Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science: ICMMCS 2021 (Advances in Intelligent Systems and Computing, 1422) 9811901813, 9789811901812

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Advances in Intelligent Systems and Computing 1422

Sheng-Lung Peng Cheng-Kuan Lin Souvik Pal   Editors

Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science ICMMCS 2021

Advances in Intelligent Systems and Computing Volume 1422

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by DBLP, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST). All books published in the series are submitted for consideration in Web of Science. For proposals from Asia please contact Aninda Bose ([email protected]).

More information about this series at https://link.springer.com/bookseries/11156

Sheng-Lung Peng · Cheng-Kuan Lin · Souvik Pal Editors

Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science ICMMCS 2021

Editors Sheng-Lung Peng Department of Creative Technologies and Product Design National Taipei University of Business Taoyuan, Taiwan

Cheng-Kuan Lin Department of Computer Science National Yang Ming Chiao Tung University Hsinchu, Taiwan

Souvik Pal Department of Computer Science and Engineering Sister Nivedita University Kolkata, West Bengal, India

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-981-19-0181-2 ISBN 978-981-19-0182-9 (eBook) https://doi.org/10.1007/978-981-19-0182-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organizing Committee and Key Members

Conference Committee Members Conference General Chair Sheng-Lung Peng, National Taipei University of Business, Taiwan

Program Conveners Hanaa Hachimi, Secretary General of Sultan Moulay Slimane University USMS of Beni Mellal, Morocco Satyendra Narayan, Sheridan Institute of Technology, Ontario, Canada Noor Zaman Jhanjhi, Taylor’s University, Malaysia

Conference Organizing Chairs D. Balaganesh, Lincoln University College, Malaysia Souvik Pal, Department of Computer Science and Engineering, Sister Nivedita University, Kolkata, West Bengal, India

Program Chairs D. Akila, Vels Institute of Science, Technology and Advanced Studies, India R. Esther Felicia, Shri Krishnaswamy College for Women, India Ahmed J. Obaid, University of Kufa, Iraq v

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Organizing Committee and Key Members

International Advisory Board Members Ahmed A. Elnger, Beni-Suef University, Egypt M. M. Awad, Mansoura University, Egypt Dac-Nhuong Le, Haiphong University, Vietnam J. M. Chang, National Taipei University of Business, Taiwan Anirban Das, University of Engineering and Management, India Kusum Yadav, University of Hail, Kingdom of Saudi Arabia Debashis De, Maulana Abul Kalam Azad University of Technology, India M. J. Diván, National University of La Pampa, Argentina Srinath Doss, Botho University, Botswana S. K. Hoskere, Shanghai United International School, China Sanjeevikumar Padmanaban, Aalborg University, Denmark Vasaki Ponnusamy, Universiti Tunku Abdul Rahman, Malaysia Bikramjit Sarkar, JIS College of Engineering, India Noor Zaman, Taylor’s University, Malaysia

Technical Chair V. R. Elangovan, A. M. Jain College, India R. K. Bathla, Desh Bhagat University, India N. Kalaivani, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India Abhishek Dhar, Swami Vivekananda University, India Kishore Ghosh, Swami Vivekananda University, India P. Sumathi, C. Kandaswami Naidu College for Men, India T. Nusrat Jabeen, Anna Adarsh College for Women, India

Technical Program Committee Members R. Amutha, Thiruvalluvar University College Arts and Science, India Ranbir Singh Batth, Lovely Professional University, India Sitikantha Chattopadhyay, Brainware University, India L. J. Hung, National Taipei University of Business, Taiwan Anand Paul, Kyungpook National University, South Korea Kamal Wadhwa, Government PG College, Pipariya, India V. Vijayalakshmi, SRM Institute of Science and Technology, Kattankulathur, India C. K. Lin, Fuzhou University, China S. Vimal, Jeppiaar Engineering College, Chennai, India K. Kavitha, Chellammal College for Women, India

Organizing Committee and Key Members

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K. J. Pai, Ming Chi University, Taiwan Ashish Mishra, Gyan Ganga Institute of Technology and Science, India N. Pradeep, Bapuji Institute of Engineering and Technology, India K. Hema Shankari, Women’s Christian College, India Mohammed Kaicer, Ibn Tofail University, Morocco S. Mathivilasini, Ethiraj College for Women, India T. Nathiya, New Prince Shri Bhavani Arts and Science College, Chennai, India T. Nagarathinam, MASS College of Arts and Science, Thanjavur, India S. M. Tang, National Defense University, Taiwan K. Kavitha, Mother Teresa Women’s University, India E. Ramaraj, Alagappa University, India R. Kalaiarasi, Tamil Nadu Open University, India R. Velmurugan, Presidency College, India D. Napoleon, Bharathiar University, India

Session Chairs Yahya-Imam Munir Kolapo, Lincoln University College, Malaysia Arthi Ganesan, PSGR Krishnammal College for Women, India Ton Quang Cuong, Vietnam National University, Vietnam Bikramjit Sarkar, JIS College of Engineering, India B. Vivekanandam, Lincoln University College, Malaysia Mathiyalagan Kalidass, Bharathiar University, India R. Sivaraman, D. G. Vaishnav College, India Midhunchakkaravarthy Janarthanan, Lincoln University College, Malaysia V. R. Elangovan, Agurchand Manmull Jain College, India Suresh Rasappan, University of Technology and Applied Sciences, Ibri, Sultanate of Oman A. Meenakshi, VELS Institute of Science Technology and Advance Studies, India M. L. Suresh, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India

Invited Speakers Prof. (Dr.) Amiya Bhaumik, Vice Chancellor and CEO, Lincoln University College, Malaysia Prof. Sun-Yuan Hsieh, Chair Professor and Dean of R&D, Department of Computer Science and Information Engineering, National Cheng Kung University, Taiwan Dr. Sanpawat Kantabutra, Associate Professor, Theory of Computation Group, Faculty of Engineering, Chiang Mai University, Thailand

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Organizing Committee and Key Members

Prof. Satyendra Narayan, Professor, School of Applied Computing, Sheridan Institute of Technology, Ontario, Canada Prof. Hanaa Hachimi, Associate Professor, Applied Mathematics and Computer Science, Secretary General of Sultan Moulay Slimane University USMS of Beni Mellal, Morocco Dr. Ramakant Bhardwaj, Associate Professor, Department of Mathematics, Amity University, Kolkata, India Prof. E. Chandrasekaran, Professor, Mathematics Department, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India Prof. R. Anandan, Department of Computer Science and Engineering, Vels Institute of Science, Technology and Advanced Studies, India Prof. A. M. S. Ramasamy, Rtrd Professor, Mathematics and Dean, Ramanujan School of Mathematical Sciences, Pondicherry University, India

Preface and Acknowledgements

The main aim of this proceedings book is to bring together leading academic scientists, researchers, and research scholars to exchange and share their experiences and research results on all aspects of intelligent ecosystems, data sciences, and mathematics. ICMMCS 2021 is a multidisciplinary conference organized with the objective of bringing together academic scientists, professors, research scholars, and students working in various fields of engineering and technology. On ICMMCS 2021, you will have the opportunity to meet some of the world’s leading researchers, to learn about some innovative research ideas and developments around the world, and to become familiar with emerging trends in science–technology. The conference will provide the authors, research scholars, listeners with opportunities for national and international collaboration and networking among universities and institutions for promoting research and developing the technologies globally. This conference aims to promote translation of basic research into institutional and industrial research and convert applied investigation into real-time application. ICMMCS 2021 has been hosted by Lincoln University College, Malaysia. ICMMCS 2021 has been held during October 29 and 30, 2021, in online mode (ZOOM Platform). The conference brought together researchers from all regions around the world working on a variety of fields and provided a stimulating forum for them to exchange ideas and report on their researches. The proceedings of ICMMCS 2021 consists of 58 best selected papers which were submitted to the conferences and peer-reviewed by conference committee members and international reviewers. The presenters have presented through virtual screen. Many distinguished scholars and eminent speakers have joined from different countries like India, Malaysia, Vietnam, Iraq, Spain, Oman, Taiwan, and Morocco to share their knowledge and experience and to explore better ways of educating our future leaders. This conference became a platform to share the knowledge domain among different countries’ research culture. The main and foremost pillar of any academic conference is the authors and the researchers. So, we are thankful to the authors for choosing this conference platform to present their works in this pandemic situation. We are sincerely thankful to the Almighty for supporting and standing at all times with us, whether it is good or tough times and given ways to concede us. Starting ix

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Preface and Acknowledgements

from the call for papers till the finalization of chapters, all the team members have given their contributions amicably, which is a positive sign of significant team works. The editors and conference organizers are sincerely thankful to all the members of Springer, especially Mr. Aninda Bose for providing constructive inputs and allowing an opportunity to finalize this conference proceedings. We are also thankful to Prof. William Achauer and Prof. Anil Chandy for their support. We are also grateful to Mr. Mani Arasan, Project Coordinator, Springer, for his cooperation. We are thankful to all the reviewers who hail from different places in and around the globe, shared their support, and stand firm toward quality chapter submission in this pandemic situation. Finally, we wish all participants every success in your presentations and networking events. Your strong supports are critical to the success of this conference. We hope that the participants not only enjoyed the technical program in the conference but also found eminent speakers and delegates in the virtual platform. Wishing you a fruitful and enjoyable ICMMCS 2021. Taoyuan, Taiwan Hsinchu, Taiwan Kolkata, India

Sheng-Lung Peng Cheng-Kuan Lin Souvik Pal

Contents

A Study About (γ, γ )α –Regular Spaces, Normal Spaces with (γ, γ )–Open Sets and α(γ, γ  )– Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Kalaivani, E. Chandrasekaran, and Hanaa Hachimi The Role of Harvesting in a Food Chain Model and Its Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. M. Vijayalakshmi, Suresh Rasappan, Pugalarasu Rajan, and Ha Huy Cuong Nguyen

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Inventory Queuing System Study Using Simulation and Birth– Death Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Vijayarangam, S. Perumal, and J. Viswanath

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Steady-State Analysis of Bulk Queuing System with Renovation, Prolonged Vacation and Tune-Up/Shutdown Times . . . . . . . . . . . . . . . . . . . S. P. Niranjan, B. Komala Durga, and M. Thangaraj

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Estimation of Public Compliance with COVID-19 Prevention Standard Operating Procedures Through a Mathematical Model . . . . . . Norazaliza Mohd Jamil and Balvinder Singh Gill

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Campus Recruitment Cost Analysis: A Roadmap for HR Managers . . . . Md Jakir Hossain Molla, Sk Md Obaidullah, Parveen Ahmed Alam, Saurabh Adhikari, Sourav Saha, and Soumya Sen

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Certain Types of Domination in Nover Top Graphs . . . . . . . . . . . . . . . . . . . R. Narmada Devi, G. Muthumari, and Suresh Rasappan

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Analysis and Classification of Physiological Signals for Emotion Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gitosree Khan, Shankar Kr. Shaw, Sonal Aggarwal, Akanksha Kumari Gupta, Saptarshi Haldar, Saurabh Adhikari, and Soumya Sen

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Wiener and Zagreb Indices for Helm and Web Graph . . . . . . . . . . . . . . . . J. Senbagamalar, M. Priyadharshini, P. Rajesh, and Hanaa Hachimi Disease Classification Using Particle Swarm Optimization with Fuzzy Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Leoni Sharmila and S. Poongothai

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A Comparison of Fuzzy and ACO-Based Fuzzy for Classification of Bio-Medical Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 S. Poongothai and S. Leoni Sharmila Secret Information Sharing Using Probability and Bilinear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Kala Raja Mohan, Suresh Rasappan, Regan Murugesan, Sathish Kumar Kumaravel, and Ahmed A. Elngar Encryption on Graph Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A. Meenakshi, J. Senbagamalar, and A. Neel Armstrong A Novel Indexing Scheme Over Lattice of Cuboids and Concept Hierarchy in Data Warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Saurabh Adhikari, Sourav Saha, Anjan Dutta, Anirban Mitra, and Soumya Sen Modeling Interactive E-book: Computational Perspective and Design Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Nguyen Tung Lam, Vu Minh Trang, Nguyen Hoa Huy, and Ton Quang Cuong Bipolar-Valued Fuzzy Subhemirings of a Hemiring Under Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 N. Kumaran, T. Gunasekar, Naresh Kumar Jothi, and A. Neel Armstrong Hybrid Phase Synchronization for Generalized Stretch, Twist, Fold Flow Chaotic System of Fractional Order . . . . . . . . . . . . . . . . . . . . . . . 159 Nagadevi Bala Nagaram, Suresh Rasappan, Regan Murugesan, Kala Raja Mohan, and Hanaa Hachimi Glucose Distribution and Drug Diffusion Mechanism in the Fuzzy Fluid Connective Tissue in Human Systems: A Mathematical Modelling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Sachindra Nath Matia, Animesh Mahata, Shariful Alam, Banamali Roy, and Balaram Manna A Generic Modelling on Neo4j to Recommend Students for Suitable Job Sectors Based on Different Skill Set Parameters . . . . . . . 179 Runa Ganguli, Md. Jakir Hossain Molla, Punyasha Chatterjee, Saurabh Adhikari, Sourav Saha, and Soumya Sen

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Rainwater Harvesting for Washing of Clothes in Washing Machine by Using Raspberry Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Bishal Chakraborty, Bidisha Paul, Abhishek Dhar, Saurabh Adhikari, and Sourav Saha A Hybrid Deep Learning Models for Hetrogeneous Medical Big Data Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A. Manikandan and R. Anandan Novel on Digital Neutrosophic Topological Spaces . . . . . . . . . . . . . . . . . . . . 213 R. Narmada Devi, Suresh Rasappan, and Ahmed J. Obaid An Approach to Solving Linear Programming Problems by Using Trapezoidal Intuitionistic Fuzzy Number and the Dual-Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Rahul Kar, Ashok Kumar Shaw, Mainak Chakraborty, Subhendu Maji, and Smriti Ghosh Restoring of Fundus Retinal Image for Detection of Diabetic Retinopathy in Presence of Blurriness of Cataract . . . . . . . . . . . . . . . . . . . . 231 Suman Bhattacharya, Anirban Mitra, Moumita Chatterjee, Sudipta Roy, Saurabh Adhikari, and Soumya Sen Digital Transformation in Higher Education: A Case Study in Vietnam from Human Rights-Based Approach . . . . . . . . . . . . . . . . . . . . . 241 Doanh-Ngan-Mac Do, Ha Thuy Mai, and Trung Tran An Unstable Flow Past in a Vertical Plate Originating from the Parabola of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 A. Neel Armstrong and Hanaa Hachimi Opinion of Faculty About the Effectiveness of Online Class During COVID Pandemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Sathish Kumar Kumaravel, Suresh Rasappan, Regan Murugesan, Kala Raja Mohan, and Vicente García-Díaz A Deterministic Replenishment Policy for Constant Deteriorating Giffen Goods with Time-Dependent Demand . . . . . . . . . . . . . . . . . . . . . . . . . 271 Saranya Palanivelu and E. Chandrasekaran Randomly Selection of Interior Points in SV Learning Algorithm Uses of Confidence Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 M. Premalatha and C. Vijayalakshmi Using Convolutional Neural Networks for Fault Analysis and Alleviation in Accelerator Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Jashanpreet Singh Sraw and M. C. Deepak

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Trust-Based Efficient Computational Scheme for MANET in Clustering Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Joydeep Kundu, Sitikantha Chattopadhyay, Subhra Prokash Dutta, Koushik Mukhopadhyay, and Souvik Pal A Comparative Approach for Solving Fuzzy Transportation Problem with Hexagonal Fuzzy Numbers and Neutrosophic Triangular Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 T. Nagalakshmi, R. Sudharani, and G. Ambika Analysis of Student’s Feedback About Online Class Using Fuzzy Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Regan Murugesan, Suresh Rasappan, Sathish Kumar Kumaravel, Nagadevi Bala Nagaram, and Dac-Nhuong Le Analysis of an Imprecise Delayed SIR Model System with Holling Type-III Treatment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Ashish Acharya, Animesh Mahata, Shariful Alam, Smriti Ghosh, and Banamali Roy Design and Application of Virtual Reality Technology in Digital Pedagogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Tran Doan Vinh Adjacent Graph for Some Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 R. Kamali and C. J. Chris Lettecia Mary Stochastic Inventory Model Using Coxian Distribution with Production and Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 D. Kanagajothi and Hanaa Hachimi Dynamics of Infected Prey–Predator System in Fuzzy Environment with Disease-Selective Predation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Narayan Mondal, Sachindra Nath Matia, Animesh Mahata, Subhendu Maji, and Shariful Alam An Analysis for Business Development by the Project Management in Moroccan Companies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Kamelia Jahnouni and Hanaa Hachimi Text Region Identification from Natural Scene Images Using Semi-Supervised MSER Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Shiplu Das, Sitikantha Chattopadhyay, Ritesh Prasad, Joydeep Kundu, and Souvik Pal Visualization of Audio Files Using Librosa . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Shubham Suman, Kshira Sagar Sahoo, Chandramouli Das, N. Z. Jhanjhi, and Ambik Mitra

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Proper Lucky Labeling of Jelly Fish Graph, Cocktail Party Graph and Crown Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 T. V. Sateesh Kumar and S. Meenakshi Topological Indices on Central Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 J. Senbagamalar, A. Meenakshi, A. Kanchana, and Hanaa Hachimi Study of Time-Delayed Fractional Order SEIRV Epidemic Model . . . . . . 435 Subrata Paul, Animesh Mahata, Supriya Mukherjee, Mainak Chakraborty, and Banamali Roy Modeling of Teaching–Learning Process of Geometrical LOCI in the Plane with GeoGebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Pham Van Hoang, Ta Duy Phuong, Nguyen Thi Bich Thuy, Tran Le Thuy, Nguyen Thi Trang, and Nguyen Hoang Vu Application of Pentagonal Fuzzy Number in CPM and PERT Network with Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 G. Ambika and T. Nagalakshmi Solving Assignment Problem Using Decision Under Uncertainty and Game Theory: A Comparative Approach with Python Coding . . . . . 471 N. Kalaivani, E. Mona Visalakshidevi, and Mostafa Ezziyyani Interval-Valued Fuzzy Dynamic Programming Approach of Capital Budgeting Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 M. Ruthara and G. Uthra Characterization of Some Standard Graphs Based on the Eccentric Distance Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 K. Deepika and S. Meenakshi A Smart Personal Assistant for Visually Challenged . . . . . . . . . . . . . . . . . . 505 Sushruta Mishra, Kunal Anand, and N. Z. Jhanjhi F-Index for Some Class of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 R. Krithika Synchronization of a Modified Colpitts Oscillator with Triangular Wave Non-linearity on Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Suresh Rasappan, K. A. Niranjan Kumar, R. Narmada Devi, and Ahmed J. Obaid T-Fuzzy Modular l-Filters in Commutative Lattice Ordered M-Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 D. Vidyadevi and S. Meenakshi The Minimum Maximal Mean Boundary Dominating Seidel Energy of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 A. Meenakshi and M. Bramila

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GSβ -Compactness of Topological Spaces with Grills . . . . . . . . . . . . . . . . . . 549 N. Kalaivani, K. Fayaz Ur Rahman, and Ahmad J. Obaid Study of Prey-Predator Model Formulation and Stability Analysis . . . . . 561 Balaram Manna, Subrata Paul, Ani mesh Mahata, Supriya Mukherjee, and Banamali Roy Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

About the Editors

Sheng-Lung Peng is Professor and Director (head) of the Department of Creative Technologies and Product Design, National Taipei University of Business, Taiwan. He received the Ph.D. degree in Computer Science from the National Tsing Hua University, Taiwan. He is Honorary Professor of Beijing Information Science and Technology University, China, and Visiting Professor of Ningxia Institute of Science and Technology, China. He is also Adjunct Professor of Mandsaur University, India. He has edited several special issues at journals, such as Soft Computing, Journal of Internet Technology, Journal of Real-Time Image Processing, International Journal of Knowledge and System Science, MDPI Algorithms, and so on. His research interests are in designing and analyzing algorithms for bioinformatics, combinatorics, data mining, and networks areas in which he has published over 100 research papers. Cheng-Kuan Lin received the B.S. degrees in Applied Mathematics from the Chinese Culture University in 2000; and received the M.S. degrees in Mathematic from the National Central University in 2002. He obtained the Ph.D. in Computer Science from the National Chiao Tung University in 2011. In 2014, he joined the School of Computer Science and Technology, Soochow University, as an Associate Professor. In December 2018, he joined the College of Mathematics and Computer Science, Fuzhou University, as a Professor. He is the Associate Professor at the Department of Computer Science at National Yang Ming Chiao Tung University. His current research interests include design and analysis of algorithms, graph theory, wireless sensor networks, wireless applications, parallel and distributed computing, and V2X. He has published over 150 research papers. Souvik Pal is an Associate Professor in the Department of Computer Science and Engineering at Sister Nivedita University (Techno India Group), Kolkata, India. Prior to that, he was associated with Global Institute of Management and Technology; Brainware University, Kolkata; JIS College of Engineering, Nadia; Elitte College of Engineering, Kolkata; and Nalanda Institute of Technology, Bhubaneswar, India. Dr. Pal received his M.Tech., and Ph.D. degrees in the field of Computer Science and Engineering from KIIT University, Bhubaneswar, India. He has more than a decade xvii

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About the Editors

of academic experience. He is author or co-editor of more than 15 books from reputed publishers, including Elsevier, Springer, CRC Press, and Wiley, and he holds three patents. He is serving as a Series Editor for Advances in Learning Analytics for Intelligent Cloud-IoT Systems, published by Scrivener-Wiley Publishing (Scopusindexed); Internet of Things: Data-Centric Intelligent Computing, Informatics, and Communication, published CRC Press, Taylor & Francis Group, USA; Conference Proceedings Series on Intelligent Systems, Data Engineering, and Optimization, published CRC Press, Taylor & Francis Group, USA; Dr. Pal has published a number of research papers in Scopus/SCI/SCIE Journals and conferences. He is the organizing chair of RICE 2019, Vietnam; RICE 2020 Vietnam; ICICIT 2019, Tunisia. He has been invited as a keynote speaker at ICICCT 2019, Turkey, and ICTIDS 2019, 2021 Malaysia. He has also served as Proceedings Editor of ICICCT 2019, 2020; ICMMCS 2020, 2021; ICWSNUCA 2021, India. His professional activities include roles as Associate Editor, Guest Editor, and Editorial Board member for more than 100+ international journals and conferences of high repute and impact. His research area includes cloud computing, big data, internet of things, wireless sensor network, and data analytics. He is a member of many professional organizations, including MIEEE; MCSI; MCSTA/ACM, USA; MIAENG, Hong Kong; MIRED, USA; MACEEE, New Delhi; MIACSIT, Singapore; and MAASCIT, USA.

