-2. Further, using Weighted Fuzzy Goal Programming-1, the same objective values are attained for different weights. Nonetheless, Fuzzy Goal Programming-2 model captures more various solutions with respective weights.

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n=8 w1

w2

1R

2R

p1

p2

w1*(p1/1R) + w2*(p2/2R)

0,1

0,9

71

2330

37

66

0,0776

0,2

0,8

71

2330

37

66

0,1269

0,3

0,7

71

2330

37

66

0,1762

0,4

0,6

71

2330

37

66

0,2254

0,5

0,5

71

2330

7

955

0,2542

0,6

0,4

71

2330

7

955

0,2231

0,7

0,3

71

2330

7

955

0,1920

0,8

0,2

71

2330

7

955

0,1608

0,9

0,1

71

2330

0

2328

0,0999

Table 5. Optimal Objective Values Obtained from Fuzzy Model-2 for n = 8 n=8 w1

w2

1R

2R

r

r*(1-w1)

r*(1-w2)

0,1

0,9

71

2330

25,556

23,0004

2,5556

0,2

0,8

71

2330

28,75

23

5,75

0,3

0,7

71

2330

32,857

22,9999

9,8571

0,4

0,6

71

2330

38,333

22,9998

15,3332

0,5

0,5

71

2330

46

23

23

0,6

0,4

71

2330

57,5

23

34,5

0,7

0,3

71

2330

76,667

23,0001

53,6669

0,8

0,2

71

2330

115

23

92

0,9

0,1

71

2330

230

23

207

Table 6. Optimal Objective Values Obtained from Fuzzy Model-1 for n = 10 n = 10 w1

w2

1R

2R

p1

p2

w1*(p1/1R) + w2*(p2/2R)

0,1

0,9

117

3300

30

122

0,0589

0,2

0,8

117

3300

20

185

0,0790

0,3

0,7

117

3300

20

185

0,0905 (continued)

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Table 6. (continued) n = 10 w1

w2

1R

2R

p1

p2

w1*(p1/1R) + w2*(p2/2R)

0,4

0,6

117

3300

20

185

0,1020

0,5

0,5

117

3300

20

185

0,1135

0,6

0,4

117

3300

20

185

0,1250

0,7

0,3

117

3300

20

185

0,1365

0,8

0,2

117

3300

20

185

0,1480

0,9

0,1

117

3300

0

3300

0,1000

Table 7. Optimal Objective Values Obtained from Fuzzy Model-2 for n = 10 n = 10 w1

w2

1R

2R

r

r*(1-w1)

r*(1-w2)

0,1

0,9

117

3300

98,889

89,0001

9,8889

0,2

0,8

117

3300

111,25

89

22,25

0,3

0,7

117

3300

116,667

81,6669

35,0001

0,4

0,6

117

3300

100

60

40

0,5

0,5

117

3300

120

60

60

0,6

0,4

117

3300

150

60

90

0,7

0,3

117

3300

174,286

52,2858

122,0002

0,8

0,2

117

3300

152,5

30,5

122

0,9

0,1

117

3300

205,556

20,5556

185,0004

5 Conclusions and Future Works This study deals with a multi-objective optimization problem in a garment manufacturing company where objectives are to minimize total set-up times and lateness in their sewing line. Since the precise aspiration level for each of the objectives is not certainly known, we resort to fuzzy goal programming technique to solve our multi-objective optimization problem. Therefore, inspired by [6], two weighted fuzzy goal programming models are proposed. By varying the weights in both models, different solutions are attained. Regarding properties of Weighted Fuzzy Goal Programming model, different effective solutions are obtained. To better analyze the results, we generated three problem instances with different size and compared their efficiency with respect to solution quality and consistency. As a result, it is demonstrated that Weighted Fuzzy Goal Programming Model 2 outperforms Weighted Fuzzy Goal Programming 1 since it captures various consistent solutions.

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As a future direction, try to propose new algorithms and methods with nice properties to deal with such a multi-objective problem could be considered as a general topic for the future study.

References 1. Anglani, A., Grieco, A., Guerriero, E., Musmanno, R.: Robust scheduling of parallel machines with sequence-dependent set-up costs. Eur. J. Oper. Res. 161, 704–720 (2005) 2. Charnes, A., Cooper, W.W.: Management models and industrial applications of linear programming. Wiley, New York (1961) 3. Gharehgozli, A.H., Tavakkoli-Moghaddam, R., Zaerpour, N.: A fuzzy-mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates. Robot. Comput.-Integr. Manufact. 25(4–5), 853–859 (2009) 4. Han, S., Ishii, H., Fujii, S.: One machine scheduling problem with fuzzy due dates. Eur. J. Oper. Res. 79, 1–12 (1994) 5. Jones, D., Tamiz, M.: Practical goal programming. Springer US, Boston, MA (2010) 6. Karakutuk, S.S., Ornek, M.A.: A goal programming approach to lean production system implementation. J. Oper. Res. Society 73, 1–14 (2022) 7. Petrovic, R., Petrovic, D.: Multicriteria ranking of inventory replenishment policies in the presence of uncertainty in customer demand. Int. J. Prod. Econ. 71, 439–446 (2001) 8. Rivaz, S., Nasseri, S.H., Ziaseraji, M.: A fuzzy goal programming approach to multi-objective transportation problems. Fuzzy Inform. Eng. 12(2), 139–149 (2020)

Fuzzy Color Computing Based on Optical Logical Architecture Victor Timchenko1

, Yuriy Kondratenko2,3 , Oleksiy Kozlov3(B) and Vladik Kreinovich4

,

1 Admiral Makarov National University of Shipbuilding, Mykolaiv 54025, Ukraine 2 Institute of Artificial Intelligence Problems of MES and NAS of Ukraine, Kyiv 01001, Ukraine 3 Petro Mohyla Black Sea National University, Mykolaiv 54003, Ukraine

[email protected]

4 University of Texas at El Paso, El Paso, TX 79968, USA

[email protected]

Abstract. This paper is dedicated to the development of intelligent techniques of optical computing for real-time decision support systems (DSS) with a large array of fuzzy input data. Currently offered as an alternative to the binary systems, fuzzy optical computing devices are very complex. To increase the efficiency of logical systems in the formation and processing of an array of input fuzzy data, we propose to use a light emitter of a certain color as a fuzzy set (FS) that is a carrier of logical information. This allows building of logical solutions and conclusions based on the additive and subtractive conversion of light radiation by appropriate color filters, measuring illumination in optical channels, and switching light emitters. The main algorithmic logical procedures based on optical gates that perform the basic logical operations of disjunction and conjunction, negation, and search for a new solution are considered. To solve a problem with a large amount of fuzzy data, we propose the architecture of networks of logical structures and inference procedures. Also, we implement optimization of the optical circuit solutions to increase the reliability of decisions and estimates. When describing color sets using a fuzzy sequence, we propose to implement a non-linear rank scale. This increases the accuracy of solutions, as shown by the decrease in root-means-square estimation error. Keywords: fuzzy color computing · optical logical architecture · ranged fuzzy sets

1 Introduction Improving the efficiency of controlling of various complex plants and processes in modern industries and different sectors of the economy is impossible without the use of intelligent DSS [1–3]. The need to develop such a class of systems for automated decision-making processes, as well as selecting, processing and evaluating information in conditions of incomplete data and uncertainty, is due to the fact that a human operator must predict the behavior of the plant in real-time and make complex decisions © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 491–498, 2023. https://doi.org/10.1007/978-3-031-39774-5_55

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that ensure its safest functioning, avoiding global accidents. The process of plant control becomes even more complicated when accidental failures or damage occur. At the same time, in extreme situations, the degree of mental and physical stress on the human operator increases significantly, which leads to an increase in the likelihood of him/her making suboptimal or even false decisions. Theoretical and applied studies conducted in different countries of the world confirm the feasibility and high efficiency of applying the principles of the theory of fuzzy logic and soft computing in the construction of DSS for the automation of plants operating in randomly changing or uncertain conditions [4–6]. The given fuzzy DSSs make it possible to approximate any non-linear multidimensional dependencies, effectively generalize fuzzy expert information of large size, train automatically based on experimental data, and form linguistic models of complex plants and processes [7, 8]. Modern realities of using fuzzy DSS, especially those operating in extreme situations when controlling complex plants, require the processing of an increasing amount of data in a very short time. In this case, there is a significant problem with the need to increase the speed of calculations and the overall speed of the DSS. One of the very promising ways to solve this problem, as well as to increase the efficiency and speed of various computer systems, is the use of optical computing [9, 10]. Modern computer systems based on optical logic have significant benefits, including high speed, the almost unlimited possibility of parallel computing of a huge amount of fuzzy data and compactness [11, 12]. One of the options for implementing optical computing into computer systems is the use of optoelectronic computational devices, in which optical components are used only to transmit information, and the traditional semiconductor gates transform optical signals into electric ones with further performing all necessary computations [13]. This approach allows in a certain way to increase the speed of the entire computer system. Even more efficient is the use of all-optical computational devices, in which all calculations are also performed optically, which allows for maximizing the speed of computational procedures [14]. However, a major limitation of this approach when using binary optical calculations is the large number of required optical logic gates that significantly complicates the design of these computer systems. In particular, fuzzy optical computing devices of such type require very complex architectures, including arrays of prisms and lenses, complex diffraction gratings, additional devices in the form of optoelectronic phase shifters and piezoelectric crystal elements, coding systems, shadow images, holograms and other [15]. To eliminate this shortcoming, as well as to improve the overall efficiency of fuzzy optical computer systems, a completely new approach has recently been applied, which consists in the formation and processing of an array of input data based on the use of light radiation of a certain color as a fuzzy variable that is a carrier of logical information [16–18]. This allows constructing logical conclusions by additive and subtractive transformation of light radiation with appropriate light-color filters, measuring of illumination in optical channels, and switching of light emitters. In [18–20], the structures of optical logic devices (coloroids) are proposed, the input and output actions of which are quanta of color radiation of a certain color. These devices allow performing basic logical operations - disjunction, conjunction, negation, inference, triggering a new solution. When applying stringent requirements to the properties of

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optical systems - monochromatic color and ideal filters, the previously proposed optical coloroids allow carrying out all logical operations. However, with an acceptable degree of blurring of optical systems, a mathematical apparatus is needed that will provide a description of logical operations in a wider range of optical characteristics. To expand the computational capabilities of optical coloroids, we offer a well-developed and multiinstrument fuzzy set apparatus. Thus, the main purpose of this work is to expand mathematical techniques for the analysis, synthesis and programming of logical coloroids based on the definition of the information quantum of the primary and secondary color (and later on to a more extended color gamut) as ranged FS.

2 Optical Color Logical Processing Architecture The architecture of optical logical coloroids for the simplest logical operations is based on the physical properties of additive and subtractive transformation of optical information in the form of a certain color. For processing fuzzy input information and logical inference, the authors proposed [19] to introduce appropriate logical estimates (with ranking by “Yes”, “No”) for primary and secondary colors: red {R} = {N} (“no”), green {G} = {YN} (“probably yes”), blue {B} = {Y } (“yes“), white {W } = {R} + {G} + {B} = {YYNN} “positive decision”, cyan {C} = {G} + {B} = {YYN} (“very probably yes”), magenta {M} = {R} + {B} = {NY } (“probably no”), yellow {Yel} = {R} + {G} = {NNY } (“very probably no”), black {Blc} = {W }−{R}−{G}−{B} = {0} (“no decision”). An elementary additive logical coloroid for the disjunction operation (Fig. 1, a) performs 7 variants of logical operations (taking into account the property of idempotence, excluding repetitive options). The elementary subtractive coloroid for the conjunction operation (Fig. 1, b; F1 , F2 , F3 are the light color filters) performs 49 variants of logical operations (also taking into account the property of idempotence) [20, 21]. An elementary logical coloroid for the negation operation {Blc}¬{W} looks like a programming device with the given fixed values of light filters.

Fig. 1. Elementary logical coloroids: a – for the disjunction; b – for the conjunction

The basic logical coloroid with 12 input ports for implementation of the primary inference procedures is presented in Fig. 2, where levels 1, 2, and 3 are the levels of evaluation; S is a white light emitter. It is used to input fuzzy tactile or expert information

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and processes more than 200 variants of logical operations as well as can serve as the basis to build a branched series-parallel coloroid neural network. An important function of the basic coloroid is that, when receiving a common output signal − the absence of a solution {Blc}, the coloroid produces a new solution output signal {W }, which is a new input signal for the coloroid with the search for a new solution with, of course, updated values of the input information for 12 inputs.

Fig. 2. Basic logical coloroid with 12 input ports

The typical operation of an example logical coloroid can be described in the following logical form: Level 1: {W } = {R} ∪ {G} ∪ {B} = {R, G, B}; Level 2: filters {R, G}, {R, B} – {R, G, B} ∩ {R, G} ∩ {R, B} = {R}, filters {R, G}, {G, B} – {R, G, B} ∩ {R, G} ∩ {G, B} = {G}, filters {R, B}, {G, B} – {R, G, B} ∩ {R, B} ∩ {G, B} = {B}, {W } = {R} ∪ {G} ∪{B} = {R, G, B}; Level 3: filters {R, G}, {G, B}, {R, B} – {R, G, B} ∩ {R, G} ∩ {G, B} ∩ {R, B} = {Blc}. Next, we consider the choice of ranked sequences with the quadratic confidence estimation, the principles of transformation of informational color quanta into fuzzy sets and logical operations with them.

3 Determination of Logical Color Information for Fuzzy Computing The transformation of a color information quantum into a fuzzy variable occurs on the basis of quantitative distribution weights of primary colors in a certain sequence. For quantitative processing of the results of the logical color input and output based on the ranking on the interval [0; 1], a linear scale of the form {Q} (γ ): {R} (0); {Yel} (0.25); {G} (0.5); {C} (0.75); {B} (1); {M} (0.5) was considered in [19]. For a comparative assessment of the average values of E and the root-mean-square assessment D when assessing the confidence of n received assessments (for example, a group of experts evaluates the safety of the situation with assessments: “safe”, “probably safe”, “not safe”) when adding a pair of primary colors (for each color in the pair there are k ratings of one color and (n − k) of another color) {R, G}, {R, B}, {G, B} we compose the

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following expressions (i, j form pairs (1, 2), (1, 3), ( 2, 3))  1 kγi + (n − k)γj ; n

(1)

2 2   1  k Ei,j − γi + (n − k) Ei,j − γj . n

(2)

Ei,j = Di,j =

Let us determine for the cores (heights) of FS in accordance with the circular scale (Fig. 3, a) {R}0 = 1/0; {G}0 = 1/0.5; {B}0 = 1/1.

Fig. 3. Circular scale and assessments: a – circular scale for primary color; b – assessments graph, X – average values E (1), Y – root-mean-square assessment D (2)

Then for FS formed for a monochromatic color (taking into account the small positive parameter σ, which determines the location of the border of color pairs), we can propose the sequences in the form: {R} =

1 1 1 + + ; −0.25 + σ 0 0.25 − σ

(3)

{G} =

1 1 1 + + ; 0.25 + σ 0.5 0.75 − σ

(4)

{B} =

1 1 1 + + . 0.75 + σ 1 −0.25 − σ

(5)

In turn, white light is defined as a logical sum (disjunction) of the form {W } = {R} ∪ {G} ∪ {B} = (6) 1 1 1 1 1 1 1 1 + + + + + + + , −0.25 + σ 0 0.25 − σ 0.25 + σ 0.5 0.75 − σ 1 0.75 + σ or, if we accept the following conditions for the exact boundaries as σ → 0 of the corresponding color pairs, taking into account (6) {W } =

1 1 1 1 1 1 + + + + + . −0.25 0 0.25 0.5 0.75 1

(7)

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The logical operation of negation, which determines the absence of light emitter, i.e. {Blc}, the empty set {0}, will be written as follows {Blc} = ¬{W } =

0 0 0 0 0 0 + + + + + = {0}. −0.25 0 0.25 0.5 0.75 1

(8)

With the distribution of estimates based on the scale (3), the analysis of root-meanssquare estimation (1, 2), for example, at n = 12, shows for the pair {R, B} 4-fold excess of the deviation values for other pairs (Fig. 3, b). To eliminate this shortcoming when forming a ranked scale, it is proposed to use estimates on a circular scale (Fig. 4, a) counterclockwise, i.e. the boundary of {R} (0) and {B} (1) is the value of (−0.5). In this case, the root-means-square estimation for all pairs will be the same (Fig. 4, b).

Fig. 4. Circular scale and assessments: a – circular scale for primary and additional color; b – assessments graph, X – average values E (1), Y – root-mean-square assessment D (2)

Then for the cores (heights) of the FS in accordance with the circular scale in Fig. 4, a we write a sequence: {R}0 = 1/0; {G}0 = 1/0.5; {B}0 = 1/1; {Yel}0 = 1/0.25; {C}0 = 1/0.75; {M }0 = 1/ − 0.25.

(9)

FS for primary {R, G, B} and additional {Yel, C, M} colors will be written as sequences (9) in the following form

{Yel} = {R} ∪ {G} =

{R} =

1 0.5 0.5 + + ; −0.25 + σ 0 0.25 − σ

(10)

{G} =

0.5 1 0.5 + + ; 0.25 + σ 0.5 0.75 − σ

(11)

{B} =

0.5 1 0.5 + + ; 0.75 + σ 1 −0.25 − σ

(12)

0.5 1 0.5 0.5 1 0.5 + + + + + ; −0.25 + σ 0 0.25 − σ 0.25 + σ 0.5 0.75 − σ (13)

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{C} = {G} ∪ {B} =

1 0.5 0.5 1 0.5 0.5 + + + + + ; 0.75 + σ 1 −0.25 − σ 0.25 + σ 0.5 0.75 − σ (14)

{M } = {R} ∪ {B} =

0.5 1 0.5 0.5 1 0.5 + + + + + . 0.75 + σ 1 −0.25 − σ −0.25 + σ 0 0.25 − σ (15)

The resulting ranked sequences (10)–(15) make it possible to establish a logical connection between the corresponding FS. To describe white light as a FS, it should be noted that white light is the sum, taking into account the property of impedance, a combination of primary (3)–(5), (10)–(12) or additional (13)–(15) colors, and thereby determine that the white light has no boundaries between the composite colors and the membership degree for each color should be 1 (9). Then we obtain for white light the equation in the form (7) and for {Blc} – (8).

4 Conclusions The paper presents the mathematical techniques for the analysis, synthesis and programming of logical coloroids, based on the definition of the information quantum of the primary and additional colors as ranged FS. When describing sets of colors using a fuzzy sequence, it is proposed to implement a non-linear rank scale. It is shown that this improves the accuracy of numerical assessments of the reliability of solutions with a decrease in the root-mean-square estimation error. The implementation of optical logical coloroids with fuzzy computing: (a) increases the speed of information processing in the direct optical channels of coloroids, (b) expands the possibilities of parallel computing for coloroid’s multilevel networks, (c) provides high reliability and noise immunity of optical processes for transmitting and processing information, and also (d) ensures the manufacturability of the computing architecture, (e) improves the visualization of the results for the man operator. The developed optical computing techniques can be effectively implemented at creating real-time intelligent DDS with a large array of fuzzy input data.

References 1. Alavi, B., Tavana, M., Mina, H.: A dynamic decision support system for sustainable supplier selection in circular economy. Sustain. Prod. Consumption 27, 905–920 (2021) 2. Solesvik, M., et al.: Fuzzy decision support systems in marine practice. In: 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–6 (2017). https://doi.org/10.1109/ FUZZ-IEEE.2017.8015471 3. Drwal, G., Sikora, M.: Fuzzy decision support system with rough set based rules generation method. In: Tsumoto, S., Słowi´nski, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 727–732. Springer, Heidelberg (2004). https://doi.org/ 10.1007/978-3-540-25929-9_92 4. Sharma, R., Kochher, R.: Fuzzy decision support system for tuberculosis detection. In: 2017 International Conference on Communication and Signal Processing (ICCSP), pp. 2001–2005 (2017). https://doi.org/10.1109/ICCSP.2017.8286753

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5. Ignatius, J., Hatami-Marbini, A., Rahman, A., Dhamotharan, L., Khoshnevis, P.: A fuzzy decision support system for credit scoring. Neural Comput. Appl. 29(10), 921–937 (2016). https://doi.org/10.1007/s00521-016-2592-1 6. Rakes, T.R., et al.: A fuzzy decision support system for pre-disaster budgeting. Int. J. Inf. Syst. Manag. (IJISAM) 1(4), 312–327 (2018) 7. Shevchenko, A., Vakulenko, M., Klymenko, M.: The Ukrainian AI strategy: premises and outlooks. In: Proceedings of the 12th International Conference on Advanced Computer Information Technologies (ACIT), pp. 511–515 (2022). https://doi.org/10.1109/ACIT54803.2022. 9913094 8. Kondratenko, Y.P., Kozlov, A.V.: Generation of rule bases of fuzzy systems based on modified ant colony algorithms. J. Autom. Inf. Sci. 51(3), 4–25 (2019) 9. Zhu, Z., Yuan, J., Jiang, L.: Multifunctional and multichannels all-optical logic gates based on the in-plane coherent control of localized surface plasmons. Opt. Lett. 45(23), 6362–6365 (2020) 10. Ma, S., Chen, Z., Dutta, N.K.: All-optical logic gates based on two-photon absorption in semiconductor optical amplifiers. Optics Commun. 282(23), 4508–4512 (2009) 11. Jung, Y.J., et al.: Reconfigurable all-optical logic AND, NAND, OR, NOR, XOR and XNOR by photonic crystal nonlinear cavities. In: Conference on Lasers and Electro-Optics, Pacific Rim, paper TuB4_3 (2009) 12. Alles, M., Sokolov, S.V., Kovalev, S.M.: Fuzzy logical control based on optical information. Autom. Control. Comput. Sci. 48(3), 123–128 (2014) 13. Azhigulov, D., Nakarmi, B., Ukaegbu, I.A.: High-speed thermally tuned electro-optical logic gates based on micro-ring resonators. Opt. Quant. Electron. 52(9), 1–16 (2020). https://doi. org/10.1007/s11082-020-02526-y 14. Amer, K.: Simulation of high quality factor all-optical logic gates based on quantum-dot semiconductor optical amplifier at 1 Tb/s. Optik 127(1), 320–325 (2016) 15. Kazanskiy, N.L., Butt, M.A., Khonina, S.N.: Optical computing: status and perspectives. Nanomaterials (Basel) 12(13), 2171 (2022). https://doi.org/10.3390/nano12132171 16. Moritaka, K., Kawano, T.: Spectroscopic analysis of the model color filters used for computation of CIELAB-based optical logic gates. ICIC Exp. Lett. Part B: Appl. 5(6), 1715–1720 (2014) 17. Kato, S., Shinomiya, I., Mori, F., Sugano, N.: Fuzzy set theoretical analysis of human membership values on the color triangle. In: Proceedings of the 3rd International Conference on Pattern Recognition Applications and Methods (ICPRAM-2014), pp. 239–246 (2014). https:// doi.org/10.5220/0004826502390246 18. Timchenko, V., Kondratenko, Y., Kreinovich, V.: Efficient optical approach to fuzzy data processing based on colors and light filter. Int. J. Probl. Control Inf. 67(4), 89–105 (2022). https://doi.org/10.34229/2786-6505-2022-4-7 19. Timchenko, V., Kondratenko, Y., Kreinovich, V.: Decision support system for the safety of ship navigation based on optical color logic gates. In: CEUR Workshop Proceedings, vol. 3347, pp. 42–52 (2022) 20. Timchenko, V., Kondratenko, Y., Kreinovich, V.: Implementation of optical logic gates based on color. In: Hu, Z., et al. (eds.) Proceedings of the the 6th International Conference on Computer Science, Engineering and Education Applications ICCSEEA 2023, Warsaw, Poland, LNDECT 181 (2023). https://doi.org/10.1007/978-3-031-36118-0_12 21. Timchenko, V.L., Kondratenko, Y.P., Kreinovich, V.: Why color optical computing? In: Phuong, N.H., Kreinovich, V. (eds.) Deep Learning and Other Soft Computing Techniques. Studies in Computational Intelligence, vol. 1097, pp. 227–233. Springer, Cham (2023). https:// doi.org/10.1007/978-3-031-29447-1_20

A Fuzzy Mixed-Integer Linear Programming Model for Aircraft Maintenance Workforce Optimization Safacan Hasancebi1,2

, Gulfem Tuzkaya1(B)

, and Huseyin Selcuk Kilic1

1 Marmara University, Maltepe, 34854 Istanbul, Turkey

{gulfem.tuzkaya,huseyin.kilic}@marmara.edu.tr 2 THY Teknik A.S., ¸ SAW E Kapisi, Pendik, 34912 Istanbul, Turkey

Abstract. Aircraft maintenance is a labor-intensive process. Hence, the majority of studies in this field focus on improving labor productivity and working conditions. Since airplanes have complex structures, the automation of their maintenance is still in its early stage. Workforce optimization is critical for both increasing airworthiness and efficient use of operator fleets. Therefore, efficient and experienced technicians are indispensable for MRO companies. This paper aims to prevent occupational accidents and TAT delays by assigning critical maintenance tasks to experienced technicians. Moreover, the rules of the authorities and the priorities of cards are also taken into account in the model and a shift-based mixed integer linear programming assignment model is created. A fuzzy rule-based system is used to determine the values of some parameters within the proposed mathematical model. Finally, the model is applied to an MRO company in Türkiye, and the data sets provided are analyzed. Keywords: Maintenance Planning · Workforce Optimization · Mixed-Integer Linear Programming · Fuzzy Rule-Based Approach

1 Introduction Labor is an important and expensive resource in the aircraft maintenance industry. Maintenance workforce planning is an important factor that aircraft maintenance organizations must manage. A good workforce plan reduces maintenance costs, reduces fleet-wide aircraft ground time, and improves flight safety by providing the inherent constraints of aircraft maintenance. Accurate determination of the workforce requirement is important for the beneficial use of the workforce, which includes different characteristics, such as experiences and skills. In aircraft maintenance applications, the demand for labor is unpredictable. Maintenance is usually scheduled for inspection purposes, and the applications/tasks required for inspection are specific. However, the findings obtained as a result of the inspections may not be predicted. This turns aircraft maintenance practices into a project whose scope and duration cannot be estimated. In such a dynamic and unpredictable environment, flexible solutions are needed for workforce planning. The unplanned absence (excuse leave, mandatory training, etc.) of personnel due to other circumstances causes not only demand but also supply to be unpredictable. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 499–506, 2023. https://doi.org/10.1007/978-3-031-39774-5_56

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Various mathematical models and heuristic algorithms have been developed for use in flexible workforce and workload environments. Optimization problems in aircraft maintenance processes are generally classified as aircraft maintenance routing, aircraft maintenance sequencing/scheduling, and task allocation. These problems relate to scheduling maintenance or optimal planning of task periods given by the authority/manufacturer. The aim is to make the right decisions in a dynamic working environment and to show the logic behind the decisions with data. It is predicted that the data-based flexible models will be more effective than workforce planning based on estimation/experience and personal predictions. Inputs and outputs of the model and sensitivity analyses will be reported to the relevant people, and workforce planning, optimization and scheduling in aircraft maintenance will become more manageable. Considering the importance of the workforce planning in aircraft maintenance, a mathematical model is developed to provide improvement within this process. The main objectives considered in the model are to reduce the man-hours spent on the workforce plan, assign maintenance work to the right amount of personnel with the right competence, and reduce the frequency of delays in Work Completion Time due to incorrect planning. The uncertainty in some of the parameter values is clarified with the help of a fuzzy rule based system. Moreover, an application is performed in an aircraft maintenance company located in Türkiye. The remainder of the study is mapped as follows: A literature review on aircraft maintenance labor optimization is given in the second section. The third section includes the problem definition and the mathematical model. Application in an aircraft company is presented in the fourth section. Finally, a conclusion is provided in the fifth section.

2 Literature Review on Aircraft Maintenance Labor Optimization A considerable number of studies focus on labor force planning in the aircraft maintenance process. Some of the related studies are explained in brief as follows: Dijkstra et al. [1] proposed a decision support system for the capacity planning of aircraft maintenance personnel. In addition, sensitivity analysis was performed for team sizes, team attributes, number of shifts per day, shift hours, and team numbers assigned to shifts. In this study, the proposed model tries to assign aircraft maintenance technicians to teams, and the smallest organizational unit to be managed as such was teams of 18 on average. It has been stated that the biggest managerial problem in workforce planning is to have the minimum amount of qualified personnel in the field capable of performing the required tasks. Similarly, Dent et al. [2] proposed a DSS for optimizing aircraft maintenance check schedule and task allocation. Heuristic and rule-based model is used for minimizing the total cost of task execution, subject to the daily available workforce. Bazargan-Lari et al. [3] proposed a simulation model for workforce planning in line maintenance applications. The main factor determining the workload in line maintenance processes is the flight schedule. Therefore, workforce planning is directly related to the flight schedule of the relevant airport. In the proposed simulation model, it is assumed that all technicians are qualified for each job. As a result of the simulation study, optimal sub-shift groups and optimal personnel assignments to these groups were created in 3 different shifts. Similarly, Iceloglu [4] developed a simulation to model the assignment

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scenarios of the personnel who will perform the maintenance operations on the InFlight Entertainment devices in line maintenance. Technicians with different skills were required for different categories and analyses were made on the simulation results. Liou & Tzeng [5] established multi-objective mathematical models for line maintenance applications. The objectives are listed as maximum profit, maximum customer satisfaction, and maximum maintenance experience. De Novo Programming and Fuzzy Multi-Purpose Programming methods were used for the solution. Ighravwe and Oke [6] used goal programming in base maintenance to minimize the number of maintenance technicians and maximize their productivity. They proposed a non-zero integer non-linear programming model for maintenance workforce sizing. Looking at the general scope in the aircraft maintenance workforce literature, one can see that proposed models are applied in two different areas; Base and Line maintenance. The term base maintenance defines the heavy maintenance applications applied in hangars. These maintenance applications have a time range from 1 day to 2–3 years. On the other hand, line maintenance applications are generally quick inspection activities performed before take-off or after landing. In the MRO sector, studies on workforce planning are investigated and some inferences are given as follows: The objective functions in studies on workforce planning are usually cost minimization or work completed maximization, and exact methods are successful when assigning a limited number of personnel and tasks. In studies with large numbers of personnel and tasks, heuristic methods have been found to be successful in assignment algorithms. After making the workforce plan, it is also critical to assign technicians to the right jobs in order to use the available workforce in the most efficient way. Especially in aircraft maintenance applications, there are constraints such as assigning inexperienced personnel with experienced personnel to gain experience and not assigning the same person to some critical jobs that should not be done by the same person. Different from most of the studies in the literature, within the scope of this study, only long-term maintenance applied in hangars will be discussed. Line maintenance practices are not included in the study. Workforce management in line maintenance applications has a different structure. In addition, a different approach to workforce planning was introduced by taking into account the experience of technicians in card assignments. In this study, only the workforce distribution of aircraft maintenance technicians will be carried out. The workforce management of engineers, technical cleaning personnel, planners, and other personnel are not within the scope of the subject. Hence, the research question of this study can be indicated as; “How can the data-based workforce planning process be carried out in aircraft maintenance applications within the limits stated?”. Accordingly, a mathematical model is developed so as to overcome the related challenge and fill in the gap in the literature about this field.

3 Problem Definition and Mathematical Model In the BASE maintenance checks, there is a long amount of Turn Around Time (TAT). Some MRO companies that deal only with one aircraft type benefit from the long time range and overcome the checks easily. These companies have standard maintenance plans and mostly standard workforce scheduling. However, there are also some companies that deal with various aircraft types including both narrow and wide bodies. It is

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necessary for these companies to employ various certified technicians and schedule them to optimize TAT for each aircraft. It is the main goal of this article to create an optimum workforce model for an MRO company with multiple types of aircraft. A workforce plan might be created in different detail levels; aircraft assignment, zone assignment, and task assignment. • Aircraft assignment: In this type of assignment plan, personnel groups are assigned to aircraft. • Zone assignment: In this type of assignment plan, personnel groups are assigned to zones such as engine, nacelle, cabin, etc. • Task assignment: This is the most detailed assignment type and it is aimed in this article to propose a task assignment model. Decision variables, parameters and indexes of the model are given below; Sets: i: Technicians j: Cards z: Zones S: Set of skilled technicians A: Set of unskilled technicians Variables: x ij = If skilled technician (i) is assigned to card (j) = 1, else = 0 Parameters: T z : Set of cards belonging to zone z Qj : Simultaneous cards for card j Pij : Preference ratio of assigning technician i to card j W j : Maximum working hours for card j Z j : Minimum working hours for card j E j : Estimated man-hour for card j m: Number of technicians n: Number of cards U i : Maximum working hour L i : Minimum working hour The objective function in this model is to maximize the assignment preference of technicians to cards. This objective function aims to assign critical cards to experienced technicians. Since critical cards are safety related and may affect maintenance time, maximization is aimed in assigning experienced technicians to critical cards. m n max z = (xij ∗ Pij ) (1) i=1

j=1

The constraints of the proposed task assignment model are given below; Equation 2 tries to guarantee that manpower assigned on cards are between minimum and maximum workload available on cards. m Wj ≥ (EJ ∗ xij ) ≥ Zj (j = 1 to n) (2) i=1

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Equation 3 tries to guarantee that each technician has a limited amount of time in a shift. Minimum and maximum working hours are restricted in the model. n (xij ∗ Ej ) ≤ Ui (i = 1 to m) (3) Li ≤ j=1

Equation 4 tries to guarantee that for every 2 unskilled technicians, there must be at least 1 skilled technician is assigned to a zone.     ( (xij )) ≥ ( (xij )) for every z (4) 2∗ j∈P

i∈S

j∈T

i∈A

Equation 5 tries to guarantee that some cards are planned to be performed simultaneously, so each technician can only be assigned to one of the simultaneous cards.  (xij ) ≤ 1 (i = 1 to m) (5) j∈Q

Equation 6 defines the binary variables. xij ∈ (0, 1) i = 1 to m, j = 1 to n

(6)

Also, some assumptions have to be made so that modeling is possible; • There are no ongoing tasks between shifts. So each card is completed within the shift. • The personnel available for overtime is specified. Extra overtime is not allowed. So that technicians work within the limits of their regular shift hours. So as to overcome the uncertainty in preference parameters, a fuzzy rule based system is utilized. Technician experience and card criticality are used as inputs and based on fuzzy rules, preference outputs are determined. Experience in aircraft maintenance is very important. Experienced technicians provide companies to shorten their TAT and for this reason, it is best to use highly experienced technicians on critical cards. On the other hand, for those cards that are not so critical, it will be wasteful to use highly experienced technicians on these cards. The steps of the fuzzy rule-based system are given in Fig. 1.

Fig. 1. Process of Fuzzy Rule Based System

For the membership functions of the inputs, 3 levels are defined; low, medium and high. Triangular numbers are used to define low and medium memberships and trapezoidal numbers are used to define high memberships. The fact that the parameters consist

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of specific small intervals and have a symmetric structure with a single vertex made it possible to use the triangular membership function. Parameters are defined based on expert opinions and as the inference system, the Mamdani type was used owing to its strength in presenting and interpreting expert judgments [7]. Finally, to obtain a crisp set, the center of gravity (centroid) method is used on fuzzy sets.

4 Application in an MRO Company In the application part, we collected data from an MRO company that works for multiple type of aircraft. A light maintenance package is chosen which contains 10 task cards in a shift. A total of 5 B1/B2/SPVR, 5 QM/CQM and 5 WUCS technicians are assigned to this maintenance package in this shift. For the zone rules (Eq. 3), assume that skilled technicians are B1/B2/SPVR and unskilled technicians are WUCS. No zone rule is given for QM/CQM. Table 1. Card Man-Hour Data Card

1

2

3

4

5

6

7

8

9

10

Estimated Man-Hour

1

1

0.5

1

1

2

1

1

2

0.25

Table 2. Card Criticality Data Card

1

2

3

4

5

6

7

8

9

10

Card Criticality

9

2

6

5

4

6

3

3

1

7

In Table 1, man-hour data for cards are given. Assume that technician-1 is assigned to card-1, this means that one man-hour will be spent by technician-1. In Table 2, the higher the criticality number, the more critical card is. So we understand that card-1 has the highest importance because it is on the critical path of the maintenance project. Table 3. Technician Experience Data Skill

B1+SPVR

Technician

1

2

3

4

5

6

QM+CQM 7

8

9

10

WUCS 11

12

13

14

15

Experience

9

8

7

8

10

6

4

5

6

5

2

3

2

1

2

Fuzzy sets (technician experience and card priority data) are defuzzified to obtain a crisp set (assignment preference) to use in the model. Experience data is given in Table 3

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and Card Priority data is given in Table 2. Fuzzy logic application is carried out in MATLAB. Fuzzy rules are given in Table 4. Basically, rules are based on assigning lower critical cards to unskilled technicians and higher critical cards to skilled technicians. Parameters are also given in Table 4. Table 4. Fuzzy Rules Preference Matrix

Card Priority

Tech. Experience

Low (0 3 5)

Medium (2 5 8)

High (5 8 10 10)

High (5 8 10 10)

Low

Medium

High

Medium (2 5 8)

Medium

High

Medium

Low (0 3 5)

High

Medium

Low

Table 5. Assignment Result Matrix Technician / 1 2 3 4 5 6 7 8 9 10 Total manhour Min Max Cards working hour working hour 1

1 0 1 0 0 1 1 0 0 1

4.75

3

8

2

1 0 1 0 0 1 1 0 0 1

4.75

3

8

3

1 0 1 1 0 1 0 1 0 1

5.75

3

8

4

1 0 1 1 0 1 0 1 0 1

5.75

3

8

5

1 0 0 0 1 1 0 0 0 1

4.25

3

8

6

1 0 1 1 0 1 0 0 0 1

4.75

3

8

7

0 1 0 1 1 0 1 1 1 0

7.00

3

11

8

1 1 1 1 0 1 1 1 1 0

9.50

3

11

9

1 0 1 1 0 1 0 0 0 1

4.75

3

11

10

0 1 1 1 1 0 0 1 1 1

6.75

3

8

11

0 1 0 0 1 0 0 1 1 0

5.00

3

8

12

0 1 0 1 1 0 1 0 1 0

6.00

3

8

13

0 1 0 0 1 0 1 1 1 0

6.00

3

8

14

0 1 0 0 1 0 1 0 1 0

5.00

3

8

15

0 1 0 0 1 0 1 1 1 0

6.00

3

8

Total Assig

8 8 8 8 8 8 8 8 8 8

Since the assignment matrix, given in Table 5, is only 15 × 10, it was possible to solve the problem using exact methods. See that all values are within the limits. Also note that, cards 5 and 6 are carried out simultaneously. So none of the technicians are assigned to both of them.

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5 Conclusion In conclusion, this paper presents a mixed-integer linear programming model for aircraft maintenance workforce optimization which takes into account the critical maintenance tasks, the rules of authorities and the priorities of cards. To address the uncertainty in some of the parameter values, a fuzzy rule-based system is used to determine the values of these parameters within the proposed mathematical model. The model is applied to an MRO company in Türkiye and the data sets provided are analyzed. The results show that the model can effectively reduce the man-hours spent on the workforce plan, assign maintenance work to the right amount of personnel with the right competence and reduce the frequency of delays in Work Completion Time due to incorrect planning. In future work, a model that works for longer time periods than one shift can be established. It can also be assumed that some tasks can be delegated between shifts in this model. The reinforcement learning method can also be examined in future studies in order to reflect the company’s experience in the solution. Overall, the proposed model provides a promising solution to the problem of workforce planning and optimization in aircraft maintenance, which is critical for increasing airworthiness and efficient use of operator fleets.

References 1. Dijkstra, M.C., Kroon, L.G., van Nunen, J.A., Salomon, M.: A DSS for capacity planning of aircraft maintenance personnel. Int. J. Prod. Econ. 23(1–3), 69–78 (1991). https://doi.org/10. 1016/0925-5273(91)90049-y 2. Deng, Q., Santos, B.F., Verhagen, W.J.: A novel decision support system for optimizing aircraft maintenance check schedule and task allocation. Decis. Support Syst. 146, 113545 (2021). https://doi.org/10.1016/j.dss.2021.113545 3. Bazargan-Lari, M., Gupta, P., Young, S.: A simulation approach to manpower planning. In: Proceedings of the 2003 International Conference on Machine Learning and Cybernetics (IEEE Cat. No. 03EX693) (2003). Published. https://doi.org/10.1109/wsc.2003.1261619 4. ˙Içelo˘glu, M.: Technician assignment for ife line maintenance in aviation MRO [Master’s Thesis, University of Bahçe¸sehir] (2018) 5. Liou, J.J.H., Tzeng, G.-H.: Airline maintenance manpower optimization from the de novo perspective. In: Shi, Y., Wang, S., Peng, Y., Li, J., Zeng, Y. (eds.) MCDM 2009. CCIS, vol. 35, pp. 806–814. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02298-2_117 6. Ighravwe, D., Oke, S.: A non-zero integer non-linear programming model for maintenance workforce sizing. Int. J. Prod. Econ. 150, 204–214 (2014). https://doi.org/10.1016/j.ijpe.2014. 01.004 7. Cevikcan, E., Kilic, H.S.: Tempo rating approach using fuzzy rule based system and westing house method for the assessment of normal time. Int. J. Ind. Eng. 23(1) (2016)

An Optimal Model for Integer Programming Problems Based on Fuzzy Membership Functions with Comparative Analysis K. Palanivel(B) Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT), Vellore 632014, TN, India [email protected], [email protected]

Abstract. All decision variables in the Integer Programming Problem (IPP) have been allowed to take certain non-negative real values, as it is very likely and relevant to have fractional values, but also to be vague across several situations. In such situations, the fractional values of variables and uncertainty may be unrealistic in the context of the real decision-making problem. Likely, the fuzzy model is the key to addressing the vagueness in such situations by considering the proposed model of IPPs, which will play a significant role in addressing the vagueness. The model begins with a problem formulation, followed by a fuzzified Gomary’s cutting plane approach with an appropriate numerical illustration, which will control the ambiguity and achieve the best optimum value. This model illustrate the problems numerically in two cases, case 1 discusses the approach in terms of fuzziness using membership functions, and the second case was approached by robust ranking. Moreover, it concluded with a detailed comparative analysis of the optimum outcome and model performance with a higher number of variables coded and executed in LINGO. At last, the comparative analysis, results and discussion reveal the complexity and cost-effectiveness of the fuzzy model, which addresses the vagueness and provides a much more optimum outcome. Keywords: Trapezoidal Fuzzy Membership Functions (TFMFs) · Robust Ranking approach · Fuzzy Integer Programming Problems (FIPP)

1 Introduction The IP mathematical model is an LP model with one additional restriction: variables must have integer values. IPPs are a particular form of LPP in which all or any of the decision variables are limited to positive integer values. It has been remembered that the problems of LP are based on the principle that the decision variables are continuous, with the consequence that they can take fractional values in the optimal solution. As a result, many situations in real life may not take this into account, and decision variables frequently take only integer values. Similarly, decision-making such as sequencing, scheduling, and routing necessarily require the use of IP models. Most of the variables must be restricted to assume only integer values in IP. There is also some zero-one integer programming © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 507–520, 2023. https://doi.org/10.1007/978-3-031-39774-5_57

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problem, which is an emerging type of such problem in which it may be necessary to reach a non-integer solution to LPPs using standard graphic or simplex methods, but instead to round off the fractional values that exist in the optimization. Though it never works in reality, deviations from the exact optimal integer values may be required to render the result unfeasible. It seems there is a need to develop a structured method to achieve an optimal solution to the situation as well as meet the overall requirements. RE Gomory’s first formal process of solving IPP was proposed in 1956. Whereas initially just pure IPPs had been addressed, the method has extensively developed to address mixed IPPs. The process utilizes a cutting plane algorithm and helps to ensure optimum integer outcome for a limited number of iterated steps. In the case of many real times, it becomes inappropriate to provide all the limitations and capitals in precisely the same way as they are in the anticipated or imprecise form. Eventually, the notion of fuzzy control comes to an end, allowing one to pursue the rules of human intuition and to make predictions dependent upon uncertain or vague information. This application deals with a scientific process for resolving issues in an uncertain environment. Fuzzy set theory is widely accepted with tremendous attention. In 1965, L.A.Zadeh advanced a fuzzy concept. The concept of a fuzzy set provides a starting point for the development of a theoretical perspective used within the case of ordinary sets. However, it is more specific than the others and can theoretically show that applications in different fields of classification and information processing are of much greater significance [1]. Bellman and Zadeh have proposed novel theories and models of decision-making in a fuzzy environment, how requirements are often connected when they are of uneven significance [2]. Tanaka et al. first introduced the idea of fuzzy linear programming (FLP) at the fundamental level [3]. The fuzzy LPP with fuzzy numbers is considered a model of decision-making problems where human perception is significant. Subsequently, several writers found different forms of FLP problems and suggested a variety of approaches to the solution of these problems. Besides, the most useful approaches are constructed on the principle of comparing fuzzy numbers using ranking functions [4]. Based on the background research studies, the purpose of this study is to develop a mathematical model for optimization in IP under uncertainty.

2 Review of Literature The mentioned scientific studies highlight the literature on fuzzy integer programming: Herrera and Verdegay investigated three models to address the uncertainty of an ambiguous nature in the context of ILP problems with fuzzy numbers. Few methods are focused on the representation theorem and also solved problems involving fuzzy numbers by ranking [5]. Fortemps and Roubens provided a comprehensive ranking method that combines adaptive geometric classification and area compensation. The implemented ranking system has some common properties with specific standard procedures and also given a comparison studies [6]. Herrera and Verdegay suggested an algorithm to solve IPPs with a fuzzy solution, which demonstrates how this technique works in the traditional parametric IPP’s [7]. Allahviranloo studied FIP with a fuzzy constant(RHS) and simplified the fuzzy problem to two IPPs. The solution was obtained by summing up the solutions and also showing associated theorems [8]. Allahviranloo and Afandizadeh developed a model for estimating optimum investment in port operations from

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a national financial point of view; through the use of an integer-programming model with fuzzy numbers are used for the results of the cargo forecast study [9]. Sudhagar described FILPP with fuzzy variables employing a ranking method that can represent decision-makers by giving an adequate range of values for fuzzy variables with illustrated examples [10]. Pandian and Jayalakshmi offered an ILPP by decomposition method for solving ILPP with fuzzy variables, and this method could significantly result in fuzzy variables [11]. Fan et al. employed the generalized FIP procedure for waste allocation and facility expansion under imprecision. Through which the stepwise interactive algorithm is introduced to address the GFIP problem and to produce solutions represented as fuzzy sets [12]. Sagnak and Kazancoglu introduced a shift scheduling method with an IP model combined with fuzzy logic due to the lack of clarity derived from the design of the decision-making process to ensure consistency in shift lengths and reduce overall labor costs [13]. Niksirat et al. proposed a FIP algorithm with fuzzy constraints for feasible and necessary measures. The described method decomposed the problem appropriately and applied it in a branch and price algorithm. This algorithm is generalized for both feasible and necessary measures, and the best outcomes are achieved [14]. Palanivel employed trapezoidal membership functions and a robust ranking index to solve the fuzzy commercial traveler problem with detailed results and discussion [15]. Dinagar and Jeyavuthin presented and described a ranking technique for fully FILPP with pentagonal fuzzy numbers [16]. Ammar and Emsimir presented a rough ILP algorithm where all the variables of the problems are rough intervals. A flow diagram is also offered to highlight the problem-solving steps, with particular examples showing the outcomes [17, 18]. Khalili Goudarzi introduced a fully FIPP, a representation of fuzzy membership function with a collaborative methodology intended for overcoming conventional multiobjective programming addressed with appropriate examples [19]. Cherfaoui et al. investigated multiobjective integer LP with a branch and cut LP algorithm, and their experimental study provides efficient solutions for the considered preference functions. Furthermore, the investigated algorithm provides efficient cuts that iteratively reduce the search domain of the MOILPP [20]. As a result of an earthquake case study, Shu-ping Wan et al. proposed a bi-objective mixed ILP using trapezoidal fuzzy numbers for distribution centre location decisions. The proposed model represents three distinct complex scenarios for large-scale emergencies. It also defines and analyses a new ranking relation and its properties for trapezoidal fuzzy numbers with α cut set. Additionally, the reported analyses demonstrate the model’s flexibility and superiority [21]. K.Palanivel and Amrit Das discussed a new fuzzy model that is proposed to overcome the uncertainty in nonlinear optimization by considering trapezoidal membership functions and a ranking function with comparative analysis [22]. H. Q.Truong C. Jeenanunta proposed a case study to optimize the variability tolerance of fuel prices under uncertainty in Thailand’s national power system. Furthermore, a sensitivity analysis on fuel price is presented for the fuzzy model developed on mixed ILP as a solution method for solving fuzzy MILP [23]. 2.1 Contributions of the Paper According to a recent research study, many fuzzy applications are widely used in numerous situations in optimization issues, but limited number of these applications are focused

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on the theoretical development of IPPs. Despite the fact that it was a primary motivation for us to conduct research using the traditional approach of IP and how it was related to uncertainty in the favor of decision makers. There is a deep desire in this paper to explore some practical problems, particularly the IPPs with uncertainty. Because the goals are to maximise the overall return while adhering to many crisp limitations, the objective function has considered as a fuzzy number. By considering the proposed model of IP using fuzzy membership functions and adopting the ranking of a fuzzy number, will help to address the vagueness in a significant role. This model focuses on FIPPs and uses TFMFs to achieve the best optimum in terms of removing vagueness and providing the best optimum outcome. It also describes the FLPP, which extends the model into a numerical illustration through two cases, the first one deals with Gomary’s cutting plane approach in terms of fuzziness, and the second one incorporates the robust ranking approach. Moreover, it concluded with a detailed comparative analysis of the optimum outcome and model performance with a higher number of variables coded and executed in LINGO. At last, the comparative analysis, results, and discussion reveal the complexity and cost-effectiveness of the fuzzy model, addressing the vagueness and providing a much more optimum outcome. The framework of this study is outlined as follows: Sect. 1 starts with an introduction, and Sect. 2 provides a brief overview of the literature and highlights the main contributions of the paper. Section 3 introduces some important preliminary concepts and ranking relations for TFMFs. Section 4 describes the construction of a fuzzy mathematical model for IPP, including the formulation of the fuzzified form of Gomary’s cutting plane approach. Section 5 presents the model justification through numerical illustration, followed by a comparative analysis with results and discussion. Finally, Sect. 6 is summarized with a conclusion and future extension.

3 Preliminaries In this section, some fundamental preliminary concepts on fuzzy membership functions and their arithmetic are outlined [15]. Following that, it appears to address some of the essential definitions, which are: ∼ Definition: A trapezoidal fuzzy number A can represent as F(=F )(=)[f1 , f2 , f3 , f4 ] and its membership function as: ⎧ x−f1 ⎪ ⎨ f2 −f1 , f1 ≤ x ≤ f2 μF (x) = 1, f 2 ≤ x ≤ f3 ⎪ ⎩ x−t4 , f ≤ x ≤ f 3 4 t3 −t4 Definition (α cut): Given a fuzzy set F in X and any real number α in [0, 1], then the α -cut or α-level or cut worthy set of F, denoted by α F is that the crisp set α F = {x ∈ X : μ (x) ≥ α} The strong a cut, denoted by α + F is that the crisp set F α+ F = {x ∈ X : μ (x) > α}. For instance, let F be a fuzzy set whose membership F function is given as above μF (x). To find the α-cut of F, allow us to set α ∈ [0, 1] [0, 1], let us set the reference functions of F to each left and right. i.e., α =

x(2) − f4 x(1) − f1 &α = f2 − f1 f3 − f4

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Expressing x to α, where x(1) = (f2 − f1 )α + f1 andx(2) = f4 + (f3 − f4 )α which provides the α− cut of F is α F = [x(1) , x(2) ] = (f2 − f1 )α + f1 , f4 + (f3 − f4 )α . Definition (Robust ranking index): The robust index of rankings satisfies compensation, homogeneity, and additive properties, and delivers results controlled by human intuition. If F might be a fuzzy number, then the robust ranking index is outlined by 

1 FαL , FαU d α, R ∈ [0, 1] R(F) = 2 R      where, FαL , FαU = x(1) , x(2) = (f2 − f1 )α + f1 , f4 + (f3 − f4 )α is the α level cut of the fuzzy number F, then the robust ranking index R(A) offers the numerical significance of the fuzzy number F.

4 The Fuzzy Model for Integer Programming Fuzzy programming (FP), which is almost a subspace of fuzzy theory, can effectively address vagueness as fuzzy numbers or fuzzy targets/restrictions. Recently, various kinds of FP approaches have been used to resolve vagueness. Additionally, existing literature on fuzzy variables in FP problems has focused primarily on specific types of fuzzy MFs (like as symmetrical, triangular, or trapezoidal). Similarly, FP is used to deal with vagueness in both constraints and decision variables, and all fuzzy sets with defined membership functions are frequently treated using the defuzzification approach. Like a modification of the formulated FIIP model, suggested under uncertainty. In detail, (i) FIP model may address for vagueness described as fuzzy numbers of defined membership functions, depending on functions whether linear or non-linear; (ii) the suggested FIP model may require vagueness to be dealt directly to the optimizing process and serves solutions; (iii) compared with other inaccurate MIP methodologies can explore the essential interrelationship among the vagueness of fuzzy parameters and such exploration can support decision-makers to sort out complexities between reliability and overall cost. Moreover, the fuzzy mathematical model for IPP was constructed in three different stages, including the formulation of the problem, the fuzzified form of approach, and hence, the numerical illustration followed by a comparative study. 4.1 Formulation of an IPP in Terms of Fuzziness The FIPP, including its imprecise parameter and objective functions represented as fuzzy numbers, might be formulated as follows: The pure IPP is to obtain the decision variables of the objective function,

  



x1(1) , x1(2) , x1(3) , x1(4) , x2(1) , x2(2) , x2(3) , x2(4) , x3(1) , x3(2) , x3(3) , x3(4) , . . . , xj(1) , xj(2) , xj(3) , xj(4)

which maximizes the objective function





 Z (1) , Z (2) , Z (3) , Z (4) (=) (c1 x1 )(1) , (c1 x1 )(2) , (c1 x1 )(3) , (c1 x1 )(4) (+) (c2 x2 )(1) , (c2 x2 )(2) , (c2 x2 )(3) , (c2 x2 )(4) 

(+) . . . (+) (cn xn )(1) , (cn xn )(2) , (cn xn )(3) , (cn xn )(4)

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subject to the linear constraints



 (ai1 x1 )(1) , (ai1 x1 )(2) , (ai1 x1 )(3) , (ai1 x1 )(4) (+) (ai2 x2 )(1) , (ai2 x2 )(2) , (ai2 x2 )(3) , (ai2 x2 )(4)



 (1)  (2)  (3)  (4)  (1) (2) (3) (4) (=) bi , bi , bi , bi (+) . . . (+) aij xj , aij xj , aij xj , aij xj

 (1) (2) (3) (4) xj , xj , xj , xj ≥ 0 & are all integers, for i = 1, 2, 3, . . . , m & j = 1, 2, 3, . . . , n. 4.2 Fuzzified Form of Gomary’s Cutting Plane Approach In this section, the model developed the Gomary’s cutting plane approach in terms of fuzziness, initially generating a new constraints in the proposed model to ensure an essential outcome for the LPP under fuzziness. Further, these new constraints do not eliminate the portion of the original feasible region that contains a feasible integer point. It also disconnects the current noninteger outcome from the LP model. First approach the problem by ignoring the integer requirement, by employing the conventional simplex algorithm under fuzzines. At last, the model presents integer values for the variables, that is the outcome of the IPP. If any of the decision variables with fractional values, (1) (2) (3) (4) then choose the variable with the largest fractional component, say [xr , xr , xr , xr ]. th Then the associated r constraint can be written as,



 (3) (4) br(1) , b(2) (=) xr(1) , xr(2) , xr(3) , xr(4) (+) r , br , br n (1) (2) (3) (4)  (1) (2) (3) (4)  arj , arj , arj , arj (∗) xj , xj , xj , xj ; j = r j=1



 (1) (2) (3) (4) (1) (2) (3) (4) br , br , br , br denotes an integral part of br , br , br , br and



 (1) (2) (3) (4) (1) (2) (3) (4) arj , arj , arj , arj denotes an integral part of arj , arj , arj , arj , then we can write 



(3) (4) (1) (2) (3) (4) b br(1) , b(2) , b , b , b , b , b (=) (+)fr , 0 < fr < 1 and r r r r r r r If



 (1) (2) (3) (4) (1) (2) (3) (4) arj , arj , arj , arj (=) arj , arj , arj , arj (+)frj , 0 < frj < 1 Therefore we get,

 

(2) (3) (4) b(1) (+)fr (=) xr(1) , xr(2) , xr(3) , xr(4) r , br , br , br

 n

(1) (2) (3) (4)  (1) (2) (3) (4) arj , arj , arj , arj (+)frj (∗) xj , xj , xj , xj ; j = r (+) j=1

 ⎞ ⎛ 



 arj(1) , arj(2) , arj(3) , arj(4) (∗) (2) (3) (4)

 ⎠ fr (+)⎝ b(1) (−) xr(1) , xr(2) , xr(3) , xr(4) (−) r , br , br , br (1) (2) (3) (4) xj , xj , xj , xj

(=)





(1) (2) (3) (4) frj (∗) xj , xj , xj , xj ; j = r

An Optimal Model for Integer Programming Problems ⎧ ⎨

Since

⎩



  (2) (3) (4) (−) xr(1) , xr(2) , xr(3) , xr(4) (−) b(1) r , br , br , br

513

 ⎫ arj(1) , arj(2) , arj(3) , arj(4) ⎬  ≥ 0.

(1) (2) (3) (4) (∗) xj , xj , xj , xj ⎭

We have,

  (1) (2) (3) (4) fr ≤ frj (∗) xj , xj , xj , xj , by introducing the slack variable, we get fr + s = i.e., s −







(1) (2) (3) (4) frj (∗) xj , xj , xj , xj



(1) (2) (3) (4) frj (∗) xj , xj , xj , xj = −fr

(4.2.1)

Under fuzziness, this becomes a new constraint called Gomary’s constraint, which represents a cutting plane. Insert this as an additional constraint in the LPP, apply the simplex algorithm, and find a new optimal outcome entirely in the form of fuzziness using the dual simplex algorithm. If the proposed model provides the optimum outcome as an integer value for all variables, we end the process; otherwise, we continue until we reach the optimality with integer values.

5 Numerical Illustration Let us explore the illustration in two different cases that will make the model for addressing the problems of FIP easier with the support of TMFs and their arithmetic operations, fuzzy Gomary ‘s constraint and its approach follows the first case. In the second case, the same problem has been addressed with the assistance of a robust ranking approach. The IPP is the form of fuzziness as described in the following: 



  (1)Maximize (1) (2) (3) (4) (1) (2) (3) (4) Z , Z (2) , Z (3) , Z (4) (=) x1 , x1 , x1 , x1 (+) x2 , x2 , x2 , x2 . Subject to the constraints, 



(1) (2) (3) (4) (1) (2) (3) (4) [0, 2, 4, 6](∗) x1 , x1 , x1 , x1 (+)[−2, 2, 3, 5](∗) x2 , x2 , x2 , x2 ≤ [2, 4, 6, 8]



(1) (2) (3) (4) [−2, 0, 2, 4](∗) x2 , x2 , x2 , x2 ≤ [−2, 2, 3, 5]

(1)

(2)

(3)

(4)

x1 , x1 , x1 , x1

  (1) (2) (3) (4) , x2 , x2 , x2 , x2 ≥ 0 and are integers.

5.1 Case (I): IPP with Fuzzy Membership Functions Let us now address the considered IPP by employing a conventional simplex algorithm under fuzziness, then the computation reaches the non-integer optimum solution in the ∼ second iteration. From the computation, the optimum outcome x2 obtained as an integer value. But x1 has a fractional value. Hence we generate a Gomary’s cut, the constraint ∼ corresponding to x1 is given below.

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x1 (+)[−2/3, 0, 2/3, 4/3](∗)S1 − [0, −1, −2/3, −1](∗)S2 (=)[6/3, 0, 0, −2/3]. [0, 2/3, 4/3, 2](∗)

Using (4.2.1), we get, [6/3, 0, 0, −2/3] < [−2/3, 0, 2/3, 4/3](∗)S1 (+)[−2/3, 0, 2/3, 4/3](∗)S2 Introducing a slack variable S, we get S(−)[−2/3, 0, 2/3, 4/3](∗)S1 (−)[−2/3, 0, 2/3, 4/3](∗)S2 (=)[2/3, 0, −2/3, −4/3] Now, the obtained Gomary’s constraint has been added to the simplex table as a iteration 3 and continue further as per the approach, at last the iteration 4 has offers an integer optimum results with all Z5 (=)Z5 − C5 ≥ 0, hence the optimum outcome of the identified IPP is  

∼ ∼ x1 (=) x1(1) , x1(2) , x1(3) , x1(4) (=)[−2, −1, 1, 2], x2 (=) x2(1) , x2(2) , x2(3) , x2(4) (=)[−2, 2, 3, 5] &



∼ Max Z (=) Z (1) , Z (2) , Z (3) , Z (4) (=)[−2, 2, 3, 5] ⎧ x + 2, −2 ≤ x ≤ −1 ⎪ ⎪ ⎨ ∼ 1, −1 ≤ x ≤ 1 For the exposed FMF of x1 is as follows, μx1 (x)(=) x−2 ⎪ −1 , 1 ≤ x ≤ 2 ⎪ ⎩ 0, x>2 To obtain the confidence of interval for each level α, then the spreads of TMFs to be signified by functions of α in a prescribed way.

(α) + 2) and α(=) x2 −2 /−1, Therefore, Here α(=)(x1(α)

 (α) (α) x1 (=) x1 , x2 ]=[α − 2, −α + 2 , below Fig. 1 shows the pictorial representation of the fuzzy decision variables.

Fig. 1. Pictorial representation of the fuzzy decision variables

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∼

Similarly, the exposed FMFs of x2 are as follows: 

(α) (α) x2 (=) x1 , x2 (=)[4α − 2, −2α + 5]. Exactly, in the same way, the fuzzy membership function for the optimum outcome is

 (α) (α) z˜ (=) x1 , x2 (=)[5α − 4, −3α + 7] The equations to be solved are 4α − 2−x1 (=)0& − 3α + 7−x2 (=)0 From the above equations, it retains only two roots of α in [0, 1] and tends to get, α(=)(x1 + 2)/4 = 0&α + 7(=)(x2 − 7)/ − 3

Therefore,⎧ x+2 , −2 ≤ x ≤ 2 ⎪ ⎪ ⎨ 4 1, 2≤x≤3 μz˜ (x)(=) x−7 , which is the required optimum outcome of the FMF ⎪ , 3 ≤x≤5 ⎪ ⎩ −3 0, x>5 for the considered IPP. 5.2 Case (II): IPP with FMFs Using the Robust Ranking Approach Let us now apply a robust ranking approach into the IPP. According to the above, the ranking index for the fuzzy number [0,2,4,6] has symbolized by R[0,2,4,6]. Besides, the membership function of the ranking index is defined and computed in the following manner: ⎧ x ,0≤x≤2 ⎪ ⎪ ⎨ 2 1, 2 ≤ x ≤ 4 μ(x)(=) x−6 , to obtain the confidence of interval for each level α, then ⎪ ⎪ −2 , 4 ≤ x ≤ 6 ⎩ 0, x > 6 the spreads of trapezoidal membership functions to be signified by functions of α in a prescribed way. x

(α)

(x

(α)

−6)

Here α(=) 12 &α(=) 2−2 ,  (1) (2)    x , x (=) (f2 − f1 )α + f1 , f4 + (f3 − f4 )α (=)[2α, −2α + 6] R(F)(=)R[0, 2, 4, 6](=)

1 2

 1 0

Therefore,

FαL , FαU d α(=)3

Likewise, extend the robust rating index to the other fuzzy numbers as described in the following: R(F)(=)R[2, 4, 6, 8](=)5; R(F)(=)R[−2, 0, 2, 4](=)1&R(F)(=)R[−2, 2, 3, 5](=)2

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By employing the above approach, the chosen problem has been reduced into the conventional problem, then the crisp problem is: Maximize Z = x1 + x2 , subject to the constraints, 3x1 + 2x2 ≤ 5; x2 ≤ 2; x1 , x2 ≥ 0&areintegers. Now continue the problem by addressing the conventional algorithm of IPP with Gomary’s constraint, then the obtained optimum outcome of the reduced problem, which is x1 = 0, x2 = 2 & the Maximum Z = 2. 5.3 Comparative Study The comparative studies of the model were highlighted in two ways in this section: the first is an optimum comparison with other reported approaches, and the second is a comparison with all different cases of the model with optimum outcomes. 5.3.1 Outcome Comparison with Other Reported Approaches Recent research has presented some new directions for addressing the problems about FIP, a decomposition method is proposed by Pandian and Jayalakshmi for solving ILPPs with fuzzy variables by using the classical approach [11]. Further, the exact problem has been compared by various researchers[10, 19] by strengthening their approaches, likewise, this model is also in the situation to compare the problem. Though, the advancement of our proposed model ensures the exact optimum outcome which is by adopting the fuzziness in the conventional algorithm of IPP. Instead of thinking about the new line of approach, the conventional algorithm with our model can address the imprecise situation effectively; this will help to decide the decision maker’s needs. Now the following table (Table 1) shows the comparison of the computational results with other reported approaches. Table 1. Outcome comparison with different approaches Comparison of different approaches and their optimum outcomes ∼

∼

∼

Approaches

x1

, x2

Pandian and Jayalakshmi [11]

(3, 4, 5)

(0, 1, 2)

(12, 19, 26)

Z

Sudhagar and Ganesan [10]

(4, 4, 4)

(0,1,3)

(16, 19, 25)

Goudarzi et al. [19]

(4, 4, 4)

(1, 1, 1)

(19, 19, 19)

Offered fuzzy model

(2, 3, 5, 6)

(−1, 0, 1, 4)

(16,18,20,22)

5.3.2 Outcome Comparison with Different Cases The below table (Table 2) provides a comparison of the optimal outcome obtained from the illustration of the model, namely.

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(i) The existing conventional approach. (ii) Offered a fuzzy model with membership functions, and (iii) A robust ranking approach, for the preferred FIPP in the above numerical illustrations. It is evident from the outcomes are shown in the table (Table 4), which attains the nearer optimum outcome for all the methodologies. Moreover, it demonstrates that the uniqueness of the model presented, besides the decision-maker, will use this type of model to eliminate the vagueness of any appropriate issues to attain the best possible outcome. In addition to the above outcomes, the presented model can be suggested instead of the conventional one, either a fuzzy membership function or a robust ranking approach, which is preferable for the decision-maker needs. Table 2. Comparative studies of optimum outcomes with different cases Comparison of optimum outcomes with all the cases ∼

∼

∼

Methodologies

x1

, x2

By using a conventional approach

0

2

MaxZ(=)2

Z

The presented model by adopting a conventional approach in the form of fuzziness

[−2, −1, 1, 2]

[−2, 2, 3, 5]

˜ MaxZ(=) [−2, 2, 3, 5]

The presented model with the support of a robust ranking approach

[−2, −1, 1, 2](=) R[−2, −1, 1, 2](=)0

[−2, 2, 3, 5](=) R[−2, 2, 3, 5](=)2

˜ MaxZ(=) [−2, 2, 3, 5](=) R[−2, 2, 3, 5](=)2

6 Model Performance Test Evaluation with Different Data Sets This section is presenting an analysis of the performance evaluation of the model proposed in the research article. And we have considered the presented model with different sets of constraints and objectives with a fuzzy coefficient and the solution is using the proposed ranking function for fuzzy. The results were obtained using lingo software and presented and discussed below. Table 3 is representing the performance index for the model with the different sizes of data set. The data are considered in fuzzy format and using the ranking function converted to its equivalent crisp model. With all these crisp converted equivalent data set, the model is designed by specifying individual constraints and objective with the coefficient presented in table 3. The LINGO which is an iterative solver for linear programming problems, non-linear, integer, and quadratic programming problems developed by Lindo system company is utilized to solve each of the individual numerical models with specified data sets in table 3. Further, it represents the obtained results in each case of the

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K. Palanivel Table 3. Model solutions with different size of data set.

Sets nos

Objective function’s coefficient

Constraint’s coefficient RHS constants

Solutions

Optimal objective value

1

[13,15,19,25], [14,16,20,26], [13,17,21,29]

[2, 4, 6, 8, 10, 12, 14]

[168,175,185,192], [133,147,155,169], [150,162,174,182]

x1 = 30, x2 = 0, x3 = 3

600

2

[13,15,19,25], [14,16,20,26], [13,17,21,29], [8,14,22,32],

[2, 4, 6, 8, 10, 12, 14]

[168,175,185,192], [133,147,155,169], [150,162,174,182], [145,178,184,197]

x1 = 28, x2 = 0, x3 = 0 ,x4 = 5

599

3

[13,15,19,25], [14,16,20,26], [13,17,21,29], [8,14,22,32], [17,29,33,41]

[2, 4, 6, 8, 10, 12, 14], [1, 3, 5, 7]

[168,175,185,192], [133,147,155,169], [150,162,174,182], [145,178,184,197], [139,157,175,189]

x1 x2 x3 x4 x5

= 0, = 13, = 0, = 0, = 41

1477

4

[13,15,19,25], [14,16,20,26], [13,17,21,29], [8,14,22,32], [17,29,33,41], [18,20,24,30]

[2, 4, 6, 8, 10, 12, 14], [1, 3, 5, 7], [3, 5, 7, 9]

[168,175,185,192], [133,147,155,169], [150,162,174,182], [145,178,184,197], [139,157,175,189], [58,76,182,196]

x1 x2 x3 x4 x5 x6

= 0, = 0, = 0, = 0, = 41, = 48

2334

5

[13,15,19,25], [14,16,20,26], [13,17,21,29], [8,14,22,32], [17,29,33,41], [18,20,24,30], [19,21,25,31]

[2, 4, 6, 8, 10, 12, 14], [1, 3, 5, 7], [3, 5, 7, 9], [6, 8, 10, 12]

[168,175,185,192], [133,147,155,169], [150,162,174,182], [145,178,184,197], [139,157,175,189], [58,76,182,196], [60,78,184,198],

x1 x2 x3 x4 x5 x6 x7

= 0, = 14, = 0, = 1, = 39, = 21, =0

1938

models, and all the solutions are global optimal solutions. It can be observed the values of the variables are also obtained in integer numbers which guaranteed the feasibility conditions of integer programming, although the instants considered in each set are in a fuzzy format the result is obtained in integer format and it will be providing much more advantages to a decision maker dealing with practical problems associated with an integer programming. 6.1 Results and Discussion By employing the presented model illustrations, the optimal outcome of the FIPP is [−2, 2, 3, 5], which might be a fresh attempt to address the vagueness. The FIPP of the optimum outcome should be increasingly greater than −2 as well as less than 5, and perhaps the greatest probable outcome is still in the range of 2 and 3. Differences in cost with considerable possibility were shown in Fig. 2. Also, obtained fuzzy optimum

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outcome x might be empirically comprehended. At this point, x describes the significance of the FIPP, as well as the perception of μmax (x) by decision-makers, where,

Fig. 2. The graphical demonstration of the fuzzy optimum outcome μ∼ (x) z

7 Conclusion Lastly, the study has been led to construct a model which enumerates the IPPs under uncertainty. The model demonstrates that the vague representation of the problem has been described in detail by a numerical illustration with two different cases, first one utilizes Gomary ‘s cutting plane approach in the form of fuzziness using membership functions, and the second one by the support of robust rankings. Membership function plays a vital role in the formation of a model to address the vagueness. A few of the modeling techniques have been explored in making the decision only the membership functions with vague goals or restrictions. This model provides an important way to address IPP in the context of vagueness. As a result, the optimum outcome intended fuzziness with model performance of a higher number of variables and its values are coded in LINGO. The performance of the model shows the detailed optimal outcomes with result and discussion. Besides, the outcome has also been demonstrated by the representation of fuzzy membership functions. Furthermore, a comparative study could be an effective manner to overcome IPPs with a vague nature. Each model seeks to highlight the vagueness and perceptions of the reality of decision-makers and this can help to meet decision-making challenges. This model may be suggested as a research direction for future studies of fuzzy optimization models to address the uncertainty of certain other types of IPPs. As a result, this model can be extended to mixed IPPs and multiobjective decision-making problems to achieve integer outcomes.

References 1. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 2. Bellman R., Zadeh, L.: Decision making in a fuzzy environment. Manage. Sci. 17(4), 141–164 (1970)

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3. Tanaka, H., Asai, K.: Fuzzy linear programming problems with fuzzy numbers. Fuzzy Sets Syst. 13(1), 1–10 (1984) 4. Campos, L., Verdegay, J.L.: Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets Syst. 32(1), 1–11 (1989) 5. Herrera, F., Verdegay, L.: Three models of fuzzy integer linear programming. Eur. J. Oper. Res. 83(3), 581–593 (1995) 6. Fortemps, P., Roubens, M.: Ranking and defuzzification methods based on area compensation. Fuzzy Sets Syst. 82(3), 319–330 (1996) 7. Herrera F., Verdegay, J.L.: Making decisions on fuzzy integer linear programming problems. In: Ruan, D. (eds.) Fuzzy Logic Foundations and Industrial Applications. International Series in Intelligent Technologies, vol. 8, pp. 147–164. Springer, Boston (1996). https://doi. org/10.1007/978-1-4613-1441-7_8 8. Allahviranloo, K.S.T., Kiani, N.A., Alizadeh, L.: Fuzzy integer linear programming problems. Int. J. Contemp. Math. Sci. 2(4), 167–181 (2007) 9. Allahviranloo, M., Afandizadeh, S.: Investment optimization on port’s development by fuzzy integer programming. Eur. J. Oper. Res. 186(1), 423–434 (2008) 10. Sudhagar, K.G.C.: Fuzzy integer linear programming with fuzzy decision variables. Appl. Math. Sci. 4(70), 3493–3502 (2010) 11. Pandian, P., Jayalakshmi, M.: A new method for solving integer linear programming problems with fuzzy variables. Appl. Math. Sci. 4(20), 997–1004 (2010) 12. Fan, Y.R., Huang, G.H., Huang, K., Jin, L., Suo, M.Q.: A generalized fuzzy integer programming approach for environmental management under uncertainty. Math. Probl. Eng. 486576, 1–16 (2014) 13. Sagnak, M., Kazancoglu, Y.: Shift scheduling with fuzzy logic: an application with an integer programming model. Procedia Econ. Finan. 26, 827–832 (2015) 14. Niksirat M., Hashemi S. M., Ghatee m.: Branch-and-price algorithm for fuzzy integer programming problems with block angular structure, Fuzzy Sets and Systems 296, 70–96 (2016) 15. Palanivel, K.: Fuzzy commercial traveler problem of trapezoidal membership functions within the sort of α optimum solution using ranking technique. Afr. Mat. 27(1), 263–277 (2016) 16. Dinagar, D.S., Jeyavuthin, M.M.: Fully fuzzy integer linear programming problems under robust ranking techniques. Int. J. Math. Appl. 6(3), 19–25 (2018) 17. Ammar, E., Emsimir, A.: A mathematical model for solving integer linear programming problems. African J. Math. Comput. Sci. Res. 13(1), 39–50 (2020) 18. Ammar, E.-S., Emsimir, A.: A mathematical model for solving fuzzy integer linear programming problems with fully rough intervals. Granular Comput. 6(3), 567–578 (2020). https:// doi.org/10.1007/s41066-020-00216-4 19. Khalili Goudarzi, F.N.: Seyed Taghi-Nezhad, Nemat: a new interactive approach for solving fully fuzzy mixed integer linear programming. Yugoslav J. Oper. Res. 30(1), 71–89 (2020) 20. Cherfaoui, Y., Moulaï, M.: Biobjective optimization over the efficient set of multiobjective integer programming problem. J. Indust. Manage. Optim. 17(1), 117–131 (2021) 21. Wan S.P., Chen Z.H., Dong J.Y.: Bi-objective trapezoidal fuzzy mixed integer linear programbased distribution center location decision for large-scale emergencies. Appl. Soft Comput. 110, 107757 (2021) 22. Kaliyaperumal, P., Das, A.: A mathematical model for nonlinear optimization which attempts membership functions to address the uncertainties. Mathematics 10, 1743 (2022) 23. Truong, H.Q., Jeenanunta, C.: Fuzzy mixed integer linear programming model for national level monthly unit commitment under price-based uncertainty: a case study in Thailand. Electric Power Syst. Res. 209,(2022)

Fuzzy Goal Programming Approach to Multi-objective Facility Location Problem for Emergency Goods and Services Distribution Mert Paldrak1(B)

, Simge Güçlükol Ergin1 and Melis Tan Taco˘glu2

, Gamze Erdem1

,

1 Industrial Engineering Department, Ya¸sar University, Bornova, ˙Izmir, Turkey

{mert.paldrak,gamze.erdem}@yasar.edu.tr 2 Logistic Management Department, Ya¸sar University, Bornova, ˙Izmir, Turkey

Abstract. Ensuring the distribution of vital goods and services during emergency or post-disaster situations is crucial for meeting the needs of those affected as quickly as possible. The challenge lies in finding suitable and pertinent locations for facilities to efficiently distribute these goods and services. In such a situation, location decisions for these facilities must be made considering multiple objectives. However, in such a real-life problem, the aspiration levels of each of the objectives are not certainly known due to unpredictable results of a disaster. Consequently, the problem is formulated as fuzzy multi-objective facility location problem where two objectives are taken into consideration. We specifically consider minimization of total cost of facilities to be opened and minimization of total distance travelled by victims. Due to the conflicting nature of these objective functions, we propose to employ two different fuzzy weighted goal programming techniques to find suitable compromise solutions for the problem. We present our developed models and provide results for three instances with different sizes. The proposed models are coded using IBM CPLEX Optimizer to obtain solutions in a reasonable amount of computational time. This paper contributes to the literature by providing two different fuzzy weighted goal programming techniques and comparing their efficiencies. Keywords: Multi-Objective Facility Location Problem · Fuzzy Logic · Fuzzy Goal Programming · Emergency Assembly Areas · Post-Disaster

1 Introduction Making a fast and reasonable decisions during post-disaster times is often challenging as the results significantly impact limiting the adverse effects of the disaster. Such decisions are generally made under time pressure, and with imperfect data. Moreover, in most cases, decision makers may not be able to accurately predict their goal aspiration due to the lack of prior experience with such situations. When making decisions in emergency or post-disaster situations, multiple conflicting objectives are to be simultaneously taken into consideration. The conditions and constraints of post-disaster situations can © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 521–531, 2023. https://doi.org/10.1007/978-3-031-39774-5_58

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rapidly change, which adds to the challenge of decision making. Therefore, such critical decisions must be made while considering the changing conditions, constraints as well as multiple conflicting objectives. One of the most critical decisions in post-disaster situations is to correctly choose the locations for post-disaster service facilities where victims can congregate. On the other hand, the distribution of essential goods and services in such situations is vital to meet the urgent needs of the victims. These facilities range from field hospitals to provide immediate medical assistance to kitchen services to distribute food, and clean water to survivors. A common practice of a post-disaster problem in urban areas is the determination of the assembly areas’ locations. In most metropolitan cities, in the event of a postdisaster situation, administration often designates specific areas for survivors to come together. Ideally, these assembly areas should be specious enough to accommodate a certain number of victims and have sufficient infrastructure, such as easy-access to health services and transportation networks. However, it is usually cumbersome to find assembly areas that can simultaneously meet all requirements in view of constraints and limitations in urban areas. Thus, these assembly areas are assumed to be natural candidates to open a facility despite their relative advantages and disadvantages. In this study, we propose a fuzzy multi-objective programming model where the objectives are to minimize total cost of facilities to be opened and total distance travelled by victims. To solve this multi-objective model, two different weighted fuzzy goal programming models are proposed. This study is organized as literature review, fuzzy multi-objective model, weighted fuzzy goal programming models, results and conclusions, respectively.

2 Literature Review Natural disasters and emergency situations can occur unexpectedly, leaving decisionmakers with time pressure and limited resources to react. One of the important decisions to make during these situations is the location of post-disaster of service facilities for distribution of essential goods and services to survivors. This decision involves multiple objectives, such as minimizing distance traveled and costs of opening facilities while covering all demand. Due to the uncertainty and vagueness associated with these objectives, fuzzy logic has been widely used in many studies to address multi-objective facility location problem (MOFLP) in emergency situations and post-disaster. Several optimization models have been proposed to address the multi-objective facility location problem in emergency situations and post-disaster. For example, a fuzzy multi-objective optimization model for post-disaster facility location that considers the total cost of opening facilities and maximization of the accessibility of facilities in Alimohammadi & Aghaei (2019) [1]. Similarly, a fuzzy goal programming model for a post-disaster facility location problem that considers the minimization of maximum weighted travelled distance and the maximization of total cost of opening facilities in Mohammaditabar et al. (2017) [2] and Mohammadi et al. (2017) [3]. Moreover, a hybrid multi-objective optimization model for the post-disaster facility location problem where TOPSIS and NSGA-II methods are combined in Yousefi et al. (2016) [4]. On the other hand, a multi-objective optimization model for postdisaster location problem where in addition to traditional objectives, minimization of

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the environmental impact of the facilities are taken into consideration Hong et al. (2018) [5]. Another approach involving fuzzy logic to solve FLPs in emergency situations is fuzzy TOPSIS in which decision maker rank alternatives based on their similarity to an ideal solution and their distance to anti-ideal solution. Several studies have shown that Fuzzy TOPSIS is an effective technique to solve such a problem when criteria are most qualitative. For example, Moradi et al. (2018) [6] proposed a multi-objective facility location model using TOPSIS to solve the problem of locating emergency shelters in disaster-prone areas. Similarly, a TOPSIS-based model for the multi-objective facility location problem in emergency situations is proposed in Tavokkali et al. (2015) [7]. To the best of our knowledge, this problem has never been modeled as a Fuzzy Weighted Goal Programming Model which is formulated as the model provided in Rivaz et al. (2020) [8] with different problem sizes and these objective functions.

3 Mathematical Modeling In this study, our problem is formulated as a fuzzy multi-objective MIP model and then two different fuzzy weighted goal programming models are proposed to find appropriate compromise solutions for such a problem with conflicting objectives. Before formulating the model, sets, indices and parameters related to our problem are given as follows: Sets I : set of neighbourhoods affected by the disaster. J : set of emergency assembly areas. Indices i: index of neighbourhoods i ∈ I = {1, 2, . . . |I|}. j: index of emergency assembly areas j ∈ J = {1, 2, . . . |J|}. Parameters dij : distance between neighborhood i and emergency assembly area j, i ∈ I, j ∈ J fj : fixed cost of opening emergency assembly area j, j ∈ J cj : capacity of emergency assembly area j, j ∈ J hi : total number of victims affected in neighborhood i, i ∈ I Dw : maximum weighted distance to be travelled by survivors Decision Variables vij : number of victims assigned from neighborhood i to emergency area j  xj =

1, 0,

 yij =

1, 0,

emergency assembly area j is opened otherwise if neigborhood i is assigned to emergency assembly area j otherwise

524

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3.1 Fuzzy Goal Programming Model In this study, we employed a fuzzy multi-objective mathematical model where the objectives are to minimize total fixed cost of opening facilities and to minimize total distance travelled by survivors to reach emergency assembly areas. To solve this model, two Weighted Fuzzy Goal Programming formulations inspired by Rivaz et al. (2020) [8] are generated. Goal programming is a modelling technique to handle multi-objective problems by reducing a complex problem into a standard single objective programming model with the help of weights associated with importance of the objectives. Most of the multi-objective problems involve conflicting objectives and it is often impossible to optimize one of them without deteriorating the others. Consequently, decision makers search for suitable compromise solutions to such problems. Goal programming is one of the most effective techniques to tackle multi-objective problems. The objective of a goal program is to minimize the sum of deviations from the aspiration levels of each of the objectives. Conventional goal programming models consider that decision makers determine a precise aspiration in advance. However, in real-life problems, especially during post-disaster situations, decision makers cannot know the aspiration levels of each of the objectives. In such cases, it is plausible to employ fuzzy goal programming technique. In this study, considering vagueness of the aspiration levels, we formulated following fuzzy multi-objective model:  fj *xj Zf (1) Min j∈J

Min Dw Zd

(2)

Constraints  (dij ∗ vij ) ≤ Dw

∀i ∈ I

(3)

∀i ∈ I , ∀j ∈ J

(4)

j∈J

vij ≤ hi ∗ yij yij ≤ xj 

∀i ∈ I , ∀j ∈ J

yij ≥ xj

∀j ∈ J

(5) (6)

i∈I



vij ≤ cj ∗ xj

∀j ∈ J

(7)

i∈I



vij = hi

∀j ∈ J

(8)

i∈I

xj ∈ {0, 1}, yij ∈ {0, 1}, vij ≥ 0

∀i ∈ I , ∀j ∈ J

(9)

Objective function as shown in Eq. (1) minimizes the total fixed cost of opening emergency assembly areas. Objective function as shown in Eq. (2) minimizes the maximum weighted distance. Constraints in Eq. (3) ensures that total weighted distance to

Fuzzy Goal Programming Approach to Multi-objective Facility Location Problem

525

reach each facility does not exceed the allowable distance. Constraints in Eq. (4) indicate that number of victims assigned to all assembly areas cannot exceed the total number of victims in each neighborhood. Constraints in Eq. (5) ensure that none of the neighborhoods can be assigned to an emergency assembly area unless it is opened. Constraints in Eq. (6) guarantee that if a facility is opened at emergency assembly area j, at least one neighborhood should be assigned to this facility. Constraints in Eq. (7) and in Eq. (8) are capacity restriction of each emergency assembly area and assignment of all victims to any assembly area, respectively. Constraints in Eq. (9) give the sign restrictions for decision variables. In this model, the symbol  is elaborated as “essentially less than” and each one of two fuzzy goals is represented by using a fuzzy set defined over the feasible set with the fuzzy membership function. Applying piecewise linear membership functions to express fuzzy goals and using weighted goal programming technique, the following model is achieved. 3.2 Fuzzy Weighted Goal Programming Model-1 In this problem, we have got two objective functions and we can write l ∈ L = {1, 2}. Our first fuzzy goal programming model minimizes the total weighted deviations from the goals of each of the objectives using fuzzy membership functions. min

2 

wl ∗

l=1

dl  lR

(10)

Constraints 

fj ∗ xj − d1 ≤ Zf

(11)

j∈J

Dw − d2 ≤ Zd dl ≤ 1,  lR dl ≥ 0,

l = 1, 2 l = 1, 2

(12) (13) (14)

Constraints Between (3) and (9) In this model, wl represents of the weight of the lth fuzzy goal which are normalized such that the sum of wl s is equal to 1. Moreover, the slack variable of the constraint dl th lR ≤ 1 equals to the degree of membership function for the l fuzzy model. 3.3 Fuzzy Weighted Goal Programming Model-2 To propose the new model, instead of using deviational variables dl , l = 1, 2 in Fuzzy Goal Programming Model-1, it is suggested to use deviational function r(1 − wk ).

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Consequently, to solve our fuzzy multi-objective model, the following fuzzy goal programming oriented model is stated as follows: min Z = r

(15)

Constraints 

fj ∗ xj − r*(1 − w1 ) ≤ Zf

(16)

j∈J

Dw − r*(1 − w2 ) ≤ Zd r*(1 − wl ) ≤ lR,

l = 1, 2

r≥0

(17) (18) (19)

Constraints Between (3) and (15) Where Zl is considered as optimal solution of the lth objective function. And  lR is determined by max Zn − Zm where n = 1, 2 & n = m. It is noteworthy that in this previous model, values of the deviational function are consistent with the priority of the objectives. In other words, larger value of wl will result in a smaller value of deviation function.

4 Numerical Examples and Computational Results To be able to demonstrate the efficiency of proposed fuzzy weighted goal programming models, we generated three instances in which number of neighborhoods and emergency assembly areas varies. The data are generated using MATLAB R2016 since it reduces problem creation effort. We classified our instances as small, medium, and large and relative information related to our instances along with optimal solutions with respect to each objective is given in Table 1 and 2. All mathematical models and algorithms are developed using ILOG IBM CPLEX Optimization Studio 20.1.0, and solved by CPLEX Optimizer 20.1.0. Furthermore, a personal computer with Intel® Core™ i5-10510U CPU, 16 GB RAM is used. In fuzzy weighted goal programming models, ten levels of wl values are utilized, and our results are given in following tables between Table 3 and Table 8: Looking at the resulting tables provided above Table 4, 5, 6 and 7, the deviational variables are consistent for all instances with the priority of objectives in both Weighted Fuzzy Goal Programming models. For example, considering our small instance solved by Fuzzy Goal Programming-1 (Table 3), when w1 = 0.1 and w2 = 0.9, the deviational values are d1 = 22 and d2 = 0, respectively. It means that the first objective function which is less preferred has the value which is further to its aspiration level. By increasing w1 , in other words by increasing the preference of the first objective function, we realize that relative di values decrease, which validates the consistency of both models. However, employing the first Weighted Fuzzy Goal Programming model, the same objective values

Fuzzy Goal Programming Approach to Multi-objective Facility Location Problem

527

Table 1. Information Related to Test Instances-1 Type

INSTANCE

OBJECTIVE

Total Fixed Cost

Total Distance

Z1

Z2

Small

|I|=15, |J|=4

Total Fixed Cost

125

386

125

4500

Total Distance

7600

4500

Total Fixed Cost

235

534

235

8000

Total Distance

15985

8000

Total Fixed Cost

485

923

485

12000

Total Distance

20000

12000

Medium

Large

|I|=22, |J|=8

|I|=35, |J|=12

Table 2. Information Related to Test Instances-2 Small Instance

Medium Instance

Large Instance

Number of Constraints

282

588

1354

Binary Variables

64

184

432

Integer Variables

60

176

420

Other Variable

1

1

1

Non-Zero Coefficients

628

1790

4259

Iterations for 1st Objective Function

214

378

332

Iterations for 2nd Objective Function

26

64

116

Table 3. Optimal Objective Values Obtained from Fuzzy Model-1 for Small Instance Small Instance, |I|=15, |J|=4 w1

w2

1R

2R

p1

p2

Z1, Z2

0,1

0,9

261

3100

22

0

147, 4500

0,2

0,8

261

3100

22

0

147, 4500

0,3

0,7

261

3100

22

0

147, 4500

0,4

0,6

261

3100

18

0

143, 4500

0,5

0,5

261

3100

18

0

143, 4500 (continued)

528

M. Paldrak et al. Table 3. (continued)

Small Instance, |I|=15, |J|=4 w1

w2

1R

2R

p1

p2

Z1, Z2

0,6

0,4

261

3100

0

200

125, 4700

0,7

0,3

261

3100

0

200

125, 4700

0,8

0,2

261

3100

0

200

125, 4700

0,9

0,1

261

3100

0

200

125, 4700

Table 4. Optimal Objective Values Obtained from Fuzzy Model-2 for Small Instance Small Instance, |I|=15, |J|=4 w1

w2

1R

2R

r

r*(1-w1)

r*(1-w2)

Z1, Z2

0,1

0,9

261

3100

24,44

22

2

147, 4502

0,2

0,8

261

3100

27,5

22

6

147, 4506

0,3

0,7

261

3100

27,8

19

8

144, 4508

0,4

0,6

261

3100

30

18

12

143, 4512

0,5

0,5

261

3100

36

18

18

143, 4518

0,6

0,4

261

3100

45

18

27

143, 4527

0,7

0,3

261

3100

60

18

42

143, 4542

0,8

0,2

261

3100

90

18

72

143, 4572

0,9

0,1

261

3100

180

18

162

143, 4662

Table 5. Optimal Objective Values Obtained from Fuzzy Model-1 for Medium Instance Medium Instance, |I|=22, |J|=8 w1

w2

1R

2R

p1

p2

Z1, Z2

0,1

0,9

299

7985

45

0

280, 8000

0,2

0,8

299

7985

45

0

280, 8000

0,3

0,7

299

7985

45

0

280, 8000

0,4

0,6

299

7985

45

0

280, 8000

0,5

0,5

299

7985

45

0

280, 8000

0,6

0,4

299

7985

20

1000

255, 9000

0,7

0,3

299

7985

0

2000

235, 10000 (continued)

Fuzzy Goal Programming Approach to Multi-objective Facility Location Problem Table 5. (continued) Medium Instance, |I|=22, |J|=8 w1

w2

1R

2R

p1

p2

Z1, Z2

0,8

0,2

299

7985

0

2000

235, 10000

0,9

0,1

299

7985

0

2000

235, 10000

Table 6. Optimal Objective Values Obtained from Fuzzy Model-2 for Medium Instance Medium Instance, |I|=22, |J|=8 w1

w2

1R

2R

r

0,1

0,9

299

7985

50

45

5

280, 8005

0,2

0,8

299

7985

56,25

45

11

280, 8011

0,3

0,7

299

7985

64,29

45

19

280, 8019

0,4

0,6

299

7985

75

45

30

280, 8030

0,5

0,5

299

7985

90

45

45

280, 8045

0,6

0,4

299

7985

112,5

45

68

280, 8068

0,7

0,3

299

7985

150

45

105

280, 8105

0,8

0,2

299

7985

225

45

180

280, 8180

0,9

0,1

299

7985

450

45

405

280, 8450

r*(1-w1)

r*(1-w2)

Z1, Z2

Table 7. Optimal Objective Values Obtained from Fuzzy Model-1 for Large Instance Large Instance, |I|=35, |J|=12 w1

w2

1R

2R

p1

p2

Z1, Z2

0,1

0,9

438

8000

34

0

519, 12000

0,2

0,8

438

8000

34

0

519, 12000

0,3

0,7

438

8000

34

0

519, 12000

0,4

0,6

438

8000

34

0

519, 12000

0,5

0,5

438

8000

0

500

485, 12500

0,6

0,4

438

8000

0

500

485, 12500

0,7

0,3

438

8000

0

500

485, 12500

0,8

0,2

438

8000

0

500

485, 12500

0,9

0,1

438

8000

0

500

485, 12500

529

530

M. Paldrak et al. Table 8. Optimal Objective Values Obtained from Fuzzy Model-2 for Large Instance

Large Instance, |I|=35, |J|=12 w1

w2

1R

2R

r

r*(1-w1)

r*(1-w2)

Z1, Z2

0,1

0,9

438

8000

37,78

34

4

519, 12004

0,2

0,8

438

8000

42,5

34

9

519, 12009

0,3

0,7

438

8000

48,57

34

15

519, 12015

0,4

0,6

438

8000

56,67

34

23

519, 12023

0,5

0,5

438

8000

68

34

34

519, 12034

0,6

0,4

438

8000

85

34

51

519, 12051

0,7

0,3

438

8000

113,33

34

79

519, 12079

0,8

0,2

438

8000

170

34

136

519, 12136

0,9

0,1

438

8000

340

34

306

519, 12306

are attained for different weights. In other words, this model fails to find a variety of efficient points. On the other hand, the second Weighted Fuzzy Goal Programming model could capture various solutions with respective to different weights. As a result, it is concluded that although both models are consistent in terms of preference among objective functions, however, the second Fuzzy Goal Programming model outperforms that of the first one in terms of solutions variability.

5 Conclusions and Future Works This study deals with a multi-objective facility location problem during emergency and post-disaster situations. The problem is modelled as a fuzzy multi-objective mathematical model since the aspiration levels of each conflicting objectives are not known precisely. The aim of the study is to find a suitable compromise solution with respect to objective functions, namely minimization of total fixed costs of opening facilities and minimization of total weighted distance travelled by victims. We proposed two different fuzzy goal programming models inspired by [8] in order to solve the problem. Three different instances with different sizes are generated to demonstrate the complexity of the problem and better analyze the efficiency of the models. Based on our results, we conclude that although both fuzzy weighted goal programming models are consistent, second one is better with respect to solution variability. As a future direction, try to propose new fuzzy goal programming algorithms, namely fuzzy Chebyshev Goal Programming model to deal with such a multi-objective problem could be considered as a general topic for a future study.

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References 1. Alimohammadi, M., Aghaei, J.: A fuzzy multi-objective optimization model for post-disaster facility location problem. J. Indust. Syst. Eng. 12(4), 9–27 (2019) 2. Mohammaditabar, D., Paydar, M.M., Tavakkoli-Moghaddam, R.: A fuzzy goal programming model for post-disaster facility location problem under uncertainty. Int. J. Disaster Risk Reduction 22, 153–163. 3. (2017) 3. Mohammadi, M., Tavakkoli-Moghaddam, R., Zahiri, B.: B: A fuzzy multi-objective programming model for emergency facility location problem. J. Indust. Syst. Eng. 10(3), 1–15 (2017) 4. Yousefi-Babadi, V., Tavakkoli-Moghaddam, R., Heydari, M.: A hybrid multi-objective optimization algorithm for post-disaster facility location problem. Ann. Oper. Res. 244(2), 461–481 (2016) 5. Hong, S., Lee, S., Lee,W.:A multi-objective optimization model for post-disaster facility location. Sustainability, 10(11) (2018) 6. Moradi, S., Noorhosseini, S.A., Mahmoodi, M.: A multi-objective model for emergency shelter location-allocation problem under uncertainty. Saf. Sci. 110, 86–97 (2018) 7. Tavakkoli-Moghaddam, R., Sahebjamnia, N., Mohammadi, M.: M: A multi-objective facility location model in emergency situations. Saf. Sci. 71, 24–33 (2015) 8. Rivaz, S., Nasseri, S.H., Ziaseraji, M.: A fuzzy goal programming approach to multi-objective transportation problems. Fuzzy Inform. Eng. 12(2), 139–149 (2020)

Social Spider Optimization for Text Classification Enhancement Fawaz S. Al-Anzi(B)

and Sumi Sarath

Department of Computer Engineering, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait {fawaz.alanzi,sumi.sarath}@ku.edu.kw

Abstract. The task of classifying text documents into the various categories for which they have been previously defined is known as text classification. The number of text documents available online exhibits exponential growth, so it is essential that a model be implemented to better organize these documents to improve the search process. The curse of dimensionality challenge is the most significant obstacles encountered during text document classification, severely affecting its performance. Identifying and reducing to a few key characteristics is a necessary first step. This study aims, first and foremost, to explore a Social Spider Optimization (SSO) method for selecting and reducing feature dimensions to enhance the text classification capabilities of machine learning models. The SSO strategy is a nature-inspired algorithm for optimization that mimics the collaborative actions of social spiders to find optimal solutions to optimization challenges. The research makes use of a standard open-source dataset to evaluate how successful the SSO algorithm is at improving the overall performance of various classifiers. These classifiers include Support Vector Machine (SVM), Logistic Regression (LR), Random Forest (RF), and Stochastic Gradient Descent (SGD) among others. The findings demonstrate that the SSO algorithm results in a substantial improvement in the classification performance of these classifiers. This research sheds light on the potential of algorithms that are inspired by natural phenomena, such as SSO, to strengthen the productivity of text classification across a multitude of domains, which includes natural language processing and information retrieval. Keywords: Social Spider Optimization · Text Classification · Stochastic Gradient Descent · Random Forest · Logistic Regression

1 Introduction The proliferation of the internet and the web has led to the digitization of a broad variety of documents that can now be accessed from anywhere in the world, regardless of location or time. Because of the exponential growth of written documents that have been published on the internet, the task of efficiently retrieving information has become significantly more difficult. Document categorization is considered as the work of organizing the text documents into several predetermined categories according to the information that is contained within them. An effective text or document classification is a crucial task for many different applications, including intent detection, subject labeling, sentiment © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 532–539, 2023. https://doi.org/10.1007/978-3-031-39774-5_59

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analysis, intent detection, question answering, language detection, news categorization, and spam detection, among others. A classification method that is based on machine learning assigns categories to texts in accordance with the observations made during the training phase. When it comes to automatic text categorization, most of the work is split between two stages: training and testing. The documents require preprocessing prior to the training procedure. In preprocessing, tasks like data cleansing, feature extraction, and document indexing are performed. It’s not possible to train or evaluate the model without a dataset that has already been classified. When training a model, features or observations are taken from the labeled training data. The documents are categorized [1] using the extracted observations. Preprocessing the text documents by eliminating stop words, capitalization, and unnecessary characters and then normalizing, stemming, or lemmatizing them can greatly enhance the classifier’s performance [2]. Then, you can get the features (a list of words) you need to describe it from the documents. Bag of words, TF-IDF, Word2Vec [3], Doc2Vec [4], and GLoVE [5] are just a few of the feature extraction techniques available for text classification. When processing text, a high dimensional feature vector boosts the margin for error, and a sufficiently large amount of data is recommended to keep the accuracy of the processing acceptable. Improved effectiveness in text classification issues requires dimensionality reduction to address the curse of dimensionality [6]. With the goal of representing the documents in a low dimensional area, it is crucial to pick the most relevant and useful characteristics. A few examples of feature selection models are [7, 8]: Chi-square statistic (CHI), Mutual Information (MI), Information Gain (IG), Latent Semantic Indexing (LSI), Singular Value Decomposition (SVD), Linear Discriminant Analysis (LDA), and Principal Component Analysis (PCA). In recent years, nature-inspired approaches to optimization like Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and so on have demonstrated encouraging results in the resolution of optimization problems [9–11]. The Social Spider Optimization (SSO) is yet another such approach for optimization. It is modeled after the gathering behavior of social spiders and has demonstrated promise in the resolution of a variety of optimization issues. In the current investigation, one of our primary goals is to investigate whether the SSO is successful in improving text categorization performance. We test the performance of SSO in text classification application using Kaggle BBC dataset [12] using various ML algorithms such as SGD, LR, SVM, and RF. The rest of the manuscript is neatly laid up as shown in the following description. In the coming segment, we will examine previous research on text classification and optimization techniques. The data sets, optimization algorithm, machine learning algorithms used, and evaluation metrics are discussed in Sect. 3 of the report. Within Sect. 4, you will find the method of study that was utilized throughout the research that was suggested. The discussion of the experimental findings and analysis can be found in Sect. 5. Last but not least, the paper is brought to a close in Sect. 6, which focuses on the conclusion of this study.

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2 Literature Review Text classification is the process of determining which category (or categories) a specific document falls under [13]. At the present, prior planning for classification of textual documents is a complex procedure that requires not only includes the model learning of frameworks, but it also consists of a large number of additional operations, such as preprocessing of textual documents, feature representation, and dimensionality reduction by selecting the relevant features which makes this process as a time-consuming endeavor [13–15]. Once you have annotated text corpus, the first stage in building a classification algorithm is to identify and represent features from the data corpus after preprocessing it. Text classification can be done with a variety of different methods of feature representation and weighting, and each model of representation comes with its own unique set of benefits and drawbacks, both of which ought to be subjected to consideration. Below, we will discuss the word embedding and N-gram based models [16], both of which belong to the category of feature extraction frameworks which are frequently utilized in document categorization architectures [15]. The construction of a classifier for text categorization makes help of a wide variety of approaches. K-Nearest Neighbor, Naive Bayes, SVM, Decision Tree, Deep Learning, and other similar methods are among them [16–19]. A big enough collection of annotated documents is used to train the classifier, and its performance is measured with a variety of metrics, including accuracy, precision, recall, and F-measure [2, 20]. In today’s world, a wide variety of optimization algorithms, many of which take their cues from natural phenomena, are utilized to tackle a wide variety of optimization challenges spanning many different subject areas, which includes data mining problems such as efficient classification and clustering. Well-known examples of such algorithms include PSO, ACO, GA, Bat Influenced Algorithm, Whale Optimization Algorithm, SSO, and many others [11]. The SSO model recreates spider’s natural behavior to solve optimization problems in a way that is both efficient and accurate, with a low margin for error. The phase of feature selection and reduction plays a significant part in determining which features are necessary for text classification based on the dataset [21]. James and Victor [22] presented a novel and promising a new metaheuristic approach to global optimization that relies on the social nature and interactions of spiders that could be employed to a broad variety of problems in the real-world situations. The evaluation outcomes show that the social spider approach outperforms other such models in terms of convergence rate and solution quality. Danillo et al. [23] proposes a novel method for detecting the illegal transfer of energy using SSO and SVM. The SVM parameters are optimized with the help of SSO, which results in a model that is more accurate and effective for detecting energy fraud. According to their findings, the SVM which is based on SSO achieves greater accuracy and better detection rates than any of the other methods.

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3 Theoretical Background 3.1 Term Frequency-Inverse Document Frequency (TF-IDF) The simplest approach, that is referred to as Term Frequency (TF), determines the frequency with which a specific term or collection of words is found in the textual document. The TF-IDF Analysis is the technique that is the most well-known and the one that is used the most frequently [24, 25]. To lessen the effect of words that are implicitly prevalent throughout the corpus, this methodology utilizes the use of IDF together with the TF [16]. The TF and IDF of a feature t in document d are calculated using Eqs. 1 and 2 respectively: Count of term t present in document d Total count of words in d

(1)

Number of entire documents in the corpus Count of documents that contain term t

(2)

tf (t, d ) = idf (t) =

TF-IDF(t, d) is determined as a multiplication of tf(t,d) with idf(t). 3.2 Random Forest Random Forest is a type of ensemble learning and it works by first building several distinct decision trees followed by compiling the outcomes of these trees into a single projection. Each decision tree receives training in light of a randomized portion of the total training corpus as well as a randomized portion of the total collection of features. This serves to prevent the model from being overfit, which in turn increases its ability to generalize. The individual trees’ forecasts are combined by either averaging the results or deciding on them, depending on the nature of the issue that needs to be addressed. 3.3 Logistic Regression Logistic regression is among the easiest models for classifying data. Given its parametric nature, it can be partially understood by examining its parameters, making it useful for experimentalists searching for correlations between variables. A parametric model can be fully specified by a vector of factors of the form β = (β0 , β1 ,…, βp ). The equation y = kx + m is a parametric model because it relies on two independent variables, k and m. Coefficients of predictor factors are the logistic regression model’s parameters, written as β0 + β1 X1 +… βp Xp . As a result, the intersection is 0. The logistic coefficient (sigmoid) function takes values between zero and one and is represented as a function of t (as in Eq. 3). F(t) =

1 1 + e−t

(3)

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3.4 Stochastic Gradient Descent SGD is a straightforward technique for modifying linear regression models and classifications to convex functions of loss, such as those utilized in LR and SVM. Both natural language processing and text classification frequently involve enormous scale and sparse ML problems, which have been successfully addressed by SGD. Consider the ith sample (xi , yi ) in the learning corpus, in which xi ∈ Rm and yi is the category identifier. The most important objective is to master the linear scoring function (f(x) = wT x + b) with model parameter, w ∈ Rm and intercept, b ∈ R. When estimating the model parameters with Eq. 4, the training error should be eliminated as much as possible. 1 n E(w, b) = L(yi, f (xi)) + αR(w) (4) i=1 n 3.5 Support Vector Machine (SVM) SVM was first presented by Vladimir Vapnik and his fellow researchers. Finding the hyperplane in the input space that provides the greatest amount of separation between the two classes is the primary goal of the SVM algorithm. The hyperplane is described as the set of points where the inner product of the given input vector and a weight vector is equal to a threshold value. In cases where the data cannot be linearly separated, the SVM model is able to make use of a kernel function to transform the space of input into a higher-dimensional space for features. Where the data to be segmented using a hyperplane. The ability to manage high-dimensional data, resilience in the face of outliers, and the capacity to generalize well to freshly collected data are the primary benefits of the SVM. 3.6 Social Spider Optimization (SSO) Algorithm The SSO algorithm is a population-centered metaheuristic optimization technique that was conceived after observing the hunting techniques of social spiders which involves cooperation and competition among individuals to locate sources of food and is presented by Mirjalili and his fellow researchers. In the SSO algorithm, the process of searching for a solution to an optimization problem is represented as a population of spiders, with each spider standing in for a different possible answer. The spiders move around the search space using a combination of social and individual behaviors, such as following the pheromone trails left by other spiders and exploring new areas. During the search process, the spiders communicate with each other by depositing pheromone trails on the best solutions found so far. This helps to guide the other spiders towards the better solutions and avoid getting stuck in local optima.

4 Proposed Model The first step in text categorization is the collection of a reasonably large, annotated training corpus so that a better classifier can be developed, implemented, and evaluated. Despite the fact that numerous researchers have presented text categorization methods for

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a variety of languages, this field is still being considered because researchers are working to improve the efficacy of the approaches by making use of forthcoming technological advances. The performance of the proposed ML procedure is primarily based on the corpus that is accessible as well as the models that are used. Figure 1 presents an overview of the overall structure of the text classification model that is being suggested.

Fig. 1. The entire architectural framework of the Proposed approach

• Preprocessing: In this stage, the text data will be cleaned and prepared for analysis by performing tasks such as stemming and removing stop words, among other things. • Feature Extraction: Text features are transferred into numerical features using TF-IDF. • Feature Selection: A subset of relevant characteristic features is extracted from the text data with the help of social spider optimization and then optimize the classification performance based on the selected features. • Hyper Parameter Tuning: The optimization algorithm may have several parameters that need to be set, such as the population size and mutation rate in the case of SSO. These parameters are tuned to optimize the effectiveness of the algorithm on the categorization application. • Training and testing: Following the training of the optimized algorithm on a labeled text dataset, the algorithm is put through its pace by having its performance evaluated using a distinct set of text documents. • Evaluation: The effectiveness of the algorithm is calculated and analyzed by employing the metrics like precision, F1 score, recall, and accuracy. • Model refinement: According to the findings of the assessment, the model may be improved by modifying the parameters or the criteria for feature selection to achieve an even higher level of performance.

5 Implementation and Experimental Findings Python was utilized to create a simulation of the suggested method. Python has a sizeable number of pre-built components that are ready to be used, which makes it easier to implement various machine learning strategies. The Kaggle BBC dataset [12] is employed so that the effectiveness of the suggested methodology can be evaluated. The Kaggle BBC dataset includes 2225 documents having textual contents, collected from the website of

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BBC News and correspond to articles that were published in 5 distinct subject categories during the years 2004–2005. These documents were obtained from the BBC News website. The text features were first extracted from the text utilizing a TF-IDF scheme by the suggested system, and then an SSO-based optimization algorithm was employed for adopting the most pertinent subset attributes from this feature representation. The proposed model is learned and evaluated using various ML techniques such as SGD, RF, SVM, and LR with and without the SSO based optimization algorithm and the system with SSO outperformed all the other approaches that were presented in this study with respect to different metrics used for evaluation, which are laid out in Table 1. Table 1. Performance Evaluation of the proposed model with respect to various metrics. Evaluation Metrics

Without SSO based Feature Selection

With SSO based Feature Selection

RF

SVM

SGD

LR

RF

SVM

SGD

LR

Accuracy

95

96

97

96

96.6

97

98

96.8

Precision

96

96

97

96

97

97

98

97

Recall

95

96

97

96

96

97

98

97

f1-Score

95

96

97

96

96

97

98

97

6 Conclusion Through the utilization of SSO, it has achieved great enhancement in the classification performance and the system is able to learn the optimal weights of features for any dataset. It can also assist in overcoming the issue of overfitting and underfitting, which is a problem that frequently arises in conventional machine learning algorithms. SSO is helpful for determining which features are the most essential and for reducing the dimensionality of the features that are input to enhance the effectiveness and precision of text categorization. The fact that it can both optimize the feature weights and decrease the dimensionality of the problem makes it a potentially useful method in NLP. The implementation outcomes demonstrated the fact that the ML algorithms with SSO based feature selection outperformed other cutting-edge algorithms without SSO. The efficiency of the proposed method can be enhanced further using the employment of various algorithms and rearrangement and the count of spiders in this proposed research.

References 1. Minaee, S., et al.: Deep Learning Based Text Classification: A Comprehensive Review (2020) 2. Kasri, M., Birjali, M., Beni-Hssane, A.: A comparison of features extraction methods for Arabic sentiment analysis. In: BDIoT 2019: Proceedings of the 4th International Conference on Big Data and Internet of Things, pp. 1–6 (2019)

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3. Mikolov, T., Chen, K., Corrado, G., Dean, J.: Efficient estimation of word representations in vector space. In: Proceedings of ICLR Workshop. Scottsdale (2013) 4. Pennington, J., Socher, R., Manning, C.D.: GloVe: global vectors for word representation. In: Empirical Methods in Natural Language Processing (EMNLP), pp. 1532–43 (2014) 5. Joulin, A., et al.: Bag of tricks for efficient text classification. In: Proceedings of the 15th Conference of the European Chapter of the Association for CL, vol. 2, pp. 427–31 (2017) 6. Fuka, A., Hanka, R.: Feature Set Reduction for Document Classification Problems. (2001) 7. Kumar, A.A., Chandrasekhar, S.: Text data pre-processing and dimensionality reduction techniques for document clustering. Int. J. Eng. Res. Technol. (IJERT) 1(5), 1–6 (2012) 8. Kim, H., Howland, P., Park, H.: Dimension reduction in text classification with support vector machines. J. Mach. Learn. Res. 6, 37–53 (2005) 9. Liu, S., Zhang, Y., Zhao, D.: Hybrid optimization algorithm for text classification. IEEE Access 7, 5092–5101 (2019) 10. Jeyapriya, M., Jeevitha, K.: Feature selection in text classification using ant colony optimization. J. Ambient. Intell. Humaniz. Comput. 11(2), 729–736 (2020) 11. Abualigah, L., et al.: Nature-inspired optimization algorithms for text document clustering—a comprehensive analysis. Algorithms 13(12), 345 (2020) 12. Greene, D., Cunningham, P.: Practical solutions to the problem of diagonal dominance in kernel document clustering. In: Proceedings of the 23rd international conference on Machine learning, ICML 2006, pp. 377–384 (2006) 13. Manning, C.D., Schutze, H., Raghavan, P.: Introduction to information retrieval. Cambridge University Press, Cambridge (2008) 14. Miro´nczuk, M.M., Protasiewicz, J.: A recent overview of the state-of-the-art elements of text classifcation. Expert Syst Appl 106, 36–54 (2018) 15. Pintas, J.T., Fernandes, L.A.F., Garcia, A.C.B.: Feature selection methods for text classification: a systematic literature review. Artif. Intell. Rev. 54(8), 6149–6200 (2021) 16. Kowsari, J.M., Heidarysafa, M., Barnes, B.: Text classification algorithms: a survey. Information 10(4), 150 (2019) 17. Hmeidi, I.I., et al.: A survey of text categorization techniques using Arabic text. In: International Conference on Information and Communication Systems, Jordan (2013) 18. Wu, H., Liu, Y., Wang, J.: Review of text classification methods on deep learning. Comput. Mater. Continua CMC 63(3), 1309–1321 (2020) 19. Jang, B., Kim, M., Harerimana, G., Kang, S., Kim, J.W.: Bi-LSTM model to increase accuracy in text classification: combining word2vec CNN and attention mechanism. Appl. Sci. 10(17), 5841 (2020) 20. Hossin, M., Sulaiman, M.N.: A review on evaluation metrics for data classification evaluations. Int. J. Data Min. Knowl. Manage. Process 5(2), 1 (2015) 21. Hosseinalipour, A., Gharehchopogh, F.S., Masdari, M., Khademi, A.: Toward text psychology analysis using social spider optimization algorithm. Concurrency Computat. Pract. Exper. 33(17), e6325 (2021) 22. Yu, J.J.Q., Li, V.O.K.: A social spider algorithm for global optimization. Appl. Soft Comput. 30, 614–627 (2015) 23. Pereira, D.R., et al.: Social-spider optimization-based support vector machines applied for energy theft detection. Comput. Electr. Eng. 49, 25–38 (2016) 24. Ahuja, R., Chug, A., Kohli, S., Gupta, S., Ahuja, P.: The impact of features extraction on the sentiment analysis. Procedia Comput. Sci. 152, 341–348 (2019) 25. Qaiser, S., Ali, R.: Text mining: use of TF-IDF to examine the relevance of words to documents. Int. J. Comput. Appl. 181, 07 (2018)

An Algorithm for Fully Intuitionistic Fuzzy Multiobjective Transportation Problem with a Goal Programming Perspective Sakshi Dhruv1(B) , Ritu Arora2 , and Shalini Arora1 1

Department of ASH, Indira Gandhi Delhi Technical University for Women, New Delhi 110006, India {sakshi052phd21,shaliniarora}@igdtuw.ac.in 2 Department of Mathematics, Keshav Mahavidyalaya, University of Delhi, New Delhi, India [email protected] Abstract. In real life, we come across the problems that are defined with greater degree of uncertainty and hesitancy due to which the decision makers fail to furnish with meticulous values of decision variables. Employing the intuitionistic fuzzy set theory the decision makers can determine the acceptance degree, rejection degree and degree of hesitancy of the desired variables. The fluctuating cost, varying demand and all other changing parameters accentuated us to formulate and solve the linear fractional transportation model under fuzzy environment. In the proposed methodology, fuzzy arithmetic operations and ordering function on fully intuitionistic fuzzy multi-objective linear fractional transportation problem (FIFMOLFTP) is applied. Then, FIFMOLFTP is transformed into its crisp equivalent multi-objective linear fractional transportation problem (MOLFTP). To solve MOLFTP, it is further transmuted to an equivalent linear programming problem (LPP) by weighted goal programming approach. The computed goals are expressed mathematically in terms of membership functions which is interpreted in the form of satisfaction level of decision makers. The weighted goal programming model is formulated and the equivalence between them is established. The computation of the obtained model is performed on easily available software. A numerical illustration is described to support our proposed methodology. Keywords: Multi-objective Transportation problem · Weighted goal programming · Fractional transportation problem · Triangular intuitionistic fuzzy number · Fuzzy arithmetic operations

1

Introduction

Transportation problem is defined as the shipment plan of the product from several sources to different destinations in a least costly manner. Hitchcock [7] preSupported by Indira Gandhi Delhi Technical University for Women. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 540–549, 2023. https://doi.org/10.1007/978-3-031-39774-5_60

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541

sented the linear programming formulation of the same and determine a solution algorithm for it. The significance of fractional objective function can be seen in the field of transportation problem too as transportation cost/transported quantity, transportation time/ profit to name a few. Swarup [14] proposed the mathematical structure of linear fractional transportation problem (LFTP) along with a solution methodology. Akram et al. [2] have considered LFTP whose parameters were defined as interval valued fermatean fuzzy numbers. The authors proposed the solution methodology which didn’t demand the transformation of problem into crisp one. A mathematical model of bilevel programming problem which contained upper level as LFTP was presented by Kaushal et al. [10]. Various reasons such as weather conditions, routes condition, inflation rate etc., are the reasons of vagueness and ambiguity in the value of parameters. This leads to uncertainty in decision variables as well. Zadeh [20] established the concept of fuzzy sets. It is quite appropriate to assimilate the uncertainty while formulating the problem. Bellman and Zadeh [3] established the concept of fuzzy constraints, fuzzy goals and fuzzy decisions which built a structure for decision making in fuzzy environment. Intuitionistic fuzzy sets (IF) were presented by Attanassov [1]. The concept of acceptance degree, rejection degree and hesitancy degree work as a greater tool in dealing uncertainty. Kaur and Kumar [9] gave a solution methodology for the transportation problem (TP) which has transportation cost defined as generalised trapezoidal fuzzy number. Bharati [6] formulated TP under interval valued intuitionistic fuzzy (IVIF) environment. A new ranking function of IVIF set was presented with an illustration of example. The decision makers often find themselves in the situation where they have to find the transportation plan which must satisfy the multiple conflicting objectives. Bharati and singh [5] described multi objective linear programming problem using IVIF sets. A solution algorithm was proposed and implemented on a production planning problem. Maity et al. [13] considered multi objective transportation problem whose all parameters were defined as interval values. A time variant along with TP cost was considered for sustainable development. Some of the existing methodologies representing a brief perspective of TP in the literature is shown in Table 1. In this paper, we have formulated and solved the mathematical structure of FIFMOLFTP under intuitionistic fuzzy environment. In real life, the demand and supply of TP will not always be defined in crisp values. Hence, to construct and solve the problem, a resilient and effective tool is required to handle uncertainty. The proposed algorithm is impregnated with three different approaches namely, variable transformation, linear membership function and weighted goal programming. Section 2 describes some fundamental definitions and properties of TIFN. Section 3 of the paper represents the formulation of the problem by proposed methodology. Section 4 defines the algorithm of proposed methodology. Section 5 illustrates the numerical to validate the algorithm.

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S. Dhruv et al. Table 1. Literature review on transportation problem

Authors

Types of objective Decision variables Parameters Methodology Single Multiple Fractional Fuzzy IF Crisp Fuzzy IF Crisp

Joshi et al. [8]



×



×

× 

×

× 

Derived upper and lower bound of total transportation cost when supply and demand are varying

Singh et al. [17]



×

×

×

× 

×

× 

Based on the concept of NWCM, LCM, VAM and MODI

Singh et al. [15]



×

×

×

 ×

×

 ×

Based on the concept of NWCM, LCM, VAM and MODI

Lee et al. [11]

×



×

×

× 

×

× 

Goal programming approach

Wahed et al. [19]

×



×

×

× 

×

× 

Fuzzy programming approach

BasirZadeh et al. [4] 

×

×

×

× 



× ×

Ranking function and Parametric method

Sayed et al. [18]

×





×

 ×

×

 ×

Charnes and Cooper variable transformation technique, Accuracy function and Zimmerman’s approach

Maruti et al. [12]

×





×

× 

×

× 

Taylor’s series approach and Fuzzy programming approach

2

Preliminaries

Some fundamental ideas and definitions have been presented in this section regarding IFS and IFNs [1,15,16]. Definition 1. An IFS F˜ in S˜ be a set of ordered triplets such that ˜ , where δ ˜ (x), ψ ˜ (x) : S˜ → [0, 1] such that 0 ≤ F˜ = {(x, δF˜ (x), ψF˜ (x))|x ∈ S} F F ˜ δF˜ (x)+ψF˜ (x) ≤ 1 , ∀x ∈ S. The function δF˜ (x) denote the degree of membership and ψF˜ (x) denote the degree of non-membership of the element x ∈ S˜ of the set F˜ . The hesitancy degree can be defined as ω(x) = 1 − δF˜ − ψF˜ for x ∈ S˜ being in F˜ . Properties like ordering, arithmetic operations on TIFNs and relational operations forms the basis of this paper.

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Definition 2. Let A˜ = (t1 , t2 , t3 ; t1 , t2 , t3 ) be a triangular intuitionistic fuzzy number (TIFN) with the membership function δA˜ and non-membership function ψA˜ defined as: ⎧ ⎧ t −x  2 x−t1 ⎪ ⎪ ⎨ t2 −t1 t1 < x ≤ t2 ⎨ t2 −t1 t1 < x ≤ t2 −x 2 = tt33−t = tx−t δA(x) t2 ≤ x < t3 ψA(x) t2 ≤ x < t3 ˜ ˜  −t 2 ⎪ ⎪ ⎩ ⎩ 3 2 0 otherwise 1 otherwise where t1 ≤ t1 ≤ t2 ≤ t3 ≤ t3 .

3

Problem Formulation

A FIFMOFTP is considered in which all the IFNs are triangular. Suppose there are ‘p’ sources and ‘q’ destinations. The optimization model of FIFMOFTP is defined as follows: p q ˜ij(a) ⊗ x ˜ij ⊕ α ˜ (a) (˜ x ) P i=1 j=1 c a ˜ a (˜ = p q x) = (P1) Min M Qa (˜ x) g˜ij(a) ⊗ x ˜ij ⊕ β˜(a) i=1

subject to

q  j=1 p 

j=1

x ˜ij  a ˜i ,

i = 1, 2, ...p

(Supply Constraints)

x ˜ij ˜bj ,

j = 1, 2, ...q

(Demand Constraints)

i=1

x ˜ij 0 2 3 where c˜ij(a) = (c1ij(a) , c2ij(a) , c3ij(a) ; c1 ij(a) , cij(a) , cij(a) ) is a TIFN denoting the cost per unit of quantity transporting from ith source to j th destination, 1 2 3 1 2 3 , gij(a) , gij(a) ; gij(a) , gij(a) , gij(a) ) is a TIFN denoting the profit g˜ij(a) = (gij(a) earned from one unit of good transporting from ith source to j th destination, 2 1 x ˜ij = (x1ij , x2ij , x3ij ; x1 ij , xij , xij ) is the decision variable i.e. the quantity to be decided to transport from ith th 1 2 3 1 2 3 ˜ source to j destination, α ˜ (a) = (α(a) , α(a) , α(a) ; α(a) , α(a) , α(a) ) & β(a) = 1 2 3 1 2 3 , β(a) , β(a) ; β(a) , β(a) , β(a) ) are the fixed TIFN cost and profit respectively (β(a) 2 3 th a ˜i = (a1i , a2i a3i ; a1 source, ˜bj = i , ai , ai ) be the available TIFN supply at i 1 2 3 1 2 3 th (bj , bj , bj ; bj , bj , bj ) be the available TIFN demand at j destination. Assumptions:

1. 2. 3. 4.

Qa (˜ x) > 0, a = 1, 2, ...N , for all the points of feasible region. a ˜i > 0 ∀i, ˜bj > 0 ∀j.   The feasibility condition: i ai ≥ j bj . Let the set of all feasible solution of (P1) be represented as S.

Definition 3. If the total intuitionistic demand equals to the total intup fuzzy q itionistic fuzzy supply i.e., i=1 a ˜i = j=1 ˜bj , then FIFMOFTP is known as balanced transportation problem otherwise it is unbalanced.

544

S. Dhruv et al.

Substituting the TIFN values of cost, supply, demand and decision variables and after performing the arithmetic operations and ordering, (P1) transforms to (P2) as follows:   ˜ a (˜ (P2) Min M x) = (Ma1 , Ma2 , Ma3 ; Ma1 , Ma2 , Ma3 ), q

subject to



x1ij  a1i ,

q



x2ij  a2i ,

q



j=1

j=1

j=1

p 

p 

p 

i=1

x1ij  b1i ,

i=1

x2ij  b2i ,

i=1

x3ij  a3i ,

x3ij  b3i ,

q



a = 1, 2, .., N 1 x1 ij  ai ,

j=1 p  i=1

1 x1 ij  bi ,

q 

3 x3 ij  ai

j=1 p 

x,3ij  b3 i

i=1

1 1 2 1 3 2 3 3 x1 ij ≥ 0, xij − xij ≥ 0, xij − xij ≥ 0, xij − xij ≥ 0, xij − xij ≥ 0 i = 1, .., p, j = 1, .., q p q 1 1 1 i=1 j=1 cij(a) xij + α(a) where Ma1 = p q 3 3 3 i=1 j=1 gij(a) xij + β(a)

Goal Programming Formulation:   , fa2 , fa3 ) be the assigned fuzzy aspiration level/ goal Let f˜a = (fa1 , fa2 , fa3 ; fa1 th ˜ a (˜ x). Since the objective function has to be minof the a objective function M ˜ a (˜ x)  f˜a and the objective will become to imized therefore, we will assume M minimize the over deviational variable. Now, to compute the value of fuzzy goals, we will consider each component of f˜a and correspondingly solve the model by ˜ a (˜ x) of (P2) along with the set of considering each component of objective M constraints of model (P2). Hence, the value of fuzzy goal will be average of maximum and minimum values of the objective. The model (P2) can be formulated as: 2 3 (F1) Find x1ij , x2ij , x3ij ; x1 ij , xij , xij , i = 1, 2, .., p, j = 1, 2, .., q ˜ a (˜ such that M x)  f˜a , a=1,2,..,N subject to all constraints of (P2)

Describing Membership Function: Now we will define the linear membership function for the fuzzy goal constraints defined in the model (F1). The restriction in the form of aspiration level specifies the shape of the membership function. The membership function is defined as follows: ⎧ ˜ ˜l ⎪ ⎪ ⎨0˜ ˜ l Ma ≤ Ma ˜ al ≤ M ˜ a ≤ f˜a ˜ a ) = M a −M a M δ˜a (M ˜l f˜a −M ⎪ a ⎪ ⎩1 ˜ a ≥ f˜a M ˜ al is the lower value of the objective function M ˜ a and f˜a is the fuzzy where M ˜ aspiration level of the objective Ma . The maximum value of membership function can be 1 which denotes the complete satisfaction of the DM. Thus, after introducing the variables va− and va+ ,

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describing the under-deviation and over- deviation respectively from the desired goals, the fuzzified inequalities of the model (F1) can be restated as: μ ˜a + va− − va+ = 1

i.e.

˜a − M ˜ al M + va− − va+ = 1 ˜l f˜a − M a

− + − + − + − + − + also va1 va1 = 0, va2 va2 = 0, va3 va3 = 0, va1 va1 = 0, va3 va3 = 0

(1)

− − − − − + + + + + , va2 , va3 , va1 , va3 , va1 , va2 , va3 , va1 , va3 ≥ 0 va1

Since all the objectives may or may be not be of equal importance, therefore, there arises a need of assigning the weights or priorities to the goals of the objective as per their importance to the DM. Let wa (a = 1, 2, .., N ) be the N weights in the normalized form i.e. a=1 wa = 1. Hence, the model (F1) is redefined as : (F2)

M in

N  a=1

subject to

+ + + + + wa (va1 + va2 + va3 + va1 + va3 )

l l Ma1 − Ma1 Ma2 − Ma2 − + − + + va1 − va1 = 1, + va2 − va2 =1 l l fa1 − Ma1 fa2 − Ma2

l  l − Ma1 Ma1 Ma3 − Ma3 − + − + + v − v = 1, + va1 − va1 =1 a3 a3 l  − M l fa3 − Ma3 fa1 a1

 l − Ma3 Ma3 − + + va3 − va3 =1  − M l fa3 a3

and set of constraints (2) along with the set of constraints of model (P2). Linearization Technique: Consider the equation (1) for linearization and the rest of the equations will be done on the same lines. After substituting the value of Ma1 , the expression (1) will be rewritten as: ⎛ ⎝ ⎛ ⎝

p  q  i=1 j=1 p  q  i=1 j=1

⎞ c1ij(a) x1ij

+

α1(a) ⎠

⎛ +

⎞

− va1

⎝

p  q  i=1 j=1

⎞ 3 gij(a) x3ij

⎛

3 3 ⎠ l gij(a) x3ij + β(a) (fa1 − Ma1 )=⎝

p  q  i=1 j=1

+

3 ⎠ β(a) (fa1

+ l − Ma1 ) − va1

⎞ 3 3 ⎠ l l gij(a) x3ij + β(a) (fa1 − Ma1 ) + Ma1

after substitution the equation is reduced as ⎛ ⎞ p  q  − + 1 ⎠ ⎝ + Va1 c1ij(a) x1ij + α(a) − Va1 = A1

(2)

i=1 j=1

+ The objective of the model (F2) includes the non-linear term. Since va1 is an + over-deviational variable which means that va1 = 1 is interpreted as full achieve+ + = 0 is interpreted as zero achievement. Therefore, 0 ≤ va1 ≤ 1 ment and va1

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Destination (b2 )

Supply

c11 = (1, 1, 2; 1, 1, 2) d11 = (1, 1, 1; 1, 1, 2) α0 = (0, 0, 0; 0, 0, 0) Origin (a) c21 = (1, 2, 4; 0, 2, 4) d21 = (1, 2, 3; 0, 2, 4) α1 = (1, 3, 4; 1, 3, 10)

c12 = (1, 2, 2; 0, 2, 2) d12 = (2, 2, 2; 1, 2, 3) β0 = (1, 2, 3; 1, 2, 4) c22 = (3, 5, 6; 1, 5, 8) (10,20,30;10,20,40) d22 = (5, 7, 10; 5, 7, 11) β1 = (2, 3, 4; 2, 3, 5)

Demand

(0,10,15;0,10,18)

(5,10,11;4,10,20)

and it is reduced as (3). Hence, the model (F2) is equivalent to (F3) + 0 ≤ Va1 −

p  q  i=1 j=1

(F3)

M in

N  a=1

+ wa (Va1

3 3 gij(a) x3ij ≤ β(a)

+

(3) + Va2

+

+ Va3

+

+ Va1

+

+ Va3 )

subject to the set of constraints (2), (3), (1) and constraint set of model (P3).

4

Proposed Algorithm

This section exhibits the methodology explained above: Step1. Consider the problem (P1)with variables in the model as TIFNs. Apply fundamental arithmetic operations and ordering on TIFNs to transmute the model (P1) into the model (P2). Step2. Calculate the aspiration level f˜a , a=1,2,...,N for each objective function and thus, formulate the goal programming model (F1). Step3. Formulate the constraints of the above obtained goal programming model by using membership function as described by the equation (1) and (2). Step4. Apply the linearisation procedure and variable transformation to formulate the weighted goal programming model(F3). Step5. Solve the obtained crisp linear programming model (F3) using any of the available software packages to find the optimal solution. Step6. Solution of the crisp linear model is substituted in model (P1) to derive the efficient solution of the given (P1).

5

Numerical Illustration

To validate the proposed algorithm, consider the fully intuitionistic fuzzy multiobjective fractional transportation problem. The demand and supply matrix has been defined in Table 2.

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The optimization model is solved using the software “LINGO - 19.0” for different weights. Table 3 presents value of decision variables and the objective functions. Table 3. Value of objective functions and decision variables for different weights Weights

6

Decision variables

Z1

Z2

w1 = 0 w2 = 1 x = (5,12,12;4,10, (0.147, 0.93, 5.82; 20) 0.04, 0.93, 11.45) y = (1.63, 8,15; 1.63, 8,18)

(0.057, 0.80, 9.34; 0.09, 0.80, 22.96871)

w1 = 1 w2 = 0 x = (5,10,11; (0.15, 0.929, 5.62; 4.037,10, 21.29982) 0.04, 0.929, 11.45) y=(1.6, 8.28,15; 1.6,8.28,18)

(0.05, 0.795, 9.2; 0.009, 0.795, 23.4)

otherwise

(0.063, 0.8, 7.9; 0.006, 0.795, 37.46)

x = (5,10,12; (0.15, 0.933, 5.39 ; 4.78,10, 20) 0.04, 0.93, 11.45) y = (2.65, 8.285,18; 0.84, 8.285,18)

Limitations

The proposed work has the following restrictions: – If the size of the problem increases, it will be difficult to handle. – If the constraints of the problem are non-linear, then it will become incommodious to operate.

7

Conclusions and Prospective Work

The aim of this study is to bring forward a structured and cost-effective approach to procure the solution of FIFMOLFTP in which all the parameters and decision variables are triangular intuitionistic fuzzy number. This model is constructed under fuzzy environment which is more effective and reliable tool for dealing with uncertain environment. In the proposed methodology, (FIFMOLFTP) is converted to (MOLFTP) which is its equivalent crisp problem. In order to solve MOLFTP, it is further transmuted to an equivalent linear programming problem (LPP) model by weighted goal programming approach. The computation of the obtained model is performed on easily available software, LINGO - 19.0. The novelty in the present methodology is that it combines three different approaches, that is, weighted goal programming, variable transformation and linear membership function. A numerical illustration with single origin and two destinations is described to support our proposed methodology. Proposed methodology has the scope for future research in the following areas:

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– Membership functions can be defined as exponential, hyperbolic, parabolic. – The value of parameters and decision variables can be defined under interval value intuitionistic fuzzy environment. Acknowledgement. The authors extend their gratitude to the Editors and the anonymous referees for their valuable suggestions and comments which helped us to improve the quality of this paper.

References 1. Atanassov, K.: Intuitionistic fuzzy sets. Int. J. Bio Autom. 20, 1 (2016) 2. Akram, M., Shah, U.M.S., Al-Shamiri, A.M.M., Edalatpanah, A.S.: Fractional transportation problem under interval-valued Fermatean fuzzy sets. AIMS Math. 7(9), 17327–17348 (2022) 3. Bellman, R.E., Zadeh, A.L.: Decision-making in a fuzzy environment. Manag. Sci. 17(4), B-141 (1970) 4. Basirzadeh, H.: An approach for solving fuzzy transportation problem. Appl. Math. Sci. 5(32), 1549–1566 (2011) 5. Bharati, S.K., Singh, S.R.: Solution of multiobjective linear programming problems in interval-valued intuitionistic fuzzy environment. Soft. Comput. 23, 77–84 (2019) 6. Bharati, S.K.: Transportation problem with interval-valued intuitionistic fuzzy sets: impact of a new ranking. Progr. Artif. Intell. 10(2), 129–145 (2021). https:// doi.org/10.1007/s13748-020-00228-w 7. Hitchcock, F.L.: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20(1–4), 224–230 (1941) 8. Joshi, V.D., Gupta, N.: Linear fractional transportation problem with varying demand and supply. Matematiche (Catania) 66(2), 3–12 (2011) 9. Kaur, A., Kumar, A.: A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 12(3), 1201–1213 (2012) 10. Kaushal, B., Arora, R., Arora, S.: An aspect of bilevel fixed charge fractional transportation problem. Int. J. Appl. Comput. Math. 6(1), 1–19 (2020). https:// doi.org/10.1007/s40819-019-0755-3 11. Lee, S.M., Moore, J.L.: Optimizing transportation problems with multiple objectives. AIIE Trans. 5(4), 333–338 (1973) 12. Maruti, D.D.: A solution procedure to solve multi-objective fractional transportation problem. Int. Ref. J. Eng. Sci 7, 67–72 (2018) 13. Maity, G.R., Kumar, S., Verdegay, L.J.: Time variant multi-objective intervalvalued transportation problem in sustainable development. Sustainability 11(21), 6161 (2019) 14. Swarup, K.: Transportation technique in linear fractional functional programming. J. Roy. Naval Sci. Serv. 21(5), 256–260 (1966) 15. Singh, S.K., Yadav, S.P.: Efficient approach for solving type-1 intuitionistic fuzzy transportation problem. Int. J. Syst. Assur. Eng. Manag. 6, 259–267 (2015) 16. Singh, S.K., Yadav, S.P.: A new approach for solving intuitionistic fuzzy transportation problem of type-2. Ann. Oper. Res. 243, 349–363 (2016) 17. Singh, S.K., Yadav, S.P.: Fuzzy programming approach for solving intuitionistic fuzzy linear fractional programming problem. Int. J. Fuzzy Syst. 18, 263–269 (2016)

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18. Sayed, E.M.A., Abo-Sinna, A.M.: A novel approach for fully intuitionistic fuzzy multi-objective fractional transportation problem. Alexandria Eng. J. 60(1), 1447– 1463 (2021) 19. Wahed, A.E., Waiel, F.: A multi-objective transportation problem under fuzziness. Fuzzy Sets Syst. 117(1), 27–33 (2001) 20. Zadeh, L.A.: Fuzzy sets. Inform. Control 8, 338–353 (1965)

Application of the Crow Search Algorithm for Dynamic Route Optimization Hubert Zarzycki(B) General Tadeusz Kosciuszko Military Academy of Land Forces in Wroclaw, Wrocław, Poland [email protected]

Abstract. Dynamic route optimization enables frequent updates and optimization of calculated routes while taking into account additional exclusion conditions. In the optimization of tasks related to the issues of dynamic route optimization, algorithms inspired by natural phenomena are widely used. Biologically inspired metaheuristics are characterized by the continuous development of methods based on analogies observed in the functioning of biological life forms. Swarm intelligence approaches, in which the source of the method are the collective behavior of living species occurring in nature, are among the most commonly used algorithms in this category. The aim of this paper is to determine the validity of the CSA algorithm in solving the hypothetical problem of dynamic route optimization. Keywords: Crow search algorithm · CSA · dynamic rote optimization · swarm intelligence · nature inspired algorithms

1 Introduction Over the last centuries, humanity has been observing the way nature works with increasing attention. It is known that, thanks to evolution, nature has developed some effective mechanisms for solving complex problems. The natural world around us is full of empirical data, solutions and even algorithms. Modern nature-inspired algorithms are computational methods that mimic the behavior and principles found in natural systems, such as biological evolution, swarm intelligence, and neural networks. These algorithms are used to solve complex problems in various fields, including optimization, data analysis, and machine learning. The use of nature-inspired algorithms has several advantages over traditional approaches. For example, they can handle large, complex problems that are difficult to solve with traditional methods. They are also adaptable and flexible, as they can be easily modified to suit different problem domains. Additionally, nature-inspired algorithms can be used to find optimal solutions in a relatively short amount of time. This is because they use parallel computing, which means that multiple solutions can be evaluated simultaneously. An example of nature-inspired algorithms is evolutionary computation, which simulates the process of natural selection to find the best solution to a given problem. It works by creating a population of possible solutions, and then evolving this population © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 550–557, 2023. https://doi.org/10.1007/978-3-031-39774-5_61

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over generations by selecting the fittest individuals and breeding them to create new solutions. This process continues until the best solution is found. Neural networks are another type of nature-inspired algorithm that mimic the structure and function of the human brain. These networks can learn from data and recognize patterns, making them useful in tasks like image recognition, speech recognition, and natural language processing. A very popular group of nature-inspired algorithms is swarm intelligence, which is based on the collective behavior of social organisms like ants [11], bees [1], bats [7], crows [5], birds [4] and even flowers [12]. These algorithms use simple rules and interactions to create complex behaviors that can be used to solve problems like routing, scheduling, and optimization. This article describes the validation of the CSA algorithm for the dynamic routing optimization problem. For this purpose, an original approach to solving the problem using the C# language was used. This paper first briefly introduces swarm intelligence (AI) methods. Then the dynamic route optimization problem is presented. The Crow Search Algorithm is described in more detail and it is demonstrated how to implement CSA to a given problem. Then, exemplary results were presented and discussed. At the end, there are short conclusions on the CSA algorithm and SI methods. 1.1 Swarm Intelligence Swarm intelligence is a type of nature-inspired algorithm that is based on the collective behavior of social organisms, such as ants, bees, fish, and birds. These organisms are able to work together in large groups to achieve complex tasks, despite having limited individual capabilities. Swarm intelligence algorithms apply this concept to solve complex problems by creating a system of individual agents that interact with each other to achieve a common goal [3]. Swarm intelligence algorithms work by dividing a problem into smaller tasks that can be performed by individual agents, or “particles”. These particles interact with each other through a set of rules that define their behavior and how they communicate with each other. This communication can be achieved through simple mechanisms such as signaling, movement, or sensing. One of the most common swarm intelligence algorithms is the ant colony optimization algorithm, which is inspired by the behavior of ants. Ants are able to find the shortest path between their nest and a food source by leaving a pheromone trail that other ants can follow. Similarly, the ant colony optimization algorithm works by simulating the foraging behavior of ants to find the best solution to a given problem. It does this by creating a population of possible solutions and then allowing the ants to explore these solutions, leaving pheromone trails behind them. Over time, the shortest path to the solution becomes the most heavily marked, and the algorithm converges to the best solution. Another example of a swarm intelligence algorithm is particle swarm optimization, which is inspired by the movement of flocks of birds. Birds are able to fly together in coordinated groups, which allows them to cover long distances and find food more efficiently. Similarly, particle swarm optimization works by creating a population of

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particles that move around the search space, following the movement of the best particle in the swarm. The author of this study has significant experience with artificial intelligence algorithms. In particular, nature-inspired algorithms [9–11], optimization methods, control systems, fuzzy numbers[7, 8]. This article will analyze the application of the Crow Search Algorithm to the dynamic route optimization problem.

2 Crow Search Algorithm The Crow Search Algorithm (CSA) is a relatively new swarm intelligence optimization algorithm that is based on the behavior of crows. It was first proposed in 2016 by Askarzadeh [2], and has been used to solve a wide range of optimization problems in various fields. The CSA is particularly useful for solving multi-objective problems, where there are multiple conflicting objectives that need to be optimized simultaneously. The CSA works by simulating the behavior of crows, which are social animals that live in groups called “murders”. Crows are able to find food by searching for it individually, and then communicating its location to the rest of the group. This allows them to work together to find food more efficiently. The CSA starts by initializing a population of crows, where each crow represents a potential solution to the optimization problem. The population is divided into several sub-populations, or “murders”, each with its own leader crow. The leaders are responsible for communicating with the other crows in their murder and sharing information about their solutions. The CSA uses three main operators: the crow movement operator, the crow encounter operator, and the crow calling operator. These operators allow the crows to search the solution space and communicate with each other to find better solutions. The crow movement operator simulates the movement of crows in search of food. It updates the position of each crow based on its current position, the position of the best crow in its murder, and the position of the global best crow. The formula for updating the position of a crow i at iteration t is given by:   (1) xi (t + 1) = xi (t) + r1 (t) ∗ (xbest (t) − xi (t)) + r2 (t) ∗ xglobal_best (t) − xi (t) where xi (t) is the position of crow i at iteration t, xbest (t) is the position of the best crow in the same murder as i at iteration t, xglobal_best (t) is the position of the global best crow at iteration t, and r1 (t) and r2 (t) are random numbers between 0 and 1 that control the movement of the crow. The crow encounter operator simulates the interaction between crows when they encounter each other in search of food. It compares the fitness of two crows and updates their positions based on their fitness values. The formula for updating the position of a crow i after encountering crow j at iteration t is given by:   (2) xi (t + 1) = xj (t) + r3 (t) ∗ xi (t) − xj (t) where r3 (t) is a random number between 0 and 1 that controls the interaction between the crows.

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The crow calling operator simulates the communication between crows in a murder. It allows the leader crow to share its solution with the other crows in its murder, which can then use this information to update their own positions. The formula for updating the position of a crow i in the same murder as the leader crow k at iteration t is given by: xi (t + 1) = xi (t) + r4 (t) ∗ (xk (t) − xi (t))

(3)

where xk (t) is the position of the leader crow k at iteration t, and r4 (t) is a random number between 0 and 1 that controls the communication between the crows. The CSA continues to iterate through these three operators until a stopping criterion is met, such as a maximum number of iterations or a minimum change in the fitness value of the population. The CSA has several advantages over other optimization algorithms. It is able to handle both continuous and discrete optimization problems, as well as multi-objective problems with multiple conflicting objectives.

3 Dynamic Route Optimization Dynamic route optimization is the process of finding the best route for a vehicle or a fleet of vehicles in real-time based on current traffic conditions and other relevant factors [13]. It is commonly used in transportation and logistics industries to optimize delivery routes, reduce transportation costs, and improve overall efficiency. The key idea behind dynamic route optimization is to use real-time data to make route adjustments on the fly. This data can include traffic congestion, road closures, weather conditions, and other relevant factors. By continuously updating the route based on this data, vehicles can avoid traffic jams and delays, and arrive at their destinations faster and more efficiently. There are several approaches to dynamic route optimization. One common approach is to use algorithms that are based on heuristics, such as the nearest neighbor algorithm or the genetic algorithm. These algorithms work by iteratively improving the route based on a set of predefined rules, such as minimizing the total distance traveled or minimizing the number of stops. Another approach is to use machine learning algorithms to predict traffic conditions and make route adjustments based on those predictions. These algorithms use historical traffic data and other relevant factors to make predictions about future traffic conditions, and then adjust the route accordingly. Dynamic route optimization has several benefits for businesses that rely on transportation and logistics. By reducing travel time and fuel costs, it can lead to significant cost savings. It can also improve customer satisfaction by ensuring timely deliveries and reducing the likelihood of missed appointments or deadlines. Additionally, it can help reduce carbon emissions by optimizing routes to minimize fuel consumption. However, there are also some challenges to implementing dynamic route optimization. One of the biggest challenges is obtaining and processing real-time data about traffic and other relevant factors. This requires sophisticated data collection and processing systems, as well as advanced analytics tools to make sense of the data. Additionally, dynamic route optimization requires a high degree of automation and coordination between vehicles, which can be difficult to achieve in practice.

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Despite these challenges, dynamic route optimization is becoming increasingly important as businesses seek to improve their transportation and logistics operations. By using real-time data and advanced algorithms, businesses can optimize their routes and improve their bottom line, while also reducing their environmental impact.

4 Solving Dynamic Route Optimization Problem with CSA Algorithm Dynamic route optimization is a complex problem that involves finding the most efficient route for a vehicle or a fleet of vehicles based on real-time traffic conditions and other relevant factors. Crow Search Algorithm (CSA) is a metaheuristic algorithm that can be used to solve dynamic route optimization problems. The basic idea behind CSA is to model the search process after the behavior of crows in nature. Crows are known for their ability to find food and resources by sharing information with other crows in their flock. Similarly, CSA uses a population-based approach where multiple candidate solutions (crows) are generated and then evaluated based on their fitness (ability to solve the problem). To use CSA for dynamic route optimization, we need to first define the problem in terms of an objective function that we want to optimize. For example, we may want to minimize the total distance traveled by a fleet of vehicles or minimize the total time taken to complete all deliveries. Next, we need to define the decision variables and constraints that govern the problem. This may include factors such as the location of delivery points, the number of vehicles available, and the capacity of each vehicle. Once the problem is defined, we can use CSA to generate a set of candidate solutions that satisfy the constraints and evaluate their fitness based on the objective function. We can then update the solutions by incorporating real-time data about traffic conditions and other relevant factors. For example, if we receive information about a road closure or heavy traffic, we can update the solutions to avoid that road or route vehicles through less congested areas. This process can be repeated continuously in real-time to optimize the routes for the fleet of vehicles. To solve dynamic route optimization problems with CSA, we need to define the problem in terms of an objective function, decision variables, and constraints. We can then use CSA to generate candidate solutions and update them based on real-time data to optimize the routes for the fleet of vehicles. Below Crow Search Algorithm steps for solving the Dynamic Vehicle Routing Problem (Fig. 1): The C# language and the Microsoft Visual Studio 2019 environment were used to create a practical solution and conduct experiments.

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1. Initialization: - Set parameters and initialize the population of crows - Evaluate the fitness of each crow

2. Searching - Identify the best crow in the population - Determine the local search radius and perform a local search around the best crow Update the position and fitness of each crow 3. Store the position and fitness of the best crow so far

4. Routing - Use the best crow's position to generate a feasible solution to the dynamic vehicle routing problem - Evaluate the fitness of the solution

5. Update - Update the population of crows with the new solution sity and prevent premature convergence - Perform mutation and crossover operations on the population to introduce diversity and prevent premature convergence - Evaluate the fitness of each new crow in the population

6. Is convergence or a maximum number of iterations reached?

7. Print the best solution found

Fig. 1. Diagram presenting the implementation of the CSA algorithm to the dynamic routing optimization problem

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5 Analysis of the Results During the study of the CSA algorithm for the problem of dynamic routing, hundreds of simulations were performed. The Table 1 below presents a simple set of exemplary results. For all presented simulations, the following parameters were used: number of nodes 15, starting node 1 ending node 15, traffic data: moderate. The data is summarized for the number of vehicles used equal to 2. Table 1. Table captions should be placed above the tables. Simulation number

Number of agents (crows)

Number of iterations

Route length

Computational time [ms]

1

20

100

31,24

52

2

30

100

32,74

117

3

40

100

30,28

230

4

20

200

29,44

127

5

30

200

30,56

224

6

40

200

30,61

441

7

20

400

29,15

234

8

30

400

30,31

429

9

40

400

30,63

820

In order to test the correctness of the application of the CSA algorithm to the problem of dynamic routing, many data sets were generated for various input parameters. The results obtained are correct and close to or equal to the optimal results. The effect of changing the number of iterations above 100 on the results is small. An increase in the number of agents (crows) above 20 also slightly affects the results. Changing these parameters affects the working time of the algorithm. Optimal adjustment of parameters means that the calculation time and results can be close to optimal at the same time. The collected results confirm the usefulness and effectiveness of the tested algorithm.

6 Conclusions The CSA algorithm compares well with other swarm intelligence algorithms in terms of solution quality and convergence speed. The results indicate that effective solutions can be obtained by applying CSA to dynamic routing optimization problems. Crow Search Algorithm as one of swarm intelligence algorithms have also several advantages over traditional algorithms. It is able to handle complex, dynamic, and uncertain problems that are difficult to solve with traditional methods. The CSA is able to provide robust and flexible solutions that can adapt to changing conditions. Additionally, CSA as other swarm intelligence algorithms are able to work in parallel, which makes them faster and more efficient than traditional methods. In further works, the author

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intend to analyze the operation of CSA and other swarm intelligence algorithms to the problem of dynamic vehicle routing with time windows. In conclusion, CSA and other swarm intelligence methods are powerful tool for solving complex problems by mimicking the collective behavior of social organisms. By creating a system of individual agents that interact with each other, swarm intelligence algorithms are able to find optimal solutions to a wide range of challenges. As a result, they are an important area of research in computer science and related fields.

References 1. Akay, B., Karaboga, D.: Artificial bee colony algorithm for large scale problems and engineering design optimization. J. Intell. Manuf. 23(4), 1001–1014 (2012) 2. Askarzadeh, A.: A novel metaheuristic method for solving constrained engineering optimization problems: crow search algorithm. Comput. Struct. 169, 1–12 (2016) 3. Engelbrecht, A.P.: Fundamentals of Computational Swarm Intelligence. Wiley (2005) 4. Kaveh, A.: Particle swarm optimization. In: Advances in Metaheuristic Algorithms for Optimal Design of Structures, pp. 9–40. Springer, Cham (2014). https://doi.org/10.1007/978-3319-05549-7_2 5. Meraihi, Y., Gabis, A.B., Ramdane-Cherif, A., Acheli, D.: A comprehensive survey of crow search algorithm and its applications. Artif. Intell. Rev. 54(4), 2669–2716 (2020) 6. Yang, X.S., He, X.: Bat algorithm: literature review and applications. Int. J. Bio-Inspired Computat. 5(3), 141–149 (2013) 7. Zarzycki, H., Czerniak, J.M., Dobrosielski, W.T.: Detecting Nasdaq Composite Index Trends ´ ezak, D. with OFNs. In: Prokopowicz, P., Czerniak, J., Mikołajewski, D., Apiecionek, Ł, Sl¸ (eds.) Theory and Applications of Ordered Fuzzy Numbers. SFSC, vol. 356, pp. 195–205. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59614-3_11 8. Zarzycki, H., Dobrosielski, W.T., Vince, T., Apiecionek, Ł: Center of Circles Intersection, a new defuzzification method on fuzzy numbers. Bull. Pol. Acad. Sci. Tech. Sci. 68(2), 185–190 (2020) 9. Zarzycki, H., Ewald, D., Skubisz, O., Kardasz, P.: A comparative study of two nature-inspired algorithms for routing optimization. In: Atanassov, K.T., et al. Uncertainty and Imprecision in Decision Making and Decision Support: New Advances, Challenges, and Perspectives. IWIFSGN BOS/SOR 2020 2020. Lecture Notes in Networks and Systems, vol. 338. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-95929-6_17 10. Zarzycki, H., Skubisz, O.: An application of ant algorithm for routing optimization problem, International Business Information Management Association, 37 IBMIA conference. Cordoba, Spain (2021) 11. Zarzycki, H., Skubisz, O.: A New Artificial Bee Colony Algorithm Approach for the Vehicle Routing Problem. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds.) INFUS 2021. LNNS, vol. 307, pp. 562–569. Springer, Cham (2022). https://doi. org/10.1007/978-3-030-85626-7_66 12. Zhang, S., Gao, D.: Flower pollination algorithm based on dynamic adjustment and collaborative search. Comput. Eng. Appl. 55(24), 46–53 (2019) 13. Zhong, Y., Cole, M.H.: A vehicle routing problem with backhauls and time windows: a guided local search solution. Transp. Res. Part E: Logistics Transp. Rev. 41(2), 131–144 (2005)

Neuro-Fuzzy Modeling

PD-Type-2 Fuzzy Neural Network Based Control of a Super-Lift Luo Converter Designed for Sustainable Future Energy Applications Ahmet Gani(B) Kayseri University, Kayseri 38280, Turkey [email protected]

Abstract. In line with sustainable future energy applications, the present study aims to model and control a super-lift Luo power converter and to convert a low direct current input voltage to a high direct current output voltage. To this end, using Matlab/Simulink environment, a super-lift luo power converter and PD-type-2 fuzzy-neural network (PD-T2FNN) were designed. A PD-T2FNN was used to analyze the super-lift performance of luo power converter against different operational conditions. Different operational conditions were considered as ramp set-point reference voltage change, step set-point input voltage change and internal disturbance (load) change. Performances of the PD-T2FNN was evaluated in terms of settling time, overshoot, undershoot and recovery time. The simulation findings clearly demonstrated that PD-T2FNN displayed an effective performance in terms of its durability against the above-mentioned operational conditions. It can be thus concluded that the super-lift luo power converter offers a feasible system for intelligent and sustainable future energy applications such as fuel cell and photovoltaic systems with a very low input voltage. Keywords: PD-Type-2 Fuzzy-Neural Network · Super-Lift Luo Converter · Intelligence and Sustainable Future Energy

1 Introduction Sustainable future energy resources in different forms of application have been analyzed from a scholarly perspective over the past two decades [1]. Direct current (DC) power converters play a significant role in sustainable future energy applications, since they provide different power applications with different levels of DC voltages [2]. Generally, conventional DC power converters such as step-down or step-up are commonly used thanks to their simplicity and ease of calculation for circuit elements [3]. Nevertheless, they may also pose important problems such as a limited voltage transfer gain due to parasitic components, coil magnetic saturation and diodes and power switches. In addition, switching duty ratio in conventional power converters increases remarkably when the intended output voltage rises, thus causing a higher number of switching losses for the converter [4]. Luo power converters that possess a very high boosting ability as a result of the voltage lift and super lift technique can be employed to eliminate the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 561–568, 2023. https://doi.org/10.1007/978-3-031-39774-5_62

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drawbacks of conventional power converters [5, 6]. The super-lift Luo (SLL) power converter benefits from super-lift technique which enables a high output voltage gain in geometric progression for various sustainable future energy applications with low output voltage such as photovoltaic energy and fuel cell [7]. The SLL power converter minimizes current and voltage ripples and limits switching element losses, high output voltage and high power conversion efficiency to a lower rate compared to other conventional DCDC power converters [8]. However, its behavior is often nonlinear due to its inherent non-linear and non-minimum phase characteristics, since it displays a more complicated control when compared with minimum phase systems. For this reason, intelligent control methods based on fuzzy and neural-network are widely preferred in many nonlinear systems [9]. Particularly, the type-2 fuzzy neural network (T2FNN) control has attracted a huge amount of capacity due to its better characterization in dealing with uncertainties [10]. When the studies [11–16], on SLL power converter are investigated, it is seen that proportional + integral + derivative (PID), sliding-mode, backstepping and discrete time controllers have been used in output voltage control of SLL power converter and outcomes of these controllers have been reported in the current literature. Unlike the studies on SLL power converter from current literature, this paper proposes the type-2 fuzzy neural network (T2FNN) working parallel with proportional + derivative (PD) controller to eliminate the input parameter uncertainties and internal disturbances on control of super lift luo (SLL) power converter. Dynamic performance of the SLL power converter using the proposed controller was evaluated for different operational conditions (rampa and step set-point changes) with MATLAB/Simulink environment. The organization of this paper is as follows. In Sect. 2, the overall proposed control structure is described in detail. In Sect. 3, the detailed simulations are conducted so as to evaluate and validate the dynamic performance of the proposed control structure. Finally, conclusions and future studies are given in Sect. 4.

2 Overall Proposed Control Structure The elementary SLL power converter includes a switching element (S), two freewheeling diodes, three energy storage elements ( L, C 1 and C 2 ), load (R) and input voltage source (Vin ).To model elementary SLL converter, it is supposed that elementary SLL power converter operates in continuous conduction mode (CCM) and all circuit elements are ideal. The selected elementary SLL power converter element values for simulation studies are given in Table 1. The overall proposed control structure that is depicted in Fig. 1 and the PD controller is used in parallel to T2FNN for elementary SLL power converter. As clearly depicted in Fig. 1, The T2FNN control structure consists of two inputs, error (e) and change in error (de), and one output, uT2FNN . Similarly, PD controller has two inputs error (e) and change in error (de), and one output, uPD . Also,the measured output voltage of elementary SLL converter is compared with a reference voltage so as to compute the value of error input (e). After that, a derivative block is applied to compute the value of change of error input (de). To minimize the number of T2FNN consequent parameters, zero-order Takagi–Sugeno Kang (ZOTSK) model is used in the inference mechanism of T2FNN [17]. The ijth rule of a ZOTSK A2-C0 fuzzy model

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Table 1. The selected elementary SLL power converter element values Parameter

Symbol

Value

Inductor

L

1mH

Capacitor

C1

22µF

Capacitor

C2

47µF

Load Resistance

R

50

Input Voltage

Vin

25V

Output Power

P0

450W

Output Voltage

V0

150V

Switching Frequency

f

50kHz

L

C1 +

C2

Vin

R

S

Vo

-

V

PWM

T2FNN e

u u

e

T2FNN

Σ

+

Π

u

+

PD

de

d dt

Vref

PD Controller

Fig. 1. Overall proposed control structure

with two input variables (e and de) and f ij ’s are the parameters of consequent part can be defined as follows:   Rij : IF e is M 1i and de M 2j , then uT 2FNN = fij i = 1, ..., I ; j = 1, ..., J

(1)

where I and J are the number of membership functions (MFs) for e and de, respectively,   and M 1i and M 2j are the type-2 fuzzy sets.For the first input (e), the upper and lower MFs are defined as μ1i and μ1i , , respectively. Similarly, for the second input (de), upper and lower MFs are represented as μ2j and μ2j , respectively. The firing strength (FS) of the Rij rule is computed as the T-norm operator of the MFs in the premise part (using a

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multiplication). W ij = μ1i (e)μ2j (de)

(2)

W ij = μ1i (e)μ2j (de)

(3)

 and W  ij in the After that, the normalized values of the upper and lower FS, W ij following forms:  = W ij

W ij I J 

(4)

W ij

i=1 j=1

 ij = W

W ij I J  i=1 j=1

(5)

W ij

After the normalization of (4–5), the output signal of the T2FNN can be obtained the following form: uT 2FNN = q

J I  

 ij + (1 − q) fij W

i=1 j=1

J I  

 W ij

(6)

i=1 j=1

Here q is the parameter that defines the sharing contribution of the lower and upper MFs. The q parameter is chosen 0.5 in this paper.The overall control input u is determined as follows: u = uPD − uT 2FNN

(7)

Here uPD and uT2FNN are the control inputs produced by the PD and the T2FNN controllers, respectively. In this paper, novel ellipsoidal type-2 fuzzy membership function (ET2FMF) is selected so as to improve control performance of PD-T2FNN [18–20].