A Study About (γ, γ )α –Regular Spaces, Normal Spaces with (γ, γ )–Open Sets and α(γ, γ  )– Open Sets N. Kalaivani, E. Chandrasekaran, and Hanaa Hachimi

α α   Abstract In the present article, the idea of γ , γ   -regular spacesand γ , γ     normal spaces has been presented with the  of γ , γ -open  set,  γ , γ -closed  support  set, α(γ ,γ  ) -open set, α(γ ,γ  ) -closed set, γ , γ -regular space, γ , γ -normal space and operators like clτ(γ ,γ  ) , intτ(γ ,γ  ) intτα γ ,γ  , clτα γ ,γ  . Moreover, some of its proper( ) ( ) ties have been premeditated with the aid of bioperation-topological spaces. The generalized open sets and generalized closed sets play a vital role in the study of general topology as well as operation topology. The α(γ ,γ  ) -generalized open sets along with α(γ ,γ  ) -generalized closed sets have been utilized to elaborate various definitions of ((γ , γ  ), α(β,β  ) )-generalized continuous mappings, ((γ , γ  ), α(β,β  ) )-generalized open mappings, ((γ , γ  ), α(β,β  ) )-closed mappings, α(γ ,γ  )(β,β  ) -generalized continuous mapping, α(γ ,γ  )(β,β  ) -generalized open mapping and α(γ ,γ  )(β,β  ) -generalized closed mapping which are provided so as to study and understand the concepts discussed in the α paper work. The corollaries provide the requirement for  current YT oS to be a β, β  -nor spa. With the aid of the definition of the ultra-nor spa, α  the concept of ultra- γ , γ  -nor spa definition has been mentioned. Applying the   α knowledge of ultra- γ , γ  -nor spa definition, certain theorems have been devel oped. The idea of ((γ , γ ), α(β,β  ) )-g -OPM (CLM) and α(γ ,γ  )(β,β  ) -g -OPMa (CLMa) is used to discuss various comparative statements as well as theorems. Keywords (γ, γ )-open set · α(γ, γ ) -open set · α(γ, γ ) -interior · α(γ, γ ) -closure · (γ, γ )α -regular spaces · (γ, γ )α -normal spaces N. Kalaivani (B) · E. Chandrasekaran Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India e-mail: [email protected] E. Chandrasekaran e-mail: [email protected] H. Hachimi Applied Mathematics and Computer Science, Sultan Moulay Slimane University, Beni-Mellal, Morocco e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_1

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1 Preface The α-open sets, operation taking  on topological spaces, τα−γ τ α−γ -interior and  place τα−γ -closure operators and α- γ , γ  -open sets were familiarized correspondingly by Njastad [1], Kasahara [2, 3], Ogata [4] also Kalaivani et al. [5, 6]. Operationscompact spaces, nearly compact spaces and almost compact spaces were discussed by Ogata [7, 8]. Operator approaches of weakly Hausdorff spaces were studied by Umehara & Maki [9]. Further deep study about the topology and general topology was done by Kuratowski [10] and Willard [11] in their books. Bioperations thought was presented by Maki & Noiri [12], Umehara α et al. [13] besides Umehara [14].  In the present article, the idea of γ , γ  -regular spaces has been announced, α  and its tracts are premeditated. Further the perception of γ , γ  -normal spaces has been familiarized, and its tracts are deliberated. The ensuing  symbolizations are practised in this work: α(γ ,γ  ) -open set symbolizes the α- γ , γ  -open set, τα(γ ,γ  ) symbolizes τα−(γ ,γ  ) -the set of entire α(γ ,γ  ) -open sets, X T O S -the topological space (X, τ ), YT O S -the topological space (Y, τ ), OS-the open set, CS-closed set, Oss-open sets, CSs-closed sets, TOS-topological space, CMacontinuous mapping, OPMa-open mapping, CL Ma-closed mapping, C-continuous and Ma-mapping.

2 Preliminaries 1. Definition A set G ⊆ X T O S is aforesaid to be an α(γ ,γ  ) -OS on condition that G ⊆ intτα γ ,γ  (clτα γ ,γ  (intτα γ ,γ  (G))). (

)

(

)

(

)

  1. Theorem Each γ , γ  -OS in X T O S is an α(γ ,γ  ) -OS. Nevertheless, the contrair need not be exact. 2. Theorem Agree τα(γ ,γ  ) be the family of α(γ ,γ  ) -OSs prevailing in X T O S . Formerly, ∪ Aα is also an α(γ ,γ  ) -OS.

α∈J

1. Remark If U, V are two α(γ ,γ  ) -OSs in X T O S , then U ∩ V need not be an α(γ ,γ  ) OS. 2. Remark The idea of α-OS and α(γ ,γ  ) -OS is self-regulating.   3. Remark If X T O S is a γ , γ  -regular space, then the idea of α(γ ,γ  ) -OS and α-OS concurs. 2. Definition A subset M of X T O S is aforesaid to be an α(γ ,γ  ) -CS on condition that X T O S − M is an α(γ ,γ  ) -OS, whichever is unvaryingly M is an α(γ ,γ  ) -CS in case that M ⊇ clτα γ ,γ  (intτα γ ,γ  (clτα γ ,γ  (M))). (

)

(

)

(

)

A Study About (γ, γ )α –Regular Spaces, Normal Spaces …

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3. Definition Let P ⊆ X T O S . Then τα(γ ,γ  ) -interior of P is the union of all α(γ ,γ  ) -OSs confined in P, represented by intτα γ ,γ  (P).

(i)

(

)

intτα γ ,γ  (P) = ∪ {S : S is an α(γ ,γ  ) -OS and S ⊆ P}. (

)

4. Remark Agree D ⊆ X T O S . At that time, the ensuing declarations grip good: (i) (ii) (iii)

intτα γ ,γ  (D) is the largest α(γ ,γ  ) -OS confined in D. ( ) D is an α(γ ,γ  ) -OS on condition that intτα γ ,γ  (D) = D. ( ) intτα γ ,γ  (intτα γ ,γ  (D)) = intτα γ ,γ  (D). (

)

(

)

(

)

3. Theorem Let E, F ⊆ X T O S . Then: (i) (ii)

If E ⊆ F then intτα γ ,γ  (E) intτα γ ,γ  (F) ( ) ( ) intτα γ ,γ  (E) ∪ intτα γ ,γ  (F) ⊆ intτα γ ,γ  (E ∪ F) (

)

(

)

(

)

4. Definition Let Q ⊆ X T S . Then τα(γ ,γ  ) -closure of Q is the intersection of all α(γ ,γ  ) -CSs encompassing Q, represented by clτα γ ,γ  (Q). ( ) clτα γ ,γ  (Q) = ∩{H : H is an α(γ ,γ  ) -CS and Q ⊆ H }. (

)

5. Remark Let X T O S be a TOS and γ , γ  be operations on τ . Then. (i) (ii)

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The clτα γ ,γ  (P) is an α(γ ,γ  ) - CS containing P. ( ) P is an α(γ ,γ  ) -CS if and only if clτα γ ,γ  (P) = P. (

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α  γ , γ  -Regular spaces

α  5. Definition A TOS X T O S is thought to stay a γ ,γ  -reg spa, in case that for each / D, there occur γ , γ  -open sets, P, Q like that d ∈ P, α(γ ,γ  ) -closed set D and d ∈ D ⊆ Q and P ∩ Q = ∅. 1. Example Agree X = {r, s, t}, τ = {ϕ, X, {r }, {t}, {r, s}, {r, t}}.  γ γ  Describe γ , γ on τ alike that, P = P ∪ {r } for every one P ∈ τ and Q = Q if Q = {r } for every single Q ∈ τ . Q ∪ {c} if Q = {r } Formerly, τ(γ ,γ  ) = {φ, X, {r }, {r, t}} besides τα(γ ,γ  ) = {φ, X, {r }, {r, s}, {r, t}}. α  Here X T O S is a γ , γ  -reg sp. α  4. Theorem Each γ , γ  -reg spa is an α(γ ,γ  ) -reg spa. Nevertheless, the contrair need not be accurate.

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Proof Trails after the Definition 5 in addition, statement of the Theorem 1. 5. Theorem The declarations stated beneath about TOS X T O S are comparable: α  (i) X T O S is a γ , γ  -reg spa;   (ii) Intended for an α(γ ,γ  ) -open set P in X T O S , there occurs a γ , γ  -OS, Q encompassing m like that m ∈ clτ(γ ,γ  ) (Q) ⊆ P. Validation (i) imples (ii) Accredit D be an α(γ ,γ  ) OS also m ∈ D. At that time, / X T O S −−D. Through the regularity of X T O S −−D is an α(γ ,γ  ) -CS  like that m ∈ X T O S , there occur γ , γ  -open sets E 1 , E 2 alike that m ∈ E 1, X T O S −−D ⊆ E 2 in addition E 1 ∩ E 2 = φ. Formerly, X T O S −−E 2 is a γ , γ  -CS confined in D besides E 1 ⊆ X T O S − E 2 ⊆ D. This infers that clτ(γ ,γ  ) (E 1 ) ⊆ X T O S − E 2 ⊆ D. Consequently, m ∈ clτ(γ ,γ  ) (E 1 ) ⊆ D. / M. Agree Q = X T O S − M. (ii) implies (i) Assume M is an α(γ ,γ  ) -CS m ∈ By means of the theory in (ii), there prevails an α(γ ,γ  ) -open set, P of m alike that clτ(γ ,γ  ) (P) ⊆ Q. At that time, P besides X T O S − clτ(γ ,γ  ) (P) are disjoint α(γ ,γ  ) -open α  sets encompassing m and M correspondingly. Henceforth, X T O S is a γ , γ  -reg spa. α  6. Theorem A TOS X T O S is a γ , γ  -reg spa on condition that for separate  m∈  / W , there prevails a γ , γ  -OSs, X T O S together with an α(γ ,γ  ) -CS, W alike that m ∈ P, Q in X T O S akin that m ∈ Q as well as W ⊆ P moreover clτ(γ ,γ  ) (Q)∩clτ(γ ,γ  ) (P) = ∅. Proof Accredit / W by Theorem 6, m ∈ X T O S and W be an α(γ ,γ  ) -CS alike that m ∈  there is a γ , γ  -OS,E akin that m ∈ E, clτ(γ ,γ  ) (E) ⊆ X T O S − M. By Theorem 6,   there is a γ , γ  -OS, Q encompassing m alike that clτ(γ ,γ  ) (Q) ⊆ E. Agree P = X T O S − clτ(γ ,γ  ) (V ). At that time, clτ(γ ,γ  ) (Q) ⊆ E ⊆ clτ(γ ,γ  ) (E) ⊆ X T O S − W infers that W ⊆ X T O S − clτ(γ ,γ  ) (E) = P. Also clτ(γ ,γ  ) (Q) ∩ clτ(γ ,γ  ) (P) = clτ(γ ,γ  ) (Q) ∩ clτ(γ ,γ  ) (X T O S − clτ(γ ,γ  ) (E)) ⊆ E ∩ clτ(γ ,γ  ) (X T O S − clτ(γ ,γ  ) (E)) ⊆ clτ(γ ,γ  ) (E ∩ (X T O S − clτ(γ ,γ  ) (E))) = clτ(γ ,γ  ) (∅) = ∅. 6. Definition A M, f M : XTOS → YTOS is defined as a (γ , γ  ), α(β,β  ) )-generalizedCMa if for any. α(β,β  ) g-CS, W . of YT O S , f M−1 (W ) is a g -CS in X T O S . 7. Definition A M, f M : XTOS → YTOS is defined as a ((γ , γ  ), α(β,β  ) )-generalized OPMa in case that for individual (γ , γ  ) g-OS, D of X T O S , f M (D) is an g-OS YT O S .  8. Definition A M, f M : XTOS → YTOS is  defined as a ((γ , γ ), α(β, β) )-generalized  CLMa conceding that for separate γ , γ g-CS E of X T O S , f M (E) is an α(β,β  ) g-CS in YT O S .

9. Definition A M, f M : XTOS → YTOS is defined as a α(γ ,γ  )(β,β  ) -generalized-CMa if for any α(β,β  ) g-CS,F of YT O S , f M−1 (F) is an α(γ ,γ  ) g-CS in X T O S .

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10. Definition A M, f M : XTOS → YTOS is defined as a α(γ ,γ  )(β,β  ) -generalized OPMa if for each α(γ ,γ  ) g-OS, G of X T O S , f M (G) is an α(β,β  ) -g-OS in YT O S . 11. Definition A M, f M : XTOS → YTOS is defined as a α(γ ,γ  )(β,β  ) -generalized CLMa if for each α(γ ,γ  ) g-CS H of X T O S , f M (H ) is an α(β,β  ) g-CS in YT O S . 7. Theorem Every single ((γ , γ  ), α(β,β  ) )-g-OPM (CLM) is an α(γ ,γ  )(β,β  ) -g-OP Ma (as well CL Ma). Proof Tracks after the Definitions 6, 7-descriptions. 8. Theorem Accredit X T O S be a TOS and γ , γ  be manoeuvres demarcated on τ . At that time, the properties stated below are comparable: α  (i) X T O S is a γ , γ  -reg spa; (ii) For any α(γ ,γ  ) -CS, E in addition any point m ∈ X T O S − E, there prevails  an Q ∈ τα(γ ,γ  ) in addition a γ , γ  -OS, V akin that m ∈ Q, E ⊆ P besides Q ∩ P = ∅;  (iii) For any α(γ ,γ  ) -CS, E of X T O S , E = ∩ clτ(γ ,γ  ) (P):E ⊆ P also P is a   γ , γ  g-OS}; (iv) For each subset M of X T O S and each α(γ ,γ ) -CS, E such that M ∩ E = ∅, there prevails a set Q ∈ τ(γ ,γ  ) and a γ , γ  g-OS, P such that M ∩ Q = ∅, E ⊆ P and Q ∩ P = ∅. Proof (i) → (ii) Trails afterwards the Definition 5 statement besides Theorem 3 statement.  (iii) Assume that  m∈ X T O S − E. Through the supposition, there prevails  (ii) → a γ , γ  -OS, Q and a γ , γ  F-OS, P such that m ∈ Q, E ⊆ P and Q ∩ P = ∅. By Theorem 3, m ∈ X T O S − clτ(γ ,γ  ) (P), and hence, m ∈ X T O S − {∩ clτ(γ ,γ  ) {(P)  and P is an α(γ ,γ  ) g − OS}. Therefore, E ⊇ ∩ clτ(γ ,γ  ) (P): E ⊆ P and P is an α(γ ,γ  ) g-OS}, and hence, F = ∩{clττ γ ,γ  (V ) : E ⊆ P and P is a α(γ ,γ  ) g-OS}. ( ) (iii) → (iv) Assume E be any α(γ ,γ  ) -CS of X T O S also M be a non-empty subset of X T O S like that M ∩ E = ∅. Take a point m ∈ M so that m ∈ X T O S − clτ(γ ,γ  ) (P). Accredit Q = X T O S − clτ(γ ,γ  ) − (P), then Q ∈ τ(γ ,γ  ) , M ∩ Q = ∅ and Q ∩ P = ∅. (iv) → (i) Let E be any α(γ ,γ  ) -CS of  X T O S and m ∈ X T O S − E. Then {m}∩ E = ∅, and there prevails Q ∈ τ(γ ,γ  ) and a γ , γ  g-OS, W such that m ∈ Q, E ⊆ W and Q ∩ W = ∅. Put P = intτ(γ ,γ  ) (W ), then by Definition 5, E ⊆ P, P ∈ τ(γ ,γ  ) and α  Q ∩ P = ∅. Therefore, X T O S . is a γ , γ  -reg spa.

4 (γ , γ  )α -Normal Spaces α  12. Definition A TOS X T O S is aforesaid to be a γ , γ  -norspa, on  condition that for any distinct α(γ ,γ  ) - CSs M and N of X T O S , there prevails γ , γ  -OSs Q, Pakin that M ⊆ Q in addition N ⊆ P besides Q ∩ P = ∅.

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2. Example Let X TOS = {m,n,o} and τ =  {∅,XTOS , {m}, {n}, {m, n}, {m, o}}, cl(M) if n ∈ /M describe a manoeuvre γ on τ alike that M γ = for every M ∈ τ . M if n ∈ M   cl(N ) i f n ∈ N The procedure γ  is on τ . Then τ is defined as N γ = for N if n ∈ /N every N ∈τ . Formerly, τα(γ ,γ  ) = {∅, X TOS , {n}, {m, n}, {m, o}}. At this juncture, X T O S is an  α γ , γ  -nor spa.    α 9. Theorem Each γ , γ  -nor spa is a γ , γ  -nor spa. Proof The verification trails as of the Definition 12 besides Theorem 1. 10. Theorem For a TOS X T O S , the declarations assumed below are alike: α  (i) X T O S is a γ , γ  -nor spa.   (ii) Consideringeverypair of γ , γ  -OSs Q and P whose amalgamation is X T O S , there occur γ , γ  g-CSs M and N alike that M ⊆ Q, N ⊆ P also M ∪ N = XT OS.   H besides exclusive γ , γ  -OS,K comprising H , (iii) For individual α(γ ,γ  ) -CS,  there prevails a γ , γ  g-OS, Q alike that H ⊆ Q ⊆ clτ(γ ,γ  ) (Q) ⊆ K .   α  Proof (i) → (ii) Agree Q then P be a couple of γ , γ  -OSs in a γ , γ  -nor spa X T O S − Q, X T O S − P are separate X T O S like that X T O S = Q ∪ P. Formerly, α  α(γ ,γ  ) CSs. Meanwhile, X T O S is a γ , γ  -nor spa, and there occur disjoint γ , γ  -Oss, U1 , V1 alike that X T O S − Q ⊆ U1 in addition X T OS − P ⊆ V1 . Accredit M = X T O S −U1 , N = X T O S − V1 . At that time, M also N are γ , γ  -CSs alike that M ⊆ Q, N ⊆ P then M ∪ N = X T O S .   K be an γ , γ  -OS comprising H . (ii) → (iii) Approve H be an α(γ ,γ ) -CS and   Formerly, X T O S − H in addition  K are γ , γ -OSs whose amalgamation is X T O S . At that time by (ii), there occur γ , γ -CSs M1 also M2 alike that M1 ⊆ X T O S − H also M2 ⊆ K in addition M1 ∪ M2 = X T O S . Besides H ⊆ X T O S − M1 , K ⊆ X T O S − M2 also (X T O S − M1 ) ∩ (X T O S − M2 ) = ∅ . Accredit  Q = X T O S − M1 also P = X T O S − M2 . At that time, Q and P are disjoint γ , γ  -OSs. By 3.10. Definition, X T O S − K ⊆ intτ(γ ,γ  ) (P). Meanwhile, Q ∩ intτ(γ ,γ  ) (P) = ∅, formerly clτ(γ ,γ  ) (U ) ∩ intτ(γ ,γ  ) (P) = ∅. This suggests that clτ(γ ,γ  ) (Q) = X T O S − intτ(γ ,γ  ) (P) ⊆ K . Henceforth, H ⊆ Q ⊆ clτ(γ ,γ  ) (Q) ⊆ K . (iii) → (i) Proof follows from the Definition 12.  α 11. Theorem A TOS X T O S is a γ , γ  -nor spa in the   casethat for any α(γ ,γ  ) -CS,  also γ , γ  -OS, Q comprising M, there prevails an γ , γ  -OS, P comprising M alike that M ⊆ P ⊆ clτ(γ ,γ  ) (P) ⊆ Q. Proof By the reason of Q is an α(γ ,γ  ) -OS containingM, X T O S − Q is an α(γ ,γ  ) -CS and M∩ (X T O S − Q) = ∅. Since X T O S is an α(γ ,γ  ) -nor spa, there happen α(γ ,γ  ) OSs P also V1 like that M ⊆ P, X T O S − Q ⊆ V1 also P ∩ V1 = ∅. Henceforth, M ⊆ P ⊆ clτ(γ ,γ  ) (P) ⊆ clτ(γ ,γ  ) (X T O S − V1 ) = X T O S − V1 ⊆ Q.

A Study About (γ, γ )α –Regular Spaces, Normal Spaces …

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Conversely, agree M, N be disjoint α(γ ,γ  ) -CSs in X T O S . At that time, M ⊆ X T O S − N , where X T O S − N is an α(γ ,γ  ) -OS in X T O S . By assumption, there is an α(γ ,γ  ) -OS, V alike that clτ(γ ,γ  ) (P) ⊆ X T O S –N , which infers that N ⊆ X T O S −   clτ(γ ,γ  ) (P) also V ∩ X T O S − clτ(γ ,γ  ) (P) = ∅ . Henceforward, M ⊆ P, N ⊆ α  X T O S − clτ(γ ,γ  ) (P). The present argument demonstrates X T O S is a γ , γ  -nor spa.   α α 12. Theorem A γ , γ  -nor spa also α(γ ,γ  ) T1 space is a γ , γ  -reg spa. / M. Subsequently, X T O S Proof Supposing that M is an α(γ ,γ  ) -CS in addition m ∈ is an α(γ ,γ  ) T1 space, through Theorem 2 statement, individual {m} is an α(γ ,γ  ) -CS in X T O S . Then X T O S is an α(γ ,γ  ) -nor spa, there occur α(γ ,γ  ) -OSs Q, P alike that {m} ⊆ Q, M ⊆ P, Q ∩ P = ∅ or m ∈ Q, M ⊆ P and Q ∩ P = ∅ infers that X T O S α is a γ , γ  -reg spa. 13. Theorem Let fM :XTOS → YTOS be an α(γ ,γ )(β,β  ) -CL, α(γ ,γ  )(β,β  ) -CMa from α onto YT O S . In case that X T O S is a γ , γ  -nor spa, at that time YT O S is a X  T O S α β, β - nor spa, wherever β, β  are open operations. Proof Accredit C1 ,D1 be separate α(β,β  ) -C subsets of YT O S . Formerly by α(γ ,γ  )(β,β  ) −1 continuity of f M , C1 = f M−1 (C  1 = f M (D) are separate α(γ ,γ  ) -C subsets of  1 ), D  α X T O S . In view of X T O S is a γ , γ -nor spa, there happen α(γ ,γ  ) -OSs Q and P akin that C1 ⊆ Q, D1 ⊆ P also Q ∩ P = ∅. By reason of f M is an α(γ ,γ  )(β,β  ) COM, f M (X T O S − Q) Q also f M (X T O S − P) are α(β,β  ) -CSs in YT S . At that time, U1 = YT O S − f M (X T O S − Q),U2 = YT O S − f M (X T O S − P) are separate α(β,β  ) OSs in YT O S comprising C,D correspondingly. This demonstrates that YT O S is an α(β,β  ) -nor spa. 14. Theorem A TOS X T O S is a c-nor spa with the proviso for specific pair M, N of disjoint α(γ ,γ  ) -CSs in X T O S , and there prevails α(γ ,γ  ) -Oss, Q, P in X T O S such that M ⊆ Q, N ⊆ P also clτ(γ ,γ  ) (Q) ∩clτ(γ ,γ  ) (P) = ∅. Proof Assume M be an α(γ ,γ  ) -CS also N be an α(γ ,γ  ) -CS not encompassing M. At that time, X T O S − N is an α(γ ,γ  ) -OS also M ⊆ X T O S − N . Formerly by Theorem 10(iii) statement, there prevails a γ , γ  -OS,C like that M ⊆ C ⊆ clτ(γ ,γ  ) (, C) ⊆ X T O S − N . Subsequently, clτ(γ ,γ  ) -cl (, C) ⊆ X T O S − N , again via 4.3.   Theorem, there prevails a γ , γ  -OS, Q encompassing C like that clτ(γ ,γ  ) (Q) ⊆ C. Consequently, M ⊆ Q ⊆ clτ(γ ,γ  ) (Q) ⊆ C ⊆ clτ(γ ,γ  ) (C) ⊆ X T O S − N . This infers that N ⊆ X T O S − τα(γ ,γ  ) -cl (C). Accredit P = X T O S − clτ(γ ,γ  ) (C). Formerly, P is an α(γ ,γ  ) -OS comprising N also clτ(γ ,γ  ) (Q) ∩ clτ(γ ,γ  ) (P) = clτ(γ ,γ  ) (Q) ∩ (X T O S − clτ(γ ,γ  ) (C)) ⊆ C ∩ clτ(γ ,γ  ) (X T O S –clτ(γ ,γ  ) (C)) ⊆ clτ(γ ,γ  ) (C∩ (X T O S − clτ(γ ,γ  ) ( C)) = clτ(γ ,γ  ) (∅) = ∅. Accordingly, Q and P are the   essential γ , γ  -OSs in X T O S . The converse fragment trails from the Definition 3 statement. 13. Definition A TOS X T O S is aforesaid to be an ultra-nor spa, in case that for individual distinct CSs M in addition N of X T O S , there prevails clopen sets Q, P akin that M ⊆ Q besides N ⊆ P in addition Q ∩ P = ∅.