3 Detailed Simulation Studies The detailed simulation studies have been developed in Matlab/Simulink environment to perform overall proposed control structure. Also, three different operational conditions are considered for the proposed controller. 3.1 Simulation Results 3.1.1 Ramp Set-Point Reference Voltage Change In this part, the dynamic performance of proposed controller is investigated with ramp set-point DC output reference (V0 ) change. After the set-point DC output reference

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200

V (V) 0

150

100

50

0

Case 3

Case 2

Case 1 0

0.5

1

1.5

Time(s)

Fig. 2. Simulation results for ramp set-point reference voltage change obtained from proposed controller

voltage reaches steady-state value, the reference command for DC output voltage is ramp changed from 150V to 160V between 0.5s and 1s. Total simulation time was 1.5 s and sampling time was taken as 0.1µs. Simulation results for ramp set-point reference voltage change obtained from proposed controller is given in Fig. 2. As clearly shown in Fig. 2, ramp set-point reference output voltage change of SLL power converter is nearly close ramp set-point reference output voltage which is changed from 150V to 160V values for case 2. Besides, proposed controller guarantees zero steady-state error for all cases. 3.1.2 Step Set-Point Input Voltage Change In this part, the dynamic performance of proposed controller is tested against to different step set-point input (Vi ) voltage values. For this purpose, Input voltage (Vi ) value is decreased from 25V to 23V at 0.25s. Total simulation time was 0.5s and sampling time 200

0

V (V)

150

100

50 Case 1 0

0

0.125

Case 2 0.25 Time(s)

0.375

0.5

Fig. 3. Simulation results for different input (Vi ) voltage values obtained from proposed controller

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was taken as 0.1µs. Simulation results for different input (Vi ) voltage values obtained from proposed controller is given in Fig. 3. As observed in Fig. 3, settling time of proposed controller is observed as 50ms for case 1. After the input voltage (Vin ) of SLL converter is reduced from 25V to 23V for case 2, the recovery time of proposed controller is calculated as 125ms. 3.1.3 Internal Disturbance (Load) Change In this part, the dynamic performance of proposed controller is also performed to analyze different load resistance (R0 ) values. For this aim, the load resistance of SLL converter is reduced from 50 to 25 at 0.25s. Total simulation time was 0.5s and sampling time was taken as 0.1µs. Simulation results for different load resistance (R0 ) values obtained from proposed controller is given in Fig. 4. 200

0

V (V)

150

100

50

0

0

0.125

0.25 Time(s)

0.375

0.5

Fig. 4. Simulation results for different load resistance (R0 ) values obtained from proposed controller

As shown in Fig. 4, after the load resistance (R0 ) of SLL converter is decreased from 50  to 25, the recovery time of proposed controller is calculated as 300ms. Moreover, overshoot and undershoot percentage values of proposed controller are calculated as 29.66% and 27.66%, respectively.

4 Conclusion This paper proposed a new control structure based on PD-T2FNN to improve the dynamic performance of a SLL converter under different operational conditions. Simulation results show that proposed controller not only provides satisfactory transient response but also gives robust rejection of disturbance inputs. In addition, proposed controller ensure nearly zero steady-state error during steady state for all different operational conditions. In the future studies, it is aimed to validate the dynamic performance of proposed controller via real-time implementation for many sustainable future energy applications such as solar and hydrogen.

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Acknowledgements. This work was financially supported by the Kayseri University, Scientific Research Projects Unit, the project entitled “Development of Advanced Fuzzy Logic Based Control Methods for High Gain DC/DC Converter” under Project No: FKB-2022–1087.

References 1. Ghasemi, F., Yazdani, M.R., Delshad, M.: Step-Up DC-DC switching converter with single switch and multi-outputs based on luo topology. IEEE Access 10, 16871–16882 (2022) 2. Chincholkar, S.H., Chan, C.Y.: Comparative study of current-mode controllers for the positive output elementary luo converter via state-space and frequency response approaches. IET Power Electron. 8(7), 1137–1145 (2015) 3. Kaliannan, P., Jyothi, N., Rangasamy, K.: Modeling and analysis of positive output luo converter in voltage control mode. Electrica 21(2), 259–271 (2021) 4. Kececioglu, O.F.: Design of type-2 fuzzy logic controller optimized with firefly algorithm for maximum power point tracking of photovoltaic system based on super lift luo converter. Int. J. Numer. Model. Electron. Networks Devices Fields 35(4), e2994 (2022) 5. Kececioglu, O.F.: Robust control of high gain DC-DC converter using type-2 fuzzy neural network controller for MPPT. J. Intell. Fuzzy Syst. 37(1), 941–951 (2019) 6. Liu, L., Li, D., Yao, L.: A family of non-isolated transformerless high step-Up DC-DC converters. Int. Trans. Electr. Energy Syst. 29(4), e2794 (2019) 7. Mahdavi, M., Shahriari-Kahkeshi, M., Abjadi, N.R.: An adaptive estimator-based sliding mode control scheme for uncertain POESLL converter. IEEE Trans. Aerosp. Electron. Syst. 55(6), 3551–3560 (2019) 8. Luo, F.L., Ye, H.: Positive output super-lift converters. IEEE Trans. Power Electron. 18(1), 105–113 (2003) 9. Hou, S., Chu, Y., Fei, J.: Robust intelligent control for a class of power-electronic converters using neuro-fuzzy learning mechanism. IEEE Trans. Power Electron. 36(8), 9441–9452 (2021) 10. Sharifian, A., Sasansara, S.F., Ghadi, M.J., Ghavidel, S., Li, L., Zhang, J.: Dynamic performance improvement of an ultra-lift luo DC–DC converter by using a type-2 fuzzy neural controller. Comput. Electr. Eng. 69, 171–182 (2018) 11. Glady, J.B.P., Manjunath, K.: Analysis-intelligent design and simulation of improved state– space modelling for control of luo converter. Comput. Electr. Eng. 75, 230–244 (2019) 12. Fallahzadeh, S.A., Abjadi, N.R., Kargar, A., Blaabjerg, F.: Nonlinear control for positive output super lift luo converter in stand alone photovoltaic system. Int. J. Eng. 33(2), 237–247 (2020) 13. Kumar, K.R., Jeevananthan, S.: Sliding mode control for current distribution control in paralleled positive output elementary super lift luo converters. J. Power Electr. 11(5), 639–654 (2011) 14. Jazi, H.N., Goudarzian, A., Pourbagher, R., Derakhshandeh, S.Y.: PI and PWM sliding mode control of POESLL converter. IEEE Trans. Aerosp. Electron. Syst. 53(5), 2167–2177 (2017) 15. Abjadi, N.R., Goudarzian, A.R., Arab Markadeh, G.R., Valipour, Z.: Reduced-order backstepping controller for POESLL DC–DC converter based on pulse width modulation. Iranian J. Sci. Technol. Trans. Electr. Eng. 43(2), 219–228 (2019) 16. Shenbagalakshmi, R., Vijayalakshmi, S.: Analysis of super lift Luo converter with discrete time controller. S¯adhan¯a 45(1), 1–5 (2020). https://doi.org/10.1007/s12046-020-1307-6 17. Acikgoz, H., Coteli, R., Dandil, B., Ata, F.: Experimental evaluation of dynamic performance of three-phase AC–DC PWM rectifier with PD-type-2 fuzzy neural network controller. IET Power Electronics 12(4), 693–702 (2019)

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18. Khanesar, M.A., Kayacan, E., Reyhanoglu, M., Kaynak, O.: Feedback error learning control of magnetic satellites using type-2 fuzzy neural networks with elliptic membership functions. IEEE Trans. Cybern. 45(4), 858–868 (2015) 19. Khanesar, M.A., Kayacan, E., Kaynak, O.: Optimal sliding mode type-2 TSK fuzzy control of a 2-DOF helicopter. In: 2015 IEEE international conference on fuzzy systems (FUZZ-IEEE), pp. 1–6. IEEE (2015) 20. Kayacan, E., Maslim, R.: Type-2 fuzzy logic trajectory tracking control of quadrotor VTOL aircraft with elliptic membership functions. IEEE/ASME Trans. Mechatron. 22(1), 339–348 (2016)

A Novel Fuzzy-Clustering-Based Deep Learning Approach for Spatio-Temporal Traffic Speed Prediction Jiyao An , Ju Fang(B)

, Xuan Zhang , and Qingqin Liu

College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China {jt_anbob,jufang}@hnu.edu.cn

Abstract. Traffic speed prediction problem is the key issue to recent intelligent transportation system, especially in urban environment. Traffic speed prediction is a challenging task duo to the complex spatial-temporal correlations between different urban road sections are uncertain and dynamic. Existing models usually use different components to capture temporal correlation and spatial correlation, respectively, ignoring the heterogeneity of spatio-temporal data in spatio-temporal dimension, while more complicated model also leads to higher computing complexity. A novel multi-view fusion spatio-temporal graph convolution network, called as MFGCN, is proposed to model various kinds of semantic correlations more comprehensively in the urban road network. In this model, a multi-graph model is constructed to represent various correlations, such as spatial correlation, temporal correlation, and dynamically generated synchronous correlation, and so on. Furthermore, the synchronous correlation is dynamically modelled as spatio-temporal heterogeneous correlation by utilizing the fuzzy C-means clustering approach, in which the number of clustering centers is adjusted according to traffic datasets size. Moreover, fuzzy correlation can also tackle the uncertain relationship between dynamically traffic road network, and all above correlations are fused with fuzzy correlation. Finally, some experiments are carried out on well-known real traffic datasets, in order to show the effectiveness and merit of the proposed model with good prediction performance. Keywords: Traffic Prediction · Deep Learning · Fuzzy Clustering · Graph Convolutional Networks · Multi-View Fusion

1 Introduction The main concern with the most recent intelligent transportation system is the prediction of traffic speed. Accurate traffic speed forecast can provide reference for people’s travel plans, thus reducing congestion. However, due to the complex temporal, spatial and semantic correlation between roads, traffic speed prediction is a difficult undertaking. Many researchers have done extensive research on traffic speed prediction, but there were still some shortcomings in these works. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 569–576, 2023. https://doi.org/10.1007/978-3-031-39774-5_63

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Firstly, some methods only consider the adjacency relationship between roads, ignoring other semantic factors that can measure the correlation between roads. Specifically, the traffic characteristics around two shopping center which are far apart, are very similar. During the weekend rush hour, these two places will encounter traffic jams at the same time. This prompts us to use multi-graph neural network to model the road network. Recent research reveals the potential of multi-graph neural network in improving prediction performance [1–3]. Secondly, in some recent research work, in order to get better prediction results, the model designed to be more complex, but it also required higher computer performance. If you want to predict a larger road network, it is worth studying to design a simpler and more effective prediction model. Based on the above limitations, in this work, our aim is to build a low-complexity model that can accurately predict the traffic speed in the future. The contributions of this paper are as follows: (1) We identify four kinds of semantic correlations among road segments in traffic speed forecast and encode them using multiple graphs. (2) We propose a novel multi-view fusion spatio-temporal graph convolution network (MFGCN) to model these semantic correlations more comprehensively in the urban network and to incorporate the global contextual information when modeling the temporal dependencies. (3) Experiments on two datasets from the real world prove the suggested method’s effectiveness and worth. The rest of this essay is arranged as follows. Section 2 contains an overview of the literature on traffic speed predictions. Our methodology is described in Sect. 3. Section 4 shows and summarizes the experiment’s results.

2 Related Research Forecasting traffic speeds has developed into a substantial and serious study topic in recent years. The importance of designing an outstanding performance and accurate traffic speed forecast model has risen. There was a vast body of work on forecasting traffic speeds. 2.1 Similarity of Temporal Sequence Similarity of temporal sequence in traffic speed forecast has been studied extensively by numerous researchers. In recent years, Dynamic time warping (DTW) [4] algorithm and Pearson correlation coefficient (PCC) have been widely used in spatio-temporal data prediction. SFTGNN [5] employed DTW to build a time chart that depicts the similarity of traffic flow nodes. T-MGCN [1] measured the similarity of historical traffic conditions between roads through DTW, and took it as the correlation between roads. HMSTGCN [2] used PCC to separate the functional correlation and time pattern correlation between road nodes. PCC was used in ATGCN [3] to quantify the similarity between two times patterns in order to find other hidden features. 2.2 Fuzzy Methods The collection of traffic speed data is hampered by significant external noise, and the deep learning model is not resilient in the face of questionable data. Fuzzy approaches are

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used in some studies to lessen the impact of data ambiguity on prediction findings [6]. To address the uncertainty in traffic data, the deep convolution network and fuzzy network was merged [7] created a gate network to cluster input data and establish fuzzy rules, and then trained the model using an expert network. Although fuzzy methods have been widely used in traffic data forecast, there were few models that use fuzzy C-means clustering to identify potential relationships among data. Fuzzy C-means clustering (FCM) is a kind of clustering, which was first proposed by Pal and Bezdek [8] and has been widely used to solve various classification and regression tasks [9]. FCM can clarify the internal relationship between distinct things more objectively by employing membership degree to define the fuzzy logical relationship between objects and grouping.

3 Methodology In this section, we formalize the learning problem of spatio-temporal traffic speed forecast and introduce our proposed model in detail. Our proposed MFGCN model is shown in Fig. 1. The model includes four modules: graph generation, time convolution, graph convolution and output module.

Fig. 1. The architecture of MFGCN.

3.1 Traffic Speed Forecast A road network is represented by the graph G = (V, E, W). Here, each edge eij in the collection of edges E reflects the correlation between the road segments vi and vj . The weighted matrix is denoted by W ∈ RN×N , where Wi,j stands for the correlation strength between vi and vj . We denote the traffic speed of ith segment at time step t as xi,t ∈ R and use Xt ∈ RN to represent all traffic speed observations at time step t. The mapping function f(·) must be learned based on the graph G and P past steps of traffic speed in

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order to predict the traffic speed of every node in the entire traffic network graph for the future Q steps,   (1) Y = f Xt−P+1:t ; GD 

3.2 Graph Generation Graph is the basis of graph convolution, so graph generation is a key step. The main task of the graph generation module is to effectively represent the spatial dependence between road sections with matrices. If the generated graph can’t encode the correlation between nodes well, it will not help model learning, and sometimes even reduce the prediction accuracy. Specifically, we establish four road graphs. Temporal Pattern Graph G D : If two road segments observe similar traffic speeds most of the time, then we have greater confidence to infer that their future traffic speeds will remain similar. Specifically, we use the daily average traffic speed of traffic speed as ti , then use DTW algorithm to generate temporal pattern graph. For a graph with N nodes, there are N temporal patterns. Suppose that these N temporal pattern sequences pattern sequence of the road segment vi . are {ti |i = 1,2, · · · N }, and ti is the temporal  Given ti = ti,1 , ti,2 , · · · ti,t , tj = tj,1 , tj,2 , · · · tj,t , dist(ti , tj ) is obtained by DTW   algorithm, and the temporal pattern graph is expressed as G T = V , E, AT , where ATij is defined as follows:  dist(ti , tj )vi = vj T (2) Aij = 0, vi = vj Fuzzy Graph G F : Roads with similar functions in urban areas usually have similar traffic patterns. Previous studies have proved that the POI (Point of Interest) distribution can measure the functions of urban areas. However, in many cases, it is not easy to obtain POI information. Informed by [10], we apply the fuzzy C-means clustering approach to find this association and simulate the number of POIs using the number of clustering centers. The fuzzy graph is represented by G F = (V , E, AF ). Assuming that the historical traffic speed sequence of all road segments is {xi |i = 1, 2, · · · N }, the cluster center of fuzzy C-means clustering is {ca |a = 1, 2, · · · m}, m is the number of clusters. xi is the historical traffic speed sequence of road segments vi , ca is the ath cluster center. Hence, Formula 3 defines the fuzzy matrix AFij element. AFij

=

 m a=1

  ui,a ∗uj,a , vi = vj 1, vi = vj

s.t. ui,a =

m k=1



d (xi , ca ) d (xi , ck )



−2 (α−1)

(3)

Aware Graph G A and Relevance Graph G R : The Wasserstein distance [11] can continuously transform one distribution into another, while maintaining the geometric characteristics of the distribution itself. Using the daily traffic speed data of each node, we express the dynamic spatial correlation between nodes by using the Wasserstein distance to capture the correlation attribute between each node’s probability distribution [12]. One-day traffic data is considered a vector for each node, and a group of multi-day

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traffic can be viewed as a vector sequence. For example, the vector sequence at node n is Xn = (wn1 , wn2 , . . . wnD ), wnd ∈ Rdt , and dt represents the number of records in a day, and D represents the number of days. pn is used to represent the probability distribution of the vector sequence of node n. As the cost function, we employ the cosine distance between traffic speed vectors. The element of aware graph matrix AAij is defined as follows:

γ (x, y)cost(wix , wjy )dxdy AAij = 1 − γ ∈ [pi ,pj ] inf

s.t.

x

y

wix 2 γ (x, y)dy = D , x=1 wix 2

wjy 2 γ (x, y)dx = D y=1 wjy 2

(4)

By changing all non-zero values in AAij to 1, we get the relevance graph matrix ARij . 3.3 Temporal Convolution Layer A road segment’s current traffic speed is connected with its past speed, and the associations change nonlinearly over time steps. In order to model these characteristics, in this paper, the time convolution module applies a set of standard extended convolution filters to extract high-level time features [13].   Hlt = tanh wf ,l  Hl−1  σ (wg,l  Hl−1 ) (5)

3.4 Multi-graph Fusion and Graph Diffusion Convolution Graph convolution involves sending messages to one-hop neighbors, which are then combined at each node to generate the following layer of embedding. The more potent and spatially confined alternate method of message transmission known as graph diffusion convolution makes use of the generalized variant of sparse graph diffusion. The correlation between roads is defined in the form of graphs, we use graph diffusion convolution (GDC) to convolution the road network to capture the interaction between nodes. We fuse multiple graphs by Formula 6 to get a fused graph, then input it into GDC. The propagation rules of GDC expressed by Formula 7. A= W 1 AF + W2 AT + W3 AA + W4 AR + b Hl =

K

(A)k Hlt W k

(6)

(7)

k=0

where A ∈ RN ×N , Wi , b are learnable parameters. Hlt represents output hidden states of temporal convolution layer in lth block, W k is parameters for depth k and K is max diffusion steps.

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4 Experiment We validate our model on actual highway traffic datasets PeMSD8 and PeMSD4 in order to assess its performance. PeMSD8 alludes to San Bernardino traffic information. The time span of PeMSD8 is from July to August in 2016, and the nodes is 170. PeMSD4 refers to the traffic data in San Francisco Bay Area. The time span of PeMSD4 is from January to February in 2018, and the nodes is 307. Both PeMSD8 and PeMSD4 were released in ASTGCN [14] after preprocessing. 4.1 Experimental Settings Metrics. In this study, Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE) are adopted to evaluate performance and merit of our model. Baselines. To validate performance of MFGCN, we compare our model with the following baselines: (1) ASTGCN [14]: It used two separate attention mechanisms to model spatial and temporal dependencies separately. (2) Graph WaveNet [15]: It combined graph convolution with 1D dilated casual convolution. (3) GMAN [16]: To represent the effect of the spatio-temporal elements on traffic conditions, a graph multi-attention network was used, which adapted an encoder-decoder architecture and both the encoder and the decoder consist of multiple spatio-temporal attention blocks. (4) DMSTGCN [13]: Dynamic and multi-faceted spatio-temporal deep learning, which proposed a novel framework to solve the traffic speed forecasting problem with multi-faceted data. Hyperparameters. To be fair, the data is divided into three parts in the same way as the baselines, i.e., 6:2:2 on PEMSD8 and PEMSD4, for training, verification and testing respectively. All the tests use an hour of history data, and the prediction horizons ranged from 5 min to an hour. On the validation set, the key hyperparameters are tweaked. The number of PEMSD8 and PEMSD4 clustering centers are calculated using the trial-anderror method.

4.2 Results and Analysis Tables 1 and 2 shows the comparison of different methods for 5 min, 15 min, 30 min and 1 h ahead predictions on datasets. As can be seen from Tables 1 and 2, apart from the latest DMSTGCN, our MFGCN has achieved the best performance in the short-term (5 min, 15 min and 30 min) prediction, and the performance is similar to that of DMSTGCN. DMSTGCN has achieved the best results in short-term prediction. On the one hand, it considers the spatial dependence among various features. On the other hand, it dynamically learns the spatial correlation and models it as a three-dimensional tensor instead of using a predefined adjacency matrix. However, MFGCN simplifies the model and, with fewer parameters and faster calculation, provides performance comparable to DMSTGCN. In addition, DMSTGCN needs additional auxiliary information, such as traffic flow. When there is no auxiliary information, the model can’t work well. GMAN is outstanding in long-term prediction, probably due to the transform attention layer, which alleviates the error propagation

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effect between different prediction time steps in long time range. However, GMAN uses node2vec algorithm to keep node structure information while implementing attention mechanism. When the connection information between nodes is not defined, these methods can’t effectively model spatio-temporal series data. MFGCN does not need to provide the connection information between nodes, it will learn from the spatio-temporal data. Table 1. MAPE of MFGCN and baselines on datasets. Method

PeMSD8

PeMSD4

5 min

15 min

30 min

1h

5 min

15 min

30 min

1h

ASTGCN

0.0234

0.0297

0.0358

0.0436

0.0225

0.0323

0.0425

0.0552

GMAN

0.0179

0.0234

0.0292

0.0355

0.0206

0.0279

0.0348

0.0416

GWNet

0.0139

0.0233

0.0319

0.0391

0.0165

0.0266

0.0356

0.0446

MTGNN

0.0146

0.0234

0.0314

0.0394

0.0173

0.0268

0.0348

0.0426

DMSTGCN

0.0132

0.0209

0.0280

0.0358

0.0161

0.0255

0.0333

0.0415

MFGCN

0.0131

0.0219

0.0294

0.0398

0.0162

0.0264

0.0348

0.0456

Table 2. RMSE of MFGCN and baselines on datasets. Method

PeMSD8

PeMSD4

5 min

15 min

30 min

1h

5 min

15 min

30 min

1h

ASTGCN

2.3076

2.9934

3.6194

4.2880

2.1089

3.1428

4.1062

5.1392

GMAN

1.8698

2.6752

3.3950

4.0524

2.0416

2.9803

3.7906

4.4791

GWNet

1.4435

2.5533

3.5327

4.2424

1.7050

2.8159

3.7640

4.5560

MTGNN

1.5000

2.5738

3.4913

4.2490

1.7548

2.8517

3.7523

4.4972

DMSTGCN

1.4217

2.4405

3.2861

4.0522

1.6929

2.7769

3.6339

4.3814

MFGCN

1.4258

2.5029

3.3861

4.2470

1.7124

2.8123

3.7738

4.5824

5 Conclusion In this study, we looked at the topic of predicting traffic speed and found its particular spatio-temporal correlations. We proposed a novel fuzzy-clustering-based deep learning model which encoded the non-Euclidean correlations among road segment using multiple graphs, and fused these graphs with learnable weights to integrate the important information in different graphs. Then we explicitly captured them using graph diffusion convolution. We further augmented the dilated convolution and gate mechanism to incorporate global information in the temporal layer. The efficacy and validity of the

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suggested strategies for forecasting short-term traffic speeds were proved by experiments on two real-world datasets. By finding more correlation, the complexity of the model can be somewhat lowered while still guaranteeing forecast accuracy. In our upcoming work, we intend to look into the following areas: (1) Think about ways to enhance the model’s representation of long-term spatio-temporal dependence in order to enhance the model’s ability to make long-term predictions. (2) Include a fuzzy inference system to make the model easier to comprehend.

References 1. Lv, M., Hong, Z., Chen, L., Chen, T., Zhu, T., Ji, S.: Temporal multi-graph convolution network for traffic flow prediction. IEEE Trans. Intell. Transp. Syst. 22(6), 3337–3348 (2020) 2. Shao, W., et al.: Long-term spatio-temporal forecasting via dynamic multiple-graph attention. arXiv preprint arXiv:2204.11008 (2022) 3. Wang, C., Zhu, Y., Zang, T., Liu, H., Yu, J.: Modeling inter-station relationships with attentive temporal graph convolutional network for air quality prediction. In: proceedings of the 14th ACM International Conference on Web Search and Data Mining, pp. 616–634 (2021) 4. Berndt, D.J., Clifford, J.: Using dynamic time warping to find patterns in times series. In: KDD workshop, pp. 359–370 (1994) 5. Li, M., Zhu, Z.: Spatial-temporal fusion graph neural networks for traffic flow forecasting. In: Proceedings for the AAAI Conference on Artificial Intelligence, vol. 35, pp. 4189–4196 (2021) 6. Chen, W., et al.: A novel fuzzy deep-learning approach to traffic flow prediction with uncertain spatial–temporal data features. Future Gener. Comput. Syst. 89, 78–88 (2018) 7. Hongbin Yin, S.C., Wong, J.X., Wong, C.K.: Urban traffic flow prediction using a fuzzy-neural approach. Transp. Res. Part C: Emerg. Technol. 10(2), 85–98 (2002) 8. Pal, N.R., Bezdek, J.C.: On cluster validity for the fuzzy c-means model. IEEE Trans. Fuzzy Syst. 3(3), 370–379 (1995) 9. Salmeron, J.L., Froelich, W.: Dynamic optimization of fuzzy cognitive maps for time series forecasting. Knowl.-Based Syst. 105, 29–37 (2016) 10. Zhang, S., Chen, Y., Zhang, W.: Spatiotemporal fuzzy-graph convolutional network model with dynamic feature encoding for traffic forecasting. Knowl.-Based Syst. 231, 107403 (2021) 11. Panaretos, V.M., Zemel, Y.: Statistical aspects of Wasserstein distances. Annu. Rev. Stat. Appl. 6(1), 405–431 (2019) 12. Lan, S., Ma, Y., Huang, W., Wang, W., Yang, H., Li, P.: DSTAGNN: dynamic spatialtemporal aware graph neural network for traffic flow forecasting. In: Proceedings of the 39th International Conference on Machine Learning, pp. 11906–11917 (2022) 13. Han, L., Du, B., Sun, L., Fu, Y., Lv, Y., Xiong, H.: Dynamic and multi-faceted spatio-temporal deep learning for traffic speed forecasting. In: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 547–555 (2021) 14. Guo, S., Lin, Y., Feng, N., Song, C., Wan, H.: Attention based spatial-temporal graph convolutional networks for traffic flow forecasting. In: proceeding of the AAAI Conference on Artificial Intelligence, pp. 922–929 (2019) 15. Wu, Z., Pan, S. Long, G., Jiang, J., Zhang, C.: Graph WaveNet for deep spatial-temporal graph modeling. In: Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, pp. 1907–1913 (2019) 16. Zheng, C., Fan, X., Wang, C., Qi, J.: GMAN: a graph multi-attention network for traffic prediction. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp. 1234–1241 (2020)

Structured Neural Network Based Quadcopter Control Under Overland Monitoring Ali Abbasov1

, Ramin Rzayev1(B)

, Tunjay Habibbayli2

, and Murad Aliyev3

1 Institute of Control Systems, Vahabzadeh Str. 9, AZ1141 Baku, Azerbaijan

[email protected]

2 Institute of Information Technology, Vahabzadeh Str. 9A, AZ1141 Baku, Azerbaijan 3 Intelpro LLC, Jabbarly Str. Globe Center 609, AZ1065 Baku, Azerbaijan

Abstract. In some areas where economic activity is carried out, the presence of mountains and forests is observed. In order to provide information support for the development of infrastructure and agriculture in these territories, in some cases the overland monitoring is required using unmanned technologies, in particular, quadcopters. To ensure autonomous maneuvering of the quadcopter under overland monitoring, it is proposed to use a structured hierarchical neural network control model, which includes two subnets: “reasonable” and “instinctive”. The training of these networks is carried out on various scenarios of the behavior of the quadcopter relative to overcoming possible obstacles in the five fields of vision. As a basic model for the formation of such scenarios, it is proposed to use a fuzzy inference system with input characteristics in the form of linguistic variables that reflect fuzzy areas of space within which the presence of obstacles and the distance to them are interpreted verbally, i.e. as the terms of the corresponding input linguistic variables. Overcoming obstacles is supposed to be carried out on the basis of fuzzy conclusions of the proposed system, formulated as output linguistic variables, reflecting changes in the angle of rotation in the horizontal plane, flight altitude and path velocity of the quadcopter. Keywords: Quadcopter · Neural Network Model · Fuzzy Set · Fuzzy Inference

1 Introduction Currently, artificial neural networks are one of the main paradigms for the formation of intelligent control systems. The spectrum of capabilities of neural networks for solving control problems is very wide. Now they are effectively used as a tool for emulating, evaluating, predicting the parameters of the control object, as well as directly in the process of their regulation. The unrelenting demand for neural network control models is due to their ability to work with both crisp and fuzzy data, which is typical for many technical control objects, including quadcopters [1–6]. One of the main advantages of neural networks is the possibility of organizing robust controllers [2], which are flexible to possible changes in the parameters of the control object and the environment, as well as to various sensor noises. The topological features of neural networks make it possible © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 577–585, 2023. https://doi.org/10.1007/978-3-031-39774-5_64

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to model complex and nonlinear dependencies. In particular, when designing various controllers, this property essentially alleviates the development of a control system, since neural network synthesis of control systems does not require the designer to have traditional knowledge about the mutual influence of internal parameters. The main expenditures of using neural networks are: the need to conduct a certain number of iterations of the controller synthesis, the high dependence of the development process on the intuition and heuristic knowledge of the developer, the need for preliminary preparation of training samples (scenarios for the behavior of control objects), which in itself is a difficult task. Nevertheless, the possibility of flexible adjustment and versatility make the use of neural network structures a powerful tool of the designer. At the same time, the construction of neural network control systems requires certain assumptions, for example, the presence of an idealized mathematical model [4], which allows to set the desired behavior of the control object. The variety of topologies of neural networks does not allow to identify in advance the most suitable of them for solving a specific practical problem, which also causes the complexity of their practical application. Basically, this problem is solved empirically, followed by a choice or based on the application of expert knowledge. The article discusses a helicopter-type unmanned aerial vehicle (quadcopter), for the control of which a structured hierarchical neural network model and the 3D trajectory formation algorithm for overland piloting in a mountainous wooded landscape are proposed, which involves autonomous maneuvering to overcome possible obstacles.

2 Neural Network Model for Quadcopter Control Figure 1 shows a hierarchical three-layer neural network consisting of input, hidden and output layers. The neural network is considered as a signal converter, which has 5 inputs and 3 outputs. The network transforms the input signals in accordance with the weights of synoptic connections and thresholds of nonlinear neurons from the hidden layer, which are adjusted during the learning process so that the network induces the desired signals at the outputs. The network learning process begins with an arbitrary choice of the specified parameters, so at the initial stage, the network outputs will differ significantly from the desired ones. The network compares the current outputs with the desired ones and adjusts its parameters until the difference between the induced outputs and the desired outputs decreases to the acceptable value. In general, this is a demonstration of how a neural network is trained by the backpropagation algorithm. Due to its approximation abilities, this neural network is flexible, that is, it is able to “intelligently” respond to untested conditions by acquiring new and changing its functions in the training process. However, this network can only respond to static input-output relationships that are fixed at a given point in time. To ensure a dynamic connection between input vectors that change over time and a series of output vectors, additional adjustment of the connection weights and thresholds of nonlinear neurons from the hidden layer is required, which in this case already depend on time. It will be necessary to have additional memory nodes where one can store behavior patterns corresponding to certain input influences. As a result, the topological structure of the network will expand significantly, which will inevitably slow down the process of its

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training. Obviously, this is unacceptable, since a quadcopter processing input patterns in real time must be capable of operational maneuvering.

Fig. 1. Three-layer neural network.

To control the behavior of the quadcopter, it is proposed to use a neural network model capable of generating various patterns of behavior in real time. Figure 3 presents a structured hierarchical neural network model that combines two types of multi-layer networks: “reasonable” and “instinctive” networks [7]. The input signals (x 1 = Front, x 2 = Left, x 3 = Right, x 4 = Higher, x 5 = Below), which as data from the quadcopter sensors are varied in the interval [0, 1]. Each input variable x k (k = 1 ÷ 5) corresponds to a sector of space horizontally or vertically in the direction of quadcopter movement, within which the presence of obstacles and the distance to them are analyzed (see Fig. 2). Influencing the output variables allows the quadcopter to avoid collision with an obstacle by changing the angle of rotation in the horizontal plane (y1 = Rotation), velocity of travel (y2 = Velocity) and flight altitude (y3 = Height).

Fig. 2. Obstacle visibility sectors: a) side view, b) top view.

The quadcopter sensor signals are fed directly to the input layer of the reasonable neural network, which checks the compliance of the input vector with a particular behavior pattern and only after it forms the quadcopter behavior model. An example of such quadcopter behavior model is: “Move forward without changing direction, altitude and velocity until the sensor detects any obstacles along the flight path”. The instinctive neural network determines the correspondence between sensory input and a series of behaviors that the quadcopter must comply with during a certain time

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Fig. 3. Structured neural network model for quadcopter control.

of its maneuvering. An example of a command issued by the instinctive network is: “Repeat the cycle of movements to the right and left until the sensor detects the presence of obstacles”. A quadcopter often has to perform such movements sequentially, so the function of the instinctive network is important when using the neural network to control the quadcopter under overland monitoring in hard-to-reach areas of a mountainous forest landscape. The instinctive network has three short-term memory nodes, two of which are mutually inhibiting. The short-term memory node maintains an excited or inhibited state for a certain period of time. As shown in Fig. 3, some of the output neurons of the reasonable network are connected to the instinctive network through the use of excitatory and inhibitory signals. The inhibitory signal from the reasonable network to the instinctive network is reset when there are no input signals from the sensors. In such cases, the excitation signal from the reasonable network is transmitted through the Short-Term Memory 2 node (STM2) to the instinctive network, which becomes active and takes control of the quadcopter. After that, the quadcopter begins to maneuver in search of the free flight zone. When receiving a signal from the sensor of positioning, the quadcopter must maneuver to avoid the obstacle. This maneuvering is the series of behaviors consisting of several successive movements, such as, for example, moving to the left for a certain distance, and then moving forward and avoiding an obstacle. To perform this action, the reasonable network sends the excitation signal to the STM2 node, which holds the excitation signal for a certain period of time. The instinctive network continues to transmit the control signal so that the quadcopter moves left (or right) and further forward, as long as the STM2 block remains active. Ultimately, this allows the quadcopter to go around obstacles.

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In those cases when in the direction of movement of the quadcopter the sensor signals the limitations of the view in the sectors of space horizontally or vertically, the instinctive network is also activated. When the quadcopter, turning left, reaches the limit of its field of view in a sector of space horizontally, the sensor signal is directly transmitted to the corresponding input neuron of the instinctive network through Short-Term Memory 1 node (STM1), which are equipped with the mutual inhibition connection. For example, the instinctive network generates a control signal to move right or left in response to the Left/Right Limit (LRL) input to turn the quadcopter away from its horizontal field of view limit. Similarly, the instinctive network generates a control signal to move up or down in response to a Higher/Below Limit (HBL) input to turn the quadcopter away from its vertical field of view limit. The quadcopter is controlled by the reasonable network in cases where sensory data prescribes typical (pre-set) behavior patterns, and controlled by the instinctive network in cases where it does not receive such sensory information, and when it is forced to perform the series of maneuvering actions within a certain period of time. At first sight, it may seem that the neural network control of the quadcopter can be carried out without using the combined hierarchical neural network structure. However, if the structured neural network control model is not used, then in the process of quadcopter autopiloting, the size of the unitary neural network will permanently increase, which will significantly complicate real-time control of the quadcopter. In addition, during operation, the number of training templates will inexorably increase, which will significantly increase the time required for training. Obviously, this approach is highly impractical. At the same time, the division of the neural network structure into two subnets involves their preliminary training on predetermined behavior patterns. The use of the bounded set of training pairs will significantly reduce the number of network connections, thereby contributing to the effective control of quadcopters.

3 Formation of Quadcopter Behavior Patterns Under Overland Monitoring In [8], an algorithm was proposed for forming the flight path of a quadcopter using a fuzzy inference system based on empirical data analysis. This algorithm provides overland autopiloting of the quadcopter equipped with an obstacle detection sensor (lidar) in five frontal viewing sectors. Based on the results obtained in [8], to determine the rules for overland autopiloting of the quadcopter in the indicated five directions in space (see Fig. 2), a bounded set of logically consistent rules was considered in [9] in the form of the information fragments, some of which are presented as follows: d 1 : “If any obstacle is not detected on the flight path of the quadcopter or it is too far away, then there is no need to change direction, height and to reduce velocity”; d 2 : “If the sensor detects an obstacle at a medium distance along the flight path of the quadcopter and the sector on the left is free, then it is necessary to lose velocity to an average value and to turn slightly to the left without changing the height”; ………… d 18 : “If an obstacle is detected at a medium distance along the flight path, obstacles are detected at a close distance to the left, right, above and below the course, then it is necessary to lose velocity to an average value without changing the course and height”;

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d 19 : “If in all sectors of the view the detected obstacles are at a close distance, then it is necessary to lose velocity to a minimum and turn sharply to the left without changing the height”. Maneuvering to the left (or to the right, which is also appropriate) from the frontal impasse caused by the presence of obstacles in all five sectors of view (see rule d 19 ), the quadcopter continues to move and, thereby, creates a new flight path for itself in accordance with the regulations established by rules d 1 ÷ d 19 . Thus, analysis of all possible scenarios of collision with obstacles made it possible to form the complete set of linguistic variables (see Table 1) and rules for forming a Fuzzy Inference System (FIS) that regulates the behavior of the quadcopter during overland piloting. Table 1. Input and output linguistic variables of the FIS and their terms. Inputs Symbol

Variable name

Universe

Term set

x1

Front: the remoteness of the obstacle in the direction of flight

[0, 1]

{X 11 = significant, X 12 = average, X 13 = insignificant}

x2

Left: the remoteness of the obstacle to the left of the direction

[0, 1]

{X 21 = significant, X 22 = average, X 23 = insignificant}

x3

Right: the remoteness of the obstacle to the right of the direction

[0, 1]

{X 31 = significant, X 32 = average, X 33 = insignificant}

x4

Higher: the remoteness of the obstacle to the higher of the direction

[0, 1]

{X 41 = significant, X 42 = average, X 43 = insignificant}

x5

Below: the remoteness of the obstacle to the lower of the direction

[0, 1]

{X 51 = significant, X 52 = average, X 53 = insignificant}

y1

Velocity: flight speed

[0, 1]

{Y 11 = full, Y 12 = average, Y 13 = zero}

y2

Rotation in the horizontal plane [-0.5, 0.5]

{Y 21 = sharply to the left, Y 22 = slightly to the left, Y 23 = is absent, Y 24 = slightly to the right, Y 25 = sharply to the right}

y3

Height: change the flight altitude

{Y 31 = sharply up, Y 32 = slightly up, Y 33 = is absent, Y 34 = slightly down, Y 35 = sharply down}

Outputs

[-0.5, 0.5]

The corresponding FIS is formed by the following rules in symbolic form:

Structured Neural Network Based Quadcopter Control

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d1 : (x1 = X11 ) ⇒ (y1 = Y11 )&(y2 = Y23 )&(y3 = Y33 ); d2 : (x1 = X12 )&(x2 = X21 ) ⇒ (y1 = Y12 )&(y2 = Y22 )&(y3 = Y33 ); .................. d18 : (x1 = X12 )&(x2 = X23 )&(x3 = X33 )&(x4 = X43 )&(x5 = X53 ) ⇒ (y1 = Y12 )&(y2 = Y23 )&(y3 = Y33 ); d19 : (x1 = X13 )&(x2 = X23 )&(x3 = X33 )&(x4 = X43 )&(x5 = X53 ) ⇒ (y1 = Y13 )&(y2 = Y21 )&(y3 = Y33 ).

To implement the FIS that provides autopiloting of the quadcopter according to the formalized schedule, it is necessary to reflect the introduced linguistic variables (LVs) to the set of their corresponding real numbers by setting membership functions, that is, fuzzify the terms of all input and output LVs. In MATLAB\FIS notation, all membership functions were established empirically. In particular, the input and output characteristics of the model are shown in Figs. 4 and 5, respectively.

Fig. 4. Terms of input LV “Front: the remoteness of the obstacle in the flight direction”

Fig. 5. Terms of output LVs: y1 – velocity, y2 – rotation on left/right, y3 – height.

As a result, the corresponding flight scenarios was generated using the interactive window of the graphical interface of the MATLAB\FIS editor. Obtained scenarios are summarized in Table 2. Table 2. The flight scenarios generated by MATLAB\FIS editor. №

Inputs

Outputs

x1

x2

x3

x4

x5

y1

y2

y3

1

0.95

0.50*

0.50

0.50

0.50

0.855

0

0

2

0.50

0.95

0.50

0.50

0.50

0.350

-0.200

0 (continued)

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A. Abbasov et al. Table 2. (continued)

№

Inputs x1

x2

x3

x4

x5

y1

y2

y3

3

0.10

0.90

0.50

0.50

0.50

0.013

-0.411

0

4

0.50

0.50

0.90

0.50

0.50

0.350

0.200

0

5

0.10

0.50

0.90

0.50

0.50

0.013

0.411

0

6

0.50

0.50

0.50

0.90

0.50

0.350

0

0.200

7

0.10

0.50

0.50

0.90

0.50

0.013

0

0.411

8

0.50

0.50

0.50

0.50

0.90

0.350

0

-0.200

9

0.10

0.50

0.50

0.50

0.90

0.013

0

-0.411

10

0.50

0.50

0.40

0.40

0.40

0.350

0

0

11

0.10

0.50

0.50

0.50

0.50

0.013

-0.411

0

12

0.50

0.10

0.50

0.50

0.50

0.350

0

0

13

0.10

0.10

0.50

0.50

0.50

0.013

0.411

0

14

0.50

0.10

0.10

0.50

0.50

0.350

0

0

15

0.10

0.10

0.10

0.50

0.50

0.013

0

0.411

16

0.50

0.10

0.10

0.10

0.50

0.350

0

0

17

0.10

0.10

0.10

0.10

0.50

0.013

0

-0.411

18

0.50

0.10

0.10

0.10

0.10

0.350

0

0

19

0.10

0.10

0.10

0.10

0.10

0.013

-0.411

0

Outputs

* The absence of obstacles determines the use of the average value 0.5 on the universe [0, 1].

4 Conclusion To control a quadcopter under overland monitoring, the hierarchical neural network structure is proposed as a basic model, consisting of two subnetworks: “reasonable” and “instinctive”, the combined functioning of which allows the quadcopter to maneuver and overcome obstacles in real time. To form the appropriate neural network behavior models, different scenarios for quadcopter maneuvering in order to overcome possible obstacles in five viewing sectors are considered. To form such scenarios and generate the 3D quadcopter trajectory under overland monitoring the corresponding algorithm was proposed based on FIS. This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant №EIF-MQM-QA-1–2021-4(41).

References 1. Jurado, F., Castillo-Toledo, B.: Stabilization of a quadrotor via Takagi-Sugeno Fuzzy Control. In: 12th World Multi-Conference on Systemics, p. 6. Cybernetics and Informatics, Orlando (2008)

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2. Nicoli, C., Macnab, C., Ramirez-Serrano A.: Robust neural network control of a quadrotor helicopter. http://www.academia.edu/8226428/ROBUST_NEURAL_NETWORK_CONT ROL_OF_A_QUADROTOR_HELICOPTER. Accessed 21 Jan 2023 3. Burka, A., Foster S.: Neato Quadcopters. http://www.web.cs.swarthmore.edu/~meeden/cs81/ s12/papers/AlexSethPaper.pdf. Accessed 12 Feb 2023 4. Shepherd, J., Tumer, K.:Robust neuro-control for a micro quadrotor. In: Genetic and Evolutionary Computation Conference, pp. 1131–1138. Portland (2010) 5. Vijaya Kumar, M., Suresh, S., Omkar, S.N., Ganguli, R., Sampath, P.: A direct adaptive neural command controller design for a nun stable helicopter. Eng. Appl. Artif. Intell. 22, 181–191 (2009) 6. Suresha, S., Sundararajan, N.: An on-line learning neural controller for helicopters performing highly nonlinear maneuvers. Appl. Soft Comput. 12, 360–371 (2012) 7. Kobersy, I., Finaev, V., Beloglazov, D., Shapovalov, I., Zargaryan, J., Soloviev, V.: Design features and research on the neuro-like learning control system of a vehicle. Int. J. Neural Networks Adv. Appl. 1, 73–80 (2014) 8. Nagata, S., Sekiguchi, M., Asakawa, K.: Mobile robot control by a structured hierarchical neural network. IEEE Control Syst. Mag. 10(3), 69–76 (1990) 9. Habibbeyli, T.: Formation of the quadcopter flight path under overland monitoring using neurofuzzy modeling methods. Math. Mach. Syst. 3, 97–107 (2022)

Drug Delivery in Chemotherapy Using an Online Wavelet-Based Neural Network Pariya Khalili1 , Mansour Ansari2 , Ali Akbar Safavi3 , and Ramin Vatankhah1(B) 1 School of Mechanical Engineering, Shiraz University, Shiraz, Iran

[email protected]

2 Department of Radio-Oncology, School of Medicine, Shiraz University of Medical Sciences,

Shiraz, Iran 3 School of Electrical Engineering, Shiraz University, Shiraz, Iran

Abstract. Intelligent prescription recommendations of chemotherapy drugs for targeted patients can be considered as an important issue for cancer treatment. This paper focuses on chemo-targeted therapy for breast cancer using online waveletbased neural networks. The basic idea, which could be generalized to other therapies as well, relies on learning from various prescriptions of chemotherapists and the conditions of patients in such a way that gradually gets closer to the best physician’s prescription. This is due to the fact that real-time models and the existence of a stream of data are the main reasons that online learning algorithms are widely used especially for dynamic processes. Wavelet-based neural network with a hierarchical learning algorithm and online learning and adaptation capabilities, which is called wave-net, has proved to be able to solve complicated problems. As shown in our results, online wave-net-based chemotherapy recommendations can help the physician with a faster and more accurate prescription. Keywords: Cancer · Chemotherapy · Intelligent prescription · Wave-net · Online learning

1 Introduction The procedure of chemotherapy drug prescription is an interesting field that has drawn the attention of researchers recently. Physicians’ needs and engineers’ solutions have facilitated this field with the aim that the final drug prescription recommendation will be given with the least error. Note that obtaining intelligent drug delivery has two main approaches: Model-based and machine-learning methods. In the former, mathematical time-based models are presented and controllers are designed to reduce cancer cells with the least side-effect on the healthy tissue [1–3]. In the latter, machine learning algorithms are utilized to calculate the drug injection rate with help of available data sets, without the observation of states during the time. The potential of the artificial neural network approach has made it a very interesting tool for modeling difficult problems. The neural networks, as a universal approximator, with the aid of online algorithms have made it possible to solve dynamic problems. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 586–593, 2023. https://doi.org/10.1007/978-3-031-39774-5_65

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This combination assures us that the approximated outputs will gradually get to the desired one, as time passes. The online learning method is capable of being prepared whenever the online inputs vary due to the numerous facts. In other words, the error of the network will gradually become smaller no matter how much the input changes. This is exactly the need that we are faced in chemotherapy treatment, knowing the fact that different patients with different characteristics are considered and a proper drug dosage prescription is to be calculated. There are numerous neural network-based methods with different purposes in order to obtain the chemotherapy drug dosage. It is important to note that the great majority of the articles utilizing neural networks, is producing data from differential equations, and various variables like cancer cells, immune cells, etc. are taken into account as inputs and outputs of the network. For instance, Zaouri first described a fractional differential equation in modeling the cancer behavior, and then applied a neural network-based adaptive back-stepping method to find the best drug dosage to reduce cancer cells [4]. A four-state model was also used as simulated patients, and then, reinforcement learning was applied to calculate a safe amount of chemotherapy [5]. A combination of neural networks and feedback linearization leads to propose of adaptive neural networks which help physicians to find the optimal chemotherapy dosage with the least side effects [6]. The capability of the neural network approach in solving real-world problems has made this field so interesting that caused great improvements in the designing of such networks in the past few years [7]. Applying rigorous mathematical frameworks to common structure of neural networks has drawn the attention of researchers in the past decades. One of these studies has led to invoke the multiresolution framework and wavelet basis function [8] to propose wave-nets in the 1990s [9]. The real-time modeling with the wave-nets proposed in [10] is one of the applicable algorithms which is based on the multi-resolution framework, and leads us to propose of the intelligence drug prescription in chemotherapy. In this paper, we have attempted to define a more realistic problem and propose an idea that is guaranteed to achieve better results compared to the common procedures. The online wave-net method is applied on the data sets produced by the real procedure performed by oncologists in the hospital. The final results shows that the recommended algorithm could be used as an alternative for the traditional approach with more accurate chemotherapy dosage suggestion. This paper is organized as follows: Sect. 2 explains the common procedure prescribed by the physician for a patient. In Sect. 3, the problem is defined and the inputs and outputs of the networks are selected. Section 4 reviews the mathematical background of online wave-net. The simulation and results are illustrated in Sect. 5. Section 6 concludes the paper.

2 Chemotherapy Treatment Procedure After a patient is diagnosed with breast cancer, several evaluations, such as image studies, biopsy, blood tests, etc. are prescribed to help the physician choose the best treatment. Every patient just does the necessary workup based on the physician’s experiment. The oncologist chooses the evaluations depends on the patient’s general condition, stage of

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the disease, and the selected modality of treatment. The common workup are based on patients’ pathology reports, immunohistochemistry (IHC) [11], and Fluorescence in Situ Hybridization (FISH) test for human epidermal growth factor receptor 2 (HER2) [12]. Since there are too many factors involved in the selection of the final treatment regimen, we have limited our study to those who were treated with targeted therapy, Trastuzumab, which was prescribed according to the IHC and FISH test results. If the IHC test for HER2 is 0 or 1+, it is described as negative and if it is 3+, it is described as positive. In some instances, in which the IHC result is 2+, it is described as equivocal and the FISH test is necessary. Therefore, cases considered in this paper are the ones who were the HER2 3+ or 2+ with a positive FISH test. The systemic treatment protocol prescribed for these patients is the combination of Trastuzumab and chemotherapy. Trastuzumab dose calculation is based on the patients’ weight, which is an 8 mg/kg loading dose (first session), followed by a 6 mg/kg maintenance dose (other sessions) every three weeks [13]. The commonly used chemotherapy regimen for breast cancer is a four-cycle of Doxorubicin plus Cyclophosphamide supervised every three weeks, which is commonly known as the AC regimen and followed by a taxane drug (Paclitaxel or Docetaxel) also for four cycles. To find the appropriate drug dose for injection, Body Surface Area (BSA), based on weight and height, must be calculated. The patients are to be injected Doxorubicin 60 mg/m2 followed by Cyclophosphamide 600 mg/m2 in every session [14]. Thus, the weight and height of the patient are the main two factors that the final dose of chemotherapy drugs is based on. The common procedure for finding the drug dosage is to calculate the BSA based on patient’s weight and height using specific apps. Afterward, the physician may change the calculated dosage based on the patient’s condition. The approximate dosage of Trastuzumab is obtained based on weight, and its final value may change as well. In this paper, we are to present a network that gives us an accurate final value compared with the common procedure.

3 Problem Definition and Data Sampling As mentioned above, the amount of the prescribed drug injection (either chemotherapy or Trastuzumab) is calculated based on the patient’s height and weight. Inspired from the common procedure, the range of 145 to 180 cm, and 45 to 105 kg for hypothetical patients’ height and weight respectively, are considered. Afterwards, the final dose of all possible combinations of height and weight based on the doctor’s prescription is calculated for the training procedure. In this case, the total number of 2196 samples are used to train and evaluate the networks. The next step is to define the proper input and output of the network. It was mentioned that height and weight are the two factors affect the injection amount. In this paper, BSA and weight are chosen as inputs. The outputs are the amount dosage of Doxorubicin, Cyclophosphamide, Trastuzumab Loading, and maintenance dose. Thus, there are two inputs and four outputs for the network.

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4 Online Wavelet-Based Neural Network After defining network inputs and outputs, online wave-net method is ready to be used. Wave-net is a hierarchical multi-resolution wavelet-based neural network with one hidden layer of nodes and localized learning. The concept of multi-resolution approximation and hierarchical learning algorithm makes this network flexible enough compared to the usual neural networks and this is mainly because of the use of wavelets as the basis functions [9]. 4.1 Wave-Net The properties such as orthonormality, localized learning ability, fast implementation, flexibility in time-frequency analysis, etc. make the wavelet family an interesting candidate for neural network basis function for approximation [9]. Wavelets are usually used in the multi-resolution framework presented by Mallat [15]. In this framework, ψm,k (x) (scaling function) and φm,k (x) (wavelet) are defined as follows, ψm,k (x) = 2

−m 2

φm,k (x) = 2

  ψ 2−m x − k m, k ∈ Z,

(1)

  φ 2−m x − k m, k ∈ Z

(2)

−m 2

F(x) as the approximation of f (x), the unknown function to be estimated, will be written as below F(x) =

+∞  k=−∞

a0,k φ0,k (x) +

0 

+∞ 

dm,k ψm,k (x)

(3)

m=−∞ k=−∞

where, m and k correspond to the dilation and translation, respectively. The parameter m determines the accuracy of the approximation. As m reduces (the resolution of the answer gets better), F(x) will be closer to f (x). For the sake of simplicity, it is usually assumed that all the calculation starts from a specific resolution, for instance, m = 0. The last step to finish the task of learning is to compute the network coefficients, a0,k and dm,k in Eq. (3). There are different methods that each is suitable for certain kinds of problems, for instance, the direct inner product is only used for orthogonal basis function [16]. In this paper, L2 learning algorithm is utilized for wave-net implementation which is discussed in details in [10]. 4.2 Online Learning Procedure with Wave-Nets The basis of the multi-resolution concept, which the wave-net is mainly based on, facilitates a way to utilize this network with the idea of online training. The error threshold is an important factor that plays an important role in online learning, which is an index that determines when the training procedure must stop. While the online data enters the network, the error is computed, and compared to the error threshold. If the former is greater, it is concluded that the dynamic of the system has changed. The algorithm of the online training performed in this paper is summarized below [10]:

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The wave-net learning algorithm is selected. The error threshold is specified (L2 , , for example). The model is trained by the new test data set. If the error is greater than the error threshold, go to (v), if not collect the online test data and go to (iii). Identify the involved wavelets in the error and replace the test data as the new training data. Update the model by retraining the part of the model involved in the error. If the error is greater than the error threshold, go to (viii), if not continue with the online wave-net model. Collect the online test data and go to (iii). Lower or upper the resolution and update the model by retraining the part of the model involved in the error. If the error is greater than the error threshold, go to (x), if not go to (xi). Retrain the whole model from the zero resolution with the train data, and evaluate the network with the test data. Go to (iii). Consider the test data set as the new training data, and the newly arrived data are considered as the new test data set for the next step of training. Collect the online test data and go to (iii).

5 Simulation and Results The network inputs and outputs are defined in detail in Sect. 3. The “software calculation” is referred to the calculated app value, and the final dosage recommended by the physician is considered as the output that the network is trained with. The comparisons between the actual and predicted targets are illustrated in Fig. 1. The results in this figure show that the outputs are predicted successfully. The evaluation error in every step of training is illustrated in Fig. 2.

Fig. 1. Online wave-net: Target vs Approximated

As it is shown the error gradually decreases for all the outputs, while the number of online data increases. This figure indicates the power of the online learning algorithm.

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The Normalized Root Mean Square Error (NRMSE) are also reported in Table 1. It is clear that the maximum online error in all outputs is smaller than the software calculation. This indicates that as the number of entered data increases, the wave-net output becomes closer to the physician’s prescription, which means the goal of the paper has been achieved. It can also be concluded that the maximum error of the calculations among all the outputs is almost 10%. This reduces to less than 0.038% using the online wave-net algorithm. This highlights the effectiveness of this online method compared to the common procedures of software calculation.

Fig. 2. NRMSE of the outputs’ vs the number of updates Table 1. Comparison of online wave-net error with the software Maximum Online Error

Final Online Error

Maximum Software Calculation Error

A (Doxorubicin)

0.020506

0.00037733

0.10670

C (Cytoxan)

0.020742

0.00037571

0.10669

Trastuzumab L

0.046501

0.00024263

0.08659

Trastuzumab M

0.046914

0.00026519

0.08886

In Fig. 3, it is illustrated that the resolution decreases because the calculated error gets larger. This means that the hierarchical learning algorithm lets us rebuild the network from a lower resolution. It could also be realized that there is no need to get into more details than resolution -4. In other words, resolution -4 can reduce the error significantly as reported earlier in Table 1.

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Fig. 3. Resolution number vs the number of updates

6 Conclusion In the common procedure performed in the hospitals for prescribing chemotherapy, an application is utilized to help the physician find the approximate dosage based on patient’s height and weight. This value changes by the physician’s experience. The main goal of this paper is to perform an online learning algorithm to recommend the amount of drug dosage which is closer to the physician’s prescription as a substitution for the mentioned common procedure. In this regard, the patients with breast cancer and positive HER2 test under targeted therapy are considered. Three drugs of Doxorubicine, Cytoxan and Trastuzumab are usually prescribed for this group of patients, which the last one has one loading and one maintenance dose. In the initial step, hypothetical patients with reasonable range of heights and weights are considered and based on the physician’s experience, the four outputs of the network are generated. In the next step, online wavenet algorithm is chosen and performed on the data. The results show that the wave-nets hierarchical learning algorithm provides a way that if the goal is not satisfied, the network improves by adding details or going to higher resolution based on the previous network. Thus, the network gets better in every step. For future studies, the idea could be utilized for proposing an application software which could be designed as an alternative for the common method of calculating the drug delivery rate for the sake of physicians’ comfort and accelerating the mentioned procedure.

References 1. Moradi, H., Vossoughi, G., Salarieh, H.: Optimal robust control of drug delivery in cancer chemotherapy: a comparison between three control approaches. Comput. Methods Programs Biomed. 112(1), 69–83 (2013). https://doi.org/10.1016/j.cmpb.2013.06.020 2. Khalili, P., Vatankhah, R.: Derivation of an optimal trajectory and nonlinear adaptive controller design for drug delivery in cancerous tumor chemotherapy. Comput. Biol. Med. 109(April), 195–206 (2019). https://doi.org/10.1016/j.compbiomed.2019.04.011 3. Khalili, P., Vatankhah, R.: Optimal control design for drug delivery of immunotherapy in chemoimmunotherapy treatment. Comput. Methods Programs Biomed. 229, 107248 (2023). https://doi.org/10.1016/j.cmpb.2022.107248 4. Zouari, F.: Neural network based adaptive backstepping dynamic surface control of drug dosage regimens in cancer treatment. Neurocomputing 366, 248–263 (2019)

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5. Padmanabhan, R., Meskin, N., Haddad, W.M.: Learning-based control of cancer chemotherapy treatment. IFAC-PapersOnLine 50(1), 15127–15132 (2017). https://doi.org/10.1016/j.ifa col.2017.08.2247 6. Floares, A., Floares, C., Cucu, M., Lazar, L.: Adaptive neural networks control of drug dosage regimens in cancer chemotherapy. In: Proceedings of the International Joint Conference on Neural Networks, vol. 1, pp. 154–159. IEEE (2003) 7. Hu, Y.H., Hwang, J.N.: Hanbook of Neural Network signal processing: Acoustical Society of America. Melville, United States (2001). https://doi.org/10.1201/9781315220413-1 8. Bakshi, G.S.B.R.: Wave-net: a multiresolution, hierarchical neural network with localized learning. AIChE J. 39(1), 57–81 (1993) 9. Safavi, J.A.R.A.A.: Application of wavelet-based neural networks to modelling and optimisation of an experimental distillation column. Eng. Appl. Artif. Intell. 10(3), 301–313 (1997) 10. Zakeri, V., Naghavi, V., Safavi, A.A.: Developing real-time wave-net models for non-linear time-varying experimental processes. Comput. Chem. Eng. 33(8), 1379–1385 (2009). https:// doi.org/10.1016/j.compchemeng.2009.02.003 11. Zaha, D.C.: Significance of immunohistochemistry in breast cancer. World J. Clin. Oncol. 5(3), 382 (2014). https://doi.org/10.5306/wjco.v5.i3.382 12. Owens, M.A., Horten, B.C., Da Silva, M.M.: HER2 amplification ratios by fluorescence in situ hybridization and correlation with immunohistochemistry in a cohort of 6556 breast cancer tissues. Clin. Breast Cancer 5(1), 63–69 (2004). https://doi.org/10.3816/CBC.2004.n.011 13. Quartino, A.L., et al.: Population pharmacokinetic and covariate analyses of intravenous trastuzumab (Herceptin®), a HER2-targeted monoclonal antibody, in patients with a variety of solid tumors. Cancer Chemother. Pharmacol. 83(2), 329–340 (2018). https://doi.org/10. 1007/s00280-018-3728-z 14. Mamounas, E.P., et al.: Paclitaxel after doxorubicin plus cyclophosphamide as adjuvant chemotherapy for node-positive breast cancer: results from NSABP B-28. J. Clin. Oncol. 23(16), 3686–3696 (2005). https://doi.org/10.1200/JCO.2005.10.517 15. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Int. 11(7), 674–693 (1989) 16. Safavi, A., Romagnoli, J.: On the learning algorithm for wave-nets, pp. 3–40. Fifth Iranian Conference on Electrical Eng., Tehran, Iran (1997)

Heuristics

Fault-Tolerant Control Using Optimized Neurons in Feed-Forward Backpropagation Neural Network-For MIMO Uncertain System: A Metaheuristic Approach Sejal Raval2 , Himanshukumar R. Patel1(B) , Vipul Shah1 , Umesh C. Rathore3 , and Paresh P. Kotak4 1 Dharmsinh Desai University, Nadiad, Gujarat 387001, India [email protected], [email protected] 2 Government Polytechnic, Ahmedabad, Gujarat, India 3 Department of Electrical Engineering, Government Hydro Engineering College Bandla, Bilaspur 174005, India 4 Government Polytechnic, Rajkot, Gujarat, India

Abstract. The problem of regulatory control of the MIMO uncertain level control system in the presence of partial actuator failure and additive process disturbances is addressed in this paper. However, the hidden layer neurons are optimized using a metaheuristic algorithm (Genetic Algorithm) to anticipate the Fault-Tolerant Control (FTC) action to overcome the actuator fault and external process disturbances. We use the passive approach for fault-tolerant control with Tilt Integral Derivative (TID) control technique to develop a fault-tolerant controller without a fault detection scheme. In addition, in the present study, we use the four residue signal features (i.e. mean, variance, skewness, and normalized data of residue signal) to train the neural network in order to overcome the difficulty caused by having less information on faults and uncertainty from residue signal. Simulations are run to demonstrate the efficacy of the proposed scheme. The residue signal was measured using both the healthy and faulty system models. Keywords: Actuator fault · MIMO uncertain system tolerant control · genetic algorithm · neural network

1

· passive fault

Introduction

Fault Tolerant Control (FTC) of industrial processes is an area of research that aims to maintain appropriate control performance and system stability in the face of faults [1]. The FTC’s main task is preventing simple faults from becoming Supported by Department of IC Engineering, Dharmsinh Desai University. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 597–609, 2023. https://doi.org/10.1007/978-3-031-39774-5_66

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serious failures, which increases system availability and reduces the risk of safety hazards [1–3]. For the last three decades, FTC has been the focus of extensive research [4,5]. Many practical industrial applications have resulted from this research effort such as [6–9]. For FTC, there are fundamentally two structured approaches. The first is active FTC, while the second is passive FTC [10,11]. Passive FTC has developed an effective controller for the system based on the predetermined circumstances and magnitude of system faults. On the contrary hand, in the active FTC approach, Fault Detection and Diagnosis (FDD) is a key component required. FDD has three essential functions that begin with the detection of the faults in the process, then isolation, and eventually identification [12,13]. The appeal of passive FTC algorithms stems from their inherent ability to consider all types of potential defects at the design level. The second significant reason is that the operating principles are comprehensible and relatively simple to explain to professionals, which appears to be a critical aspect during the industry’s adoption of a new control arrangement. Several applications in industry and chemical process level control processes are frequently employed for a variety of reasons (such as liquid product transportation, storage, disposal, packaging, and so on). As a result, extensive research has been conducted to control the single input single output (SISO), and multi input multi output (MIMO) level control processes under different circumstances. Some advanced control algorithms have previously been tested on benchmark SISO and MIMO level control processes under different risks (e.g., actuator fault, system component fault, sensor fault, and process disturbances). A failure occurrence in sensor and system elements is a critical problem in the control of SISO and MIMO level control processes [14]. Nevertheless, a partial breakdown of the actuator may result in a dangerous situation and, as a result, system instability with loss of control action. As a result, time, capital, and even some manpower are wasted. As a result, to improve the system’s reliability while performing the task in the presence of a partial failure or fault in the actuator, we employ analytical redundancy in the form of advanced control strategies such as the Passive Fault-Tolerant Controller (PFTC). As a result, FTC is critical in the event of a partial breakdown of the main actuator. The author of a recent [15] used an innovative hybrid control scheme to design a passive FTC algorithm, combining a conventional PID controller with a fuzzy logic-based controller for SISO and MIMO level control with all possible faults. As mentioned in [16,17], Passive FTC algorithms have matured substantially, especially for linear systems. However, dealing with nonlinear systems presents a number of challenges. There are still issues with nonlinear process modeling, state estimation, and fault identification or fault tolerant control [18,19]. Several passive FTC methods have been developed to date for SISO/MIMO level control systems, such as PID + type-1 fuzzy logic control, PID + type-2 fuzzy logic control, fractional order PID + type-1 fuzzy logic control, and fractional order PID + type-2 fuzzy logic control [17–19]. Without any question, neural networks are the most frequently employed models. They are very adaptable, universal, and readily used when no accurate mathematical model of the

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process exists [15]. Although neural network-based control is not a new concept [14–17]. Nevertheless, we used FFBPNN to design PFTC, and the performance of the MIMO uncertain level control system under partial actuator fault and additive process disturbances is examined [20–24]. Our paper’s primary contributions are as follows. (1) The fault-tolerant controller does not include a fault detection scheme. (2) We create a healthy and faulty model for generating the residue signal for the MIMO level control process. (3) To analyze the residue signal, determine four parameters (mean, variance, skewness, and residue signal normalized data). We train the FFBPNN using these four parameter data to produce the controller output as an output signal. (4) In practice, the external disturbances are assumed to be unknown. PFTC takes care of these uncertainties. The following is how the work is structured. Section 2 follows the Introduction with a description of the MIMO uncertain level control process for which the fault tolerant control system was designed. The Passive FTC design is then proposed in Section 3 using FFBPNN and a bioinspired optimization method employing passive FTC. Section 4 displays and discusses the simulation results and implementation of the proposed scheme. The conclusions and future scope are included in the final section.

2 2.1

Uncertain MIMO Level Control System Model Uncertain MIMO Level Control System

The process is based on the industry-standard Two-Tank Level Control System (TTLCS) and includes two cylindrical water tanks, one sump tank, a pneumatic control valve, and one electric pump. The system has been incorporated [9]. The controlled variable of the system is the height of the tank h1 and manipulated variable is inlet flow control by control valve CV . A tank, a storage vessel, and a control valve with a positioner, a pump, and transducers to measure process variables comprise the system. The tank is realized as a horizontally placed cylinder, which is which introduces significant nonlinearity into the system’s static characteristics. Matlab/Simulink software was used to create the level control system laboratory stand. Real data from the physical installation was used to successfully validate the simulation model. The entire process variable description is provided in [9]. The simulator has a benefit over real processes in that it enables you to test the behavior of the proposed advanced control in the presence of potentially faulty circumstances.

3

Proposed Methodology for Passive FTC

The suggested technique for Passive Fault Tolerant Control (PFTC) is divided into four stages: data generation, pre-processing, training, and control output prediction. Figure 1 depicts the proposed methodology, which will be expanded on in the subsections.

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Bioinspired Optimization Methods

Genetic Algorithms for Optimization. The following cycle conveys the simple genetic algorithm in pseudo-code. The modified GOA with parameter adaptation is modified, where the main difference with respect to original GOA is the calculation of the new crossover K1 and mutation rate K2 .

Fig. 1. Suggested approach for Passive FTC with fuzzy bioinspired optimization algorithm [10]

Algorithm 1. Pseudo code for Genetic Algorithm Optimization 0: Generate the initial population of individuals aleatorily P (0). while (Number Generations f it x f it xc g+1 i i i xi = (9) xi g , otherwise Here, p = q = r = i, p, q, r ∈ [1, n], F ∈ [0, 2] is the scaling factor and pc ∈ [0, 1] is the crossover rate.

Software Test Suite Minimization Using Hybrid Metaheuristics

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629

Hybrid Bat Differential Evolution (BADE)

Bats utilise echolocation to identify prey and distinguish between different sorts of insects even in the dark, i.e., they are sensitive to noises and generally eat insects that can make noise. A swarm of bats living in a given habitat, may forage for food at the same time. As a result, bats may be vulnerable to noise and interference from their prey and companions. These interference are not included in the basic BA. So, in this work, this concept has been embedded to make virtual bats look like actual bats. The insects’ interference for the bats may be visually reproduced as a stochastic choice by combining the mutation operator in the DE/rand/1/bin scheme with the bat algorithm. The insects only cause problems for the bats when rand (0, 1) is less than pc, the crossover probability of DE. Let us say, three different individuals get in the way of the simulated bats. If the interference is powerful enough that the virtual bats are unable to differentiate the targets on their own, they will follow the interference’s suggestions. Otherwise, they will continue to use their own methods to find their targets. On the other hand, the swarm’s mean velocity is recreated by simulating the effects of the other bats on the bat. According to Eqs. (1), (2), and (3), the bat if the swarm’s mean velocity is ignored. However, if bat i will be placed at xg+1 i i continues in this manner, it will be far from the best solution. The situation will improve if the mean velocity of the opposing swarm is taken into account. When we utilise the Eqs. (1), (12), and (3) to update the new location, then might be closer to the ideal solution. xg+1 i vi g+1 = w ∗ (vi g − w2 ∗ v¯) + (xi g − xbest ) ∗ fi

(10)

here w ∈ (0, 1) and w2 ∈ (−1, 1) are two uniform random numbers. Furthermore, if the outcome does not change for δ attempts then the pulse emission rate is set too high and loudness is reset again to search globally.

4 4.1

Proposed Work Improved BADE Algorithms for Test Suite Minimization

IBADE with Adaptive Crossover Probability (IBADEc). In the IBADE algorithm, it is considered that the insects interference might affect the bat to find the food. However, the intensity of the interference is not taken into account. So, the improved version which uses adaptive crossover probability ranging between pcmin and pcmax . If the interference is high and the fitness of the bats is low then increase the crossover probability to explore the search space and vice versa.      g+1 > f it x pcmax , pcmax + rand ∗ 0.1 f it xcg+1 i i (11) pc = pcmin , pcmin + rand ∗ 0.1, otherwise

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Secondly, the local search has been updated using the Gaussian distribution of the loudness to improve the exploitation phase using the following equations:

  (12) xi g+1 = xbest ∗ 1 + {randn 0, σ 2 } σ 2 = |Ai g − mean (Ag )| + 

(13)

IBADE with Elite Solutions (IBADEe). The information theory and prior experience helped to exploit the search space efficiently [1]. Therefore, the solutions’ direction is chosen to help the local search by adding information flow between the bats with the help of elite and best bats, thereby enhancing the exploitation at the early stages. It is done by selecting two random solutions from the top 20% local best solutions and the one solution with higher fitness value is considered as information communicator xe .  xbest ∗ η + (1 − η)xe g < 0.5 gmax g+1 xi = (14) xbest + ε ∗ mean (A) , otherwise

5

Results and Analysis

Table 1 presents the programs under test taken from open-source software infrastructure repository (SIR) [11]. The results are compared with GA BA, and DE. The efficiency and effectiveness of these algorithms are validated with the help of following performance metrics: 1) Test Suite Size Reduction Percentage (TSR): It is the ratio of the reduced test suite to the original test suite size. 2) Cost Reduction Percentage (TCRP): It is the proportion of reduced cost to the total cost of the test suite. Table 1. Subject Programs Program Version Number of Test Cases jmeter

5

97

jtopas

4

209

ant

7

878

Table 2 shows the average test suite size reduction percentages and reduced suite costs of the programs under test. It is observed that the proposed IBADEe performed better than all the other algorithms. Observations also depict that the TCRP of IBADEc and IBADEe is more than all other algorithms and the values are proportionate to the TSR. Overall, IBADEe surpass IBADEc as it includes the elite solutions.

Software Test Suite Minimization Using Hybrid Metaheuristics

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Table 2. Performance metrics comparison for test suite minimization Algorithms TSRP TCRP

6

RS

80.045 86.785

GA

85.678 87.979

BA

82.456 86.980

DE

83.289 87.989

BADE

86.099 87.808

IBADEe

86.789 88.679

IBADEc

87.919 90.562

Conclusion

Regression test suite minimization problem is solved by using a hybrid of Bat and Differential Evolution (BADE) algorithm which takes into account the interference due to prey and companions of the bat. Its improved versions IBADEc and IBADEe are introduced are also proposed by varying the interference intensity and selecting the elite solutions for early exploitation. The proposed algorithms have been compared with the existing nature-inspired algorithms. The experimental results show that IBADEe outperformed other algorithms. In the future, the experiments will be performed on more real-world case studies. Moreover, the proposed algorithms will be explored for cost-effective test case selection and prioritization.

References 1. Yang, Q., Dong, N., Zhang, J.: An enhanced adaptive bat algorithm for microgrid energy scheduling. Energy 121014 (2021) 2. Bajaj, A., Sangwan, O.P.: A survey on regression testing using nature-inspired approaches. In: 2018 4th International Conference on Computing Communication and Automation, pp. 1–5 (2018) 3. Bajaj, A., Abraham, A.: Prioritizing and minimizing test cases using dragonfly algorithms. Int. J. Comput. Inf. Syst. Ind. Manag. Appl. 13, 062–071 (2021) 4. Anwar, Z., Afzal, H., Bibi, N., Abbas, H., Mohsin, A., Arif, O.: A hybrid-adaptive neurofuzzy inference system for multi-objective regression test suites optimization. Neural Comput. Appl. 31(11), 7287–7301 (2019) 5. Ma, Y., Zhao, Z., Liang, Y., Yun, M.: A usable selection range standard based on test suite reduction algorithms. Wuhan Univ. J. Nat. Sci. 15(3), 261–266 (2010) 6. Bajaj, A., Sangwan, O.P.: Discrete and combinatorial gravitational search algorithms for test case prioritization and minimization. Int. J. Inf. Technol. 13(2), 817–823 (2021). https://doi.org/10.1007/s41870-021-00628-8 7. Zhang, Y.K., Liu, J.C., Cui, Y.A., Hei, X.H., Zhang, M.H.: An improved quantum genetic algorithm for test suite reduction. In: 2011 IEEE International Conference on Computer Science and Automation Engineering, vol. 2, pp. 149–153 (2011)

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8. Sugave, S.R., Patil, S.H., Reddy, B.E.: DDF: diversity dragonfly algorithm for costaware test suite minimization approach for software testing. In: International Conference on Intelligent Computing and Control Systems, pp. 701–707 (2017) 9. Sugave, S.R., Patil, S.H., Reddy, B.E.: DIV-TBAT algorithm for test suite reduction in software testing. IET Softw. 12(3), 271–279 (2018) 10. Bajaj, A., Sangwan, O.P.: 7 nature-inspired approaches to test suite minimization for regression testing. In: Computational Intelligence Techniques and their Applications to Software Engineering Problems, pp. 99–110. CRC Press (2020) 11. Do, H., Mirarab, S., Tahvildari, L., Rothermel, G.: The effects of time constraints on test case prioritization: a series of controlled experiments. IEEE Trans. Softw. Eng. 36(5), 593–617 (2010) 12. Bajaj, A., Sangwan, O.P.: Tri-level regression testing using nature-inspired algorithms. Innov. Syst. Softw. Eng. 17(1), 1–16 (2021). https://doi.org/10.1007/ s11334-021-00384-9 13. Bajaj, A., Sangwan, O.P.: Test case prioritization using bat algorithm. Recent Adv. Comput. Sci. Commun. 14(2), 593–598 (2021) 14. Ali, M., Pant, M., Abraham, A.: Simplex differential evolution. Acta Polytechnica Hungarica, 6(5), 95–115 (2009) 15. Izakian, H., Tork Ladani, B., Zamanifar, K., Abraham, A.: A novel particle swarm optimization approach for grid job scheduling. In: Prasad, S.K., Routray, S., Khurana, R., Sahni, S. (eds.) ICISTM 2009. CCIS, vol. 31, pp. 100–109. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00405-6 14 16. Dasgupta, S., Das, S., Biswas, A., Abraham, A.: On stability and convergence of the population-dynamics in differential evolution. AI Commun. 22(1), 1–20 (2009) 17. Xhafa, F., Abraham, A.: Metaheuristics for Scheduling in Industrial and Manufacturing Applications. SCI, vol. 128. Springer, Cham (2008). https://doi.org/10. 1007/978-3-540-78985-7

Decision Making

Net-Zero Policy Performance Assessment South America Countries Through DEA Method Mehtap Dursun(B) and Rana Duygu Alkurt Industrial Engineering Department, Galatasaray University, Ortakoy, Istanbul, Turkey [email protected]

Abstract. Addressing climate change and its environmental impact is one of the greatest challenges of modern life. In fact, most of today’s sustainable strategies are essentially related to the intent of reducing overall carbon dioxide emissions, which is subsystem of carbon footprint. According to the Paris Agreement, which was signed in 2015 and entered into force in 2016, to reduce climate change, countries aim to balance the greenhouse gases released and captured until 2050. Thus, this study aims to measure the performance of selected countries based on carbon emissions in 2050. To measure performance, the Data Envelopment Analysis (DEA) method is used. In order to use this method, decision-making units (DMUs), inputs and outputs are determined. DMUs are selected from the countries that signed the Paris Agreement. Input is identified as Primary Energy Consumption. Outputs are selected as Gross Domestic Product (GDP), Carbondioxide Emission (CO2 ), Nitrous Oxide (N2 O). The countries’ primary energy consumption and total CO2 emission data are available between 1980 and 2019. The countries’ GDP and total N2 O data are available between 1990 and 2019. For each selected country, these data are fore-casted in R Studio using the Arima method with a 95% confidence interval until 2050. Performance measurements are conducted across South America. Keywords: Climate change · CO2 emission · Data envelopment analysis · Net zero policy · Performance management

1 Introduction Addressing climate change and its environmental impact is one of the greatest challenges of modern life. In fact, most of today’s sustainable strategies are essentially related to the intent of reducing overall carbon dioxide emissions which is subsystem of carbon footprint. Carbon footprint is the amount of greenhouse released to atmosphere due to specific human activities To prevent them, international negotiations began in the late 1980s [1]. Kyoto Protocol was organized between 40 countries in 1997 in order to reduce greenhouse gases such as carbon dioxide, methane etc. However, the protocol entered into force in 2005. Every year, the countries participating in the protocol hold meetings. At the 15th meeting, developed countries such as the United States played a major role in developing the Copenhagen Agreement which commits to limiting global temperature rise to 2 °C to prevent dangerous anthropogenic interference with the climate system [2]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 635–640, 2023. https://doi.org/10.1007/978-3-031-39774-5_70

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As a result of the inadequacy of the Kyoto protocol, the Paris Conference was organized in 2015. After the conference, Paris Agreement was signed but it entered into force in 2016. The target of agreement is limit global mean temperature rising less than 2 °C, even intended to rise less than 1.5 degrees. In recent years, researchers have been conducting carbon emission analyzes specific to regions. Zhou et. al. [3] divided the world into 8 regions such as OECD, Africa. They made a carbon, energy density comparison of 8 regions using the DEA method. On the other hand, some researchers have focused on specific countries. Guo et al. [4] conducted a study on 29 provinces in China. They aimed to estimate CO2 emissions reduction in provinces using multiple dea methods. Millot et al. [5] evaluated Sweden and France energy consumption. According to the evaluations, it has been revealed that it is important to turn to renewable energy sources. Although renewable energy sources are beneficial to environment, they require high investment and have many uncertainties. For example, not every country receives the same intensity of sun for solar energy. This study aims to investigate how well the South America countries can comply with the aims of the Paris Agreement. We have tried to predict the year 2050 based on input and outputs employing ARIMA forecasting method.

2 Materials and Methods In this study, input indicators are chosen as Primary Energy Consumption and output indicators are chosen as Nitrous Oxide Emission, CO2 emissions and GDP. Input and output indicators are given in Table 1. CO2 emissions, primary energy consumption, GDP and Nitrous Oxide Emission data have been found from Our World in Data (OWID) website [6]. This website has open-access data tools. T = {(K, D, U , N ) : (K) produce(D, U , N )

(1)

Table 1. Input Output Indicators Table Input Indicators

Energy Input

Primary Energy Consumption (K)

Output Indicators

Desirable Output

GDP (D)

Undesirable Output

CO2 Emission (U) N2 0 Emission (N)

The CO2 emissions amount of countries between 1980 and 2019 are found for CO2 consumptions. The energy consumption amount (Twh) of countries between 1980 and 2019 are found for primary energy consumption. Primary energy consumption include renewable and non-renewable. The GDP of countries between 1990 and 2019 are found for Gross Domestic Product. The Nitrous Oxide Emission amount of countries between 1990 and 2019 are found for Nitrous Oxide Emission. In order to evaluate the performance of 2050, these data should be estimated for the year 2050.