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α  14. Definition A TOS X T O S is said an ultra- γ , γ  -nor spa, with the condition   to be  that  each and every distinct γ , γ - CSs M and N of X T O S , and there occurs  for  γ , γ -clopen sets Q, P akin that M ⊆ U together with N ⊆ V also Q ∩ P = ∅. 15. Theorem Supposing that f M :X T O S → YTOS is a contra α α(γ ,γ  ) -continuous, closed injective mapping along with YT O S is an ultra- γ , γ  -nor spa, later X T O S α  is a γ , γ  -nor spa. Proof Assume K 1 and K 2 be separate closed subsets of X T O S . By reason of f M is a C besides injective Ma, f M (K 1 ) and also f M (K 2 ) are disjoint C subsets of α YTOS . In view of YT O S is an ultra- γ , γ  -nor spa, f M (K 1 ) together with f M (K 2 ) is separated by disjoint γ , γ  -clopen sets V1 and V2 correspondingly. Henceforward, Fi ⊆ f M−1 (Vi ), f M−1 (Vi )∈τα(γ ,γ  ) for i = 1, 2 in addition f M−1 (V1 ) ∩ f M−1 (V2 ) = ∅. α  Consequently, X T O S is a γ , γ  -nor spa.     -irresolute, γ , γ  , β, β  16. Theorem Let f M : XTOS → YTOS be an          α γ , γ , β, β -g-closed surjection as well as X T O S is a γ , γ -nor spa, formerly  α YTOS is a β, β  -nor spa.   Proof Accredit A1 ,B1 be disjoint β, β  -closed subsets of YT O S . Formerly, M =   f −1 (A ),N = f M−1 (B1 ) are disjoint γ , γ  -closed subsets of X T O S . Later f M is a  α M 1   γ , γ  , β,β  -irresolute Ma. Meanwhile, X T O S is a γ , γ  -nor spa, and there occur γ , γ  -Oss Q also  P  akin  that M ⊆ Q, N ⊆ P along with Q ∩ P = ∅.  is a γ , γ , β, β -g CMa, f M (X T O S − Q) and (X T O S − P) Subsequently, f M   U are β, β  -CSs in YTOS . At that time,  1 = YT oS − f M (X T O S − Q),U2 =  YT oS −− f M (X T O S −−P) are disjoint β, β  -OSs  in YTOS comprising A1 ,B1 , respectively. This demonstrates that YTOS is a β, β  –nor spa.     1. Corollary Let f M : XTOS → YTOS be a γ , γ  , β, β        α irresolute, γ , γ  , β, β  -C surjection and X T O S is a γ , γ  -nor spa, then    α YT oS is a β, β -nor spa. Proof The verification tracks after the Theorem 16.     17. Theorem Let f M : XTOS → YTOS be a γ , γ  , β, β  -continuous mapping,     α  γ , γ  , β, β  -g-closed surjection also X T O S is a γ , γ  -nor spa, at that time   α YT oS is a β, β  -nor spa.   Proof Accredit G 1 , H1 be disjoint β, β  -closed subsets of YT S . Subsequently,     f is a γ , γ  , β, β  -CMa, M = f M−1 (G 1 ), N = f M−1 (H1 ) are disjoint α   M  subsets of X T O S . Meanwhile, X T O S is a γ , γ  -nor spa, and there γ , γ -closed   occurs γ , γ  -open sets F in additionE like  that M  ⊆ F, N ⊆ E as well as F ∩ E= ∅. Subsequently, f M isa γ , γ  , β, β  -COMa, f M (X T O S − F) besides f M (X T O S − E) are β, β  -closed sets in YT O S . At that  O1 =  time, YT oS − f M (X T oS − F), O2 = YTOS − f M X T O S − E) are disjoint β, β  -open sets α  in YTOS comprising G 1 , H1 , respectively. This evidences that YT oS is a β, β  -nor spa.

A Study About (γ, γ )α –Regular Spaces, Normal Spaces …

9

    2. Corollary Let fM : XTOS → YTOS be a γ , γ  , β, β  -C mapping,    α   γ , γ  , β, β  -C surjection besides X T oS is a γ , γ  -nor spa, at that time YT oS   α is a β, β  -nor spa. Proof The validation trails after the Theorem 17. Application of the concept

 α Urysohn Lemma can be studied in γ , γ  -normal spaces also. Further the bioperations approach on ideal theory will be including interesting approaches and facts. This study of ideal theory and their operation approaches can be extended in digital topology. Conclusion Utilizing the concept of bioperations, we had already introduced a new category α of open sets called α(γ ,γ  ) -open sets. In this present work, the concept of γ , γ    α regular spa γ , γ  -normal spa  hasbeen announced, and their tracts are elaborately deliberated with the help of γ , γ  -open sets, α(γ ,γ  ) -open sets, various combinations of continuous mappings,  generalized   continuous mappings and  contra   contin     , β, β -continuous mappings, γ , γ , β, β  uous mappings like γ , γ     closed surjection, γ , γ  , β, β  -generalized  closed  surjection, contra α(γ ,γ  )  continuous, closed injective mapping, γ , γ  , β, β  -irresolute and α(γ ,γ  ) T1 α α   spaces. Theorems related to the existence of a γ , γ  -reg spa and γ , γ  -nor spa are conferred.

References 1. Njastad, O. 1965. On some classes of nearly open sets. Pacific I. Math 15: 961–970. 2. Kasahara, S. 1973. Characterizations of compactness and countable compactness. Proceedings of the Japan Academy Series A, Mathematical Sciences 49: 523–524. 3. Kasahara, S. 1979. Operation—Compact spaces. Mathematica Japonica 24: 97–105. 4. Ogata, H. 1991. Operation on topological spaces and associated topology. Mathematica Japonica 36: 175–184. 5. Kalaivani, N., and G Sai Sundara. Krishnan. 2013. Operation approaches on α-γ-open sets in topological spaces. International Journal of Mathematical Analysis 10: 491–498. 6. Kalaivani, N., D. Saravanakumar, and G Sai Sundara Krishnan. 2014. On α-(γ, γ )-open sets and α-(γ, γ )-Ti spaces. In Proceedings of the International Conference on Applied Mathematical Models, 25–29, 3–5 January. Coimbatore, India 7. Ogata, H. 1991. Remarks on operation-compact spaces. Memoirs of the Faculty of Science Kochi University Series A Mathematics 15: 51–63. 8. Ogata, H., and T. Fukutake. 1991. On operation-compact, operation-nearly compactness and operation-almost compcatness. Bulletion Fukuoka University Edition Part III 40: 45–48. 9. Umehara, J., and H. Maki. 1990. Operator approaches of weakly Hausdorff spaces. Memoirs of the Faculty of Science. Series A. Mathematics Kochi University 11: 65–73. 10. Kuratowski, K. 1966. Topology. New York: Academic Press. 11. Willard, S. 2004. General Topology. New York: Dover Publications.

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12. Maki, H., and T. Noiri. 2001. Bioperations and some separation axioms. Scientiae Mathematicae Japonicae 4: 165–180. 13. Umehara, J., H. Maki, and T. Noiri. 1992. Bioperation on topological spaces and some separation Axioms. Memoirs of the Faculty of Science. Series A. Mathematics Kochi University 13: 45–59. 14. Umehara, J. 1994. A certain bioperation on topological spaces. Memoirs of the Faculty of Science. Series A. Mathematics Kochi University 15: 41–49.

The Role of Harvesting in a Food Chain Model and Its Stability Analysis G. M. Vijayalakshmi, Suresh Rasappan, Pugalarasu Rajan, and Ha Huy Cuong Nguyen

Abstract We have explored a food chain continuous time model in this research in which it is composed of functional response named as Holling type II with the impact of harvesting on prey species. Throughout Holling’s studies on predator–prey interaction, the phrase “functional response” was created to explain that a predator’s rate of prey consumption varied as the density of prey varies. Holling’s simple premise depends on the two phases of predator‘s period: “hunting for” and “dealing of” prey to establish his three types of functional features. The slope of Holling’s type II functional response declines steadily with increasing prey density, gradually saturating at a steady prey consumption value. We have consider a chain model here, where the predator predates on prey and super predator predates on predators. To obtain the strategies for controlling this food chain dynamical prey-predator system we have given illustrations on the harvesting effort as control to the system. Stable, unstable periodic, limit cycles and bifurcation has been identified in this model using biological plausible parameter values. That is, we present results on the equilibrium points, existence, stability, and local bifurcations. The food chain system has been achieved global asymptotic stability by establishing Lyapunov function. The nonlinear nature of these models is also explored and its response is determined for being extremely perceptive to the parameter variations along with everyday life parameters. The G. M. Vijayalakshmi (B) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamilnadu 600062, India e-mail: [email protected] S. Rasappan Department of Mathematics, University of Technology and Applied Sciences, Ibri, Sultanate of Oman P. Rajan Department of IT, University of Technology and Applied Science, Ibri, Sultanate of Oman e-mail: [email protected] H. H. C. Nguyen The University of Da Nang, Da Nang, Vietnam e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_2

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model’s bifurcation diagrams and Lyapunov function have been shown to demonstrate that when the parameters are changed, the system interacts chaotically. The nonlinear nature of the system is observed to be hyper delicate to parameter values as well as real life parameter. The bifurcation of prey species with varying parameters of prey growth rate and carrying capacity has been discussed elaborately. To explain the analytical conclusion, with various sets of parameters, computer simulations were carried out, particularly MATLAB simulations are used. Keywords Prey-predator · Food chain systems · Harvesting · Global stability · Lyapunov function · Limit cycles

1 Introduction According to the universal occurrence and importance, the complex chaotic oscillations of the prey-predator mathematical system is retained among the most popular subjects in eco-system. It is universally recognized that chaotic systems are highly unpredictable, with serious consequences for species management. In recent years, mathematics has been used to describe and explain the concepts nonlinear nature of the differential equations. The Lotka-Volterra predator-prey model is an old model with strong mathematical logics that is used in many population dynamical systems. Two-dimensional interaction between one prey and one predator is well studied [1–5]. Using Lyapunov functions, Paul Georgescu [3] showed global stability for two species mutualisms. As the number of dimensions grows, the analysis becomes more complicated. A recent study focused into the interaction of more than two species models, particularly three. Employing any one of the three forms of Holling’s response functions, several researchers have investigated the diseased prey-predator model [6–8]. Weimingwang et al. [4] have introduced Allee effect on prey-predator system. Many studies has been identified the harvesting of two species and three species prey-predator models like Refs. [2, 9–12]. Numerous publications have recently explored on the behaviour of the three species food chain system [2, 6–11, 13–20]. Several mathematical models of three species food chain harvesting have assumed that the species are only influenced by harvesting. To the extent of the author’s perception, no approach has yet been made to investigate a prey species harvesting model in the existence of predator and a super predator as second and third species that are not harvested. In view of this, we have examine food chain system Michaelies-Menten (i.e. Holling’s 2nd type) response function with harvested prey and emphasizing on the stability analysis globally. Through enhancing the Lyapunov’s scalar function, we were able to verify the major conclusions as well as the boundedness of our system. The remaining work is organized as follows: We proposed our model in the Sect. 2. In Sect. 3, the proposed system’s equilibrium points are identified, and stability criteria are examined by illustrating the system’s

The Role of Harvesting in a Food Chain Model and Its Stability …

13

boundedness. The construction of the Lyapunov function in Sect. 4 established global stability. Numerical simulations and conclusions are covered in Sects. 5 and 6.

2 Description of the Model A three species predator–prey dynamical food chain system includes a prey breed species (x1 ), a predator (x2 ) and a super predator (x3 ) involving type II Holling behavioural function by assuming that prey species (x1 ) is subjected to harvesting effort. Now we can write the following nonlinear differential equations:    x1  α2 x2 − − Eq x˙1 = x1 r 1 − k β2 + x1   e2 α2 x1 α3 x3 − − d2 x˙2 = x2 β2 + x1 β3 + x2   e3 α3 x2 x˙3 = x3 − d3 β3 + x2

(1)

where r -velocity of prey reproduction, k- ability to carry, αi , βi , ei , and di , i = 2, 3, are predation rate at a threshold level, mid-consumption level, efficiency and extinction rates of x2 and x3 , respectively, and q is the catchability coefficient of the prey species and E is the harvesting effort.

3 Analysis of the Model 3.1 Boundedness, Equilibria Assume that x1 (t) > 0, x2 (t) > 0, and x3 (t) > 0 are positive initial conditions. The (2.1) system’s uniform boundedness is now verified by the following theorem. Theorem 1 With favourable primary conditions, x1 (0), x2 (0) , and x3 (0), entire solution of the model (2.1) are bounded uniformly. Proof Let x1 (t) > 0, x2 (t) > 0, and x3 (t) > 0 at the time t = 0, can be any system‘s solution with positive screening states. Consider the following function. (t) = x1 (t) + x2 (t) + x3 (t) We get the below derivative ˙ (t) = x˙1 (t) + x˙2 (t) + x˙3 (t)

(2)

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  r k 2 rk − ϑ = − x1 − + k 2 4

(3)

where ϑ = min(Eq, d2 , d3 ) and we prefer e2 = e3 = 1. rk d ≤ − ϑ dt 4 d rk + ϑ ≤ ≈ dt 4 d + ϑ =  (Say) dt

(4)

The theory of differential inequality is now being applied below (Birkoff and Rota 1982) (x1 (0), x2 (0), x3 (0))  + ϑ eϑt  As t → ∞, we get 0 ≤ (x1 , x2 , x3 ) ≤ ϑ 0 ≤ (x1 , x2 , x3 ) ≤

(5)

As a result, all of the system’s (1) solutions occur into the region B where,   3 + ε, foranyε > 0 B = (x1 , x2 , x3 )εR+ : 0 ≤  ≤ ϑ The theorem is now completed. The following four equilibrium points can be derived from (1). (a)

The Trivial equilibrium state P0 (0, 0, 0),



−

−



The Boundary equilibrium states P1 (0, x 2 , x 3 ) and P2 x 1 , 0, x 3 and

 (c) The Interior equilibrium state P3 x1∗ , x2∗ , x3∗ .

r −Eq d2 β2 where x2 = β2 (rα−Eq) 1 + ; x = − 3 α3 α2 2 (b)

d2 β2 d3 β3 ; x3 = e2 α2 − d2 e3 α3 − d3    −A1 + A21 − 4 A2 ; x1∗ = 2   e2 α2 x1∗ d3 β3 e3 β3 ∗ ∗ x2 = ;x = − d2 e3 α3 − d3 3 e3 α3 − d3 β1 + x1∗ x1 =

here A1 = β2 + rk (Eq − r ); A2 = β2 (Eq − r ) +

kα2 d3 β3 r (e3 α3 −d3 )

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3.2 Localite Stability Criteria of the Stable Points The variational matrix V (x1 , x2 , x3 ) of (1) is evaluated is as follows ⎡

1 − 2 rk x1 −

⎢ e α2 β2 x2 V(x1 ,x2 ,x3 ) = ⎣ (β2 2 +x 2 1) 0

α2 β2 x2 (β2 +x1 )2

x1 − Eq − βα22+x 1

⎤

0

β3 x3 e2 α2 x1 − (βα3+x 2 β2 +x1 3 2) e3 α3 β3 x3 (β3 +x2 )2

x2 − d2 − βα33+x 2 e3 α3 x2 β3 +x2

⎥ ⎦ (6)

− d3

Lemma 1 If P0 (0, 0, 0) exists with r < Eq, then P0 (0, 0, 0) is locally asymptotically stable. Proof At P0 (0, 0, 0), the Eigen values are r − Eq, −d 2 , −d 3 . Therefore, except r − Eq all the Eigen values are negative. Hence the lemma. Lemma 2 If P1 (0, x 2 , x 3 ) exists with r < asymptotically in a limited scale.

α2 x 2 β2

+ Eq, then P1 (0, x 2 , x 3 ) is stable

Proof At P1 (0, x 2 , x 3 ), one of the Eigen values is r − αβ2 x2 2 − Eq. This Eigen value depends on r whether it is weaker or stronger than αβ2 x2 2 + Eq. The below quadratic form gives remaining two Eigen values.   e3 α3 x 2 α3 β3 x 2 λ + 2d3 − + β3 + x 2 (β3 + x 2 )2   α3 d3 e3 x 2 α3 β3 x 3 2 − + d3 = 0 λ+ β3 + x 2 β3 + x 2 2

 In (3) the sum of the roots is equal to − 2d3 −

(7) 

β3 x 2 e3 α3 x 2 which is always + (βα3+x 2 β3 +x 2 3 2) α3 β3 d3 x 3 α3 d3 e3 x 2 − β3 +x 2 + d32 which is β3 +x 2

negative and the product of the root is equal to non-negative. Thus P1 (0, x 2 , x 3 ) is locally asymptotically stable only if with r < α2 x 2 + Eq. β2



 3x2 Lemma 3 If P2 x 1 , 0, x 3 exists with d3 > eβ3 α+x , then P2 x 1 , 0, x 3 is locally 3 2 asymptotically stable.

 3x2 Proof At P2 x 1 , 0, x 3 , one of the Eigen values is eβ3 α+x = −d3 . This Eigen value 3

depends upon d3 whether it is weaker or stronger than

2

e3 α3 x 2 . β3 +x 2

The below quadratic form will gives remaining two Eigen values. λ2 + B1 λ + B2 = 0  where B1 = − 2rk x 1 −

e2 α2 x 1 β2 +x 1

+

α2 β2 x 2 β2 +x 1

+ Eq + d2 − r

(8) 

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d1 α2 β2 x 1 2r e2 α2 x 1 Eqe2 α2 x 1 −

− + β2 + x 1 k β2 + x 1 β2 + x 1 β2 + x 1 2r d1 − d1 r + x 1 + Eqd2 k

B2 =

e2 α2 r x 1

In (8) the sum of the roots is equal to B1 which is always negative and the product of the root is equal to B2 which is usually positive.

 3x2 . Hence, P2 x 1 , 0, x 3 is asymptotically stable on a limited scale only if d3 > eβ3 α+x 3



 ∗



2

 ∗

Lemma 4 If P3 x1∗ , x2∗ , x3 exists with a1 a3 > a2 , then P3 x1∗ , x2∗ , x3 is stable asymptotically in a limited scale.

 Proof At P3 x1∗ , x2∗ , x3∗ the characteristic equation is: λ3 + a 1 λ2 + a 2 λ + a 3 = 0

(9)

where   2r x1∗ + e2 v + 2d3 − u − w + γ a1 = − r − k   2r a2 = (e2 v − w − d3 ) r − x1∗ − u + γ − d3 k   2r + r − x1∗ − u (γ − d3 ) + e3 γ w + e2 uv k   2r ∗ a3 = − r − x1 − u (e2 v − w − d3 )(γ − d3 ) k    2r ∗ + r − x1 − u γ w + 2uve3 (γ − d3 ) k here u = and

α2 β2 x2∗

(β2 +x1∗ )2

x1∗

;v=

=

x3∗ =

α2 x1∗ ; β2 +x1∗

−A1 +

w=



α3 β3 x3∗

(β3 +x2∗ )2

A21 − 4 A2

2 

e2 α2 x1∗ e3 β3 e3 α3 − d3 β1 + x1∗

;γ =

α3 x2∗ β3 +x2∗

d3 β3 > 0; x2∗ = > 0; e3 α3 − d3  − d2 > 0

Here, it is clear that under above conditions a1 > 0, a2 > 0 and a3 > 0.

 Thus from Routh Hurwitz criteria, P3 x1∗ , x2∗ , x3∗ will be locally asymptotically [19] if a1 a3 > a2 which completes the lemma.

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4 Analysis of the Model By using the Lyapunov function to derive the following theorem,

 we were able to determine the criteria for inner equilibrium point P 3 x ∗1 , x ∗2 , x ∗3 global stability.

 Theorem 2 The inner stable point P3 x1∗ , x2∗ , x3∗ global asymptotic stability exists if e2 = e3 = 1. Proof Consider the following Lyapunov scalar function. V (x1 , x2 , x3 ) =

3  

xi −

xi∗



−

i=1

xi∗ log

xi xi∗

 (10)

Then its derivative becomes x˙1 x˙2 x˙3 V˙ (x1 , x2 , x3 ) = z 1 + z 2 + z 3 x1 x2 x3   



here z 1 = x1 − x1∗ , z 2 = x2 − x2∗ , and z 3 = x3 − x3∗ . Using the set of Eqs. (1) and (11) we get α2 z 1 z 2 e2 α2 z 1 z 2 α3 z 2 z 3 e3 α3 z 2 z 3 z2 + V˙ (x1 , x2 , x3 ) = −r 1 − − + k β2 + z β2 + z 1 β3 + z 2 β3 + z 2

(11)

(12)

Now choosing e2 = e3 = 1, we get 2

z (13) V˙ (x1 , x2 , x3 ) = −r 1 ≤ 0 k

∗ ∗ ∗ This illustrates that the point 3 x 1 , x 2, x 3 is asymptotically stable. Hence based

P on the Lyapunov theorem, P3 x1∗ , x2∗ , x3∗ is globally asymptotically stable.

5 Numerical Simulation Under this section, numerical simulations are performed for various parameter values to delving the dynamic performance of the model (1), and bisection diagrams of prey species in terms of r , time scale plots, and phase trajectories for the model (1) are provided, each of which confirm our theoretical predictions. Figs. 1, 2, and 3 exhibits the bifurcation diagram of the prey species by varying r = −1 to 1 and keeping k = 1.5 fixed and varying x1 , x2 , α2 , β2 , E , and q. Chaotic bands, periodic oscillations, and the model is settling into a stable state succeeding r ≥ 0 are portrayed in Fig. 3.

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Fig. 1 x1 = 0.25, x2 = 0.95, α2 = 1.6, β2 = 0.65, E = 0.1, q = 0.005

Fig. 2 x1 = 1.65, x2 = 0.95, α2 = 0.1, β2 = 7, E = 5, q = 0.1

In Figs. 4, 5, and 6, we observe steady state for 0 < r < 1, gradually forming for r > 3, there seems to be periodic transient along with messy behaviour for r>3.5 with fixed parameter r = 0 to 4, k = 1.5 , and q = 0.1. Figs. 7, 8, and 9 represents the variation of the populations against time beginning with initial conditions x1 = 0.2, x2 = 0.9, x3 = 0 and with fixed parameters r = 0.32, α2 = 1.6, α3 = 0.5, d2 = 0.84, d3 = 0.002, e2 = 2.1, e3 = 0.5, E = 0.001, q = 0.1. The limit cycles for the fixed parameters r = 2, k = 3.5, α2 = 1.9, α3 = 0.5, d2 = 0.2, d3 = 0.1, e2 = 0.9, e3 = 2, E = 0.009, q = 0.005, x1 = 0.2, x2 = 0.9 are shown in Figs. 10, 11, and 12. Further the phase diagram in Fig. 12 stipulates the aggressive behaviour of the system accompanying the above parameter values with harvesting effort E = 0.009.

The Role of Harvesting in a Food Chain Model and Its Stability …

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Fig. 3 x1 = 4.65, x2 = 1.95, α2 = 0.1, β2 = 7, E = 5, q = 0.1

Fig. 4 x1 = 0.5, x2 = 1.95, α2 = 1.6, β2 = 7, E = 4

6 Conclusion On a continuous time food chain system, we have analysed the structural behaviour and harvesting significantly effect on prey species accompanying by two dominant predators involving Holling type II functional response by taking into account that predators predates on prey species and super predator predates on predator species. The non-negative equilibria were determined to achieve the local stability conditions. For food chain models, the difficulty of developing Lyapunov function has been investigated. The Lyapunov scalar function is adapted in order to verify the food chain system’s asymptotic global stability with harvested prey. Finally, simulation results were performed to verify our conclusions. Bifurcation diagrams, time series

20 Fig. 5 x1 = 0.5, x2 = 0.4, α2 = 1.6, β2 = 7, E = 4

Fig. 6 x1 = 0.5, x2 = 1.95, α2 = 0.001, β2 = 0.64, E = 0.1

Fig. 7 Varying k = 2, β2 = 1, β3 = 2 , Eigen values = (0.9159, 0.9159, 0.0511)

G. M. Vijayalakshmi et al.

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Fig. 8 Varying k = 4, β2 = 1, β3 = 2 , Eigen values = (0.9095, 0.9095, 0.0512)

Fig. 9 Varying k = 4, β2 = 5, β3 = 9 , Eigen values = (0.7237, 0.0152, 0.0014)

Fig. 10 r = 2, β2 = 2, β3 = 2.5, Eigen values = (0.3310, 0.3310, 0.1672)

plots, and limit cycles are generated for various parameter values by using MATLAB software, and these graphs have been thoroughly discussed. The bifurcation of prey species has been extensively discussed, with varied parameters of prey growth rate and carrying capacity.

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Fig. 11 r = 1, β2 = 2, β3 = 2.5, Eigen values = (0.9673, 0.1420, 0.1420)

Fig. 12 r = 0.465, β2 = 5.6, β3 = 9, Eigen values = (0.3835, 0.3835, 0.1204)

References 1. Azar, C., J. Holmberg, and K. Lindgren. 1995. Stability analysis of harvesting in a predator-prey model. Journal of Theoretical Biology 174 (1): 13–19. 2. Kar, T.K., and H. Matsuda. 2007. Global dynamics and controllability of a harvested prey– predator system with Holling type III functional response. Nonlinear Analysis: Hybrid Systems 1 (1): 59–67. 3. Georgescu, P., and H. Zhang. 2014. Lyapunovfunctionals for two-species mutualisms. Applied Mathematics and Computation 226: 754–764. 4. Wang, W., Y.N. Zhu, Y. Cai, and W. Wang. 2014. Dynamical complexity induced by Allee effect in a predator–prey model. Nonlinear Analysis: Real World Applications 16: 103–119. 5. VijayaLakshmi, G.M. 2020. Effect of herd behaviour prey-predator model with competition in predator. Materials Today: Proceedings 33: 3197–3200. 6. Johri, A., N. Trivedi, A. Sisodiya, B. Sing, and S. Jain. 2012. Study of a prey-predator model with diseased prey. International Journal Contemp. Mathematics Sciences 7 (10): 489–498. 7. Sahoo, B., and S. Poria. 2014. Diseased prey predator model with general Holling type interactions. Applied Mathematics and computation 226: 83–100. 8. Bhattacharya, S., M. Martcheva, and X.Z. Li. 2014. A predator–prey–disease model with immune response in infected prey. Journal of Mathematical Analysis and Applications 411 (1): 297–313. 9. Kar, T.K., and K.S. Chaudhuri. 2004. Harvesting in a two-prey one-predator fishery: a bioeconomic model. The ANZIAM Journal 45 (3): 443–456. 10. Agarwal, M., and R. Pathak. 2013. Influence of non-selective harvesting and prey reserve capacity on prey–predator dynamics. International Journal of Mathematics Trends and Technology 4: 295–309.

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11. Agarwal, M., and R. Pathak. 2017. Harvesting and Hopf Bifurcation in a prey-predator model with Holling type IV functional response. International Journal of Mathematics and Soft Computing 2 (1): 99. 12. Vijaya Lakshmi, G.M., S. Rasappan, and P. Rajan. 2021. A study of stochastic ecological model with prey harvesting as a tool of disease control. In Proceedings of First International Conference on Mathematical Modeling and Computational Science: ICMMCS, August 14–15, pp. 11–22. Springer: Singapore (1292). 13. Al-Khedhairi, A. 2009. The chaos and control of food chain model using nonlinear feedback. Applied Mathematical Sciences 3 (12): 591–604. 14. Sahoo, B., and S. Poria. 2014. The chaos and control of a food chain model supplying additional food to top-predator. Chaos, Solitons and Fractals 58: 52–64. 15. Deng, B. 2004. Food chain chaos with canard explosion. Chaos: An Interdisciplinary Journal of Nonlinear Science 14 (4): 1083–1092. 16. Chiu, C.H., and S.B. Hsu. 1998. Extinction of top-predator in a three-level food-chain model. Journal of Mathematical Biology 37 (4): 372–380. 17. Kar, T.K., and A. Batabyal. 2010. Persistence and stability of a two prey one predator system. International Journal of Engineering, Science and Technology 2 (2): 174–190. 18. Guo, L., Z.G. Song, and J. Xu. 2014. Complex dynamics in the Leslie-Gower type of the food chain system with multiple delays. Communications in Nonlinear Science and Numerical Simulation 19 (8): 2850–2865. 19. Gakkhar, S., and B. Singh. 2007. The dynamics of a food web consisting of two preys and a harvesting predator. Chaos, Solitons and Fractals 34 (4): 1346–1356. 20. Hsu, S.B., T.W. Hwang, and Y. Kuang. 2003. A ratio-dependent food chain model and its applications to biological control. Mathematical Biosciences 181 (1): 55–83.

Inventory Queuing System Study Using Simulation and Birth–Death Process J. Vijayarangam, S. Perumal, and J. Viswanath

Abstract Inventory problem and queuing problems are two of the most widely studied domains in operation research. But when we are applying these ideas in real systems, we are facing some issues related to their form as such in the system which usually is a combination of more than one model under the operations research category of models. So, a combinational study of models becomes an essential part of implementation and Inventory queuing system is one such combinational model where we combine ideas of a queuing model and an inventory model to suit the necessities of real life implementation. This is a more realistic one for study and in this paper we are using a well-established line of a Birth–Death process to study which both is interesting and a fruitful line of study. To add another dimension to the study we have added simulation, a relatively easy and most sought-after avenue of studying practical implementation of models which is usually used when we have no compact analytical solutions is available by theoretical modeling. We first study an inventory queuing system as a two-dimensional Birth–Death process and derive stationary probabilities analytically. As the solutions obtained have not turned out to be as compact as we usually have in an inventory set up, we moved into the simulation domain and employed two lines of simulation, one using the usual newsboy type and the other inspired by the BD type study and we have obtained a reasonable set measures to make some nice conclusions. Keywords Newsboy · Simulation · Inventory · Queue · Birth–Death process

J. Vijayarangam (B) Sri Venkateswara College of Engineering, Sriperumbudur, Chennai, India e-mail: [email protected] S. Perumal New Horizon College of Engineering, Bangalore, India J. Viswanath Vel Tech Rangarajan Dr, Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_3

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1 Introduction Queuing system analysis is one with an abundant literature and this paper is one more avenue in which the aspect of reality is incorporated by way of viewing a queuing system as a system which serves people for a particular purpose and we assume that to be selling a product wherein the inventory aspect of the product is our reality incorporation. Ref. [1] is one of the best books for operations research topics. It discusses every OR topic in detail and is a good book to learn. Ref. [2] is a book every individual trying to learn discrete event simulation would love to go to. Ref. [3] is a nice book about queuing theory. Ref. [4] is a book which discusses general topics of social studies to enhance our understanding of a problem. Ref. [5] is a nice read regarding queuing theory and how it can be applied to real world problems. Ref. [6] is a good study about applying queuing theory in problems regarding port congestions as are [7] which are applications of queuing theory in real life. Ref. [8] is a paper discussing application of simulation in a queuing problem discussion in real life. Ref. [9] is a simple study about simulation application in an inventory problem. Ref. [10–18] are different avenues of addressing inventory problems which are helping a researcher find his variation. Ref. [19] is a paper which discusses an inventory system with two types of customers and [20] is a paper discussing multiproduct inventory. Ref. [21] is a very nice paper discussing the queuing inventory concept which is actually the one which has inspired us to this paper. Ref. [22] is a paper which has sort of opened the avenue of two-dimensional BD process as a possible discussion line for our paper. We are treading a path of studying an inventory queuing system viewing that as a two-dimensional Birth–Death process. A queuing system is based on its two basic entities, customers and servers and hence the type of service. If the point of view is changed and a queuing system is viewed in alignment with its service, through the inventory system concepts, it is bound to make the study all the more realistic and relevant and useful. So, we study an inventory queuing system where a queuing system is studied with its service viewed through the inventory.

2 Inventory Queuing System as a Birth–Death Process We describe this as a two-dimensional system [C(t), P(t)] where C(t) is the process counting the number of customers in the system at time ‘t’ and P(t) is the process counting the number of inventory available at time ‘t’. Viewing this as a twodimensional Birth–Death process, given in general form by Fig. 1, we proceed to analyze it. Since the above is very difficult to analyze in its general form, we start from a simplified model with the following assumptions: Assumption1: The maximum number of products available at any time is 2. Assumption2: We cannot have more than two customers at any time. Assumption3: Any customer who enters serves himself and only him and definitely makes a purchase of exactly one product. The

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Fig. 1 General two-dimensional BD process

state diagram of such a two-dimensional BD process for the inventory queuing system with arrival rate α and service rate β, is given by Fig. 2 The flow balance equations for the above state diagram are:

Fig. 2 State diagram of a two product inventory two-dimensional BD process

π02 α = π12 (α + β)

(1)

π12 β = π01 α

(2)

π12 α = π22 β

(3)

π11 β = π01 α + π22 β

(4)

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π00 = π11 β

(5)

Simplifying the above four equations and solving them along with the normalizing constraint. π00 + π01 + π11 + π02 + π12 + π22 = 1

(6)

Which gives the solution π00 = απ02 β π02 α+β

(8)

α π02 β

(9)

α π02 α+β

(10)

α α π02 β α+β

(11)

π01 =

π11 = π12 = π22 =

(7)

  α α β α α α + + =1 π02 1 + α + + β α+β α α+β β α+β

(12)

The above is clearly not a compact representation to interest further study. So, we altered the assumptions like a customer can either buy a product or leave without buying and studied it but again we obtained a solution which is not compact. We tried increasing the number of products and studied with no great improvement in compactness of the solution. So, after losing a bit of hair we have decided to look for alternate avenues like simulation.

3 Method 1 3.1 Newsboy M|M|1 Model View a M|M|1 service facility, basically viewed as a queuing system, with an added parameter of the inventory size. In other words we are considering what we wish to brand as a “newsboy’s M|M|1 model-N B M M1 model”. From practical experience, one knows or believes that when there is a single server in the system and the stock level is high like it is the morning scene in a newsboy problem, the time of service is expected to be small as compared to the same system at a later time of the day,

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assuming the demand to be uniform, having half the inventory level or the quarter level. If we incorporate this idea through the service time, then that will be a better approximation of the system than the normal way of simulation. So, we assume that the service time when the inventory level [M, M/2] is x, ST when inventory level [M/2, M/4] is 1.5x and during [M/4, 0] is 1.75x, where 1.5 and 1.75 are logic inspired assumptions. We also make this model more realistic as possible by introducing the quantity of purchase by every served customer to identify the current inventory level.

3.2 Methodology Step1: Generate Markovian Inter-arrival time and service time using Monte Carlo Simulation. Step2: Simulate a M|M|1 model before and after incorporating the inventory based variation. Step3: For the parameters used in step 1, obtain the steady state solutions using theoretical models. Step 4: Compare the step 2 and 3 measures to make inferences.

3.3 Results The results obtained along with the input distributions are in Table 1, Fig. 3. Table 1 Input distributions, measures for the newsboy M M1 model Newsboy M M1 model IAT (mts)

Probability

ST (mts)

Probability

Units purchased

Probability

1

0.4

0.25

0.35

1

0.6

2

0.4

0.5

0.5

2

0.2

3

0.2

1

0.15

3

0.2

Comparison of measures Simulation M M1

N B M M1

Theoretical M M1

Wq

0.69

0.81

3.49

Ws

1.92

1.82

3.97

Lq

1.25

1.46

6.28

Ls

3.45

3.27

7.16

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Fig. 3 Measures of an inventory queuing system comparison

4 Method 2 To simulate such a system, we needed a clear point of view of the movement of the entities in the system. We kept the original idea of a system with self-serving mechanism, no great restrictions in the number of inventory level as we are working in a different medium but each customer’s transactions are independent. But for simplicity, we fixed a number max P(t) = 25 and terminating condition for the simulation as P(t) = 0. But we kept the other assumptions like each customer purchases only one product and a customer can go in only after the previous one leaves which is something similar to recording a number of independent M|M|1 systems. The simulation is performed for 25 runs. Data collected and we obtained three measures, number of waiting customers, average waiting time and time of simulation. They are in Table 2 and Figs. 4, 5 and 6. The average number of waiting customers for the many simulation runs made came out as 15 with an average simulation time of 6.564732 min and the average waiting time for random customers was obtained as 0.208496 min. This is depicted in Table 3. Table 2 Simulation of an inventory queuing system Simulation run

Number of cust simulated

Number of cust wait

Avg WT

Time of simulation

1

25

18

0.1602

4.8526

2

25

23

0.5586

4.9476

3

25

20

0.3616

4.7923

4

25

12

0.144

6.8653

… 25 Runs

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Fig. 4 Number of waiting customers of an inventory queuing system simulated

Fig. 5 Average waiting time of a customers for the many simulation runs

Fig. 6 Total simulation time of the simulations Table 3 Average waiting time and time of simulation

Averages No: customers who wait

Average waiting time

Time of simulation

15

0.208496

6.564732

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5 Conclusion The simulation 1, using the Newsboy idea gives measures which are almost similar with respect to the simulation but differ significantly when compared with the theoretical values. So, this method of simulation seems not serving our purpose for the present problem under discussion. But the simulation 2, using the point of view of a Birth–Death process is much more reasonable as we have moved to a realistic measures of average waiting time and average waiting customers which can implicitly help us visualize a system for any inventory level and for any number of customers. It kind of tells us that when we go for avenues like simulation, it is more reasonable to obtain measures suiting their mode and not as per se like W q and Lq with theoretical models as our purpose is served once we obtain such measures. So, we could conclude that inventory queuing models for which we could not obtain compact analytical results can be solved using simulation.

References 1. Taha, Hamdy. 2010. Operations Research, 5th edn. Pearson. 2. Banks, J., J.S. Carson, B.L. Nelson, et al. 2010. Discrete Event System Simulation, 5th edn. Pearson. 3. Cooper, R.B. 1981. Introduction to Queuing Theory. North Holland. 4. Prabhu, N.U. 1997. Qualitative research practice. A Guide for Social Science Students and Researchers. London: Sage Publications. 5. Sztrik, János. 2010. Queuing theory and its applications, a personal view. In International Conference on Applied Informatics, Proceedings of the 8th Conference, vol. 1, 9–30. Eger: Hungary. 6. Bolanle, Amole Bilqis. 2011. Application of queuing theory to port congestion problem in Nigeria. European Journal of Business and Management 3 (8): 2011. 7. Mwangi, Sammy Kariuki, and Thomas Mageto Ombuni. 2015. An empirical analysis of queuing model and queuing behaviour in relation to customer satisfaction at Jkuat students finance office. American Journal of Theoretical and Applied Statistics 4 (4): 233–246. 8. Ehsanifar, Mohammad, Nima Hamta, and Mahshid Hemesy. 2017. A simulation approach to evaluate performance indices of fuzzy exponential queuing system (An M/M/C model in a banking case study). Journal of Industrial Engineering and management Studies 4 (2): 35–51. 9. Vijayarangam, J. 2015. Simulation based analysis of a newsboy problem. Journal Technology Advances and Scientific Research 13: 8–10. 10. Nair, Anoop N., and M.J. Jacob. 2015. An (s, S) production inventory controlled self-service queuing system. Journal of Probability and Statistics 2015 (2): 1–8. 11. Arivarignan, G., C. Elango, and N. Arumugam. 2002. A continuous review perishable inventory control system at service facilities. Advances in Stochastic Modelling, Notable Publications, NJ, USA, 29–40. 12. Berman, O., and E. Kim. 1999. Stochastic models for inventory management at service facilities. Communications in Statistics-Stochastic Models 15 (4): 695–718. 13. Berman, O., E.H. Kaplan, and D. Shimshak. 1993. Deterministic approximations for inventory management at service facilities. IIE Transactions 25 (5): 98–104. 14. Deepak, T.G., A. Krishnamoorthy, V.C. Narayan, and K. Vineetha. 2008. Control policies for inventory with service time. Annals of Operations Research 160: 191–213.

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15. Kalpakam, S., and S. Shanthi. 2001. Perishable inventory system with modified (S-1, S) policy and arbitrary processing times. Computers and Operations Research. 28 (5): 453–471. 16. Kalpakam, S., and S. Shanthi. 2000. A perishable system with modified base stock policy and random supply quantity. Computers and Mathematics with Applications 39 (12): 79–89. 17. Levi, R., G. Perakis, and J. Uichanco. 2015. The data-driven newsvendor problem. New Bounds and Insights: Journal Of Operations Research 63 (6): 1294–1306. 18. Manuel, P., B. Sivakumar, and G. Arivarignan. 2008. A perishable inventory system with service facilities and retrial customers. Computers and Industrial Engineering 54 (3): 484–502. 19. Karthick, T., B. Sivakumar, and G. Arivarignan. 2015. An inventory system with two types of customers and retrial demands. International Journal of Systems Science Operations and Logistics 2 (2): 90–112. 20. Vijayarangam, J., B. Navin Kumar, A. Vasudevan, and Pandiyarajan. 2020. Simplified analysis of a multiproduct newsboy problem using simulation. IOP Conference Series, Materials Science and Engineering. https://doi.org/10.1088/1757-899X/988/1/012089. 21. Krishnamoorthy, A., B. Lakshmi, and R. Manikandan. 2011. A revisit to queuing-inventory system with positive service time. Opsearch 48 (2): 153–169. 22. Mode, C.J. 1962. Some multi-dimensional birth and death processes and their applications in population genetics. Biometrics 18 (4): 543–567.

Steady-State Analysis of Bulk Queuing System with Renovation, Prolonged Vacation and Tune-Up/Shutdown Times S. P. Niranjan, B. Komala Durga, and M. Thangaraj

Abstract This paper investigates bulk arrival and batch service queueing system with server failure, prolonged vacation and tune-up time/shutdown times. On service completion epoch, if the queue length is less than ‘a’, then the server leaves for vacation (secondary job). When the server gets breakdown whilst serving customers service process will not be interrupted, it will be continued for present batch of customers by doing some precaution arrangements. The server will be repaired after the service process during renewal time of the server. In vacation queueing models, the server will be idle after the vacation due to insufficient number of customers, but in this model, if the server wishes to go another type of vacation, it will be allowed with probability α or the server will go to dormant period with probability 1 − α. For the proposed model, probability generating function of the queue size distribution at an arbitrary time epoch is obtained by using supplementary variable technique. Various performance measures are derived. Keywords Batch service · Renovation period · Prolonged vacation · Supplementary variable technique · Tune-up/shutdown times

1 Introduction Mathematical analysis of queueing systems with vacations has gain tremendous attention amongst researchers because it has lot of application in communication networks, inventory management, manufacturing industries and production line systems. Vacation period indicates unavailability of the server or idle time of the server. In some cases, the server cannot be able to start the service due to server loss and/or insufficient number of customers to start the service in case of bulk service S. P. Niranjan (B) · B. Komala Durga Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India e-mail: [email protected] M. Thangaraj Jain (Deemed-To-Be University), Bangalore, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_4

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systems. During vacation period, the server wishes to do some supplementary works such as preservation works or serving secondary customers. The main objective of queueing model with server vacation is effective utilization of idle time of the server. Therefore, vacation queueing model tends to minimize the total average cost of the system. Modelling and analysis of queueing systems are taken into an account by many researchers in past years, which include a single server queueing models with vacations by Takagi [1], vacation queueing models by Tian and Zhang [2]. Neuts [3] classified bulk queues with Poisson input. Arumuganathan and Jeyakumar [4] analyzed analysis of M X /G(a, b)/1 queueing system with multiple vacations, setup times, closedown times and N-policy. They derived steady-state condition and some performance measures for the queueing system. Lee et al. [5] analyzed M x /G/1 queue with N-policy and multiple vacations. Recently, Jeyakumar and Senthilnathan [6] introduced multiple working vacations for bulk arrival and batch service queueing system. Madan [7] introduced optional deterministic server vacations in a single server queue providing two types of first essential service followed by two types of additional optional service. Recently, Ayappan and Deepa [8] modelled M [X ] /G(a, b)/1 queuing system with additional optional service, multiple vacation and setup time. Bulk arrival and batch service queueing model with vacation interruption have been analyzed by Haridass and Arumuganathan [9]. In the above model, they defined steady-state condition by using supplementary variable technique. Agarwal [10] derived an optimal N-policy for finite queue with server breakdown and state dependent rate. Queueing model with server breakdown have been explained by many of the authors which include Madan et al. [11] studied steady-state analysis of two M X /M(a, b)/1 queue models with random breakdowns. They considered that the repair time is exponential for one model and deterministic for another one. But in most of the situations, it is not possible to disturb the server before completing its batch of service. This simulates some authors to model bulk arrival and batch service system with breakdown without service interruption. In their model, they described as when the server got breakdown, there is no need to stop its service, it will be continued for some time by doing some technical arrangements. After completed its batch service, server will be repaired before starting another service that period is called renovation time. In some situations, the server may prolonged its vacation period to utilize his idle time effectively. This simulates the authors to model bulk arrival and batch service queuing model with breakdown without service interruption, setup time, close down time and prolonged vacation. Only few authors have analyzed about queue with server breakdown which includes the study of reliability analysis of M/G/1 queueing system with server breakdowns and vacations by Li et al. [12]. Nishimura [13] studied an M/G/1 vacation model with two service modes. Also, Nobel [14] found an optimal control for an MX/G/1 queue with two service modes. Madan et al. [11] studied steady-state analysis of two M X /M(a, b)/1 queue models with random breakdowns. They considered that the repair time is exponential for one model and deterministic for another one. In the literature of queueing models with server breakdown, it is observed that all the

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authors except Madan et al. [11], who deal the server breakdown in the bulk service queueing model, deal with the server which can serve only one customer at a time. It is clear that if breakdown occurs, the server is allowed to interrupt immediately. But in most of the situations, it is not possible to disturb the server before completing its batch of service. Breakdown without service interruption in a bulk arrival and batch service queueing model is also studied by Jeyakumar and Senthilnathan [15]. They modelled with closedown time and derived probability generating function of service completion epoch, vacation completion epoch and renovation completion epoch. Wu et al. [16] analyzed an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility. Choudhury and Deka [17] studied batch arrival queue with two phases of service, such as Bernoulli vacation schedule and server breakdown.