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637

There are many time series forecast methods such as Exponential Smoothing, AutoRegressive Integrated Moving Average (ARIMA) etc. In this study, ARIMA method is chosen to forecast these data. For each input and output, the data of the countries have been taken from the same year. All forecasts are obtained using R Studio v.2022.12.0. An ARIMA (p, d, q) model is created when the parameters are merged with the integration (differencing) term, where p, d, and q stand for the orders of autoregression, differencing, and moving average, respectively [7]. The concept can be expressed numerically as: (1−β)d Xt =

λ(β) πt φ(β)

(2)

After finding the “p, d and q” values, the data sets have been estimated until 2050 with the forecast function in R Studio. The confidence interval has been taken as 95%. DEA is a nonparametric technique to measure the relative efficiency of performance measure within a set of homogeneous decision units (DMUs) with inputs, desirable outputs and undesirable outputs [8]. The first DEA model, which named as CCR (Charles, Cooper and Rhodes), developed by Charnes et al. [9]. There are two CCR DEA methods according to the change in objective functions. These are input-oriented and outputoriented which have desirable inputs and desirable outputs. The input-oriented CCRDEA model evaluate the relative efficiency of DMUs. This evaluation is done by maximizing the ratio of the total weighted output to the total weighted input, This model constraint is output-to-input ratio of each DMU should be less than or equal to unity. The input-oriented CCR-DEA model can be represented as follow; s uk ykjo max Ejo = k=1 m i=1 vi xijo Subject to s uk ykj k=1 m i=1 vi xij uk , vi ≥∈,

≤1

∀j,

(3)

∀k, i,

On the other hand, Outputs or inputs can be undesirable like CO2 Emission or GHG Emission. The economic justification for using undesirable output variables as inputs in DEA models. Inputs and undesirable outputs cost a DMU money, therefore DMUs often try to minimize both sorts of variables. To deal with undesirable outputs, new mathematical models have been developed by changing the constraints. For example, Tyteca [10] developed model which name is Pure Environmental Performance Index. Min θ Subject to K k=1

zk xnk ≤ xn0

∀n,

(4)

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K k=1

zk ymk ≤ ym0

∀m,

zk ujk = θ uj0

∀j,

K k=1

K k=1

zk ≤ 1

Zk ≥ 0, k = 1, .., K. In above linear model is consist of an input vector xk whose nth component xnk is the amount of input n consumed by DMUk , output vector yk whose mth component ymk is the amount of desirable output m yield by DMUk , an output vector ujk whose jth component ujk is the amount of undesirable output j yield by DMUk .

3 Case Study The case study is performed in 11 South America countries. As mentioned in the previous section, forecasts have been made for each input and each output until 2050. Forecasts for 2050 are given in Table 2. Table 2. South America Input & Output 2050 Forecast Data Countries

K(Twh)

D

U

N

Argentina

1199.4

22066.1

234037143

64316297

12430.039

37330067

23806721

Bolivia Brazil

146.7181

19144.82

693732962

298465444

Chile

779.9786

41128.13

1481153048

10181286

Colombia

816.1198

21240.17

137981294

29193096

Ecuador

348.4132

14519.68

60773801

8143041

22165.4

2730134

1455151

17040.82

12870042

16333575

Guyana

5358.412

11.15125

Paraguay

226.1752

Peru

725.0964

20583.21

84399631

14656207

Suriname

14.03193

18911.89

2682870

381911

Uruguay

98.56587

34805.63

6489942

7840000

K:Primary Energy Consumption D:GDP U:CO2 Emission N:N20 Emission

Due to the fact that forecasts are in different units, input and each outputs are needed normalized within themselves. To normalize forecasts, each country data is divided by the highest country data at the relevant input or output (Linear Normalization). For example, the normalized gdp value of Argentine has been found by dividing the gdp value of Chile, which is the highest gdp value (Table 3).

Net-Zero Policy Performance Assessment South America Countries

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Table 3. South America Input & Output 2050 Forecast Normalized Data Countries

K

D

U

N

Argentina

0.223835

0.536521

0.15801

0.21549

Bolivia

0.027381

0.302227

0.025203

0.079764

Brazil

1

0.465492

0.468374

1

Chile

0.145562

1

1

0.034112

Colombia

0.152306

0.516439

0.093158

0.097811

Ecuador

0.065022

0.353035

0.041031

0.027283

Guyana

0.002081

0.538935

0.001843

0.004875

Paraguay

0.042209

0.414335

0.008689

0.054725

Peru

0.135319

0.500465

0.056982

0.049105

Suriname

0.002619

0.459829

0.001811

0.00128

Uruguay

0.018395

0.846273

0.004382

0.026268

K:Primary Energy Consumption D:GDP U:CO2 Emission N:N20 Emission

To measure performance, Pure Environmental Performance Index mathematical model has been coded in General Algebraic Modeling System v.42.5.0 (GAMS). In this model, it was run by putting normalized values. Models results are shown in Table 4. Table 4. South America Performance Measurement Countries

Results

Argentina

0.0222

Bolivia

0.0433

Brazil

1

Chile

0.0395

Colombia

0.0361

Ecuador

1

Guyana

1

Paraguay

1

Peru

0.0640

Suriname

1

Uruguay

1

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4 Concluding Remarks According to performance results, Argentina is the country with the lowest performance. The result is an indication that the desirable output of the Argentina is not good enough compared to the undesirable input. Although Bolivia, Chile, Colombia and Peru have higher performances than Argentina, their results were not good. Brazil, Ecuador, Guyana, Paraguay, Suriname and Uruguay seem to have paid more attention to the Paris Agreement in 2050 than other countries on their continents. However, this assessment is made only among countries. All South America countries should continue to take precautions for a sustainable world. Future researchers can do this study specific to continents. Thus, the performance of countries in other continents can be measured. Necessary measures can be taken according to these measurements. Acknowledgement. This work is financially supported by Galatasaray University Research Fund Project FOA-2022–1092.

References 1. Vlassopoulos, C.A.: Competing definition of climate change and the post-Kyoto negotiations. Int. J. Clim. Change Strat. Manage. 4(1), 104–118 (2014) 2. Lau, L.C., Lee, K.T., Mohamed, A.R.: Global warming mitigation and renewable energy policy development from the Kyoto protocol to the Copenhagen accord—a comment. Renew. Sustain. Energy Rev. 16(7), 5280–5284 (2012) 3. Zhou, P., Ang, B.W., Poh, K.L.: Measuring environmental performance under different environmental DEA technologies. Energy Economics 30(1), 1–14 (2008). https://doi.org/10.1016/ j.eneco.2006.05.001 4. Guo, X.-D., Zhu, L., Fan, Y., Xie, B.-C.: Evaluation of potential reductions in carbon emissions in Chinese provinces based on environmental DEA. Energy Policy 39(5), 2352–2360 (2011) 5. Millot, A., Krook-Riekkola, A., Maïzi, N.: Guiding the future energy transition to net-zero emissions: lessons from exploring the differences between France and Sweden. Energy Policy 139, 111358 (2020) 6. https://ourworldindata.org/ 7. Taneja, K., Ahmad, S., Ahmad, K., Attri, S.D.: Time series analysis of aerosol optical depth over New Delhi using Box–Jenkins ARIMA modeling approach. Atmos Pollut. Res. 7(4), 585–596 (2016) 8. Martín-Gambo, M., Iribarren, D.: Coupled life cycle thinking and data envelopment analysis for quantitative sustainability improvement. Methods in Sustainability Science: Elsevier (2021). 295–320 9. Charnes, A., Cooper, W.W., Rhodes, E.L.: Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2(6), 429–444 (1978) 10. Tyteca, D.: On the measurement of the environmental performance of firms— a literature review and a productive efficiency perspective. J. Environ. Manage. 46(3), 281–308 (1996)

Sustainable Supplier Selection in Fuzzy Environment: A Case Study in Turkey Ilgaz Cerit(B) and Tuncay Gürbüz Galatasaray University, Istanbul, Turkey [email protected], [email protected]

Abstract. In today’s business world, the supply chain performances of companies determine their competitiveness. To have a seamless supply chain management system, companies are seeking sustainable suppliers. That’s because, in the business world we live in now, selecting sustainable suppliers has a critical effect on the competitiveness of the entire supply chain network [1]. To determine whether the supplier is sustainable or not, three main criteria of sustainability are taken into account which are called environmental, fiscal, and social sustainability. A supplier which can give the best overall performance according to these criteria should be selected. However, the selection process of the suppliers is consisting of many linguistic terms and their evaluation contains abundant errors due to the human element. Most of the time, linguistic terms might cloud human judgment and make it challenging to select the most appropriate choice. In this study, a sustainable supplier selection has been made by using the newly proposed Spherical Fuzzy the Order of Preference by Similarity to the Ideal Solution (SF-TOPSIS) Method based on the weights determined by a selected expert group of ten people, who are supply chain professionals working for the same company, were tasked to determine the interdependencies of fifteen criteria. This study aims to show that SF-TOPSIS is prosperous to minimize human error and can give the best result. Keywords: Sustainability · Supplier Selection · Fuzzy MCDM · Spherical Fuzzy · TOPSIS

1 Introduction Through time and developed business conducting methods, there are several explanations came up for the term called sustainability. In general, sustainability means being perpetual to provide continuity [2]. This term is used in many forms which compensates both inner and outer factors of a company. According to Brundtland Report, sustainability is defined as “The ability of future generations to meet their current needs without compromising their capacity to meet their own needs” [3]. However, the term sustainability evolved and changed circumstances. Researchers and business professionals also came up with broader definitions of it. Dyllick and Hockerts state that, the long-run sustainability of businesses is possible not only by managing economic factors but also by managing social and environmental factors simultaneously [4]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 641–651, 2023. https://doi.org/10.1007/978-3-031-39774-5_71

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For being fiscally sustainable, suppliers must be focusing on their financials. Fiscal or economic sustainability is defined by several conventional criteria, which are determined by researchers who work on supply chain management, which includes transaction cost economics and a resource-based approach to business management [5]. Environmental criteria are mainly consisting of pollution and management based. To fulfill the requirements, suppliers should be able to minimize their pollutants and/or find a way to manage their waste. To do that, they can implement ecological design methods while building facilities or determining cleaner routes for their logistics [6]. To be socially sustainable, manufacturers must consider the most common areas which are worker’s health, human rights, equity, diversity, and other social & securityrelated to become competitive in the market [7]. As is aimed in this study, when it is desired to make a selection among alternatives that have criteria listed according to hierarchy, MCDM methods can be used, such as TOPSIS. It is proposed by Hwang and Yoon to solve MCDM problems in 1981 [8]. After this proposal, he and other researchers work on TOPSIS to improve the original idea. In 1987, Yoon published an upgraded version of TOPSIS [9]. After Yoon, further developments continued, and in 1993, Hwang et.al. Published the final form of the TOPSIS algorithm [10]. All these research and developments made TOPSIS a widely used MCDM algorithm. After the increased popularity of decision-making, new branches of mathematics started to get involved in it to increase the accuracy of decision-making processes. One of them is fuzzy sets, which are introduced in the mid-1960s by Zadeh [11]. After Zadeh’s introduction, several researchers developed further extensions to improve the ordinary fuzzy set approach. These extensions are listed in Table 1. Table 1. Types of Fuzzy Extensions Name of Extension

Explanation

Introduced By

Type 2 Fuzzy Sets

Used to determine exact membership function for a fuzzy set

Zadeh – 1975 [12]

Intuitionistic Fuzzy Sets Both membership and non-membership can Atanassov – 1975 [13] be defined Hesitant Fuzzy Sets

Allows potential degrees of membership of an element to a set

Torra – 2010 [14]

Pythagorean Fuzzy Sets Membership degree and a non-membership Yager – 1980 [15] degree satisfying the condition that the square sum of its membership degree and non-membership degree is equal to or less than one

The uniqueness of this study is its’ application area. There isn’t any research on this scale in literature as of now. There are few studies including both sustainability and supplier selection terms however, none of them use Spherical Fuzzy TOPSIS. By using

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643

SF-TOPSIS, decision-makers can be able to show their hesitancy by using linguistic terms in the decision-making process. The remaining parts of this paper are defined as follows: The second section is consisting of details of the Sustainable Supplier Selection Problem and a Literature Review about it. The third one is consisting of Spherical Fuzzy TOPSIS. The fourth section will be the application of it. Finally, the last section is consisting of concluding remarks.

2 Sustainable Supplier Selection Problem Literature Review There are more than 100 papers about supplier selection problem, most of them including hybrid methods [16]. However, including SF-TOPSIS, hospital selection [17], construction company selection [18] and package tour provider problem [19] are the notable ones. In addition to those, there are several researches that include sustainability terms without making any supplier selection. Those articles mainly consider the impacts of these terms to several areas of the business [6, 7].

3 Spherical Fuzzy TOPSIS Before explaining steps of the SF-TOPSIS, two necessary equations will be shared. First one is multiplication of SF numbers and second one is Spherical Weighted Arithmetic Mean (SWAM) operator, given below [20]; ⎧ ⎫ 1/2 ⎬ μA˜ s μB˜ s , (v2˜ + V ˜2 − v2˜ V ˜2 ) , ∼ ⎨ Bs AS ˜ B=    AS Bs  (1) A⊕ 1/2 ⎩ ( 1 − V 2 π 2 + 1 − v2 π 2 − π 2 π 2 ) ⎭ ˜ ˜ ˜ ˜ ˜ ˜ Bs

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

As

AS

Bs

n

As Bs

to, w = (w  1 , w2 , . . . , wn ); wi ε[0, 1]; i=1 wi = 1 SWAMw A˜ S1 , . . . , A˜ sn = w1 A˜ s1 + · · · + wn A˜ sn wi  21

n  2 ⎪ 1 − 1 − μ , ⎪ i=1 ˜ ⎪ ⎪  wi As ⎪

w 1/2 ⎪ 2 ⎩ n vwi , [ n − ni=1 (1 − μ2˜ − π 2˜ ) i ] i=1 ˜ i=1 1 − μ ˜ AS

As

As

(2)

Asi

At the first step of SF-TOPSIS, DM’s fill decision and criteria evaluation matrices by using linguistic terms given in Table 2. Then, by using SWAM operator given in Eq. 2, aggregate judgements of DM’s and criteria weights. Then construct SF decision matrix. After having SF-Decision Matrix, use aggregated weights to construct aggregated weighted SF-Decision Matrix by using Eq. 1. Later, defuzzify that matrix by using Eq. 3 given below, to calculate scores which will be used to determine SF positive and negative values.   ijw 2 ijw 2 ) − (vijw − ) Score Cj (Xiw ) = (2μijw − 2 2

(3)

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In next step determine SF-PIS and SF-NIS by using Eqs. 4 and 5 with respect to each other.     X ∗ = Cj , maxi < Score Cj (Xiw ) > |j = 1, 2 . . . , n (4)     X  = Cj , mini < Score Cj (Xiw ) > |j = 1, 2 . . . , n

(5)

Then calculate distances between alternative Xi and SF-PIS and SF-NIS with respect to each other. Normalized Euclidean Distance is used in our problem [20]   n 1    D Xi , X ∗ =  ((μxi − μx∗ )2 + (vxi − vx∗ )2 + (πxi − πx∗ )2 ) (6) 2n i=1

  n 1  D(Xi , X ) =  ((μxi − μx )2 + (vxi − vx )2 + (πxi − πx )2 ) 2n

(7)

i=1

Finally, Classical Closeness Ratio must be calculated by using Eq. 8 and alternatives must be ranked in decreasing closeness ratio order. ξ (Xi ) =

D(Xi , X ) D(Xi , X ∗ ) + D(X i , X )

(8)

4 Application of Problem To find the most appropriate supplier, three suppliers selected for this problem (S1, S2, S3). Based on detailed literature review and expert opinion, under three main criteria group, fifteen sub criteria were determined. Main criteria; Fiscal Sustainability (FC), Environmental Sustainability (EC) and Social Sustainability (SC). For Fiscal Sustainability, sub criteria are; Logistics Expenses (FC01), Financial Stability (FC02), Daily Delivery Capacity (FC03), Infrastructure Expenses (FC04), and Maintenance Expenses of Infrastructure (FC05).For Environmental Sustainability, sub criteria are; Proximity to Supplier’s Depot to Companies’ Facilities (EC01), Zero Waste Goals (EC02), Carbon Emissions (EC03), Share of Renewable Energy in Total Consumption (EC04), and Impact on Biodiversity (EC05).For Social Sustainability, sub criteria are; Customer Service (SC01), Past Experience (SC02), Company Image (SC03), Diversity Among Employees (SC04), and Health and Safety Regulations (SC05).There are ten decision makers (DM1…DM10), who are selected according to their experience level and all have different weights on decision making process which given in starting from DM1 to 10 (0,22/0,22/0,12/0,12/0,12/0,08/0,04/0,04/0,02/0,02). At first, importance weights assigned by DM’s and DM’s judgements are gathered which are given in Table 3 and Table 4 and Table 5 with respect to each other. Judgments shown in Table 4 and Table 5 are aggregated by using SWAM operator which is given in Eq. 2 and Decision Matrix is formed shown in Table 6. By using weights in Table 3 and aggregated decision matrix in Table 6, weighted decision matrix is formed by using

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Eq. 1, given in Table 7. To determine Score Function values, Table 7 and Eq. 4 and 5 are used and score values are given in Table 8. Highest values in score function represents Positive Ideal Solution (PIS) and lowest values represents Negative Ideal Solution (NIS). By using PIS and NIS values in Score Function, SF-PIS and SF-NIS are shown in Table 9. By using Eq. 6 and Eq. 7, distances between SF-PIS and SF-NIS and alternative Xi is calculated. Instead of X, term S is used as a referring to Supplier. Finally, by using Eq. 8 closeness ratios are calculated and shown in Table 10. Also ranking order, which is from highest to lowest closeness ratio value, is given in Table 10. The overall ranking is S3 > S2 > S1 . Table 2. Linguistic Terms and Their Corresponding Spherical Fuzzy Numbers [17] Linguistic Terms

(μ, v, π )

Linguistic Terms

(μ, v, π )

Absolutely Important (AI)

(0.9, 0.1,0.1)

Slightly Unimportant (SU)

(0.4, 0.6, 0.6)

More Important (MI)

(0.8, 0.2, 0.2)

Unimportant (U)

(0.3, 0.7, 0.7)

Important (I)

(0.7, 0.3, 0.3)

Strongly Unimportant (STU)

(0.2, 0.8, 0.8)

Slightly Important (SI)

(0.6, 0.4, 0.4)

Absolutely Unimportant (AU)

(0.1, 0.9, 0.9)

Equally Important (EI)

(0.5, 0.5, 0.5)

Table 3. Importance Weights of Criteria by Decision Makers SF Format SubCriteria μ



SF Format π

SubCriteria μ



SF Format π

SubCriteria μ



π

FC01

0,74 0,26 0,27 EC01

0,66 0,34 0,35 SC01

0,44 0,57 0,59

FC02

0,66 0,34 0,35 EC02

0,54 0,47 0,47 SC02

0,77 0,24 0,26

FC03

0,78 0,22 0,25 EC03

0,63 0,38 0,39 SC03

0,57 0,43 0,44

FC04

0,57 0,44 0,44 EC04

0,42 0,58 0,60 SC04

0,30 0,71 0,73

FC05

0,32 0,71 0,75 EC05

0,50 0,51 0,52 SC05

0,71 0,30 0,32

Table 4. Judgements by Decision Makers FS01

FS02

FS03

FS04

FS05

ES01

ES02

ES03

S1

MI

I

AI

I

AU

I

EI

I

S2

I

I

MI

EI

U

I

EI

I

S3

EI

SI

AI

EI

STU

I

I

SI

S1

AI

MI

AI

EI

STU

EI

EI

I

DM1

DM2 (continued)

646

I. Cerit and T. Gürbüz Table 4. (continued) FS01

FS02

FS03

FS04

FS05

ES01

ES02

ES03

S2

MI

I

I

S3

I

EI

EI

EI

STU

I

I

SI

EI

U

EI

I

SI

S1

MI

EI

MI

I

AU

EI

I

EI

S2

I

SI

I

I

AU

EI

SI

SI

S3

I

MI

EI

SI

AU

I

SI

SI

S1

I

I

MI

EI

U

I

SI

I

S2

MI

I

MI

SI

U

SI

I

I

S3

I

I

MI

SI

STU

I

I

I

S1

SI

MI

I

SI

SU

SI

EI

EI

S2

EI

I

I

SI

SU

EI

EI

EI

S3

I

EI

MI

I

EI

I

SI

SI

S1

SI

SI

SI

I

SU

SI

SU

SI

S2

I

SI

EI

SI

U

EI

U

SI

S3

I

I

SI

EI

EI

SU

SU

I

S1

I

I

MI

EI

EI

EI

SU

MI

S2

I

MI

I

EI

EI

EI

EI

I

S3

I

MI

SI

EI

EI

SU

EI

MI

S1

MI

I

EI

SI

STU

EI

I

SI

S2

I

SI

SI

SI

SU

EI

I

EI

S3

MI

I

MI

EI

U

SI

SU

SI

S1

EI

AI

I

EI

AU

SI

SU

EI

S2

SI

MI

I

SI

STU

I

SU

SI

S3

I

I

MI

SI

U

SI

SU

SU

S1

MI

AI

MI

EI

AU

I

EI

SU

S2

MI

MI

MI

I

U

EI

SU

U

S3

MI

I

MI

I

U

SI

U

U

DM3

DM4

DM5

DM6

DM7

DM8

DM9

DM10

Table 5. Judgements of Decision Makers (Rest of Table 4) ES04

ES05

SC01

SC02

SC03

SC04

SC05

EI

U

EI

I

EI

U

I

EI

U

EI

I

EI

U

I

SU

SU

SI

MI

SI

U

I

DM1

(continued)

Sustainable Supplier Selection in Fuzzy Environment Table 5. (continued) ES04

ES05

SC01

SC02

SC03

SC04

SC05

SU

SU

SI

MI

SI

SU

I

U

U

SI

MI

EI

U

MI

U

U

EI

MI

EI

STU

MI

U

SU

MI

I

SU

STU

I

U

SU

SI

MI

SU

AU

AI

U

SU

EI

I

U

STU

MI

SU

EI

U

AI

U

U

EI

SU

U

EI

AI

U

U

EI

U

EI

SU

MI

U

SU

SI

U

EI

SU

SI

U

STU

SI

U

EI

EI

EI

SU

STU

I

U

EI

I

SI

STU

AU

SI

SU

EI

I

SI

EI

AU

SI

U

SU

EI

I

U

STU

I

SU

SU

SU

SI

U

U

MI

EI

STU

SU

MI

U

U

I

SU

U

U

I

EI

EI

MI

EI

U

U

SI

SU

SU

MI

EI

STU

U

SI

STU

SU

I

U

STU

U

I

AU

EI

SI

U

EI

U

MI

STU

EI

MI

U

U

U

SI

U

SU

SI

U

U

EI

I

EI

EI

EI

EI

STU

EI

EI

U

STU

EI

SU

AU

EI

EI

EI

U

EI

EI

U

SI

I

STU

U

MI

U

AU

SI

I

U

STU

SI

DM2

DM3

DM4

DM5

DM6

DM7

DM8

DM9

DM10

647

648

I. Cerit and T. Gürbüz Table 6. Decision Matrix FC01 μ

FC02 

π

μ

FC03 

π

μ

FC04 

π

μ

FC05 

π

μ



π

S1 0,79 0,21 0,23 0,73 0,27 0,29 0,83 0,17 0,19 0,61 0,39 0,40 0,26 0,76 0,80 S2 0,73 0,28 0,29 0,69 0,31 0,32 0,73 0,27 0,28 0,58 0,43 0,43 0,30 0,72 0,74 S3 0,68 0,33 0,34 0,65 0,36 0,37 0,76 0,25 0,29 0,56 0,44 0,45 0,33 0,70 0,73 EC01

EC02

EC03

EC04

EC05

S1 0,61 0,40 0,41 0,55 0,46 0,47 0,65 0,35 0,37 0,41 0,59 0,60 0,40 0,61 0,62 S2 0,62 0,38 0,40 0,59 0,41 0,44 0,63 0,38 0,39 0,38 0,63 0,64 0,35 0,66 0,66 S3 0,63 0,37 0,39 0,64 0,37 0,39 0,63 0,37 0,38 0,35 0,66 0,66 0,41 0,60 0,62 SC01

SC02

SC03

SC04

SC05

S1 0,57 0,45 0,48 0,75 0,26 0,27 0,46 0,55 0,57 0,30 0,71 0,73 0,66 0,34 0,35 S2 0,53 0,47 0,48 0,76 0,24 0,26 0,43 0,58 0,60 0,30 0,72 0,74 0,75 0,26 0,27 S3 0,53 0,47 0,49 0,75 0,25 0,27 0,43 0,58 0,62 0,28 0,73 0,75 0,74 0,27 0,28

Table 7. Weighted Decision Matrix FC01 μ

FC02 

π

μ

FC03 

π

μ

FC04 

π

μ

FC05 

π

μ



π

S1 0,59 0,33 0,34 0,48 0,43 0,42 0,65 0,28 0,30 0,35 0,56 0,52 0,08 0,89 0,44 S2 0,54 0,37 0,37 0,46 0,45 0,43 0,57 0,35 0,36 0,33 0,58 0,53 0,09 0,87 0,49 S3 0,50 0,41 0,40 0,43 0,48 0,46 0,59 0,33 0,36 0,32 0,59 0,53 0,10 0,86 0,50 EC01

EC02

EC03

EC04

EC05

S1 0,40 0,51 0,48 0,29 0,62 0,55 0,41 0,50 0,48 0,18 0,75 0,58 0,20 0,73 0,60 S2 0,41 0,50 0,47 0,32 0,59 0,54 0,40 0,51 0,49 0,16 0,78 0,58 0,18 0,76 0,60 S3 0,42 0,49 0,47 0,34 0,57 0,53 0,40 0,51 0,49 0,15 0,79 0,58 0,20 0,72 0,59 SC01

SC02

SC03

SC04

SC05

S1 0,25 0,68 0,60 0,57 0,34 0,36 0,26 0,66 0,58 0,09 0,87 0,50 0,47 0,44 0,43 S2 0,23 0,69 0,59 0,58 0,34 0,35 0,25 0,68 0,59 0,09 0,87 0,49 0,53 0,38 0,39 S3 0,23 0,69 0,59 0,58 0,34 0,35 0,25 0,68 0,60 0,09 0,88 0,48 0,52 0,39 0,40

Sustainable Supplier Selection in Fuzzy Environment

649

Table 8. Score Function Values Based on SWAM Operator FC01

FC02

FC03

FC04

FC05

EC01

EC02

EC03

S1

0,981

0,526

1,308

0,101

-0,450

0,247

-0,020

0,269

S2

0,761

0,429

0,900

0,049

-0,393

0,270

0,032

0,227

S3

0,600

0,332

0,980

0,034

-0,373

0,295

0,087

0,229

EC04

EC05

SC01

SC02

SC03

SC04

SC05

S1

−0,211

−0,172

−0,106

0,910

−0,079

−0,378

0,467

S2

−0,233

−0,206

−0,129

0,961

−0,108

−0,389

0,720

S3

−0,249

−0,171

−0,128

0,919

−0,108

−0,403

0,683

Table 9. SF-PIS and SF-NIS Values Based o SWAM Operator FC01 μ

FC02 

π

μ

FC03 

π

μ

FC04 

π

μ

FC05 

π

μ



π

S* 0,42 0,43 0,42 0,35 0,50 0,47 0,47 0,39 0,40 0,25 0,61 0,53 0,08 0,88 0,48 S- 0,36 0,48 0,46 0,31 0,54 0,49 0,41 0,44 0,43 0,23 0,63 0,54 0,06 0,90 0,42 EC01

EC02

EC03

EC04

EC05

S* 0,28 0,58 0,52 0,23 0,64 0,55 0,27 0,58 0,53 0,12 0,79 0,56 0,14 0,76 0,57 S- 0,27 0,59 0,52 0,20 0,68 0,56 0,26 0,59 0,53 0,10 0,82 0,55 0,12 0,79 0,57 SC01

SC02

SC03

SC04

SC05

S* 0,13 0,76 0,57 0,32 0,55 0,53 0,14 0,74 0,57 0,05 0,90 0,44 0,29 0,57 0,54 S- 0,13 0,77 0,56 0,31 0,55 0,53 0,13 0,76 0,57 0,05 0,90 0,42 0,28 0,58 0,54

Table 10. Closeness Ratio and Ranking Order of Alternatives D(Xi,X*)

D(Xi,X’)

Closeness Ratio

Rank

S1

0,591

0,398

0,402

3

S2

0,393

0,460

0,539

2

S3

0,402

0,559

0,582

1

5 Conclusion Spherical Fuzzy Sets and Spherical Fuzzy TOPSIS is a recently proposed method by Kutlu Gündo˘gdu and Kahraman [21] and it is applied to a Sustainable Supplier Selection Problem. The main objective was to show that SF-TOPSIS can be applied to large-scale problems such as this one. To decrease the human element in errors, this method can be applied due to its usage of linguistic variables and a wide range of corresponding

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fuzzy numbers for these linguistic terms. For further research, other techniques such as AHP [22], TOPSIS [23], VIKOR [24], PROMETHEE [25], etc. can be used in this problem to compare their performances. Also, other Spherical Fuzzy MCDM methods can be applied to this problem such as Interval Values SF-TOPSIS [26], SF-VIKOR [27], and SF-WASPAS [28]. Another option can be, increasing the amount of criteria and/or number of decision makers can be increased to make further research. Finally, a layered criteria structure can be assembled such as using weight values for both main and sub-criteria and calculating a global criterion for sub-criteria instead of using them as groups as it is demonstrated in this application.

References 1. De Boer, L., Labro, E., Morlacchi, P.: A review of methods supporting supplier selection. Eur. J. Purch. Supply Manag. 7, 75–89 (2001) 2. Saban, M., Küçüker, H., Küçüker, M.: Kurumsal sürdürülebilirlik ile ilgili raporlama çerçeveleri ve sürdürülebilirlik raporlamasında muhasebenin rolü. ˙I¸sletme Bilimi Dergisi 5(1), 101–115 (2017) 3. Karcıo˘glu, R., Öztürk, S.: Sürdürülebilirlik muhasebesi ve raporlaması. Gazi Kitabevi (2021) 4. Dyllick, R., Hockerts, K.: Beyond the business case for corporate sustainability. Bus. Strateg. Environ. 11(2), 130–141 (2002) 5. Baskaran, V., Nachiappan, S., Rahman, S.: Indian textile suppliers’ sustainability evaluation using the Grey approach. Int. J. Prod. Econ. 135, 647–658 (2012) 6. Handfield, R., Walton, S., Sroufe, R., Melnyk, S.: Applying environmental criteria to supplier assessment: a study in the application of the Analytical Hierarchy Process. Eur. J. Oper. Res. 141, 70–87 (2002) 7. Sutherland, J., Richter, J., Hutchins, M.: The role of manufacturing in affecting the social dimension of sustainability. CIRP Ann. 65(2), 689–712 (2016) 8. Hwang, C.-L., Yoon, K.: Methods and applications a state-of-the-art survey. In: Multiple Attribute Decision Making. Springer, Heidelberg (1981). https://doi.org/10.1007/978-3-64248318-9 9. Yoon, K.: A reconciliation among discrete compromise situations. J. Oper. Res. Soc. 38(3), 277–286 (1987) 10. Hwang, C.-L., Lai, Y., Liu, T.: A new approach for multiple objective decision making. Comput. Oper. Res. 20(8), 889–899 (1993) 11. Zadeh, L.A.: Fuzzy Sets. Inf. Control 8, 338–353 (1965) 12. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8, 199–249 (1975) 13. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986) 14. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010) 15. Yager, R.: Pythagorean fuzzy subsets. In: Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, pp. 57–61 (2013) 16. Junyi, C., James, N.K., Liu, E., Ngai, W.T.: Application of decision-making techniques in supplier selection: a systematic review of literature. Exp. Syst. Appl. 40(10), 3872–3885 (2013) 17. Kahraman, C., Kutlu Gundogdu, F., Cevik Onar, S., Oztaysi, B.: Hospital selection using spherical fuzzy TOPSIS. In: Proceedings of the 11th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), pp. 77–82 (2019)

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18. Gurmani, S.H., Chen, H., Bai, Y.: Multi-attribute group decision-making model for selecting the most suitable construction company using the linguistic interval-valued T-spherical fuzzy TOPSIS method. Appl. Intell. (2022) 19. Nguyen, P.H.: Spherical fuzzy decision-making approach integrating Delphi and TOPSIS for package tour provider selection. Math. Prob. Eng. 2022 (2022) 20. Lin, C.T., Tsai, M.C.: Location choice for direct foreign investment in new hospitals in China by using ANP and TOPSIS. Qual. Quant. 44(2), 375–390 (2010) 21. Kutlu Gündo˘gdu, F., Kahraman, C.: Spherical fuzzy sets and spherical fuzzy TOPSIS method. J. Intell. Fuzzy Syst. 36(1), 337–352 (2019) 22. Pr˚uša, P., Hruška, R., Babi´c, D.: The use of AHP method for selection of supplier. Transport 29, 195–203 (2014) 23. George, J., Singh, A., Kumar Bhaisare, A.: Implementation of TOPSIS technique for supplier selection. Int. Res. J. Eng. Technol. 5(6), 2582–2585 (2018) 24. Sanayei, A., Farid Mousavi, S., Yazdankhah, A.: Group decision making process for supplier selection with VIKOR under fuzzy environment. Expert Syst. Appl. 37(1), 24–30 (2010) 25. Abdullah, L., Chan, W., Afshari, A.: Application of PROMETHEE method for green supplier selection: a comparative result based on preference functions. J. Indust. Eng. Int. 15(2), 271–285 (2018) 26. Kutlu Gündo˘gdu, F., Kahraman, C.: A novel fuzzy TOPSIS method using emerging interval valued spherical fuzzy sets. Eng. Appl. Artif. Intell. 85, 307–323 (2019) 27. Kutlu Gündo˘gdu, F., Kahraman, C.: A novel VIKOR method using spherical fuzzy sets and its application to warehouse site selection. J. Intell. Fuzzy Syst. 37, 1–15 (2019) 28. Kutlu Gündo˘gdu, F., Kahraman, C.: Extension of WASPAS with spherical fuzzy sets. Informatica 30(2), 269–292 (2019)

Evaluation of Mechanical Energy Storage Technologies in the Context of a Fuzzy Environment Ahmet Sarucan , Mehmet Emin Baysal(B)

, and Orhan Engin

Industrial Engineering Department, Konya Technical University, Konya, Turkey {asarucan,mebaysal,oengin}@ktun.edu.tr

Abstract. Clean energy sources are becoming increasingly important as traditional energy sources are rapidly depleted around the world. The energy produced must be stored to make this orientation more meaningful. One of the very different storage technologies used is mechanical energy storage. Mechanical energy makes up a large percentage of the energy storage capacity in the world. Keeping track of current mechanical energy storage technology developments is a key task for policymakers. From this point of view, it is necessary to have an evaluation of different mechanical energy storage techniques under certain criteria and in consideration of the opinions of decision makers. Therefore, a fuzzy multi-criteria decision-making model is proposed. In the case study, 3 different mechanical energy storage techniques, pumped hydro, compressed air, and flywheel energy storage alternatives were evaluated with the fuzzy grey relation analysis for common criteria, power capacity, lifetime, energy density, efficiency, response time of these alternatives. Growing use of energy storage in every aspect of life, from electric vehicles to mobile applications, from smart buildings to cities, makes these studies relevant. Keywords: Fuzzy Environment · Mechanical Energy Storage · Multi-Criteria Decision-Making

1 Introduction Electricity is of vital importance to human life today. Because people always need electricity to continue their daily lives. Energy storage technologies are developed to store electricity for the later demand response. Selection of energy stor-age technology is very important for demanded energy response. There are some studies in the literature about the selection of energy storage technology. These are summarized below. Montignac et al. [1] proposed the MACBETH multi-criteria evaluation approach to evaluate and compare hydrogen storage technologies. Daim et al. [2] evaluated pumped hydro energy storage, compressed air energy storage, and sodium sulfur battery storage as a multicriteria decision problem. They used the fuzzy Delphi method to select the evaluation criteria. To derive the relative weights of the evaluation criteria, they used Analytic Hierarchy Process (AHP). Li et al. [3] used a multi-objective optimization method based © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 652–659, 2023. https://doi.org/10.1007/978-3-031-39774-5_72

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on an augmented ε constraint method with the economic and environmental objectives for the optimal selection of energy storage technology. Ren and Ren [4] developed a multi-attribute decision analysis framework for energy storage technologies. They used ten criteria in four categories that are economic, performance, technological and environmental. To determine the weights of the evaluation criteria, they used fuzzy numbers for establishing the comparative judgments. They studied five energy storage technologies. Zhang et al. [5] proposed MULTIMOORA-IFN2 technique for multicriteria decision making to select energy storage technology. They compared their results with TOPSIS and VIKOR methods. Nagaraju et al. [6] proposed fuzzy AHP with fuzzy VIKOR to determine the best technology for energy storage. They evaluated it as five energy stores. In this study, a fuzzy multi-criteria decision making method is adapted to evaluate the 3 different mechanical energy storage techniques, pumped hydro, compressed air and flywheel energy storage alternatives. As a contribution to the literature, the evaluation of mechanical energy storage techniques in a fuzzy environment with gray relation analysis is performed for the first time. The paper is organized as follows. Section 2 explains mechanical energy storage technologies. Section 3 explains the methodology. Section 4 gives a case study on the selection of mechanical energy storage technology. Section 5 gives conclusions and future study.

2 Mechanical Energy Storage Technologies Energy storage technologies are classified according to their applications and technical characteristics. Generally, these can be divided into five categories based on the energy stored. These are mechanical, electrochemical, electrical, chemical and thermal storage [3]. The mechanical energy storage technologies are known as pumped hydro energy storage, compressed air energy storage, and flywheel energy storage technologies. These technologies are briefly described as follows. Pumped Hydropower Energy Storage Technology (PHEST): In PHEST, potential energy is stored by pumping water from a lower level reservoir to a higher level reservoir. This situation; It is done by using the electrical energy taken from the grid to operate the water pumps during the periods when the energy consumption is low. The water, which is pumped and stored in the upper reservoir, is sent to the water turbine in periods when the electrical energy requirement is high, and mechanical energy is obtained. The mechanical energy obtained is converted into electrical energy with the help of a generator and transferred to the electrical grid [7]. PHEST has been widely used for more than two decades. It is preferred for the high storage capacity. For large amounts of cheap energy during off-peak period, it is very useful for arbitrage storage. Compressed Air Energy Storage Technology (CAEST): In the hours when the load on the electrical system is low and the electricity price is cheap, in case of running the compressor, the air in the environment is compressed and stored in the underground impermeable caves. When there is a need for energy, turbines are operated with the help of compressed air and electrical energy is produced [7]. Also, CAEST has high storage capacity and it is useful for arbitrage storage of energy, during off-peak period.

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Flywheel Energy Storage Technology (FEST): The flywheel stores the energy in the form of kinetic energy, thanks to the motor rotating at high speed, when the demand for electrical energy is not intense. Later, when the demand for electrical energy is high, kinetic energy is converted into electrical energy and used. When energy is taken from the system, the number of revolutions decreases due to the law of conservation of energy. Similarly, when the system is energized, the number of revolutions increases [7]. FEST is applicable to frequency regulation and load following. It stores small to medium amounts of energy.

3 Method Fuzzy AHP and Fuzzy Grey Relational Analysis (GRA) were used in the study [8]. These methods are preferred because they allow decision makers to overcome uncertainty and lack of data in the evaluation of criteria and alternatives through the use of linguistic terms. 3.1 Fuzzy AHP Fuzzy AHP method uses verbal expressions to evaluate all alternatives by subjective and objective criteria. One of its main advantages is the ease of comparing multiple evaluation criteria. Since the method is based on the judgment of decision makers, it requires the use of fuzzy numbers. To determine the weighting of the criteria, it is necessary to generate an fuzzy pairwise comparison matrix. Then, fuzzy geometric mean and fuzzy weight of each criterion are calculated using geometric mean technique. Finally, the weights of the criteria are obtained by defuzzification of the weight values [8]. 3.2 Fuzzy GRA The Fuzzy GRA method ranks the mechanical energy storing technologies by performing the following steps [8]: Step 1: Prepare Dataset and Create Decision Matrix. Table 1 is used for this matrix. Step 2: Generate the normalized decision matrix. The Fuzzy GRA normalization process is performed using Eq. (1) when the criteria are benefit-based and Eq. (2) when the criteria are cost-based.   lij mij uij , , , i = 1, 2, · · · , m; j = 1, 2, · · · , n; uj+ = max rij , e˘ger J ∈ F r˜ij = i uj+ uj+ uj+ (1)  − − − lj lj l j r˜ij = , , , i = 1, 2, · · · , m; j = 1, 2, · · · , n; lj− = min lij , e˘ger J ∈ M i uij mij lij (2) Step 3: Determine the reference series. In this step, the alternatives that express the target state or contain the values closest to the target state in the normalized decision

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655

Table 1. Linguistic scale rating alternatives for each criterion [9]. Linguistic Term

Fuzzy Numbers

Too Inadequate (TI)

(1, 2, 3)

Inadequate (I)

(2, 3, 4)

Quite Inadequate (QI)

(4, 5, 6)

Adequate (A)

(5, 6, 7)

Quite Adequate (QA)

(7, 8, 9)

Definitely Adequate (DA)

(8, 9, 10)

matrix are determined one by one on the basis of criteria. When creating the reference series, the largest value of the criterion is used if the objective function is maximization, and the smallest value of the criterion is used if the objective function is minimization. Step 4: Generate the distance matrix. Each value of the distance matrix is calculated using Eq. (3). It is then substituted into the distance matrix in Eq. (4). The distance matrix is the matrix that shows the distance of the values in the normalized decision matrix from the reference series.    ∼  1 (3) d A, B˜ = (l1 − l2 )2 + (m1 − m2 )2 + (u1 − u2 )2 3 ⎡ ⎤ 01 (1) 01 (2) · · · 01 (n) ⎢ 02 (1) 02 (2) · · · 02 (n) ⎥ ⎢ ⎥ (4) 0i = ⎢ . ⎥ .. . . .. ⎣ .. ⎦ .. . 0m (1) 0m (2) · · · 0m (n) Step 5: Generate the Gray Relational Coefficient Matrix. After the grey relational coefficients are calculated using Eq. (5), the grey relational coefficient matrix is obtained in Eq. (6). The parameter ζ is called the discriminant coefficient. It can take values between [0, 1]. It has been observed in the literature that this coefficient usually takes the value of 0.5 [10]. γ0i (j) =

min + ζ max , max = max max 0i (j), min = min min 0i (j) i j i j 0i (j) + ζ max ⎡ ⎤ γ01 (1) γ01 (2) . . . γ01 (n) ⎢ γ02 (1) γ02 (2) . . . γ02 (n) ⎥ ⎢ ⎥ γ0i = ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . .

(5)

(6)

γ0m (1) γ0m (2) . . . γ0m (n)

Step 6: Calculate the grey relational degrees. The last step is the determination of the values of the amount of difference of the alternatives from the reference series. These values are the result of the multiplication of the weight values of each criterion. These

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A. Sarucan et al.

values have a direct impact on the ranking and evaluation of the alternatives and on the final decision process. Equation (7) is used to determine them. n   0i = Wi (j)γ0i (j) i = 1, 2, · · · , m; j = 1, 2, · · · , n (7) j=1

4 Case Study It is a very complex process to select different mechanical energy storage technologies. Therefore, the multi-criteria decision making method is very well suited to deal with these problems [11]. The uncertainty is a problem for multi-criteria decision making method. Generally, fuzzy logic is used for this situation in multi-criteria decision making method [12–14]. In this study, a grey relational analysis in a fuzzy environment is used to evaluate mechanical energy storage techniques. Grey system has been developed by Deng in 1982 [15]. An expert consulting in this sector aims to rank mechanical energy storage technologies according to their technical characteristics in order to help decision makers in the energy sector. There are 3 different (PHEST, CAEST and FEST) mechanical energy storage technologies that are used as alternatives in the study [16]. The criteria used in the study were as follows: Efficiency (E): Efficiency of an ESS is defined as the energy output during discharging/the energy input during charging. Energy Density (ED): It is the amount of energy that can be stored per unit volume of storage material. Power Density (PD): It is the maximum usable power that can be stored. Cycles (C): The number of full charge and discharge cycles a storage system can run over its lifetime. Lifetime (L): The lifetime of a storage system is calculated as the number of years that the storage system can reach its rated capacity relative to its rated power. Capital Cost (CC): The value of the mechanical energy storage technology in e/kWh is referred to as the investment cost. Table 2 gives an overview of the technologies for the mechanical storage of energy. A closer look at the table reveals that some values that belong to criteria ED and PD are not identifiable. Fuzzy GRA was used to eliminate this disadvantage. Table 2. Overview of different mechanical energy storage technologies using 75% quarter from if not indicated differently [17, 18]. Technology

E

ED

PD

C

L

CC

PHEST

85

-

1.5

50

60

500

CAEST

88

6

-

20

40

300

FEST

87

82

6700

93

20

1543

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Table 3 shows the weights of the criteria obtained with the Fuzzy AHP method. It is observed that capital cost (CC) has the highest decision impact. These values will be in use in step 6 of the fuzzy GRA method. Table 3. Calculated weights of the criteria.

Wi

E

ED

PD

C

L

CC

0.091

0.091

0.056

0.049

0.289

0.424

The fuzzy GRA method was applied after the weight values of the criteria were calculated. Step 1: By taking Table 1 as a reference and consulting the expert opinion, a linguistic scale decision matrix as shown in Table 4 is constructed. Then, this table is converted into fuzzy numbers, and the second step is carried out. Table 4. Linguistic terms used in fuzzy GRA. Technology

E

ED

PD

C

L

CC

PHEST

I

I

TI

QI

A

A

CAEST

A

QI

TI

I

QI

QA

FEST

QI

A

QA

A

I

TI

Step 2–3: The normalized fuzzy decision matrix and the reference series are obtained. Refer to the Table 5: Table 5. Normalized values. Technology

E

ED

PD

C

L

CC

PHEST

(.286,.429,.571)

(.286,.429,.571)

(.111,.222,.333)

(.571,.714,.857)

(.714,.857,1.000)

(.556,.667,.778)

CAEST

(.714,.857,1.000)

(.571,.714,.857)

(.111,.222,.333)

(.286,.429,.571)

(.571,.714,.857)

(.778,.889,1.000)

FEST

(.571,.714,.857)

(.714,.857,1.000)

(.778,.889,1.000)

(.714,.857,1.000)

(.286,.429,.571)

(.111,.222,.333)

Reference Series

(.714,.857,1.000)

(.714,.857,1.000)

(.778,.889,1.000)

(.714,.857,1.000)

(.714,.857,1.000)

(.778,.889,1.000)

Step 4–5: The distance matrix is calculated from the reference series. The grey relation coefficient matrix is calculated by Eq. (5). The parameter ζ is assumed to be 0.5. Table 6 shows the matrix. Step 6: The last step is to compute the grey relation degrees using Eq. (7). Table 7 shows the values of difference amount of alternatives from reference series. These values are multiplied by each criterion weight found by fuzzy AHP. The fuzzy GRA values of the alternatives are 0.676, 0.821 and 0.527, respectively (Table 7). The ranking is CAEST > PHEST > FEST.

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A. Sarucan et al. Table 6. Grey relational matrix of the coefficients.

Technology

E

ED

PD

C

L

CC

PHEST CAEST

0.438

0.438

0.333

0.700

1.000

0.600

1.000

0.700

0.333

0.438

0.700

1.000

FEST

0.700

1.000

1.000

1.000

0.438

0.333

Table 7. Grey relational degrees and rankings. Technology

E

ED

PD

C

L

CC

GRA

Rank

PHEST

0.040

0.040

0.019

0.034

0.289

0.254

0.676

2

CAEST

0.091

0.064

0.019

0.021

0.202

0.424

0.821

1

FEST

0.064

0.091

0.056

0.049

0.126

0.141

0.527

3

5 Conclusions and Future Study The depletion of traditional energy resources, risks such as pandemics and wars, and advances in technology are factors that are increasing the importance of energy storage technologies. Energy storage technologies in different structures have an important role in establishing the supply and demand balances in energy. These technologies ensure the sustainability of the energy system with the help of information system applications and smart grids. It also helps the energy system to serve reliably, flexibly and efficiently. Therefore, governments support investments and studies in the field of storage technologies. While making investment decisions, alternative energy storage technologies are evaluated according to qualitative or quantitative criteria. The comparison of these technologies is a multi-criteria decision making problem that involves uncertainty and imprecise data. This study adapts a fuzzy multicriteria decision making method to evaluate mechanical energy storage technologies in a fuzzy environment with grey relational analysis. These mechanical energy storage techniques are pumped hydro, compressed air and flywheel. The study showed that CAEST performs better under the investigated criteria. The effect of the capital cost criterion was high in the selection of CAEST. Future studies can extend the problem with new fuzzy extensions such as q-rung fuzzy sets, spherical fuzzy sets and compare the results with this study.

References 1. Montignac, F, Noirot, I., Chaudourne S.: Multi-criteria evaluation of on-board hydrogen storage technologies using the MACBETH approach. Int. J. Hydrogen Energy 34, 4561–4568 (2009) 2. Daim, T.U., Li, X., Kim, J., Simms, S.: Evaluation of energy storage technologies for integration with renewable electricity: quantifying expert opinions. Environ. Innov. Soc. Trans. 3, 29–49 (2012)

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3. Li, L., Liu, P., Li, Z., Wang, X.: A multi-objective optimization approach for selection of energy storage systems. Comput. Chem. Eng. 115, 213–225 (2018) 4. Ren, J., Ren, X.: Sustainability ranking of energy storage technologies under uncertainties. J. Clean. Prod. 170, 1387–1398 (2018) 5. Zhang, C., Chen, C., Streimikiene, D., Balezentis, T.: Intuitionistic fuzzy MULTIMOORA approach for multi-criteria assessment of the energy storage technologies. Appl. Soft Comput. J. 79, 410–423 (2019) 6. Nagaraju, D., et al.: Semantic approach for evaluation of energy storage technologies under fuzzy environment. Adv. Fuzzy Syst. 2022, 1–11 (2022) 7. Kocaman, B.: Energy storage technologies. ˙Iksad Publishing House, Türkiye (2021) 8. Sarucan, A., Demircan, L.: Fuzzy analytic hierarchy process and fuzzy gray relational analysis methods an employee competencies analysis. J. Soc. Hum. Sci. Res. (JSHSR) 5(11), 4698– 4708 (2018) 9. Gumus, A.T., Yayla, A.Y., Çelik, E., Yıldız, A.: A combined Fuzzy-AHP and Fuzzy-GRA methodology for hydrogen energy storage method selection in Turkey. Energies 6, 3017–3032 (2013) 10. Kuo, M.S., Liang, G.S.: Combining VIKOR with GRA techniques to evaluate service quality of airports under fuzzy environment. Expert Syst. Appl. 38(3), 1304–1312 (2011) 11. Baysal, M.E., Kahraman, C., Sarucan, A., Kaya, I., Engin, O.: A two phased fuzzy methodology for selection among municipal projects. Technol. Econ. Dev. Econ. 21(3), 405–422 (2015) 12. Koçar, O., Dizdar, E.: A risk assessment model for traffic crashes problem using fuzzy logic: a case study of Zonguldak Turkey. Int. J. Transp. Res. 14(5), 492–502 (2022) 13. Sarucan, A., Baysal, M.E., Engin, O.: A spherical fuzzy TOPSIS method for solving the physician selection problem. J. Intell. Fuzzy Syst. 42, 181–194 (2022) 14. Baysal, M.E., Sarucan, A., Kahraman, C., Engin, O.: The selection of renewable energy power plant technology using fuzzy data envelopment analysis. Proc. World Congr. Eng. (WCE 2011) 2, 1140–1143 (2011) 15. Sarucan, A., Baysal, M.E., Kahraman, C., Engin, O.: A hierarchy grey relational analysis for selecting the renewable electricity generation technologies. In: Proceedings of the World Congress on Engineering (WCE 2011), vol. II, pp. 1149–1154 (2011) 16. Joshi, G.K., Rongali, B., Biswal, M.: A review on mechanical energy storage technology. In: 2022 International Conference on Intelligent Controller and Computing for Smart Power (ICICCSP), pp. 1–5. IEEE, Hyderabad, India (2022) 17. Baumann, M., Weil, M., Peters, J.F., Chibeles-Martins, N.: A review of multi-criteria decision making approaches for evaluating energy storage systems for grid applications. Renew. Sustain. Energy Rev. 107, 516–534 (2019) 18. Bulut, M., Özcan, E.: A novel approach towards evaluation of joint technology performances of battery energy storage system in a fuzzy environment. J. Energy Storage 36, 102361 (2021)

Selection of Fighter Aircraft for Turkish Air Forces Under Uncertain Environment Yusuf Çifçi , Akın Metin , Tarık Tu˘gra Arslan , and Fatma Kutlu Gündo˘gdu(B) Department of Industrial Engineering, Turkish Air Force Academy, National Defence University, 34149 Istanbul, Turkey [email protected]

Abstract. Modern warplanes are assessed by military forces according to a multitude of factors. The production of fighter aircraft for the Turkish Air Forces, National Combat Aircraft (NCA) project currently being carried out by Turkish Aerospace Industries under the auspices of the defense industry presidency. In the present conditions, purchasing a new combatant warplane is a faster solution than producing a new one. According to the findings of the literature review, a total of 13 criteria should be taken into consideration for the selection of fighter aircraft for the Turkish Air Forces, including purchasing cost, installation and maintenance cost, the suitability of the existing weapon systems of the Turkish Armed Forces, and the political climate of relevant countries. Since some of the criteria cannot be expressed numerically due to the shortage of information, fuzzy logic can be utilized to overcome this issue. Especially, hesitant fuzzy sets can be a good candidate to express the vagueness with the assigning more than one membership. To determine the weights of the criteria, the hesitant fuzzy DEMATEL (Decision Making Trial and Evaluation Laboratory) method has been integrated z-number theory considering reliability functions of the decision-makers. Then, the classical VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) mothod has been used to select the alternative aircraft type deemed most suitable for the Turkish Air Force. To show reliability of the paper, the sensitivity analysis has been performed. Keywords: Hesitant Fuzzy DEMATEL · Fighter Aircraft Selection · Turkish Air Forces · VIKOR · Z-number Theory

1 Introduction Jet aircraft developed for military purposes are generally called as destroyer aircraft. These aircrafts are deployed in many military tasks, including achieving air superiority in combat zones, performing reconnaissance missions, destroying air defense systems, and conducting air strikes. Generally, destroyer jet aircraft are built to be fast and extremely maneuverable while also being outfitted with powerful weaponry, electronic warfare technology, and other war equipment. The F-16 Fighting Falcon, the F-35 Lightning II, and the Su-35 are notable examples of these type aircrafts. Such aircraft development and deployment are essential elements of contemporary military strategy and constitute a sizable investment in the capabilities of the country’s defense. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 660–669, 2023. https://doi.org/10.1007/978-3-031-39774-5_73

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Fighter aircraft selection is a crucial decision for any nation, with implications for the defense industry, national security, and the economy. Fighter jets are essential for strengthening a nation’s military capability, deterring potential aggressors, and assisting ground forces in air-to-air and air-to-ground missions. The selected aircraft must align with the strategic defense requirements and doctrine of the nation, operate effectively in the nation’s environment and terrain, and contribute to the overall defense strategy. In addition, selecting fighter jets can create opportunities for technology transfer, domestic production, and export, thereby boosting the economy and reducing reliance on foreign suppliers. This decision must be made after careful consideration and investigation. The selection of a fighter aircraft has significant implications for Türkiye’s national defense. Comparing the inventory of our nation to that of other nations reveals that our organization possesses a limited number and variety of fighter aircraft. By deterring possible aggressors and providing air support in the event of a confrontation, fighter planes can enhance Türkiye’s military capabilities and defense posture. As a result of the retirement of the F4-E Phantom from its inventory, the Turkish Air Forces would encounter a difficult scenario in terms of aircraft variety. Hence, there is a need for a new fighter aircraft, and in order to fulfill this need, multi-criteria decision-making methods will be used to select the most suitable aircraft for purchase. Fuzzy logic is a mathematical approach that can be used in an election survey to address issues arising from complex, uncertain, and incomplete data [1]. It can also model subjective human reasoning and decision-making processes, allowing for the inclusion of voters’ opinions and attitudes that may not be quantifiable. As the possible values of the membership degree in hesitant fuzzy sets are random, the HFS represents fuzziness and vagueness more naturally than any other extensional kind of fuzzy set. In this study, fuzzy logic was utilized with the opinions of three highly reliable experts to overcome the knowledge gaps in aircraft evaluations and uncertainties in expert opinions. This paper focuses on the selection of fighter aircraft to be incorporated into the inventory of the Air Force Command. To make this selection, the criteria weights have been determined by utilizing the hesitant fuzzy DEMATEL method to account for the relationships between criteria [2]. Subsequently, fighter jet aircraft has been chosen based on the classical VIKOR method. The rest of this chapter is organized as follows. Section 2 present a literature review on aircraft and criteria selection. Section 3, the paper presents a detailed account of the methodology employed, specifically focusing on the use of hesitant fuzzy DEMATEL and classical VIKOR techniques. Section 4 present the application of the study in Türkiye. This chapter is closed with the conclusion section.

2 Literature Review 2.1 Studies on Aircraft Selection The literature primarily focuses on aircraft selection, and the multi-criteria decisionmaking techniques used are presented in Table 1. Generally, the criteria used in this study were derived from aircraft selection problems for commercial airlines. To the best of our knowledge, apart from Yousaf et al. [7], there is no study conducted on the selection of fighter aircraft. The distinctive feature of our study is its hybrid

662

Y. Çifçi et al. Table 1. The Literature review on aircraft selection

Reference

Year

Purpose of the Paper

Methodology

Soner and Önüt [3]

2006 Commercial Aircraft Selection

Analytical Hierarchy Process (AHP) and Fuzzy Analytic Hierarchy Process (FAHP)

Kiracı and Akan [4]

2007 Selection of Business Jet Models in the Civil Aviation Industry

AHP and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)

Wang and Chang [5] 2007 Selection of Training Aircraft

TOPSIS

Doži´c and Kalic [6]

2015 Comparison of Different Aircraft Selection Methods

AHP and Even Swaps Method (ESM)

Yousaf et al. [7]

2017 Selection of Fighter Jet Aircraft AHP to be Used Against Terrorist Organizations in Pakistan

Doži´c et al. [8]

2018 Selection of Passenger Aircraft

Logarithmic fuzzy preference programming (LFPP), FAHP

Kiracı and Bakır [9]

2018 Commercial Aircraft Selection

AHP

Durmaz [10]

2019 Aircraft Selection Practice in Defense Industry

Stochastic multi-criteria acceptability analysis (SMAA), Stepwise Weight Assessment Ratio Analysis (SWARA)

Bakır et al. [11]

2021 Selection of Regional Aircraft

Fuzzy Pivot Pairwise Relative Criteria Importance Assessment (F- PIPRECIA), Fuzzy Measurement Alternatives and Ranking according to the Compromise Solution (F-MARCOS)

Haneol Lee [12]

2022 Purchase of Two Different Aircraft for the Republic of Korea

Heuristic methods

approach, which examines the relationships between criteria and creates a consensus solution. This approach integrates unstable fuzzy sets and takes uncertainties into account in the selection process in the field. After a thorough examination of the literature, the hesitant fuzzy DEMATEL with z-number and the classical VIKOR techniques were integrated to select a fighter aircraft for Türkiye. 2.2 Determination of Criteria for the Selection of Fighter Aircraft The criteria were collected based on a literature review. The study incorporates 13 criteria, along with their references, which are listed in Table 2.

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663

Table 2. The Selected Criteria and Related References CRITERIA

EXPLANATION

REFERANCE

Cost of Acquisition

Aircraft procurement cost refers to the amount paid to the manufacturer and country after an aircraft is selected for purchase

Doži´c et al. [8]; Kiracı and Bakır [9]; Doži´c and Kali´c [6]

Installation and Maintenance Costs

Installation and maintenance Doži´c et al. [8].; Kiracı and costs during the inventory Bakır [9]; Soner and Önüt inclusion and utilization process [3]; Bakır et al. [11] of an aircraft after its acquisition

Average Operating Cost

The total cost of fuel Doži´c et al. [8]; Kiracı and consumption, personnel salaries, Bakır [9]; Kiracı and Akan [4] and materials used during a single sortie operation, including important items necessary for the aircraft during the operation, such as ammunition

Service Area Capacity

The capacity of the aircraft to cover the maximum distance with a full fuel tank

Özdemir et al. [11]; Kiracı and Akan [4]; Ali et al. [7]; Wang and Chang [5]

Required Runway Length for Full Landing

The minimum distance required for the aircraft to come to a complete stop after landing

Özdemir et al. [11]; Wang and Chang [5]

Capacity of Transported Useful Load

The maximum weight that an Özdemir et al. [11]; Ali et al. aircraft can carry during the [7]; flight, including important items necessary for the operation, such as fuel, ammunition, etc

Maximum Speed

The maximum speed an aircraft can reach under International Standard Atmosphere (ISA) conditions, measured in Mach units

Ali et al. [7]; Kiracı and Akan [4]; Wang and Chang [5]

Maneuverability

The ability of an aircraft to perform sharp and sudden maneuvers is referred to as maneuverability

Soner and Önüt [3]; Ali et al. [7]

(continued)

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Y. Çifçi et al. Table 2. (continued)

CRITERIA

EXPLANATION

REFERANCE

Compliance with Turkish Armed Forces Standards

(Turkish Armed Forces Ali et al. [7]; Durmaz [10] Weapon-System Compatibility): The compatibility between the aircraft and the systems to be purchased with the capabilities and infrastructure of the Turkish Air Force

Radar Stealthiness

Radar invisibility refers to an Soner and Önüt [3]; Durmaz aircraft’s ability to avoid [10] detection by radar using features such as its paint, electronic systems, and software

Country Relations

The degree of political, social, Expert Opinions and commercial relations between the Republic of Türkiye and the countries that are potential sellers of alternative aircraft

Autonomous Attack Capability to Targets

Autonomous attack capability Soner and Önüt [3]; Durmaz refers to the degree to which the [10] aircraft can recognize a designated target and attack it without the need for pilot control

Sensitivity of Aircraft Radar

The sensitivity of the aircraft’s radar is related to its range, viewing angle, and technology level

Soner and Önüt [3]; Durmaz [10]

3 Methodology DEMATEL was created at The Battelle Memorial Institute and was first used in 1972. Considering Zadeh’s fuzzy logic and executive multi-criteria decision-making techniques, while remaining convinced that decision makers’ verbal and linguistic expressions can be introduced to problem-solving scenarios. By using this approach, causation may be expressed both in terms of a model and the quantitative relationships between the various elements. Lin and Wu proposed that the DEMATEL approach can be applied in conjunction with its logic for this reason. By using linguistic variables to represent uncertainty, the decision-making group can increase the applicability and compatibility

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of their judgments with the real world by operating the integrated logic of the DEMATEL technique, which forms the basis of the logic[13]. The Z-fuzzy number allows computing the uncertainty regarding the reliability of the existing information by proposed by Zadeh [15]. For the multi-criteria optimization of complex systems, Opricovic and Tzeng created the VIKOR approach [14]. It is one of the most popular approaches for making multicriteria decisions since it lists the options and chooses the one that comes the closest to the ideal compromise [16]. In this study, the two-stage hybrid decision-making methodology has been proposed to select a fighter jet aircraft. For this aim, the hesitant fuzzy DEMATEL method will be integrated with z-number theory to determine the relationships and weights of the criteria. After that, to order of the alternatives will be computed by using VIKOR method considering the weights of the criteria. The steps of the methodology are given in pseudocode as follows:

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4 Application The relationship between the criteria was linguistically evaluated by 10 different experts. It was found the missing of some evaluations. Therefore, three reliable experts were consulted, and they were asked to reach a consensus on a single linguistic expression among themselves. The interrelationships and corresponding weights of the criteria were determined by utilizing hesitant fuzzy DEMATEL methodology. Table 3 indicates the aggregated results, and the ranking of the criteria. The most important criteria are given in order as follows: C1 - Cost of Acquisition, C9 - Compliance with TurAF Standards, C2 - Installation and Maintenance Costs. The criteria weights obtained from DEMATEL were utilized in VIKOR methodology to rank the alternatives. The ranking of the alternatives is presented in Table 4. Sensitivity

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Table 3. Result of the hesitant fuzzy DEMATEL method Criteria

D

R

Aggregated Results

Normalized values

Rank

C1-Cost of Acquisition

2.6118

2.9359

5.5572

0.0958

1

C9-Compliance with TurAF Standards

1.5668

3.1268

4.9460

0.0852

2

C2-Installation and Maintenance Costs

1.9699

2.7290

4.7599

0.0820

3

C6-Capacity of Transported Useful Load

2.3670

2.2816

4.6494

0.0802

4

C3-Average Operating Cost

1.9978

2.5463

4.5771

0.0789

5

C7-Maximum Speed

2.6066

1.8705

4.5372

0.0782

6

C4-Service Area Capacity

2.2248

2.2066

4.4316

0.0764

7

C5-Required Runway Length for Full Landing

2.3335

1.9124

4.2668

0.0736

8

C12-Autonomous Attack Capability to Targets

2.2679

1.8829

4.1687

0.0719

9

C10-Radar Stealthiness

2.2147

1.8379

4.0701

0.0702

10

C13-Sensitivity of Aircraft 2.3391 Radar

1.6646

4.0602

0.0700

11

C11-Country Relations

2.1994

1.8228

4.0398

0.0697

12

C8-Maneuverability

2.022009

1.903878

3.92766409

0.067727489

13

analysis has been performed by changing threshold value (q). The threshold value is 0 or 0.5, the optimal alternative will be Eurofighter while the threshold value is 1, the optimal alternative will be General Dynamics F-16 Fighting Falcon. Table 4. Result of the classical VIKOR Method Alternatives

Evaluation

Threshold Values

q = 0,00 q = 0,50 q = 1,00 q = 0,00 q = 0,50 q = 1,00

Rank

General Dynamics F-16 Fighting Falcon 0.2797

0.1398

0.0000

3

2

1

Eurofighter Typhoon

0.0000

0.0222

0.0445

1

1

2

Dassault Rafale

0.0689

0.4067

0.7446

2

3

6

Saab JAS 39 Gripen

0.5628

0.7813

1.0000

5

6

7

Lockheed Martin F-35 Lightning II

1.0000

0.6450

0.2900

6

5

3

Suhoy Su-57

0.4030

0.5038

0.6047

4

4

5

Suhoy Su-35

1.0000

0.7983

0.5967

6

7

4

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Y. Çifçi et al.

5 Conclusion Due to its geographical location, Türkiye has a significant number of neighboring countries, which increases the potential threats it faces. Therefore, Türkiye must use its air power as an indispensable force in an effective and deterrent manner. This study focuses on the selection of a fighter aircraft for Türkiye using fuzzy multi-criteria decisionmaking methods. The distinct feature of this study is the examination of the relationships between the criteria, the use of uncertain linguistic expressions of expert opinions with the help of fuzzy logic, and the hybridization of two different methods to create a consensus solution. The relationships between the criteria were revealed using hesitant fuzzy DEMATEL with the reliability functions of experts. The most important criteria were identified as the acquisition cost, compatibility with existing weapon systems, and installation & maintenance costs, respectively. Based on the criteria and weights identified by hesitant fuzzy DEMATEL, the classical VIKOR method was used to compare and select the best alternatives. By adjusting the threshold value, sensitivity analysis has been performed (q). The optimal option when the threshold value is 0 or 0.5 is the Eurofighter, however when the threshold value is 1, the optimal alternative is the General Dynamics F-16 Fighting Falcon. For future research, the results of this study can be enhanced by enlarging the dataset, obtaining the opinions of more experts, incorporating more criteria, and employing alternative MCDM methodologies and integrations with other fuzzy extensions.

References 1. Buckley, J.J.: Fuzzy hierarchical analysis. Fuzzy Sets Syst. 17(3), 233–247 (1985). https:// doi.org/10.1016/0165-0114(85)90090-9 2. Gök, A.C., Perçin, P.D.S.: Elektronik Alı¸sveri¸s (E-alı¸sveri¸s) Sitelerinin E-hizmet Kalitesi Açısından De˘gerlendirilmesinde DEMATEL-AAS-VIKOR Yakla¸sımı. Anadolu Üniversitesi Sos. Bilim. Derg. 16(2), 131–144 (2016). https://doi.org/10.18037/AUSBD.389223 3. Soner, Y., Önüt, S.: Uçak Seçm Krterlernn De˘gerlendrlmesnde Ahp Ve Bulanik Ahp Uygulamasi Yüksek Lsans Tez (2006) 4. Kiracı, K., Akan, E.: Aircraft selection by applying AHP and TOPSIS in interval type-2 fuzzy sets. J. Air Transp. Manag. 89, 101924 (2020). https://doi.org/10.1016/J.JAIRTRAMAN. 2020.101924 5. Wang, T.C., Chang, T.H.: Application of TOPSIS in evaluating initial training aircraft under a fuzzy environment. Expert Syst. Appl. 33(4), 870–880 (2007). https://doi.org/10.1016/J. ESWA.2006.07.003 6. Doži´c, S., Kali´c, M.: Comparison of two MCDM methodologies in aircraft type selection problem. Transp. Res. Procedia 10, 910–919 (2015). https://doi.org/10.1016/j.trpro.2015. 09.044 7. Ali, Y., Muzzaffar, A.A., Muhammad, N., Salman, A.: Selection of a fighter aircraft to improve the effectiveness of air combat in the war on terror: Pakistan air force - a case in point. Int. J. Anal. Hierarchy Process 9(2), 2017–1936 (2017). https://doi.org/10.13033/IJAHP.V9I2.489 8. Doži´c, S., Lutovac, T., Kali´c, M.: Fuzzy AHP approach to passenger aircraft type selection. J. Air Transp. Manag. 68, 165–175 (2018). https://doi.org/10.1016/J.JAIRTRAMAN.2017. 08.003

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9. Kiracı, K., Bakır, M.: Havaaracı Seçim Problemlerinde Çok Kriterli Karar Verme Yöntemlerinin Kullanılması ve Bir Uygulama. JTL J. Transp. Logist. 3(1), 13–24 (2018). https://doi. org/10.26650/JTL.2018.03.01.02 10. Durmaz, K.˙I., Stokastik çok kriterli karar vermede yeni bir yöntem: SWARA-SMAA-2 ve savunma sanayinde uçak seçimi uygulaması. December 2019. Accessed 12 Oct 2022. https:// acikbilim.yok.gov.tr/handle/20.500.12812/362202 11. Bakır, M., Akan, S., ¸ Özdemir, E.: Regional aircraft selection with fuzzy piprecia and fuzzy Marcos: a case study of the Turkish airline industry. Facta Univ. Ser. Mech. Eng., 19(3 Special Issue), 423–445 (2021). https://doi.org/10.22190/FUME210505053B 12. Lee, H.: Evaluating civil-military relationship for effective procurement decision-making: the case of two fighter jet procurements of the Republic of Korea. Def. Secur. Anal. (2022). https://doi.org/10.1080/14751798.2022.2088331 13. Thakkar, J.J.: Decision-making trial and evaluation laboratory (DEMATEL). In: MultiCriteria Decision Making Studies in Systems, Decision and Control. SSDC, vol. 336, pp. 139–159. Springer Singapore (2021). https://doi.org/10.1007/978-981-33-4745-8_9 14. Opricovic, S., Tzeng, G.H.: Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. Eur. J. Oper. Res. 156(2), 445–455 (2004). https://doi.org/10.1016/ S0377-2217(03)00020-1 15. Gul, M., Celik, E, Aydin, N., Taskin Gumus, A., Guneri, A.F.: A state of the art literature review of VIKOR and its fuzzy extensions on applications. Appl. Soft Comput. 46, 60–89 (2016). https://doi.org/10.1016/J.ASOC.2016.04.040 16. Opricovi´c, S.: An extension of compromise programming to the solution of dynamic multicriteria problem. Optim. Tech. 508–517 (2006). https://doi.org/10.1007/BFB0036431 17. Ilhan, M., Gundogdu, F.K.: Evaluation of spaceport site selection criteria based on hesitant z-fuzzy linguistic terms: a case for Turkiye. Int. J. Inf. Technol. Dec. Mak. vol: Inpress (2022)

Analysis of Suppliers’ Resilience Factors Under Uncertainty Fatma Cayvaz Parlak1

, Huseyin Selcuk Kilic2

, and Gulfem Tuzkaya2(B)

1 Industrial Engineering Department, Istanbul Medeniyet University, 34700 Istanbul, Turkey

[email protected]

2 Industrial Engineering Department, Marmara University, 34722 Istanbul, Turkey

{huseyin.kilic,gulfem.tuzkaya}@marmara.edu.tr

Abstract. Selecting a resilient supplier is an essential component in supply chain management. Resilient suppliers can tolerate supply chain disruptions brought on by unforeseen events like natural catastrophes, economic downturns, and other unforeseen occurrences. Organizations may reduce the risks of supply chain disruptions and enhance the overall stability and sustainability of their supply chains by selecting resilient suppliers. To select a resilient supplier, organizations should take into account a variety of aspects, such as the supplier’s financial stability, risk management capabilities, degree of operational and supply chain redundancy, and dedication to ongoing development. In this study, it is aimed to reveal and assess the resilience factors in the selection of suppliers. Accordingly, the literature is searched and the most frequently examined resilience factors in the literature are determined via Pareto analysis. Afterward, Pythagorean Fuzzy DEMATEL method is used to evaluate the determined resilience factors. As a result, from the most important to the least, the resilience factors are found to be agility flexibility, responsiveness, visibility, robustness, redundancy, surplus inventory, and cooperation. Keywords: Resilience Factors · Resilient Supplier · Supplier Selection · Pareto Analysis · Pythagorean Fuzzy DEMATEL

1 Introduction Supply chain resilience is the ability of a supply chain to react to disruptions, adapt to unexpected occurrences, and recover from them while maintaining the required level of connection and control over the design and operation of the supply chain [1]. The supply chain disruptions must consequently be understood. After the disruptions are understood, measures can be taken against them. Supply chain disruptions were compared with each other and it was determined that the most important disruption factor was the shortage of raw materials [2]. Thus, resilient suppliers should be selected to avoid the shortage of raw materials. Due to the growth of global supply chains and strategic outsourcing, the selection of appropriate suppliers or a group of suppliers based on one or more criteria, commonly known as the supplier selection problem, has emerged as a crucial issue [3]. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 670–677, 2023. https://doi.org/10.1007/978-3-031-39774-5_74

Analysis of Suppliers’ Resilience Factors Under Uncertainty

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selection of suppliers is an essential aspect of supply chain (SC) management that plays an important role in maintaining the competitive advantage [4] and the suppliers were addressed with many different resilience factors [5]–[8]. For a resilient supplier selection process, the meanings and scopes of the factors should be considered correctly. In this study, the literature was reviewed and the resilient factors considered in the selection of resilient suppliers were determined. The most important resilient factors were analyzed using Pareto analysis and identified as Responsiveness (Res), Flexibility (Flx), Agility (Agl), Surplus Inventory (SI), Robustness (Rob), Redundancy (Red), Cooperation (Coop) and Visibility (Visb). To address the research gap, a Pythagorean Fuzzy Decision Making Trial and Evaluation Laboratory (PF-DEMATEL) methodology is used to evaluate the relative importance of various resilience factors for selecting resilient suppliers through a comparative analysis. Fuzzy linguistic expressions were used by industry experts to evaluate the importance of resilience factors on each other. The paper is structured as follows: Section two provides a review of the literature, section three explains the Pythagorean Fuzzy DEMATEL methodology, section four presents the methodology’s application, and the paper concludes with final remarks.

2 Literature Review Some of the recent resilient supplier selection and evaluation studies are summarized as following: By using interval-valued fuzzy sets (IVFSs) and fuzzy possibilistic statistical concepts, Foroozesh et al. [9] proposed a new multi-criteria group decision-making model. Under uncertainty in supply chain networks, a new weighting method is presented for supply chain experts or decision makers (DMs). Mohammed et al.’s [5] effective and resilient supplier selection methodology is based on a fresh, all-encompassing framework that makes it possible to pinpoint crucial features of flexibility as well as conventional business standards. The study suggests purchasing departments to strictly monitor sourcing selections in order to build a supply chain that is resilient to any unanticipated interruptions. By using the DEMATEL technique, the relative weight of conventional business criteria and resilience pillars was determined. To demonstrate the viability of the suggested strategy, a genuine sourcing issue for a steel manufacturing firm was resolved. Supply chain managers may benefit from the research by creating robust supply networks that lower sourcing costs and the risk of losses from disruptive threats. The Pythagorean fuzzy set based DEMATEL approach was applied by Giri et al. [10] to resolve a supplier selection issue in sustainable supply chain management. The identification of the system’s cause-and-effect components is done using a method that Pythagorean fuzzy sets have provided. In another article, Green supply chain management (GSCM) and resilience in the face of global supply chain hazards are examined by Xiong et al. [11]. In order to choose the best resilient-green provider, the paper applied a hybrid technique that combines the Best-Worst method (BWM), Weighted Aggregated Sum-Product Assessment (WASPAS), and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS).

672

F. C. Parlak et al.

3 Pythagorean Fuzzy DEMATEL DEMATEL presents a straightforward but effective methodology for constructing structural frameworks, which encompass interconnected system components [12]. The DEMATEL technique analyzes the impact of two factors on each other in the system and can determine whether the factor is cause or effect. In this paper, Pythagorean fuzzy DEMATEL (PF-DEMATEL) is used. Pythagorean fuzzy sets were introduced by Yager [13]. The objective of Pythagorean fuzzy sets is to effectively manage imprecision and elucidate ambiguity with a considerable degree of dependability. A Pythagorean fuzzy set P is an object as follows: P = {x, P(μ P (x), v(x))|x ∈ X } and the degree of indeterminacy of x to P is denoted by πP (x) = 1 − (μP (x)2 − vP (x)2 ). The following steps [14] of PF-DEMATEL are applied within the study. Step 1: Determination of linguistic variables. In this study, linguistic terms and pythagorean fuzzy numbers given in Table 1 are used to articulate the cause-and-effect relationships between resilience factors. Table 1. Linguistic terms and Pythagorean fuzzy numbers [14] Linguistic terms

Pythagorean fuzzy sets

Very low influence (VLI)

(0.15, 0.85)

Low influence (LI)

(0.25, 0.75)

Moderately low influence (MLI)

(0.35, 0.65)

Medium influence (MI)

(0.50, 0.45)

Moderately high influence (MHI)

(0.65, 0.35)

High influence (HI)

(0.75, 0.25)

Very high influence (VHI)

(0.85, 0.15)

Step2: The direct-relation matrix. The second step is to determine a n × 2n direct-relation matrix, as shown in Eq. 1. Using the linguistic terms listed in Table 1, decision makers (DMs) in the relevant field evaluated the cause and effect relationships between resilience factors in order to determine how closely related they were to one another. ⎡

   =  M rij n×2n

⎤ (0, 0) (μ12 , ν12 ) · · · (μ1n , ν1n ) ⎢ (μ21 , ν21 ) (0, 0) · · · (μ2n , ν2n ) ⎥ ⎢ ⎥ =⎢ ⎥, i, j ∈ {1, 2, . . . , n} .. .. .. .. ⎣ ⎦ . . . . (μn1 , νn1 ) (μn2 , νn2 ) · · ·

(1)

(0, 0)

Step 3: Aggregation of DMs’ views. A unified matrix is created after DMs have finished building the direct-relation matrix. The average Pythagorean fuzzy decision matrix is then calculated using Eq. 2,

Analysis of Suppliers’ Resilience Factors Under Uncertainty

where k denotes the evaluation of each DM, with k = 1, 2, . . . , m. ⎞⎤ ⎡⎛  m  m    2  m1   1    = a˜ ij 1 − μij vij m ⎠⎦ = ⎣⎝1 − , M n×2n k=1

k=1

673

(2) n×n

Step 4: Defuzzification. The mean crisp matrix is obtained by using the pythagorean fuzzy state aggregation matrix Eq. 3. The resilience preference coefficient (ϕ) ranges from 0 to 1. Craij = 0.5(1 + μij − vij + (ϕ − 0.5) × πij ) As shown in Eq. 4, the mean crisp matrix is obtained.   C = Craij n×n

(3)

(4)

Step 5: Normalization. The subsequent stage involves generating the normalized mean crisp matrix N, which is determined through  the use of Eq. 5. It is important to note that the sum of each row is represented by nj=1 Cr aij . N=

max

1 n

j=1 Craij

× C, i ∈ {1, 2, . . . , n}

(5)

Step 6: Constructing the total-relation matrix. Let’s assume that matrix I serves as the representation of the unit matrix. The whole connection matrix is obtained by multiplying the inverse of (I - N) by the normalized mean crisp matrix N, as shown in Eq. 6. This complete connection matrix sheds light on both the direct and indirect causal linkages between several resilience-related factors.   T = N (I − N )−1 = tij n×n , i, j ∈ {1, 2, . . . , n} (6) Step 7: Developing a causal diagram. To obtain the causal diagram, it is necessary to sum the rows (D) and columns (R), as shown in Eqs. 7 and 8. Di =

n 

tij i = 1, 2, . . . , n

(7)

tij j = 1, 2, . . . , n

(8)

j=1

Rj =

n  i=1

Step 8: The weights assigned to the factors are determined by normalizing their (Di + Rj ) values [15].