2 Model Description In this paper, bulk arrival and batch service queueing model with breakdown without service interruption, tune-up time, shutdown time and prolonged vacation are considered. Customers are arriving into the system in bulk according to Poisson arrival rate λ. Server affords service in batches with minimum batch size ‘a’ and maximum batch size ‘b’ according to general bulk service rule. After completing a batch of service, if the server finds at least ‘a’ customers waiting for service say ξ, it serves a batch of min (ξ, b) customers, where b > a. On the other hand, if the queue length is at the most ‘a–1’, the server undergone with shutdown process and leaves for a secondary job (vacation) of random length. After completing a vacation, if the queue length is still less than ‘a’ and the server wishes to go another type of vacation, it will be allowed (prolonged vacation) with probability α or with probaility (1 − α) and the server remains idle (dormant period) until the queue length reaches the value ‘a’. The server extends its vacation period even after the vacation process is called prolonged vacation. In the vacation completion epoch or prolonged vacation period completion, if the queue length reaches the value ‘a’, then the server begins tune-up process to initiate service. After the tune-up process, the server serves a batch of customers waiting in the queue. In batch service completion epoch, if the server is breakdown with probability β, then the renovation of service station will be considered. After completing a renovation of service station or if there is no breakdown with probability(1–β), if the queue length is less than ‘a’, then the server performs shutdown works and leaves for vacation. During dormant period, if the queue length reaches the threshhold value ‘a’, then the server is undergone with tune-up process to start its service. Schematic representation of the pr oposed queueing model is depicted below. For the proposed model, the probability generating function of the steadystate queue size distribution at an arbitrary time is obtained. Various performance measures are derived. The above system is modelled using the supplementary variable technique (Fig. 1).

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Fig. 1 Schematic representation

2.1 Notations X–Group size random variable of the arrival. λ–Poisson arrival rate. gk –Probability that ‘k’ customers arrive in a batch. X (z)–Probability generating function of X. Nq (t)–Number of customers waiting for service at time t. Ns (t)–Number of customers under the service at time t. Table 1 Table 1 Notations Cumulative distribution function

Probability density function

Laplace-Stieltjes transform

Remaining time

Service

A(x)

a(x)

A0 (x)

Vacation

V (x)

v(x)

˜ A(θ) ˜ V (θ)

Renovation

R(x)

r (x)

Shutdown time

F(x)

Tune-up time

S(x)

V 0 (x)

f (x)

˜ R(θ) ˜ F(θ)

F 0 (x)

s(x)

˜ S(θ)

S 0 (x)

R 0 (x)

Steady-State Analysis of Bulk Queuing System with Renovation …

39

⎧ 0, when the server is busy with service ⎪ ⎪ ⎪ ⎪ ⎪ 1, when the server is on vacation ⎪ ⎪ ⎪ ⎪ ⎨ 2, when the server is on tune - up process C(t) = 3, when the server undergone shutdown process ⎪ ⎪ ⎪ 4, when the server is under renewal process ⎪ ⎪ ⎪ ⎪ 5, when the server is on extended vacation ⎪ ⎪ ⎩ 6, when the server is on dormant period The state probabilities are defined as follows:   Pi j (x, t)dt = Pr Ns (t) = i, Nq (t) = j, x ≤ A0 (t) ≤ x + dt, C(t) = 0 , a ≤ i ≤ b − 1; j ≥ 1   Q n (x, t)dt = Pr Nq (t) = n, x ≤ V 0 (t) ≤ x + dt, C(t) = 1 , 0 ≤ n ≤ a − 1,   Sn (x, t)dt = Pr Nq (t) = n, x ≤ S 0 (t) ≤ x + dt, C(t) = 2 , n ≥ a,   Fn (x, t)dt = Pr Nq (t) = n, x ≤ F 0 (t) ≤ x + dt, C(t) = 3 , n   Rn (x, t)dt = Pr Nq (t) = n, x ≤ R 0 (t) ≤ x + dt, C(t) = 4 , n ≥ 0;   Un (x, t)dt = Pr Nq (t) = n, x ≤ U 0 (t) ≤ x + dt, C(t) = 5 , 0 ≤ n ≤ a − 1, is   Tn (t) = Pr Nq (t) = n, C(t) = 6 , 0 ≤ n ≤ a − 1

3 Steady-State Analysis The steady-state system equations are obtained by using supplementary variable technique introduced by Cox, [18]. 0 = −λT0 + (1 − α)Q 0 (0) 0 = −λTn + (1 − α)Q n (0) +

n  k=1

Tn−k λgk 1 ≤ n ≤ a − 1

(1)

(2)

40

S. P. Niranjan et al.

⎛ −

⎞

b

⎜ (1 − β) m=a Pmi (0) + Ri (0) ⎟ d ⎟a(x)a ≤ i ≤ b (3) Pi0 (x) = −λPi0 (x) + ⎜ ⎠ ⎝ a−1 dx + Tm λgi−m + Si (0) m=0

− −

j  d Pi, j−k (x)λgk a ≤ i ≤ b − 1, j ≥ 1 Pi j (x) = −λPi j (x) + dx k=1

d Pbj (x) = −λPbj (x) dx  + (1 − β)

b 

(4)

 Pm,b+ j (0) + +Sb+ j (0) + Rb+ j (0) a(x)

m=a

+

j 

Pb, j−k (x)λgk j ≥ 1

(5)

k=1

−

d Q 0 (x) = −λQ 0 (x) + F0 (0)v(x) + R0 (0)v(x) dx

(6)

 d − Q n (x) = −λQ n (x) + Q n−k (x)λgk dx k=1 n

+ Fn (0)v(x)1 ≤ n ≤ a − 1 b  d R0 (x) = −λR0 (x) + β Pm0 (0)r (x) dx m=a

(8)

b b   d Rn (x) = −λRn (x) + β Pmn (0)r (x) + Rn−k (x)λgk n ≥ 1 dx m=a m=a

(9)

−

−

(7)

d U0 (x) = −λU0 (x) + α Q 0 (0)u(x) dx

(10)

b  d Un (x) = −λUn (x) + α Q n (0)u(x) + Un−k (x)λgk n ≥ 1 dx m=a

(11)

− −

−

b  d Fn (x) = −λFn (x) + (1 − β) Pmn (0) f (x) dx m=a

+

n  k=1

Fn−k (x)λgk + Rn (0) f (x)n ≤ a − 1

(12)

Steady-State Analysis of Bulk Queuing System with Renovation …

41

 d Fn (x) = −λFn (x) + Fn−k (x)λgk n ≥ a dx k=1 n−a

−

 d − Sn (x) = −λSn (x) + ((1 − α)Q n (0) + Un (0))s(x) + Sn−k (x)λgk dx k=1

(13)

n

(14)

Taking Laplace-Stieltjes transform on both sides of the Eq. (1) through (14), we have −λT0 = (1 − α)Q 0 (0) 0 = λTn − (1 − α)Q n (0) −

n 

(15)

Tn−k λgk 1 ≤ n ≤ a − 1

(16)

k=1

θ P˜i0 (θ ) − P˜i0 (0) = λ P˜i0 (θ ) − (1 − β)

b 

˜ )a ≤ i ≤ b Pmi (0)A(θ

m=a

−

a−1 

˜ ) − Si (0)A(θ ˜ ) ˜ ) − Ri (0)A(θ Tm λgi−m A(θ

(17)

m=0

θ P˜i j (θ ) − Pi j (0) = λ P˜i j (θ ) −

j 

Pi, j−k (θ )λgk

(18)

Pb, j−k (θ )λgk j ≥ 1

(19)

k=1

θ P˜bj (θ ) − Pbj (0) = λ P˜bj (θ ) −

j  k=1

⎛

⎞ b a−1 − β) P T λg + (1 (0) m,b+ j m b+ j−m ⎠ ˜ −⎝ A(θ ) m=a m=0 +Rb+ j (0) − Sb+ j (0) θ Q˜ 0 (θ ) − Q 0 (0) = λ Q˜ 0 (θ ) − F0 (0)V˜ (θ ) θ Q˜ n (θ )−−Q n (0) = λ Q˜ n (θ ) −

n 

Q˜ n−k (θ )λgk − Fn (0)V˜ (θ )

(20)

(21)

k=1

θ R˜ 0 (θ ) − R0 (0) = λ R˜ 0 (θ ) − β

b 

˜ ) Pmn (0) R(θ

(22)

m=a

θ R˜ n (θ ) − Rn (0) = λ R˜ n (θ )−−β

b  m=a

˜ )− Pmn (0) R(θ

b  m=a

R˜ n−k (θ )(t)λgk

(23)

42

S. P. Niranjan et al.

θ F˜n (θ ) − Fn (0) = λ F˜n (θ )−−(1 − β)

b 

˜ ) Pmn (0) F(θ

m=a

−

n 

˜ )n ≤a−1 F˜n−k (θ )λgk − Rn (0) F(θ

(24)

k=1

θ F˜n (θ ) − Fn (0) = λ F˜n (θ ) −

n 

F˜n−k (θ )λgk n ≥ a

(25)

k=1

θ U˜ 0 (θ ) − U0 (0) = λU˜ 0 (θ ) − α Q 0 (0)U˜ (θ )

(26)

θ U˜ n (θ )−−Un (0) = λ Q˜ n (θ ) − α Q n (0)U˜ (θ ) −

n 

U˜ n−k (θ )λgk 1 ≤ n ≤ a − 1

(27)

k=1

˜ ) − Un (0) S(θ ˜ )n ≥ a θ S˜n (θ )−−Sn (0) = λ S˜n (θ )−−(1 − α)Q n (0) S(θ

(28)

4 Probability Generating Function P˜ i (z, θ ) =

∞ 

P˜ i, j (θ)z j P i (z, 0) =

j =0

˜ Q(z, θ) =

a−1 

∞  j =0

˜ n (θ )z n Q(z, 0) = Q

a−1 

n=0

Q n (0)z n T (z) =

n=0

˜ R(z, θ) =

a−1 

˜ n (θ )z n R(z, 0) = R S˜ n (θ )z n S(z, 0) =

n=a

 F(z, θ ) =

Fn (θ )z

a−1  n=0

∞ 

R n (0)z n

∞ 

Sn (0)z n

n=a n

F(z, 0) =

n=0

˜ U(z, θ) =

Tn z n a ≤ i ≤ b

n=0

∞ 

a−1 

a−1  n=0

n=0

˜ S(z, θ) =

P i j (0)z j

∞ 

F n (0)z n

n=0

U˜ n (θ )z n U(z, 0) =

a−1 

U n (0)z n

n=0

 (θ )F(z, 0)0 ≤ n ≤ a − 1 Q(z, θ ) = Q(z, 0) − V (θ − λ + λX(z)) 

(28)

Steady-State Analysis of Bulk Queuing System with Renovation …

43

(θ − λ + λX(z)) P˜ i (z, θ ) = P i (z, 0)

 ) − A(θ

⎫ ⎧ b ⎪ ⎪ ⎪ Pmi (0) + Si (0) + Ri (0) ⎪ ⎬ ⎨ (1 − β) m=a

a−1 ⎪ ⎪ ⎩+ Tm λgi−m

(29)

⎪ ⎪ ⎭

m=0

(θ − λ + λX (z)) P˜b (z, θ ) = Pb (z, 0) ⎡



b

⎤

b b−1

Pm (z, 0) − Pm j (0)z ⎥ ⎢ (1 − β) ⎢  m=a  m=a j=0  ⎥ ⎢ ⎥ ⎢ ⎥ a−1 b−m−1 ⎢ +λ T (z)X (z) − ⎥ m j gjz Tm z  )⎢ ⎥ A(θ ⎢ ⎥ m=0 j=1 −− b ⎢ ⎥ z ⎢ b−1 ⎥ n ⎢ +S(z, 0) − ⎥ R (0)z n ⎢ ⎥ n=0 ⎢ ⎥ b−1 ⎣ ⎦ n +R(z, 0) − Rn (0)z j

(30)

n=0

˜ θ ) = R(z, 0) − β (θ − λ + λX (z)) R(z,

b 

 ) Pm (z, 0) R(θ

(31)

m=a

˜ θ ) = S(z, 0)−−((1 − α)Q(z, 0) + U (z, 0)) S(θ ) (θ − λ + λX (z)) S(z,

(32)

⎡

⎤ b n − β) P (1 (0)z mn ⎢ ⎥ m=a ⎥ ˜ θ ) = F(z, 0)−− F(θ ˜ )⎢ (θ − λ + λX (z)) F(z, ⎣ a−1 ⎦ − Rn (0)z n

(33)

n=0

(θ − λ + λX (z))U˜ (z, θ ) = U (z, 0) − α Q n (0)U˜ (θ )

(34)

4.1 Probability Generating Function of Queue Size The probability generating function of queue size at an arbitrary time epoch can be obtained by using the below given equation ⎛ b−1 ⎞ ˜ 0) i (z, 0) + P˜b (z, 0) + Q(z, P ⎠ P(z) = ⎝ i=a ˜ 0) + S(z, ˜ 0) + U˜ (z, 0) + R(z, ˜ 0) + F(z,

44

S. P. Niranjan et al.

Substituting θ = λ − λX (z) in Eqs. (29)–(35), after doing some algebra, probability generating function of queue size is defined below, we get 

⎛ 



⎞

a−1 b−1 i + m=0 Tm λgi−m ω(z) i=a di + s ⎜    ⎜ a−1 ˜ − λX (z)) − 1 + γ (z) F(λ dn z n ⎜+ n=0    ⎝

⎟ ⎟ ⎟  ⎠ ˜ − λX (z)) − 1 β A(λ ˜ − λX (z)) + A(λ ˜ − λX (z)) − 1 +ϕ(z) R(λ  P(z) =  ˜ − λX (z)) − β A(λ ˜ − λX (z)) R(λ ˜ − λX (z)) [−λ + λX (z)] z b − (1 − β) A(λ

where ⎛ ϕ(z) = λ⎝T (z)X (z) − ⎛



a−1 

⎛ ⎝Tm z m

m=0

b−m−1 

⎞⎞ g j z j ⎠⎠ −

b−1  (dn + sn )z n n=0

j=1

˜ − λX (z)) − β A(λ ˜ − λX (z)) R(λ ˜ − λX (z)) z b − (1 − β) A(λ



⎞

⎜ ⎛ ⎞⎟     ⎜ ⎟ ˜ − λX (z)) − 1 V˜ (λ − λX (z)) − 1 + S(λ ⎜ ⎟ ⎜ ×⎝  ⎠⎟    ⎜ ⎟ ˜ ˜ ˜ γ (z) = ⎜ × V (λ − λX (z)) (1 − α) + U (λ − λX (z)) − 1 α V (λ − λX (z)) ⎟ ⎜ ⎟   ⎜ ⎟ ˜ − λX (z)) S(λ ˜ − λX (z)) + (1 − α) R(λ ⎜ ⎟ ⎝     ⎠ ˜ ˜ ˜ × R(λ − λX (z)) − 1 + β A(λ − λX (z)) + A(λ − λX (z)) − 1   ⎛ ⎞ ˜ − λX (z)) − β A(λ ˜ − λX (z)) R(λ ˜ − λX (z)) z b − (1 − β) A(λ  ⎟  ⎜ ⎜ × z b − (1 − β) A(λ ⎟ ˜ − λX (z)) − β A(λ ˜ − λX (z)) R(λ ˜ − λX (z)) ⎟ ⎜  ⎞⎟ ⎜ ⎛ ⎟ ω(z) = ⎜ ˜ ˜ A(λ − λX − 1 − β) + β R(λ − λX (z)) (1 (z)) ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎜ + A(λ ⎟⎟ ˜ − λX (z))⎜ ˜ − λX (z)) +β R(λ ⎝ ⎠⎠ ⎝   ˜ − λX (z)) − 1 β A(λ ˜ − λX (z))(1 − β) + R(λ

dn = (1 − β) pn + Rn + vn

4.2 Steady-State Condition The probability generating function P(z) has to satisfy P(1) = 1. In order to satisfy this condition, applying hospital’s rule and evaluating lim P(z) and equating the z→1

expression to 1, it is derived that ρ < 1 is the condition to be satisfied for the existence of steady state for the model under consideration, where ρ=

λE(X )[E(A) + β E(R)] b

Steady-State Analysis of Bulk Queuing System with Renovation …

45

5 Performance Measures 5.1 Expected Queue Length The mean queue length E(Q) (i.e. mean number of customers waiting in the queue) at an arbitrary time epoch is obtained by differentiating P(z) at z = 1 and is given by E (Q) = lim P  (z) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ E(Q) =

z→1





a−1 b−1 m=0 Tm λgi−m × (B1 + F1 − F2 ) i=a di + si + A−1 + n=0 dn (2F5 (B2 + B3 + B4 + B5 ) − 2F3 F6 − 3T2 F7 )   ndn 2(B4 + B5 + B4 + B6 )F5−2F4 F6 −3F1 F7 + a−1 n=0 + a−1 n=0 n(n − 1)dn × ((B5 + B3 + B6 )2F5 − 2F6 F2 )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

+2F5(B1 + B2 + B3 + B4 + B5 + B6 + B7 ) − 2F6 F3 F4 − 3F7 F1 T7 24T 12

5.2 Expected Waiting Time in the Queue The mean waiting time of the customers in the queue E(W ) can be easily obtained using Little’s formula E(W ) =

E(Q) λE(X )

5.3 Expected Length of Busy Period Let B be the busy period random variable. Let T be the residence time that the server is rendering service or under repair. Then, E(T ) = E( A) + β E(R) E(T ) P(J = 0) E(T ) = a−1 − β)ai + Ri (1 i=0

E(B) =

46

S. P. Niranjan et al.

5.4 Probability that the Server is on Vacation Let P(V ) be the probability that the server is on vacation at time t. ˜ Q(z, 0) =

a−1    1 ˜ ˜ dn z n V(λ − λx(z)) − 1 F(λ − λX (z) (−λ + λx(z)) n=0

Therefore, ˜ 0) P(V ) = lim Q(z, z→1

=

a−1   1 V˜ (λ − λx(1)) − 1 dn − = λ + λx(1) n=0

where dn = (1 − β)an + Rn .

6 Numerical Illustration The following assumptions are made to obtain the numerical results: Batch service time distribution is 4-Erlang with parameter μ1 . Batch size distribution of the arrival is geometric with mean 2 Vacation time is exponential with parameter εl Prolonged vacation is exponential with parameter γ . Tune-up time is exponential with parameter ϕ. Shutdown time is exponential with parameter sb . Renewal time is exponential with parameter η Figs. 1, 2 and 3 Table 2.

7 Conclusion In this paper, bulk arrival and batch service queueing model with breakdown without service interruption, shutdown time, tune-up times and prolonged vacation are analyzed. This model so considered is unique in the sense that prolonged vacation is introduced for M X /G(a, b)/1 queueing system. Probability generating function of steady-state queue size at an arbitrary time epoch is obtained by using supplementary variable technique. Various performance measures are also derived in the paper with numerical results.

Steady-State Analysis of Bulk Queuing System with Renovation …

47

Fig. 2 Arrival rate versus performance measures

Fig. 3 Renewal rate versus performance measures Table 2 Arrival rate versus performance measures λ 0.5

E(Q) 7.4639

E(W )

E(B)

15.9238

11.4322

1.0

8.3268

18.3126

13.3351

1.5

10.9761

19.2391

13.9923

2.0

13.9632

21.6392

16.1123

2.5

14.5310

23.5812

17.8876

3.0

16.3417

24.9623

20.2341

3.5

17.9340

26.3421

22.1765

4.0

19.3214

29.2852

23.9231

4.5

21.4523

31.7623

25.7232

48

S. P. Niranjan et al.

References 1. Takagi, H. 1991. Vacation and priority systems, vol. 1. North Holland. 2. Tian, N. and Z.G. Zhang. 2006. Vacation Queueing Models: Theory and Applications, vol. 93. Springer Science & Business Media. 3. Neuts, M.F. 1967. A general class of bulk queues with Poisson input. The Annals of Mathematical Statistics 38 (3): 759–770. 4. Arumuganathan, R., and S. Jeyakumar. 2005. Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Applied Mathematical Modelling 29 (10): 972–986. 5. Lee, H.W., S.S. Lee, J.O. Park, and K.C. Chae. 1994. Analysis of the Mx/G/1 queue by N-policy and multiple vacations. Journal of Applied Probability 31 (2): 476–496. 6. Jeyakumar, S., and B. Senthilnathan. 2016. Steady state analysis of bulk arrival and bulk service queueing model with multiple working vacations. International Journal of Mathematics in Operational Research 9 (3): 375–394. 7. Madan, K.C. 2018. On optional deterministic server vacations in a single server queue providing two types of first essential service followed by two types of additional optional service. Applied Mathematical Sciences 12 (4): 147–159. 8. Ayyappan, G., and T. Deepa. 2019. Analysis of batch arrival bulk service queue with additional optional service multiple vacation and setup time. International Journal of Mathematics in Operational Research 15 (1): 1–25. 9. Haridass, M., and R. Arumuganathan. 2012. Analysis of a MX/G (a, b)/1 queueing system with vacation interruption. RAIRO-Operations Research 46 (4): 305–334. 10. Agrawal, P.K., M. Jain, and A. Singh. 2017. Optimal N-policy for finite queue with server breakdown and state-dependent rate. International Journal Computer Engineering Research 7: 61–68. 11. Madan, K.C., W. Abu-Dayyeh, and M. Gharaibeh. 2003. Steady state analysis of two queue models with random breakdowns. Information and Management Science 14 (3): 37–51. 12. Li, W., D. Shi, and X. Chao. 1997. Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations. Journal of Applied probability 34 (2): 546–555. 13. Nishimura, S., and Y. Jiang. 1995. An M/G/l vacation model with two service modes. Probability in the Engineering and Informational Sciences 9 (3): 355–374. 14. Nobel, R.D., and H.C. Tijms. 1999. Optimal control for an MX/G/1 queue with two service modes. European Journal of Operational Research 113 (3): 610–619. 15. Jeyakumar, S., and B. Senthilnathan. 2012. A study on the behaviour of the server breakdown without interruption in a MX/G (a, b)/1 queueing system with multiple vacations and closedown time. Applied Mathematics and Computation 219 (5): 2618–2633. 16. Wu, W., Y. Tang, and M. Yu. 2015. Analysis of an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility. Opsearch 52 (4): 670–691. 17. Choudhury, G., and M. Deka. 2016. A batch arrival unreliable server queue with two phases of service and Bernoulli vacation schedule under randomised vacation policy. International Journal of Services and Operations Management 24 (1): 33–72. 18. Cox, D.R. 1955, July. The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Mathematical Proceedings of the Cambridge Philosophical Society 51(3): 433–441. Cambridge University Press.