674

F. C. Parlak et al.

4 Evaluation of Suppliers’ Resilience Factors Organizations must create resilient supply chains in light of the increase in supply chain disruptions, and a resilient supplier is an essential part of this. The paper reviews the literature on the topic and determines main resilience factors via Pareto Analysis. The factors are responsiveness, flexibility, agility, surplus inventory, robustness, redundancy, cooperation, visibility and reliability. Pythagorean fuzzy sets have been recognized as a powerful tool for handling uncertainty in decision-making problems. This article applied the integration of Pythagorean fuzzy sets with the Decision-Making Trial and Evaluation Laboratory (DEMATEL) method, known as Pythagorean fuzzy DEMATEL, to solve complex decision-making problems. Step 1: In the first step, the linguistic terms in Table 1 were taken into account. Step 2: In the second step, the relation matrix is collected from the decision makers. In this study, 5 decision makers from the industry and academia made relation the Direct-Relation matrix is created using Eq. 1. Step 3: In the third step, aggregation matrix is created. Matrices collected from DMs are converted into a single matrix using Eq. 2. The aggregation matrix is shown in Table 2. Table 2. Aggregation Matrix Res

Flx

Agl

SI

Rob

Red

Coop

Visb

Res

0.00 0.00 0.67 0.33 0.80 0.20 0.75 0.25 0.53 0.46 0.54 0.50 0.53 0.50 0.45 0.59

Flx

0.41 0.61 0.00 0.00 0.81 0.20 0.60 0.41 0.60 0.40 0.68 0.34 0.41 0.58 0.49 0.54

Agl

0.76 0.25 0.81 0.20 0.00 0.00 0.53 0.50 0.71 0.29 0.57 0.44 0.41 0.58 0.47 0.53

SI

0.67 0.33 0.70 0.31 0.57 0.41 0.00 0.00 0.49 0.50 0.51 0.50 0.23 0.79 0.30 0.71

Rob

0.65 0.34 0.74 0.26 0.57 0.41 0.29 0.71 0.00 0.00 0.52 0.49 0.49 0.50 0.36 0.64

Red

0.64 0.38 0.77 0.23 0.75 0.26 0.40 0.63 0.52 0.46 0.00 0.00 0.44 0.54 0.31 0.69

Coop 0.46 0.55 0.60 0.39 0.70 0.31 0.27 0.73 0.52 0.46 0.52 0.46 0.00 0.00 0.80 0.20 Visb

0.70 0.31 0.56 0.46 0.65 0.34 0.63 0.39 0.57 0.41 0.36 0.64 0.75 0.25 0.00 0.00

Step 4: The aggregation matrix is transformed into the mean crisp matrix using Eq. 3. The matrix form in Eq. 4 is obtained. The mean crisp matrix presented in Table 3. Step 5: The crisp matrix is normalized using Eq. 5. The normalized mean crisp matrix is shown in Table 4. Step 6: The total-relation matrix was constructed using Eq. 6 and new matrix is given in Table 5. Step 7: Di and Rj values were calculated using Eqs. 7 and 8. Step 8: The factors’ importance weights is calculated based on the normalization of (Di + Rj ) values and the results are shown in the Table 6. According to the results, three factors are identified as cause group. Those are flexibility, agility and robustness. Also five factors, responsiveness, surplus inventory, redundancy, cooperation and visibility, are in the effective group. After calculating the

Analysis of Suppliers’ Resilience Factors Under Uncertainty

675

Table 3. The Mean Crisp Matrix Res

Flx

Agl

SI

Rob

Red

Coop

Visb

Res

0.50

0.67

0.80

0.75

0.53

0.52

0.51

0.53

Flx

0.40

0.50

0.80

0.59

0.60

0.67

0.41

0.60

Agl

0.75

0.80

0.50

0.51

0.71

0.56

0.41

0.71

SI

0.67

0.69

0.58

0.50

0.50

0.51

0.22

0.50

Rob

0.65

0.74

0.58

0.29

0.50

0.52

0.50

0.50

Red

0.63

0.77

0.74

0.39

0.53

0.50

0.45

0.53

Coop

0.45

0.61

0.70

0.27

0.53

0.53

0.50

0.53

Visb

0.69

0.55

0.65

0.62

0.58

0.36

0.75

0.58

Table 4. Normalized Mean Crisp Matrix Res

Flx

Agl

SI

Rob

Red

Coop

Visb

Res

0.101

0.136

0.160

0.152

0.108

0.105

0.103

0.108

Flx

0.081

0.101

0.162

0.119

0.121

0.135

0.083

0.121

Agl

0.152

0.162

0.101

0.103

0.143

0.113

0.083

0.143

SI

0.135

0.140

0.117

0.101

0.100

0.102

0.044

0.100

Rob

0.132

0.150

0.117

0.059

0.101

0.104

0.100

0.101

Red

0.127

0.156

0.149

0.078

0.106

0.101

0.091

0.106

Coop

0.091

0.122

0.140

0.055

0.106

0.106

0.101

0.106

Visb

0.140

0.110

0.132

0.124

0.117

0.072

0.151

0.117

Table 5. Total-Relation Matrix Res

Flx

Agl

SI

Rob

Red

Coop

Visb

Res

1.528

1.735

1.762

1.35

1.464

1.363

1.228

1.464

Flx

1.444

1.628

1.689

1.261

1.415

1.334

1.159

1.415

Agl

1.621

1.809

1.757

1.343

1.541

1.41

1.249

1.541

SI

1.377

1.532

1.515

1.15

1.281

1.198

1.026

1.281

Rob

1.403

1.577

1.552

1.132

1.313

1.229

1.11

1.313

Red

1.477

1.67

1.67

1.216

1.392

1.294

1.16

1.392

Coop

1.312

1.491

1.513

1.081

1.268

1.184

1.069

1.268

Visb

1.548

1.689

1.716

1.309

1.456

1.315

1.266

1.456

676

F. C. Parlak et al. Table 6. Final Scores and Importance Weights Di

Rj

Di +Rj

Di -Rj

w

Responsiveness

11.71

11.89

23.6

−0.18

0.132

Flexibility

13.13

11.34

24.47

1.788

0.136

Agility

13.17

12.27

25.44

0.902

0.142

10.36

20.2

−0.52

0.113

Robustness

11.13

10.63

21.76

0.502

0.121

Redundancy

10.33

11.27

21.6

−0.95

0.120

10.19

19.46

−0.92

0.108

11.76

22.89

−0.63

0.128

Surplus Inventory

Cooperation Visibility

9.842

9.268 11.13

importance weights of the factors, agility has the highest importance. The order of importance of resilience factors is flexibility, responsiveness, visibility, robustness, redundancy, surplus inventory, and cooperation.

5 Conclusion Selecting resilient suppliers is essential for businesses to ensure their continuity in the face of unforeseen disruptions. To achieve this, companies need to implement a rigorous selection process that considers both the supplier’s operational capabilities and their risk management strategies. Businesses should also develop a good relationship with their suppliers, build trust, and communicate regularly to enhance collaboration and increase the supplier’s commitment to their success. In this study, resilience factors were evaluated via PF-DEMATEL and their importance levels were calculated. According to the results, while agility factor has the highest degree of importance, cooperation has the least importance. The results provide insight to companies in the selection of resilient suppliers. In future studies, resilience factor analyses can be applied for different stakeholders in the supply chain and used for specific industries. Additionally, different MADM techniques can be used for the evaluation of factors. Finally, the weighted resilience factors can be used in different studies.

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A Framework of Directed Network Based Influence-Trust Fuzzy Group Decision Making Nor Hanimah Kamis1,2(B) , Adem Kilicman2,3 , Norhidayah A Kadir1 , and Francisco Chiclana4,5 1

School of Mathematical Sciences, College of Computing, Informatics and Media, Universiti Teknologi MARA (UiTM), 40450 Shah Alam, Selangor, Malaysia [email protected],[email protected], [email protected] 2 Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia [email protected] 3 Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Malaysia 4 Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester, UK [email protected] 5 Andalusian Research Institute on Data Science and Computational Intelligence (DaSCI), University of Granada, Granada, Spain

Abstract. Daily life requires individuals or groups of decision-makers to engage in critical decision-making processes. Fuzzy set theory has been integrated into group decision-making (GDM) to address the ambiguity and vagueness of expert preferences. Social Network Group Decision Making (SNGDM) is a newly emerging research area that focuses on the use of social networks to facilitate information exchange and communication among experts in GDM. Moreover, Social Influence Group Decision Making (SIGDM) has been initiated, which considers social influence as a factor that can impact experts’ preferences during interactions or discussions. Studies in this area have proposed innovative measurements of social influence, including the use of trust statements to explicitly influence experts’ opinions. In this study, a new trust index called TrustRank is proposed, which acts as an additional weightage of experts’ importance and is embedded in the influence network measure that represents the strength of the expert’s influence degree. These values are then utilized as the order-inducing variable in the IOWA-based fusion operator to obtain the collective preference and ranking of alternatives. The proposed framework, which is the directed network-based Influence-Trust Fuzzy GDM, is presented along with its implementation, results, and discussion to showcase its applicability. Keywords: Influence · Trust · Directed Network · Social Network Analysis · Fuzzy Group Decision Making Supported by Prof. Yoo Hang Kim Young Women Scientists Award 2023, Association of Academies and Societies of Sciences in Asia (AASSA). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 678–688, 2023. https://doi.org/10.1007/978-3-031-39774-5_75

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1 Introduction Group Decision Making (GDM) is a ubiquitous phenomenon in both individual and collective decision-making contexts. The process involves a set of decision makers, referred to as experts, who express their preferences for a set of alternatives or criteria and engage in discussions to determine the optimal solution that satisfies the group’s consensus. To tackle the challenges posed by vagueness and ambiguity in expert preferences, Fuzzy Set Theory [1] has been introduced and integrated into the GDM framework. Over the years, Fuzzy GDM has gained significant attention from researchers, with numerous studies investigating its efficacy and applicability. A new area of study, known as Social Network Group Decision Making (SNGDM), has recently emerged. SNGDM leverages social networks to facilitate the exchange of information and communication among decision-making experts. During their interactions, experts with greater experience, knowledge, or trustworthiness may exert a greater influence on their peers, shaping the decision-making process. Researchers have made significant progress in developing SNGDM frameworks to enhance decisionmaking outcomes. Recent efforts by Kamis et al. [2, 3], Shang et al. [4], Hua et al. [5], and others have presented promising approaches for implementing SNGDM. Opinions expressed by experts within a social network can be explicitly conveyed in the form of trust statements, as trust enables the assessment of an expert’s reputation based on their past actions or behavior [6]. In recent years, researchers have proposed novel methodologies for trust-based GDM. Notable efforts have been made by Ahlim et al. [7], Gai et al. [8], Liu et al. [9] and many more. Recent GDM studies introduced social influence, giving rise to a new research area, named as Social Influence Group Decision Making (SIGDM). SIGDM involves both intra- and interpersonal interactions among experts with varying degrees of influence, in order to revise, exchange, or persuade others towards achieving a final agreed decision [11]. It is widely acknowledged that experts’ preferences can be influenced by social factors during interactions, discussions, or opinion exchange in a network. In recent times, several SIGDM models have been proposed, such as Capuano et al. [10], Sedek et al. [11] and Yao and Gu [12]. In this paper, the GDM, Fuzzy GDM, SNGDM, trust-based GDM and SIGDM are introduced in Sect. 1. This study proposes a new trust measure, known as TrustRank for the purpose of determining the most trusted expert in the directed trust network. The proposed notion is embedded in Kamis et al. [3] work and the consecutive steps are elaborated in Sect. 2. The implementation and results are explained in Sect. 3. The framework of Influence-Trust Fuzzy GDM is presented in Sect. 4 and finally, the conclusion and future works are drawn in Sect. 5.

2 Methodology In this section, a new definition of TrustRank index is introduced. This notion is then integrated in the IOWA-based fusion operator and extended to the resolution process for achieving the final agreed solution. The proposed work is implemented in Fuzzy GDM model by Kamis et al. [3] for the purpose of validating the necessity of the proposal.

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A New Trust Measure

The notion of trust is a powerful concept in GDM, as it promotes collaboration, improves communication and interactions, increases efficiency, and builds strong relationships among experts. In addition, trust also can influence the GDM process because trust creates a sense of confidence and credibility that can change expert’s decisions. Definition of trust relation, which adapting the concept of trust defined by Wu et al. [13] is shown as: Definition 1. A trust relation, T is a mapping in E × E with membership function, μT : E × E → [0, 1], where μT (ei , e j ) = ti j represents the trust degree of expert ei towards expert e j . The associated semantics of trust relation is presented as follows: ⎧ if ei completely trust e j ⎨1 ti j = ti j ∈ (0, 1) if ei proportionally trust e j ⎩ 0 if ei completely distrust e j for all i, j = 1, 2, . . . , m. For expert trust degree over himself/herself, tii is assumed equal to 1. Note that the higher the ti j , the higher the trustworthiness of expert ei towards expert e j . Experts’ trust relations can be represented as a trust sociomatrix, T = (ti j )m×m [13]: ⎡ ⎤ t11 t12 . . . t1m ⎢ t21 t22 . . . t2m ⎥ ⎢ ⎥ T =⎢ . . . . ⎥ ⎣ .. .. . . .. ⎦ tm1 tm2 . . . tmm In order to measure the trust index of each expert in a trust sociomatrix, a new definition is introduced. The trust index, known as TrustRank is presented. This notion represents the in-degree and out-degree trust relationships. The in-degree trust indicates the reliance of an expert to other expert, whereas out-degree trust shows how strongly the expert is relied upon. Definition 2. The TrustRank index of each expert, Z(e j ) is: m

Z(e j ) =

∑ (ti j )

i=1 m

∑

i=1

m

(ti j ) + ∑ (ti j )

.

j=1

 Example 1. Assume that the trust sociomatrix of 8 experts, E = e1 , e2 , . . . , e8 is presented below. The TrustRank, Z(e j ) can be determined using Definition 2 and the values

A Framework of Directed Network Based Influence-Trust Fuzzy GDM

are shown as follows: ⎡

1 ⎢ 0.7 ⎢ 0 ⎢ ⎢ 0.7 T =⎢ ⎢ 0.4 ⎢ ⎢ 0.8 ⎣ 0 0.75

0.65 1 0.75 0.8 1 0.95 0.7 0.85

0 0.8 1 0.75 0.65 0.75 0.9 0.8

0.8 0.4 0.8 1 0.5 0 0.8 0.5

0.9 0.8 1 0.75 1 0.8 1 0.8

0.9 0.75 0.7 0.8 0.75 1 0.5 0.85

0.7 0.9 0.8 0.8 0.75 0.85 1 1

⎤

⎡

681

⎤

0.7 0.4350 0.85⎥ ⎢0.5194⎥ ⎢0.4829⎥ 1 ⎥ ⎢ ⎥ ⎥ ⎢0.4248⎥ ⎥ 0.9 ⎥ ⎢ ⎥ Z = ⎢0.5465⎥ 0.8 ⎥ ⎢ ⎥ ⎥ ⎢0.5040⎥ 1 ⎥ ⎣ ⎦ ⎦ 0.8 0.5440 1 0.5184

An expert with a high TrustRank index is considered to have a higher level of trust than the experts they have trusted and who have trusted them. On the other hand, an expert with a low TrustRank index is considered to have a lower level of trust compared to the experts they have trusted and who have trusted them. 2.2 Directed Network Based Influence-Trust Fuzzy GDM  Initially, a group of experts E = e1 , e2 , . . . , em expresses their preferences over a set of alternatives, A = {A1 , A2 , . . . , An } (n > 2). Step 1: Experts evaluate all alternatives based on the reciprocal fuzzy preference relation (RFPR). Definition 3. [14] A fuzzy binary relation μP : A × A −→ [0, 1] that associated each pair of alternatives (Ai , A j ) a value μP (Ai , A j ) = pi j verifying the reciprocity property pi j + p ji = 1 (∀i, j) and the interpretations of (i) pi j = 0.5 if Ai and A j are equally preferred (indifference), (ii) pi j ∈ (0.5, 1) if Ai is slightly preferred to A j , (iii) pi j = 1 if Ai is absolutely preferred to A j . The RFPR of alternatives, A can be represented as P = (pi j ) with dimension n × n. Step 2: Transform the RFPR into the intensity preference vector (IPV). Definition 4. [15] The intensity preference vector (IPV) , V ∈ Rn(n−1)/2 can be expressed as:  

V = p12 , p13 , . . . , p1n , p23 , . . . , p2n , . . . , p(n−1)n = v1 , v2 , . . . , vr , . . . , vn(n−1)/2 . All individual experts’  preferences in terms of IPVs toward the set of alternatives A can be denoted by V = V 1 ,V 2 , . . . ,V m . Step 3: Determine the experts’ preference similarity indexes. Definition 5. [2] Given V as the set of experts’ IPVs toward the set of alternatives A. A preference similarity measure is a fuzzy subset of V × V with membership function S : V × V → [0, 1] and it verifies property (S (V a ,V a ) = 1) and the

bthe areflexive  a b symmetric property S V ,V = S V ,V .

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Definition 6. [2] The cosine preference similarity index of pair of experts, ea and eb is:  a b · v v ∑ i i i=1  . n(n−1)/2    2 (vai )2 ·  ∑ vbi

n(n−1)/2

  Sab = S V a ,V b =  n(n−1)/2  

∑

i=1



i=1

Step 4: Construct the Similarity Social Influence Network (SSIN). Definition 7. [3] A Similarity Social Influence Network (SSIN) is an ordered 3-tuple,  G = E, L, Sη , where E is a set of experts’ nodes, L is a set of links of pairs of experts and Sη is a set of row normalized preference similarity weights attached to L. Step 5: Determine the influence-trust index of expert. The influence-trust index represents the strength of expert’s influence and trust degree in a group, where the most influential and trusted expert in the network can be identified. This index can be computed by: Definition 8. Let Sη be a set of row normalized preference similarity weights in SSIN G, σ be the relative importance of endogenous (network connections) over exogenous indexes. The influence-trust (external) effects, and Z = (z)m×1 be a set of TrustRank  index or δ -centrality of experts E, Y = y1 , . . . , ym is represented by: −1

Y = I − σ STη Z. Step 6: Normalize the influence-trust index in [0, 1] values. YN =

Y i − min(Y ) . max(Y ) − min(Y )

Step 7: Aggregate all individual experts’ preferences into a collective one. By referring to the influence-trust indexes, the aggregation step can be carried out. The influence-trust  indexes areused as the order inducing variable of the experts’ preference evaluations, p1i j , . . . , pm i j in the IOWA-based fusion operator [16]. The aggregation operator, known as the δ -IOWA operator can be defined as: Definition 9. The δ -IOWA operator of dimension m, ΦWδ , is an IOWA-based operator with the order inducing variable from  the set of expert’s normalized influence-trust

index in the network, YN = y1 , . . . , ym . All aggregated preferences are then presented in a matrix, known as an influence-trust based collective preference matrix, Pc . Step 8: Rank the alternatives and choose the best alternative. We utilize the Quantifier Guided Dominance Degree (QGDD) based on the OWA operator, guided by the linguistic quantifier, Q, as defined as folows:

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  Definition 10. [17] Consider a collective preference relation Pc = pcij on a set of alternatives X = {x1 , x2 , . . . , xn }, the quantifier guided dominance degree, QGDD (xi ), quantifies the degree of dominance that an alternative xi has over all the other alternatives in a fuzzy majority sense as:

 QGDD (xi ) = ΦQ pcij , j = 1, . . . , n, j = i , where ΦQ is an OWA operator guided by the linguistic quantifier Q representing the fuzzy majority concept. From this definition, the elements of the set is generated: X QGDD = {x | x ∈ X, QGDD (x) = supx∈X QGDD (x)} , indicates that the maximum dominance elements of the fuzzy majority of X quantified by Q.

3 Implementation and Results In order to implement the proposed work, a numerical example from Shamudin [18] is referred. Let E ={e1 , e2 , ..., e8 } be a group of 8 experts who expressing their preferences on 7 alternatives, A = {A1 , A2 , . . . , A7 } in the form of RFPR format. Their evaluations are presented as follows: ⎡

0.5 0.6 0.7 0.2 0.3 0.8 0.7

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ e1 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡

0.5 0.6 0.7 0.2 0.8 0.9 0.5

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ e4 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

⎡

⎤

⎡

0.4 0.5 0.6 0.8 0.2 0.1 0.4

0.3 0.4 0.5 0.7 0.1 0.8 0.6

0.8 0.2 0.3 0.5 0.9 0.7 0.9

0.7 0.8 0.9 0.1 0.5 0.2 0.4

0.2 0.9 0.2 0.3 0.8 0.5 0.1

0.3⎥ ⎢0.5 ⎢ ⎥ 0.6⎥⎥⎥ ⎢⎢⎢0.7 ⎢ ⎥ ⎥ 0.4⎥⎥ ⎢⎢⎢0.9 ⎥ e = ⎥ 0.1⎥ 2 ⎢⎢⎢0.5 ⎢ ⎥ 0.6⎥⎥⎥ ⎢⎢⎢0.6 ⎢ ⎥ 0.9⎥⎥⎦ ⎢⎢⎣0.2 0.5 0.3

0.4 0.5 0.1 0.3 0.1 0.6 0.2

0.3 0.9 0.5 0.4 0.7 0.6 0.8

0.8 0.7 0.6 0.5 0.3 0.9 0.8

0.2 0.9 0.3 0.7 0.5 0.1 0.4

0.1 0.4 0.4 0.1 0.9 0.5 0.7 0.3 0.5 0.6 0.8 0.6 0.2 0.7

⎡

0.5 0.7 0.7 0.3 0.4 0.8 0.6

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ e7 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

⎡

⎤

⎡

0.3 0.5 0.4 0.8 0.6 0.7 0.5

0.1 0.6 0.5 0.9 0.2 0.8 0.4

0.5 0.2 0.1 0.5 0.9 0.8 0.1

0.4 0.4 0.8 0.1 0.5 0.3 0.8

0.8 0.3 0.2 0.2 0.7 0.5 0.3

0.7⎥ ⎢0.5 ⎢ ⎥ 0.5⎥⎥⎥ ⎢⎢⎢0.9 ⎢ ⎥ ⎥ 0.6⎥⎥ ⎢⎢⎢0.2 ⎥ e = ⎥ 0.9⎥ 3 ⎢⎢⎢0.6 ⎢ ⎥ 0.2⎥⎥⎥ ⎢⎢⎢0.7 ⎢ ⎥ 0.7⎥⎥⎦ ⎢⎢⎣0.1 0.5 0.3

0.5⎥ ⎢0.5 ⎢ ⎥ 0.8⎥⎥⎥ ⎢⎢⎢0.2 ⎢ ⎥ 0.2⎥⎥⎥ ⎢⎢⎢0.4 ⎥ 0.2⎥⎥ e5 = ⎢⎢⎢0.4 ⎢ ⎥ 0.6⎥⎥⎥ ⎢⎢⎢0.5 ⎢ ⎥ 0.3⎥⎥⎦ ⎢⎢⎣0.5 0.5 0.1

0.8 0.5 0.6 0.8 0.7 0.6 0.2

0.6 0.4 0.5 0.3 0.1 0.9 0.3

0.6 0.2 0.7 0.5 0.9 0.7 0.8

0.5 0.3 0.9 0.1 0.5 0.2 0.4

0.5 0.4 0.1 0.3 0.8 0.5 0.9

0.3 0.4 0.5 0.9 0.6 0.4 0.8

0.2 0.8 0.6 0.7 0.6 0.5 0.6

0.4⎥ ⎢0.5 ⎢ ⎥ 0.3⎥⎥⎥ ⎢⎢⎢0.7 ⎢ ⎥ ⎥ 0.2⎥⎥ ⎢⎢⎢0.9 0.1⎥⎥⎥ e8 = ⎢⎢⎢0.6 ⎢ ⎥ 0.5⎥⎥⎥ ⎢⎢⎢0.5 ⎢ ⎥ 0.4⎥⎥⎦ ⎢⎢⎣0.4 0.5 0.4

0.3 0.5 0.4 0.6 0.8 0.7 0.9

0.7 0.2 0.1 0.5 0.7 0.3 0.9

0.6 0.4 0.4 0.3 0.5 0.4 0.5

⎤

⎡

⎤

0.1 0.5 0.7 0.5 0.7 0.8 0.8

0.8 0.3 0.5 0.3 0.8 0.6 0.3

0.4 0.5 0.7 0.5 0.9 0.2 0.7

0.3 0.3 0.2 0.1 0.5 0.6 0.5

0.9 0.2 0.4 0.8 0.4 0.5 0.8

0.7⎥ ⎥ 0.2⎥⎥⎥ ⎥ 0.7⎥⎥⎥ 0.3⎥⎥⎥ ⎥ 0.5⎥⎥⎥ ⎥ 0.2⎥⎥⎦ 0.5

0.9⎥ ⎢0.5 ⎢ ⎥ 0.8⎥⎥⎥ ⎢⎢⎢0.7 ⎢ ⎥ 0.7⎥⎥⎥ ⎢⎢⎢0.6 ⎥ 0.2⎥⎥ e6 = ⎢⎢⎢0.1 ⎢ ⎥ 0.6⎥⎥⎥ ⎢⎢⎢0.3 ⎢ ⎥ 0.1⎥⎥⎦ ⎢⎢⎣0.3 0.5 0.4

0.3 0.5 0.6 0.9 0.8 0.8 0.2

0.4 0.4 0.5 0.7 0.2 0.1 0.8

0.9 0.1 0.3 0.5 0.4 0.7 0.7

0.7 0.2 0.8 0.6 0.5 0.9 0.6

0.7 0.2 0.9 0.3 0.1 0.5 0.8

0.6⎥ ⎥ 0.8⎥⎥⎥ ⎥ 0.2⎥⎥⎥ 0.3⎥⎥⎥ ⎥ 0.4⎥⎥⎥ ⎥ 0.2⎥⎥⎦ 0.5

0.1 0.6 0.5 0.2 0.3 0.5 0.6

0.6 0.3 0.5 0.8 0.7 0.5 0.7

0.6⎥ ⎥ 0.1⎥⎥⎥ ⎥ 0.4⎥⎥⎥ 0.6⎥⎥⎥ ⎥ 0.8⎥⎥⎥ ⎥ 0.3⎥⎥⎦ 0.5

0.4 0.4 0.8 0.5 0.7 0.2 0.4

0.5 0.2 0.7 0.3 0.5 0.3 0.2

⎤

⎤

Next, Step 2 and 3 are implemented, thus the similarity of experts’ opinions, S can be presented below:

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⎡

1 ⎢ 0.8142 ⎢ 0.6575 ⎢ ⎢ 0.7914 S=⎢ ⎢ 0.8318 ⎢ ⎢ 0.7456 ⎣ 0.9035 0.7671

0.8142 1 0.7379 0.7267 0.8220 0.7737 0.7482 0.8263

0.6575 0.7379 1 0.6830 0.8102 0.7374 0.7592 0.8717

0.7914 0.7267 0.6830 1 0.7949 0.7450 0.8140 0.7681

0.8318 0.8220 0.8102 0.7949 1 0.8028 0.7949 0.8363

0.7456 0.7737 0.7374 0.7450 0.8028 1 0.8197 0.7853

0.9035 0.7482 0.7592 0.8140 0.7949 0.8197 1 0.8474

⎤

0.7671 0.8264 ⎥ 0.8717 ⎥ ⎥ 0.7681 ⎥ ⎥ 0.8363 ⎥ ⎥ 0.7853 ⎥ ⎦ 0.8474 1

From Step 4, the row normalized preference similarity weights matrix, SN is shown below: ⎡ ⎤ 0.1536 0.1250 0.1010 0.1215 0.1278 0.1145 0.1388 0.1178

⎢ 0.1263 0.1551 0.1144 0.1127 0.1275 0.1200 0.1160 0.1281 ⎥ ⎢ 0.1051 0.1179 0.1598 0.1092 0.1295 0.1179 0.1213 0.1393 ⎥ ⎢ ⎥ ⎢ 0.1252 0.1149 0.1080 0.1582 0.1257 0.1178 0.1287 0.1215 ⎥ ⎢ ⎥ SN = ⎢ ⎥ ⎢ 0.1243 0.1228 0.1211 0.1188 0.1494 0.1199 0.1188 0.1250 ⎥ ⎢ 0.1163 0.1207 0.1150 0.1162 0.1253 0.1560 0.1279 0.1225 ⎥ ⎣ ⎦ 0.1351 0.1119 0.1135 0.1217 0.1189 0.1226 0.1495 0.1267 0.1145 0.1233 0.1301 0.1146 0.1248 0.1172 0.1264 0.1492

Based on SN , we visualize the similarity social influence network (SSIN), depicted in Fig. 1. For simplicity, only several nodes are shown.

Fig. 1. The similarity social influence network (SSIN) that consists of eight experts’ nodes

Referring to Step 5, the influence-trust score is calculated, where the value of Z is a set of TrustRank indexes and the scalar σ is fixed as 0.5. Y is then normalized as in Step 6 and the results are given below:

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⎡0.1536 0.1250 0.1010 0.1215 0.1278 0.1145 0.1388 0.1178⎤−1 ⎡0.4350⎤⎞ 0.1263 0.1551 0.1144 0.1127 0.1275 0.1200 0.1160 0.1281 0.5194 ⎜ ⎢0.1051 0.1179 0.1598 0.1092 0.1295 0.1179 0.1213 0.1393⎥ ⎢0.4829⎥⎟ ⎜ ⎢ ⎥ ⎢ ⎥⎟ ⎜ 0.1252 0.1149 0.1080 0.1582 0.1257 0.1178 0.1287 0.1215⎥ ⎢0.4248⎥⎟ Y = ⎜I − (0.5) ⎢ ⎢0.1243 0.1228 0.1211 0.1188 0.1494 0.1199 0.1188 0.1250⎥ ⎢0.5465⎥⎟ ⎜ ⎣0.1163 0.1207 0.1150 0.1162 0.1253 0.1560 0.1279 0.1225⎦ ⎣0.5040⎦⎟ ⎠ ⎝ ⎛

0.1351 0.1119 0.1135 0.1217 0.1189 0.1226 0.1495 0.1267 0.1145 0.1233 0.1301 0.1146 0.1248 0.1172 0.1264 0.1492

0.5440 0.5184

  YN = 0.9821 0 0.9939 0.9877 0.9946 0.9935 1 0.991 Based on YN , the most influential expert in this influence-trust network is expert 7 with the highest influence-trust score and the least influential expert is expert 2. In Step 7, the collective preference relation Pc for all individual experts is obtained and presented below: ⎡ ⎤ 1

0.3696 0.3993 0.6572 0.5248 0.4092 0.5530

⎢ 0.6304 1 0.4699 0.2838 0.4038 0.5354 0.4612 ⎥ ⎢ ⎥ ⎢ 0.6007 0.5301 1 0.4033 0.5501 0.4738 0.3708 ⎥ ⎢ ⎥ c P = ⎢ 0.3428 0.7162 0.5967 1 0.2925 0.5403 0.2124 ⎥ ⎢ 0.4752 0.5962 0.4499 0.7075 1 0.6043 0.5476 ⎥ ⎢ ⎥ ⎣ 0.5908 0.4646 0.5262 0.4597 0.3957 1 0.3286 ⎦ 0.4470 0.5388 0.6292 0.7876 0.4524 0.6714

1

Finally, Step 8 is carried out and the final ranking of the alternatives are: A8  A5  A4  A1  A3  A2  A6 . It is obtained that the best alternative to be chosen is alternative 8 and the worst one is alternative 6. This result is the agreed decision of the group, considering the influence-trust element in the decision making process.

4 The Framework of the Proposed Model Based on the methodology presented above, the framework of the proposed model is shown in Fig. 2. The new idea is proposed and successfully integrated in Fuzzy GDM model by Kamis et al. [3].

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Fig. 2. The framework of Influence-Trust Fuzzy GDM

5 Conclusion and Future Works The level of trust in an expert can significantly impact their influence in decision making. Experts who are trusted and respected in their field are more likely to shape decisions compared to those who lack trust or respect. Additionally, influence can contribute to building trust when an individual is capable of positively influencing others through offering helpful advice or making decisions that benefit all parties involved. This positive relationship between trust and influence suggests that building trust can enhance an individual or organization’s influence in decision making. This study proposes a new definition of trust and integrates it into an influencebased fuzzy GDM, offering promising ideas for future research and the development of new fuzzy GDM models based on trust. This research has the potential to expand its scope to encompass more sophisticated algorithms and be applied to practical scenarios for real-world problem-solving. Acknowledgement. I would like to extend my sincerest appreciation to the President of the Association of Academies and Societies of Sciences in Asia (AASSA) and the chair of the Women in Science and Engineering (WISE) Special Committee under AASSA for bestowing upon me the prestigious Prof. Yoo Hang Kim Young Women Scientists Award 2023. I am

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immensely grateful for the sponsorship provided by this award, which covered all expenses related to my presentation at INFUS 2023 in Istanbul, Turkey.

References 1. Zadeh, L.H.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 2. Kamis, N.H., Chiclana, F., Levesley, J.: Preference similarity network structural equivalence clustering based consensus group decision making model. Appl. Soft Comput. 67, 706–720 (2018) 3. Kamis, N.H., Chiclana, F., Levesley, J.: An influence-driven feedback system for preference similarity network clustering based consensus group decision making model. Inf. Fusion 52, 257–267 (2019) 4. Shang, C., Zhang, R., Zhu, X., Liu, Y.: An adaptive consensus method based on feedback mechanism and social interaction in social network group decision making. Inf. Sci. 625, 430–456 (2023) 5. Hua, Z., Jing, X., Mart´ınez, L.: Consensus reaching for social network group decision making with ELICIT information: a perspective from the complex network. Inf. Sci. 627, 71–96 (2023) 6. Artz, D., Gil, Y.: A survey of trust in computer science and the semantic web. J. Web Semantics 5(2), 58–71 (2007) 7. Ahlim, W.S.A.W., Kamis, N.H., Ahmad, S.A.S., Chiclana, F.: Similarity-trust network for clustering-based consensus group decision-making model. Int. J. Intell. Syst. 37(4), 2758– 2773 (2022) 8. Gai, T., et al.: Consensus-trust driven bidirectional feedback mechanism for improving consensus in social network large-group decision making. Group Decis. Negot. 32(1), 45–74 (2023) 9. Liu, Y., Liang, H., Dong, Y., Cao, Y.: Multi-attribute strategic weight manipulation with minimum adjustment trust relationship in social network group decision making. Eng. Appl. Artif. Intell. 118, 105672 (2023) 10. Capuano, N., Chiclana, F., Herrera-Viedma, E., Fujita, H., Loia, V.: Fuzzy group decision making for influence-aware recommendations. Comput. Hum. Behav. 101, 371–379 (2019) 11. Sedek, A.N.S.A.B., Kamis, N.H., Kadir, N.A., Mohamad, D., Chiclana, F.: Social influence in fuzzy group decision making with applications. In: Kahraman, C., Tolga, A.C., Cevik Onar, S., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) Intelligent and Fuzzy Systems: Digital Acceleration and The New Normal-Proceedings of the INFUS 2022 Conference, vol. 505, pp. 272–279. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-09176-6 32 12. Yao, S., Gu, M.: An influence network-based consensus model for large-scale group decision making with linguistic information. Int. J. Comput. Intell. Syst. 15, 1–17 (2022) 13. Wu, J., Chiclana, F.: A social network analysis trust-consensus based approach to group decision-making problems with interval-valued fuzzy reciprocal preference relations. Knowl.-Based Syst. 59, 97–107 (2014) 14. Herrera-Viedma, E., Herrera, F., Chiclana, F., Luque, M.: Some issues on consistency of fuzzy preference relations. Eur. J. Oper. Res. 154(1), 98–109 (2004) 15. Gonz´alez-Arteaga, T., de Andr´es Calle, R., Chiclana, F.: A new measure of consensus with reciprocal preference relations: the correlation consensus degree. Knowl. Based Syst. 107, 104–116 (2016) 16. Yager, R.R., Filev, D.: Operations for granular computing: mixing words and numbers. In: 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98CH36228), vol. 1, pp. 123–128. IEEE (1998)

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17. Chiclana, F., Herrera, F., Herrera-Viedma, E., Poyatos, M.C.: A classification method of alternatives for multiple preference ordering criteria based on fuzzy majority. J. Fuzzy Math. 4, 801–814 (1996) 18. Shamudin, N.A.A.: Visualisation of cause-effect criteria in preference similarity social influence network group decision making model (Unpublished master dissertation). Universiti Teknologi MARA (UiTM) (2022)

Rough Data Envelopment Analysis: An Application to Indian Agriculture Alka Arya1 , Adel Hatami-Marbini2(B) , and Pegah Khoshnevis3 1

Indian Institute of Management Kashipur, Kashipur 244713, India [email protected] 2 Huddersfield Business School, University of Huddersfield, Huddersfield, UK [email protected] 3 Sheffield University Management School, University of Sheffield, Sheffield, UK [email protected]

Abstract. In an uncertain world, nothing is definite. Measuring a person’s effectiveness in such a volatile world is inevitable. For a traditional data envelopment analysis (DEA) approach for precisely evaluating the relative efficacy of homogenous decision-making units (DMUs), precise input and output quantity data are required. However, it’s possible that precise knowledge of the data won’t be available in many real-world applications. This kind of situation can be handled using rough set theory. In order to quantify uncertainty, this work attempts to build a rough DEA model by merging traditional DEA with rough set theory with optimistic and pessimistic confidence levels of rough variables. In the proposed method, a unified production frontier is created for all DMUs using the same set of restrictions, allowing one to precisely evaluates each DMU’s efficiency in the presence of rough data. Additionally, a ranking system based on the approaches put out is offered. The paper examines the effectiveness of the Indian fertiliser supply chain for over a decade in the face of uncertain circumstances. The results of the suggested models are contrasted with those of the current DEA models to show how policymakers might improve the performance of the Indian fertiliser industry’ supply chains. (An earlier version of this study was published in Arya, A., & Hatami-Marbini, A. (2023). A new efficiency evaluation approach with rough data: An application to Indian fertilizer. Journal of Industrial and Management Optimization, 19(7), 5183–5208).

Keywords: Data envelopment analysis Fertilizer industries

1

· Frontiers · Rough data ·

Introduction

Supply chain management (SCM) is a key component of boosting an organization’s sales and profitability in today’s business environment. SCM was first

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 689–696, 2023. https://doi.org/10.1007/978-3-031-39774-5_76

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developed in the 1990s, and it involved managing and scheduling manufacturing/production, shipping, and distribution to satisfy customers. Despite the fact that there is no one universal definition of SCM, a supply chain is a network of partners involved in the many processes and activities that produce value in the form of goods and services in the hands of the final consumer (Agrell and Hatami-Marbini, 2013). Fertiliser factories are prominent in India because of the country’s heavy reliance on agriculture. The goal of the fertiliser industry is to determine the ideal flow and quantity of components for crop productivity. Due to India’s highly advanced fertilizer-producing businesses, the food sector there has become a benchmark. India has a large fertiliser industry, both state and private. Data envelopment analysis (DEA) is the most widely used non-parametric tool first published by Charnes et al. (1978) to quantify the relative efficiency of a set of enterprises, commonly known as decision-making units (DMUs). However, some observed data in several real-world situations is ambiguous, uncertain, or inaccurate. To address practical DEA issues, we used the crude set theory in this study. In supply chain scenarios, the rough set theory, introduced by Pawlak (1982), is a useful technique to handle unclear data. To ascertain the performance efficiency in the furniture manufacturing sector, Xu et al. (2009) proposed the rough DEA model. Their method uses a distinct constraint set (production possibility sets) to estimate each DMU’s efficiency. Comparisons between efficiencies may not be accurate and impartial if the border is not united, which reduces the significance of the comparisons. In order to create a unified production frontier for all DMUs that can be used to objectively evaluate each DMU’s performance in the presence of imperfect input-output data, the suggested method uses the same set of constraints. To the best of our knowledge, no research on supply chain and rough DEA with unified frontier has been done in order to provide the aforementioned contributions. The following is how the paper is set up: There is a literature review in Sect. 2. The suggested DEA is provided with rough data and a unified frontier in Sect. 3. The use of the suggested techniques is shown in Sect. 4 with reference to the Indian fertiliser sector. The findings and the paper’s future focus are summarised in the final section.

2

Literature Review

There are 57 sizable fertiliser facilities in India, as well as a few individual businesses. The 12th Plan report from India’s Department of Fertilisers estimates the need for food grains and fertilisers from 2020 to 2025. 10 million more tonnes of food grains and 3 million more tonnes of fertiliser will be required up until 2025. Therefore, enhancing fertiliser firm performance should be a priority for the country’s research and development policy. The importance of chemical fertilisers in raising agricultural output has long been established. Some contend that fertiliser was just as important to the success of the Green Revolution as the crops it was used to change. It may be responsible for up to 50% of Asia’s yield growth, per (Hopper 1993). In order to make the best use of these ingredients, fertiliser firms must be effective in their operations. Therefore, assessing the

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performance of the fertiliser business is crucial. (Charnes et al., 1978) suggested conventional DEA to assess the comparative effectiveness of a group of businesses. The majority of the time, traditional DEA models call for deterministic input and output values, or (Charnes et al., 1978). A large number of crucial elements that go into evaluating supply chain performance are dynamic. Because the traditional DEA method requires data to be deterministic variables, fluctuations in data make it challenging for conventional DEA to address the supply chain problem. (Charnes et al., 1978). For more than ten years, the rough set approach has been applied in DEA to capture the variation in the use of inputs and outputs. To estimate the performance efficiency in the furniture manufacturing sector, Xu et al. (2009) suggested a rough DEA model. To deal with the uncertainty present in supply chains, Shafiee (2017) added rough variables in a two-stage SBM model. In order to deal with the inherent uncertainties and assess the supply chain performance of 17 commercial and public fertiliser industries in India over a ten-year period, this study used the rough set theory in DEA with unified frontier. The rough DEA model was published in Xu et al. (2009) to evaluate the effectiveness of the furniture manufacturing sector’s performance. Each DMU in their method has a unique set of producing possibilities. All DMUs’ efficiencies could vary if the border is not united, making efficiency comparisons pointless and unnecessary. In order to build a uniform production frontier for all DMUs and to be able to objectively evaluate each DMU’s performance in the face of imperfect input-output data, we suggested a rudimentary DEA model with the same set of constraints.

3

Proposed DEA Model

To build a single production frontier for all DMUs using this strategy, we present the same set of restrictions (production possibility sets) that can be used to objectively evaluate each DMU’s performance in the presence of rough input s  yrj vr r=1 and output data. Assume that Ej =  represents the jth DMU effim xij ui i=1

ciency. We create the following set of interval DEA models to handle various production frontiers for determining the lower and upper limit efficiencies of DM Uo , using the optimistic and pessimistic data generated from rough data:

s  sup(β) max Ej

=

r=1 m  i=1

sup(β)

yrj

inf(β)

xij

s 

vr

ui

inf(β) max Ej

=

inf(β)

r=1 m  i=1

yrj

vr

sup(β)

xij

ui

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Sub. to

inf(β) Ej

inf(β)

r=1 m 

=

i=1

yrj

sup(β)

xij

s 

vr ui

≤ 1, Sub. to

inf(β) Ej

=

inf(β)

r=1 m  i=1

yrj

vr

sup(β)

xij

ui

≤ 1,

(1) (2) Where all the variables are greater than or equal to zero. After applying the transformation (Charnes and Cooper, 1962), the dual of the aforementioned pair of DEA models (1)–(2) is created: sup(β)

Eo

inf(β)

= min θ n  sup(β) inf(β) sub. to xij λj ≤ xio θ, ∀i,

= min θ n  sup(β) sup(β) sub. to xij λj ≤ xio θ, ∀i,

n 

n 

j=1

j=1

inf(β) yrj λj

≥

sup(β) yro ,

∀r.

(3)

Eo

j=1

j=1

inf(β) yrj λj

inf(β)

≥ yro

, ∀r.

(4)

When the other DMUs are assumed to be operating at their lowest level of production activity while DM Uo appears to be operating at their highest level, inf(β) Eo in model (4) aims to assess the most favourable relative efficiency of sup(β) DM Uo . On the other hand, the goal of Eo in model (3) is to determine the lower bound of the best relative efficiency of DM Uo under the worst production activity scenario for all DMUs. For a given betain[0.6, 1], the optimistic model (3) and pessimistic model (4) are referred to as the rough DEA (RDEA) models. Remarkably if β ∈ (0, 0.5], the models (1) and (2) can be translated into models inf(β) sup(1−β) sup(β) inf(1−β) = Eo and Eo = Eo . (3) and (4), respectively using Eo sup(β)∗ inf(β)∗ Let Eo and Eo represent the best values for the models (3) and (4). Based on the trust value, efficient, weakly efficient, and inefficient DMUs for any β ∈ [0.6, 1] can be described as follows: sup(β)∗

= 1, DM Uo is efficient, – If Eo inf(β)∗ sup(β)∗ – If Eo = 1 and Eo < 1, DM Uo is weakly efficient, inf(β)∗ – If Eo < 1, DM Uo is inefficient. In this research, we offer models (3) and (4) where the constraint sets are similar for all DMUs and for both optimistic and pessimistic evaluations of all DMUs. sup(β)∗

inf(β)∗

and Eo be the optimal values of models (3) and Theorem 1. Let Eo sup(β)∗ inf(β)∗ (4), respectively. Then, Eo ≤ Eo . sup(β)

Theorem 2. Eo

inf(β)

and Eo

lie within (0, 1].

The decision-maker frequently tries to differentiate between the performance of efficient DMUs, however the efficient DMUs cannot be fully graded. A superefficiency DEA model was initially created by Andersen and Petersen (1993) to enhance the discriminatory power of DEA. Here, we suggest the following

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super-efficiency RDEA models to rank the efficient DMUs in the presence of rough data for a given β ∈ [0.6, 1]. sup(β)

SEo

sub. to

= min θ n  sup(β) inf(β) xij λj ≤ xio θ, ∀i,

inf(β)

SEo

sub. to

j=1,=o n  j=1,=o

inf(β)

yrj

sup(β)

λj ≥ yro

, ∀r,

= min θ n  sup(β) sup(β) xij λj ≤ xio θ, ∀i,

j=1,=o n  j=1,=o

inf(β)

yrj

inf(β)

λj ≥ yro

, ∀r,

λj ≥ 0, ∀j, θ is unrestricted. λj ≥ 0, ∀j, θ is unrestricted. (5) (6) Xu et al. (2009) proposed the rough DEA model, but the models were not proposed for finding the super-efficiencies. According to their method, the production frontier varies for each DMU and for both optimistic and pessimistic models. Comparisons of efficiency are therefore useless. In the suggested method, each DMU’s effectiveness is assessed using the same production frontier.

4

Indian Fertilizer’s Industries

The systems that deliver fertiliser are vital to increasing India’s access to food. There is no agreement in the literature or among practitioners about the choice of inputs and outputs, which is a key phase in the DEA process. This case study comprises the three inputs and three outputs (Kumar et al., 2017) that are considered to estimate the respective technology of the fertiliser supply chain, without losing the scope of applicability (see Fig. 1). The data for this application were gathered from the Centre for Monitoring Indian Economics (CMIE), and it takes into account 17 Indian fertiliser supply chains over a ten-year period. Let’s say the β trust level is 0.6. The rough variables β optimistic and beta pessimistic values are first calculated, and the resulting rough variables are then employed in the suggested models (3 and 4). Given that the data are in interval sup(β) inf(β) , Eo ]. Models (3) form, it should arrive to the interval efficiencies [Eo and (4) are used to calculate the efficiencies. According to efficiency results, the efficiency of the fertiliser supply chains F6, F1, and F6 are [1, 1], indicating that no improvement in their data is necessary, while the interval efficiency of the fertiliser supply chain F1 associated with β = 1 is [0.3425, 1], indicating that some improvement in the optimistic side’s data is necessary to reach the frontier. Only when viewed positively, the F15 fertiliser supply chain is efficient. For β values, the super-efficiency models (5) and (6) are proposed, and the corresponding findings are shown in Fig. 2. The efficient supply chains may be seen to take a value greater than 1, which gives us strong discriminatory power. The fertiliser supply chain F1 contains super-efficiency measures greater than 1 in addition to the optimistic perspective of β = 1, as shown in Fig. 2. The largest private fertiliser supply chain in Mumbai, F6, has the highest efficiency for all pre-determined βs, as shown in Fig. 2. Even after combining the pessimistic and optimistic viewpoints, the fertiliser supply chain F6 performs the best as expected. The second most effective supply

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Fig. 1. Internal structure of supply chain

Fig. 2. 17 fertiliser supply chains’ super-efficiencies

chain is the F15 fertiliser supply network, which is run by a private company in Gujarat (on India’s west coast). The government and the states of Andhra Pradesh and Mumbai, respectively, are home to the F8 and F17 supply networks, which are the least effective of the 17 supply chains. The efficiency of the fertiliser supply chain might be increased by further optimising the production process, and this study can provide some insight into potential solutions. Compare the rudimentary DEA technique established in this work with an original DEA method to see how efficient the supply chain of the fertiliser companies performed under unpredictable conditions. When there is uncertainty, supply chain performance analysis cannot typically be performed using the traditional DEA method. In order to deal with uncertainty, the suggested rough DEA findings are obtained by using optimistic and pessimistic β trust values for rough variables. By aggregating the rough input-output data values, the rough data are transformed into crisp ones, and the crisp data are then employed in the original DEA approach to assess the efficacy of fertiliser supply networks. To determine if there is a statistically significant difference between two independent samples, the non-parametric Kruskal-Wallis test is frequently used (Lehmann and D’Abrera, 1975). The asymptotic p-value for the Kruskal-Wallis test is 0.0, which is less than the significance level (0.05), and it has 1 degree of

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freedom. Therefore, we draw the conclusion that there is no proof that the findings are comparable in terms of efficiency measurements. Overall, we would argue that the proposed method’s evaluation results are more appropriate, accurate, and true to reality because the fertiliser supply chain F6 is the most competitive and productive among other fertiliser sectors. According to the description above, some general DEA qualities include (i) handling ambiguous information in the supply chain and (ii) handling outliers. Therefore, when dealing with some supply chain applications in ambiguous circumstances, a rough DEA is more appropriate.

5

Conclusion

In this investigation, DEA models were created while rough variables were present. The performance of the supply chain network is evaluated using the provided rough DEA models. The suggested rough DEA creates a single production frontier for all DMUs using the same set of restrictions (production possibility sets), allowing one to precisely estimate each DMU’s performance in the presence of rough input and output data. In order to demonstrate the usability and effectiveness of the created method in the situation of sparse data, we lastly implemented the suggested methods to the Indian fertiliser supply chains. According to the data, the Indian fertiliser supply chain’s efficiency scores range from 0.0059 to 0.5425, indicating that it is often relatively inefficient. The most competitive and effective fertiliser industry is the F6 (private) fertiliser supply chain. The second-most effective fertiliser delivery system is run by a private company called F15 in Gujarat. The F8 and F17 supply chains, which are run by the government, are the least effective of the 17 supply chains. It would be fantastic to further optimise the production process in order to increase the effectiveness of the fertiliser supply chain in India. The operational efficacy and efficiency of the Indian fertiliser supply chain can be significantly increased with the assistance of decision-makers thanks to the proposed evaluation approach. The operational efficacy and efficiency of the Indian fertiliser supply chain can be significantly increased with the assistance of decision-makers thanks to the proposed evaluation approach. The Kruskal-Wallis test shows that there is no proof that the efficiency measures from the rough DEA and conventional DEA are equivalent. Uncertain data are only taken into account as rough variables as a research restriction, and efficiency are only calculated in cases when homogenous units are available. Researchers may in the future focus on solving this problem. The suggested rough technique can be applied to the Malmquist productivity index, the Cross-efficiency, the network DEA, and others.

References Agrell, P.J., Hatami-Marbini, A.: Frontier-based performance analysis models for supply chain management: state of the art and research directions. Comput. Ind. Eng. 66(3), 567–583 (2013)

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Andersen, P., Petersen, N.C.: A procedure for ranking efficient units in data envelopment analysis. Manage. Sci. 39(10), 1261–1264 (1993) Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Naval Res. Logistics Q. 9(3–4), 181–186 (1962) Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2(6), 429–444 (1978) Hopper, W.: Indian agriculture and fertilizer: an outsider’s observations. In: Keynote Address to the FAI Seminar on Emerging Scenario in Fertilizer and Agriculture: Global Dimensions. FAI, New Delhi (1993) Kumar, P., Singh, R.K., Shankar, R.: Efficiency measurement of fertilizermanufacturing organizations using fuzzy data envelopment analysis. J. Manage. Analytics 4(3), 276–295 (2017) Lehmann, E.L., D’Abrera, H.J.: Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, San Francisco (1975) Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982) Shafiee, M.: Supply chain performance evaluation with rough two-stage data envelopment analysis model: noncooperative Stackelberg game approach. J. Comput. Inf. Sci. Eng. 17(4) (2017) Xu, J., Li, B., Wu, D.: Rough data envelopment analysis and its application to supply chain performance evaluation. Int. J. Prod. Econ. 122(2), 628–638 (2009)

Finance

Analysis of Various Portfolio Allocation Decision-Making Techniques in Crypto Assets Using Fuzzy Sets Murat Levent Demircan1(B)

and Tayfun Dirinda2

1 Faculty of Engineering and Technology, Galatasaray University, Ortaköy, ˙Istanbul, Turkey

[email protected]

2 QNB Finansbank, ˙Istanbul, Turkey

[email protected]

Abstract. To achieve success in financial markets, no matter how much data is obtained and trading markets develop, it will continue to be a fundamental problem for professional portfolio managers, academics, and society. The modern portfolio theory proposed by the American economist Harry Markowitz to find solutions to the problems in portfolio allocation (1952) made its mark in history as the pioneering work of the period. Afterward, many studies have been carried out to design the best portfolio distribution method based on this study. The resulting studies provided solutions to the missing points in previous studies in parallel with the development and maturation of the stock market. Advances in different academic fields (e.g., behavioral finance, operations research) and technological advances (e.g. big data, fast computers) have influenced these studies. There are several approaches to improving the portfolio allocation process. The primary purpose of this study is to examine the effects of different portfolio allocation techniques, whose first emergence dates back to the middle of the 20th century, on new-generation risky assets by using linguistic labels (fuzzy terms) to express the preferences of decision-makers due to the uncertainty of information in the financial markets and the complexity of the decision-making problem. It is expected that serious differences will be observed in performance measurements due to the structure of the stocks they are working on and the price structure of the crypto assets used in this study. The success of crypto-assets is expected to fail older-generation investment instruments (stocks, commodities) visibly. Keywords: Multi-Criteria Decision Making · Fuzzy Sets · Portfolio Allocation

1 Introduction In its most basic form, investment ensures that securities (shares, funds, etc.) appreciate in value over time. In short, investment is a way to build wealth for our future by preventing our currency from depreciating in the face of inflation. Of course, the job does not involve just choosing the sector and the stock you will invest in. The process of selecting a portfolio may be divided into two stages. The first stage starts with observation and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 699–707, 2023. https://doi.org/10.1007/978-3-031-39774-5_77

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experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the portfolio choice. In this study, crypto assets, a relatively new investment area, were used, unlike the sampling of the techniques popularly used in the literature. To make complex distribution decisions that a portfolio manager should take as accurately as possible, the success levels expressed verbally in the financial world are supported by fuzzy triangular sets. A highly complex process awaits decision-makers in investment, like many other issues. Stock, historical data, opinions of other investors, macroeconomic data, the situation of the country to be invested in, the wishes of the investor, the investment period and amount, and finally, the risk perception. Since it would be difficult to compare all these variables, it would be much more accurate for the decision maker to decide between linguistic options and benefit from fuzzy number sets instead of giving a clear numerical value. Triangular fuzzy numbers that are chosen as a method in this study not only can be used to express the vagueness and the uncertainty of information but also can be used to represent fuzzy terms in information processing. Besides being integrated with decisionmaking, the triangular fuzzy number has been applied in many fields, such as risk evaluation, performance evaluation, forecast, matrix games, and space representation. Modern portfolio theory (MPT) is a practical method for selecting investments to maximize their overall return within an acceptable level of risk. An important component of the theory is diversification. Most investments are either high risk and high return or low risk and low return. Markowitz argues that investors can achieve the best results by choosing the most appropriate mix based on assessing their risk tolerance. Because the “expected stock returns” used in MPT are challenging to predict, the estimated values often differ significantly from the actual ones. These estimation errors lead to suboptimal portfolio composition and, thus, to poor portfolio performance. Instead, the GMVP (Global Minimum Variance Portfolio) assumes all stocks have equal expected returns. Under this assumption, all stock portfolios differ only in their risk but not in expected returns. Therefore, the only productive stock portfolio is the one with the least risk. SIM (Single Index Model) is a simple asset pricing model for measuring both the risk and return of a stock. The formula has market exposure (beta), firm-specific exposure (alpha), and a firm-specific unexpected component (residual). The performance of each stock is related to the performance of a market index. The shrinkage method helps investors to produce the most suitable portfolio in terms of earning higher profits with lower risk levels compared to the portfolio of the traditional SCM (Sample Covariance Matrix) method. The technique’s success comes down to choosing a suitable shrinkage target, but generally, investors have struggled to find it. According to Black-Litterman, asset allocation is a decision faced by an investor who has to choose how to distribute their portfolio across a range of asset classes. It assumes that the initially expected returns are necessary before the asset allocation can be made. The user only needs to indicate how their assumptions about expected returns differ from the markets and indicate the degree of confidence in the alternative assumptions. This is where the difficulties of decision-making we mentioned above come into play. Instead

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of making a clear numerical comparison, the verbal answers of the decision-maker were digitized by using triangular fuzzy number sets. From this, the Black-Litterman method calculates the desired (mean-variance efficient) asset allocation [12–16], At this point, the next step after creating the portfolio is to measure the return of the portfolio. So, is the portfolio with the best rate of return the best portfolio? In a successful investment, it is impossible to ignore the risk we take to obtain this return. Treynor suggested that risk has two components—risk from fluctuations in the stock market and risk from fluctuations in individual securities. The Sharpe ratio risk measure uses the standard deviation of the portfolio instead of only considering the systematic risk represented by beta. The Information Ratio (IR) is often used to measure a portfolio manager’s skill and ability to generate excessive returns. However, it attempts to determine the consistency of performance by including a tracking error or standard deviation component in the calculation [17]. The first step that the portfolio manager will take to increase his success while investing will be to focus on instruments with a historical background. However, the older a market is, the lower the risk and the lower the return volatility. In relatively new markets, the risks are high because the investor pool could be more solid and stable. At their core, cryptocurrencies are decentralized digital currencies usually designed to be used over the internet. In summary, in the following sections of the article, you will see the applications of the 5 portfolio management techniques we mentioned above using two different investment tools. To increase the success of the Black-Litterman technique, the verbal questions asked to the investors will be interpreted by converting them into fuzzy triangular numbers. Cryptocurrencies and American stocks will be our primary materials. Later, the 3 measurement coefficients that we mentioned in the first section will be applied in all 3 portfolios and will be compared. As we said at the end of all these measurements, we expect the success order to be Stock > Mixed > Crypto because the techniques were previously found, and the crypto asset market is at an early stage.

2 Literature Review An essential real-world issue in finance is portfolio creation. The conventional method involves assuming customarily distributed returns and building a portfolio with a minimal amount of risk (as determined by the standard deviation of portfolio returns) for a predetermined (and least acceptable) return. In reality, returns are heavily tailed and not regularly distributed. The normalcy assumption thus grossly underestimates risk. For the creation of the most stable portfolio, Pinsky and Vasiukevich adopt stable elliptical distributions. The key findings show that stable portfolios are often equivalent to traditional Markowitz portfolios [1]. An expert’s judgment and investment data are considered in a two-stage portfolio selection methodology that Dalkiliç & Akba¸s devised. The constrained Fuzzy Analytic Hierarchy Process is employed in the initial stage. In the subsequent stage, a fuzzy logic technique is used to resolve the fuzzy linear programming (FLP) issue that was generated using the weights of the given criteria [2]. The efficacy of optimum portfolios and their comparative benefits over the naive, equal-weight portfolio (also known as the 1/N rule) have been hotly contested. DeMiguel

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[3] tests fourteen portfolio methods on seven datasets in their landmark work and discovers none consistently beat the naive approach. The enormous volume of citations for their study demonstrates the influence it has had on academia and industry, even though their results are considerably inflated (e.g., see arguments in Kirby and Ostdiek [4] and Kan [5]). When used on a multi-asset dataset, Bessler [6] demonstrates that a strategy built on the Black and Litterman [7] framework performs better than the naive technique. Branger [8] proposes a grouping approach in which the optimization of portfolios is carried out on groups of similarly weighted stocks. They demonstrate that their technique outperforms many current strategies that seek to handle estimate challenges. The shrinkage models of Kan and Zhou [10] and Tu and Zhou [11] are both improved by Han [9], who discovered that the suggested model performs better than both the naive strategies and the current shrinkage models. A unique approach of defuzzification based on the statistical Beta distribution mean value and an algorithm for ranking fuzzy numbers based on the crisp number ranking system using R are proposed by Rahmani and Lotfi [18]. Several numerical examples are used to explain the algorithm, then contrasted with some of the other approaches described in the literature. The article by Kumar De and Beg [19] discusses a brandnew defuzzification technique for massive fuzzy collections. The dense fuzzy set for triangular fuzzy integers is first defined. After then, new defuzzification techniques were developed with precise convergence checks.

3 Methodology and Applications 3.1 Preliminaries of Randomness Test First, the volatile nature of crypto assets seems unpredictable to investors and as risky as gambling. With the randomness analysis to be made, two portfolios consisting of the stock market and crypto assets will be compared. While doing this study, 10 crypto assets and stocks were selected. While stock selections were made, 10 stocks with the highest trading volume on NYSE and NASDAQ (May 22) were selected. While choosing crypto assets, care was taken to make two different portfolios. The first 10 coins with the highest market volume (May 22) were brought together. After the selected assets were determined, the weights were determined using a random number generator for both the two stock portfolios and the two crypto portfolios. The 1-year returns of these 10 portfolios were calculated. A return matrix was created for each portfolio, and 10 random weightings were compared. 3.2 Fuzz Sets and Defuzzification As mentioned in the previous sections, the portfolio manager decides whether the investment instruments will perform better or worse than the average to apply the BlackLetterman technique more accurately. As shown in the table below, the success order is divided into 9 degrees. Afterward, each entity was asked one by one to the experts on the subject. The results were included in the calculation after defuzzification (Table 1). According to Rahmani and Lotfi [18], the beta distribution technique is used to defuzzify the chosen number. Consider the triangular fuzzy number Á = (l, m, u). We

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Table 1. Fuzzy Set Values Verbal Meaning

Number

L

M

U

Result

May generate extremely higher-than-average returns

1

0,850

0,950

1,000

0,9333

May generate significantly higher-than-average returns

2

0,750

0,800

0,900

0,8167

May generate noticeably higher-than-average returns

3

0,625

0,660

0,850

0,7117

May generate returns slightly more than average

4

0,525

0,575

0,650

0,5833

May produce returns equal to the average

5

0,450

0,500

0,550

0,5000

May generate returns slightly less than average

6

0,325

0,400

0,475

0,4000

May generate noticeably lower-than-average returns

7

0,175

0,250

0,350

0,2583

May generate significantly higher-than-average returns

8

0,095

0,150

0,185

0,1433

May generate extremely higher-than-average returns

9

0,000

0,075

0,125

0,0667

first project Á on the interval (0,1), which will be in the form of Áı = ((l-l)/(u-l),(ml)/(u-l),(u-l)/(u-l)) = (0,(m-l)/(u-l),1). Then we define the parameter corresponding to the Beta distribution as follows: α=

m−l +1 u−l

(1)

In the Beta distribution corresponding to the projection of fuzzy number Á = (l,m,u), we have xmod = (m-l)/(u-l), and by using the above equation and substituting it, we get: β=

u−m +1 u−l

(2)

We use these two equations, as shown below, to calculate the mean value of Beta distribution corresponding to the fuzzy number: μ =

m + u − 2l α = α+β 3(u − l)

(3)

´ which is obtained (as shown below) by transferring μ from The real number μA, the interval (0,1) to the interval (l,u), is considered as the real number corresponding to the fuzzy number Á = (l,m,u): 

´ = µ (u − l) + l μA

(4)

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3.3 Main Study Portfolios As mentioned in the introduction, 3 portfolios were used in the study. First, the assets to be included in the portfolio consisting of Crypto assets were selected. While creating the stock portfolio, American stocks were chosen as the source. It was chosen among the stocks with the highest volume traded (November 22) on the S&P 500 index, which was chosen to be used in the techniques accompanying us in our study. While creating the portfolio created as a mix of these two portfolios, an equal amount of assets were taken from both crypto and stocks. While selecting these assets, the highest volume assets in their two portfolios were taken to give a more consistent result. After the assets in all portfolios were selected, the studies were carried out using Excel. To apply the techniques, 2-year price data were provided daily. 30 June 2019 was chosen as the beginning, and 30 June 2021 as the end. Investing.com websites were selected for stocks, and coinmarketcap.com websites were selected for crypto assets. After the techniques were applied and the weights of the assets were determined, a 1year investment horizon was determined, and the investment made between July 1, 2021, and July 1, 2022, was adhered to. No sale or purchase has been made. Then, the 1 July 2022 values of the investment were determined, and the returns were calculated. While applying the techniques, no restrictions were made on short selling. For stock portfolio; APPL, AMCR, BA, CSX, CTRA, F, FDX, FTNT, GE, GM, GOOGL, GPS, HST, JPM, KO, MO, MRNA, MSFT, NEM, NFLX, NKE, OXY, PFE, T, TSLA, UAA, WBA, WBD are used. For crypto portfolio; BAT, BTCH, BTC, BNB, ADA, LINK, DASH, MANA, DOGE, EOS, ETC, ETH, IOTA, LTC, LRC, MKR, XMR, NEM, NEO, MATIC, QTUM, XLM, XTZ, THETA, TRX, VET, XRP, ZCASH are used.

4 Results 4.1 Randomness Test Interpretations As mentioned in the previous section, 10 random portfolios created for both crypto assets and stocks were compared with each other in terms of return generation. When stock portfolios are compared, return yield differentials in the 15%/-21% (NYSE) and 27%/22% (NASDAQ) range in the band. When the portfolios of crypto assets are compared, the yield differences vary in the band range for the Top 10 cryptocurrencies (40%/-26%) and in the band range for the mixed (32%/-28%). These results clearly show us that the risks taken by a portfolio manager who wants to invest in crypto assets are much higher than investing in stocks if they are uninformed. This shows that someone who enters the market with knowledge is more likely to get relatively more consistent results from a random investment. The randomness analysis shows that in accordance with the purpose of the study, cryptocurrencies are a market that needs to be studied. 4.2 Return Rates The way to measure the success of any investment made is to examine the year-end returns. The higher the return, the more successful the portfolio is. The table below shows the returns of the 5 techniques for all portfolios. Average portfolio returns, Stock P: %-12, Hybride P: -%25; Crypto P: -%56, are also added for comparison. While preparing the average portfolio, the weight distribution was made equally (Table 2).

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Table 2. Return Rates of 5 Technique Returns

Efficient

GMVP

Single Index

Shrinkage

Black-Litterman

Stock

20%

-1%

-7%

1%

22%

Hybrid

15%

8%

-9%

0%

-1%

Crypto

-355%

-37%

-61%

-80%

-11%

4.3 Best Performances in Each Technique by Each Measurement Oftentimes, investors consider themselves successful when they achieve high rates of return. But in fact, experienced portfolio managers can maximize the return for the risk they take. An evaluation is made by considering risk and risk distribution. The most successful technique-portfolio pairs according to the three measurement methods are shown in Table 3. Table 3. Top 5 Performance by Each Measurement Rank

Treynor

Sharpe

Information Ratio

1

Hybrid-Single ˙Index

Stock-Efficient

Stock-Efficient

2

Hybrid-Shrinkage

Stock-Black-Litterman

Stock-Black-Litterman

3

Crypto-Efficient

Hybrid-Efficient

Hybrid-Efficient

4

Hybrid-GMVP

Crypto-Efficient

Hybrid-GMVP

5

Stock-Efficient

Crypto-Black-Litterman

Stock-GMVP

5 Conclusion and Discussion First, when we look at the yield table, stocks provide more returns by far. However, considering the price realities of the study history, stocks generally show a much better price performance than crypto assets. In this case, it would be misleading to look only at the rate of return. While the return of the average stock portfolio was -12%, it outperformed this rate in 5 techniques. Regarding crypto assets, the same success is not in question. While the average return is -56%, only GMVP, and Black-Litterman achieve more than that. While Shrinkage and Single Index performed worse than average. The Efficient portfolio suffered a severe failure of -355%. The hybrid portfolio, on the other hand, yielded an average of -25% return on both, while the techniques showed extraordinary success accordingly. In this section, when comparing the additional returns created by the techniques with respect to the average, one of the points to be considered should be the application of the techniques. Unlike other techniques, Black-Litterman, the only technique with “Analyst Opinion” in its structure, has achieved severe success in stocks and cryptos. In light of

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the evaluations shaped according to the answers received using fuzzy sets, the technique was applied, and the level of return was increased significantly. Due to the analyst’s more significant investment and market knowledge in crypto assets compared to stocks, higher returns were produced, especially in crypto assets, compared to the average. Here, the analyst has to consider more than one variable when comparing assets with each other. Using fuzzy sets while estimating the success/failure of the assets both made it easier for the person to make comparisons and increased the success accuracy. As we mentioned in the previous sections, rates of return do not always lead us to the most successful portfolio. At this point, we will be evaluating the return provided for the risk taken, thanks to the success measurement methods we have looked at. The success order of the portfolios within the three metrics was Stock > Hybrid > Crypto. Of course, considering that the techniques have been adapted for stocks, this result is expected. In addition, since the fluctuating price structure of crypto assets increased the risks, high standard deviations were revealed. The interesting point is that the hybrid portfolio has almost the same success as stocks in 3 measurement tools. In the continuation of the study, the most suitable techniques for each investment tool are shown. To summarize, crypto assets are still risky assets that have hardly been studied and perhaps even refrained from investing. This is understandable since their historical background is new and they are subject to high price movements throughout their market life. These 5 techniques, which were developed using stocks as raw materials, gave the most successful results in stocks. As our thesis says, the application to crypto assets has been relatively unsuccessful. Unfortunately, today’s most popular investment instruments are not yet suitable for applying these techniques. However, thanks to the bold success of the hybrid portfolio, it is seen that the price structure that settles over time gives much more successful results. In future studies, these results should be tested using different variables, and the results should be compared. The number of selected stocks should be increased, and the stocks should be selected from markets of different countries. When using fuzzy sets, the applied measurements and defuzzification method should be changed. The knowledge level of responding investors should be diversified. Crypto projects of different volumes should also be included. Price samples should be taken as widely as possible and varied to account for different macroeconomic equilibria. Analyzes should be made on the investment returns made in periods such as 1 year, 3 years, 5 years, and 10 years.

References 1. Vasiukevich, A., Pinsky, E.: Constructing portfolios using stable distributions: the case of S&P 500 sectors exchange-traded funds. Mach. Learn. Appl. 10, 100434 (2022) 2. Akba¸s, S., Dalkiliç, T.E.: A hybrid algorithm for portfolio selection: an application on the Dow Jones Index (DJI). J. Comput. Appl. Math. 398, 113678 (2021) 3. Demiguel, M., Garlappi, L.: Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Rev. Financ. Stud. 22(5), 1915–1953 (2009) 4. Kirby, C., Ostdiek, B.: It’s all in the timing: simple active portfolio strategies that outperform naïve diversification. J. Financ. Quant. Anal. 47(2), 437–467 (2012) 5. Kan, R., Wan, X.: On the Economic Value of Alphas, Papers SSRN (2011)

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6. Bessler, W., Opfer, H., Wolf, D.: Multi-asset portfolio optimization and an out-of-sample performance: an evaluation of black-letterman, mean-variance, and naïve diversification approaches. Eur. J. Finance 23, 1–30 (2014) 7. Black, F., Litterman, R.: Global portfolio optimization. Financ. Anal. J. 48(5), 28–43 (1992) 8. Branger, N., Weissenstainer, A.: Optimal granularity for portfolio choice. J. Empir. Financ. 50, 125–146 (2019) 9. Han, C.: How much should portfolios shrink? Financ. Manage. 49(3), 707–740 (2019) 10. Kan, R., Zhou, G.: Optimal portfolio choice with parameter uncertainty. J. Financ. Quant. Anal. 42(3), 621–656 (2007) 11. Tu, J., Zhou, G.: Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. J. Financ. Econ. 99(1), 204–215 (2011) 12. Markowitz, H.: Portfolio selection. J. Finance 7, 77–91 (1952) 13. Alexander, K., Christoph, M.: On the Estimation of the Global Minimum Variance Portfolio. CFR Working Paper, No. 05–02 (2005) 14. Ledoit, O., Wolf, M.: Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J. Empir. Financ. 10(5), 603–621 (2003) 15. Ledoit, O., Wolf, M.: Honey, I shrunk the sample covariance matrix. J. Portfolio Mgmt. 30(4), 110–119 (2003) 16. Nguyen, N., Nguyen, T.: Shrinkage model selection for portfolio optimization on Vietnam stock market. J. Asian Financ. Econ. Bus. (2020) 17. Frank Reilly’s Investment Analysis and Portfolio Management (10th Edition), Chapter 25 18. Rahmani, A., Hosseinzadeh Lotfi, F., Rostamy-Malkhalifeh, M., Allahviranloo, T.: A new method for defuzzification and ranking of fuzzy numbers based on the statistical beta distribution. Adv. Fuzzy Syst. 2016, 1–8 (2016) 19. De Kumar, S., Beg, I.: Triangular dense fuzzy sets and new defuzzification methods. J. Intell. Fuzzy Syst. 31, 469–477 (2016)

Intelligent Software for Optimizing Adaptive Control of Regional Investment Projects Andrey Shorikov1

and Elena Butsenko2(B)

1 Institute of Economics, Urals Branch of the Russian Academy of Sciences, Ekaterinburg,

Russia 2 Ural State University of Economics, Ekaterinburg, Russia

[email protected]

Abstract. The paper describes the functionality and structure of the intelligent computer software package developed by the authors for modeling the solution of optimization problems for adaptive control of regional investment design processes. The results of the work are based on the method of network formalization and optimization of adaptive project management, using network economic and mathematical modeling and feedback principles (adaptive management). The created intelligent computer software package is designed to automate the modeling of regional investment design processes and optimize adaptive decision-making management during their implementation based on network economic and mathematical modeling. In the initial data for the software package, when modeling the processes of optimizing the adaptive management of regional investment projects, the existing specific technical and economic conditions and information support are taken into account. The basis of the described software package is the use of the method of network formalization and optimization of adaptive project management, modernized to solve the problems of regional investment design. The intelligent computer software package created by the authors can serve as the basis for creating new tools to support managerial decision-making in practical problems of the regional economy in the implementation of investment design processes. Keywords: Intelligent computer software package · optimization of investment projecting · adaptive control · regional investment projects · economic and mathematical modeling · network model

1 Introduction Modern conditions for the development of regional economic systems require solving the problems of optimizing the management of the relevant processes. Given the complexity of regional economic processes, there is a need to use more advanced tools that meet emerging needs. To develop and create decision support systems in the management of regional economic systems, economic and mathematical models and methods are used in the work, which serve as the basis for solving such problems. The results obtained are based on studies [1–7]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 708–715, 2023. https://doi.org/10.1007/978-3-031-39774-5_78

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The article describes the structure and functional purpose of the structural elements of a software system designed to solve the problems of optimizing the management of regional socio-economic systems. The software elements described in the article are designed to solve the problem of optimizing control in deterministic systems that describe specific regional investment projects. The developed computer software can serve as a basis for creating tools for decision support in practical problems of the regional economy. Thus, the development of intelligent software to optimize the management of investment projects in the region is an important and urgent task of the modern economy, the solution of which makes it possible to increase the efficiency of making responsible decisions. The first three parts of the created software package are presented in [6]. Let’s consider the functional purpose of software components that were not presented earlier, as well as the corresponding algorithms for their implementation and the possibility of using them.

2 Research Results The development of a network economic-mathematical model of the process of managing a regional investment project for an economic entity is presented in [4]. Algorithmization and computer simulation of solving the task of optimizing the adaptive control of a regional investment project is carried out in accordance with the method proposed in [5]. The formation of a network model corresponding to the list of processes of regional investment design is carried out on the basis of solving the task of structural planning (network modeling). To solve this task, the user is prompted to indicate for all operations (works) of a specific investment project - the duration of their implementation, the sequence of their implementation and mark the operations that can be implemented in parallel, as well as the operations that determine the initial and final events. In the process of project implementation, delays in the execution of specific operations are introduced. The software implementation of the network model optimization process is carried out on the basis of the implementation of the adaptive project management optimization strategy [5] and represents the formation of the current positions (states) of the project execution - optimal or non-optimal, with the obligatory consideration by the user of the resulting delays in the execution of operations (if they are). The execution of this procedure is carried out taking into account its influence on the duration of the critical path time of the model, in which the delay value of the operation exceeds its full time reserve for execution. If there is a delay in the execution of some work-operation in the optimal position, the calculation of its added duration is realized and the formation of an additional work-operation, which may also be part of the original one. The value of the full reserve of the operation acts as the added duration. For an additional operation, the duration is set to the difference between the delay values of the operation and its total lead time for execution. After the formation of a new network model of the project that corresponds to the current position of the project execution, this network model is considered the initial model for continuing the implementation of the adaptive control optimization process.

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Delays in the execution of operations that do not affect the critical path of the model must also be taken into account. Therefore, after detecting such delays, it is necessary to take them into account and carry out a new construction of the critical path of the network model of the project. Taking into account the delay for an operation, the total reserve of which does not exceed the calculated value, means setting for this work-operation the added duration of its execution, equal to the value of the duration of the operation delay. The input and at the same time intermediate variables for the procedures for implementing the process of optimizing the adaptive control of regional investment projecting are the following: – – – – – – – – – – – – – – – – – – – – – – – – – – – –

ListOfJobs – list of network model operations. Type: TObjectList; ListOfEvents – list of network model events. Type: TObjectList; Duration – the duration of the operation execution. Type: Integer; Description – description of the operation. Type: String. Delay – operation execution delay. Type: Integer; Early – early date of event implementation. Type: Integer; Late – the late date of the event implementation. Type: Integer; FullTR - full reserve of time for the execution of the operation. Type: Integer; FreeTR - free time reserve for the execution of the operation. Type: Integer; ITR – independent time reserve for the execution of the operation. Type: Integer; CriticalTime – critical time of the network model. Type: Integer; SearchCP is a parameter whose true value indicates that the critical time of the network model has been determined. Type: Boolean. CriticalPath is a parameter for each operation of the network model, the value of which is true indicates that the operation is critical. Type: Boolean. LOfJ – list of critical operations of the network model. Type: TObjectList; LOfE - a list of events through which the critical path of the network model passes. Type: TObjectList; ListOfCriticalPaths – list of critical paths of the network model. Type: TObjectList; Chart – depiction of non-fictitious operations of the network model. Type: TBitmap; AxisX – image of the abscissa axis. Type: TBitmap; AxisY – image of the y-axis. Type: TBitmap; ChNW – length of the y-axis of the Gantt chart. Type: Integer; ChNH – Gantt chart abscissa height. Type: Integer; Factor – a multiplier that corrects the lengths of various images forming the Gantt chart. Type: Real; DGXS is the value of the coordinate along the abscissa axis for the beginning of the display of the operation on the Gantt chart. Type: Integer; SizeJob – the value of the coordinate along the abscissa axis for the end of the part of the image that displays the duration of the operation on the Gantt chart. Type: Integer; DGXF – coordinate value along the abscissa axis of the end of the operation image on the Gantt chart. Type: Integer; Division – step of marking the abscissa axis of the Gantt chart. Type: Integer; ScaleValueX – Gantt chart scale value along the abscissa axis. Type: Real; ScaleValueY – Gantt chart scale value along the y-axis. Type: Real;

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– MassivOfMarks – an array, the length of which is equal to the value of the critical time of the model, and the true value of each of its cells indicates the need to plot on the abscissa axis of the Gantt chart the value that is equal to the index of a certain cell of this array. Type: Array of Boolean; – Realized – parameter of each operation of the network model, the true value of which indicates the completion of the execution of the operation. Type: Boolean. – Delayded – a parameter of each operation of the network model, the true value of which indicates that the delay in execution of this operation has led to an increase in the critical path time of the model. Type: Boolean. – AddedDuration – additional duration of operation execution. Type: Integer; – NewNumber – parameter of each operation of the network model indicating its number in the network model. Type: Integer; – NewNumber – parameter of each network model event indicating its number in the network model. Type: Integer. These variables of the software package are not only input, but also intermediate, since after the formation of the current position (optimal or non-optimal) of the control process, i.e. corresponding network model, it becomes the starting point for the formation of the next position. Intermediate variables: – LinkStart – initial event for executing the operation. Type: Pointer; – LinkFinish – final event for the execution of the operation. Type: Pointer. Output (calculated) variables: – ListOfOptimizedPositionOfModels – list of optimal network model positions. Variable type: TObjectList. Next, we present the methods that are used to implement the procedure for optimizing the adaptive control of regional investment design and display the corresponding results: – MenuOfOptimization (V: boolean; FillTable: boolean = true) – a method of the Models class that displays and closes the optimization menu. This menu is necessary for entering delays in the execution of operations. The options are: – V is a parameter whose true value indicates that the optimization menu should be placed in the enabled state. Type: Boolean. – FillTable – parameter, the true value of which indicates that when the optimization menu switches to the enabled state, it is necessary to fill in the table of delays in the execution of operations; – OptimizeModel (SetPosition: boolean = true; MakeCopy: boolean = true) – a method of the Models class that starts the optimization process of adaptive management of regional investment design. The options are: – SetPosition – parameter, the true value of which indicates the need to move to the last optimal position of the model after the implementation of the process of optimization of adaptive control of regional investment projecting; – MakeCopy – parameter, the true value of which indicates the need to copy the model’s tabular data before moving to its new (optimal or non-optimal) position;

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– CreateOptimizedPositionOfModel (Job: Jobs) is a recursive method of the Models class that creates one of the current positions of the network model, where Job is an operation whose execution delay affects the duration of the critical time of the network model. – CreateObject(TableMode: boolean = true) – constructor of the Models class that creates an instance of this class, which, as a result of the CreateOptimizedPositionOfModel method, will serve as the optimal position of the model. – CompletePreviousJobs – method of the Jobs class indicating that all operations on the way to the operation calling this method, as well as the operation itself, are implemented; – ShowSystemOfOptimization – a method of the Models class that displays one of the current positions of the network model; – CreateJob(LinkStart,LinkFinish: Pointer; ShowSet: boolean) – constructor of the Jobs class that creates a network model operation. The options are: – LinkStart – the initial event of the operation; – LinkFinish – final event of the operation; – ShowSet – parameter, the true value of which indicates that after the creation of the operation, a form should be opened on which the user needs to set the values for the duration of its execution and description for this operation. Within the OptimizeModel method, the CreateEvent method is called with the given parameter set to false; – CreateEvent(x1, y1: integer; Add,Draw: boolean) – constructor of the Events class that creates a network model event, where. – x1 – x coordinate of the event position on the network diagram. Type: integer; – y1 – y coordinate of the event position on the network diagram. Type: integer; – Add – parameter, the true value of which indicates automatic addition of the created event to the end of the ListOfEvents list; – Draw – parameter, the true value of which indicates the display of the event on the network diagram when it is created. Within the OptimizeModel method, the CreateEvent method is called with the given parameter set to false. Consider the process of modeling the optimization of adaptive management of regional investment projects in a network model. To start this process, the procedure OptimizeModel(SetPosition: boolean = true; MakeCopy: boolean = true) corresponding to the current control period is executed. This procedure is performed in a loop over the variable i, which allows us to analyze all the j-th operations of the network model remaining by the considered time period. In the loop over j, the possible delay of the i-th operation is compared with its total reserve time. If there is a delay in the i-th operation and its value does not exceed the total reserve time for this operation, then it is taken into account and a new critical path of the project model is formed. If the delay exceeds the full slack for this operation, then the CreateOptimizedPositionOfModel(Job: Jobs) procedure is executed and the loop exits. The i-th operation is specified as a parameter for this procedure: CreateOptimizedPositionOfModel(Jobs(ListOfJobs[i])). Ending the loop over j, without iterating over the next operations, is necessary because the CreateOptimizedPositionOfModel(Job: Jobs) procedure is recursive. Let’s consider how the CreateOptimizedPositionOfModel(Job: Jobs) procedure works. The parameter of this procedure is the i-th operation belonging to the original optimal network model. When creating the current position of the control process

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(optimal or non-optimal), a search for new delays occurs, which is carried out among the operations not of the original network model, but only among the operations that have not yet been executed. Note that the generated initial network model and any part of it that forms the optimal position have the same structure, since they are instances of the same Models class, therefore, an instance of each optimal position has a ListOfOptimizedPositionOfModels list. When forming the current positions of the control process, the generated models are stored in the ListOfOptimizedPositionOfModels list only for the initial optimal network model. Before describing the algorithm of the method used for the formation of the current position of the process of optimizing the adaptive control of investment projecting, we introduce the following conventions: – NJob – a copy of the operation on the basis of which the current position is formed; – AJob is an additional operation generated by the delay in the execution of the NJob operation; – NEvent – a copy of the event, which is the final one for the operation, on the basis of which the current position is formed; – AEvent is an additional event that is the final event in the current position for the NJob operation and the initial event for the AJob operation. Algorithm for the implementation of the process of optimizing the adaptive control of investment projecting Step 1. The following steps are performed: – An instance of the Models class is created and written to the Model variable: Model: = Models.CreateObject; – The created instance of Model is added to the list of realized (optimal or non-optimal) positions of the model ListOfOptimizedPositionOfModels, the operations of which belong to the generated initial optimal model: Models(ListOfModels[NumberOfModel]).ListOfOptimizedPositionOfModels.Add(Model); Step 2. Copies of the lists of operations and events of the original network model are created and written for the Model. Also, as part of this process, the corresponding copies are written to the NJob and NEvent variables; Step 3. For the NJob operation, the value of its full reserve is set as the added duration: NJob.AddedDuration: = NJob.FullTR; Step 4. A new AEvent event is created and added to the Model’s ListOfEvents list before the NEvent: Model.ListOfEvents.Insert(Model.ListOfEvents.IndexOf (NEvent),Events.CreateEvent(NEvent.X-50,NEvent.Y-50),False,False)); Step 5: Create a new AJob between the AEvent event and the NEvent event by adding it to the Model’s ListOfJobs list after the NJob operation: Model.ListOfJobs.Insert(Model.ListOfJobs.IndexOf(NJob) + 1,Jobs.CreateJob (AEvent, NEvent,False)); Step 6. The value of the difference between the delay time of the NJob operation and the value of the full slack is set as the execution duration for the AJob operation: AJob.Duration: = NJob.Delay-NJob.FullTR; Step 7. The AEvent event is set as the final event for the NJob operation: NJob.LinkFinish: = AEvent;

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Step 8. The fact of completion is established for the NJob operation, as well as for those operations that are previous to it in the initial network model: NJob.CompletePreviousJobs; Step 9. For the created current position (optimal or non-optimal) of the control process, using the FindPath(Mode: integer = 0) procedure, the critical time of the project model is calculated. This procedure is called without specifying the parameter value: Model.FindPath(); Step 10. If it is established in the current position that the delays of all the operations of the original network model remaining for execution do not affect the duration of the corresponding critical path, then the value “minus one” is set to the variable a: a: = –1; Step 11. Looping over the i variable, looping through all network model operations described by the Model variable, starting with the next operation regarding the NJob operation: i is set to the value of the NJob operation number in the ListOfJobs list, incremented by one: i: = Model.ListOfJobs.IndexOf (NJob) + 1; Step 12. If the value of the i variable is equal to the number of operations in the ListOfJobs list of the Model variable, then go to step 15; Step 13. If the value of the delay of the i-th operation exceeds the value of its total slack: – variable a is set to the value of variable i: a: = i; – go to step 15. Otherwise, if the delay value of the i-th operation is greater than zero: – set as an additional duration for the i-th operation the value of the delay of this operation: Jobs(Model.ListOfJobs[i]).AddedDuration: = Jobs(Model.ListOfJobs[i]).Delay; – the formation of a critical path for the network model described by the Model variable is performed: Model.FindPath(0). Step 14. The following actions are performed: – the value of the variable i increases by one: i: = i + 1; – go to step 12. Step 15. If, as a result of executing the loop over the variable i, an operation was found whose delay increases the critical time of the model (that is, the value of the variable a > –1), then the next current position (optimal or non-optimal) is formed: the CreateOptimizedPositionOfModel(Job: Jobs) specifying as a parameter of the operation from the ListOfJobs list under the number a of the network model described by the Model variable: CreateOptimizedPositionOfModel(Jobs(Model.ListOfJobs[a])); Step 16. End of the optimization procedure. As a result of the execution of this algorithm, the initial optimal network model is formed, in accordance with the implementation of the optimization strategy for adaptive control of regional investment projecting, the current positions (optimal and non-optimal) of the control process, the optimal timing for the execution of all work-operations of the project and the optimal time for the execution of the project as a whole are formed.

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3 Conclusion The intellectual computer software package presented in the article makes it possible to implement an original algorithm for solving the task of optimizing the control of regional investment projects. This software package can be used as a tool to support managerial decision-making on the feasibility of implementing regional investment projects and to optimize their implementation. The developed software package can also be useful in the development of computer information systems for optimizing the management of complex regional economic and technical systems [1–3]. Mathematical models of such systems are considered, for example, in [2, 3, 7]. Acknowledgments. The work was carried out in accordance with the Research plan of the Institute of Economics of the Ural Branch of the Russian Academy of Sciences.

References 1. Akberdina, V.: Smart specialization and sustainable regional growth. E3S Web Confer. 250, 04009 (2021). https://doi.org/10.1051/e3sconf/202125004009 2. Aksyonov, K., Aksyonova, O., Ziomkovskaya, P., Kai, W.: Development of a process resource balancing method based on integration of network planning and multi-agent approach. IOP Conf. Ser.: Mater. Sci. Eng. 971(4), 042095 (2020). https://doi.org/10.1088/1757-899X/971/ 4/042095 3. Chentsov, A.G., Chentsov, A.A., Sesekin, A.N.: On the problem of sequential traversal of megalopolises with precedence conditions and cost functions depending on a list of tasks. In: Proceedings of the Steklov Institute of Mathematics, T. 315(Suppl. 1), pp. S67-S80 (2021). https://doi.org/10.1134/S0081543821060067 4. Shorikov, A.F., Butsenko, E.V.: Computer intelligent software system for optimization of adaptive management of investment design based on deterministic and stochastic network formalization. Int. J. Risk Assessment and Management, vol. 24, no. 2/3/4, pp. 196–213 (2021). https:// doi.org/10.1504/IJRAM.2021.10051399 5. Shorikov, A.F., Butsenko, E.V. Development of a Computer Intelligent System for Optimizing Adaptive Control of Business Planning Processes. J. Phys.: Conf. Ser. 1794(1), 012005 (2021)https://doi.org/10.1088/1742-6596/1794/1/012005 6. Butsenko, E.V., Surina, A.A., Tyrsin, A.N., Tyulyukin, V.A., Shorikov, A.F.: Structure and functional purpose of software package modules for solving tasks of optimizing the control of regional socio-economic systems. AMUR-2019, pp. 71–79 (2019) 7. Shorikov, A.F., Tyulyukin, V.A.: Minimax optimization of forecasting and management of socio-economic development of the region. Chapter 6 In: Chereshneva, V.A., Tatarkina, A.I., Yu, S. (eds.) Glazyev. Ekaterinburg, Forecasting the socio-economic development of the region. Ed. Publ. Institute of Economics of the Ural Branch of the RAS, pp. 183–213 (2011)

Q-Rung Orthopair Fuzzy Benefit/Cost Analysis Eda Boltürk(B) Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey [email protected]

Abstract. Cotton is very important product in our daily lives. Periodically cotton is harvested from farms with machines. Engineering economic techniques have been used in investment analysis by researchers. Benefit/cost analysis is an engineering economics technique and extended with some fuzzy sets. Human opinions can contain vagueness, and indefiniteness and they should be taken into calculations for investment assessments. Fuzzy sets is used in uncertain conditions to remove the effect in the mathematical models and extended its extensions. In this paper, an extension of benefit/cost analysis with q-rung orthopair fuzzy sets with detailed explanations and formulations. In addition, an illustrative application is given in cotton harvesting machine selection. In conclusion, future recommendations are shared. Keywords: Benefit/Cost Analysis · Q-Rung Orthopair Fuzzy Sets · Cotton Harvesting Machine

1 Introduction Investment analysis is critical for every investor to do in efficient way. Although some methods have been developed for investment analysis, factors and methods including these situations should be considered in cases involving multiple criteria and uncertainties. When we look at it as methods, we see that there are methods such as decision tree analysis, present value analysis, benefit/cost analysis, but there are issues such as fuzzy sets and its extensions in uncertainty issues. Fuzzy sets is proposed by Zadeh in 1965 [1] and extended its extensions. Q-rung orthopair fuzzy sets (Q-ROF) is one of the extensions and introduced by Yager [2]. QROF is defined with membership (µA : X → [0,1]) and non-membership degrees (vA : X → [0,1]). A Q-ROF in a finite universe of discourse X is defined as in Eq. (1). A = {x, µA (x), vA (x)|x ∈ X }

(1)

where µA and vA of the element x ∈ X to the set A. The relationship of µA and vA is 0 ≤ (µA q + vA q ) ≤ 1, (q ≥ 1). One of the approaches used to evaluate the alternatives is the benefit/cost analysis method and provides information to investors about the feasibility of investments for engineering economics problems. It is seen that fuzzy sets are used while making the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 716–720, 2023. https://doi.org/10.1007/978-3-031-39774-5_79

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717

evaluations because there may be uncertainty in the ideas of the investors. Benefit/cost analysis has been extended with some fuzzy sets [3–8] in different areas. To the best of our knowledge, Q-ROF benefit/cost analysis has not been developed in the literature until now. The contribution and the originality this paper is to introduce the Q-ROF benefit/cost analysis method and show the method in cotton harvesting machine selection evaluation problem. The application of the extended method is given for a cotton machine evaluation problem. This research is organized as follows: The preliminaries of Q-ROF sets are given in Sect. 2 with equations. Q-ROF benefit/cost analysis with formulas is given in Sect. 3. In Sect. 4, cotton harvesting machine alternative evaluation is given with details. In Sect. 5, conclusion is given with future recommendations.

2 Q-Rung Orthopair Fuzzy Sets In this section, preliminaries of Q-ROF aggregation are given with definitions. Definition 1. Let e˜ k = μk , vk (k = 1, 2, . . . , n) is a set of Q-ROF numbers. Q-ROF weighted averaging operator (q-ROFWA): Mn → M , is shown in Eq. (2) [9]. q − ROFWA(˜e1 , e˜ 2 , . . . , e˜ n ) = w1 e˜ 1 ⊕ w2 e˜ 2 ⊕ · · · ⊕ wn e˜ n

(2)

T where M is the set of all Q-ROF numbers n & w = (w1 , w2 , . . . , wn ) is weight vector of (˜e1 , e˜ 2 , . . . , e˜ n ) where 0 ≤ wk ≤ 1 & k=1 wk = 1. The Q-ROF aggregation equation is shown in Eq. (3) as follows:

q − ROFWA(˜e1 , e˜ 2 , . . . , e˜ n ) = (1 −

n 

1/q q w

(1 − μk ) k )

k=1

,

n 

vk wk 

(3)

k=1

Definition 2. Score function (S) & normalization function (N ) for k = (μ, ϑ) are shown in Eq. (4) and (5) [9].   (4) S(k) = μq − ϑ q , S(k)[0, 1] S(k) N (k) = n i=1 S(k)

(5)

3 Q-Rung Orthopair Fuzzy Benefit/Cost Analysis Q-ROF cost/benefit analysis (Q-BCA) formulations are given in this section with expla˜ Q , AC ˜ Q , SV ˜ Q , n˜ Q and ˜iQ . ˜ Q , AB nations. The parameters that shown in Q-BCA are FC ∼

∼

∼

Q-ROF first cost = FCQ , Q-ROF annual benefit = ABQ , Q-ROF annual cost = ACQ , ∼

∼

Q-ROF investment life = n Q , Q-ROF interest rate = i Q , and Q-ROF salvage value = ∼

SVQ . Experts who share their opinions for Q-BCA are given in Eqs. (6–11).   ˜ Q = fci , QFN m FC i 

(6)

718

E. Boltürk ∼   ABQ = abi , QFN m i 

(7)

  ˜ Q = aci , QFN m AC i 

(8)

  ˜ Q = svi , QFN m SV i 

(9)

  n˜ Q = ni , QFN m i 

(10)

  ˜iQ = ni , QFN m i 

(11)

where 0 ≤ i ≤ m, m is the number of opinions given by experts. Benefit/cost ratio for Q-ROF is given in Eq. (12). If the result of the Eq. (12) is bigger than 1, the investment can be said as feasible. In other terms, the investment can be realized. ˜ Q ( P , ˜iQ , n˜ Q ) AB B˜ Q A = ˜ Q + AC ˜ Q ( P , ˜iQ , n˜ Q ) − SV ˜ Q ( P , ˜iQ , n˜ Q ) C˜ Q FC A F

(12)

4 Application In this study, an illustrative example is given in order to show the application process of the method. 3 experts are trying to assess a cotton harvesting machine for a textile manufacturer in Turkey. The experts are Senior Machine Engineer (Expert 1), and R&D manager (Expert 2), and Machine Technician Worker (Expert 3) with their decision weights are given as 0.4, 0.4, and 0.2, respectively. q is taken as 3 in the application. Possible values of the alternative are given in Table 1 by the experts. The expert’s opinions are aggregated with Eq. (3). The aggregation results are firstly calculated with Eq. (4) and normalized with Eq. (5), respectively. Defuzzified values of parameters for alternative are given in Table 2 and these values are to calculate benefit/cost ratio as inputs. The ratio is calculated as 1.11 which tells cotton harvesting machine investment is feasible. ∼ ∼ p P ,∼ ,∼ 624625.02( A , i, n) ABQ ( A i Q nQ) BQ   = 1.11 = ∼ = ∼ ∼ ∼ ∼ ∼ ∼ ∼ P , i, n − 1268.05 P , i, n P, 4212638.28 + 977.86 A CQ FCQ + ACQ ( A i Q , n Q ) − SVQ ( FP , i Q , n Q ) F

Table 1. Opinions for cotton harvesting machine alternative given by experts. Parameters

Values

MembershipExpert 1

MembershipExpert 2

MembershipExpert 3

W = 0.4

W = 0.4

W = 0.2

e4, 500, 000

0.1, 0.1

0.1, 0.1

0.5, 0.4

e3, 500, 000

0.4, 0.5

0.7, 0.1

0.7, 0.1

e5, 000, 000

0.7, 0.1

0.7, 0.1

0.6, 0.1

e4, 000, 000

0.3, 0.3

0.3, 0.3

0.1, 0.1

Weights First Cost

(continued)

Q-Rung Orthopair Fuzzy Benefit/Cost Analysis Table 1. (continued) Parameters

Values

MembershipExpert 1

MembershipExpert 2

MembershipExpert 3

W = 0.4

W = 0.4

W = 0.2

e400, 000

0.3, 0.7

0.2, 0.1

0.1, 0.1

e500, 000

0.3, 0.2

0.5, 0.3

0.7, 0.1

e600, 000

0.3, 0.7

0.1, 0.1

0.3, 0.3

e900, 000

0.7, 0.1

0.4, 0.5

0.7, 0.1

e1000

0.5, 0.4

0.1, 0.1

0.3, 0.3

e1200

0.5, 0.4

0.7, 0.1

0.5, 0.4

e900

0.7, 0.1

0.5, 0.4

0.5, 0.4

e800

0.2, 0.3

0.5, 0.4

0.1, 0.1

1500

0.3, 0.3

0.1, 0.1

0.2, 0.3

1000

0.6, 0.1

0.5, 0.4

0.7, 0.1

2000

0.5, 0.4

0.6, 0.1

0.3, 0.3

500

0.7, 0.1

0.5, 0.4

0.5, 0.4

10%

0.2, 0.3

0.6, 0.1

0.1, 0.1

12%

0.5, 0.4

0.7, 0.1

0.2, 0.3

14%

0.2, 0.3

0.3, 0.3

0.5, 0.4

Weights Annual Benefit

Annual Cost

Salvage Value

Interest

Life

16%

0.7, 0.1

0.6, 0.1

0.7, 0.1

30 year

0.7, 0.1

0.7, 0.1

0.5, 0.4

32 year

0.4, 0.5

0.5, 0.4

0.7, 0.1

33 year

0.4, 0.5

0.5, 0.4

0.7, 0.1

35 year

0.2, 0.3

0.5, 0.4

0.2, 0.3

Table 2. Defuzzified fuzzy values of parameters. Parameters

Defuzified Values

First Cost

4,250,477.23

Annual Benefit

631,762.14

Annual Cost

986.98

Salvage Value

1,205.10

Interest

13.11%

Life

32.28 year

719

720

E. Boltürk

5 Conclusion Cotton harvesting machine evaluation is a crucial investment decision for factories that produce cotton. Q-ROF has been used in many areas like decision making and engineering economics studies and results for decision makers in uncertain environment. In this paper, benefit cost analysis is extended with q-rung fuzzy sets and an investment analysis decision making on cotton machine is evaluated with Q-BCA. 3 experts’ opinions are taken into consideration in application. It is shown that the steps of the methodology is shown and explained. The application results of Q-BCA showed that the investment is feasible for the cotton machine investment analysis. For further studies, benefit/cost analysis can be extended with decomposed fuzzy sets.

References 1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 2. Yager, R.R.: Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 25(5), 1222–1230 (2016) 3. Boltürk, E.: Picture fuzzy benefit/cost analysis in digital transformation for an IT firm. In: Cengiz Kahraman, A., Tolga, C., Onar, S.C., Cebi, S., Oztaysi, B., Sari, I.U. (eds.) Intelligent and Fuzzy Systems: Digital Acceleration and The New Normal - Proceedings of the INFUS 2022 Conference, Volume 1, pp. 508–516. Springer International Publishing, Cham (2022). https://doi.org/10.1007/978-3-031-09173-5_60 4. Kahraman, C., Kaya, ˙I.: Fuzzy benefit/cost analysis and applications. In: Kahraman, C. (eds) Fuzzy Engineering Economics with Applications. Studies in Fuzziness and Soft Computing, vol 233. Springer, Berlin, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70810-0_8 5. Kahraman, C.: Fuzzy versus probabilistic benefit/cost ratio analysis for public work projects. Int. J. Appl. Math. Comput. Sci. 11(3), 705–718 (2001) 6. Kahraman, C., Tolga, E., Ulukan, Z.: Justification of manufacturing technologies using fuzzy benefit/cost ratio analysis. Int. J. Prod. Econ. 66(1), 45–52 (2000) 7. Kahraman, C., Cevik Onar, S., Oztaysi, B.: A comparison of wind energy investment alternatives using interval-valued intuitionistic fuzzy benefit/cost analysis. Sustainability 8(2), 118 (2016) 8. Wang, M.J., Liang, G.S.: Benefit/cost analysis using fuzzy concept. Eng. Econ. 40(4), 359–376 (1995) 9. Liu, P.D., Wang, P.: Some q-Rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst. 33, 259–280 (2018)

Cash Replenishment and Vehicle Routing Improvement for Automated Teller Machines Deniz Orhan1(B)

and Müjde Erol Genevois2

1 Institute of Science, Galatasaray University, 34349 Istanbul, Turkey

[email protected]

2 Department of Industrial Engineering, Galatasaray University, 34349 Istanbul, Turkey

[email protected]

Abstract. Logistics has emerged as a crucial component in various business domains, playing a significant role in ensuring efficient operations. In addition to traditional applications, logistics principles are also being applied in the financial sector, specifically in the management of Automated Teller Machines (ATMs). ATMs offer a self-service and time-independent mechanism, providing financial institutions with an efficient means of serving their customers. However, the network design of cash distribution poses several challenges that necessitate an optimized solution. This solution aims to fulfill customer demands for ATMs while simultaneously minimizing losses for banks. This paper proposes a combined approach to address these challenges, integrating the demand forecasting with the vehicle routing problem. The replenishment policy begins with forecasting cash withdrawals, utilizing various methods such as statistical methodologies (e.g., ARIMA and SARIMA) and machine learning techniques (e.g., Prophet and DNN). To determine optimal routes for armored trucks and minimize costs based on the forecasted data, the VRP Spreadsheet Solver tool is implemented. By developing a decision support system, several methods are applied to facilitate ATM visitation using inventory control methodologies and vehicle routing techniques. This integrated approach seeks to achieve a balance between meeting ATM customer demands and optimizing the utilization of resources in cash replenishment and distribution. Overall, this research presents a comprehensive solution for addressing the challenges in cash network design for ATMs. By combining forecasting methods with vehicle routing optimization, it offers a decision support system that enhances the efficiency of ATM operations while minimizing costs and ensuring customer satisfaction. Keywords: Inventory Management · Forecasting · Vehicle Routing · Decision Support System · ATM

1 Introduction Meeting the diverse expectations of clients and ensuring the timely delivery of highquality goods poses a significant challenge in logistics. Among the crucial service points in our daily lives, Automated Teller Machines (ATMs) stand out as indispensable components. Achieving optimal delivery is essential to enhance service efficiency while © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 721–729, 2023. https://doi.org/10.1007/978-3-031-39774-5_80

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minimizing costs [1]. In our research, we address this challenge by tackling two interconnected aspects: developing a cash replenishment policy and designing the vehicle routing problem for cash-carrying trucks. It is ineffective to study these aspects independently as the constraints of the models are interrelated [2]. Therefore, the main challenge of this problem lies in harmoniously combining these two elements. The initial step involves generating a forecast for daily withdrawal demands. The forecast accuracy ensures customer demands are met without an excessive amount of cash in ATMs which are deprived from the over-night interest. Hence, it is urgent to construct a well-balanced cash management plan that optimizes resource allocation [3]. To evaluate the performance of various forecast methods, the results are subjected to comparison using the Mean Absolute Percentage Error (MAPE) metric. After identifying the most accurate forecasting method we proceeded with determining optimal routes using the Vehicle Routing Problem (VRP). VRP has emerged as a prominent approach for addressing routing and delivery challenges in cash management problems [4][5]. Over the past decade, numerous countries have witnessed a substantial increase in the volume of currency circulating within their economies [6]. This growth necessitates the development of effective strategies for cash management, including the optimization of routes for trucks responsible for cash transportation. By applying VRP principles, we aim to enhance the efficiency of the cash delivery process and mitigate associated logistical challenges in the service sector. The remainder of this paper is structured as follows. The second section presents the relevant literature, summarizing previous research conducted in this field. In the third section, we outline the proposed algorithms and provide their mathematical formulations, elucidating the underlying principles and methodologies. Subsequently, in the fourth section, we present a detailed case study to illustrate the practical application of the proposed approach. The fifth section presents the findings derived from our study, offering insights and analysis of the results obtained. Finally, in the sixth section, we draw conclusions based on the research findings, highlighting key implications and potential areas for further research.

2 Literature Review 2.1 Forecasting Problem Accurately forecasting the appropriate quantity of cash to meet the daily customer demand for ATMs poses a significant challenge. Ensuring that the minimum amount of cash is consistently available until the next replenishment is a complex task. To address this challenge, forecasting techniques like ARIMA and SARIMA [7, 8], neural networks [9, 10], LSTM [11], and convolutional networks [12] are employed in the literature. By adopting a data-driven approach, these techniques use historical data to estimate the correct cash amount required for an individual or a set of ATMs. This data-driven approach offers a reliable way of determining optimal cash levels, enabling financial institutions to efficiently manage cash resources while ensuring uninterrupted service for customers.

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2.2 Cost Minimization The service sector of today confronts numerous challenges in its quest to deliver high-quality services while maintaining cost-effectiveness. Addressing these challenges requires innovative research and solutions. Bolduc et al. (2010) [13] introduced a novel model with the objective of minimizing overall transportation and inventory costs. By carefully considering the allocation of resources and optimizing delivery routes, their model contributed to cost reduction and improved efficiency in the service sector. Anbuudayasankar et al. (2012) [14], developed three heuristic models to solve bi-objective vehicle routing problems with forced backhauls, namely minimizing the total routing cost and minimizing the span of travel tours. Van Anholt et al. (2016) [15] proposed a complex multiperiod inventory-routing problem inspired by ATM replenishment operations in the Netherlands. These studies provide valuable insights into the complexities and challenges associated with managing cash supply chains and optimizing logistics operations in the service sector. By incorporating real-world data and employing innovative models, these researchers contribute to the development of effective strategies for enhancing cash management, reducing costs, and improving overall service quality. 2.3 ATM Clustering Based on Location Implementing a cluster replenishment strategy can yield significant cost savings for ATMs located in close geographic proximity. This strategy involves combining adjacent ATMs and calculating an average of their daily withdrawals. The results of studies have indicated that a cluster-based forecasting approach outperforms individual forecasts, demonstrating its effectiveness in improving performance [2]. Different location patterns can be explored to address this problem. The cash depot, which serves as a central hub for vehicle routing, plays a crucial role in the optimization process. By prioritizing the depot as a key element, a reasonable approach can be developed. One heuristic method like the nearest neighbor algorithm involves assigning each customer to its closest depot and subsequently designing routes for the customer sets associated with each depot [16]. This heuristic method optimizes the allocation of customers and efficiently designs routes based on the proximity of depots, contributing to enhanced operational efficiency in the replenishment process.

3 Suggested Algorithms and Mathematical Formulations 3.1 Prophet Method Forecasting plays a critical role in various business aspects such as production planning schedules, workload prediction, and identifying areas of improvement. In this context, the Prophet forecasting model has emerged as a valuable tool. Prophet utilizes a decomposable time series model comprising three fundamental components: trend, seasonality, and holidays. These components are combined using the following equation, representing the mathematical formulation of the Prophet model [17]. y(t) = g(t) + s(t) + h(t) + ε

(1)

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Cash withdrawals are analyzed to find suitable forecasting method ARIMA, SARIMA, DNN and Prophet models are performed using python MAPE for 13 ATMs within 56 days is calculated (Prophet had the lowest error) Prophet forecasted data is used for VRP Fig. 1. Selection process of forecast method

where g(t) is the trend, s(t) is the seasonal change, h(t) is the holiday factor and ε is the error term. The trend parameters in the Prophet forecasting model can be discussed in two ways. The first way is through nonlinear growth, which describes a pattern where the trend gradually increases but eventually reaches a saturation point. This implies that the growth rate of the trend slows down over time, eventually leveling off or stabilizing. The Prophet model accommodates such nonlinear trends by capturing the underlying patterns and dynamics of the data, allowing for more accurate forecasting and prediction of future trends (Fig. 1). g(t) =

C(t)    1 + exp(−(k + a(t)T δ t − m + a(t)T γ )

(2)

where C(t) is the time-varying capacity, k is the base rate, m is offset, k + a(t)T δ is a growth rate of time varying, to connect the endpoints of segments m + a(t)T γ is adjusted as an offset parameter and δ is the change in the growth rate. For linear growth, a piece-wise constant rate of growth ensures an efficient model most of the time.     g(t) = k + a(t)T δ t + m + a(t)T γ (3) Where k is the growth rate, δ is rate adjustments, m is offset and γ is to make the function continuous. Seasonality models must be expressed as periodic functions of t to anticipate seasonal impacts. To imitate seasonality, Fourier terms are used in regression models by using sine and cosine terms.     N  2π nt 2π nt an cos + bn sin (4) s(t) = n=1 P P P is the regular period that is expected, 365 for annual data and 7 for weekly data. N = 10 and N = 3 for yearly and weekly seasonality work well for most cases. Some holidays are on certain dates in the year but some of them can vary every year such as religious holidays. Therefore, the model needs to fit this change. Resuming that the

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impacts of vacations are independent. Assign each holiday a parameter κi , which is the corresponding change in the prediction, and an indicator function showing whether time t is during holiday i. h(t) = Z(t)κ

(5)

3.2 VRP Spreadsheet Solver The VRP Spreadsheet Solver is an open-source Excel-based tool that is introduced in this study as a real-world application. Due to their dynamic nature, the distance and duration data must be continuously collected from a Geographical Information System (GIS). Selective client visits, simultaneous pickups and deliveries, time windows, fleet composition, distance restrictions, and the destination of the vehicles are all aspects of the VRP Spreadsheet Solver [18]. We used Bing Maps Key for real-time traffic data to calculate the duration and visualization of routes. Every truck should return to the depot at the end and routes need to be finished in the specific timeslot. After then, locations of ATMs are added to the system. Service time is determined as 15 min by workers. Capacity and distance limits are determined as infinite to allow the system to make an optimal decision about visiting. The cost of the trip includes vehicle rent cost, driver wage, and fuel cost of the vehicle per km. In the solution sheet, the system gives the results according to parameters and preferences about “don’t visit”, “must be visit” or “may be visit”. Total cost is calculated by the sum of operational cost for each ATM, fixed cost per trip, cost per km multiplied by the distance traveled, and penalty cost of the time window. The objective function of Solver is changed based on our model since minimizing the cost is aiming to find instead of profit. A variant of the LNS (Large Neighborhood Search) algorithm is applied within VRP Spreadsheet Solver.

4 Case Study The NN5 Competition dataset composed of daily time series originated from the observation of daily withdrawals at 111 randomly selected different cash machines [19]. We matched the data into 13 ATMs located randomly in the Besiktas area of Istanbul. These ATMs are owned by one of the biggest banks in Turkey. Also, the location of the depot is determined according to the bank headquarters building. For analyzing the results effectively and finding the optimal solution, several scenarios are applied to the data by using VRP Spreadsheet Solver. In the first scenario, all ATMs have no stock on the first day and they are filled with forecasted data at the beginning of the day. In the following days, the solver decides to visit the ATM or not according to their end of the day stock level. At the end of 15 days, the cost value is obtained for comparison. We aimed for a 90% service level, thus, we decided to keep 10% of the forecasted demand amount in the ATMs every day. To calculate the delivery amounts, pickup amounts, and cost of these operations, linear programming of the general aggregate planning problem is applied. The mathematical formulation of the problem is taken from Nahmias (2013) [20], some of the

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variables and parameters are modified based on our model. Cost parameters and given information are Dt withdrawal rate in period t, P forecast of demand level in period t. Variable is I t Inventory level in period t, It = It−1 + Pt − Dt for 1 ≤ t ≤ T

(6)

Constraint (6) ensures the inventory balance of each ATM. In the second scenario, all ATMs are visited on the first day with two days forecast data loaded but on the second day, ATMs are not visited. It is a cost saving option since the trip costs of a vehicle are expensive. In the third scenario, all ATMs are visited every 3 days. The forecasted value for the three days is filled into the ATMs (Fig. 2). Forecast amount of the day subtracted from the stock amount of the previous day. If the stock amount is negative, it is replaced by 0. It means that no cash is left in the ATM and we should load a certain day forecasted amount.

To calculate the pickup amount, the forecasted amount is subtracted from the real withdrawal amount showing remaining money at the end of the day in the ATM.

Cost calculation is made from the stock value. If the stock value is positive, it is multiplied by the interest rate of the bank (%9 yearly). If the stock amount of the ATM is negative, it is multiplied by the penalty cost (double interest rate) This means that customers cannot find cash in the ATM and there should be a penalty cost for this undesirable situation.

VRP Spreadsheet Solver is runned every day, at the end of every day tomorrow’s data would be changed because for each ATM at least 10% cash of demand rate should be kept in the ATM to prevent the shortage. If the stock level is placed below this amount, option “must be visit” is selected, if not “may be visit” option is selected, and solver made the decision based on the cost value.

Stock amount and cost value are changed when ATM is not visited. ATM use remaining balance if it is not visited today.

Fig. 2. Steps of calculating delivery amounts, pickup amounts, and cost

After applying these scenarios, we grouped the ATMs according to the nearest neighbor methodology. To assign the ATMs to the vehicles, the average amount of forecasted data is calculated for each ATM, after then a total of these 13 ATM averages is divided into three since only three vehicles are used. In addition, each ATMs contribution to the final total is calculated as a percentage. To find the replenishment amount, the contribution percentage of the ATM is multiplied by the total demand of ATMs in the group. Moreover, the same scenarios are also applied to grouping ATMs (Fig. 3).

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Fig. 3. Groups of ATMs

5 Findings We decided to run the model in terms of forecast the last eight days by using the first 48 days’ data. Thanks to that, whole data become compatible data set among each other. Forecasting methods comparison with real data can be seen at Fig. 4. Arima-Sarima-DNN-Prophet 15000

10000 5000 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Real Data NN5-103

Arima NN5-103

Sarima NN5-103

DNN NN5-103

Prophet NN5-103 Fig. 4. Comparison of methods for ATM 103 which is randomly selected.

We used ARIMA, SARIMA, DNN, and Prophet models and reached MAPE scores of 23.33%, 22.82%, 21.67%, and 18.52%, respectively. In the second part, we compared the results of scenarios stated in the case study by using Prophet forecasts which yielded to best MAPE scores. Results show that visiting the ATMs almost every day has the minimum cost value and is followed by a visit every 2 days and a visit every 3 days strategy. Cost values are 20.852, 26.950, and 36.325 respectively. This study once again showed that the logistics process needs to be considered from all perspectives to find an optimal strategy.

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In the next stage of calculations which is clustering the ATMs are the same as the previous strategy of individuals. A comparison of three scenarios of grouping ATMs is made by analyzing cost values for 15 days. Once again, it showed that visiting almost every day is the most economical alternative with a cost value of 22.570 and is followed by a visit every 2 days and a visit every 3 days strategy with respect to 23.750 and 33.358. In the end, two strategies are analyzed. Results showed that it is more profitable to deal with ATMs individually. The decision support system finds out that the optimized model becomes examining the ATM individually and visits the ATMs almost every day.

6 Conclusion and Future Study The decision support system is constructed for ATM cash replenishment and vehicle routing policy. To create this system, the Prophet forecasting method and the VRP Spreadsheet Solution, are used in this study. Several forecasting methods are implemented in the data set and the optimal one is selected for the next stage. Various strategy is applied to the routing part, and we found out that analyzing the ATMs individually gives a more effective result than clustering ATMs. Delivery amount calculations are made for group after then it split to the ATMs. This different calculation method may have caused the difference between two strategies. On many benchmark situations from the literature, we demonstrated the importance of the problem and why it should be done optimally. As we know that humanity always need a cash therefore this problem is always open to improvement. Running the problem for 56 days can be a future study. In addition, as future work, time window specifications may change for each ATM. Acknowledgment. The authors acknowledge that this research was financially supported by Galatasaray University Research Fund (Project Number: FOA-2022–1128).

References 1. Ekinci, Y., Serban, N., Duman, E.: Optimal ATM replenishment policies under demand uncertainty. Oper. Res. Int. J. 21(2), 999–1029 (2019). https://doi.org/10.1007/s12351-019-004 66-4 2. Ekinci, Y., Lu, J.C., Duman, E.: Optimization of ATM cash replenishment with group-demand forecasts. Expert Syst. Appl. 42(7), 3480–3490 (2015) 3. Cedolin, M., Erol Genevois, M.: District Performance of the ATMs by alternative DEA techniques. In: 2019 3rd International Conference on Data Science and Business Analytics (ICDSBA) (pp. 61–64). IEEE (2019) 4. Talarico, L., Sorensen, K., Springael, J.: Metaheuristics for the risk constrained cash-in-transit vehicle routing problem. Eur. J. Oper. Res. 244(2), 457–470 (2015) 5. Gubar, E., Zubareva, M., Merzljakova, J.: Cash flow optimization in ATM network model. Contrib. Game Theory Manage. 4, 213–222 (2011) 6. Xu, G., Li, Y., Szeto, W.Y., Li, J.: A cash transportation vehicle routing problem with combinations of different cash denominations. Int. Trans. Oper. Res. 26(6), 2179–2198 (2019) 7. Cedolin, M., Erol Genevois, M.: An averaging approach to individual time series employing econometric models: a case study on NN5 ATM transactions data. Kybernetes 51(9), 2673– 2694 (2022)

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8. Khanarsa, P., Sinapiromsaran, K.: Multiple ARIMA subsequences aggregate time series model to forecast cash in ATM. In: 2017 9th International Conference on Knowledge and Smart Technology (KST) (pp. 83–88). IEEE (2017) 9. Simutis, R., Dilijonas, D., Bastina, L.: Cash demand forecasting for ATM using neural networks and support vector regression algorithms. In: 20th International Conference, Euro Mini Conference Continuous Optimization and Knowledge-Based Technologies, 416–421 (2008) 10. Simutis, R., Dilijonas, D., Bastina, L., Friman, J.: A flexible neural network for ATM cash demand forecasting. Cimmacs ‘07: WSEAS International Conference on Computational Intelligence, Man-Machine Systems and Cybernetics, 163–168 (2007) 11. Asad, M., Rafi M.: A long-short-term-memory based model for predicting ATM replenishment amount. In: 21st International Arab Conference on Information Technology (ACIT) (2020) 12. Poorzaker Arabani, S., Ebrahimpour Komleh, H.: The improvement of forecasting ATMS cash demand of Iran banking network using convolutional neural network. Arab. J. Sci. Eng. 44(4), 3733–3743 (2019) 13. Bolduc, M.-C., Laporte, G., Renaud, J., Boctor, F.F.: A tabu search heuristic for the split delivery vehicle routing problem with production and demand calendars. Eur. J. Oper. Res. 202(1), 122–130 (2010) 14. Anbuudayasankar, S.P., Ganesh, K., Koh, S.C.L., Ducq, Y.: Modified savings heuristics and genetic algorithm for bi-objective vehicle routing problem with forced backhauls. Expert Syst. Appl. 39(3), 2296–2305 (2012) 15. Van Anholt, R.G., Coelho, L.C., Laporte, G., Vis, I.F.: An inventory-routing problem with pickups and deliveries arising in the replenishment of automated teller machines. Transp. Sci. 50(3), 1077–1091 (2016) 16. Bae, H., Moon, I.: Multi-depot vehicle routing problem with time windows considering delivery and installation vehicles, pp. 6536–6549 (2016) 17. Taylor, S.J., Letham, B.: Prophet: forecasting at scale, pp. 37–45 (2017) 18. Erdo˘gan, G.: An open source spreadsheet solver for vehicle routing problems. Comput. Oper. Res. 84, 62–72 (2017) 19. http://www.neural-forecasting-competition.com/NN5/ 20. Nahmias, S.: Production and Operation Analysis (6th ed.). Mc Graw Hill. (2013)

A Dynamic Feature Selection Technique for the Stock Price Forecasting Mahmut Sami Sivri1 , Ahmet Berkay Gultekin1 , Alp Ustundag1 Omer Faruk Beyca1 , Omer Faruk Gurcan2(B) , and Emre Ari1

,

1 Istanbul Technical University, Istanbul, Turkey {sivri,gultekinah,ustundaga,beyca,ari18}@itu.edu.tr 2 Sivas Cumhuriyet University, Sivas, Turkey [email protected]

Abstract. Stock market prices are inherently volatile, and accurate forecasting is challenging. An accurate prediction of stock prices helps traders and investors to decide timely buy or sell, so an optimal investment strategy can be built, decreasing investment risks. Traditionally, linear and non-linear methods have been applied to stock market prediction. Many studies on stock market prediction have recently employed machine learning and deep learning models with the proliferation of big data and rapid development in artificial intelligence. On the other hand, previous prediction studies mostly overlooked key indicators and feature engineering in the models. The feature selection can help to develop better prediction models. The stock price prediction requires a dynamic feature selection due to its time-dependent characteristics. There is no optimal set of technical indicators for stocks that perform well in all market scenarios. We propose a stock price prediction model focusing on dynamic feature selection in this study. The model uses technical, operational, and economic indicators besides price and volume data. The feature selection process has two stages. In the first stage, the importance of features for stocks is found by an ensemble learning algorithm. The final importance score is calculated by multiplying feature importance values with the next day’s model return which is the performance of the prediction method. In the second stage, a regression analysis is made daily for each feature using feature importance scores to track their performance in terms of average importance and slope (importance movement) dynamically. The proposed model enables better interpretability of features on stock price behavior and makes better stock price predictions. Keywords: dynamic · feature · selection · stock · prediction · time-series · ensemble learning

1 Introduction The stock market connects individuals and businesses through financial movements. Profitability and accessibility of the stock market attract both firms needing funding and investors looking for profitable returns. Fast growth in electronic trading platforms and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 730–737, 2023. https://doi.org/10.1007/978-3-031-39774-5_81

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computing technologies has increased the effectiveness of stock trading and generated a vast amount of data. As a result, big data on stock prices has become a hot topic for researchers that try to understand stock price behavior [1]. Stock prices are complex, volatile, noisy, non-linear, and irrational [1]. On the other hand, several external factors influence stock markets, such as industry-specific attributes, the economy, government policy, social news, public sentiment, or the psychology of investors [2]. This dynamic nature makes stock price and movement prediction a challenging task. Stock forecasting is a time series problem. The ability to precisely forecast the direction of a stock price allows traders and investors to select when to purchase or sell at a particular price, which makes it feasible to develop an investing strategy that minimizes risk. Numerous methods exist for researching the stock market and forecasting stock price behavior. Some statistical techniques (e.g., regression analysis, ARMA, ARIMA, ARCH, GARCH) have been applied extensively in stock price studies. These techniques rely on stock markets being linear, normal, and stationary [3]. Because of the properties of stock data, these statistical models do not accurately represent the complexity of the stock price data. Many researchers have used machine learning and deep learning approaches to investigate the stock market since the introduction of big data and the rapid growth of artificial intelligence-based algorithms in recent decades [1]. Machine learning has grown in importance for stock price forecasting studies because of its ability to analyze large amounts of data, detect patterns and trends, and make accurate predictions based on historical data. The following are some of the main advantages of applying machine learning to stock price forecasting: In contrast to conventional approaches, machine learning algorithms learn from historical data and spot patterns and trends that can be used to predict future stock prices. Investment decisions can be made more rapidly and with greater knowledge because of machine learning algorithms’ speedy analysis of large volumes of data and real-time prediction capabilities. Investors can reduce the risk of making bad investing decisions by using machine learning algorithms to help them make more data-driven decisions based on accurate predictions. Investors can better manage their risk exposure and reduce potential losses by using machine learning algorithms to discover and assess potential risks related to certain investments. High-dimensional datasets can reason several challenges in machine learning problems. These problems include the curse of dimensionality, overfitting, computational complexity, and difficulty interpreting model results. It is crucial to carefully preprocess and prepare high-dimensional datasets, use the proper feature selection and dimensionality reduction approaches, and carefully choose machine learning algorithms that are scalable and effective for high-dimensional data to solve these problems [4–6]. Feature selection in machine learning is selecting a subset of relevant features (predictors/variables) for use in a model. The aim is to improve model accuracy and reduce dimensionality by decreasing the number of redundant or irrelevant features, which can reason overfitting, decreased model interpretability, and longer training times.

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Feature selection can be performed using a variety of techniques, including filter methods (e.g., correlation analysis), wrapper methods (e.g., forward/backward selection), and embedded methods (e.g., Lasso regression). These techniques evaluate the importance of each feature by measuring its relevance to the target variable or its ability to reduce the model error. Ultimately, the choice of feature selection technique will depend on the problem and dataset at hand, and may require experimentation and iteration to identify the optimal set of features for the given task. Similar to other machine learning tasks, stock prediction calls for choosing an appropriate set of features from a given set to feed into a prediction model [10]. Many existing studies apply a single feature selection method in stock forecasting, which can oversight some substantial assumptions. In this study, we employ a two-stage feature selection model to obtain a more optimal feature set and improved the stock price prediction performance. Stocks are selected from Borsa ˙Istanbul. This paper is organized into five sections. Section 2 gives related studies; Sect. 3 explains the application; experimental results are outlined in Sect. 4, followed by discussion and conclusions in the last section.

2 Related Works Researchers have presented numerous forecasting models to successfully forecast the stock market and generate large profits. Recently machine learning algorithms and hybrid approaches have been used extensively. Choosing appropriate features from a given set and utilizing them to train a prediction model is similar to other machine learning challenges when it comes to stock forecasting. The goal of successful stock prediction is to get the highest forecast accuracy with the least amount of input and the least complicated learning model [3]. Naik and Mohan [7] applied Brouta Algorithm for feature selection, where the random forest finds important technical indicator features. The authors classified the stock price movement, and data is collected from the Indian stock market. The prediction performance of an artificial neural network, support vector machines, and a deep learning model are compared. Haq et al. [3] applied multiple feature selection methods and a deep model to predict daily stock trends. Feature importances are calculated by Support Vector Machine, logistic regression, and random forests. Then clusters are generated to obtain the optimal feature set. Peng et al. [8] built various deep neural networks to predict stock price movement from seven markets. The authors applied Tournament Screening, Sequential Forward Floating Selection, and LASSO as feature selection methods. Yun et al. [9] proposed a hybrid Genetic Algorithm – an XGBoost-based feature selection model to predict stock price direction. The authors used KOSPI data and applied a three-stage feature engineering process. The classification performance of XGBoost is compared to other machine learning classifiers, and XGBooost gives the highest accuracy. SHAP and LIME techniques are used for the interpretability of the proposed model. Similarly, Yun et al. [1] hybridized Genetic Algorithm with tree-structured machine-learning algorithms (wrapper method) and then applied the filter method to select optimal features.

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Ji et al. [10] predict stock price movements by improving technical indicators that apply wavelet denoising and adaptive feature selection methods. The feature selection is designed based on permutation importance. The best feature set is chosen after trying to construct feature sets with various temporal dimensions according to the size-variable time windows. Random forest is used for prediction. Data are collected from four stock markets. Using daily stock data from six market indices, Kumari and Swarnkar [2] first calculated an extended collection of 83 technical indicators, and then a hybrid normalization was made. The authors generated a feature set with various feature selection methods: information gain (IG), forward feature selection (FFS), and LASSO. The performance of classifiers and feature selection methods are compared. The proposed model presents that intersection of features from various feature selection methods gives better accuracy. Yan [11] proposed a hybrid model that predicts stock index feature price using AdaBoost regressor for feature selection and LSTM predictor. Data are collected from Shanghai-Shenzhen 300 stock index.

3 Application The primary objective of this study is to develop a model for predicting the direction of stock prices using a two-stage feature engineering procedure in order to maximize prediction accuracy. Figure 1 presents flowchart of the proposed model.

Fig. 1. Proposed model.

This study provides a comprehensive approach for the collection and preparation of data from the BIST stock market, which includes utilizing a range of technical, operational, and economic indicators, as well as historical price and volume data. The data preparation process includes essential steps such as normalization and cleaning, which ensure the suitability of the data for subsequent analysis. The data pre-processing stage involves using various combinations of feature selection and machine learning models to obtain the best base model. Different techniques such as LASSO, HSIC LASSO, SHAP, and correlation values are utilized to select relevant features from a large pool of variables. Additionally, ensemble models such as XGBoost, Random Forest, and LightGBM are employed to optimize the predictive power of the model. The subsequent stage of the analysis involves adaptive feature selection, which employs a rolling method to calculate dynamic feature importances. This method involves two steps: first, the base model is applied at each rolling window to extract

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feature importances specified by the model. Then, the final importance score is weighted by the next day’s model return, which serves as a performance indicator of the prediction method. Features with higher importances in successful models are retained. Table 1. Stock and Analysis information for January 2023. Stock

Predictor

# of Estimator

# of Training Days

Test Period

# of Features

# of Selected Features

BIMAS

XGB

200

400

02.01.2023–27.01.2023

35

4

DOHOL

LGB

20

500

02.01.2023–27.01.2023

60

41

EKGYO

XGB

200

200

02.01.2023–27.01.2023

30

20

FROTO

LGB

50

400

02.01.2023–27.01.2023

40

18

GARAN

LGB

20

250

02.01.2023–27.01.2023

40

28

HALKB

XGB

20

400

02.01.2023–27.01.2023

45

21

ISCTR

XGB

100

150

02.01.2023–27.01.2023

35

25

KCHOL

RF

50

250

02.01.2023–27.01.2023

35

29

KOZAA

XGB

50

250

02.01.2023–27.01.2023

35

23

KOZAL

XGB

50

250

02.01.2023–27.01.2023

45

21

MGROS

LGB

100

250

02.01.2023–27.01.2023

30

28

PETKM

LGB

200

500

02.01.2023–27.01.2023

30

26

SAHOL

LGB

50

200

02.01.2023–27.01.2023

35

31

SISE

LGB

100

250

02.01.2023–27.01.2023

40

33

TAVHL

LGB

200

500

02.01.2023–27.01.2023

35

33

TCELL

XGB

50

500

02.01.2023–27.01.2023

35

27

THYAO

LGB

20

500

02.01.2023–27.01.2023

40

19

TKFEN

RF

200

400

02.01.2023–27.01.2023

55

38

TOASO

XGB

50

500

02.01.2023–27.01.2023

60

28

TSKB

XGB

200

200

02.01.2023–27.01.2023

35

12

TTKOM

XGB

10

500

02.01.2023–27.01.2023

40

16

TUPRS

LGB

100

500

02.01.2023–27.01.2023

30

24

YKBNK

LGB

200

200

02.01.2023–27.01.2023

50

32

In the second stage, the average importance and trend of importance change are monitored to eliminate unnecessary features. This approach ensures that only the most relevant features are included in the subsequent analysis, thus enhancing the accuracy and effectiveness of the prediction model. The selected features are used to make predictions for the period spanning from December 2022 to February 2023.

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This study provides a robust methodology for the collection, preparation, and selection of relevant features from BIST stock market data, with significant implications for the development of more accurate and effective prediction models in the field of stock market analysis. Table 1 presents stocks and some of the parameters used in the analysis for January 2023.

4 Results

Table 2. The model returns differences after feature selection. Stock BIMAS DOHOL

January

February

December

3,309364955

1,79869411

16,74756642

3,672974353

18,25449583

−10,30017179

EKGYO

12,43344602

25,07319285

0,107779947

FROTO

8,87858926

−6,434336081

2,191926098

GARAN

−2,211283592

−3,68657989

HALKB

−8,12978888

−11,38803014

−0,01660641

ISCTR

−3,33842463

13,08785165

−0,85140792

KCHOL

3,712272108

KOZAA

2,46983163

KOZAL MGROS PETKM

11,57418 2,262884163 −0,8042074

PGSUS SAHOL SISE

−7,04094558 3,11866719

0,022636371

−6,790949793 −6,358111055 4,98168738 6,504672782

33,43268301 −28,70446274 21,96521992

−2,84182286

20,81960773

−0,52663488

−25,90300299

−5,372494473 −15,29830005 −6,757209947

−2,111424022 1,485154182 −0,97766682

TAVHL

2,72325511

TCELL

−18,49864395

THYAO

10,20776662

TKFEN

−6,452732147

TOASO

11,64196618

−4,17644881

−14,52952403

12,29801645

−0,35217427

TSKB

−14,37469436

7,583124236 −4,511128668

1,00767508

13,79044278 6,500024034

TTKOM

35,33545678

20,66336219

−1,54621425

TUPRS

−1,748716267

22,86125555

4,61877973

YKBNK

7,30508922

16,17236559

6,59313169

Average difference

2,430064222

2,97675486

2,241369436

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Table 2 presents model returns differences for selected months in terms of stocks. Average model returns are increased after feature selection for three months. Table 3 compares the average of model returns without feature selection and after feature selection. According to the results, average model returns increased substantially for three months. Feature selection enabled performance improvement for prediction. Table 3. Comparison of average model returns. Months

average model returns after feature selection

before feature selection

January

7,325716396

4,895652174

February

4,854027588

1,877272727

December

1,428869436

−0,8125

5 Discussions and Concluding Remarks The dynamic nature of stock price patterns demands an agile approach to measure the impact of parameters on stock prices. Therefore, this research proposes a novel method to measure the dynamic feature importances that affect stock returns. By applying developed method, we identified and eliminated unnecessary features, which resulted in increased accuracy and investment returns. Our approach ensures that the prediction model remains adaptable to changes in the stock market, enabling investors to make informed decisions in a timely manner.Based on the investment returns presented in the table, it is clear that the model developed by our research team has generated higher returns compared to the previous model. In December, the after-model return was 1.43%, which was a positive return, while the before-model return was negative at -0.81%. This indicates that our model has the potential to produce positive returns even in unfavorable market conditions. In January, the after-model return was 7.33%, which was significantly higher than the before-model return of 4.90%. This suggests that our model has a higher potential for generating returns and could provide investors with a competitive advantage in the BIST stock market. Furthermore, the outperformance of our model continued in February, with an aftermodel return of 4.85% compared to a before-model return of only 1.88%. This indicates that our model is more effective in identifying profitable investment opportunities and generating returns for investors. Overall, these results suggest that our model has the potential to improve investment performance and provide investors with a competitive advantage in the BIST stock market. However, further research is needed to validate these findings and explore the potential of our model in different market conditions and investment strategies.

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References 1. Yun, K.K., Yoon, S.W., Won, D.: Interpretable stock price forecasting model using genetic algorithm-machine learning regressions and best feature subset selection. Expert Syst. Appl. 213, 118803 (2023) 2. Kumari, B., Swarnkar, T.: Forecasting daily stock movement using a hybrid normalization based intersection feature selection and ANN. Procedia Comput. Sci. 218, 1424–1433 (2023) 3. Haq, A.U., Zeb, A., Lei, Z., Zhang, D.: Forecasting daily stock trend using multi-filter feature selection and deep learning. Expert Syst. Appl. 168, 114444 (2021) 4. Li, J., et al.: Feature selection: a data perspective. ACM Comput. Surv. (CSUR) 50(6), 1–45 (2017) 5. Huang, Q., Xia, T., Sun, H., Yamada, M., Chang, Y.: Unsupervised nonlinear feature selection from high-dimensional signed networks. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp. 4182–4189. AAAI Press, USA (2020) 6. Gürcan, Ö.F., Beyca, Ö.F., Do˘gan, O.: A comprehensive study of machine learning methods on diabetic retinopathy classification. Int. J. Computat. Intell. Syst. 14(1), 1132–1141 (2021) 7. Naik, N., Mohan, B.R.: Stock price movements classification using machine and deep learning techniques-the case study of indian stock market. In: Macintyre, J., Iliadis, L., Maglogiannis, I., Jayne, C. (eds.) EANN 2019. CCIS, vol. 1000, pp. 445–452. Springer, Cham (2019). https:// doi.org/10.1007/978-3-030-20257-6_38 8. Peng, Y., Albuquerque, P.H.M., Kimura, H., Saavedra, C.A.P.B.: Feature selection and deep neural networks for stock price direction forecasting using technical analysis indicators. Mach. Learn. Appl. 5, 100060 (2021) 9. Yun, K.K., Yoon, S.W., Won, D.: Prediction of stock price direction using a hybrid GAXGBoost algorithm with a three-stage feature engineering process. Expert Syst. Appl. 186, 115716 (2021) 10. Ji, G., Yu, J., Hu, K., Xie, J., Ji, X.: An adaptive feature selection schema using improved technical indicators for predicting stock price movements. Expert Syst. Appl. 200, 116941 (2022) 11. Yan, W.L.: Stock index futures price prediction using feature selection and deep learning. North Am. J. Econ. Finance 64, 101867 (2023)

Blockchain: Architecture, Security and Consensus Algorithms Taher Abouzaid Abdel Aty Abdel Bary1(B) , Basem Mohamed Elomda1,2 , and Hesham Ahmed Hassan1,3 1 Egyptian E-Learning University, 33 El Messaha St., Dokki, Giza, Egypt [email protected], {bmohamed,hhassan}@eelu.edu.eg, [email protected] 2 Agriculture Research Center (ARC), Cairo, Egypt 3 Faculty of Computers and Information and Artificial Intelligence, Cairo, Egypt [email protected]

Abstract. Blockchain is an innovative technology that is making a high impression on current society through its precision, decentralization, and high-security assets. In this research, we provide a survey of the different architecture, security, and consensus algorithms of Blockchain technology. We also provide a comparison between different Blockchain frameworks and classification of consensus algorithms that are mostly used as well as analyzing security risks and cryptographic primitives currently being used in the Blockchain. This paper also elaborates on important future directions, creative application cases, and unresolved research difficulties that could be investigated by scientists to enhance this subject. Keywords: Blockchain · Consensus algorithm · Cryptocurrency · Smart contract · Blockchain security

1 Introduction Blockchain technology is a cutting-edge database system that facilitates the interchange of transparent data within a company’s internal network. Data is stored in a blockchain database as blocks that are connected in a chain [1]. Due to the inability to eliminate or modify the series without compromising network compatibility, the data are chronologically consistent. Using blockchain technology, you can establish an immutable or immutable ledger for recording orders, payments, accounts, and other transactions [2]. These transactions are portrayed consistently by the system’s built-in features, which also prevent the submission of fraudulent transactions. The documentation of financial transactions using conventional database systems presents numerous challenges. To prevent potential legal issues, transactions must be monitored and validated by a trustworthy third party. This centralized authority not only complicates the transaction but also introduces a single point of failure. The compromise of the primary database may have repercussions for both parties. By establishing a decentralized, impenetrable © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 738–753, 2023. https://doi.org/10.1007/978-3-031-39774-5_82

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method for transaction recording, blockchain mitigates these issues. In a real estate transaction, the blockchain establishes a single ledger for the buyer and seller [1]. Every transaction in each party’s ledger must be automatically updated in real-time and approved by both parties. Any alteration of previous transactions will contaminate the entire ledger. This blockchain technology is now utilized in a variety of industries, including the creation of digital currencies like Bitcoin. Therefore, blockchain technology is one of the technologies made available by the global distribution of processing power [3]. The blockchain is the digital ledger where transactions are recorded. For instance, Bitcoin and other cryptocurrencies are publicly and historically recorded [4]. The objective of this paper is to provide a summary of the several blockchain architectures, security protocols, and consensus algorithms. In addition, we provide a comparison of various blockchain frameworks, a classification of the most popular consensus algorithms, and an analysis of the security risks and cryptographic primitives presently used in the blockchain. Also, we focus on the significant future directions, innovative application examples, and unresolved research challenges that scientists could investigate to advance this field. 1.1 Blockchain History In 1982, it was the first known individual to suggest “a blockchain-like protocol was Chaum in his Ph.D. thesis” [5]. A secured chain of blocks was described cryptographically by Haber and Stornetta in 1991[6]. Merkle trees were introduced into the design by Bayer et al. in 1993 [7]. In 1998, Szabo [8] developed "bit gold", a decentralized digital currency mechanism. In 2008, Satoshi Nakamoto [9] unveiled the Bitcoin peerto-peer electronic currency system. In 2008, the term "blockchain" was first applied to the distributed ledger that enables Bitcoin transactions. Buterin proposed Ethereum in his 2013 whitepaper [10]. Crowdfunding for the establishment of Ethereum began in 2014, and the network went live on July 30, 2015. When Ethereum debuted, it was thought that blockchain 2.0 had arrived because, unlike other blockchain projects that had focused on creating altcoins (currencies similar to Bitcoin), Ethereum enabled users to communicate via unauthorized distributed applications on its blockchain. Bitcoin, on the other hand, was designed for a distributed ledger. The Ethereum 2.0 revision aims to enhance the network’s performance, scalability, efficiency, and security. The Hyperledger initiative, which is open-source blockchain software, was announced in 2015 by the Linux Foundation. Hyperledger blockchain frameworks, unlike Bitcoin and Ethereum, are designed to construct enterprise blockchains [11]. There are eight blockchain frameworks under the Hyperledger umbrella, and five Hyperledger tools,. Figure 1 depicts a timeline of the development of blockchain. 1.2 Blockchain Evolution The evolution of Blockchain technology comes with a revolution like the origin of the internet. The next three stages or three generations of blockchain development can be identified in the next sections, and we make the comparison between the three generations in Table 4 in the last section of this paper.

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1982 • Initiation of proposed blockchainworking as protocol 1991 • First paper published that described a crypto-graphically secure chain of block 1993 • Incorporated Merkle trees + proof of work (PoW) consensus proposed by Hal Finney 1998 • Designed Bit gold and using cryptography. 2008 • Bitcoin term of Blockchain + Distributed ledger + Permissionless (public) Blockchain 2013 • Ethereum foundation + Distributed ledger + Permissionless (public) Blockchain 2015 • Hyperledger project + Smart contract + Permissioned (Private) Blockchain 2022 • Distributed ledger + Smart contract + Ethereum 2.0

Fig. 1. Timeline of the development of blockchain

1.3 Blockchain Technology Although blockchain technology has great potential for the development of future Internet services, it must surmount several technical obstacles. Scalability is an important concern, to begin with. Currently, a Bitcoin block cannot exceed 1 MB in size, and one is mined approximately every ten minutes. As a consequence, the Bitcoin network is incapable of handling high-frequency trading, as it can only process seven transactions per second [10]. However, larger blocks require more storage space and propagate over the network more slowly. As fewer individuals are willing to maintain such a large blockchain, centralization will ensue. Therefore, achieving a balance between block size and security has proven to be difficult [12]. Second, it has been demonstrated that selfish mining strategies may allow miners to earn more than their reasonable share. For future financial gain, miners conceal their extracted blocks. In that instance, branches could occur frequently, which would impede the blockchain’s development [13]. Therefore, it is necessary to propose solutions to this problem. Moreover, it has been demonstrated that privacy leakage can occur in blockchain transactions even when users only use their private and public keys [14]. In addition, current consensus techniques, such as work proof and stake proof, have several significant drawbacks. For instance, the proof of stake consensus mechanism may cause the wealthy to become wealthier because proof of work consumes excessive amounts of electricity. Thus, blockchain technology is described as an encrypted information system based on a decentralized database distributed across the entire network infrastructure that tracks all transactions and transaction revisions, ensuring that all parties have access to accurate data. High levels of decentralization and transparency are the two primary advantages of blockchain. These include payments and bank transfers, maintaining records of real estate ownership and national identification, transferring assets and documentation, and other activities. Blockchain employs a decentralized approach to data construction and storage, in contrast to conventional databases. 1.4 Blockchain Types The different types of Blockchains are based on their handling and distinct attributes are 1) Public blockchains 2) Private blockchains and 3) Consortium blockchains 4) Hybrid

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blockchains. In terms of management and rights, the consortium Blockchain, also known as "Federated Blockchain", lies halfway between public and private blockchains [15]. Participants can join a public Blockchain network without prior authorization. This blockchain is truly decentralized because anyone can participate in the consensus process, view, and submit transactions, and maintain the shared ledger [16]. Private or permissioned blockchains are designed specifically for a single organization. Users are permitted to join the network and participate in specified duties that support the Blockchain’s decentralized operation only upon invitation [17]. Consortium blockchains are private blockchains despite being designed for multiple enterprises. The network can only be joined and maintained by members who have been invited and are reputable [1]. Finally, Hybrid blockchain incorporates public and private blockchain characteristics. It combines the positive attributes of each blockchain; specifically, a hybrid blockchain inhibits the benefits of private blockchains for privacy and public blockchains for transparency when required [5].

2 Consensus Algorithms Before incorporating a transaction into a block, the procedure of ensuring that it is legitimate (not fraudulent, double spending, etc.) is called the block trust algorithm. Through consensus algorithms, it is agreed that a block should be added to the blockchain. These consensus algorithms capitalize on the fact that the majority of Blockchain users share an interest in preserving the blockchain’s integrity [1]. A blockchain system uses a consensus method to establish trust and effectively store transactions on blocks. Consider consensus algorithms to be the minds of all blockchain transactions. In essence, a consensus protocol is a set of rules that each participant must follow. Blockchain requires a distributed consensus method because it is a distributed technology lacking universal trust so that all parties can concur on the blockchain’s current state. Consensus on the blockchain is based on the notion that having more of a scarce resource gives you greater control over how it is utilized [7]. Some popular consensus algorithms are detailed as the following: 1) Proof of Work (PoW): PoW chooses a problem that can only be guessed at. When creating and validating a full block, for instance, the challenge is to come up with a nonce value such that, when a hash function is used with the transaction data and nonce value as inputs, the difficulty of the hash output, such as starting with four leading zeros, must be met[9]. 2) Proof of Stake (PoS). It is the second-most prevalent consensus method requires fewer calculations than PoW to mine. PoS eliminates the time and energy consumption issues associated with PoW, as discovering a nonce requires electricity and requires miners to spend time. Nodes must stake funds to be designated as the next block creator in a PoS network. The creator of each selected block will receive the associated transaction fees. If a block victor attempts to add an invalid block, they will lose their stake. During the initial phase of the Ethereum 2.0 upgrade, the "world computers" of the blockchain transition from the PoW to the PoS consensus algorithm.

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3) Delegated Proof-of-Stake (DPoS). Every token holder in DPoS has the ability to cast multiple ballots for delegates and delegate voting authority to other users. The quantity of tokens a token holder possesses determines their voting power. In order to safeguard the network, delegates are responsible for validating transactions and blocks [8]. In DPoS, token holders can vote on who mines new blocks and only the finest miners are rewarded, as opposed to PoW and PoS, where the miner with the most computing power or tokens wins. EOS is a blockchain system that uses the DPoS algorithm [8]. 4) Proof of Elapsed Time (PoET). Intel Corporation developed PoET to provide an alternative method for selecting a block miner [8]. Each potential PoET validation node requires a random delay time generated on a dependable computing platform, such as Intel SGX. The validation winner is the node that finishes waiting for the allotted period of time first and can add the new block, and every node has an opportunity to win [18]. 5) Practical Byzantine Fault Tolerance (PBFT). The purpose of Byzantine Fault Tolerance (BFT) is to reach a suitable consensus while resolving the well-known problem of dishonest generals. The consensus algorithm known as PBFT optimises BFT [19]. In PBFT, the blockchain system will reach consensus on the current state of the blockchain if less than one-third of the nodes are malicious or antagonistic. The number of nodes in a blockchain system increases its security. PBFT is currently employed by Hyperledger Fabric. 6) Directed Acyclic Graph (DAG). In contrast to earlier consensus procedures, DAGs consist of vertices and edges (the lines connecting them). The edges and vertices are acyclic because they do not loop back on themselves and are directed since they all point in the same direction [20]. Each vertex in the structure represents a transaction. Here, there is no concept of blocks, and adding transactions is not dependent on mining. Each transaction is constructed on top of another rather than being gathered into blocks. Still, when a node submits a transaction, a little PoW operation is performed. This confirms that earlier transactions are legitimate and prevents network spam. DAG is the consensus algorithm used by IOTA [1]. Finally, we make a comparison between all algorithms as shown in Table 1 at the last section to describe the pros. and cons. of them.

3 Blockchain Security This section examines in depth a variety of security issues and related research developments. Scalability, Securer software codes, Privacy preservation, Quantum computing impact, Regulation and standard issues, and IOTA Security are some of the Blockchain security issues that will be discussed. Current surveys have revealed the potential future applications or trends for blockchain technology, which include smart contracts, preventing the tendency towards centralization, big data analytics, blockchain applications, and artificial intelligence. This section would highlight the problems and research trends listed below in addition to those that are valid trends and scopes.

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3.1 Scalability Compared to contemporary centralized approaches, blockchain technology is more scalable. However, there have been instances in which a greater number of nodes resulted in the technology performing at a lower level. This remains a significant obstacle, particularly for applications involving network security where a large number of users require service, and the network is expanding swiftly. In addition, the dynamic nature of the system makes scaling issues more likely because nodes must frequently submit transaction updates. Both the Hyperledger and Ethereum platforms provide explicit scalability guarantees. However, the operational experiments reveal that the two systems still have room for improvement in certain scalability-related areas. 3.2 Securer Software Codes We are aware that attacks on software code and smart contracts occur almost annually. Security must be a non-negotiable component of all software related to assets. Since smart contracts involve valuable information such as cryptocurrencies, tokens, and other digital assets, their security is crucial. The transactions generated by smart contracts are irreversible, and it is very difficult to alter or modify their software codes if a flaw is discovered. Several restrictions on smart contracts are in place to safeguard the blockchain ecosystem from attackers. Because of the cryptographically hashed sequences, the accounts and transactions are also secure and immutable. In 2020, research on Flash Boys 2.0 revealed the dangers of smart contracts and how Ethereum may be gravely imperiled by arbitrage bots and miners that can extract the value of smart contracts’ transaction-ordering dependencies. One of the current challenges for blockchain is ensuring the security of smart contracts, as it is exceedingly difficult to ensure the security of smart contract code, per Reference [21]. 3.3 Privacy Preserving Obscuro was introduced in 2018 to prevent payers and payees from being linked to achieving anonymous payments. It provides a safe and efficient Bitcoin mixer. BITE was developed in 2019 to facilitate privacy-preserving requests from light clients, and Ouroboros Cryptinous was created to investigate the privacy-preserving PoS protocol. In 2020, it was demonstrated that Zexe could attain privacy-preserving equivalents of popular applications. In 2020, remote side-channel attacks on receiver privacy were introduced. This point is important because we will focus on it and specific Transaction Privacy Leakage for all users in public blockchain networks regarding IOT networks. Therefore, our solution will be the proposed model is a hybrid system that allows several users to make transactions simultaneously with multiple inputs and outputs. Transaction inputs will not be able to map to their corresponding outputs, and it will be difficult to trace the identity of the sender. Furthermore, it will be difficult to accomplish blockchain scalability securely by permitting sophisticated authorization of data for different users.

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3.4 Quantum Computing Impact on Blockchain IBM, Intel, Google, Rigetti, D-Wave, IonQ, Microsoft, and other major corporations and governments are actively developing quantum computing. Using a quantum algorithm disclosed by Shor in 1994 [22], the most prevalent public key cryptography methods can be broken, and an improved version of Shor’s algorithm may be able to defeat ECDSA. In July 2020, the NIST selected 15 postquantum cryptography algorithms from a pool of 26, and these 15 algorithms are currently undergoing the third round of public review. Due to the preimage resistance of the hash function, it is mathematically impossible to reverse-engineer the public key given the P2PKH address. If the quantum computer’s public key is unknown, it is unable to derive its private key. However, as soon as any amount of money is sent from a particular P2PKH address, its public key will be made public to verify the transaction’s digital signature, rendering its private key vulnerable to quantum computing. The blockchain community will also discuss the impact of quantum computing on the technology. Quantum attacks can only be defended against postquantum cryptography. One of the trends in research is the investigation of methods to use post-quantum cryptography to construct blockchain systems that are secure and resistant to quantum physics. The organization will then be required to construct a new blockchain that is distinct from the current blockchain and utilizes the new post-quantum cryptography protocol, such as blockchain 3.0. 3.5 Regulation and Standard Issue First, it is anticipated that cryptocurrencies will acquire popularity, which will facilitate and reduce the cost of financial transactions. Additionally, it undermines the nation’s financial management and policies. Second, the number of global blockchain applications is growing. To validate COVID19 vaccination injection certificates, for instance, blockchain-based solutions are employed. There must be rules and agreements in place for various nations to accept injection certificates maintained on blockchain systems. Thirdly, it may be challenging to establish a single or international standard because multiple parties must agree to use the blockchain as a shared infrastructure, even within the same nation. Consequently, regulation and standards will be one of the obstacles to the ubiquitous implementation of blockchain systems. 3.6 IOTA Security As Bitcoin and Ethereum-based cryptocurrencies struggle with scalability and transaction fees, IOTA may be a viable alternative due to its radically different nature structure of vertices and edges using DAGs rather than blocks chain. IOTA asserts that its Tangle technology makes it infinitely scalable and free of transaction fees. Concerns exist, however, regarding Tangle’s inability to correctly store transaction orders and Curl, an IOTA hash function designed by Tangle’s developers that is susceptible to vulnerabilities. In Table 3, we compare and contrast the characteristics of numerous frameworks. IOTA must surmount these obstacles. When the technology is mature, it is anticipated that it will be widely adopted in the IoT industry, a swiftly expanding and enormously promising field..

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4 Summary for Survey Works There are many surveys paper that talked about Blockchain in many fields as security, architecture, and Consensus Algorithms. So, we will now go to make different comparisons of consensus algorithms, as shown in Table 1. Now we will focus on the key features of Public, Private, and Consortium blockchain. There are some similarities between private and consortium blockchain, while public blockchain is totally different from the previous two types. I will use the key features as shown in Table 2 to compare between three types. Recently, a few additional interesting blockchain frameworks, such Corda and Ripple, have been presented. It can be difficult to choose a framework that will meet application needs when there are so many different ones available. Although there may be other criteria, we chose the ones listed in Table 3 to best serve the demands of the users[23]. Table 3 compares various development frameworks in terms of various features and attributes. It concludes the comparative analysis of the characteristics of various frameworks. There are three distinct stages or generations of blockchain development (Blockchain1.0, Blockchain2.0, and Blockchain3.0). Education, healthcare, smart cities, the legal industry, the music industry, eGovernment to combat poverty and corruption, tax administration, and a multitude of other industries that rely on third parties to establish trust have significant blockchain application potential. Consequently, blockchain is expanding beyond its initial applications in financial services and money to a vast array of other domains, referred to as blockchain version 3.0. Table 4 provides a summary of emerging application domains for blockchain technology. Table 5 summarises the relevant survey research and our work in this paper. Additionally, our contributions to this paper are evident. First, we enumerate the categories of papers that are or are not associated with blockchain technology. In addition, some survey papers on blockchain security examined potential vulnerabilities and security at the process, data, and infrastructure levels, whereas the focus of this paper is the security of blockchain itself, which the majority of previous surveys only partially presented or did not present. Last but not least, Table 5 displays the key findings of other survey papers described in these articles.

High

open

Relatively high cost of entry, but high returns

Public permissionless/Private blockchain

No

Bitcoin

calability

Node identity management

Cost of entry and returns

Setup

Susceptible to Sybil attack

Examples

Ethereum

Yes

Public permissionless/Private blockchain

Low cost of entry, but low returns

open

High

Low

High

Power consumption

Bitshares

Yes

Public/Private blockchain

Lower cost and lower returns than PoS

open

Low

Low

< 51% validators

< 51% stack

Fault Tolerance Low

DPoS Voting to elect witness node

Computational power

Main feature

PoS

Stake (amount of coins)

Pow

Consensus Algorithms

permissioned

Low

Low

33% replicas

Reach consensus

PBFT

Hyperledger

No

Private permissioned/ permissionless blockchain

Hyperledger Fabric

Yes

Private permissioned blockchain

Very low cost of All participate entry, but low with no return returns

open

Low

High

Yes

Lottery based election

PoET

Table 1. Comparison of Consensus Algorithms – Reference [1, 7]

IOTA

No

Public permissioned non-blockchain

All participate with no return

permissioned

Very High

High

Not applicable

Consensus for IOT blockchain

DAG

746 T. A. A. A. Abdel Bary et al.

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Table 2. Public, Private, and Consortium Comparison and key features - referenced in[2, 31] PUBLIC

PRIVATE

CONSORTIUM

Users

• Decentralized • Without Authorization • Secret identity • Probability of harmful source

• Multiple organization • Authorization • Real identity • Trusted source

• • • •

Permission

• Permissionless

• Permissioned

• Permissioned

Collective consensus and confirmation mechanism

• Pow, PoS, PoET, etc • High power consumption

• Consensus Algorithms • Low power consumption

• Consensus Algorithms • Low power consumption

Transaction

• Long in confirmation • Short in • Short in Time confirmation Time confirmation Time • Low < 100 • High > 100 • High > 100 Transaction /second Transaction/second Transaction/second

Main feature

• Decentralized, no need for third party to complete transaction

one organization Authorization Real identity Trusted source

• High security and transparency with reducing in cost and time • Reduce data redundancy

• High security and transparency with reducing in cost and time • Reduce data redundancy

Validator

• Node or miner

• Authorized nodes

• Authorized nodes

Throughout Network

• High

• Low

• Medium

Governance Type

• Public

• Consensus is managed by single node

• Consensus is managed by consortium of participation

Examples

• Bitcoin – Ethereum

• MONAX - Quorum • R3 Corda, HyperLedger

Table 3. Comparative analysis of the Characteristics of Various Frameworks – Referenced [23] Framework

Ethereum

Hyperledger

MultiChain

Quorum

IOTA

Development community

Yes

Yes

Yes

Yes

Yes

Enterprise Activity

Yes

Yes

Yes

No

No

Roadmap

Yes

Yes

Yes

Yes

Yes (continued)

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Framework

Ethereum

Hyperledger

MultiChain

Quorum

IOTA

Ease of programming

Yes

No

Yes

Yes

No

Support model

No

No

Yes

Yes

Yes

License

Yes

Yes

Yes

Yes

Yes

Reliable Backing

No

Yes

Yes

Yes

No

Table 4. Evolving blockchain application domains – Referenced [23] Blockchain 1.0

Blockchain 2.0

Blockchain 3.0

Initial application

Currency

Banking & financial services, smart contracts, economics, and financial market

Beyond Blockchain 1.0 and Blockchain 2.0

Examples

Bitcoin, Litecoin, Ethereum, etc

Smart contracts, Smart property, and asset

Domain name, digital identity, eGovernment, IoT, smart cities, Industry 4.0, online electronic voting, among others

Table 5. Summaries of previous survey papers belongs to Blockchain Reference Blockchain category Blockchain Security Studies points [7] 2022

Yes

Yes

• It assesses the blockchain security from risk analysis to derive comprehensive blockchain security risk categories • It provides the challenges and research trends are presented to achieve more scalable and securer blockchain systems for the massive deployments (continued)

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Table 5. (continued) Reference Blockchain category Blockchain Security Studies points [1] 2021

Yes

Yes

• It presents a comparative analysis of frameworks, classification of consensus algorithms, and analysis of security risks & cryptographic primitives that have been used in the Blockchain so far • It elaborates on key future directions, novel use cases and open research challenges, which could be explored by researchers to make further advances in this field

[24] 2021 Yes

Yes

• It presents the operational mechanisms of these and other consensus protocols • It analyzes and compares their advantages and drawbacks

[25] 2021 Yes

No

• It provides an overview of blockchain interoperability • It discusses supporting technologies, standards, use cases, open challenges, and future research directions, paving the way for research in the area

[26] 2021 Yes

Yes

• It analyzes other surveys of blockchain in healthcare and compare both the positive and negative aspects of their papers • Additionally, it summarizes the methods used in healthcare per application area and show their pros and cons

[27] 2021 Yes

No

• Ensuring that blockchain is widely available through public and open-source code libraries and tools will help to ensure that the full potential of the technology is reached • It focusses on the developments which can be made concerning the long-term goals of blockchain enthusiasts (continued)

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Reference Blockchain category Blockchain Security Studies points [28] 2020 Yes

Yes

• It presents a review of the main consensus procedures, including the new consensus proposed by Algorand: Pure Proof-of-Stake (Pure PoS) • It provides a framework to compare the performances of PoW, PoS and the Pure PoS, based on throughput and scalability

[15] 2019 Yes

Yes

• It presents a comparative study of the trade-offs of blockchain • It explains the taxonomy and architecture of blockchain, provides a comparison among different consensus mechanisms and discusses challenges, including scalability, privacy, interoperability, energy consumption and regulatory issues • In addition, it notes the future scope of blockchain technology

[29] 2018 Yes

Yes

• Analysing consensus protocols in Blockchain which already proposed and their feasibility and efficiency in meeting the characteristics they propose to provide

[30] 2017 Yes

Yes

• It presents a comprehensive overview on blockchain technology • It provides an overview of blockchain architecture and compare some typical consensus algorithms used in different blockchains • It provides technical challenges and recent advances are briefly listed Also focus on possible future trends for blockchain

5 Conclusion A blockchain is a decentralised, transparent, and immutable distributed ledger that organises a growing list of transaction data into blocks. We emphasise on the key aspects of the blockchain (The evolution of Blockchain technology, Architecture of Blockchain, how it operates, its various types, and security-related challenges and future trends). This paper

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provides the groundwork for scholars, developers, and the broader blockchain community to comprehend the current state of consensus algorithms, blockchain types, trends, and future work in the blockchain domain. So, our findings of our paper are that the capacity of the blockchain solution to strike a balance between scalability, security, and decentralisation will determine whether it can transform use cases in industries spanning from finance to agriculture. In addition, the purpose of developing more practical and efficient industrial applications that may fully benefit from the usage of blockchain and accomplish the intended goals. Also, there are still a number of unresolved challenges that require further research and analysis, which are unresolved problems that related to security, privacy, scalability, energy use, integration with other systems, and, more particularly, regulatory challenges. To solve these problems and bridge the gaps for more effective, scalable, and secure blockchain industrial applications, further research in this area is needed. Finally, our suggestions for future research are as the following: 1. It is recommended to enhance the blockchain’s usability in order to surmount technological limitations. As a result of the blockchain’s decentralised decision-making process, it is sluggish and contains more data than the centralised system. For this, we require scalable solutions like Plasma, Lightning network, and optimistic roll-up. 2. It is necessary to acquire evaluation data for cyber defence. As discussed in a previous section, the blockchain will be impacted by additional restrictions in cyber defence in contrast to commercial networks. We presented a number of studies on this topic, but there is not nearly enough information to implement the blockchain in earnest. Obtaining evaluated data from blockchain in defence environments is therefore a significant future undertaking. 3. For future surveys, a comprehensive investigation of the ongoing government lead blockchain initiative is required with more analysis criteria due to linguistic constraints and policies on the disclosure of information.