Estimation of Public Compliance with COVID-19 Prevention Standard Operating Procedures Through a Mathematical Model Norazaliza Mohd Jamil and Balvinder Singh Gill

Abstract Despite the enforcement of control plan and preventive measures, the transmission of COVID-19 is still ongoing and yet to be contained successfully. Hence, this study aimed to determine the level of compliance of the public with the standard operating procedures for COVID-19 prevention in Malaysia. A compartmental model with new formulations of timely dependent epidemiological parameter for COVID-19 outbreaks was developed. The model, consisting of ordinary differential equations, was solved by the 4th order Runge–Kutta method. The model representation is in the form of graphical user interface (GUI) built in MATLAB. The estimation of the level of compliance of the population with the control measures was done by fitting the model curve to the actual data in the GUI. The result shows that the current compliance level of the public to the control measures is at an unsatisfactory level that leads to repeated lockdown. The compliance level estimation is important to policymakers and health officials as they can infer the effectiveness of intervention strategies. Additionally, this study revealed how individual responsibility to adherence the control measures will affects the number of cases. Further action to increase public compliance to a satisfactory level is required to halt the pandemic successfully. Keywords Compliance · Modelling and simulation · COVID-19 · Malaysia · Compartmental model

N. M. Jamil (B) Centre for Mathematical Sciences, College of Computing and Applied Sciences, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Kuantan, Pahang, Malaysia e-mail: [email protected] B. S. Gill Institute for Medical Research (IMR), Ministry of Health Malaysia, 50588 Kuala Lumpur, Malaysia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_5

49

50

N. M. Jamil and B. S. Gill

1 Introduction COVID-19 first discovered in Wuhan, China in December 2019 was subsequently affirmed a pandemic by the World Health Organization (WHO) on 11 March 2020 [1]. More than 124 million cases have been reported with over 2.7 million deaths worldwide [2]. In Malaysia, the first wave of COVID-19 resulted from imported cases on 25 January 2020 with a total of 22 cases [3]. A larger second wave started, following a 4-day mass gathering event at Sri Petaling, Kuala Lumpur on 27 February 2021 [4]. This resulted in the largest COVID-19 cluster during the second wave with 3375 confirmed cases with 34 deaths reported [5, 6]. Currently, Malaysia is facing the third COVID-19 wave which started after the Sabah state electoral process on 26 September 2020 [7]. Understanding the evolving epidemiology of COVID-19 during this pandemic may better inform prevention and control policies, hence it is important to systematically characterise the epidemiology of COVID-19 [8]. Epidemiological indicators, estimated via a surveillance system, are critical in developing timely interventions. However, constraints in the surveillance process, especially in the developing world, have meant these epidemiological indicators are inadequately or not estimated [9]. Whilst indicators such as incidence rate, growth rate and doubling time may be useful for outbreak monitoring, it is unable to estimate compliance level of the public towards the control measures. Estimating compliance level during an epidemic is important as it provides an aggregated measure of the effectiveness of these control measures. Therefore, compliance levels are effective and useful indicators for control purposes. Malaysia implemented the first national lockdown, namely Movement Control Order (MCO) on 18 March 2020. The second nationwide lockdown known as MCO 2.0 was imposed on 13 January 2021. The increasing number of cases have forced the government to enforce MCO 3.0 on 12 May 2021. The Ministry of Health (MoH) introduced the standard operating procedures (SOPs) based on avoiding crowded places, confined spaces and close conservation to slow down the propagation of the infection. The public must exercise physical distancing and practise the 3Ws (wash hands, wear a facemask and warn self and others). In spite of the rules and efforts exerted by the government, the spread of the disease was perceived to be growing occasionally, which required the implementation of repeated lockdowns. One reason for this can be attributed to the inadequate compliance of the public towards the SOPs. As cases of COVID-19 increase in Malaysia, there was a need to develop a procedure to determine the level of compliance. Hence, this paper aims to develop an automated interactive visual component to generate the outbreak progression of COVID-19 based on compliance level. In this paper, we aim to estimate the SOPs compliance level in Malaysia using a compartmental SIRD model and to develop a GUI to automate the model outputs. Findings from this study will inform policymakers and health officials and assist them infer the effectiveness of intervention

Estimation of Public Compliance with COVID-19 Prevention …

51

strategies. In addition, this study will increase public awareness about their individual responsibility and role in fighting against the pandemic.

2 Mathematical Model We established a new mathematical formulation to describe the spread of the disease in a population using the SIRD compartmental model originated from the work of Kermack and McKendrick in 1927 [10]. This compartmental model comprises of four groups which are S (susceptible), I (infectious), R (recovered) and D (death). Essentially, S(t) is the total number of individuals at risk of being infected by the disease measured from day 1 to day t. I (t) is the total number of individuals who have been infected and can spread the disease to group S from day 1 to day t. The total number of individuals who have been infected with the disease and have recovered from the illness from day 1 to day t is labelled as R(t). The final category is D(t), which denotes the total number of people who have died from the disease and counted from day 1 to day t. The model was developed based on the following assumptions. (a) (b)

The total population is constant whilst the natural birth and death rates are ignored The disease is spread by infected individuals coming into close contact with susceptible individuals (the sporadic case is not considered).

A set of ordinary differential equations formulates the mathematical model as follows, I dS = −β(t) S dt N

(1)

dI I = β(t) S − γ (t)I − μ(t)I dt N

(2)

dR = γ (t)I dt

(3)

dD = μ(t)I dt

(4)

where β(t) is the function of infection rate of the disease, γ (t) is the function of recovery rate and μ(t) is the function of death rate. N is the constant total population which can be formulated as N = S(t) + I (t) + R(t) + D(t)

(5)

Constant values of infection, recovery and death rates would only produce the result of the model for one wave of the outbreak. Hence, to model multiple waves of

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outbreaks in a single formulation and initial value, we used the piecewise functions in formulating the functions of epidemiological parameters. The piecewise functions divide the time intervals based on the intervention measures implemented. Different functions will be adopted to the time prior to lockdown, during the lockdown and the reopening period after the lockdown. Assuming tlock and tlock 2 be the day when a country imposed the first and second lockdown, respectively. Whilst tlift and tlift2 indicate the day of easing of lockdown rules after the first and second lockdown, respectively. At the time of the writing, Malaysia is currently in MCO 3.0, and the future case trends are unknown, hence our model only considered the time for MCO 1.0 and MCO 2.0. In this work, new formulations for time-dependent epidemiological function parameters, β(t), γ (t) and μ(t), were constructed as follows. MCO 1.0 γ (t) = γ1 , μ(t) = μ1

(6)

⎧ β,  t < tlock ⎪ ⎪  ⎨ 1 (t−tlock ) , tlock ≤ t < tlift β exp − 1 β(t) = τβ1   ⎪ ⎪ ⎩ (1 − r)β1 + rβ1 exp − (t−tlock ) , t ≥ tlift τβ1

(7)

γ (t) = γ1 , μ(t) = μ1

(8)

MCO 2.0

⎧ β1 ,  t < tlock ⎪ ⎪  ⎪ ⎪ (t−tlock ) ⎪ ⎪ tlock ≤ t < tlift β1 exp − τβ1 , ⎪ ⎪   ⎨ (tlift −tlock ) , tlift ≤ t < tlock2 2 (t − tlift ) + β1 exp − β(t) = β     τβ1 ⎪ ⎪ lock2 ) ⎪ ⎪ β2 (tlock2 − tlift ) + β1 exp − (tliftτ−tβ1lock ) exp − (t−tτβ2 , tlock2 ≤ t < tlift2 ⎪ ⎪   ⎪ ⎪ (t−tlock ) ⎩ t ≥ tlift2 (1 − r)β1 + rβ1 exp − τβ1 , (9) The recovery and death rates depend on the disease and the individual and do not vary as a function of t and the lockdown. Specifically, 1/γ is the infectious period. Hence, the recovery rate, γ (t) and death rate, μ(t) were assumed to be a constant value as in Eqs. (6) and (8). The infection rate depends on the rate of contact between infected and susceptible individuals. Lockdown is a promising way to reduce the contact rate between individuals. Hence, the infection rate is formulated as a time-dependent function. Factors for intervention measures, such as lockdown, social distancing, quarantine, healthcare system, the percentage of people who follow the SOPs, hospitalisation of the infected individuals and treatment were considered in formulating the

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infection rate. Equations (7) and (9) formulated the infection rate for MCO 1.0 and MCO 2.0, respectively. At the start of the outbreak (t < tlock ), the population was highly mobile, and the infection rate was assumed to be constant, β1 . When the first lockdown was introduced (tlock ≤ t < tlift ), the infection rate decayed due to restricted movement of the population and social distancing practices. This behaviour was described by an exponential function in Eq. (7) with τβ1 as the characteristic time of transmission during lockdown 1. To measure the SOP compliance level after MCO 1.0 (t ≥ tlift ), a new parameter r which indicates the fraction of compliance of the public to the SOPs was introduced. In other words, r is the percentage of the public who follow the SOPs, even after the lockdown was lifted. If 0% of public compliance level (r = 0), the infection rate followed the trend at the beginning of the outbreak when there was no lockdown implemented. If 100% of public compliance level (r = 1), the infection rate followed the trend when the lockdown was implemented. The mathematical model for MCO 2.0 in Eq. (9) followed the same manner as in the model for MCO 1.0 by extending the formulation for tlift ≤ t < tlock2 . During the second lockdown (tlock2 ≤ t < tlift2 ), the infection rate was decreased, and it was described by an exponential function with τβ2 as the characteristic time of transmission during lockdown 2. The behaviour of infection rate after the lockdown was lifted is depended on the percentage of people who followed the SOPs. We used MCO 1.0 as a reference to measure the compliance level after MCO 2.0 (t ≥ tlift2 ), hence the infection rate was similar to formulation for t ≥ tlift . A 4th order Runge–Kutta method coded in MATLAB was used to numerically solve the mathematical model. A parameter fitting technique called Nelder-Mead algorithm was employed to find the value of all unknown parameters β1 , β2 , γ1 , μ1 , τβ1 and τβ2 . The Nelder-Mead algorithm performed an adaptive process that searched for points to satisfy function minimization that provides the best-fit trajectory of the model with the actual epidemic data. A graphical user interface (GUI) simulation tool for COVID-19 outbreaks was developed in MATLAB. At the beginning of the programming process, the user is required to fill two input boxes as shown in Figs. 1and 2 by entering the lockdown easing date and the percentage of people who follow the SOPs. The two parameters are activated with a push-button labelled ‘run the simulator’ resulting in two projection graphs appearing. The first graph will show the value of the effective reproduction number, and the second graph visualizes the number of active cases of COVID-19. The black square symbol plots the actual observed-case numbers.

3 Results and Discussion The dynamics of the COVID-19 outbreak in Malaysia were investigated by applying this new developed model and the GUI interface. The total population of Malaysia was assumed to be constant (N = 32 million). The initial conditions I (0) = 3,

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Fig. 1 Compliance level estimation after MCO 1.0 with 40% compliance (fit the model curve to the data)

Fig. 2 Compliance level estimation after MCO 2.0 with 55% compliance (fit the model curve to the data)

Estimation of Public Compliance with COVID-19 Prevention … Table 1 Parameter values

Parameter

55

Value

β1

0.146556534767631

β2

0.000854504525536

γ1

0.051036351893306

μ1

0.000668710997547

τβ1

38.321191281261029

τβ2

16.621055127092735

S(0) = N − I (0) − R(0) − D(0) and R(0) = D(0) = 0 were obtained from the actual data reflecting the first case of COVID-19 detected in Malaysia which was on 25th January 2020 (day 1). Denoting tlock as the day when Malaysia imposed the first lockdown termed as the Movement Control Order (MCO) on 18 March 2020 (day 54). tlift is described as the lifting time that refers to the date 22 July 2020 when schools in Malaysia was reopened (day 180). The second lockdown known as MCO 2.0 was then implemented on 13 January 2021 (day 355). Daily COVID-19 case data for Malaysia are from the Crisis Preparedness and Response Centre (CPRC) of the Malaysian Ministry of Health and from press releases on the official Ministry of Health (MOH) Website at http://www.moh.gov.my/. The Nelder-Mead algorithm estimated the six unknown parameters by fitting the model to actual COVID-19 case data during the period between 25th January 2020 and 31st March 2021. Table 1 lists the resulting parameter values. Health authorities are facing enormous challenges in controlling the spread of COVID-19. The introduction of non-pharmaceutical control measures such as travel restrictions, physical distancing, face masks, hand hygiene, quarantine and isolation, aims to decrease the effective reproduction number, thus slowing the spread of COVID-19 and sustains the healthcare delivery systems. However, individuals are required to adopt new behaviours and required to be compliant to the prescribed SOPs stringently for prolonged durations, which are affecting the social and economic wellbeing of the population. This balance between lives and livelihood is crucial to the success in combating the COVID-19 pandemic. Public compliance with the control measures proposed by the authorities is a crucial factor in controlling the disease. This situation requires in the ability to measure the levels of SOP compliance of the public. Past studies have attempted measuring the compliance levels of the public to the SOPs with mixed results. Kayrite et al. [11] measured the compliance level of the people in Ethiopia by investigating food and drink establishments via face-to-face interviews. Plohl and Musil [12] used 525 online surveys and developed a multivariate model to identify different responses to COVID-19 prevention compliance. In this study, we used a novel method to estimate the time-varying epidemiological parameters to the compartmental SIRD model, wherein we explored the effect of SOP compliance levels on the dynamics of the disease outbreak in Malaysia. The model proposed in our study is represented in the form of a GUI, and in addition, the degree of compliance can be estimated by fitting the model curve to the data.

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Malaysia launched its first national lockdown, called the Movement Control Order (MCO) on 18 March 2020. Interstate and interdistrict travel were not allowed. In Fig. 1, the result shows that a 40% compliance fits well to the data. This determines that only 40% of the public adheres to the SOPs. As described by this model, the strict MCO 1.0 had successfully flattened the curve, however, poor compliance to the SOPs by the public after the easing of the movement restrictions led to an increased number of positive COVID-19 cases. The mass gathering during the Sabah state elections on 26 September 2020 can be attributed as a leading cause in the sharp rise of COVID-19 cases in Malaysia [13]. To address the worsening surge of COVID-19 cases, Malaysia imposed its second nationwide lockdown, called Movement Control Order 2.0 on 13 January 2021. The implementation of MCO 2.0 had successfully decreased the number of COVID19 cases at an early stage. After the MCO 2.0 was lifted on 4 March 2021, the observed data showed a subsequent increase in caseloads in early April 2021, indicating a decrease in compliance to the SOPs. As shown in Fig. 2, we fitted the model curve to the actual data in the GUI and concluded that only 55% SOP compliance was achieved. In this simulation, we set the lockdown easing date on 08/03/2021, marking the reopening of primary schools in Malaysia. The model measured that only 55% of the public followed the SOPs after the relaxation of MCO 2.0 which resulted in the observed rise in daily cases. With the introduction of new variants with higher infection rates and the lack of SOP compliance, the government was forced to introduce MCO 3.0 on 12 May 2021, to curb the rising number of COVID-19 cases. These findings disclosed that the overall compliance level of the public to the SOPs was inadequate, which lead to the introduction of repeated lockdowns. Authorities took action over non-compliance with the SOPs such as failure of individuals to comply with physical distancing measures, over-crowding, not registering movement details, not complying with temperature screening and travel restrictions. Non-compliance to the SOPs may occur due to various factors such as lack of self-discipline and awareness of the public. Willingness to comply depends on the individuals’ capacity to obey the rules, education, moral support and social norms [14]. Individuals who live in poverty and have economic constraints may not be unable or unwilling to adhere to the SOPs due to lack of income [15]. In addition, workers who live in crowded dormitories may have difficulty complying with the prevention guidelines. Public compliance towards these SOPs are crucial to curb the COVID-19 pandemic.

4 Conclusion This paper developed a novel approach to estimate the public compliance in Malaysia with COVID-19 prevention standard operating procedures by using a mathematical model. In addition, a novel formulation for the time-dependent epidemiological parameters for the SIRD model was also proposed, and a graphical user interface

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(GUI) was created. The GUI provides a user-friendly interface to assist in the simulation of the effect of SOPs compliance on the spread of the outbreak. The overall results show that the compliance level to the COVID-19 prevention SOPs in Malaysia is inadequate, resulting in the implementation of repeated lockdowns. This study revealed the importance of individual responsibility to the adherence to SOPs as a crucial factor to win the battle against the pandemic in Malaysia. Finally, in regards to the modified SIRD model outcome, this study showed that the proposed model is able to measure the SOP compliance levels during this pandemic. Since, there is limited relevant research on measuring the SOP compliance level in Malaysia; hence, this study provides significant findings and knowledge on the dynamics of the COVID-19 pandemic in Malaysia. Acknowledgements We would like to thank the Director General of Health Malaysia for his permission to publish this article. Funding This research was funded by a grant from Universiti Malaysia Pahang, Internal Grant RDU 210329. (Ref: UMP.05/26.10/03/RDU210329).

References 1. Cucinotta, D., and M. Vanelli. 2020. WHO declares covid-19 a pandemic. Acta Bio Medica: Atenei Parmensis 91 (1): 157–160. 2. WHO Coronavirus Disease (Covid-19) Dashboard (https://covid19.who.int/). 3. Khor, V., A. Arunasalam, S. Azli, M.G. Khairul-asri, and O. Fahmy. 2020. Experience from Malaysia during the covid-19 movement control order. Urology 141: 179–180. 4. New Straits Times (https://www.nst.com.my/news/nation/2020/04/583127/sri-petaling-tab ligh-gathering-remains-msias-largest-Covid-19-cluster). 5. Chong, Y.M., I.C. Sam, J. Chong, M. Kahar Bador, S. Ponnampalavanar, S.F. Syed Omar, A. Kamarulzaman, V. Munusamy, C.K. Wong, F.H. Jamaluddin, Y.F. Chan. 2020. SARS-CoV2 lineage B.6 was the major contributor to early pandemic transmission in Malaysia. PLoS Neglected Tropical Diseases 14(11): e0008744 6. Mat, N.F.C., H.A. Edinur, M.K.A.A. Razab, and S. Safuan. 2020. A single mass gathering resulted in massive transmission of covid-19 infections in Malaysia with further international spread. Journal of Travel Medicine 27 (3): 1–4. 7. MOH. Situation of covid-19 pandemic in Malaysia (http://Covid-19.moh.gov.my/). 8. Lipsitch, M., D.L. Swerdlow, and L. Finelli. 2020. Defining the epidemiology of covid-19— studies needed. New England Journal of Medicine 382 (13): 1194–1196. 9. Ibrahim, N.K. 2020. Epidemiologic surveillance for controlling covid-19 pandemic: Types, challenges and implications. Journal of Infection and Public Health 13 (11): 1630–1638. 10. Kermack, W.O., and A.G. McKendrick. 1927. A contribution to the mathematical theory of epidemics. Proceedings of the royal society of London. Series A, Containing Papers of a Mathematical and Physical Character 115 (772): 700–721. 11. Kayrite, Q.Q., A.A. Hailu, T.N. Tola, T.D. Adula, and S.H. Lambyo. 2020. Compliance with covid-19 preventive and control measures among food and drink establishments in BenchSheko and West-Omo Zones, Ethiopia, 2020. International Journal of General Medicine 13: 1147–1155. 12. Plohl, N., and B. Musil. 2021. Modeling compliance with covid-19 prevention guidelines: The critical role of trust in science. Psychology, Health and Medicine 26 (1): 1–12.

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13. Lim, J.T., K. Maung, S.T. Tan, S.E. Ong, J.M. Lim, J.R. Koo, and B.S.L. Dickens. 2021. Estimating direct and spill-over impacts of political elections on Covid-19 transmission using synthetic control methods. PLOS Computational Biology 17 (5): 1–15. 14. Van Rooij, B., A.L. de Bruijn, C. Reinders Folmer, E. Kooistra, M.E. Kuiper, M. Brownlee, and A. Fine. 2020. Compliance with covid-19 mitigation measures in the United States. Amsterdam Law School Research Paper 2020–21: 1–40. 15. Wright, A.L., K. Sonin, J. Driscoll, and J. Wilson. 2020. Poverty and economic dislocation reduce compliance with covid-19 shelter-in-place protocols. Journal of Economic Behavior and Organization 180: 544–554.

Campus Recruitment Cost Analysis: A Roadmap for HR Managers Md Jakir Hossain Molla, Sk Md Obaidullah, Parveen Ahmed Alam, Saurabh Adhikari, Sourav Saha, and Soumya Sen

Abstract Recruitment of proper human resource is a challenge for many organizations. In order to properly conduct the resources, HR managers need to consider many parameters. The recruitment process get complex when mass recruitment is conducted. Generally, this happens for the fresher candidates, and for that, the recruitment team needs to travel throughout the country and it is a costly affair. HR managers want to reduce this cost and try to recruit more number of quality resources as per the requirement. No standardized method is there for the HR managers to optimize the cost. In this research work, different cost parameters (both tangible and intangible) associated with the recruitment are identified, and a cost matrix is proposed to guide the HR managers. A case study is also presented based on the proposed methodology. Keywords Cost analysis · Recruitment · Campus drive · Student placement · HR managers

1 Introduction Recruitment is an inevitable for organizations across the world. The recruitment of resources can be categorized in two types. First type is the recruitment of the fresher (joining the job for first time), and another type is the recruitment of experienced people. The recruitment process of experienced people is not generally in large M. J. H. Molla · S. M. Obaidullah · P. A. Alam Aliah University, Kolkata, India S. Adhikari · S. Saha Swami Vivekananda University, Kolkata, India e-mail: [email protected] S. Saha e-mail: [email protected] S. Sen (B) University of Calcutta, Kolkata, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_6

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numbers, and the recruitment for these people generally takes place in the office of the concerned organization in one-to-one mode or may be even conducted in online mode such as video conferencing, phone call, etc. Whereas, the recruitment of fresher candidates is complex when mass recruitment takes place. The large IT companies in India often recruit around 100,000 fresh candidates per year, and for that, they need to plan visit of the different campuses of the educational Institutes across India. This recruitment process pan-India is a costly process. It is not only an issue for India rather for all the countries where the size is relatively large and the institutes span across the country. Henceforth, the decision of how to carry out the recruitment process is a challenging issue for the HR managers who plan and conduct the recruitment. There are many issues which drive the decision. There are certain tangible factors which can be computed in terms of money such as transportation cost, lodging cost, and daily allowances. These factors are easy to compute. However, there are other intangible factors such as quality of the students, rating of the institutes, number of students actually join after the selection, etc. These intangible factors are difficult to compute. Even the recruitment process gets complex when a company is planning to hire the people with different skill levels and henceforth offer variable salary. Sometimes, they have the plan to recruit people with different skill/expertise level for the different types of jobs in their organization. In the case of fresher recruitment as the HR managers have to travel across the country to hire, they need to plan the recruitment process. This planning is often multistage. At first, they choose a region to recruit (A country is demographically divided into many regions). Now in the region, they can plan to visit an institute to recruit the students of that institute only (this is called on-campus recruitment) or they may organize pool-campus recruitment where they visit a particular institute but recruit the students from many institutes. A company organizes one or multiple on-campus and pool-campus recruitments as per their needs. It is the decision of the HR managers how to plan the recruitment process based on the number of institutes in the region and number of candidates to appear. This planning also involves sensitive decision like for which institute they want to organize on-campus recruitment (generally good institutes are selected for on-campus) and for pool-campus what are the different institutes to call. As there are many factors (both tangible and intangible), it is a complex process to determine an optimized plan for recruitment. No such work exists to provide a computational tool for the HR managers to derive the cost factors including both tangible and intangible parameters. In this research work, the authors identify both tangible and intangible parameters associated with recruitment and derive a cost matrix to compare the cost of every region. In the next section, the process of campus recruitment is explained along with the basic cost calculation. Section 3 presents a survey work on the placement and recruitment procedures. In Sect. 4, objective and contributions of this research work are presented. In Sect. 5, a methodology is proposed to standardize the cost matrix and evaluate and compare the cost region wise. A case study is presented in Sect. 6 based on the proposed methodology. Finally in Sect. 7, conclusion is drawn and the future scope of this research work is outlined.