References 1. Bhutta, M.N.M., et al.: A survey on blockchain technology: evolution, architecture and security. IEEE Access 9, 61048–61073 (2021) 2. Rashid, M.A., Pajooh, H.H.: A security framework for IoT authentication and authorization based on blockchain technology. IEEE Int. Conf. On Trust, Secur. Priv. Comput. Commun. 9, 264–271 (2019) 3. Ahram, T., Sargolzaei, A., Sargolzaei, S., Daniels, J., Amaba, B.: Blockchain technology innovations, IEEE Technol. Eng. Manage. Conference (TEMSCON), pp.137–141 (2017) 4. Rateb, J.: Blockchain for the Internet of Vehicles: A Decentralized IoT Solution for Vehicles Communication and Payment using Ethereum. HESAM, Paris (2021) 5. Chaum D. L.: Computer systems established, maintained and trusted by mutually suspicious groups. Electronics Research Laboratory, University of California (1979) 6. Haber, S., Stornetta, W.S.: How to time-stamp a digital document. J. Cryptol. 3(2), 99–111 (1991). https://doi.org/10.1007/BF00196791 7. Guo, H., Yu, X.: A Survey on blockchain technology and its security, blockchain: research and applications. Journal 3(2), 100067 (2022) 8. Guo, H., Yu, X.: Blockchain: research and applications. Journal 3(2), 100067 (2022)

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Risk Assessment

Fuzzy Logic Approach in Failure Mode and Effects Analysis: Glass Industry Application Irem Düzdar Argun1

and Tugce Ozdemir2(B)

1 Duzce University, Duzce 81000, Turkey 2 Duzce Cam, Duzce 81000, Turkey

[email protected]

Abstract. The competitive environment in market circumstances, invites wide notice from glass factory managers about quality complications. There is a direct connection between quality and production efficiency. Increasing quality increases efficiency in production, reduces costs and increases market share. The company, which has to compete with many companies in glass production, aims to increase its competitive advantage with the quality and production efficiency. The desired feature in glass production is durability and a world-class production quality. Defects occurring in the glass in the production of flat glass in the Düzce Glass Factory lead to loss of production. For this reason, the company carries out the necessary studies in order to achieve quality in production and to ensure that the established quality management system is in operation. In the study, Pareto analysis and Failure Mode and Effects Analysis (FMEA) technique are examined in the quality improvement process. In this study, Pareto analysis of the glass produced and sold according to the customer complaint form is applied, FMEA is handled and the processes is examined. The aim of this study is to be encountered the defects in the glass production process are analyzed according to the FMEA technique. As a result of the analysis, it has been shown that the technique provides success in improving the quality functions of the company. Keywords: Quality Control · Fuzzy FMEA · Glass

1 Intrduction Customer orientation is the priority of every company in today’s glass industry. This is possible by being able to offer the goods or services that customers want at the exact time and in the exact quantity where customers want them. Quality management, which plays one of the most important roles here, needs to function smoothly. It is indispensable now to have the ability to foresee situations and obstacles that may cause disruptions in quality processes and take precautions in the flow of quality goods and services to customers. At this stage, a proactive approach can be demonstrated by determining the priority of possible problems and preventing them from occurring with Failure Mode and Effect Analysis (FMEA), which is the most well-known of the defect prevention analyses [1]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 757–766, 2023. https://doi.org/10.1007/978-3-031-39774-5_83

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FMEA is widely used in scientific research in models created for the purpose of solving complex problems. It provides great benefits to researchers in the field of manufacturing, especially by revealing meaningful relationships on quality data and effective calculation methods. [2] In this study, applications that perform highly successful efficiency estimates and various classifications along with FMEA fuzzy logic, which provides great convenience to glass manufacturers and researchers thanks to the decision support systems used in decision-making and evaluation processes, have been included and will support glass manufacturers. [2, 3]. When looking at the company as a whole, it is divided into varieties by its use in the production process, system and service areas. The ease of application and the fact that it is usable for all sectors makes the FMEA technique more advantageous than other techniques. In this study, the FMEA technique was used in the glass sector. [3] First of all, customer feedback forms were examined, complaints were determined, and qualities were examined. FMEA application was started with Pareto analysis using simultaneous Engineering techniques. After that, FMEA was applied to the production process in a glass manufacturing factory and fuzzy results were evaluated.[4].

2 Literature Review The study of Quality Control Circles, (1984) pareto analysis, one of the seven techniques of statistical process control, was examined. Pareto analysis is a simple but extremely effective technique in the diagnosis and analysis of the problem. [5]. The study of Kobu (1987), the topic of modern quality management, which has been adopted and started to be applied especially in the industrial and service sectors of Turkey in recent years, is examined. Ozcan (2001) has an application of Pareto analysis has been made in the cement manufacturing industry. Sivas Cement Factory has been selected as the application site.[6]. The study of VDA (1996) is about the quality of products and services in quality management and the combined result of activities at each stage of the total business process at all times. In addition, it describes the prioritization of planning and prevention activities in order to determine relationships and interdependencies, appropriate measures to prevent the occurrence of nonconformities.[7].

3 Material and Method The fuzzy logic approach is widely preferred in cases where one or more of the components are not clearly known but can be classified. For this reason, considering the related advantages, the fuzzy logic approach to the FMEA method is considered as an application that will contribute [5]. As a first step, the activity subject to the FMEA risk analysis should be examined; data should be collected for analysis. It is expected that the defect types will be decided as a result of data collection. Then, the O, S, D status should be evaluated for the Fuzzy FMEA risk analysis of the relevant defect types. It would be appropriate to prefer a team consisting of 3–6 people who have knowledge about the study process to make the evaluation [3, 21]. For each defect type, it may be possible to apply three different ways to evaluate the O, S, and D inputs [13]. The method used

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within the scope of this study is that each specialist evaluates O, S, D individually. The score value of the O, S, D factors is obtained by the geometric mean method. The FRPN value is obtained by following the fuzzy logic algorithm with the obtained factors [9]. It is estimated that it would be appropriate to take preventive action for defect types with an FRPN value above the average FRPN value [26]. The fuzzy RPN value, which is the output of the fuzzy logic algorithm, is calculated as a product of the parameters O, S, D and mathematically expressed by Eq. 1[33]. FRPN = O x S  x D

(1)

In order to obtain a fuzzy RPN (FRPN) within the scope of the fuzzy FMEA application, the addition process will be applied to obtain the total fuzzy O, S, D and relative weights of the risk parameter values. The mathematical expression of the related process is included in Eq. 2 [30]. O = {(fO)agg}o; S = {(fS)agg}s; D = {(fD)agg}d

(2)

(fO)agg: The total fuzzy value of the probability. (fS)agg: The total fuzzy value of violence. (fD)agg: The total fuzzy value of discoverability. The addition process is stated as the process of combining several fuzzy sets in a desired way to produce a single fuzzy set. In other words, ‘n’ fuzzy sets can be reduced to a single fuzzy set by the addition process [21]. Accreditation of institutions and organizations performing conformity assessment activities, international standards specific to the relevant activity, industry-specific criteria, guidelines, etc. It is carried out within the framework of the requirements [1]. With the current version of the standard, there is a question of adopting a “Risk-based thinking approach” in laboratories. The standard requires laboratories to address their risks [2]. Along with this requirement, risk assessment and risk analysis are gaining great importance in laboratories. In the following, the types of defects that may occur in laboratories and their effects have been determined. These nonconformities were examined and repeated nonconformities were eliminated. Subsequently, the obtained nonconformities are classified according to the standard articles and are intended to be understandable by expressing them in the standard manner. As a result, a total of 41 types of defects that pose a risk to laboratories have been identified. A common value was obtained by taking the geometric average of the obtained expert opinions [6]. Fuzzy FMEA method is also preferred because it offers a more realistic, practical and flexible structure for combining O, S, D parameters; Fuzzy FMEA is preferred because it allows to reveal the relative importance for the relevant parameters. The O, S, and D states that constitute the input to the fuzzy FMEA have been associated with linguistic variables. Table 1 it is expressed in [19].

4 Application In this study, the quality classes of glass production were mentioned. The complaint requests from the customer are detailed and listed by sales and marketing team. And according to this list, research was carried out and the quality classes were detailed with

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Linguistic Variable

1

Very Low

2

Low

3

Middle

4

High

5

Very High

the ISRA device located in the cutting department in those months and further scaled up with the microscope device. Thus, all defects that occurred during the determined time period were taken into account. What these defects consist of, their dimensions, which class they are in, the main reasons for the class they are in, and the improvement processes of these reasons have been examined. The most important defects were determined according to these results by applying Pareto analysis to the general defects first. Defects were prioritized by applying FMEA to these defects. Improvements have been made according to this priority list. By detailing these improvements, an attempt has been made to reduce the RÖS rating. 4.1 Application of Pareto Analysis In the production of flat glass at the Düzce Glass Factory, defects in the glass cause a loss of production. Pareto analysis was used to determine which defects should be given priority. Thanks to this analysis, it is planned to classify defects and give weight to studies on defects. The device that informs the optimization computer about the class, size and location of defects on the glass surface is the ISRA device (Fig. 1). After determining the severity and magnitude of the defect in the ISRA device, the quality of the glass (A1, A2, A3, A4, A5) is determined. The Pareto analysis applied to the defect groups for the last 3 months is as shown in Fig. 2. In flat glass production, the most defect occurring in glass is A3 group defect with 32.2%. The total of A3, A2 and A4 defect groups is 77.7%. In this case, the defect groups A3, A2 and A4 should be refined. The defects and defect percentages that have occurred in the ISRA device in the last 3 months at the Düzce Glass Factory are taken from the ISRA device. The Pareto analysis of the defects received from the ISRA device in the last 3 months at the Düzce Glass Factory and customer complaint notifications and customer feedback during the same period are given in Fig. 3 and Fig. 4. 4.2 Fuzzy FMEA Application A scale of 10 was used to determine the severity score, to determine the detectability, to determine the occurrence. When determining the values, it was tried to use the historical data as much as possible, and the resulting weight values were determined by the members [22]. Fuzzy Logic Design Toolbox included in MATLAB program has been used

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Fig. 1. ISRA Device

Fig. 2. Pareto Analysis for the Last 3 Months

for calculation of RPN values with fuzzy logic approach. Since it does not require any discretion and therefore it can work in an analytically defined inference model, it uses learning algorithms, the model was created according to the Mamdani method within the scope of the Fuzzy FMEA study; the COG rinsing method was used with the Mamdani type inference mechanism [31]. In the established Fuzzy FMEA model, symmetric triangular membership functions were used for the inputs O, S, D, which were determined as inputs (Fig. 5). These sub regions, represented by symmetric triangle membership functions, are; Very low; Low; Medium; High and Very High, respectively.

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Fig. 3. General Defects of Pareto Analysis

Fig. 4. Return Received Defect

The triangular membership functions for the inputs are shown in Fig. 6. In the established Fuzzy FMEA model, asymmetric triangular membership functions were used for the FRPN, which was determined as the output. For the output variable, the scale between 1–125 points, which is the minimum and maximum December Decadency, is divided into 5 different parts. These sub regions, represented by asymmetric triangle membership functions, are; Very low; Low; Medium; High and Very High respectively. The triangular membership functions for the output are shown in Fig. 7.

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Fig. 5. Fuzzy FMEA Model

(A)

(B)

(C)

Fig. 6. A, B, C Input Variables Membership Functions

Fig. 7. The Membership Function for the Output Variable

A rule base consisting of 125 rules has been created in the model in order to express all possible situations. The rules on the screen shown in Fig. 8 have been prepared by evaluating all possible situations one by one.

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Fig. 8. Rule Base

5 Conclusions and Recommendations The RPN value, which is the classical FMEA analysis output for 41 different defect types evaluated in the application, and the FRPN values, which are the fuzzy FMEA output, were compared graphically and statistically. In graphical comparison, the compatibility of RPN and FRPN values with each other are seen in Fig. 9.

150 100 50 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 DEFECT

RPN

FRPN

Fig. 9. Graphical Comparison of RPN and FRPN

Accordingly, fuzzy identification of ambiguous O, S, D inputs related to defect types, fuzzy FMEA method results close to the classical FMEA method are obtained. In this context, the fuzzy FMEA method is considered as an appropriate approach in the analysis of risks. As a result of these values, a comparison was made with the defect rates given by the ISRA device. The accuracy of the probability, detection and non-detection values obtained by comparing the RPN and FRPN values has been proven, and all values have resulted in the values on the surface and the ISRA device. Thus, it was decided to check the monthly data of the ISRA device and apply FMEA monthly to all defects. In this

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way, both the accuracy of the ISRA device and the faulty products sent to customers have started to be determined more accurately from what causes. The highest FRPN and RPN values are observed in devitrification defect. The reason is due to the fact that its detectability is too high. It is very easy to detect because the defect is seen very clearly on the ISRA device. The cause of devitrification is the formation of crystals in glass due to very low temperature or unstable temperature conditions, local blend separation, inappropriate glass composition, glass viscosity and surface evaporation. After the found RPN and FRPN values, the unstable temperatures in the oven have been prevented and have started to be constantly controlled with thermal cameras. In the studies to be conducted in the future, it is proposed to test the fuzzy FMEA application in different areas, different membership functions and scales, and interpret the results. Determining the relevant defect types as a risk for all laboratories; it is estimated that the risk management process will not be an appropriate approach, since it is specific to the laboratory, it is recommended to consider them together. Within this framework, laboratories, personnel training, devices, methods, etc. it is recommended to intervene in the risks by determining the preventive activities in the relevant issues. Researchers who will work in this field should pay attention to the departments they survey. The ratings given by the departments in quality are very important.

References 1. Türk Akreditasyon Kurumu. https://web.turkak.org.tr/Sayfa/4, (Eri¸sim tarihi: 13.04.2019) 2. Lipol, L., Haq, J.: Risk analysis method: FMEA/FMECA in the organizations. Int. J. Basic Appl. Sci., No: 5, Sayfa, 74–82 (2011) 3. Soykan, Y., Kurnaz, N., Kayık, M.: Sa˘glık ˙I¸sletmelerinde Hata Türü ve Etkileri Analizi ile Bula¸sıcı Hastalık Risklerinin Derecelendirilmesi, Organizasyon ve Yönetim Bilimleri Dergisi, No: 1, Sayfa, 172–183 (2014) 4. Carlson, C.S.: Effective FMEAs Achieving Safe, Reliable and Economical Products and Processes Using Failure Mode and Effects Analysis. A John Wiley & Sons, Inc. Publication (2012) 5. Stamatis, D.H.: Failure mode and effect analysis – FMEA from theory to execution. ASQC Quality Press (1995) 6. Pillay, A., Wang, J.: Modified failure mode and effects analysis using approximate reasoning, Reliab. Eng. Syst. Safety. No:79, Sayfa, 69, 85 (2003) 7. Özkan, M., Bircan, H.: Bulanık Hedef Programlama ile Ürün Hedef Optimizasyonu:Yang, Ignizio ve Kim Modeli, ˙Istanbul Üniversitesi ˙I¸sletme Fakültesi Dergisi, No: 2, Sayfa, 109–119 (2016) 8. Salicone, S.: The mathematical theory of evidence and measurement uncertainty expression and combination of measurement results via the random-fuzzy variables. IEEE Instrum. Measur. Mag., No:17, Sayfa, 36–44 (2014) 9. Gilles, M., Lasserre, V., Foulloy, L.: A fuzzy approach for the expression of uncertainty in measurement. Measurement, No: 29, Sayfa, 165–177 (2001) 10. Ferrero, A., Salicone, S.: An Innovative approach to the determination of uncertainty in measurements based on fuzzy variables. IEEE Trans. Instrum. Measur., N:52, Sayfa, 1174– 1181 (2003) 11. Adlassing, K.P.: Fuzzy set theory in medical diagnosis. IEEE Transactions On Systems, Man, And Cybernetics, No:2, Sayfa, 260–265 (1986)

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Market Risk Assessment by Expert Knowledge Compilation Using a Fuzzy Maximin Convolution Ramin Rzayev1(B)

, Elmar Aliev2

, Jamirza Aghajanov3

, and Inara Rzayeva4

1 Institute of Control Systems, Vahabzadeh Str. 9, AZ1141 Baku, Azerbaijan

[email protected]

2 FINOKO Non-Bank Credit Organization, Jabbarli Str., Globus C., AZ1065 Baku, Azerbaijan 3 Baku State University, Khalilov Str. 23, AZ1148 Baku, Azerbaijan 4 Azerbaijan State University of Economics, Istiglaliyyat Str. 6, AZ1101 Baku, Azerbaijan

Abstract. Permanent volatility in the economy requires any commercial bank to conduct effective market risk management. Every year, the volume of banking operations associated with market risks increases appreciably, which causes further complication of financial instruments. Based on this premise, the importance and relevance of the task of managing market risks from the position of finding the optimal ratio between income, risk and bank liquidity becomes obvious. The above applies to small and medium-sized commercial banks, whose assets do not have the necessary expert and analytical resources to analyze trends and predict dynamics in the securities market, exchange operations and interest rates. Actually, this has become the main subject of this study, which discusses the methods of expert evaluation of market risks using the example of a hypothetical commercial bank. As an analytical method for estimating MR, a fuzzy maximin convolution method is used, where the qualitative evaluation criteria are described by appropriate fuzzy sets. Keywords: Market Risks · Expert Evaluation · Fuzzy Set · Maximin Convolution

1 Introduction Fuzzy logic L. Zadeh has a wide practical application in various fields of knowledge, including the banking sector of the economy. In the field of banking risk management, along with rating and expert systems, methods of scoring analysis and econometrics, numerous developments are also used based on the theory of fuzzy sets. However, choosing an appropriate method and software to support the banking decision-making process is still the quite difficult problem. Based on this premise, it becomes obvious the importance and relevance of studying various methods for assessing banking risks, conducting their comparative analysis and developing recommendations for their use in a commercial bank. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 767–775, 2023. https://doi.org/10.1007/978-3-031-39774-5_84

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Commercial banks are systematically improving their own risk management systems that take into account all types of risks, among which the most significant are credit, market, operational, etc. In the decision-making process, a commercial bank (CB) preliminarily conducts a comprehensive assessment of all alternative options with an allowance for risk. At the same time, it is almost impossible to completely level the risks. Therefore, the most important task of market risk management is to keep an acceptable balance between risk, profitability and liquidity, which acts as the main factor in making managerial decisions. According to the “Risk Management Rules in Banks” approved by the Board of the Central Bank of the Azerbaijan Republic dated 06.09.2010 No. 24 [1], market risk (MR) is understood as the risk of losses due to changes in the market of interest rates, exchange rates, prices of securities and goods. According to this document and the recommendations of the Basel Committee [2], MR has the following subcategories: interest rate risk – the risk of losses owing to unfavorable changes in interest rates; currency risk – the risk of losses owing to unfavorable changes in foreign exchange; capital risk – the risk arising owing to unfavorable changes in the cost of capital and securities. This risk affects capital as well as subsidiaries of capital used for hedging and speculation purposes; commodity risk – the risk arising owing to adverse changes in the cost of goods on the market.

2 Problem Definition Suppose that for a certain CB “X” for the current date the analysis of income, expenses and results on operations subject to market risks has been carried out. As a result of this analysis, objective characteristics of MR were established and summarized in Table 1. From the point of view of bank risk management within the risk management system, the quantitative characteristics of MR given in Table 1 must be converted into a unified indicator R, which would aggregate the relative influence of the considered number of listed factors x i (i = 1 ÷ 4) using an imaginary function R = R(x 1 , x 2 , x 3 , x 4 ). Estimates obtained on the basis of an expert analysis of the data given in Table 1, banking operations subject to MRs, assets and liabilities of CB “X” by types of currencies, as well as summary data on its open foreign exchange positions are used as initial information. Table 1. Detailing of market risk on the example of a hypothetical bank “X” Factor

The share of the indicator in the aggregate capital of the bank (%)

Share of indicator in market risk (%)

Market risk, total:

32.3

100

- interest rate risk (x 1 )

19.6

60.68

- currency risk (x 2 )

4.9

15.17

- capital risk (x 3 )

3.6

11.15

- commodity risk (x 4 )

4.2

13

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3 Determination of the Weighted Level of MR Based on Expert Evaluations The requirements for the competence of approaches to the formation and assessment of the MR level implies the study of its constituent components as influences. In other words, the estimation of MR is the multicriteria procedure that implies the application of the compositional aggregation rule for each of the factors x i (i = 1 ÷ 4). Suppose that to conduct a weighted examination relative to the assessment of the MR, the CB “X” invited 15 specialized specialists. At the initial stage, each expert was asked to rank the variables x i according to the following principle: designate the most important with the number “1”, the next less important – with the number “2” and then in descending order of importance. As a result of the independent questioning, expert rank estimates of factors x i were established and summarized in Table 2. Table 2. Ranking of MR variables in orders of experts’ preferences. Expert

Estimated MR variables and their ranks (r ij ) x1

x2

x3

x4

01

1

2

3

4

02

1

3

2

03

1

2

04

1

3

05

1

06 07 08

Expert

Estimated MR variables and their ranks (r ij ) x1

x2

x3

x4

09

1

2

3

4

4

10

2

1

4

3

3

4

11

1

3

2

4

2

4

12

2

1

4

3

3

4

2

13

1

2

4

3

1

4

3

2

14

1

3

2

4

1

3

2

4

1

2

3

4

1

3

2

4

15  r ij

17

37

43

53

Before identifying the weights of the MR factors, it is necessary to determine the presence of consistency of expert opinions in the form of a degree demonstrating the acceptable multiple rank correlation of expert conclusions. For this, the Kendall concordance coefficient is used [4], which is calculated by the formula:    (1) W = 12 · S/ m2 n3 −n where m is the number of experts; n is the number of MR variables; S is the deviation of expert opinions  from the variable, which is calculated by the average rank of the MR 2 , where r is the rank of i-th factor r − m(n + 1)/2] formula [4]: S = ni=1 [ m ij j=1 ij established by j-th expert. In our case, i.e. for n = 4, m = 15, r ij ∈ {1, 2, 3, 4} (i = 1 ÷ 4; j = 1 ÷ 15) the deviation of expert opinions is S = 691. According to (1) the Kendall concordance coefficient is calculated as following: W = 12·691 / [152 (43 – 4)] = 0.6142. For the totality of expert

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rank estimates, the value of the Kendall concordance coefficient exceeds the threshold value of 0.6 (W = 0.6142 > 0.6), which indicates the fairly strong consistency of expert opinions relative to the degrees of influences x i on the level of MR as a whole. To identify the generalized weights of the MR variables, assume that at the preliminary stage of the independent survey, the experts were asked to set the values of the estimates α ij for the generalized weights of the variables x i in accordance with the ranks  r ij and the condition 4i=1 αij = 1, j = 1 ÷ 15. Suppose that as a result of the survey of experts, the corresponding estimates were obtained and summarized in Table 3. Table 3. Expert estimates of the generalized weights of the MR variables. Expert

Estimated MR variables and normalized values of their weights (α ij ) x1

x2

x3

x4

01

0.350

0.250

0.245

0.155

02

0.400

0.200

0.300

03

0.360

0.240

04

0.450

0.200

05

0.500

06 07 08

Expert

Estimated MR variables and normalized values of their weights (α ij ) x1

x2

x3

x4

09

0.450

0.300

0.200

0.050

0.100

10

0.300

0.350

0.100

0.250

0.225

0.175

11

0.350

0.200

0.300

0.150

0.250

0.100

12

0.300

0.400

0.050

0.250

0.150

0.100

0.250

13

0.400

0.300

0.100

0.200

0.400

0.150

0.200

0.250

14

0.450

0.150

0.300

0.100

0.500

0.100

0.350

0.050

0.350

0.250

0.225

0.175

0.350

0.250

0.300

0.100

15  α ij

5.910

3.490

3.245

2.355

Next, preliminary calculations are carried out for the subsequent identification of the weights of the MR variables based on the data presented in Table 3. To do this, group estimates of variables x i (i = 1 ÷ 4) and numerical characteristics are determined that characterize the “degrees of competence” of experts. To calculate the average value α i for the i-th group of normalized estimates of the generalized weights of variables x i , the following equation is used [5]: 15 wj (t)αij , (2) αi (t + 1) = j=1

where wj (t) is the indicator characterizing the competence of the j-th expert at time t. According to [5], at each iteration step, these indicators are updated by follows: ⎧ 4 ⎪ ⎨ wj (t) = (1/η(t)) αi (t) · αij (j = 1, 14), i=1 , (3)  15 14 ⎪ ⎩ w15 (t) = 1 − wj (t), wj (t) = 1, j=1

j=1

where η(t) is a normalizing factor that ensures the transition to the next iteration step and calculated by the formula 4 15 αi (t)αij , (4) η(t) = i=1

j=1

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The process of finding group estimates of the normalized values of the generalized weights of variables x i (i = 1 ÷ 4) is completed under condition max{|αi (t + 1) − αi (t)|} ≤ ε,

(5)

where ε is the feasible accuracy of calculations, which is set by the user. Assume that the feasible calculation error is ε = 0.001. Then, assuming at the initial stage (t = 0) the number w(0) = 1/15 to be the same degree of competence for all experts, in the 1st approximation the average values of the normalized estimates of the generalized weights of the variables  15 x i are established from the particular case of the w (0)α = (2): αi (1) = 15 j ij j=1 j=1 αij /15, as following numbers: α 1 (1) = 0.3940; α 2 (1) = 0.2327; α 3 (1) = 0.2163; α 4 (1) = 0.1570. It is easy to see that the condition (5) for the 1st approximation of the group estimates of the generalized weights of the to the next iteration step is carried out using variables x i is not satisfied. So, the transition   nd the normalizing factor η(1): η(1) = 4i=1 15 j=1 αi (1)αij = 4.2123. Then, for the 2 iteration step, according to (3) the experts’ competence indicators are: w1 (1) = 0.0649; w2 (1) = 0.0676; w3 (1) = 0.0650; w4 (1) = 0.0697; w5 (1) = 0.0695; w6 (1) = 0.0653; w7 (1) = 0.0721; w8 (1) = 0.0657; w9 (1) = 0.0708; w10 (1) = 0.0619; w11 (1) = 0.0648; w12 (1) = 0.0620; w13 (1) = 0.0666; w14 (1) = 0.0695; w15 (1) = 0.0646. Further, having updated indicators of expert competence, it is already possible to calculate the average estimates of the generalized  weights for groupsndof variables x i . According to (4) or more specifically: αi (2) = 15 j=1 wj (1)αij , in the 2 approximation 0.3967, α2 (2) = 0.2304, α3. (2) = 0.2181, these are the following numbers: α1 (2) =  α4. (2) = 0.1549, which also correspond to 4i=1 αi (2) = 1. . However, checking these values of the generalized weights of the MR variables for the fulfillment of condition (5) and making sure that it is not satisfied again: max{|αi (2) − αi (1)|} = 0.00268 > ε, according to (4) we proceed to the calculation of the factor: η(2) = 4.2208. As a result, based on (3) the updated expert competence indicators for the 3rd step are the corresponding numbers: w1 (2) = 0.0649; w2 (2) = 0.0677; w3 (2) = 0.0650; w4 (2) = 0.0698; w5 (2) = 0.0695; w6 (2) = 0.0653; w7 (2) = 0.0724; w8 (2) = 0.0657; w9 (2) = 0.0708; w10 (2) = 0.0616; w11 (2) = 0.0648; w12 (2) = 0.0618; w13 (2) = 0.0665; w14 (2) = 0.0697; w15 (2) = 0.0646. The average estimates of the generalized weights for groups of variables x i (i = 1 ÷ 4) in the 3rd approximation are established from the particular case of formula (4): α1 (3) = 0.3968; α2 (3) = 0.2302; α3 (3)= 0.2183; α4 (3) = 0.1547, which satisfy requirement  4 i=1 αi (3) = 1. Moreover, the obtained values ensure the fulfillment of conditions (5), which is confirmed by the following: max{|α i (3)-α i (2)|} = 0.00014 < ε. Thus, the values of group estimates in the 3rd approximation {α 1 (3), …, α 4 (3)} will be considered as generalized weights of variables x i that affect the total level of MR. The method of expert assessments involves discussing the influence of factors x i (i = 1 ÷ 4) on the total level of MR for the particular CB “X”. Thus, each expert is invited to individually assess the risks by factors x i in terms of their impact on the level of MR based on the following five-point scoring system: 5 – non- existent risk; 4 – risky situation is unlikely to occur; 3 – nothing definite can be said about the possibility of risk; 2 – risky situation is likely to occur; 1 – a risky situation will surely come. Further, expert assessments of risk situations

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are analyzed for their consistency (or inconsistency) according to the rule: the maximum allowable difference between two expert opinions on any type of risk relative to x i (i = 1 ÷ 4) should not exceed 3. This rule allows to filter out unacceptable deviations in expert assessments of the probability of risk occurrence for the separate MR factor. The determination of the weighted MR index, ranging from 0 to 100, is carried out using the formula: 4 i=1 αi ei R= × 100, (6) 4 max{ i=1 αi ei } i

where α i is the specific weight of the variable x i as a component of the MR; ei is the consolidated expert evaluation of the probability of the risk situation occurring according to the x i on a five-point system. At the same time, the minimum index means the maximum risk, and vice versa, the maximum index means the minimum risk. Suppose that based on the analysis of the data presented in Table 1, banking operations subject to market risks, assets and liabilities of CB “X” by type of currency, as well as summary data on its open currency positions, relatively consistent expert assessments of risk situations by components MR according to the above-accepted five-point system. These expert estimates are summarized in Table 4, which also shows the aggregated MR indexes and Average Score (AS) obtained using formula (6). Table 4. Expert description of MR according to a five-point scoring system. Expert Weights of MR variables

MR Expert Weights of MR variables MR index α 1 (3) α 2 (3) α 3 (3) α 4 (3) α 1 (3) α 2 (3) α 3 (3) α 4 (3) index

01

4

3

2

3

68.69 09

5

4

4

2

88.34

02

5

4

3

2

83.62 10

3

5

2

4

73.42

03

3

5

4

3

79.51 11

4

4

4

3

83.10

04

4

4

4

4

86.45 12

5

3

3

2

78.64

05

5

3

3

3

81.99 13

4

4

4

4

86.45

06

3

4

2

3

65.10 14

4

4

3

3

78.39

07

3

4

4

4

77.87 15

3

5

2

2

66.73

08

4

5

3

3

83.36 AS

3.93

4.07

3.13

3.00

78.78

4 MR Estimation Using the Maximin Convolution Method The expert assessments of risk situations presented in Table 4 for the components x i (i = 1 ÷ 4) will be considered as alternative solutions of the problem of assessing the total level of MR. In this case, CB “X” must make a rational choice of one of the 10 alternative expert solutions. The rational choice is due to the desire of the CB management to obtain a

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solution characterized by acceptable estimates for all components of the MR. At the same time, the significance of risk situations by components x i when assessing the total level of MR, established according to the above method in the form of weights α i (i = 1 ÷ 4), are acceptable for CB. Then, assuming expert assessments of risk situations for the MR components in the form of alternative solutions ej ( j = 1 ÷ 10) (see Table 4), we denote the establishment of the aggregate level of MR as a multi-criteria assessment procedure, where the variables x i are the qualitative evaluation criteria. Then we reflect each of these criteria as a fuzzy subset of a finite set of evaluated alternative expert decisions {e1 , e2 , …, e10 } in the form of Xi = {μXi (e1 )/e1 , μXi (e2 )/e2 , .., μXi (e10 )/e10 }, where μXi (ej ) ( j = 1 ÷ 10) is the degree of belonging the expert decision ej relative to the risk situation for the component x i to the fuzzy set X i . As an appropriate membership function the function of the Gaussian type is chosen: μXi (ej ) = exp[−(eji − 5)2 /σi2 ], where eji is the assessment of the j-th expert of the risk situation for the i-th component of the MR on a five-point scale for its compliance with the level as “non-existent”; σ i is the standard deviation chosen as the same for all cases of fuzzification as equal to 2. Then, assuming the MR components x i to be linguistic variables, one of their values – the term “non- existent risk” can be reflected in the form of the fuzzy sets X i : X1={0.7788/e1, 1/e2, 0.3679/e3, 0.7788/e4, 1/e5, 0.3679/e6, 0.3679/e7, 0.7788/e8, 1/e9, 0.3679/e10}; X2={0.3679/e1, 0.7788/e2, 1/e3, 0.7788/e4, 0.3679/e5, 0.7788/e6, 0.7788/e7, 1/e8, 0.7788/e9, 1/e10}; X3={0.1054/e1, 0.3679/e2, 0.7788/e3, 0.7788/e4, 0.3679/e5, …, 0.3679/e8, 0.7788/e9, 0.1054/e10}; X4={0.3679/e1, 0.1054/e2, 0.3679/e3, 0.7788/e4, 0.3679/e5, …, 0.3679/e8, 0.1054/e9, 0.7788/e10}.

To identify the most rational solution relative to the MR level, the use of the fuzzy maximin convolution method involves the construction of a set of so-called optimal alternatives [6]. This procedure is carried out by finding the intersection of fuzzy sets X i : X = X1 ∩ X2 ∩ X3 ∩ X4 , containing alternative expert solutions according to the NON-EXISTENT RISK criterion. According to the maximin convolution method, the most rational solution is the alternative e* , which has the largest value of the membership function: μX (e∗ ) = max{μX (ej )}. According to [7], the fuzzy set intersection operation corresponds to the choice of the minimum value for the alternative ej ( j = 1 ÷ 10): μX (ej ) = min{μXi (ej )}. In our case, the set of optimal alternatives is formed as follows: A={min{0.7788, 0.3679, 0.1054, 0.3679}; min{1.0000, 0.7788, 0.3679, min{0.3679, 1.0000, 0.7788, 0.3679}; min{0.7788, 0.7788, 0.7788, min{1.0000, 0.3679, 0.3679, 0.3679}; min{0.3679, 0.7788, 0.1054, min{0.3679, 0.7788, 0.7788, 0.7788}; min{0.7788, 1.0000, 0.3679, min{1.0000, 0.7788, 0.7788, 0.1054}; min{0.3679, 1.0000, 0.1054, 0.7788}.

0.1054}; 0.7788}; 0.3679}; 0.3679};

Then the resulting priority vector of alternative expert decisions relative to the risk situations has the following form [6]: max{μX (ej )} = max{0.1054, 0.1054, 0.3679, 0.7888, 0.3679, 0.1054, 0.3679, 0.3679, 0.1054, 0.1054}.

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Regarding the estimating of the MR level, the best alternative solution is the estimate of the 4-th expert (e4 ), which corresponds to the maximum value of 0.7788. The remaining expert decisions are divided into two groups, characterized by the values: • 0.3679, which corresponds to expert solutions e3 , e5 , e7 , e8 ; • 0.1054, which corresponds to expert solutions e1 , e2 , e6 , e9 , e10 . MR components have different importance, so their contribution to the overall solution can be represented as a weighted intersection: X = X1α1 ∩ X2α2 ∩ X3α3 ∩ X4α4 , where α 1 = 0.39676, α 2 = 0.23023, α 3 = 0.21826, α 4 = 0.15475 are the weights of the corresponding criteria for assessing the total level of MR. Then, we have:

A={min{0.77880.39676, 0.36790.23023, 0.10540.21826, 0.36790.15475}; min{1.00000.39676, 0.77880.23023, 0.36790.21826, 0.10540.15475}; min{0.36790.39676, 1.00000.23023, 0.77880.21826, 0.36790.15475}; min{0.77880.39676, 0.77880.23023, 0.77880.21826, 0.77880.15475}; 0.39676 0.23023 0.21826 0.15475 min{1.0000 , 0.3679 , 0.3679 , 0.3679 }; min{0.36790.39676, 0.77880.23023, 0.10540.21826, 0.36790.15475}; min{0.36790.39676, 0.77880.23023, 0.77880.21826, 0.77880.15475}; min{0.77880.39676, 1.00000.23023, 0.36790.21826, 0.36790.15475}; 0.39676 0.23023 0.21826 0.15475 min{1.0000 , 0.7788 , 0.7788 , 0.1054 }; min{0.36790.39676, 0.23023 0.21826 0.15475 1.0000 , 0.1054 , 0.7788 }.

The resulting priority vector of alternative expert decisions relative to the risk situations has the following form: max{μX (ej )} = max{0.6120, 0.7060, 0.6725, 0.9056, 0.7944, 0.6120, 0.6725, 0.8039, 0.7060, 0.6120}, where the best expert decision relative to the MR level is also the expert’s conclusion e4 , which corresponds to the value of 0.9056. Further, expert estimates are ranked in descending order of the corresponding values of the membership functions: e8 → 0.8039, e5 → 0.7944, (e2 , e9 ) → 0.7060, (e3 , e7 ) → 0.6725, (e1 , e6 , e10 ) → 0.6120.

5 Conclusion At the current level of development of information technologies, the tools for regulating MR are distinguished by their diversity. Many banks use international experience, but approaches to managing MR are constantly updated based on their own resources. The article proposes an alternative approach to the assessment of MR based on expert knowledge. For a hypothetical CB “X”, expert evaluations of risk situations for all four MR factors are considered and the method for identifying their generalized weights is adapted, taking into account their relative impact on the total level of MR. As a criterion for a weighted assessment of the MR, the index is used that aggregates five-point expert assessments regarding risk situations by MR components. In Table 4, using criterion (6), for the CB “X”, the unified MR index is established as the weighted value R = 78.78 obtained by expert e4 based on the five-point risk gradation scale. A similar result regarding the choice of the best expert decision was obtained using the fuzzy maximin convolution method, both in the case of the same importance of the MR components, and in the case of their different importance.

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References 1. Risk management rules in banks (Central Bank of the Republic of Azerbaijan). https://contin ent-online.com/Document/?doc_id=30951076#pos=0;38 2. International convergence of capital measurement and capital standards: a revised framework. Basel Committee on Banking Supervision, June (2004) 3. Agajanov, J.: Organization of risk management in a commercial bank based on a fuzzy cognitive map. Math. Mach. Syst. 3, 77–90 (2022) 4. Lin, A., Hedayat, S., Wu, W.: Statistical Tools for Measuring Agreement. Springer, New York (2012) 5. Mardanov, M., Rzayev, R.: One approach to multi-criteria evaluation of alternatives in the logical basis of neural networks. In: Aliev, R.A., Kacprzyk, J., Pedrycz, W., Jamshidi, M., Sadikoglu, F.M. (eds.) ICAFS 2018. AISC, vol. 896, pp. 279–287. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-04164-9_38 6. Andreichenkov, A.V, Andreichenkova, O.N.: Analysis, synthesis, planning decisions in the economy. Finance and Statistics, Moscow (2000) (in Russian) 7. Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8(3), 199–249 (1975)

Decomposed Fuzzy Set-Based Failure Mode and Effects Analysis for Occupational Health and Safety Risk Assessment Selcuk Cebi(B) Department of Industrial Engineering, Yildiz Technical University, 34349 Besiktas ˙Istanbul, Turkey [email protected]

Abstract. The Failure Mode and Effects Analysis (FMEA) is utilized to evaluate the level of risk and determine appropriate measures to reduce the magnitude of those risks. In order to determine the risk level in a work environment, the method employs three parameters; probability, severity, and detectability. However, evaluating these parameters accurately is not an easy task since it involves randomness in the estimation of these parameters. For instance, when an accident occurs, it could result in minor injuries as well as fatalities, making it challenging to predict the outcome of the situation and creating uncertainty in the evaluation process. To address this uncertainty, a new approach is proposed in this study that utilizes decomposed fuzzy sets. Decomposed fuzzy sets are a type of fuzzy set extension that takes into account evaluations made from both a pessimistic and an optimistic perspective. In this study, the FMEA method is first extended to incorporate decomposed fuzzy sets, and the proposed approach is then applied to the production of chemicals and chemical products. Based on the results, the risks associated with chemical spills and releases, exposure to toxic or hazardous chemicals, and machinery and equipment failures are found to be higher compared to other failures. Keywords: Fuzzy sets and extensions · Decomposed fuzzy sets · FMEA · OHS

1 Introduction Risk assessment involves recognizing and assessing potential dangers in the work environment, estimating the possibility and consequence of harm that may arise from these risks, and choosing suitable measures to manage or eradicate them. Regarding occupational health and safety, risk assessment is a vital aspect of any competent safety program. It enables employers to identify workplace hazards, assess the risk of harm to employees, and determine the appropriate control measures to reduce the risk of injury or illness. The risk assessment procedure usually includes several phases, such as recognizing hazards, analyzing risks, controlling and mitigating risks, and reviewing and revising the assessment. During the risk analysis phase, the level of risk to workers is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 776–786, 2023. https://doi.org/10.1007/978-3-031-39774-5_85

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determined by evaluating the likelihood and severity of harm from each identified hazard. However, this is not always a straightforward process as assessing the probability and consequence of harm from hazards can be challenging due to the lack of statistical data and reliance on expert opinions. Therefore, subjectiveness is often a factor in the assessment process [1, 2]. Fuzzy set-based risk assessment methods are commonly used in the literature to deal with subjectiveness, but these studies don’t address bias in human judgments. Experts may provide different evaluations depending on their optimistic or pessimistic perspectives. Additionally, the consequence of harm can vary significantly, leading to uncertainty. For instance, falling from a height can result in either injury or fatality, making it challenging to determine the most likely outcome [1–3]. To tackle these uncertainties and address the subjectivity issue, this study proposes using the decomposed fuzzy set as a new extension of intuitionistic fuzzy sets. This method takes into account the diverse evaluations of the same event made from optimistic and pessimistic viewpoints. Additionally, this study aims to apply the DFS-based FMEA method to develop a more effective risk assessment approach. The following sections of this paper are organized as follows: Sect. 2 provides a literature review of the FMEA method. Section 3 offers an introduction to decomposed fuzzy sets and the FMEA method. The proposed method is explained in Sect. 4, followed by an example in Sect. 5. Finally, concluding remarks are presented in the Conclusion section.

2 Literature Review Evaluating risks associated with occupational health and safety is a common practice in the industry, and the FMEA method is widely used as one of the techniques for this purpose. FMEA is also commonly used in engineering, manufacturing, and quality management to proactively identify potential problems and take corrective measures to prevent or minimize their impact. FMEA is also used to evaluate occupational health and safety failures. Some theoretical developments on FMEA can be illustrated as follows; Zahed et al. [4] evaluated the critical risks and their effects on each other in a tunnel construction project using the FMEA method, as well as the William Fine method to study health hazards affecting employees. Karamustafa and Cebi [1] proposed a neutrosophic set-based FMEA to handle inconsistencies, subjectivities, and indecisions in the risk assessment process. The proposed method uses membership degrees of truth, indeterminacy, and falsity to account for uncertainties in expert assessments of risk. Azizi et al. [5] employed a variety of methods including document analysis, expert interviews, and the Delphi technique and questionnaire to identify 17 risks across four categories: environmental, health, safety, and occupational. The FMEA method was utilized to prioritize these risks. 6.Saranjam et al. [6] compared four methods (ETBA, HEMP, HOSHRA, and AHP) for assessing hazards at a hospital in Iran using the FMEA method. The study found that the hazard risks identified by ETBA were different from FMEA, while HEMP and HOSHRA had similar results to FMEA for certain types of hazards. Cavaignac et al. [7] proposed an application of FMEA with an occupational safety approach in the maintenance of a water supply network in a medium-sized city in Brazil. The results show that FMEA was an effective tool for risk prioritization in

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work processes and can identify and prioritize risks based on a preliminary risk index obtained. Jahangoshai Rezaee et al. [8] developed a hybrid approach based on linguistic FMEA, Fuzzy Inference System (FIS), and Fuzzy Data Envelopment Analysis (DEA) to prioritize HSE risks in organizations. Mete [9] proposed a new risk assessment approach, called the FMEA-based AHP-MOORA integrated approach under Pythagorean fuzzy sets, for assessing occupational risks in a natural gas pipeline construction project. Mutlu and Altuntas [10] proposed an approach that integrates the FMEA method with the fault tree analysis (FTA) method and belief in the fuzzy probability estimations of time (BIFPET) algorithm to improve the accuracy and reliability of risk analysis. The approach is applied to assess potential risks for a finishing process in the fabric dyeing department of a textile company. Sadidi et al. [11] evaluated tower crane safety indicators using the FMEA risk assessment model. A total of 30 failure modes were analyzed, with the base sections, trolley body, and jib retaining split pins having the highest RPNs due to the high work height and heavy weight of the loads. Lotfolahzadeh et al. [12] developed adapted FMEA tables that evaluate risks and calculate insurance rates based on the actual conditions of a cement factory. Data on machinery faults and human accidents were collected and used to design adapted FMEA insurance tables. Sang et al. [13] proposed an FMEA approach for risk analysis and assessment in rainfed lowland rice production in Sarawak, Borneo. The study focused on environmental issues, as well as health and safety risks for farmers and consumers. The authors developed a fuzzy FMEA model using data and information from experienced farmers and identify musculoskeletal disorders as the most significant occupational health hazard. The literature review shows that FMEA has been the subject of quite different applications in the field of occupational health and safety. Furthermore, the use of FMEA in fuzzy environments has also been on the rise, as evidenced by the examples provided. Figure 1 provides a comparison between the FMEA and Fuzzy FMEA studies that have been published in the literature. According to the figure, the number of fuzzy FMEA studies has increased in parallel with classical FMEA studies and the gap between them has almost closed in the 2020s.

Fig. 1. A comparison between studies on FMEA and Fuzzy FMEA

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3 Preliminaries 3.1 Decomposed Fuzzy Sets (DFS) The DFS approach was proposed to deal with inconsistencies in expert preferences by incorporating optimistic (O) and pessimistic (P) viewpoints. This method evaluates uncertainty in decision-making by examining the decision-maker’s responses to questions about functional and dysfunctional scenarios to measure consistency. The decision maker’s inconsistency is expressed as I = 1 − |O − P|. The following are the fundamental mathematical concepts and operations of DFS [14, 15]. Definition. Consider a universe of discourse X. A Decomposed Fuzzy Set (DFS)  denoted by A˜  is a structure the form  of     O O ∈ X where the functions , A˜ = x, O μ ˜ (x), ϑ ˜ (x) , P μP˜ (x), ϑ P (x) x A A A A˜ μA˜ (x) : X → [0, 1], νA˜ (x) : X → [0, 1] represent the degree of membership and non-membership of x in the optimistic and pessimistic sets, O and P, respectively. It must O P P satisfy the conditions 0 < μO ˜ (x) + ϑ ˜ (x) ≤ 1, 0 < μ ˜ (x) + ϑ ˜ (x) ≤ 1. The A

A

A

A

P 2 inconsistency of the judgment is represented by CIA = 1 − (((μO ˜ (x) − ϑ˜ (x)) + I

I

(x))2 + (1 − μO (x) − ϑ˜O (x))2 + (1 − μP (x) − ϑ˜P (x))2 )/2) 2 where 0 (ϑ˜O (x) − μP I I˜ I˜ I I˜ I ≤ CIA ≤ 1 and 0 ≤ μO (x) + ϑ O (x) + μP (x) + ϑ P (x) ≤ 2. A DFS A˜ has maximum A˜

A˜

A˜

1

A˜

consistency when CIA = 1 and a DFS A˜ has maximum inconsistency if CIA = 0 [14, 15]. Definition 3.2. Consider DF numbers A˜ = {O(a, b), P(c, d )}, α˜ 1 = {O(a1 , b1 ), P(c1 , d1 )}, and B˜ = {O(a2 , b2 ), P(c2 , d2 )}. The addition, multiplication, and power for these numbers are given by the following expressions, respectively [14, 15]:  

a1 + a2 − 2a1 a2 b1 b2 A˜ ⊕ B˜ = O , P(c1 + c2 − c1 c2 , d1 d2 ) (1) , 1 − a1 a2 b1 + b2 − b1 b2

  d1 + d2 − 2d1 d2 c1 c2 A˜ ⊗ α˜ 2 = O(a1 a2 , b1 + b2 − b1 b2 ), P (2) , c1 + c2 − c1 c2 1 − d1 d2  

 λa b λ · A˜ = O , P 1 − (1 − c)λ , d λ for λ > 0 (3) , (λ − 1)a + 1 λ − (λ − 1)b

    λd c ˜Aλ = O aλ , 1 − (1 − b)λ , P , for λ > 0 (4) λ − (λ − 1)c (λ − 1)d + 1

3.2 Failure Mode and Effect Analysis (FMEA) FMEA is a structured approach to identify and analyze potential failures or risks that may occur in a product, process, or system, and their potential effects or consequences [16]. FMEA which involves a multidisciplinary team of experts typically involves identifying potential failure modes and their causes, assessing the severity of their effects, and

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determining the likelihood of their occurrence. Based on this analysis, appropriate actions can be taken to prevent or mitigate the identified risks. The FMEA process typically begins with defining the scope of the analysis, identifying the relevant inputs and outputs, and determining components of the product or process being analyzed. The team then brainstorms potential failure modes, and for each failure mode, they identify its potential causes, the effects or consequences on the product or process, and the likelihood of its occurrence. By considering the severity, likelihood, and detectability of potential failure modes, FMEA helps organizations prioritize their risk mitigation efforts and allocate their resources effectively. The parameters, severity, detectability, and likelihood are typically determined on a scale from 1 to 10, with 10 representing the most severe effects, hard to detection of failure, and the highest likelihood of occurrence. The severity (S), likelihood (L), and detectability (D) values are typically multiplied together to calculate a Risk Priority Number (RPN), which can be used to rank the potential failure modes [16]. R=L×S ×D

(5)

4 Proposed Approach: Decomposed Fuzzy FMEA The main objective of this section is to extend FMEA to the Decompose Fuzzy Sets (DFS) which is a new extension of intuitionistic fuzzy sets. The steps of the decomposed fuzzy FMEA (DF-FMEA) method are as follows: Phase 1. Pre-analysis processes: This phase consists of the identification of the process that will be analyzed, the establishment risk assessment team, the identification of potential hazards, and the gathering data. Step 1.1. Assemble the team: Form a team of experts with relevant knowledge and experience to conduct the FMEA since the effectiveness of the risk analysis heavily relies on the knowledge and proficiency of the designated team. Step 1.2. Identify potential failure modes and effects: All potential failure modes that could occur in the process or system should be identified. Then, the effects of each failure mode on the system or process should be determined. Step 1.3. Assign severity rankings: A severity ranking is assigned to each failure mode based on the impact of its effects. Following functional (FQ) and dysfunctional questions (DQ) should be answered by the risk assessment team for each failure mode: – FQ: What are the most and the least likely expected severity of the failure mode before implementing the control measure? – DQ: What are the most and the least likely expected severity of the failure mode after implementing the control measure?   O The answer of each expert is defined by Cik = O μO ˜S (x), ϑS˜ (x) ,   P (x) where i denotes the number of failure mode (i = 1, . . . , m), and k P μP ϑ (x), ˜ ˜ S

S

represents the number of experts k = 1, . . . , K, μO (x) and ϑ O (x) presents the memS˜ S˜ bership and non-membership degrees of the most and the least likely expected severity,

Decomposed Fuzzy Set-Based Failure Mode and Effects Analysis p

781

p

respectively before suitable control measures while μ ˜ (x) and ϑ ˜ (x) denotes the memS S bership and non-membership degrees of the most and the least likely expected severity after suitable control measures. Step 1.4. Assign likelihood rankings: A likelihood value for each failure mode based on the probability of occurrence is assigned considering potential causes for each failure mode. Following FQ and DQ should be answered by the risk assessment team for each failure mode: – FQ: What is the likelihood that the failure mode will occur and not occur before implementing the control measure? – DQ: What is the likelihood that the failure mode will occur and not occur after implementing the control measure?   O (x) , = O μO ϑ The answer of each expert is defined by Lki (x), L˜ L˜   P (x) where i denotes the number of risk (i = 1, . . . , m), and k represents P μP ϑ (x), ˜ ˜ L

L

the number of experts k = 1, . . . , K, μO (x) and ϑ ˜O (x) presents the membership and L˜ L non-membership degrees of the likelihood that the failure mode will occur and not occur, p p respectively before implementing control measures while μ ˜ (x) and ϑ ˜ (x) denotes the L L membership and non-membership degrees that the failure mode will occur and not occur respectively after implementing control measure. Step 1.5. Assign detectability rankings: A detectability value for each failure mode based on the likelihood of detection is assigned considering the existing controls that are in place to prevent or detect each failure mode. Detectability is evaluated based on the number of repetitions of the task and whether the activity is routine or not. The following functional and dysfunctional questions are answered by the risk assessment team for each harm for the frequency of the harm: – FQ: What are the most and the least likely detectability of the failure mode during routine operation? – DQ: What are the most and the least likely detectability of the failure mode during the control phase?   O (x) , ϑ The answer of each expert is defined by Fik = O μO (x), D ˜ D   P P P μ ˜ (x), ϑ ˜ (x) where i denotes the number of risk (i = 1, . . . , m), and k represents D

D

O (x) presents the membership and the number of experts k = 1, . . . , K, μO ˜ (x) and ϑD ˜ D non-membership degrees of the most and the least likely detectability of the failure mode p p during routine operation, respectively while μ ˜ (x) and ϑ ˜ (x) denotes the membership D D and non-membership degrees of the most and the least likely detectability of the failure mode during the control phase, respectively. During the evaluation process, the decision maker assigns a numerical value between 0–1 to each question, which represents their judgment, instead of using a linguistic scale. Phase 2. Analysis: In this phase aggregated decision matrix is obtained and the risk magnitude for each risk is obtained.

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Step 2.1. Aggregation of expert evaluations: Calculate the aggregated decision matrix, using the Decomposed Weighted Arithmetic Mean (DWAM) formula [14, 15]. ⎫ ⎧  n n λi ai bi ⎪ ⎪ i=1 i=1 ⎪ ⎬ ⎨ O 1+n λ a − ai , n bn−1 λ (1−b )+n b , ⎪ i i=1 i i n i=1 i i i=1 i  DWAM (α˜ 1 , α˜ 2 . . . . . . ., α˜ n ) = n n   λi ⎪ ⎪ ⎪ ⎪ P 1 − (1 − ci )λi , di ⎭ ⎩ i=1

i=1

(6) where λi = (λ1 , λ2 . . . . . . ., λk ); λi ∈ [0, 1],

n  i=1

λi = 1 and α˜ i = Lki , Cik , Fik

Step 2.2. Calculate the Risk Priority Number (RPN): The RPN for each failure mode by multiplying the severity, likelihood, and detectability rankings is obtained by Eq. 7 RPNi = LDWAM ⊗ SiDWAM ⊗ DiDWAM i

(7)

Let α˜ 1 = {O(a1 , b1 ), P(c1 , d1 )}, and α˜ 2 = {O(a2 , b2 ), P(c2 , d2 )} be a decomposed fuzzy set. If none of them has maximum consistency or maximum inconsistency, the multiplication operator is defined as follows [14, 15].

  d1 + d2 − 2d1 d2 c1 c2 (8) α˜ 1 ⊗ α˜ 2 = O(a1 a2 , b1 + b2 − b1 b2 ), P , c1 + c2 − c1 c2 1 − d1 d2 Since the calculated values are in DFS, the score index given in Eq. 9 is used to compare obtained results. The score index (SI) of DFN (α˜ = {O(a, b), P(c, d )}) is proposed as follows [14, 15]:  ˜ (a+b−c+d ).CI (α) , SI (α) ˜ >0 2.k SI (α) ˜ = (9) 0, SI (α) ˜ ≤0 where k is the linguistic scale multiplier. The value k is obtained by Eq. 10.     a + b − c + d  .CI α (10) k= 2      = O a , b , P c , d  is the maximum value for the used linguistic scale. where α The consistency index (CI) of decomposed fuzzy number (α˜ = {O(a, b), P(c, d )}) is defined as [14, 15]; ⎛ ⎞ 2 2 2 2 + − c) + − a − b) + − c − d − d (b (1 (1 ) (a ) ⎠, CI (α) ˜ = 1 − Iα = 1 − ⎝ 2 (11) 0 ≤ CI (α) ˜ ≤1 Phase 3. Prioritize and monitor corrective actions: In this phase, failure modes are ranked from the most important to the least important. Then, the corrective actions

Decomposed Fuzzy Set-Based Failure Mode and Effects Analysis

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are planned and monitored whether they have reduced or eliminated the identified failure modes. Step 3.1. Prioritize actions: The corrective actions based on RPN scores are prioritized and failure modes that need to be addressed first are determined. Step 3.2. Implement and monitor corrective actions: The corrective actions are implemented, and their effectiveness is monitored to ensure that they have reduced or eliminated the identified failure modes.

5 Case Study: Risk Assessment in Electric Arc Welding Job Shop In this section, we aim to illustrate the proposed approach by analyzing the production of chemicals and chemical products. The production of chemicals and chemical products poses various health and safety failures to workers involved in the production process. These failures can include ergonomics (F1), noise exposure (F2), lack of per-sonal protective equipment (F3), machinery and equipment (F4), chemical spills and releases (F5), exposure to toxic or hazardous chemicals (F6), and fire and explosion (F7). Each of these failures can have serious health and safety consequences for workers, nearby communities, and the environment. Therefore, it is crucial for a business to implement effective health and safety measures to protect their workers and the environment. Phase 1. Pre-analysis processes: Phase 1 of the risk assessment process involves three important steps: forming a risk assessment team, identifying potential hazards, and collecting relevant data. For this study, the risk assessment team comprises three specialists in occupational health and safety and evaluates the failures in the process by using the linguistic scale given in Table 1. The expert preferences are given in Table 2. Table 1. Linguistic scale used for the evaluation µ

ϑ

0.8

Symbol

Likelihood (L)

Detectability (D)

Severity (C)

0.2

Almost certain

Not detectable

Catastrophic

0.65

0.35

Likely

Very low

Critical

0.5

0.5

Possible

Low

Serious

0.35

0.65

Unlikely

Moderate

Marginal

0.2

0.8

Rare

High

Negligible

Phase 2. Analysis: The preferences given in Table 2 are aggregated by using Eq. 6 (Table 3). Then, the risk magnitude of each failure is obtained by using Eqs. 7 and 8 and to rank the obtained risk Eqs. 9–11 are utilized. The obtained results are given in Table 3.

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S. Cebi Table 2. Expert preferences on the determined risks

Table 3. Aggregations of expert preferences and risk magnitudes Likelihood (L)

Severity (S)

Detectability (D)

Risk Magnitude

CI

F1

((0.5,0.5),(0.2,0.8))

((0.65,0.35),(0.2,0.8))

((0.8,0.2),(0.5,0.5))

((0.26,0.74),(0.13,0.87))

0.39

SI 0.42

F2

((0.5,0.5),(0.2,0.8))

((0.5,0.5),(0.5,0.5))

((0.8,0.2),(0.8,0.2))

((0.2,0.8),(0.19,0.81))

0.39

0.40

F3

((0.65,0.35),(0.65,0.35))

((0.65,0.35),(0.2,0.8))

((0.8,0.2),(0.8,0.2))

((0.34,0.66),(0.21,0.79))

0.54

0.54

F4

((0.5,0.5),(0.5,0.5))

((0.65,0.35),(0.5,0.5))

((0.8,0.2),(0.35,0.65))

((0.36,0.6),(0.2,0.8))

0.58

0.57

F5

((0.8,0.2),(0.2,0.8))

((0.8,0.2),(0.2,0.8))

((0.8,0.2),(0.5,0.5))

((0.39,0.58),(0.21,0.79))

0.62

0.60

F6

((0.65,0.35),(0.2,0.8))

((0.65,0.35),(0.2,0.8))

((0.8,0.2),(0.35,0.65))

((0.42,0.55),(0.11,0.89))

0.54

0.59

F7

((0.5,0.5),(0.5,0.5))

((0.65,0.35),(0.35,0.65))

((0.65,0.35),(0.5,0.5))

((0.34,0.61),(0.24,0.76))

0.60

0.56

Phase 3. Chemical spills and releases (F5) and exposure to toxic or hazardous chemicals (F6) are identified as the highest-risk failures, followed by machinery and equipment (F4), fire and explosion (F7), lack of personal protective equipment (F3), noise exposure (F2), and ergonomics (F1) failures, respectively. If the problem were analyzed using the traditional FMEA method, F5 would be classified as high risk, F2 as low risk, and the remaining failures as moderate risk. However, the proposed method evaluates both the current situation and the precautions taken using functional and dysfunctional questions, revealing differences in the level of risk associated with each failure. Therefore, it is evident that not all failures carry the same level of risk, and the proposed method can provide a more comprehensive risk assessment.

6 Conclusion The FMEA method is commonly utilized to evaluate and reduce risks in the workplace, utilizing factors such as likelihood, severity, and detectability to determine the level of risk associated with a particular hazard. However, it can be challenging to accurately assess these factors due to the wide range of possible outcomes in the event of an accident, ranging from minor injuries to fatal incidents. To address this issue, a new approach involving decomposed fuzzy sets has been proposed, allowing for both pessimistic and

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optimistic evaluations to be considered. This study extends the FMEA method to incorporate decomposed fuzzy sets and applies it to the production of chemicals and chemical products. The findings indicate that chemical spills and releases and exposure to toxic or hazardous chemicals, and machinery and equipment failures present a greater risk than the others. In future studies, fuzzy inference based on decomposed sets, which simulates human reasoning, could be integrated into the FMEA method to determine the magnitude of risk more accurately.

References 1. Karamusta, M., Cebi, S.: A new model for the occupational health and safety risk assessment process: Neutrosophic FMEA. J. Fac. Eng. Archit. Gazi Univ. 38(1), 29–43 (2023) 2. Karamustafa, M., Cebi, S.: Extension of safety and critical effect analysis to Neutrosophic sets for the evaluation of occupational risks. Appl. Soft Comput. 110, 107719 (2021) 3. Cinar, U., Cebi, S.: A novel approach to assess occupational risks and prevention of hazards: the house of safety prevention. J. Intell. Fuzzy Syst. 42(1), 517–528 (2022) 4. Zahed, M.A., Seidi, F., Salehi, S., Pardakhti, A.: Simultaneous assessment of health, safety, and environmental risks using William Fine and FMEA methods based on OHSAS 18001: 2007 standard in the Alborz tunnel. Iran, Geomechanik und Tunnelbau 16(1), 103–113 (2023) 5. Azizi, H., Agha, M.M.A., Azadbakht, B., Samadyar, H.: Identification and assessment of health, safety and environmental risk factors of chemical industry using Delphi and FMEA methods (a case study). Anthropogenic Pollution 6(2), 39–47 (2022) 6. Saranjam, B., Naghizadeh, L., Rahimi, E., Etemad, M., Babaei-Pouya, A.: Assessment of Health and Safety Hazards in Hospitals using five methods and comparing the results with the FMEA method. Pakistan J. Med. Health Sci. 14(2), 813–817 (2020) 7. Cavaignac, A.L.O., Uchoa, J.G.L., Dos Santos, H.F.O.: Risk analysis and prioritization in water supply network maintenance works through the failure modes and effects analysis: occupational safety FMEA application. Brazilian J. Oper. Prod. Manage. 17(1), e2020887 (2020) 8. Jahangoshai Rezaee, M., Yousefi, S., Eshkevari, M., Valipour, M., Saberi, M.: Risk analysis of health, safety and environment in chemical industry integrating linguistic FMEA, fuzzy inference system and fuzzy DEA. Stoch. Env. Res. Risk Assess. 34(1), 201–218 (2019). https://doi.org/10.1007/s00477-019-01754-3 9. Mete, S.: Assessing occupational risks in pipeline construction using FMEA-based AHPMOORA integrated approach under Pythagorean fuzzy environment. Hum. Ecol. Risk Assess. 25(7), 1645–1660 (2019) 10. Mutlu, N.G., Altuntas, S.: Risk analysis for occupational safety and health in the textile industry: integration of FMEA FTA, and BIFPET methods. Int. J. Ind. Ergon. 72, 222–240 (2019) 11. Sadidi, J., Gholamnia, R., Gharabagh, M.J., Mosavianasl, Z.: Original article evaluation of safety indexes of the tower crane with the FMEA model (a case study: tower cranes mounted in the city of Mashhad in 2016). Pakistan J. Med. Health Sci. 13(2), 607–612 (2009) 12. Lotfolahzadeh, A., Lavasani, M.M., Dehghani, A.: Application of failure mode effects analysis (FMEA) to calculate the insurance rate to remaining risk. Pakistan J. Med. Health Sci. 12(4), 1777–1783 (2018) 13. Sang, A.J., Tay, K.M., Lim, C.P., Nahavandi, S.: Application of a genetic-fuzzy FMEA to rainfed lowland rice production in sarawak: environmental, health, and safety perspectives. IEEE Access 6(8543779), 74628–74647 (2018)

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14. Cebi, S., Gündo˘gdu, F.K., Kahraman, C.: Operational risk analysis in business processes using decomposed fuzzy sets. J. Intell. Fuzzy Syst. 1–18,(2022). https://doi.org/10.3233/ JIFS-213385 15. Cebi, S., Gündo˘gdu, F.K., Kahraman, C.: Consideration of reciprocal judgments through decomposed fuzzy analytical hierarchy process: a case study in the pharmaceutical industry. Applied Soft Comput., 110000 (2023) 16. Stamatis, D.H.: Failure mode and effect analysis: FMEA from theory to execution. ASQC Quality Press (2003)

A Novel Risk Assessment Approach: Decomposed Fuzzy Set-Based Fine-Kinney Method Selcuk Cebi1(B)

and Palanivel Kaliyaperuma2

1 Department of Industrial Engineering, Yildiz Technical University, 34349 Besiktas ˙Istanbul,

Turkey [email protected] 2 Department of Mathematics – School of Advanced Sciences, Vellore Institute of Technology (VIT), Vellore 632014, Tamil Nadu, India

Abstract. The Fine-Kinney approach is used to assess the risk level and determine appropriate steps to mitigate risk magnitude. The method uses the probability, severity, and frequency parameters while assessing the risk magnitude in the work environment. However, it is not easy to evaluate these parameters precisely. When an accident occurs, it is possible to result in death as well as minor abrasions. This makes it difficult to predict the outcome of the situation and creates uncertainty in the evaluation. In order to address the uncertainty illustrated here, the newly proposed decomposed fuzzy sets will be used. The decomposed fuzzy set is one of the fuzzy set extensions that take into account the evaluations made from a pessimistic and optimistic point of view. Therefore, in the scope of this study, the Fine&Kinney method is firstly extended to decomposed fuzzy sets. The proposed approach is applied to the electric arc welding process to illustrate the applicability of the method. According to the findings, harmful radiation and gas exposure present a greater risk compared to electric shock, fire and explosion, and musculoskeletal disorders. Keywords: Fuzzy sets and extensions · Decomposed fuzzy sets · Fine-Kinney

1 Introduction Risk assessment is the process of identifying and evaluating potential hazards in the workplace, assessing the likelihood and severity of harm that could result from these hazards, and determining appropriate measures to control or eliminate them. In terms of occupational health and safety, risk assessment is a critical component of any effective safety program. It helps employers identify workplace hazards, evaluate the risk of harm to workers, and determine appropriate control measures to eliminate or reduce the risk of injury or illness. The risk assessment process typically involves several steps, including hazard Identification, risk analysis, risk control and risk mitigation, and reviewing and revising. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 787–797, 2023. https://doi.org/10.1007/978-3-031-39774-5_86

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In the risk analysis step, the likelihood and severity of harm from each identified hazard are assessed to determine the level of risk to workers. However, assessing the likelihood (probability) and severity (consequence) of harm from identified hazards is not always straightforward [1, 3]. Furthermore, since statistical data is lacking to make accurate assessments, the process relies on expert opinions. Therefore, the assessment process includes subjectiveness [1–3]. To cope with the subjectiveness fuzzy set-based risk assessment methods are widely used in the literature. However, none of them copes with the bias in human judgments. Experts may find different evaluations depending on the optimistic and pessimistic points of view in their evaluations. For example, when asked about the possibility of harm occurring, his answer may be different from the one he gave when asked about the possibility of not occurring. However, the answers to functional and dysfunctional questions asked for the same situation should complement each other. Another important question that must be addressed is the prediction of the harm’s severity, which can range widely. For instance, falling from a height can lead to either fatality or injury. In this context, the severity is determined based on the most likely outcome. However, the possibility of diverse outcomes introduces uncertainty. Therefore, in this study, the decomposed fuzzy set, which is presented in the literature as a new extension of the intuitionistic fuzzy sets, will be used to handle the uncertainties in expert evaluations and to evaluate the indeterminacy arising from different results of the same event. For this, this study aims to develop the DFS-based Fine&Kinney method to address these issues. The rest of this paper is organized as follows; Sect. 2 describes a literature review on the Fine&Kinney method. Section 3 presents the preliminaries on decomposed fuzzy sets and the Fine&Kinney method. The proposed approach is given in Sect. 4 and an illustrative example is handled in Sect. 5. Finally concluding remarks are given in Conclusion Section.

2 Literature Review The Fine&Kinney method is one of the widely used techniques in the industry for evaluating risks related to occupational health and safety. Recent studies on the method can be summarized as follows: Wang et al. [4] proposed a new framework for occupational risk in the natural gas pipeline construction project using the Fine&Kinney model. The proposed approach employs a prioritized weighted average operator for complex spherical fuzzy numbers (CSFNs) to take into consideration the priority levels of experts and is based on the compromise ranking of alternatives based on the distance to the ideal solution technique. To account for the influence of interacting risk factors, the Choquet integral for CSFNs is also incorporated into the framework. Gökler et al. [5] put out a brand-new hybrid risk assessment technique based on Fine&Kinney and the machine learning approach. Using a three-stage hybrid technique based on the Fine-Kinney, AHP, and MULTIMOORA methodologies to handle uncertainty in risk evaluation, Seker [6] proposed a novel risk assessment method for subway construction. Wang et al. [7] integrated the cumulative prospect theory, the weighted power average operator, and the ORESTE method with modified Fine&Kinney model to incorporate reference dependence effects, linkages between risks, and deviations in risk evaluation data. The model

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is examined using the production of automotive parts. Li et al. [8] proposed an extended Fine&Kinney framework for risk assessment by incorporating ORESTE, Choquet integral, and Probabilistic Linguistic Term Sets. Wang et al. [9] proposed a hybrid risk prioritization approach for the Fine&Kinney model by integrating the GLDS method, interval type-2 fuzzy sets (IT2FSs), and Maclaurin symmetric mean (MSM) operator. The use of the suggested method was illustrated with a numerical example of potential risks in a construction excavation. Güney and Karaman [10] conducted a risk assessment in environmental research laboratories using the Analytic Hierarchy Process (AHP) and Fine&Kinney Method (FKM). Tang et al. [11] put up a hybrid risk prioritizing strategy for the Fine&Kinney method by combining generalized TODIM, best-worst, and the interval type-2 fuzzy set. Dagsuyu et al. [12] created a novel application of the Fine&Kinney method to assess the risk level of classes using Manhattan and Euclidean distances by combining K-Means and hierarchical clustering techniques. A mediumsized textile company’s workshop was given the new method. Yang et al. [13] examined the danger of human error in collision accidents in coastal areas. A priority algorithm was used to mine association rules for maritime accidents in Zhejiang coastal area, and statistical software was used to compare and analyze the contribution value of each factor and confidence value in association rules. Ersoy et al. [14] integrated the Fine&Kinney method, grey relation analysis, and Borda count method for the evaluation risks in a marble quarry’s block excavation process. Wang et al.[15] proposed a new approach that combines the Fine&Kinney method with triangular fuzzy numbers, the MULTIMOORA method, and the Choquet integral. A case study of ballast tank maintenance is utilized to demonstrate and evaluate the efficacy of the proposed approach. In the literature, there are numerous studies in which the Fine&Kinney method is combined with different methods and applied in various fields. However, none of the studies have considered both the uncertainty arising from the decision maker’s point of view and the indeterminacy arising from the situation. it is the first time the Fine&Kinney method has been extended to DFS.

3 Preliminaries 3.1 Decomposed Fuzzy Sets (DFS) DFS is a technique introduced in the literature to address inconsistencies in expert judgments by considering both optimistic (O) and pessimistic (P) perspectives. It computes uncertainty in the decision environment by measuring the consistency of the decision maker’s judgments, using responses to functional and dysfunctional questions. The inconsistency of the preference is defined by I = 1-|O-P|. The mathematical definition and basic operations of DFS are given below [16, 17]. ˜ Definition 3.1. Let X be a universe and a decomposed fuzzy set (DFS)     A is an  of  discourse   O O P P object having the form, A˜ = x, O μ ˜ (x), ϑ ˜ (x) , P μ ˜ (x), ϑ ˜ (x) x ∈ X A A A A where the function μA˜ (x) : X → [0, 1], νA˜ (x) : X → [0, 1] are the degrees of membership, non-membership of x to O and P, respectively where O and P are optiO mistic and pessimistic sets, satisfying the conditions 0 < μO ˜ (x) + ϑ ˜ (x) ≤ 1, 0 < A

A

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O P A 2 μP˜ (x) + ϑ P ˜ (x) ≤ 1, and inconsistency in the judgment is I = (((μ˜ (x) − ϑ˜ (x)) + A

I

A

I

(ϑ˜O (x) − μP (x))2 + (1 − μO (x) − ϑ˜O (x))2 +(1 − m˜P (x) − ϑ˜P (x))2 /2) 2 where 0 I I˜ I˜ I I I ≤ I A ≤ 1 and 0 ≤ μO (x) + ϑ O (x) + μP (x) + ϑ P (x) ≤ 2. A DFS A˜ has maximum A˜

A˜

A˜

1

A˜

inconsistency if I A = 1 and a DFS A˜ has maximum consistency if I A = 0 [16, 17]. ∼ Definition 3.2. Let A˜ = {O(a, b), P(c, d )}, α 1 = {O(a1 , b1 ), P(c1 , d1 )}, and B˜ = {O(a2 , b2 ), P(c2 , d2 )} be DF numbers. The addition, multiplication, multiplication by a scalar, and power are as follows, respectively [16, 17]:

  a1 + a2 − 2a1 a2 b1 b2 (1) A˜ ⊕ B˜ = O , P(c1 + c2 − c1 c2 , d1 d2 ) , 1 − a1 a2 b1 + b2 − b1 b2

  d1 + d2 − 2d1 d2 c1 c2 ˜A ⊗ α˜ 2 = O(a1 a2 , b1 + b2 − b1 b2 ), P (2) , c1 + c2 − c1 c2 1 − d1 d2  

 λa b , P 1 − (1 − c)λ , d λ for λ > 0 (3) λ · A˜ = O , (λ − 1)a + 1 λ − (λ − 1)b

    λd c ˜Aλ = O aλ , 1 − (1 − b)λ , P , for λ > 0 (4) λ − (λ − 1)c (λ − 1)d + 1

3.2 Fine and Kinney Method The Fine&Kinney method is a quantitative risk assessment method that was developed by Fine and Kinney in 1976 to assess risks by considering risk factors such as likelihood (l), frequency (f), and consequence (c). The method utilizes Eq. 5 and a linguistic scale (given in Table 1) for each factor that provides the score and corresponding definition, and prioritizes risks based on the calculated risk score (R) [18]. R = l ×f ×c

(5)

4 Proposed Approach: Decomposed Fuzzy Fine&Kinney The main objective of this section is to extend Fine&Kinney Method to the decompose fuzzy sets which is a new extension of intuitionistic fuzzy sets. The steps of the decomposed fuzzy Fine&Kinney method are as follows: Phase 1. Pre-analysis processes: This phase consists of the establishment risk assessment team, identification of potential hazards, and gathering of data. Step 1.1. Establish the risk assessment team: A risk assessment team is formed by gathering a group of experts with diverse backgrounds in the related application area. Establish the risk assessment team is vital since the effectiveness of the risk analysis heavily relies on the knowledge and proficiency of the designated team. For this reason, it’s important to carefully select team members from various backgrounds and positions that are relevant to the specific issue at hand.

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Table 1. Linguistic scale used for the evaluation Likelihood

Value

Frequency

Value

Consequence

Value

Probably to be predicted

10

Continuous

10

Catastrophe

100

Quite probable

6

Frequent (daily)

6

Disaster

40

Unusual but probable

3

Occasional (weekly)

3

Very serious

15

Slightly probable

1

Unusual (monthly)

2

Serious

7

Extremely unlikely

0.5

Rare (a few per year)

1

Important

3

Virtually impossible

0.2

Very rare (yearly)

0.5

Noticeable

1

Almost impossible

0.1

Risk score

Risk situation

> 400

Very high risk: stop operations

(200, 400]

High risk: urgent repair

(70. 200]

Substantial risk: need correction

[20, 70]

Possible risk: caution advised

< 20

Risk; maybe acceptable

Step 1.2. Determine potential hazards: In this step, fundamental identifications have been carried out. For this intention, potential hazards that may arise from operations are identified. Then, the potential harms caused by identified hazards are determined. Step 1.3. Collect data: The likelihood (L), frequency (F), and consequence (C) of each harm are assessed by using the linguistic scale given in Table 2. The following factors are considered when determining likelihood: workflow, condition of the required equipment for the task, environmental conditions, previous accidents, employee experience, Unsafe behaviors, and current safety measures in the working environment. The following functional and dysfunctional questions are answered by the risk assessment team for each harm for the likelihood of the harm: Functional question: What is the likelihood of the harm occurring before suitable control measures have not been implemented? Dysfunctional question: What is the likelihood of harm occurring after suitable control measures have not been implemented?   O = O μO The answer of each expert is defined by Lki ˜L (x), ϑL˜ (x) ,   P (x) where i denotes the number of risk (i = 1, . . . , m), and k represents P μP ϑ (x), ˜ ˜ L

L

the number of experts k = 1, . . . , K, μO (x) and ϑ ˜O (x) presents the membership and L˜ L non-membership degrees of likelihood of the harm occurring before suitable control p p measures while μ ˜ (x) and ϑ ˜ (x) denotes the membership and non-membership degrees L L of likelihood of the harm occurring after suitable control measures. When determining consequences, the issues including the nature of the activity performed, parts of the body that can be affected, possible degree and duration of damage,

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and a number of exposed workers are considered. The following functional and dysfunctional questions are answered by the risk assessment team for each harm for the consequence of the harm: Functional question: What is the expected consequence of the harm before suitable control measures have not been implemented? Dysfunctional question: What is the expected consequence of the harm after suitable control measures have not been implemented?   O (x) , The answer of each expert is defined by Cik = O μO ϑ (x), C˜ C˜   P μP˜ (x), ϑ P where i denotes the number of risk (i = 1, . . . , m), and k repre(x) ˜ C

C

F

F

sents the number of experts k = 1, . . . , K, μO (x) and ϑ O (x) presents the membership C˜ C˜ and non-membership degrees of potential consequence and rare consequence of the p p harm occurring, respectively before suitable control measures while μ ˜ (x) and ϑ ˜ (x) C C denotes the membership and non-membership degrees of potential consequence and rare consequence of the harm occurring after suitable control measures. Frequency is evaluated based on the number of repetitions of the task and whether the activity is routine or not. The following functional and dysfunctional questions are answered by the risk assessment team for each harm for the frequency of the harm: Functional question: What is the potential to be a frequently repeated activity/behavior? Dysfunctional question: What is the potential to be a rarely occurring activity/behavior?   O (x) , ϑ The answer of each expert is defined by Fik = O μO (x), F˜ F˜   P P P μ ˜ (x), ϑ ˜ (x) where i denotes the number of risk (i = 1, . . . , m), and k represents

the number of experts k = 1, . . . , K, μO (x) and ϑ O (x) presents the membership and F˜ F˜ p non-membership degrees of weakly occurrence frequency of the task while μ ˜ (x) and F p ϑ ˜ (x) denotes the membership and non-membership degrees of annually occurrence F frequency of the harm. During the evaluation process, the decision maker assigns a numerical value between 0–1 to each question, which represents their judgment, instead of using a linguistic scale. Table 2 defines the specific judgment associated with each numerical value. Table 2. Illustrative linguistic scale used for the evaluation µ

ϑ

Likelihood (L)

Frequency (F)

Consequence (C)

0.8

0.2

Probably to be predicted

Continuous

Catastrophe

0.65

0.35

Quite probable

Frequent

Disaster

0.5

0.5

Unusual but probable

Occasional

Serious

0.35

0.65

Slightly probable

Unusual

Important

0.2

0.8

Practically impossible

Rare

Noticeable

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Phase 2. Analysis: In this phase aggregated decision matrix is obtained and risk magnitude for each risk is obtained. Step 2.1. Aggregation of expert evaluations: Calculate the aggregated decision matrix, using the Decomposed Weighted Arithmetic Mean (DWAM) formula (Cebi et al., 2022; 2023) ⎧  ⎫ n n ⎪ ⎪ λi ai i=1 i=1 bi ⎨  ∼ ∼     O ,⎬ ai , n n n ∼ n−1 λ 1+ a − b λ b (1−b )+ i i i i i i=1 i=1 i=1 n i DWAM α 1 , α 2 . . . . . . ., α n =     ⎪ ⎪ ⎩ ⎭ P 1 − ni=1 (1 − ci )λi , ni=1 diλi (6) n

where λi = (λ1 , λ2 . . . . . . ., λk ); λi ∈ [0, 1],

i=1 λi

∼

k

k

= 1 and α i = Lki , C i , F i

Step 2.2. Step 2.2. Obtain risk magnitude: Risk magnitude obtained by fallowing formula DWAM

RM i = LDWAM ⊗ Ci i ∼

DWAM

⊗ Fi

(7)

∼

Let α 1 = {O(a1 , b1 ), P(c1 , d1 )}, and α 2 = {O(a2 , b2 ), P(c2 , d2 )} be a decomposed fuzzy set. If none of them has maximum consistency or maximum inconsistency, the multiplication operator is defined as follows (Cebi et al. 2022; 2023)

  d1 + d2 − 2d1 d2 c1 c2 ∼ ∼ α 1 ⊗ α 2 = O(a1 a2 , b1 + b2 − b1 b2 ), P (8) , c1 + c2 − c1 c2 1 − d1 d2 Since the calculated values are in DFS, score  to compare ∼index given in Eq. 9 is used α= {O(a, b), P(c, d )} is proposed as obtained results. The score index (SI) of DFN follows (Cebi et al. 2022; 2023); ⎧ ∼ ∼ ⎨ (a+b−c+d ).CI α ∼ ⎪ α >0 , SI 2.k SI α = (9) ∼ ⎪ ⎩ 0, SI α ≤ 0 where k is the linguistic scale multiplier. The value k is obtained aby Eq. 10.   ∼   a + b − c + d .CI α k=

2

(10)

∼

where α = {O(a , b ), P(c , d )} is the maximum value for  ∼ the used linguistic scale. The consistency index (CI) of decomposed fuzzy number α= {O(a, b), P(c, d )} is defined as (Cebi et al. 2022; 2023); ⎛ ⎞ ∼ ∼ 2 2 2 2 + − c) + − a − b) + − c − d − d (b (1 (1 ) (a ) α ⎠, 0 ≤ CI α ≤1 CI α = 1 − I = 1 − ⎝ 2

(11)

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Phase 3. Rank and eliminate/reduce the risks: In this phase, risks are ranked from the most important to the least important. Then, to manage risks, a plan is developed to implement control measures that either eliminate the risks or reduce their magnitude to an acceptable level within the working environment. Subsequently, the effectiveness of these measures is monitored to ensure they are adequate.

5 Case Study: Risk Assessment in Electric Arc Welding Job Shop In this section, we aim to illustrate the proposed approach by analyzing the electric arc welding process. Electric arc welding is a welding process that uses an electric arc to join metal workpieces. The electric arc is created between an electrode (a metal rod or wire) and the metal workpiece, which melts the metal and fuses it when it cools. The welding process typically involves applying an electric current to the electrode and moving it along the metal workpiece to create the arc. Electric arc welding poses several occupational health and safety risks. It is important for employers to identify and mitigate these risks to ensure the safety and health of workers performing electric arc welding. This can be done through the use of appropriate personal protective equipment (PPE), proper ventilation and exhaust systems, and by implementing safe work practices and training programs. The steps of the proposed approach are as follows. Phase 1. Pre-analysis processes: This stage involves three key steps as mentioned above: forming a risk assessment team, identifying potential hazards, and collecting relevant data. In this study, risk assessment team consists of three occupational health and safety specialists.

Fig. 1. Electric arc welding job shop

Figure 1 presents a caption from an electric arc welding Job Shop. Some of the inappropriate situations, in this case, could include. The potential hazards are as follows;

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• Appropriate personnel protective equipment has not been used. Eye protection, such as welding helmets or goggles with tinted lenses that filter out harmful UV and IR radiation should be used. In addition, the skin should also be covered with protective clothing, such as welding jackets or gloves, to prevent burns and skin damage. Furthermore, welding screens or curtains have not been used to protect other workers in the vicinity from harmful radiation. So, there is a risk of exposure to harmful radiation (R1) • There is a lack of ventilation. There is not any ventilation equipment in the welding area, which could percent in the accumulation of harmful fumes and gases that can cause respiratory problems. Furthermore, there are not any other prevention that protects other workers from harmful gases. So, there is a risk of exposure to harmful gases (R2). • The proper equipment is not used: Welding equipment has not been properly maintained and checked regularly for any damage or malfunction. Furthermore, there is no use of properly grounded and insulated equipment. Therefore, welders can be exposed to electric shock from the welding equipment, which can cause serious injury or death. So, there is a risk of electric shock (R3) • There are flammable materials in the working environment, and it has the potential to start fires or cause explosions since the welding generates sparks, hot slag, and molten metal. So, there is a risk of fire and explosion (R4) • In the working environment, the welder works in awkward and repetitive postures, and this has the potential to develop musculoskeletal disorders. So, there is a risk of musculoskeletal disorders (R5). The risk assessment team evaluates the likelihood (L), consequence (C), and frequency (F) of each hazardous event. In this example, since all of the hazardous events belong to the electrical arc welding the frequencies of the hazardous events are assumed to be equal. The expert preferences are given in Table 3. Table 3. Expert preferences on the determined risks Expert-1 L O

Expert-2 C

F

L P

O

Expert-3 C

P

O

P

O

R1 VH L

H

L

VH VL VH L

P

O

F P

VH M

O

L P

O

C P

O

F P

VH VL VH VL VH M

O

P

VH VL

R2 VH VL VH VL VH VL VH VL VH VL VH VL H

L

VH VL VH VL

R3 M

M

L

VH VL VH VL

R4 H

VL M

VL VH VL H

R5 H

L

L

VH VL VH VL H M

L

VH VL VH VL H

VL H

VH VL VH L

M

VL VH VL H L

VL M

VH VL VH L

H

VL VH VL L

VH VL

Phase 2. Analysis: The preferences given in Table 3 are aggregated by using Eq. 6 (Table 4). The risk magnitude of each risk is obtained by using Eqs. 7 and 8 and to rank the obtained risk Eqs. 9–11 are utilized. The obtained results are given in Table 4.

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S. Cebi and P. Kaliyaperuma Table 4. Aggregations of expert preferences and risk magnitudes Likelihood

Consequence

Frequency

Risk Magnitude

CI

SI

R1

((0.8,0.2),(0.3,0.7))

((0.75,0.23),(0.45,0.55))

((0.8,0.2),(0.2,0.8))

((0.23,0.45),(0.55,0.8))

0.39

0.56

R2

((0.75,0.23),(0.25,0.75))

((0.8,0.2),(0.2,0.8))

((0.8,0.2),(0.2,0.8))

((0.2,0.2),(0.8,0.8))

0.43

0.55

R3

((0.6,0.39),(0.4,0.6))

((0.8,0.2),(0.2,0.8))

((0.8,0.2),(0.2,0.8))

((0.2,0.2),(0.8,0.8))

0.52

0.46

R4

((0.65,0.35),(0.2,0.8))

((0.55,0.44),(0.2,0.8))

((0.8,0.2),(0.2,0.8))

((0.44,0.2),(0.8,0.8))

0.64

0.35

R5

((0.75,0.23),(0.35,0.65))

((0.55,0.44),(0.35,0.65))

((0.8,0.2),(0.2,0.8))

((0.44,0.35),(0.65,0.8))

0.55

0.42

Phase 3. Exposure to harmful radiation and gases has higher risks than the risks of electric shock, fire and explosion, and musculoskeletal disorders. The preventative measures that should be taken to minimize the risks of exposure to harmful radiation and gases during electric arc welding include proper ventilation, using personal protective equipment, limiting exposure time, keeping the work area clean, and providing proper training for workers. These measures can help reduce the concentration of harmful gases and radiation, prevent the accumulation of metal dust, and ensure the safety and health of workers. If the problem considered in this study had been handled with the classical Fine&Kinney method, all risks would have been equal to each other as very high risk. Since the proposed method evaluates both the current situation and the precautions taken with functional and dysfunctional questions, it has revealed that there are differences among the considered risks and that not all of them carry the same level of risk.

6 Conclusion The Fine & Kinney method is used to assess and mitigate risks in the work environment, using parameters such as probability, severity, and frequency to determine the level of risk. However, it can be difficult to evaluate these parameters accurately since accidents can result in a range of outcomes from minor injuries to fatalities. To address this uncertainty, the newly proposed decomposed fuzzy sets have been used to consider both pessimistic and optimistic evaluations. In this study, the Fine&Kinney method is extended to include decomposed fuzzy sets and applied to the electric arc welding process. The results show that exposure to harmful radiation and gases poses a higher risk than electric shock, fire and explosion, and musculoskeletal disorders. In future studies, instead of scalar calculation in the Fine&Kinney method, decomposed-based fuzzy inference, which has an inference similar to human logic, can be added to the method to obtain the risk magnitude.

References 1. Karamusta, M., Cebi, S.: A new model for the occupational health and safety risk assessment process: Neutrosophic FMEA. J. Fac. Eng. Archit. Gazi Univ. 38(1), 29–43 (2023) 2. Cinar, U., Cebi, S.: A novel approach to assess occupational risks and prevention of hazards: The house of safety prevention. J. Intell. Fuzzy Syst. 42(1), 517–528 (2021)

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3. Karamustafa, M., Cebi, S.: Extension of safety and critical effect analysis to Neutrosophic sets for the evaluation of occupational risks. Appl. Soft Comput. 110, 107719 (2021) 4. Wang, W., Wang, Y., Fan, S., Han, X., Wu, Q., Pamucar, D.: A complex spherical fuzzy CRADIS method based Fine-Kinney framework for occupational risk evaluation in natural gas pipeline construction. J. Pet. Sci. Eng. 220, 111246 (2023) 5. Gökler, S.H., Yı, D., Ürük, Z.F., Boran, S.: A new hybrid risk assessment method based on Fine-Kinney and ANFIS methods for evaluation spatial risks in nursing homes. Heliyon 8(10), e11028 (2022). https://doi.org/10.1016/j.heliyon.2022.e11028 6. Seker, S.: A novel risk assessment approach using a hybrid method based on Fine-Kinney and extended MCDM methods under interval-valued intuitionistic fuzzy environment. Int. J. Inf. Technol. Decis. Mak. 21(5), 1591–1616 (2022) 7. Wang, W., Ding, L., Liu, X., Liu, S.: An interval 2-Tuple linguistic Fine-Kinney model for risk analysis based on extended ORESTE method with cumulative prospect theory. Inf. Fus. 78, 40–56 (2022) 8. Li, H., Liu, S., Wang, W.: The probabilistic linguistic term sets based ORESTE method for risk evaluation in Fine-Kinney model with interactive risk factors. J. Intell. Fuzzy Syst. 43(3), 3493–3512 (2022) 9. Wang, W., Jiang, W., Han, X., Liu, S.: An extended gained and lost dominance score method based risk prioritization for Fine-Kinney model with interval type-2 fuzzy information. Hum. Ecol. Risk Assess. 28(1), 154–183 (2022) 10. Güney, G., Kahraman, B.: Implementation of the analytic hierarchy process (AHP) and FineKinney method (FKM) against risk factors to determine the total cost of occupational health and safety precautions in environmental research laboratories. Int. J. Occup. Saf. Ergon. 28(4), 2606–2622 (2022) 11. Tang, J., Liu, X., Wang, W.: A hybrid risk prioritization method based on generalized TODIM and BWM for Fine-Kinney under interval type-2 fuzzy environment. Hum. Ecol. Risk Assess. 27(4), 954–979 (2021) 12. Dagsuyu, C., Oturakci, M., Essiz, E.S.: A new Fine-Kinney method based on clustering approach. Int. J. Uncertainty, Fuzziness Knowl.-Based Syst. 28(3), 497–512 (2020) 13. Yang, B.-C., Zhao, Z.-L., Chen, H.-L.: Human error risk analysis of coastal water collision accidents based on Fine-Kinney method of fuzzy rule base. Dalian Haishi Daxue Xuebao/J. Dalian Maritime Univ. 45(1), 40–46 (2019) 14. Ersoy, M., Çelik, M.Y., Ye¸silkaya, L., Çolak, O.: Combination of Fine-Kinney and GRA methods to solve occupational health and safety problems. J. Fac. Eng. Archit. Gazi Univ. 34(2), 751–770 (2019) 15. Wang, W., Liu, X., Qin, Y.: A fuzzy Fine-Kinney-based risk evaluation approach with extended MULTIMOORA method based on Choquet integral. Comput. Ind. Eng. 125, 111–123 (2018) 16. Cebi, S., Gündo˘gdu, F.K., Kahraman, C.: Operational risk analysis in business processes using decomposed fuzzy sets. J. Intell. Fuzzy Syst. 1–18 (2022). https://doi.org/10.3233/ JIFS-213385 17. Cebi, S., Gündo˘gdu, F.K., Kahraman, C.: Consideration of reciprocal judgments through decomposed fuzzy analytical hierarchy process: a case study in the pharmaceutical industry. Appl. Soft Comput., 110000 (2023) 18. Kinney, G.F., Wiruth, A.D.: Practical risk analysis for safety management (No. NWC-TP5865). Naval Weapons Center China Lake, CA (1976)

Risk Assessment on the Grinding Machine with SWARA/ARAS and Visual PROMETHEE Based on Unstable Fuzzy Linguistic Terms Turgay Duruel(B)

and Bahadır Gülsün

Industrial Engineering Yildiz Technical University, 34353 Be¸sikta¸s, Turkey [email protected], [email protected]

Abstract. Occupational health and safety is now an extremely important scientific field all over the world. The most important argument in the field of occupational health and safety is risk assessment. In this study, unlike traditional risk assessment, multi-criteria decision-making methods as well as unstable fuzzy linguistic terms are used to assess the risks in grinding machines, one of the hazardous machine tools used in the machinery sector. In the study, first of all, hazards and risks were identified by taking into account the opinions of 3 experts experienced in the field and literature studies in this field. The identified risks are weighted by the SWARA method based on Undecided fuzzy linguistic terms. In the next stage, risks are ranked by the ARAS method based on Undecided Fuzzy Linguistic terms. In this method, expert weights were determined by the CRITIC method, and risk parameters were weighted by probability, severity, and frequency AHP method to determine the importance of risks. Then, using the weights of the risks we found previously, the risk assessment was completed by ranking the alternative measures using the Visual PROMETHEE software program. The results of the risk assessment with the new method were compared with the risk assessment with the classical method and the results were discussed. Keywords: Occupational Health and Safety · Risk Assessment · Unstable Fuzzy Linguistic Terms

1 Introduction Today, the field of occupational health and safety has become a discipline that has gained importance all over the world. An important element of occupational safety is risk assessment. There are dangers in every job according to the nature of the work. Hazards create risks. In this study, an assessment of the risks of the grinding machine, which is one of the most dangerous machines in the machinery sector and which makes the surfaces of the parts smooth in the machinery sector, was carried out. The aim of this study is to propose a new risk assessment model. In this risk assessment, unlike the classical risk assessment, an integrative risk assessment was made with the Visual PROMETHEE T. Duruel—This study is produced from the doctoral dissertation of the first-ranked author. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 798–807, 2023. https://doi.org/10.1007/978-3-031-39774-5_87

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academic software program with the undecided fuzzy SWARA and ARAS methods. The results of the study were evaluated by GAIA visual method and PROMETHEE and Fuzzy PROMETHEE methods. The results were compared with classical risk assessment. 3 experts were assigned to the risk assessment study. The first expert was the occupational safety specialist, and the second expert was the workshop supervisor and the third expert was the experienced employee working on the grinding machine. As a result of the opinions of experts and the literature study, 6 risks were identified in the grinding machine. In order to prevent these risks, 5 alternative measures have been identified risk are represented by the letter ‘R’ and sub-alternatives by ‘P’. After the weights of the identified risks were determined by Undecided fuzzy SWARA, the ranking of the risks was done by the Undecided fuzzy ARAS method. Here, expert weights were determined by the CRITIC method, and risk assessment parameters probability, severity, and frequency were determined by the Superdecision software program. In the following sections, the study is concluded with a brief summary of the methods, the findings of the risk assessment, a discussion and a conclusion.

2 Literature There is a limited number of risk assessment studies in the field of occupational safety with multi-criteria decision-making, fuzzy logic, and unstable fuzzy linguistic terms. Studies have gained momentum in recent years. Panagiotis et al. conducted a risk assessment in the chemical industry by integrating AHP and HAZOP methods in their study in 2022 [8]. In their 2016 study, Liu et al. conducted a risk assessment in the healthcare sector by integrating the Fuzzy linguistic terms method and the FMEA method [9].

3 Method In this study, risk assessment was carried out using a total of six methods with multicriteria decision-making and fuzzy logic and undecided fuzzy logic methods. The grinding machine and parts of the risk assessment are shown in Fig. 1 below.

Fig. 1. Grinding Machine image [1]

Figure 1 shows the grinding machine and its parts. Grinding has an important place in manufacturing methods. Its task is to make the workpieces press-free with the desired

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surface quality and size [2]. One of the most important uses of this method is for machining parts that do not have the desired surface quality on lathes and milling machines [3]. The reason why these machines are considered to be the most dangerous in the field of machinery is that the rotating stone is made up of many small particles. When this stone explodes, thousands of particles are scattered around like lead. Therefore, it is necessary to be very careful when working on these benches. The steps of the methods used in risk assessment on the grinding machine are briefly given below. 3.1 Unstable Fuzzy SWARA The concept of Undecidable Fuzzy Linguistic Term (UFLT) Sets, which has gained importance in recent years, provides decision-makers with a more flexible perspective by using multiple linguistic terms instead of using a single linguistic term in solving decision problems. SWARA (Step-Wise Weight Assessment Ratio Analysis) method, which is one of the multi-criteria decision-making methods, is a method used in criteria weighting. In this study, the weighting of risks is used by combining the fuzzy method with the fuzzy uncertainty method. Unstable Fuzzy SWARA steps are briefly summarized below [4]. The steps are as follows; Step.1: Determination of the objectives and criteria to be achieved in the decision problem, identification of decision makers from experts in the field Step.2: Determining the most important criteria by taking into account expert opinions Step.3: Determining the importance of the criteria according to the most important criterion Step.4: Integrating the views of experts who are decision-makers Step.5: Calculation of criterion points Step.6: Calculation of criterion coefficient and importance vectors Step.7: Calculation of criteria weights Unstable Fuzzy ARAS In 2010, it was introduced to the literature by Zavadskas and Turskis. The ARAS method (Additive Ratio Assesment) is a decision-making method that starts from a reference alternative and evaluates other alternatives based on the ideal alternative. In this study, the ARAS method is combined with the Undecided Fuzzy Linguistic (UFL). In this context, UFL-ARAS was proposed by Lio et al. in 2015 [5]. The steps of the UFL-ARAS method are briefly summarized below [6]; Step.1: Determination of Alternatives, Criteria and Criteria weights Step.2: Creation of UFL decision matrices Step.3: Normalization of the UFD decision matrix Step.4: Integration of normalized decision matrices Step.5: Weighting of the integrated decision matrix Step.6: Calculating the total performance value for each alternative Step.7: Ranking of alternatives

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AHP (Analytical Hierarchy Process) It is one of the most widely used Multi-Criteria Decision Making methods. Developed in 1970 as a result of the work of Thomas L. Saaty. It is used for criteria weighting by making pairwise comparisons. The summary steps of the AHP are as follows and those who want to learn more can benefit from this resource [5]; 1.Step: Hierarchical structure is designed. 2.Step: Pairwise comparison matrices are prepared. 3.Step: Determination of the eigenvector (Relative Importance Vector). 4.Step: Calculate the Consistency of the Eigenvector 5.Step: Obtaining the Overall Result of the Hierarchical Structure CRITIC (CRiteria Importance Through Intercriteria Correlation) This method is a weighting method used in Multi-Criteria Decision Making techniques. The objective determination of criteria weights is the most important feature of this method. It was discovered by Diakoulaki et al. in 1995. Briefly, the steps are given below. The CRITIC method consists of four steps [6]; Step.1: Creating and Normalizing the Decision Matrix Step.2: Determination of Relationships between Criteria Step.3: Determination of correlation levels between criteria Step.4: Calculation of Criteria Weights PROMETHEE and Fuzzy PROMETHEE (Preference Ranking Organization Method for Enrichment Assessment) It is a method used in the ranking of alternatives from multi-criteria decision-making methods The fact that this method is a software program and that GAIA (Geometrical Analysis for Interactive Aid) includes the visual method has brought this method to the forefront. The process steps in Fuzzy PROMETHEE are almost the same as in PROMETHEE. Fuzzy PROMETHEE is based on fuzzy numbers and the process of stabilization. For more detailed information, please refer to this source [7].

4 Findings The team of three experts first identified 6 major risks that could occur on the grinding machine and 5 alternative measures that could be taken. It is given in Table 1 below. The expert team conducted the risk assessment in 5 stages. In stage 1, the weighting of risks was determined by the Undecided Fuzzy SWARA method. In the 2nd stage, the CRITIC method was used to weigh the experts with criteria such as experience in this field and age. In stage 3, the probability, severity, and frequency parameters used in risk assessment were weighted with the AHP method using the Superdecision software program. In the 4th stage, risks were ranked according to their importance with the Unstable Fuzzy ARAS method. In Stage 5, alternative measures were ranked with Fuzzy PROMETHEE and PROMETHEE Visual PROMETHEE software programs. A comparison was made

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between the two methods. The findings obtained as a result of the analysis are presented in order. 1. Stage: The risks identified in Table 1 below have been weighted with Undecided Fuzzy SWARA. Table 1. Risks and alternative measures to be taken Risk Code

Precaution ary Code

R1

The explosion of stone

P1

Education

R2

Burr escapes to the eye

P2

Virtual Reality Education

R3

The ejection of the piece

P3

Taking the machine to a separate section

R4

Chemical risk caused by coolant

P4

Employee specific clothing

R5

Noise

P5

Unbreakable glass sper

R6

Risk of Hand and Arm injury

Table 2 below shows the risk ranking obtained as a result of the analysis made with Unstable Fuzzy SWARA. Table 2. The result of the analysis with Unstable Fuzzy SWARA Criteria

Score

R1

Coefficient

Importance Vector

Criteria Weight

Ranking

1,0000

1,0000

0,4039

1

R3

0,857

1,8571

0,5385

0,2175

2

R2

0,405

1,4048

0,3833

0,1548

3

R4

0,429

1,4286

0,2683

0,1084

4

R6

0,595

1,5952

0,1682

0,0679

5

R5

0,429

1,4286

0,1177

0,0476

6

Table 2 shows that the most important risk is R1. The risk ranking was found as R1 > R3 > R2 > R4 > R4 > R6 > R5. 2. Stage: At this stage, the CRITIC method was used and the calculation of the expert weights by taking into account the criteria of the employees’ years of working in the grinding sector, age, experience in the field of occupational safety, and years of working in the enterprise is given in Table 3 below. Here e1; OHS expert e2; Workshop supervisor e3; Experienced employee. In the expert weights, the most effective expert was the experienced employee working on the grinding machine.

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Table 3. Expert weights score value

expert weights

e1

1,79

0,035

e2

3,55

0,069

e3

45,95

0,896

3. Stage: At this stage, probability, severity, and frequency parameters were weighted with the AHP method using the Superdecision software program. The results are shown in Fig. 2 below. The consistency ratio is also below 0.10.

Fig. 2. AHP risk parameter weighting results

4. Stage: At this stage, with the Undecided Fuzzy Linguistic ARAS method, the experts evaluated the risks using the linguistic terms in Table 4. The evaluation data are given in Table 5. Table 4. Linguistic Term Scale [5] Linguistic Term

Si

Extremely High (EH)

S8

Very High (VH)

S7

High(H)

S6

Lıttle High (LH)

S5

Medium(M)

S4

A Litle Low (LL)

S3

Low (L)

S2

Very Low (VL)

S1

Extremely Low (EL)

S0

The results of the evaluation are given in Table 6 below. The result of the risk ranking is found as R4 > R2 > R3 > R5 > R5 > R6 > R1 according to Table 6.

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Alternatives

E1

E2

E3

Criteria

Criteria

C1: C2: Probability Frequency

C3: Violence

C1: Probability

C2: Frequency

C3: Violence

R1

between VL LL and LH

EH

{S3,S4,S5} {S1}

R2

M

between M M and H

{S4}

R3

between L and M

between L and M

H

{S2,S3,S4} {S2,S3,S4} {S6}

R4

between M LH and H

between M and H

{S4,S5,S6} {S5}

R5

M

H

M

{S4}

{S6}

{S4}

R6

between L and M

L

LH

{S2,S3,S4} {S2}

{S5}

R1

between M L and H

EH

{S4,S5,S6} {S2}

{S8}

R2

L

VL

M

{S2}

{S3}

{S4}

R3

L

VL

LH

{S2}

{S1}

{S5}

R4

between M LH and H

LH

{S4,S5,S6} {S5}

{S5}

R5

M

M

M

{S4}

{S4}

{S4}

R6

between L and M

L

LH

{S2,S3,S4} {S2}

{S5}

R1

between L and M

VL

Minimum {S2,S3,S4} {S1} VH

{S7,S8}

R2

M

LL

M

{S4}

{S3}

{S4}

R3

between L LL and LH

LH

{S3,S4,S5} {S2}

{S5}

R4

LH

LH

LH

{S5}

{S5}

{S5}

R5

between L and M

M

M

{S2,S3,S4} {S4}

{S4}

R6

LL

L

LH

{S3}

{S5}

{S8}

{S4,S5,S6} {S4}

{S2}

{S4,S5,S6}

5. Stage: At this stage, the ranking of alternative measures was done by experts using PROMETHEE and Fuzzy PROMETHEE methods using the Visual PROMETHEE software program. A 5-point scale was used. The scale used is given in Table 7. The results of the PROMETHEE and Fuzzy PROMETHEE analysis are given in Fig. 3 below, with the evaluation of the experts using the scale given in Table 7.

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Table 6. Undecided Fuzzy ARAS risk ranking P(i)

Q(i)

Ranking

R0

0,12665

0,14829

0,1655

0,14682

1,00000

R1

0,09807

0,11518

0,1277

0,11365

0,77413

6

R2

0,16730

0,14616

0,1304

0,14795

1,00774

2

R3

0,12667

0,13624

0,1428

0,13525

0,92119

3

R4

0,22523

0,20046

0,1815

0,20241

1,37869

1

R5

0,12397

0,13597

0,14383

0,13459

0,91673

4

R6

0,12746

0,11470

0,1046

0,11560

0,78740

5

Table 7. Fuzzy numbers corresponding to linguistic expressions and 5-point scale [5] linguistic expressions

the fuzzy number equivalent

5-point scale

Too bad

(0;0;0,25)

1

bad

(0;0,25;0,50)

2

middle

(0,25;0,50;0,75)

3

good

(0,50;0,75;1,00)

4

very good

(0,75;1,00;1,00)

5

Fig. 3. PROMETHEE and Fuzzy PROMETHEE preventive measure ranking.

When we examine Fig. 3, it is concluded that the ranking made by both methods has not changed. The best measure ranking was found as P1 > P4 > P5 > P2 > P3. Below in Fig. 4 GAIA plane comparisons are made.

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Fig. 4. Comparison of alternative measures with PROMETHEE and Fuzzy PROMETHEE in GAIA plane

5 Result In the risk assessment of the risks identified in the grinding machine with the integration of Undecided Fuzzy SWARA/ARAS and AHP/CRITIC/PROMETHE/Fuzzy PROMETHEE methods by the experts, the risk ranking was found as R4 > R2 > R3 > R5 > R6 > R1. The ranking of alternative measures to be taken was also done with PROMETHEE and Fuzzy PROMETHEE and Visual PROMETHEE academic software program. It was found that the ranking was the same in both methods. In the analyses performed in the flow table, it was found that the measure order was the same in both methods as P1 > P4 > P5 > P2 > P3. In the analysis made in the GAIA plane, it was seen that the sub-measures closest to the decision bar were P1 and P4 in Fuzzy PROMETHEE and P1 and P4 in the PROMETHEE method. In future studies, it can be applied in more than one machine and in different sectors. Acknowledgment. The paper was produced within the scope of TYU BAP-related project and the study was supported by the BAP Unit.

References 1. https://www.yandex.com.tr/gorsel/search?text. Accessed 20 Mar 2023 2. Demir, H., Güllü, A., Çiftçi, ˙I., Seker, ¸ U.: An investigation into the influences of grain size and grinding parameters on surface roughness and grinding forces when grinding. Stroj. Vestn-J. Mech. E 56, 447–454 (2010) 3. Kalpakjian, S.: Manufacturing Process for Engineering Materials, pp. 120–121. AddisonWesley. New York (1991) 4. Ecer Akta¸s, C.: Weighting of country evaluation criteria using the swara method based on unstable fuzzy linguistic terms. In: Fuzzy Multi-Criteria Decision-Making Methods - MS Excel ® and Software-Solved Applications, pp. 217–229. Nobel Publishing House, Ankara (2021) 5. Duruel, T., Gülsün, B.: Integrated risk assessment with unstable fuzzy INTERMEDIATE, AHP and CRITIC methods in machine tooling workshops In: 10th International Engineering Architecture and Design Congress, December, Istanbul, pp. 24–25–26 (2022)

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6. Diakoulaki, D., Mavrotas, G., Papayannakis, L.: Determining objective weights in multiple criteria problems: the critic method. Comput. Oper. Res. 22(7), 763–770 (1995) 7. Da˘gdeviren, M., Eraslan, E.: Supplier selection by PROMETHEE sorting method. Gazi Univ. J. Faculty Eng. Arch. 23(1), 69–75 (2008) 8. Panagiotis, K., Marhavilas, M.F., Koulinas, G.K., Koulouriotis, D.E.: Safety-assessment by hybridizing the MCDM/AHP & HAZOP-DMRA techniques through safety’s level colored maps: Implementation in a petrochemical industry. Alexandria Eng. J. 61, 6959–6977 (2022) 9. Liu, H.C., You, J.X., Li, P., Su, Q.: Failure mode and effect analysis under uncertainty: an integrated multiple criteria decision making approach. IEEE Trans. Reliab. 65(3), 1380–1392 (2016)

An Intelligent Fuzzy Functional Resonance Analysis Model on System Safety and Human Factors Esmaeil Zarei1,2(B) , Mohammad Yazdi3 , Brian J. Roggow1,2 , and Ahmad BahooToroody4 1 Department of Safety Science, College of Aviation, Embry-Riddle Aeronautical University,

Prescott, Arizona, USA 2 Robertson Safety Institute (RSI), Embry-Riddle Aeronautical University,

Prescott, Arizona, USA [email protected] 3 School of Computing, Engineering and Physical Sciences, University of the West of Scotland, London, UK [email protected] 4 Department of Mechanical Engineering, Marine Technology, Research Group on Safe and Efficient Marine and Ship Systems, Aalto University, Espoo, Finland [email protected]

Abstract. System safety and human factors analysis are pivotal in critical operations from healthcare to aviation and aerospace. Conventional safety and human factors models cannot model emergent risks that can seriously threaten organizational life. To this end, this research has developed a novel Interval-Valued Spherical Fuzzy Sets (IVSFS)-Functional Resonance Analysis (IVSFS-FRA) model to analyze system safety and human factors in critical socio-technical systems. The proposed model deals with critical concerns in knowledge acquisition and captures emerging risks in critical operations. An actual industrial maintenance cycle has been investigated to demonstrate the model’s capabilities and effectiveness. The model can recognize evolving and emerging critical resonances, address uncertainty, and reveal tight couplings in complex systems. Accordingly, the proposed model can prevent system disruption and enhance system resilience by allocating timely and cost-effective safety countermeasures. Keywords: Intelligence systems · Soft Computing · Risk Analysis

1 Introduction The Functional Resonance Analysis Method (FRAM) is established on four fundamental principles, including [1] a) the equivalence of successes and failures, b) the approximate adjustments, c) the emergence, and d) the functional resonance. Accordingly, performance variability is emergent rather than resultant and produced in non-linear combinations. Researchers often investigate human and system safety performance from two distinctive perspectives of human reliability analysis (HRA) or human factors. The former mainly emphasizes predicting human performance (e.g., error probability), while © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 808–815, 2023. https://doi.org/10.1007/978-3-031-39774-5_88

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improving human performance by optimizing system or task design is a major concern in the latter perspective. Furthermore, they often produced specific probability values regarding human-oriented functions while failing to deliver deep insights concerning the sources of vulnerability, and complex resonance mechanisms resulted in adverse [2]. Moreover, accidents have been recognized due to combined variabilities instead of a single or group components’ failure, while conventional techniques focus on predicting human error in each task’s steps and independently threat highly estimated values to prevent human error. As a result, the focus of safety-driven investigations should be exploring the tight couplings and complex dynamic dependencies that can firmly explain the adverse outcome [1]. Subsequently, decision makings in designing for safety are according to system thinking focus which will monitor performance variability and dampen critical ones according to the four underlying principles of FRAM [1]. Several researchers have made genuine attempts to improve the FRAM and its application from healthcare to [3] to oil and gas [4]. However, FRAM requires more attention to deal with quantification concerns which help to establish a solid system modeling. Accordingly, the present research integrated advanced intelligent techniques into FRAM to improve the model capabilities and effectiveness for the system safety assessment in a risk-based manner. Accordingly, this research is among the first systematic attempt aimed at proposing a new intelligent model in system safety of complex systems. The rest of paper is followed by introducing the proposed Fuzzy FRAM model in Sect. 2, while key results and discussion are provided in Sect. 3. Final section is allocated to Conclusion of the present research.

2 Methodology We first utilized the recently proposed Performance Variability Shaping Factors (PVSF) taxonomy [5] to reveal which factors contributed to the performance resonance in system functions, including human, organizational, and technical functions. Secondly, a novel Interval-Valued Spherical Fuzzy Sets (IVSFS) [5] was employed to quantify the performance variability magnitude arising from the influence of PVSFs. In the following, Dempster- Shafer Evidence theory (DSET) and Monte Carlo Simulation (MCS) methods, and Bayesian networks (BNs) [6] were employed to model the system performance resonance probabilistically. It is a new hybrid model based on Safety-II, which is used to quantitatively model system safety performance under uncertainty. A practical approach estimates the relationship among VSFs, which captures their inter-dependencies [6]. Finally, a risk-based perspective was used to rank the function (performance) variability to prioritize safety countermeasures to dampen the critical variabilities before leading to major system disruption. To illustrate a methodology for somewhat, we used Table 1 to collect five subject experts’ knowledge about the magnitude and probability of performance variabilities in the studied function. After that, a novel Interval-valued Spherical Weighted Arithmetic Mean (IVSWAM) is utilized to aggregate experts’ knowledge       ∼ (Eq. 1). Given that δ j =  αj , βj , γj , δj , εj , ζj  is an IVFSS and weighted arithmetic Mean of n sets concerning weight (zj ), that denotes the experts’ weight, who give their knowledge using linguistic terms presented in Table 1, Zj = (z1 , z2 , . . . , zn ); z ∈ [0, 1],  and nj=1 zj = 1.

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∼ ∼ ∼  ∼  IVSWAM δ 1 , δ 2 , . . . , δ n = nj=1 zj . δ j = ⎫    z  21  z  21  n ⎪ zj , n δ zj , , ⎪ 1 − nj=1 1 − αj 2 j , 1 − nj=1 1 − βj 2 j , γ ⎬ j=1 j j=1 j  1  1  







 ⎪ ⎪ 2 2 z z z z n n n n 2 j − 2 2 j 2 j − 2 2 j ⎪ ⎪ ⎭ ⎩ , j=1 1 − aj j=1 1 − aj − εj j=1 1 − βj j=1 1 − βj − ζj ⎧ ⎪ ⎪ ⎨

(1) The numbers of membership degrees are indicated with α (lower) and β (upper), non-membership (β ∼ ) with γ and δ, whereas hesitancy (γ ∼ ) by ε and ζ. After that, AS

AS

the score function is employed to the defuzzification of IVSFS number a˜ is estimated as Eq. 2. ∼ Score δ = S(δ) =

 2 2 α 2 +β 2 −γ 2 −δ 2 −( 2ε ) − ζ2 2

(2)

Finally, Eq. 3 is proposed to estimate the Variability Magnitude Index (VMI) as:   n  VMIx = µi αi (3) i=1

where μi denotes how much VSFsi (i = 1, 2, …, n) influence on (Eq. 2) function x performance, αi means the importance level of VSF i (i = 1, 2, …, n) which is calculated using the Best-Worst Method. We would like to refer enthusiastic readers to primary references where all techniques are clearly explained for the sake of conference page limitation [5, 6]. The proposed model has been tested on overhaul maintenance in the oil and gas industry to model system safety and human factors as per Safety Resilience engineering and Safety-II perspective.

3 Results and Discussion As per FRAM procedure, we identified thirty-one functions (activities) in maintenance operation, including nineteen humans (gray colored), five organizational (yellow colored), and seven technological functions (blue colored). Figure 1 illustrates those functions and their coupling. As can be seen, there are numerous tight relationships interand intra-three function groups which reflect how much system functions influence their performance and, subsequently, system safety. After that, we asked five subject expert matters to learn about the magnitude and probability of performance variability in each function duo to influence PVSFs via Table 1. The magnitude and probability of performance variability of each organizational, human, and technological function are demonstrated in Table 2. The proposed VMI means severity, while normal probability distribution (Mean ± SD) indicates the probability of variability. Accordingly, system functions can be ranked in both aspects of the risk concept. For instance, human-related activities concerned F18 “Conducting the PreStartup Safety Review (PSSR) and run operation”, F15 “Performing the required maintenance”, and F9 “Depressurizing, draining, and purging” contributed most to the performance resonances. Furthermore, F22 “Establishing the Radar system to improve the

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Table 1. Linguistic terms and the IVSFS used in this research PVMs

Linguistic terms

Score

Interval-valued   [a, b], [c, d ], e, f

T4: Omission P4: Wrong

Absolutely more influence /probably

9

[0.85, 0.95], [0.10, 0.15], [0.05, 0.15]

Very high influence / probably

8

[0.75, 0.85], [0.15, 0.20], [0.15, 0.20]

T3: Too Late P3: Imprecise

High influence/ probably

7

[0.65, 0.75], [0.20, 0.25], [0.20, 0.25]

Slightly more influence / probably

6

[0.55, 0.65], [0.25, 0.30], [0.25, 0.30]

Slightly low influence/ probably

5

[0.50, 0.55], [0.45, 0.55], [0.30, 0.40]

Low influence/ probably

4

[0.25, 0.30], [0.55, 0.65], [0.25, 0.30]

Very low influence/ probably 3

[0.20, 0.25], [0.65, 0.75], [0.20, 0.25]

Absolutely low influence / probably

2

[0.15, 0.20], [0.75, 0.85], [0.15, 0.20]

No influence / probably

1

[0.10, 0.15], [0.85,0.95], [0.05, 0.15]

T2: Too Earlier P2: Acceptable

T1: On Time P1: Precise

PVMs: Performance variability manifestations, T: Time, P: Precision

spirit of team working, mutual communication and safety culture” and F20 “Establishing and holding the crew training programs” from organizational functions, while F27 “Depressurizing, draining, and purging system” and F25 “Pressure and leak test system” from technological functions let to, respectively, most critical variability in the maintenance operation. The prior probability of variability caused by each PVSF is estimated using the experts’ knowledge and DSET. It followed by using the BNs to conduct probabilistically predictive resonance to estimate each function’s variability. Given the normal distribution, 100,000 iterations were conducted by MCS to estimate the expected value and standard deviation for performance variability presented in Table 2. As presented, F9 “Depressurizing, draining and purging”, F17 “Preparing for start-up and conducting the pressure tests”, and F10 “Performing mechanical and electrical isolation” from Human, F21 “Providing the required hardware, software, and legal support” from Organization and F27 “Depressurizing, draining, and purging system” from technical functions probabilistically imposed the highest variability in their performance. The proposed approach captured the variance in variability characterization, which addressed the uncertainty associated with performance variability using the DSET and MCS. We estimated the risk level of performance variability by multiplying the magnitude and probability of variability for each function, as illustrated in Fig. 2. Risk characterization revealed that F9 (R = 5.2541), F17 (R = 4.8580), F10 (R = 4.7057), F4 (R

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Fig. 1. FRAM model of maintenance operation from system safety perspective

= 4.3487), and F18 (R = 4.0671) have the highest, respectively, performance resonance. Per the resilience engineering principle, those functions should be prioritized to dampen critical variability, prevent system disruption, and improve system safety and resilience. This is among the rare research that proposed a strong risk-based approach to dampen critical system resonance because previous research either concentrated on the probability facet or employed a qualitative risk estimation for performance variability. Furthermore, Fig. 3 demonstrates which functions lead to variability in the F9 : Depressurizing, draining, and purging, and which functions (F11 “Performing pressure and isolation leak test” and F14 “Confirm PTW and monitor its validity” have been impacted by the propagation of performance variability in F#9. For instance, two organizational and three technological functions, apart from two human functions associated with chemical, mechanical, and electrical system isolation, are tightly coupled to the F9 as upstream functions. However, F9 output acts as a Precondition over “Confirming the permit to work” and Input on “Performing pressure and leak test” as downstream functions. Accordingly, we can assign appropriate interventions before (on upstream functions) or after (on downstream functions) this critical performance resonance as per operational circumstances, providing more flexibility in adopting dampening strategies. This indicates system safety investment areas and effective decision-making to improve system resilience.

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Table 2. Performance variability magnitude index (VMI) and probability distribution (Mean) F#

VMI

Mean

SD

F#

OVI

Mean

SD

F1

0.5367

1.1284

0.3795

F17

0.5760

1.8921

0.6754

F2

0.5367

1.079

0.3103

F18

0.5947

1.6993

0.6941

F3

0.4530

1.0486

0.238

F19

0.5730

1.5021

0.6854

F4

0.4998

1.7672

0.6936

F20

0.6737

1.1801

0.4663

F5

0.5635

1.6345

0.6879

F21

0.6275

1.4921

0.6965

F6

0.4910

1.6627

0.6906

F22

0.6996

1.4514

0.6783

F7

0.5246

1.1409

0.4016

F23

0.6572

1.3719

0.6235

F8

0.5375

1.4225

0.6734

F24

0.6230

1.1375

0.4026

F9

0.5782

1.9871

0.657

F25

0.6781

1.4766

0.6989

F10

0.5620

1.8565

0.6782

F26

0.6628

1.2588

0.5718

F11

0.5331

1.4215

0.6542

F27

0.6807

1.6317

0.707

F12

0.5077

1.2237

0.5195

F28

0.6756

1.533

0.7054

F13

0.5417

1.2704

0.5509

F29

0.6228

1.4012

0.6683

F14

0.5291

1.5744

0.6861

F30

0.6629

1.2098

0.5231

F15

0.5829

1.6864

0.6976

F31

0.6560

1.1732

0.4653

F16

0.5647

1.3472

0.6204

6.00

Upper Risk Level

Lower Risk Level

Risk Levels

5.00 4.00 3.00 2.00 1.00 0.00 F1

F3

F5

F7

F9 F11 F13 F15 F17 F19 F21 F23 F25 F27 F29 F31

Maintenance Functions Fig. 2. Risk level interval of performance variability in maintenance functions

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Fig.3. Upstream-downstream performance variability in maintenance operation

4 Conclusion This research employed an advanced risk-based model for system safety analysis in socio-technical systems. This is among the first scholarly studies which quantitatively deal with performance variability from a risk perspective. The model handles subjective and objective uncertainties in model inputs, parameters, and risk characterization. It also quantitatively estimated internal, external, and upstream-downstream verbalities which can be dampened earlier before hazardous propagation entire the system. The proposed model is in line with the system thinking where all system functions and their potential couplings with a primary focus on human factors have been identified. The case study on maintenance activities proved the model’s capability and effectiveness in identifying how we can improve system safety and resilience in a cost and time-effective approach for Resilience engineering and Safety-II perspective. The proposed approach can also be employed in other complex systems where system safety should be carefully analyzed, and timely actions should be in place to establish safe and resilient operations. It should be noted that modeling upstream-downstream variability using the proposed approach can be ideal research demand for future studies.

References 1. Hollnagel, E.: FRAM, the functional resonance analysis method: modelling complex sociotechnical systems: Ashgate Publishing, Ltd. (2012) 2. Zarei, E., Khan, F., Abbassi, R.: How to account artificial intelligence in human factor analysis of complex systems? Process. Saf. Environ. Prot. 171, 736–750 (2023) 3. Salehi, V., Hanson, N., Smith, D., McCloskey, R., Jarrett, P., Veitch, B.: Modeling and analyzing hospital to home transition processes of frail older adults using the functional resonance analysis method (FRAM). Appl. Ergon. 93, 103392 (2021)

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4. Aguilera, M.V.C., da Fonseca, B.B., Ferris, T.K., Vidal, M.C.R., de Carvalho, P., Rodrigues, V.: Modelling performance variabilities in oil spill response to improve system resilience. J. Loss Prev. Process. Ind. 41, 8–30 (2016) 5. Zarei, E., Khan, F., Abbassi, R.: An advanced approach to the system safety in socio-technical systems. Saf. Sci. 158, 105961 (2023) 6. Zarei, E., Khan, F., Abbassi, R.: A dynamic human-factor risk model to analyze safety in socio-technical systems. Process. Saf. Environ. Prot. 164, 479–498 (2022)

Vehicle and Pedestrian Crash Risk Modeling in Arabian Gulf Region Sharaf AlKheder(B) Civil Engineering Department, College of Engineering and Petroleum, Kuwait University, P.O. Box 5969 SAFAT, 13109 Kuwait, Kuwait [email protected]

Abstract. Road traffic accidents are a major world health issue that exhausts the financial status of countries and threatens millions of lives every day. In this work a five-and-a-half-year period data from Abu Dhabi, UAE was obtained and categorized. Random Parameter logit model-that assessed the unobserved heterogeneity of risk factors-was used with four dependent tolerable outputs. Correlation tests were conducted to assess the significance of the model estimated results for different injury levels. Results indicated that in vehicle-only accidents; old age drivers, male drivers, falling off vehicle accidents, dry paved covered with sand, night weak light condition, speeding, driving under the influence of drugs, exhaustion and sleepiness were associated with higher crash severity. However, for pedestrian accidents; results showed a highly positive correlation with pedestrian standing on the median, using pedestrian crosswalk at intersection, using pedestrian crosswalks where there is no intersection, and not using the designated pedestrian crosswalks.

1 Introduction The overall impact of the increase of fatality rates across the world every year leads to lower economic stability and reduces the growth opportunities. This requires decisionmakers to maximize the contributions in road safety developments by incorporating education, enforcements, and engineering countermeasures in order to maintain a lower fatality rate in road accidents, and to sustain a lower rate of road traffic injuries (RTI) that cause disability and lower productivity of individuals involved in the crash undoubtedly anticipated in safer roadways. In order to maintain a high level of accidents prevention, a wide range of statistical models are used that provide the critical relationship between accident severity and factors. Accordingly, the model used in this study is the Random Parameter (or mixed) logit model, which is a statistical model that takes into consideration the heterogeneity among the population over time. The model assesses the unobserved heterogeneity of risk factors that include driver behavior, time of collision, location of accident, roadway and environmental factors that would anticipate more frequently in road accidents severity.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 816–829, 2023. https://doi.org/10.1007/978-3-031-39774-5_89

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Gulf Cooperation Council (GCC) countries suffer from high road traffic accidents, and high road deaths and injury severity rate. In order to confine a better understanding of the circumstance that leads to traffic accidents and fatality, an essential approach could help in mitigating the road safety issues encountered by road users across GCC countries, by accounting for the unobserved heterogeneity using statistical modeling that identifies factors contributing in road accidents severity levels.

2 Literature Review It is clear that despite the many attempts in previous researches to control traffic accidents, its growth is continuously increasing. Thus, instigating a sense of urgency to gear the researches in traffic safety toward injury severity studies. Researches on injury severity became more oriented in identifying the factors which affect the severity of accidents such as specific travel modes, specific crash type, or by considering different lighting conditions, aim to obtain more useful analysis and results which leads to appropriate assessment and thus more productive specification of the prevention of factors affecting severity levels (Abrari Vajari et al. 2020, Azimi et al. 2020, Mashhadi et al. 2018). The GCC region had many studies that investigated the injury severity to enhance the safety of road users and reduce the fatality numbers. For example, the research of Alkheder et al. (2019) analyzed the significance of accidents through two types of variables and showed that both the location and time were significant variables related to traffic accident type’s occurrence. Zhang et al. (2000) found that younger drivers are less likely to be killed or seriously injured in crashes than elderly drivers. Pahukula et al. (2015) showed that female drivers are more vulnerable to severe and fatal injury. The modeling system of the latent class (finite mixture) preserves heterogeneity by utilizing a semi-parametric, discrete allocation, allowing parameters to deviate across unattended population groups. On the other hand, many previous works introduced the Random Parameters Ordered Probit model such as Fountas et al. (2018) which addressed the heterogeneity in conformity with the provisions of unobserved variables, the results change over time randomly. Similarly, the multivariate models which accounted for the consequences of the vehicle characteristics and the driver behavior by a significant empirical contribution in the joint analysis of injury severity and the heterogeneity unobserved countermeasures outcomes (Russo et al. 2014). Alternatively, another heterogeneity model is the Markov’s switching models which are statistically superior to the traditional multinomial logit (single-state) (Xiong et al. 2014).

3 Methodology Figure 1 shows a flow chart for the methodology of this work.

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Fig. 1. Methodology and flow chart

The data studied in this research is similar in nature to that by Benhood and Mannering (2019) where they arrived at the conclusion that using a random parameters (mixed) logit model provides the most statistically superior model to study the factors affecting injury severity in large-truck accidents. In this paper, the dependent variables are the four reported injury severity levels, which are minor injury, moderate injury, severe injury, and fatal, while the independent variables are the ones shown in the first column of the model tables in the next section. The equation for the mixed effect logit model used in this study fits the following form: g{E(y |X, u)} = Xβ + Zu, y ∼ F y is the n × 1 vector of responses from the distributional family F, X is an n × p design/covariate matrix for the fixed effects β, and Z is the n × q design/covariate matrix for the random effects u. The Xβ + Zu part is called the linear predictor, and it is often denoted as η. The linear predictor contains the offset or exposure variable when offset() or exposure() is specified. g(·) is called the link function and is assumed to be invertible such that: E(y|X , u) = g −1 (X β + Zu) = H (η) = μ For notational convenience here and throughout this manual entry, we suppress the dependence of y on X. Substituting various definitions for g(·) and F results in a wide array of models. If g(·) is the logit function and y is distributed as Bernoulli, we have logit{ E(y)} = X β + Zu, y ∼ Bernoulli To account for the unobserved heterogeneity in the means and variances of random parameters, βkn is treated as a vector of estimable parameters that varies across crashes

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as Seraneeprakarn et al. (2017) and Behnood and Mannering (2017b): βkn = βk + θkn Zkn + σkn EXP(ωkn Wkn )vkn where βk is the mean parameter estimate across all observed accidents, Zkn is a vector of explanatory variables that captures heterogeneity in the mean that affect injury severity level k, θkn is a corresponding vector of estimable parameters, Wkn is a vector of explanatory variables that captures heterogeneity in the standard deviation σkn with corresponding parameter vector ωkn , and vkn is a disturbance term. After the forms are filled and the data is collected and transcribed in an excel sheet, it was coded using the statistical analysis software. The key indicator for a relation between variables is the p-value, and wherever a low p-value exists it means that changes in the predictor’s value are related to changes in the response variable. If the p-value is larger this suggests that there is no correlation and changes in the predictor are not affected by changes in the variable. Then Wald test is used to find out if explanatory variables in a model are significant. The chi-square formula used in this report is: χ 2 = (O-E)2 /E; Where, χ2 is Chi square; O is the Observed Frequency in each category; E is the Expected Frequency in the category. Also, there is the link test which adds the squared independent variable to the model and tests for significance of the model used if: P value of _HAT < 0.05; P value of _HATSQ > 0.05.

4 Results and Discussion Accident data were converted into binary data, to be understood and analyzed through the Stata. Accident data for vehicle only accidents and pedestrian-involved (runover) accidents were analyzed separately and are thus represented in two different sections in this chapter. 4.1 Summary of Variables Used in Models By studying the mean and standard deviation of the logic variables, Table 1 summarize the major reasons for accidents of the observed data. Table 1. Percentage of runovers and vehicle-only accidents due to several variable Runover Accidents

Vehicle-Only Accidents

Variable

Percentage

Variable

Percentage

Not paying attention

42%

Crossing red light

43%

Crossing red light

19%

Tailgating

13% (continued)

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S. AlKheder Table 1. (continued)

Runover Accidents

Vehicle-Only Accidents

Variable

Percentage

Variable

Percentage

Speeding without consideration for road ahead

11.1%

Driving outside lane

7.8%

Not giving priority to pedestrians

6.5%

Speeding without consideration for road ahead

7.5%

No pedestrian crossing

21%

Old Age (>50)

8.5%

Old Age (>50)

13.6%

Middle Age (31 – 50)

39%

Middle Age (31 – 50)

38%

Young Age (< 31)

52.4%

Young Age (< 31)

48.4%

Sideswipe

36%

Side-impact

22%

Rear-end collision

21%

Pileup accident

6.6%

4.2 Regression Analysis for Vehicle-Only Accidents 4.2.1 Regression Model Analysis Between the Different Injury Levels and Driver Characteristics in Vehicle-Only Accidents To be able to properly assess the characteristics of the drivers involved in non-pedestrian accidents, the data in this section was categorized for drivers only. The resulting number of observations is 2,115. The relation between the injury levels resulting from the accidents and the characteristics of the drivers were modeled and the results summery are shown in Table 2. From the p-value for the _hat and _hatsq it can be said that using random parameters mixed logit model is a good fit for analyzing driver characteristics affecting injury levels in accidents. A quick glance at the results shows that there is higher chance of injury to be more severe and the probability of a fatal accident if the driver was male or not wearing seatbelt. It can be noticed that there doesn’t exist any significant correlation between a driver’s age and the severity of injuries in vehicle-only accidents. 4.2.2 Regression Model Analysis Between the Different Injury Levels and Accident Types in Vehicle−Only Accidents For the models in this subsection and the remaining subsections relating to vehicle-only accidents the reported accident data for passengers was added to the 2,115 observations exclusive to drivers, resulting in a total number of 4,185 observations. In this section the results of the random parameter mixed logit model are analyzed for injury levels as dependent variables and the different accident types as independent variables. The results in Table 3 show that there is a significant relation between pile-up accidents and minor injuries, whereby the occurrence of a pile-up accident will most likely result in only minor injuries. There was no significant correlation observed between the different

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Table 2. Regression model results between the different injury levels and driver characteristics in vehicle-only accidents Defined for

Minor Injury

Variables Coefficient

Moderate Injury P > z Coefficient

P>z

Severe Injury Coefficient

Fatal Injury P > z Coefficient

P>z

Old Age (>50)

−0.0488946 0.769 −0.2052055

0.262

0.606839

Middle Age (31 – 50)

−0.120803

0.238 0.1281741

0.231

0.1292683 0.542 −0.3032002

Young Age (18 – 30)

0.0571264

0.529 −0.0630505

0.515 −0.1390485 0.458

0.3296764

0.265

Gender (1 for Male)

−0.6004572 0.000 0.2834849

0.033

1.306266

0.033

Seatbelt on

0.9881655

0.000 −0.6701273 0.001 −1.104007

0.000

0.000 −0.7312629

1.656378

0.038

0.000

0.445389

0.328 0.381

Model stats: Wald chi2

108.04

58.61

30.88

23.90

_hat

0.7968865

0.000 0.7762944

0.000

0.1408053 0.037

0.0693232

0.111

_hatsq

−0.3813393 0.081 0.2725396

0.102

0.0156087 0.667

0.0062185

0.247

accident types and moderate injuries, however one significant correlation was observed for severe injuries as a result of falling off a vehicle. Since fatal injuries have a low percentage from the total number of observed injuries the correlation that exist with the accident types are negative correlations. The 3 types of accidents that increase the likelihood of not observing a fatality are rear-end collisions, side-impact collisions and sideswipe collisions. The results help in understanding which types of accidents result in more severe injuries, and accordingly help those responsible for traffic planning to take this kind of information into consideration. Although the results obtained from the model didn’t have many correlations the ones that exist do make sense. All the link-tests resulted in a good fit for the model by having all the p-values for _hat be significant, while those of _hatsq insignificant. 4.2.3 Regression Model Analysis Between the Different Injury Levels and Road, Weather, and Light Conditions in Vehicle-Only Accidents Table 4 represents three significant correlations for minor injuries which are when the road is dry paved, when the weather is clear and the third and unexpected result is when the weather is foggy. The opposite is true for moderate injuries and having dry and paved surface of roads, whereby the negative correlation implies that it is less likely

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that there will be a moderate injury when these conditions are met. For more serious injuries there were two interesting observed positive correlations, the first is severe injury when the surface of the road is covered with sand. The second is fatal injuries when having a weak light at night. Again, all the models showed a good fit under the link test, further cementing the robustness of the random parameter logit model for analyzing traffic accident data. The results of this regression are very beneficial as they prove the importance of maintaining road surfaces and road lighting in avoiding severe and fatal injuries occurring in accidents. Table 3. Regression model results between the different injury levels and accident types in vehicle-only accident Defined for

Minor Injury

Variables

Coefficient

Moderate Injury

Collision with fixed object on road

−0.1540043 0.662

0.0629748 0.867

1.330013

0.201 −0.7969502 0.255

Falling off vehicle

−0.6505876 0.235

0.117783

2.190667

0.048 −0.5108647 0.667

Head-on collision

0.5084899

0.148 −0.3759747 0.316

0.7734823 0.460 −2.442426

0.312

Pileup accident

0.9298628

0.036 −0.8938179 0.067

0.9130066 0.422 −14.75653

0.983

Rear-end collision

0.4953634

0.138 −0.3342021 0.348

0.630938

0.538 −2.19486

Rollover

−0.2863654 0.423

0.2146329 0.572

1.292181

0.217 −0.7396832 0.300

Run-off road collision

−0.2293741 0.555

0.3119391 0.446

0.2702903 0.817 −0.3483466 0.645

P > z Coefficient

Severe Injury

P > z Coefficient

0.839

Fatal Injury P > z Coefficient

P>z

0.002

Side-impact 0.3269897

0.326 −0.244757

0.491

0.9179079 0.369 −1.762297

0.008

Sideswipe

0.180 −0.3127955 0.374

0.5930637 0.561 −1.511415

0.017

0.4427197

Model stats: Wald chi2

57.91

_hat

0.2629038

0.004

28.03 0.2864749 0.016

20.91

0.1408053 0.007 0.0333155

26.21 0.000

_hatsq

−0.0312255 0.714

0.0454776 0.620

0.0156087 0.667 0.0014968

0.317

4.2.4 Regression Model Analysis Between the Different Injury Levels and Accident Reasons in Vehicle-Only Accidents Table 5 shows that the linktest results for the minor injury regression model have a significant p-value for both _hat and _hatsq. This implies that the model is not considered

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Table 4. Regression model results between the different injury levels and road, weather, and light conditions in vehicle-only accidents Defined for

Minor Injury

Variables

Coefficient

Moderate Injury P > z Coefficient

Severe Injury

P > z Coefficient

Dry −1.108796 covered with sand

0.193 −0.1899808 0.806

Dry paved

0.6801091

0.002 −0.6942843 0.002

Foggy

1.589793

0.020 −1.430416

Night weak light

−0.5897766 0.227 0.5110654

2.080477

Fatal Injury P > z Coefficient P > z 0.017 −12.88633 0.996

−0.2897721 0.500 0.4053088

0.689

0.054 −14.0089

0.982 14.61998

0.992

0.317 −13.24112

0.985 1.775651

0.021

Model stats: Wald chi2

23.42

16.22

15.96

11.44

_hat

0.242162

0.000 0.2670306

0.042

0.1157752 0.000 0.0165404

0.018

_hatsq

−0.0203565 0.696 0.0359217

0.635

0.0059297 0.462 0.000653

0.134

a good fit for this type of regression but it shows a good fit for the three remaining injury levels and thus more stress should be given towards the significant correlations observed under these models. It can be observed that minor injuries are related to crossing a red light and tailgating. However, moderate injuries are highest likely to occur in accidents where the driver did not stop at a stop sign. Speeding and driving under the influence have a strong positive correlation with severe injuries. While, stronger correlation exists between speeding and fatal injuries, proving that speeding is the number one cause of death in car accidents. The other significant cause of death based on the results of the model is exhaustion and sleepiness. Table 5. Regression model results between the different injury levels and accident reasons in vehicle-only accidents Define for

Minor Injury

Variables

Coefficient

Crossing red light

Moderate Injury P > z Coefficient

Severe Injury

P > z Coefficient

Fatal Injury P > z Coefficient

P>z

0.8151056 0.000 −0.820322 0.000 −0.365131 0.160 −0.231512 0.646

(continued)

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S. AlKheder Table 5. (continued)

Define for

Minor Injury

Variables

Coefficient

P > z Coefficient

P > z Coefficient

Didn’t stop at stop sign

−0.33155

0.098 0.452094

0.026 −0.259578 0.586 −14.22525

Exhaustion and sleepiness

Moderate Injury

Severe Injury

0.3471431 0.411 −0.841894 0.097 0.3134005

Not paying attention

−0.487588

0.032 0.2510493

Speeding

−1.228235

0.000 −0.184399 0.485 1.497829

Tailgating

0.281 0.6530654

Fatal Injury P > z Coefficient

P>z

0.684 1.742569

0.043

0.107 0.802562

0.278

0.000 2.851232

0.000

0.5767707 0.000 −0.463790 0.002 −0.468491 0.153 −14.22525

Model stats: Wald chi2 _hat _hatsq

195.74

121.15

0.2377129 0.000 0.222766 −0.014360

0.000 0.0133777

65.62

-

0.022 0.0956273

0.001 0.0595625

0.000

0.627 0.0050036

0.954 0.0025747

0.114

4.3 Regression Analysis for Pedestrian-Involved (Runover) Accidents The number of observations for pedestrian involved accidents that were reported were 1,176. The data were analyzed similar to what was done for vehicle-only accidents. Results showed that road, weather and lighting conditions play no role in the severity of injuries sustained in runover accidents. 4.3.1 Regression Model Analysis Between the Different Injury Levels and Driver Characteristics in Runover Accidents The regression model for driver characteristics is done for minor and moderate injury levels only due to the small number of observations where the drivers sustained an injury that was considered a serious injury in comparison to those sustained by pedestrians. Therefore, a more relevant conclusion regarding the characteristics of drivers involved in runover accidents can be obtained from the previous tables. The model shows that if the driver was male, it is highly probable that the injuries sustained are moderate injuries and less likely to be minor ones. As expected, the injuries sustained when the seatbelt is fastened are most likely to be minor ones not moderate. The linktest implies that both models are a good fit. 4.3.2 Regression Model Analysis Between the Different Injury Levels and The Flow of Pedestrians In Runover Accidents The parameters in this subsection help in understanding the position of the pedestrian when a runover accident occurs. Since the models for both moderate and severe injuries

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show a positive correlation when the pedestrian was crossing at a point where there was no pedestrian crossing, the more accurate model is chosen based on the result of the link test which implies that the result in moderate injuries is a good fit not like the results of the severe injuries as shown in Table 6. Another parameter shows that high probability of death resulted when the pedestrian was not using the existing pedestrian crossing to cross. It is interesting to observe the fact that none of the seven independent variables have a positive significant correlation with minor injuries, meaning runover accidents in nature regardless of the position of the pedestrian, result in more serious injuries than minor ones. This kind of information is very important for traffic planners as better safety measures can be installed for protecting pedestrian standing on medians, for example by installing more protective poles. Table 6. Regression model results between the different injury levels and the flow of pedestrians in runover accidents Defined for Minor Injury

Moderate Injury

Severe Injury

Fatal Injury

Variables

Coefficient

P > z Coefficient

P > z Coefficient P > z Coefficient

P>z

Crossing red light

1.51334

0.000 −1.153489

0.000 −1.113616 0.028 −0.9974636 0.054

Didn’t stop at stop sign

1.392397

0.048 −1.220888

0.133 0.0952972

0.931 −15.06194

0.994

Driving outside lane

0.7564937 0.048 −0.6078805 0.127 0.1524456

0.807 −1.202257

0.262

Not giving priority to pedestrians

−0.546773

0.133 0.0706696

0.863 0.496043

0.234

Speeding

−0.3197189 0.493 −0.8845638 0.065 0.2130583

0.760 1.715129

0.001

0.031 0.3564541

Model stats: Wald chi2

112.00

66.55

15.37

38.09

_hat

0.1504657 0.000 0.2403674

0.000 0.0421355

0.000 0.1105731

0.000

_hatsq

0.0001472 0.931 0.015805

0.086 0.0018677

0.000 0.0052328

0.651

4.3.3 Regression Model Analysis Between the Different Injury Levels and Accident Reasons in Runover Accidents Similar to the same parameters studied in vehicle-only accidents, all the parameters apply for runover accidents with the addition of lack of priority given to pedestrians by drivers. Unlike the results observed in the models for accident reasons in vehicle-only accidents, the models for accident reasons and injury levels in runover accident provided different correlations which prove that vehicle-only accidents and runover accidents should not be assessed together as one accident reason that might be considered fatal

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in runover accidents, might be less dangerous in vehicle-only accidents. The results in Table 7 show that minor injuries were correlated to not stopping at stop sign, driving outside the designated lanes, and crossing when the pedestrian signal was red. There was one significant result obtained from the regression model with fatal injuries which is speeding. Speeding hence is found to be the leading cause for fatalities in both vehicleonly and runover accidents (Table 8). Table 7. Regression model results between the different injury levels and accident reasons in runover accident Minor Injury

Moderate Injury P > z Coefficient

Severe Injury

P > z Coefficient

Fatal Injury P > z Coefficient

P>z

Variables

Coefficient

No pedestrian crossing

−0.9868486 0.000

0.7091356 0.000

0.577192

Not using pedestrian crossing

−1.078698

0.000

0.6975988 0.000

0.5003988 0.066

1.076535

Standing on the side of the road

−0.7174042 0.013

0.5593373 0.050

0.0440921 0.936

0.9241033 0.110

Standing on median

−2.391381

0.025

1.434806

0.000

Using −0.8425673 0.039 pedestrian crossing no intersection

Using −1.57307 pedestrian crossing at intersection

0.044 −11.99651

0.027

0.5835454 0.126

0.002

0.980

2.216872

0.008

0.9239804 0.000

−0.0777005 0.875

1.875701

0.000

0.5185153 0.193

0.0639082 0.932

1.390193

0.036

Model stats: Wald chi2

82.20

_hat

0.2554378

0.000

37.82 0.242483

0.014

0.1101691 0.009

6.93

29.09 0.2230832 0.023

_hatsq

0.032057

0.466

0.0049289 0.971

0.0060751 0.017

0.0294598 0.104

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Table 8. Summary of Results with good fit and positive correlation Model Name

Minor

Moderate

Severe

Fatal

Driver Characteristics in Vehicle Only accidents

-UAE citizen -Seatbelt on

-Gender (Male)

-Old age (>50) -Gender (Male)

Not a good fit

Accident types in vehicle-only accidents

-Pileup accident

None

-Falling off vehicle accident

Negative correlation

Road, weather, and light conditions in vehicle-only accidents

-Dry paved -Clear weather -Foggy weather

Negative correlation

-Dry paved covered with sand

-Night weak light

Accident reasons Not a good fit in vehicle-only accidents

Negative correlation

-Driving under influence -Speeding

-Exhaustion and sleepiness -Speeding

Driver Characteristics in runover accidents

-Seatbelt on

-Gender (Male)

Not conclusive

Not conclusive

Flow of pedestrians in runover accidents

Negative correlation

-No pedestrian crossing -Not using pedestrian crossing -Standing (on side of road) -Standing on median -Using pedestrian crossing at intersection

Not a good fit

-Not using pedestrian crossing -Standing on median -Using pedestrian crossing at intersection -Using pedestrian crossing no intersection

Road, weather, and light conditions in runover accidents

None

Not a good fit

None

None

Not a good fit

Speeding

Accident reasons -crossing red light Negative in runover -Didn’t stop at stop Correlation accidents sign -Driving outside lane

5 Conclusion and Recommendation The study team found that the best model for analyzing traffic accidents is the randomparameter logit model. The models for driver characteristics in both vehicle-only and runover accidents showed that there is a strong correlation between injury severity and

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wearing the seatbelt or not. Moreover, there is a correlation between the driver being male and the probability of a severe and fatal accident. Falling off a vehicle was found to be correlated with severe injuries, while rear-end collisions, side-impact collisions and sideswipe collisions are the least likely to cause fatal accidents. Regarding road, weather and light conditions no significant correlations were found with runover accidents, whereas in vehicle-only accidents, it was observed that there is a correlation between severe injuries and when the road surface is covered with sand, and when the road is weakly lit and fatal accidents. The results of the model studying accident reasons and their effect on injury levels in vehicle-only accidents showed that while crossing red light will likely result in minor injuries, speeding is the most dangerous parameter along with exhaustion and sleepiness resulting in a high probability of fatal accidents. It was also found that accidents occurring due to driving under influence have a significant correlation with severe injuries. The results of the models for runover accidents showed that speeding again was the most dangerous parameter resulting in severe or fatal accidents. While crossing red lights will most likely result in minor injuries. Minor injuries are also correlated with accidents where the driver is driving outside the designated lane, these types of accident normally cause injuries to the feet of pedestrians. A very informative model obtained from this study is the model studying the position of the pedestrian when the accident occurred and the injury levels sustained during these accidents. The results of this model showed that crossing where it is illegal to cross results in the highest number of accidents. Crossing where there is no pedestrian crossing will most probably result in a moderate injury while not using the pedestrian crossing where there is one, will likely result in death. This model also showed that there exists a strong correlation with fatal injuries when the pedestrian is runover at a standing position in the median or on the side of the road, and when using the pedestrian crossing at intersections. These results show that pedestrian safety needs to be improved to decrease the probability of fatal accidents happening in runover accidents. Acknowledgement. The author would like to thank Abeer Abdullah Alenezi, Mariam Ali Alwohaida, Shaikah Saleh Almutairi, and Nouf Khaled Alsaqabi for their great help in data collection and analysis.

References Abrari Vajari, M., Aghabayk, K., Sadeghian, M., Shiwakoti, N.: A multinomial logit model of motorcycle crash severity at Australian intersections. J. Safety Res. 73, 17–24 (2020). https:// doi.org/10.1016/j.jsr.2020.02.008 Alkheder, S., Alrukaibi, F., Aiash, A.: Road safety modeling in Kuwait. Int. Conf. Ind. Eng. Oper. Manag. (Feb. 14, 2020) Azimi, G., Rahimi, A., Asgari, H., Jin, X.: Severity analysis for large truck rollover crashes using a random parameter ordered logit model. Accid. Anal. Prev. 135, 105355 (2020). https://doi. org/10.1016/j.aap.2019.105355 Behnood, A., Mannering, F.: Time-of-day variations and temporal instability of factors affecting injury severities in large-truck crashes. Anal. Methods Accid. Res. 23, 100102 (2019). https:// doi.org/10.1016/j.amar.2019.100102

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Fountas, G., Anastasopoulos, P.C., Abdel-Aty, M.: Analysis of accident injury-severities using a correlated random parameter ordered probit approach with time variant covariates. Anal. Methods Accid. Res. 18, 57–68 (2018) Pahukula, J., Hernandez, S., Unnikrishnan, A.: A time of day analysis of crashes involving large trucks in urban areas. Accid. Anal. Prev. 75, 155–163 (2015) Mashhadi, M., Wulff, S., Ksaibati, K.: A comprehensive study of single and multiple truck crashes using violation and crash data. Open Transp. J. 12(1), 43–56 (2018). https://doi.org/10.2174/ 1874447801812010043 Russo, B., Savolainen, P., Schneider, W., Anastasopoulos, P.C.: Comparison of factors affecting injury severity in angle collisions by fault status using a random parameter bivariate ordered probit model. Anal. Methods Accid. Res. 2, 21–29 (2014) Xiong, Y., Tobias, J., Mannering, F.: The analysis of vehicle crash injury-severity data: a Markov switching approach with road-segment heterogeneity. Transp. Res. Part B 67, 109–128 (2014) Zhang, J., Lindsay, J., Clarke, K., Robbins, G., Mao, Y.: Factors affecting the severity of motor vehicle traffic crashes involving elderly drivers in Ontario. Accid. Anal. Prev. 32, 117–125 (2000)

Correction to: The Use of Z-numbers to Assess the Level of Motivation of Employees, Taking into Account Non-formalized Motivational Factors Alekperov Ramiz Balashirin

Correction to: Chapter “The Use of Z-numbers to Assess the Level of Motivation of Employees, Taking into Account Non-formalized Motivational Factors” in: C. Kahraman et al. (Eds.): Intelligent and Fuzzy Systems, LNNS 758, https://doi.org/10.1007/978-3-031-39774-5_42

In the original version of this chapter, the reference [1] was incorrect. The correct version is given below: Lambovska, M.: A fuzzy logic model for evaluating the motivation for high-quality publications: evidence from a Bulgarian university. Manag. J. Contemp. Manag. Issues 27(2), 87–108 (2022). https://doi.org/10.30924/mjcmi.27.2.6 The Original chapter has been corrected.

The updated original version of this chapter can be found at https://doi.org/10.1007/978-3-031-39774-5_42 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, p. C1, 2023. https://doi.org/10.1007/978-3-031-39774-5_90

Author Index

A A Kadir, Norhidayah 678 Abbasov, Ali 577 Abdel Bary, Taher Abouzaid Abdel Aty Abraham, Ajith 626 Aghajanov, Jamirza 767 Akın, Erdem 307 Akkaya, Beril 365 Al-Anzi, Fawaz S. 532 Aliev, Elmar 767 Aliyev, Murad 577 AlKheder, Sharaf 816 Alkurt, Rana Duygu 635 Amador-Angulo, Leticia 318 Amar, Hichem 454 Amézquita, Lucio 471 Amir, Monir 454 An, Jiyao 569 Ansari, Mansour 586 Argun, Irem Düzdar 757 Ari, Emre 730 Arora, Ritu 540 Arora, Shalini 540 Arslan, Tarık Tu˘gra 660 Arya, Alka 689 Asan, Umut 152 Atanassov, Krassimir 11, 243, 259 Atik, Furkan 462 Avramova-Todorova, Gergana 220 Ayber, Sezen 284 Aydın, Ahmet 284 B Babes, Badreddine 454 BahooToroody, Ahmad 808 Bajaj, Anu 626 Balashirin, Alekperov Ramiz 354 Barakath, A. Jamal 144 Basaran, Alparslan Abdurrahman 77 Basaran, Murat Alper 77 Baysal, Mehmet Emin 652

738

Beyca, Omer Faruk 730 Bibiloni-Femenias, M. D. M. 94 Blesi´c, Andrija 119 Boltürk, Eda 716 Bozhenyuk, Alexander 160, 168 Burda, Michal 102 Bureva, Veselina 220, 227, 259 Butsenko, Elena 407, 708 C Can, Yaren 480 Castillo, Oscar 318, 471 Çaylı, Gül Deniz 110 Cebi, Selcuk 440, 776, 787 Cerit, Ilgaz 641 Ceylan, Sude Dila 480 Chiclana, Francisco 678 Choi, Seung Hoe 619 Çifçi, Yusuf 660 ˇ Comi´ c, Lidija 119 Cortes-Antonio, Prometeo 471 D Dare, V. R. 204 Demir, Mehmet Ozer 77 Demircan, Murat Levent 699 Dhruv, Sakshi 540 Dirinda, Tayfun 699 Dobrev, Pavel 212 Duran, Devin 480 Durcheva, Mariana 85 Dursun, Mehtap 635 Duruel, Turgay 798 E Edward, Jency Leona 426 Elomda, Basem Mohamed 738 Engin, Orhan 652 Erdem, Gamze 521 Ergin, Simge Güçlükol 521, 610 Erginel, Nihal 234, 284

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Kahraman et al. (Eds.): INFUS 2023, LNNS 758, pp. 831–833, 2023. https://doi.org/10.1007/978-3-031-39774-5

832

Author Index

Erol Genevois, Müjde 721 Erol, Ba¸sak 480 Esmi, Estevão 47 F Fang, Ju 569 G Gani, Ahmet 561 Ghodbane, Hatem 454 Gökçe, Mahmut Ali 610 Göksel, Gökhan 284 Gülbay, Murat 376 Gülsün, Bahadır 798 Gultekin, Ahmet Berkay 730 Gündo˘gdu, Fatma Kutlu 660 Gürbüz, Tuncay 641 Gurcan, Omer Faruk 730 H Habibbayli, Tunjay 577 Haktanır, Elif 267 Hamouda, Noureddine 454 Hasancebi, Safacan 499 Hassan, Hesham Ahmed 738 Hatami-Marbini, Adel 689 I Ili´c, Vladimir 119 Indrati, Ch. Rini 293 Intepe, Gizem 396 Ivanova, Malinka 85 J Jansirani, N.

204

K Kacprzyk, Janusz 3 Kahraman, Cengiz 275, 326, 337, 346, 365, 376, 389 Kalayathankal, Sunny Joseph 293 Kaliyaperuma, Palanivel 787 Kamis, Nor Hanimah 678 Khalili, Pariya 586 Khoshnevis, Pegah 689 Kilic, Huseyin Selcuk 499, 670 Kilicman, Adem 678 Kim, Mi Young 619 Knyazeva, Margarita 160, 168

Kocaman, Hatice 152 Kondratenko, Yuriy 491 Kosenko, Olesiya 168 Kotak, Paresh P. 597 Kozlov, Oleksiy 491 Kreinovich, Vladik 491 Kuzu, Tunahan 480 L Lee, Nan-Hi 619 Lepskiy, Alexander 69 Liu, Qingqin 569 Lupiáñez, Francisco Gallego

65

M Metin, Akın 660 Miñana, J.-J. 94 Moˇckoˇr, Jiˇrí 136 Murinová, Petra 102 N Nasir, V. Kamal

144

O Onar, Sezi Cevik 275, 326, 346, 389 Orhan, Deniz 721 Örnek, Mustafa Arslan 415, 480 Ozdemir, Tugce 757 Özok, Ahmet Fahri 35 Öztay¸si, Basar 275, 307, 326, 346, 389 P Palanivel, K. 426, 440, 507 Paldrak, Mert 415, 480, 521 Parlak, Fatma Cayvaz 670 Patel, Himanshukumar R. 597 Paunovi´c, Marija 119 Pavliska, Viktor 102 Perfilieva, Irina 22 Petrov, Petar 259 R Ralevi´c, Nebojša 119 Rathore, Umesh C. 597 Raval, Sejal 597 Ribagin, Simeon 243, 252 Roggow, Brian J. 808 Rosyida, Isnaini 293 Rozenberg, Igor 168

Author Index

Rzayev, Ramin 577, 767 Rzayeva, Inara 767 S Safavi, Ali Akbar 586 Sambyal, Nitigya 626 Samoylov, Leontiy 160 Sarath, Sumi 532 Sarı, Elvin 480 Sarucan, Ahmet 462, 652 Sentürk, ¸ Sevil 234 Shah, Vipul 597 Shahidi, Mina 47 Shorikov, Andrey 407, 708 Singh, Koushal 54 Singh, Surender 54 Sivri, Mahmut Sami 730 Soria, Jose 471 Sotirov, Sotir 243, 252 Sotirova, Evdokia 212, 252 T Taco˘glu, Melis Tan 415, 521 Timchenko, Victor 491 Todorov, Milen 227 Torra, Vicenç 39

833

Toufani, Samrand 396 Tranev, Stoyan 185, 194 Traneva, Velichka 185, 194 Tüysüz, Nurdan 337 Tuzkaya, Gulfem 499, 670 U Ucal Sari, Irem 396 Ünver, Mustafa 177, 284 Ustundag, Alp 730 V Valero, O. 94 Vatankhah, Ramin 586 Vijayaraghavan, N. 204 Y Yazdi, Mohammad 808 Yörür, Bahadır 234 Z Zadro˙zny, Sławomir 3 Zarei, Esmaeil 808 Zarzycki, Hubert 550 Zhang, Xuan 569 Zhang, Yinsheng 128