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2 Campus Recruitment Process and Cost Computation In this section, we present the different steps that are being conducted for performing campus recruitment. The steps include: • Identifying positions to be filled or hiring needs—Identify which positions will be filled by entry-level hires and how many open positions are available. • Determining the budget—Be aware of how much of the recruitment budget can be allocated for the campus drive. Knowing the budget that will help to determine how far recruiters can go for conduct the drive and whether or not they have funds to assist in relocating new hires from out of state. • Considering diversity requirements—Review the strategy of recruiting diversity. Ensure that the company meets the internal diversity requirements as part of the campus drive strategy. • Identifying academic programs—It is important to learn which academic programs each university or college specializes in and select the programs that best fit with the skill sets recruiters are looking for to find in employable candidates. • Location—Questions like whether the location of the institutes, i.e., the distance to campus justifies the time and money involved to recruit from there or whether the distance will create relocation and retention issues, needs to be addressed. • Graduation dates—When will the candidates be available for joining? • Competitive environment—Can the organization meet the student’s expectation? • Internal opinion of the Institute—What is the overall opinion of the university within the company? Would the university or college be accepted as part of the campus drive program? • Number of colleges at that zone—Before planning for a campus drive, if we get an idea about the number of colleges we are going to target to fulfill our criterion, then it will be very helpful for the recruiters to plan a drive at a minimum cost possible, and for that, the first thing need to do is to select a perfect area for our company to go and conduct a recruitment program. By this, we can target maximum number of universities or colleges at a time with the minimum cost possible, i.e., cost optimization is an important criteria. And with this planning, we can easily get the students who will fit perfectly for the organization. The above factors are often considered by different HR managers, and they compute the cost issues on their own. This may vary from company to company as it is perceived by their HR managers. Even in the same company, this planning may be different based on the different regions. Cost Computation  (External Campus Costs)  + (Internal Campus Costs)

Overall Cost =

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External Costs: External costs variable comprises all the sources of spending outside of the organization on recruiting efforts during the time period in question. Internal Costs: Internal costs variable comprises all the sources of the internal resources and the costs used for staffing efforts during the time period in question.

3 Related Work An enterprise resource planning (ERP) tool [1] was proposed to manage the campus placement in colleges. Here, the main focus is on managing student’s data, digitizing the student practices, streamline the hiring process. Knowledge-intensive business services (KIBS) [2] play an important role in many business processes, and it includes outsourcing as an alternative to the standard business process. Outsourcing the recruitment process results in better in terms of quality, cost, and time. In [2], a case study is conducted on the outsourcing of recruitment process for effective HR management. For the last two decades information technology (IT) and information technology enables services (ITES) emerged as the most important job generation verticals. Many of the recruitment policies are aligned toward these two verticals. IT is used effectively across all business domains. It is even used to manage employment screening [3]. This includes pre-employment screening, online background screening, generating screening report, application tracking, generating offer letters, etc. A framework [3] was developed to digitize the recruitment process under IT framework. Software engineering [4] is considered as one of the basic parameters for recruiting in many of the organizations. This is the area where the skill enhancement is required to meet up the problems of real-life business cases. It is also identified that the students lack industry skill after the end of their course. In order to bridge this gap, many companies plan internships [5] for the graduates to enrich their technical skills, nurturing the domain knowledge, etc. The effectiveness of internships is discussed in [5]. As the probability factors are associated with the recruitment, analytical models [6] are used to predict the campusing. In [7], k-nearest neighbor classifier is used to predict the placement system. In modern days, social media plays an important role in recruitment process. The challenges of recruitment using social media are laid down in [8]. In [9], a recommendation model for HR managers to select the candidates using social media is discussed. The parameters such as legality, reliability, standardization, and job relevance are considered in [9] for the recommendation. Big data analysis is also applied for recruitment analysis. Social media data are analyzed for big data applications [10], and methods like correlation matrix and regression analysis are used for better management of recruitment process. In another research work, [11] academic skills and soft skills of the candidates are considered and compute the employability and hiring trends of the candidates of UAE. This will help the HR mangers to choose the institutes for recruitment. However, no work is found in the area of campus drive and placement to guide the HR managers to optimize the cost associated with the recruitment process.

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4 Objectives and Contributions Campus drive throughout the country is a complex and costly process as it involves the movement of many human resources of own organization at different parts of country for recruiting suitable human resources. The different parameters which drive the cost are not about the money rather evaluating human resources as well as the institutes where from the recruitment will take place. No standard methodology is available for the HR managers to readily compute the cost including many tangible and intangible parameters. This research work is focused to generate a cost matrix that will guide the HR managers to evaluate and compare the cost factors. Contributions: 1. 2. 3.

Identify the parameters that determine the cost associated with the recruitment process. Generating a cost matrix that will guide the HR managers to compare and select regions for recruitment. A standardized method to guide the recruitment process.

5 Proposed Methodology There are many factors that drive the recruitment process. Here, we select the important parameters that will be used to generate the cost matrix. (i) Overall cost: Total cost associated with the recruitment process. Here, the cost for a region is considered as a recruitment team generally moves to a region to conduct many on-campus and pool-campus selection on a single trip. (ii) Average cost: Overall cost may not be always correct parameter to select a region. Because, it may be possible that for a region, cost is less but the number of candidates to appear in the selection process is also less. Hence, the company may not be able to recruit many resources as per their requirements. Hence, average cost per candidate will be a better estimate for the cost matrix. (iii) Rating of the institutes: Any organization wants to recruit good candidates for their organization. Therefore, they want to visit an institute with higher rating so that they can recruit better candidates. Hence, the rating of the institute is important for all the recruiters. (iv) Past recruitment experience: Numbers of candidates recruited from an institute/region are also an important parameter to select a campus or region. However in many cases, all the selected students do not join the organization. This count is also to be considered. Another issue that may be considered is the salary offered to the students. Nowadays, companies offer different packages of salary based on the quality of the institutes as well as for same institute they offer different salary to the students based on their performance.

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Based on the above issues, the cost matrix is generated. The parameters for this cost matrix are explained, and the computation is also shown: (i) Overall Cost (OVC):  (External Campus Costs) +  (Internal Campus Costs) External Campus Costs: It includes the parameters such as travel cost, staying cost, and daily allowances for the recruiters. Say the distance of the area of placement from the company is ‘x’ km and the cost to travel per km is ‘y.’ Travel_cost (T ) = 2 * x * y (considering returning back to the home location). Say ‘z’ numbers of recruiters are going to conduct the campus drive for ‘d’ number of days and their per day hotel cost is ‘s.’ Staying_cost (S) = z * d * s. Daily allowances for the recruiter is ‘a.’ Hence, for ‘z’ numbers of recruiters and for ‘d’ number of days, total allowance is calculated as. Allowance_Cost (A) = z * d * a. External Cost = T + S + A. Internal Campus Costs: Internal Campus Cost per student is say ‘b.’ Say, N is the numbers of students to appear in the campus drive from a region. Internal Cost = N * b. Therefore, Overall Cost(OVC) = T + S + A + N ∗ b

(1)

Organizations with a low budget and planning to recruit few candidates will chose zone with the lowest budget. However, companies want to recruit more people and want to analyze more parameters will consider the following points also. (ii) Average Cost Per Assessment of Students (AVC): The companies aiming to recruit more numbers of candidates will plan to reduce the average cost per candidate. It is computed as AVC = OVC/N

(2)

(iii) Rating Region Wise (RRW): Different institutes have different ratings in the region. Most of the companies want to hire more number of candidates from higher ranked institutes. The following computation helps to identify the rating per student. Companies will prefer higher value of this parameter. Say ‘n’ is the numbers of institutes in a region

Campus Recruitment Cost Analysis: A Roadmap for HR Managers

RRW =

N  Ratingi ∗ number of students of ith institute Number of Students in the region(N ) i=1

65

(3)

(iv) Past Recruitment Weightage: If a company already conducted campus drive in a region, that experience will also help them in the planning of recruitment for current year. For this, two parameters are identified. (a)

Number of candidates recruited (CR): If higher numbers of candidates are recruited, then that institute/region is preferred. Number of candidates not joined (CNJ): Some of the candidates do not join the organization after the selection. This is not desirable for an organization.

(b)

Based on the above parameters recruitment factor (RF) is calculated. Companies will prefer higher values of RRW and CR and lower value of CNJ and AVC. RF =

RRW ∗ CR (RRW ∗ CNJ) − AVC AV

(4)

If RF is high for a region, HR will prefer that region. HR can rank the regions based on the descending values of RF. (v) Variable Salary: If a company has offered different salary to the students then the following clauses is considered and Eq. (4) is modified. Two factors are considered. (a) Average salary offered (AVS) (b) Average salary of students not joined (AVSN) Based on the above two parameters, Eq. (4) is modified. RF =

(RRW ∗ CR ∗ AVS) (RRW ∗ CR ∗ AVSN) − AVC AVC

(5)

6 Case Study Case 1: A company wants to conduct a campus drive in a place which is 1000 km away from them, and for that, a team of ‘4’ no. of recruiters go to the city for ‘2’ days, where they have to pay a travel cost of ‘3’ Rs per km. The hotel cost for each person for each day is Rs. 5000. The company gives Rs. 2000 as allowance for each recruiter for each day. In the campus drive, there are ‘20’ institutes among which some institutes are of rating ‘i’ >= 5, and total ‘700’ students appeared, and they also have to bear a cost of Rs 300 for each candidate. Find the RF for this drive.

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Let suppose x = 1000 km, y = 3 Rs, z = 4, d = 2, s = 5000 per day, a = 2000 Rs, b = 300, N = 700, n = 20, CR = 200 and CNJ = 20. i.e. T = 4 ∗ (2 ∗ 1000 ∗ 3) = 24000 Rs., S = 4 ∗ 2 ∗ 5000 = 40000 Rs A = 4 ∗ 2 ∗ 2000 = 16000 Rs External Cost = 24000 + 40000 + 16000 = 80000 Rs Internal Cost = 700 ∗ 300 = 210000 Rs Therefore from Eq. (1) Overall Cost (OVC) = 80000 + 210, 000 = 290000 Rs. From Eq. (2) AVC = 290000/700 = 414.3 From Eq. (3) RRW = (4500 + 2000)/700 = 9.28 [Where, we assume number of 5 rating colleges 15 and 6 rating colleges 5 with 300 and 400 students accordingly]. Therefore, from Eq. (4) RF = [(9.28 ∗ 200)/414.3]−[(9.28 ∗ 20)/414.3] = 4.48−0.45 = 4.03

(6)

Case 2: For a campus drive, a team of ‘4’ recruiters from a company go to a region that is ‘1500’ km away for ‘2’ days, where they have to pay a travel cost of Rs 4 per km. The hotel cost for each person for each day is Rs 5000. The company gives Rs. 2000 as allowance for each recruiter for each day. In the campus drive, there are ‘25’ institutes among which some institutes are of rating i >= 5, total 900 students appeared, and they also have to bear a cost of ‘300’ Rs for each candidate. Find the RF for this drive. Let suppose x = 1500 km, y = 4 Rs, z = 4, d = 2, s = 5000 per day, a = 2000 Rs, b = 300, N = 900, n = 25, CR = 250 and CNJ = 20. i.e. T = 4 ∗ (2 ∗ 1500 ∗ 4) = 48000 S = 4 ∗ 2 ∗ 5000 = 40000 Rs A = 4 ∗ 2 ∗ 2000 = 16000 Rs External Cost = 48000 + 40000 + 16000 = 104000 Rs Internal Cost = 900 ∗ 300 = 270000 Rs Therefore from Eq. (1)

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Overall Cost (OVC) = 104000 + 270000 = 374000 Rs. From Eq. (2) AVC = 374000/900 = 415.5 From Eq. (3) RRW = (8000 + 2500)/900 = 11.67 [Where, we assume number of 5 rating colleges 20 and 6 rating colleges 5 with 400 and 500 students accordingly]. Therefore, from Eq. (4) RF = [(11.67 ∗ 250)/415.5]−[(11.67 ∗ 20)/415.5] = 7.02 − 0.56 = 6.46

(7)

From Eqs. (6) and (7), we can say RF is high for the region in the second case study in spite of long distance, so HR will prefer the region as described in case study-2. HR can rank the regions based on the descending values of RF.

7 Conclusion This research work proposes a novel methodology for recruitment process by proposing a cost matrix. The necessary parameters are identified for the formation of the cost matrix. The proposed method will standardize the recruitment costing and therefore uniformity will be followed. HR managers can use this model for recruitment of different types of fresher candidates having different backgrounds from a particular region. It is also possible to consider the recruitment of multiple candidates with different salary ranges. The research work could be extended to include more intangible parameters such as time of publication of results, relevancy of the syllabus to the industry requirements, training of students, facility of the pool-campus recruitment, etc. Along with that, analytical methods could be included to analyze the historical data of previous placements records to evaluate the students. It will be an interesting work to identify the students with different skills from different regions. Skill parameter can be tuned to meet the different types of recruitment of a company from a single campus drive in the region.

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References 1. Kendle, A.S., Nagare, M.S., Patre, H.G., Zanwar, R.S., Kottawar, V.G., and Deskhmukh, P.B. 2021, May. TnP vision: Automation and analysis of campus placements in colleges. In 2021 5th International Conference on Computer, Communication and Signal Processing (ICCCSP), 1–6. IEEE. 2. Laumer, S., Blinn, N., and Eckhardt, A., 2012, January. Opening the black box of outsourcing knowledge intensive business processes—A longitudinal case study of outsourcing recruiting activities. In 2012 45th Hawaii International Conference on System Sciences, 3827–3836. IEEE. 3. Muderedzwa, M., and Nyakwende, E. 2010, December. A framework for improving the effectiveness of it in employment screening. In 2010 IEEE Student Conference on Research and Development (SCOReD), 133–138. IEEE. 4. Heggen, S., and Myers, C. 2018, June. Hiring millennial students as software engineers: A study in developing self-confidence and marketable skills. In Proceedings of the 2nd International Workshop on Software Engineering Education for Millennials, 32–39. 5. Sivananda, S., Sathyanarayana, V., and Pati, P.B. 2009, February. Industry-academia collaboration via internships. In 2009 22nd Conference on Software Engineering Education and Training, 255–262. IEEE. 6. Ghosh, P., Sadhu, D., and Sen, S., A Real-Time Business Analysis Framework Using Virtual Data Warehouse. 7. Giri, A., Bhagavath, M.V.V., Pruthvi, B., and Dubey, N. 2016, August. A placement prediction system using k-nearest neighbors classifier. In 2016 Second International Conference on Cognitive Computing and Information Processing (CCIP), 1–4. IEEE. 8. Landers, R.N., and Schmidt, G.B. 2016. Social media in employee selection and recruitment: An overview. Social Media in Employee Selection and Recruitment 3–11 9. Davison, H.K., Bing, M.N., Kluemper, D.H., and Roth, P.L. 2016. Social media as a personnel selection and hiring resource: Reservations and recommendations. In Social Media in Employee Selection and Recruitment, 15–42. Springer, Cham. 10. Ikram, A., Su, Q., Fiaz, M., and Khadim, S. 2017, March. Big data in enterprise management: Transformation of traditional recruitment strategy. In 2017 IEEE 2nd International Conference on Big Data Analysis (ICBDA), 414–419. IEEE. 11. Rajan, S., and Pandita, A. 2019, March. Employability and hiring trends of engineering job aspirants in UAE. In 2019 Advances in Science and Engineering Technology International Conferences (ASET), 1–6. IEEE.

Certain Types of Domination in Nover Top Graphs R. Narmada Devi, G. Muthumari, and Suresh Rasappan

Abstract Neutrosophicover set (Nover) was introduced by smarandache. Due to some real-time situation, decision-makers deal with uncertainty and inconsistency to identify the best result. Connectivity will be very important in neutrosophic graph. In this research study, we introduced the certain types of domination which are so-called perfect NOverTop-dominating set (perf Noverdom set), connected perfect NOverTop-dominating set (CONN perfNoverdom set), total perfect NOverTop-dominating set (Tot perf Noverdom set), connected total perfect NOverTop-dominating sets (CONN Tot perf Noverdom set), and also properties of domination numbers are established with necessary examples. Further, those relationship are discussed. Keywords NOverTop-dom set · Perfect NOverTop-dom set · Connected perfect NOverTop-dom set · Total perfect NOverTop-dom set · Connected total perfect NOverTop-dom set and connected total perfect NOverTop-dom number

1 Introduction Zadeh [1] introduced the concept of fuzzy set in 1965. Rosenfield added a fuzzy graph to the equation. The use of fuzzy graphs for obtaining answers has become increasingly popular in the areas of traffic congestion, decision-making, networking, privacy and security, and so on. Smaradche [2, 3] defined “neutrosophic logic” as a generalization of intuitionstic fuzzy logic. Also [4, 5], he defined neutrosophic over set. Broumi [2, 3] introduced numerous fascinating concepts, such as intervalvalued neutrosophic graphs, single-valued neutrosophic graphs and their applications R. N. Devi (B) · G. Muthumari Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamilnadu 600062, India e-mail: [email protected] S. Rasappan Department of Mathematics, University of Technology and Applied Sciences-Ibri, Ibri 466, 516, Sultanate of Oman © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_7

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[6–8]. The notions of neutrosophic topological area are introduced. Ore [9] was the source of dominance. Many studies afterwards acknowledged domination and kinds of domination in fuzzy graphs, including Somasundaram [10], Nagoor Gani [11], Revathi [1, 12] and others. The notions of neutrosophic complex N-continuity and minimal domination via neutrosophic over graphs were presented by Narmada Devi [13–15]. Nover’s top dominating set and its numerous types are discussed in this article. Along with the examples, there are also interesting characterizations of various types of Nover top graphs.

2 Preliminaries Definition 2.1 A Nover graph is a pair ψ = (A, B) of a crisp graph ψ∗ = (V , E) where A is Nvertex over set on V and B is a Nedge over set on E such that TB (xy) ≤ (TA (x) ∧ TA (y)), IB (xy) ≤ (IA (x) ∧ IA (y)), FB (xy) ≥ (FA (x) ∨ FA (y)). Definition 2.2 Let ψ be a Nover top graph. Let x, y ∈ V. Then x dominates y in ψ if edge xy is effective edge TB (xy) = (TA (x) ∧ TA (y)), IB (xy) = (IA (x) ∧ IA (y)), FB (xy) = (FA (x) ∨ FA (y)). / DN , A subset DN of V is said to be a Nover top dom set in ψ if each vertex V ∈ there exists u ∈ DN such that u dominates V. Definition 2.3 A dom set DN of Nover top graph is called a minimal Nover top dom set if no proper subset of DN is dom set. Definition 2.4 Minimum cardinality of a Nover top dom set in a Nover top graph ψ NOT NOT is said to be Nover top dom number of ψ and is represented by γ (ψ) (or) γ .

3 Types of Nover Top Domination Graphs NOT

Definition 3.1 A subset P of V is said to be perf. Nover top dom set of ψ if for NOT NOT each vertex v ∈ / P is dominating by exactly one vertex u of P . NOT

Definition 3.2 A perfNover top dom set P of a Nover top graph ψ is called minimal perfNover top dom set for v ∈ P NOT , P NOT − {v} is not a perfNover top dom set of ψ. Definition 3.3 The minimal cardinality of a minimal perfNover top dom set of ψ is NOT said to be the perfNover top dom number of ψ. It is represented by γpf (ψ). Definition 3.4 The maximal cardinality of a minimal perfNover top dom set of ψ is NOT said to be the upper perfNover top dom number of ψ. It is represented as pf (ψ).

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Example 1

Let a, b, c, d , e, f , g and h denote the vertices and (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2) and (0.3, 0.5, 1.2) denote the edges which are labelled fu (0.3, 0.5, 1.2) = (a, b), fu (0.3, 0.5, 1.2) = (b, c), fu (0.3, 0.5, 1.2) = (c, d ), fu (0.3, 0.5, 1.2) = (d , e), fu (0.3, 0.5, 1.2) = (e, f ), fu (0.3, 0.5, 1.2) = (f , g), fu (0.3, 0.5, 1.2) = (g, h). Let X = {a, b, c, d , e, f , g, h, (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2), (0.3, 0.5, 1.2)} be a topological space defined by the topology τ = {∅, X , {a}, {b}, {c}, {d , e}, {f , g, h}, {a, b}, {a, c}, {a, d , e}, {a, f , g, h}, {b, c}, {b, d , e}, {b, f , g, h}, {c, d , e}, {c, f , g, h}, {d , e, f , g, h}, {a, b, c}, {a, b, d , e}, {a, b, f , g, h}, {b, c, d , e}, {b, c, f , g, h}, {c, d , e}, {c, f , g, h}, {d , e, f , g, h}, {a, b, c, d , e}, {a, b, c, f , g, h}, {a, b, d , e, f , g, h}, {a, b, c, e, f , g, h}, {a, b, c, d , f , g, h}, {a, b, c, d , e, g, h}, {a, b, c, d , e, f , h}, {a, b, c, d , e, f , g}, {a, b, c, d , g, h}, {a, b, c, d , f , h}, {a, b, c, d , e, f , g}, {a, b, c, d , g, h}, {a, b, c, d , f , h}, {a, b, c, d , f , g}, {a, b, c, d , e, h}, {a, b, c, d , e, g} {a, b, c, e, f , g, h}, {a, b, c, d , f , g, h}, {a, b, c, d , e, g, h}, {a, b, c, d , e, f , h}, } Here for every x ∈ X , {x} is open or closed. By the definition of Nover top graph, we have |∂(A)| ≤ 2 and ∂(a) = {b}, ∂(b) = {a, c}, ∂(c) = {b, d }, ∂(d ) = {c, e}, ∂(e) = {d , f }, ∂(f ) = {e, g}, ∂(g) = {f , h},

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∂(h) = {g} with ∂(ai ) = 2 where i = 1, 2, · · · , 8. Hence, this graph is Nover top graph. Now, consider a perfect effective dominating Nover top graph ψ = (A, B) where V = {a, b, c, d , e, f , g, h} and TA ,IA ,FA are given by TA : V → [0, ], IA : V → [0, ] and FA : V → [0, ] where TA (a) = min[TB (a, b)] = min[0.3] = 0.3 IA (a) = min[IB (a, b)] = min[0.5] = 0.5 FA (a) = max[FB (a, b)] = max[1.2] = 1.2 TA (b) = min[TB (b, c), TB (b, a)] = min[0.3, 0.3] = 0.3 IA (b) = min[IB (b, c), IB (b, a)] = min[0.5, 0.5] = 0.5 FA (b) = max[FB (b, c), FB (b, a)] = max[1.2, 1.2] = 1.2 Similarly, TA (c) = 0.3, IA (c) = 0.5, FA (c) = 1.2 TA (d ) = 0.3, IA (d ) = 0.5, FA (d ) = 1.2 TA (e) = 0.3, IA (e) = 0.5, FA (e) = 1.2 TA (f ) = 0.3, IA (f ) = 0.5, FA (f ) = 1.2 TA (g) = 0.3, IA (g) = 0.5, FA (g) = 1.2 TA (h) = 0.3, IA (h) = 0.5, FA (h) = 1.2 Here a dominates b because TB (a, b) ≤ TA (a) ∧ TA (b), 0.3 ≤ 0.3 ∧ 0.3 IA (a, b) ≤ IA (a) ∧ TA (b), 0.5 ≤ 0.5 ∧ 0.5 FA (a, b) ≥ FA (a) ∨ TA (b), 1.2 ≥ 1.2 ∨ 1.2 Here b dominates c because TB (b, c) ≤ TA (b) ∧ TA (c), 0.3 ≤ 0.3 ∧ 0.3 IB (b, c) ≤ IA (b) ∧ IA (c), 0.5 ≤ 0.5 ∧ 0.5 FB (b, c) ≥ FA (b) ∨ FA (c), 1.2 ≥ 1.2 ∨ 1.2 Similarly, c dominates d , d dominates e, e dominates f , f dominates g, g dominates h Here perfNover top dom sets are {b, e, h}, {a, d , g}, {b, d , g} and {b, e, g}. Minimal perfNover top domset P NOT = {b, e, h}. NOT PerfNover top domnumber γpf (ψ) = 1, 2.

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NOT

Upper perfNover top domnumber pf (ψ) = 1, 2. NOT

Definition 3.5 A perfNover top dom set Pc is said to be CONN perfNover top NOT dom set if the Nover top induced subgraph Pc is CONN. NOT

Definition 3.6 A perfNover top dom set Pc of a Nover top graph G is called a NOT NOT minimal CONN perfNover top dom set if for each vertex v in Pc , Pc − {v} is not a CONN perfNover top dom set of ψ. Definition 3.7 The minimum cardinality of a CONN perf Nover top dom set of a Nover top graph ψ is called the CONN perf Nover top dom number of ψ and is NOT represented as γcpf (ψ). Definition 3.8 The maximum cardinality of a minimal CONN perf Nover top dom set ψ is called the upper CONN perf Nover top dom number of ψ and is represented NOT by cpf (ψ). NOT

Definition 3.9 A perf Nover top dom set PT of Nover top graph ψ is said to be Tot perfNover top dom set if each vertex v ∈ V in ψ is dominated to at least one vertex NOT in PT . Definition 3.10 The minimum cardinality of a Tot perfNover top dom set is Tot NOT perfNover top dom number and is represented as γTPF (ψ). NOT

Definition 3.11 A Tot perfNover top dom set PT of Nover top graph ψ is called a NOT NOT minimal Tot perfNover top dom set if for each vertex v in PT , PT − {v} is not a Nover top dom set of ψ. Definition 3.12 The maximum cardinality of a minimal Tot perfNover top dom set of ψ is called the upper Tot perfNover top number of Nover top graph ψ and is NOT represented by TPF (ψ). Example 2

By the previous example, similarly we can find the Nover top conditions.

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Hence, this graph is Nover top graph. Here Nover top dom sets are {a, d }, {b, c}. Tot perfNover top domset = {a, d }. NOT Minimal Tot perfNover top dom number γcpf = 1. NOT

Upper Tot CONN perfNover top dom number pf (ψ) = 1. Theorem 3.1 Every CONN perf Nover top dom set is a Tot perfNover top dom set. Assume that ψ is a Nover top graph with no isolated vertices. By using Definition 3.8 CONN perfNover top dom set, if the induced subgraph NOT PC is CONN in the CONN perf Nover top dom set of ψ. It is clear that, if each NOT vertex in ψ is dominated to atleast one vertex in PC which is the definition of Tot perfNover top dom set of ψ. Therefore, every CONN perf Nover top dom set is a Tot perfNover top dom set of ψ. Remark 1 The converse of the theorem is not true. Example 3 In this Example 2, the collection of the sets {a, d }, {c, d }, {b, d }, {a, c} are Tot perf Nover top dom sets, but they are not CONN perf Nover top dom sets. The set {a, b, c} is CONN Perf Nover top set and Tot perfNover top dom set respectively. CONN Tot perf Nover top dom set in Nover top graph NOT

Definition 3.13 A Tot perfNover top dom set Pct is said to be a CONN TotperfNover NOT top dom set if the induced subgraph Pct is CONN. NOT

Definition 3.14 A perfNover top dom set Pct of a Nover top graph ψ is said to be NOT NOT minimal CONN Tot perfNover top dom set if for each vertex v in Pct , Pct − {v} is not a CONN Tot perfNover top dom set of ψ. Definition 3.15 The minimum cardinality of a CONN Tot perfNover top dom set of a Nover top graph ψ is called the CONN Tot perfNover top dom number of ψ and NOT is represented as γctp (ψ). Definition 3.16 The maximum cardinality of a minimal CONN Tot perfNover top dom set of ψ is said to be the upper CONN Tot perfNover top dom number of ψ and NOT is represented by ctp (ψ).

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Example 4

By the previous example, similarly we can find the Nover top condition. CONN Tot perfNover top dom sets are {a, b, c, d , e, f }, {b, c, d , e, f , g}. Minimal CONN Tot perfNover top domset = {a, b, c, d , e, f }. CONN Tot perfNover top number = 2.1. Upper CONN Tot perfNover top number = 2.1. Theorem 3.2 Every CONN Tot perfNover top dom set is a Tot perf Nover top dom set of ψ. Suppose ψ be a Nover top graph without isolated vertices. By using Definition 2.17. NOT CONN Tot perfNover top set, if the induced subgraph Pct is CONN in the CONN Tot perfNover top dom set of a Nover top graph ψ. It is clear that, if each NOT vertex in ψ is dominated to at least one vertex in Pct which is the definition of Tot perfNover top dom set of ψ. Therefore, every CONN Tot perfNover top dom set is a Tot perfNover top dom set of ψ. NOT

Theorem 3.3 For any CONNNover top graph G, then γp NOT γctp (ψ).

NOT

(ψ) ≤ γtp (ψ) ≤

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Example 5

CONN Tot perfNover top dom set = {b, c, d , e}. NOT CONN Tot perfNover top domnumber γctp (ψ) = 0.4. Tot perfNover top domset = {b, c, d , e}. NOT Tot perfNover top domnumber γtp (ψ) = 0.4. Perf Nover top dom set = {b, d , f }. Perf Nover top dom number = .3. NOT NOT NOT Therefore, γp f (ψ) ≤ γtp (ψ) ≤ γctp (ψ). Theorem 3.4 If H is a CONN spanning subgraph of a Nover top graph G, then NOT γ NOT ctp (ψ) ≤ γctp (H ). However, every CONN Tot perfNover top dom set of H is also CONN Tot perfNover top dom set of ψ. Example 6

By the previous example, similarly we can find the Nover top condition. Hence, this graph is Nover top graph. CONN Tot perfNover top dom set {a, b, c} NOT γctp (ψ) = 0.15

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Similarly, we can find out Nover top spanning subgraph (H ). ∴ CONN Tot perfNover top dom set {a, b, c} NOT γctp (H ) = 0.15 NOT NOT Therefore, γctp (ψ) ≤ γctp (H ).

Remark 2 The interrelation among Nover top dominating sets is

References 1. Revathi, S., P.J. Jayalakshmi, and C.V.R. Harinarayanan. 2013. Perfect dominating sets in fuzzy graphs. IOSR Journal of Mathematics 8 (3): 43–47. 2. Broumi, S., M. Talea, A. Bakali, and F. Smarandache. 2016. Single valued neutrosophic graphs. Journal of New Theory 10: 86–101. 3. Smarandache, F., 2016. Neutrosophic over set, neutrosophic under set, neutrosophic off set. Pons Editions, Brussels. 4. Smarandache, F., 1999. A Unifying Field in Logics: Neutrosophic Logic. American Research Press, pp.1–141. 5. Somasundaram, A., and S. Somasundaram. 1998. Domination in fuzzy graphs–I. Pattern Recognition Letters 19 (9): 787–791. 6. Salama, A.A., 2013. Neutrosophic crisp points and neutrosophic crisp ideals. Neutrosophic Sets and Systems 50–53.

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7. Salama, A.A., F. Smarandache, and V. Kroumov. 2014. Neutrosophic crisp sets and neutrosophic crisp topological spaces. Neutrosophic Sets and Systems 25–30. 8. Salama, A.A., F. Smarandache, and V. Kromov, 2014. Neutrosophic closed set and neutrosophic continuous functions. Neutrosophic Sets and Systems 4–8. 9. Ore, O., 1962. Theory of graphs. American Mathematical Society Colloquium Publications 38. 10. Zadeh, L.A., 1965. Fuzzy sets. Information and Control 8 (3): 338–353. 11. Nagoorgani, A., and V.T. Chandrasekaran, 2006. Domination in fuzzy graph. Advances in Fuzzy Sets and System 17–26. 12. Revathi, S., C.V.R. Harinarayanan, and R. Muthuraj. 2015. Connected perfect domination in fuzzy graph. Golden Research thoughts 5: 1–5. 13. Devi, R.N., N. Kalaivani, S. Broumi, and K.A. Venkatesan, 2018. Characterizations of strong and balanced neutrosophic complex graphs. Infinite Study. 14. Devi, R.N., 2017. Neutrosophic complex N-continuity. Infinite Study. 15. Devi, R.N., 2020. Minimal domination via neutrosophic over graphs. AIP Conference Proceedings, November 2277: 100019.

Analysis and Classification of Physiological Signals for Emotion Detection Gitosree Khan, Shankar Kr. Shaw, Sonal Aggarwal, Akanksha Kumari Gupta, Saptarshi Haldar, Saurabh Adhikari, and Soumya Sen

Abstract Detecting and classifying emotions using several physiological signals has become a pivot area of research nowadays. The most popular method for analysis of emotion recognition is the use of physiological sensors. This paper focuses on physiological signal-based emotion recognition, including analysis of emotional physiological datasets and classifier models. The study helps human computer interaction (HCI) research immensely. The acquisition of the signals through heterogeneous datasets is done through several physiological sensors like PPG, GSR, EEG, etc., to detect human emotions automatically by selecting best-fit algorithm. The signals in terms of training datasets are extracted once the analysis of the pre-processed data is over and is validated using data validation model. The trained and test datasets are classified based on some machine learning models that improved the overall performance factor in compare to other classifier model. These steps help us in finding the correlation between variables and enable us to predict the classified output variable based on the predictor variables. Keywords PPG · GSR · EEG · Emotion · Detection · Classification

1 Introduction In recent days, emotion detection plays a pivot role in both human physiological and psychological status. The identification and classification of human emotions is a challenging paradigm nowadays.

G. Khan · S. Kr. Shaw · S. Aggarwal · A. K. Gupta · S. Haldar B. P. Poddar Institute of Management and Technology, Kolkata, India S. Adhikari Swami Vivekananda University, Kolkata, India S. Sen (B) University of Calcutta, Kolkata, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_8

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To describe the physiological signal-based emotion recognition, the internal physical signals blood pressure, heart rate, respiration, electrocardiogram (ECG) is given as input in both research and real-life applications. The signals are collected through various physiological sensors, which helps to monitor users’ dynamic activity associated with emotional states in real-time scenario. The exact emotion recognition allows to understand humans’ emotions and interact with humans in accordance with physiological states for better communications. Mostly, emotion recognition works on the input information logically, whereas human emotions does not. Emotion recognition approaches have achieved high efficiency and reliability, allowing the development of various physiological signal-based applications. This advancement has taken place mainly in the cognitive environment and real-time applications. However, with the increase of computing paradigm of several physiological sensors and their in-built sensors, have provide them an environment in which the task of identifying human emotions can be performed at least as effectively and efficiently. In general, each cognitive emotion is identified based on the analysis of five key physiological components by analyzing five main components of emotion such as expressions, cognitivebased appraisals, behavioral tendencies, subjective based feelings, and physiological reactions but only the first four component scan be retrieved automatically. These components indicate the human physiological state during human computer interaction, without any further delay. The last component subjective-based feelings are recovered based on self-assessment techniques. Usually, the human emotion recognition is done dynamically by measuring certain key parameters such as human expression, human speech, human gesture, and human posture. Further, it analyzes the changes arise during the process of cognitive emotion recognition. The key techniques of measuring those human physical attributes are electroencephalography (EEG), galvanic skin resistance (GSR), pressure, rate of heart, photoplethysmogram (PPG), electrocardiogram (ECG), and respiratory rate analysis (RR). This paper proposed to recognize human emotions based on physiological signals using machine learning algorithms. The objective is to provide a basis for the emotion recognition technique in recent research of human computer interaction compared to the traditional emotion classification method.

2 Review of Related Work The emotion recognition in the way human communicates may help us to understand human relating processes, such as concentration, memory, and information processing skills. Machine learning and deep learning techniques are used consistently to recognize emotions experienced by human beings through several human physiological activities. The models should have capability to work against different high dimensional heterogeneous data with unstable time ambiguity issues. Creating models and predicting the state of human emotions over the years is not a key issue because continuous datasets labeling is costly and not always convenient in recent days. This is a problem of importance in real-world scenario, because the labeling of

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the features data is scattered and mostly highlights on important emotional occasions. The author created a model in [1] in which stress-related data were collected from different groups of people. The study in [2] presented a new technique for recognizing emotions based on ECG and GSR signals. A model was created to detect emotions from GSR data [3], which can utilize information from the sensors to detect the emotional event of the responder. Once verified, support vector machine (SVM) is trained using a dataset. In the work [4], the author has focuses mostly on classification of AEE methods. In paper [5], the author highlights on GSR device detection results, whether the results show an effort or a different scenario in context to relaxation of a success rate of 90.97%. The study [6] focuses on emotion recognition framework using various physiological signals gathered from different sensors associated with human emotion identification. In concern with the mobile nature of emotion recognition of smart wearables [7], the physiological data can be read in a mobile in a discreet manner. The paper [8] reviewed the importance of human emotion recognition approach based on physiological and speech signals. Paper [9] highlights on the extension of multi-modal emotion recognition technique based on the processing of extracted physiological datasets. The convolutional networks compared to the most recent algorithms of machine learning described better results in terms of performance and reliability in the context emotion detection method [10]. In paper [11], the author describes the process of extracting high-level features of two separate views of real-world datasets that are correlated among each other highly. The author used deep Canonical correlation methodology for the process. Paper [12] highlights on a novel technique of emotion recognition and classification analysis using EEG signals. Further, several researchers focus on deep learning, physiological signals enhancement, and EEG-based BCI switch using different fuzzy models [13].The author in work [14] describes on the variability in heart rate for analysis of the physical state of a patient and finding an alternative method to diagnose the treatment of cardiopathies as well. It has been observed that the PPG technique has got more utilization over ECG in bio-medical fields. The paper [15] provides a non-invasive technique developed for the photoplethysmography signal estimation of the blood glucose level (BGL). Detection and recognition of human emotion is a major issue nowadays. There are a few limitations in classifying human emotions these days. The limitations are: (a) Data description is an expensive, time consuming, and errorprone task since it requires clear evidence data. (b) Subject-dependent algorithms perform better than human-independent algorithms because even for the materials that draw out same emotions, the extracted emotions depend on the surroundings, human culture, present mood status, human personality, and perception of the subject considered. Therefore, additional research must be done to acquire more generic algorithms. (c) Many recent research shown that with the increase in number of trained datasets, there is an increase in human emotion recognition rate; however till yet there is no clear evidence which physiological signals are the important ones.

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Fig. 1 Workflow of classification model

3 Proposed Work In this work, we have analyzed and classified various pre-processed data collecting from different sensors like PPG, GSR, EEG, etc., which helps to detect human emotions dynamically using best-fit algorithm. The following are the key steps of the proposed work: (a) Firstly, the pre-processed data is acquired from SEED and DEAP datasets, using which the analysis of data is done. (b) Secondly, feature extraction is done over the pre-processed data by removing the redundant data present in the trained sets. (c) Data Validation on training datasets is being performed once the physiological signals are retrieved. (d) The training and test datasets are classified based on classifier models considered in this paper. (e) Finally, the data classification steps are performed over the test datasets.

3.1 Workflow of the Classifier Model In this subsection, a workflow of the classifier model is described using various physiological signals like PPG, GSR, EEG, etc., that helps to detect human emotions automatically. Figure 1 depicts the workflow of the classifier model.

3.2 Analysis of Pre-processed Datasets In this subsection, the analysis of pre-processed data using statistical techniques and visualizations is done. This analysis gives an overview of the quality of data and its

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Fig. 2 Analysis of statistical values for GSR data

key properties such as mean, standard deviation, mode and distribution of the data. Figures 2, 3 and 4 show the analysis of pre-processed data for GSR, EEG, and PPG.

Fig. 3 Analysis of signal (alpha, beta, gamma, theta, delta) of EEG data

Fig. 4 Analysis of statistical values for PPG data

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Fig. 5 Extracted features visualization

3.3 Feature Selection, Extraction, and Validation Feature extraction is the method of reducing the features from the existing ones associated in trained datasets and improved the emotion recognition performance. It plays a vital role in the recognition technique. Figure 5 shows the extracted feature visualizations of the proposed model. The extracted features help to describe certain information associated in the original datasets. Lots of researches focuses on several feature extraction methodology like fast Fourier transform analysis, variability of heart rate, adaptive time-space analysis method, etc. In Fig. 5, three types of emotions are being considered. They are positive, calm, and negative. Positive emotions improve work efficiency and performance, while calm and negative emotions cause health issues. Empirical analysis so far in this paper describes that emotion work has both positive and negative effects on human physical condition. For human emotional dissonance, the negative effect was found and is shown in Fig. 5. Here, emotions are not discrete in nature but rather continuous ones. In order to extract feature for estimation, the correlation between the single-feature vectors and the self-assessed arousal/valence vector was taken into consideration. The feature having high absolute correlation coefficient is chosen as an effective one.

3.4 Classification with Machine Learning Algorithms In this section, feature extraction, feature selection, binary classification, and performance evaluation are done over pre-processed dataset. Here, three emotional dimensions of positive, negative, and calm are being done over SEED dataset, and four emotional dimensions based on Russel’s Circumflex Models in which arousal and valence are categorized into emotion of happy, sad, anger, and fear over DEAP dataset. (a) K-Nearest Neighbor (KNN) Algorithm for Machine Learning

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K-nearest neighbor is one of the prime supervised-based machine learning approaches nowadays. The algorithm considers the similarity index between the new case data and existing case data and includes into categories similar to the existing ones. (b) Support Vector Machine (SVM) Algorithm for Machine Learning A support vector machine is a supervised machine learning model that uses twogroup classification problems in addressing the challenges of classification and regression. Here, the decision boundary is created that divide the n-dimensional space into several classes so that the new data point can be placed in correct position. (c) Random Forest (RF) Algorithm for Machine Learning This supervised machine learning algorithm approach is used as one of the classifier models in this paper for identifying emotion detection. The approach supports the concept of ensemble learning, which combines multiple classifiers together to improve the performance of the trained model.

3.5 Performance Analysis of Classification This section discusses about the performance analysis of several classifier models including accuracy percentage as discussed in Sect. 3.4. Tables 2 and 4 highlight on classification report of KNN and RF model for EEG signals. Tables 5 and 6 focus on classification report of KNN and RF model for GSR signals. Tables 8 and 9 describe the classification report for PPG signals. Figures 6, 7, 9, 10, 11, and 12 study the nature of the learning curves of KNN, SVM, and RF algorithm. Tables 1, 4, and 7 show the accuracy values of EEG, GSR, and PPG signals for the different algorithms for the 70–30 train test split and for the 80–20 train test split. We do not consider SVC further as the result of SVC is poor KNN and RF (Fig. 8 and Table 3). Fig. 12Learning curve of RF algorithm Fig. 6 Learning curve for KNN algorithm for EEG signal

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Fig. 7 Learning curve of SVM algorithm

Table 1 Result for KNN, SVC, and RF algorithms for EEG signal Classifier

Train test split 70–30 (%)

80–20 (%)

KNN

90

91

RF

96

98

SVC

56

55

Table 2 Classification report—KNN model for EEG signal Precision

Recall

F1 score

Support

−1

0.88

0.87

0.88

20,812

0

0.89

0.90

0.90

20,555

1

0.91

0.92

0.92

21,762

0.90

63,129

Accuracy Macro average

0.90

0.90

0.90

63,129

Weighted average

0.90

0.90

0.90

63,129

Fig. 8 Learning curve of RF algorithm

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Fig. 9 Learning curve for KNN algorithm for GSR signal

Fig. 10 Learning curve of RF algorithm for GSR signal

Table 3 Classification report—RF model for EEG signal Precision

Recall

F1 Score

Support

−1

0.98

0.98

0.98

5770

0

0.97

0.98

0.97

3000

1

0.97

0.97

0.97

3104

Accuracy

0.98

11,874

Macro average

0.97

0.98

0.97

11,874

Weighted average

0.98

0.98

0.98

11,874

Table 4 Result for KNN and RF algorithms for GSR signal

Classifier

Train test split 70–30 (%)

80–20 (%)

KNN

75

72

RF

83

80

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Table 5 Classification report—KNN model for GSR signal Precision

Recall

F1 score

0

0.83

0.62

0.71

8

1

0.40

1.00

0.57

2

2

1.00

0.89

0.94

9

3

0.60

0.60

0.60

5

0.75

24

Macro average

0.71

0.78

0.71

24

Weighted average

0.81

0.75

0.76

24

Support

Accuracy

Support

Table 6 Classification report—RF model for GSR signal Precision

Recall

F1 score

0

0.83

1.00

0.91

1

0.40

1.00

0.57

2

2

1.00

0.80

0.89

10

3

1.00

0.71

0.83

7

0.83

24

Accuracy

5

Macro average

0.81

0.88

0.80

24

Weighted average

0.92

0.83

0.85

24

Table 7 Result for KNN and RF algorithms for PPG signal Classifier

Train test split 70–30 (%)

80–20 (%)

KNN

71

83

RF

86

88

Table 8 Classification report—KNN model for PPG signal Precision

Recall

F1 score

0

0.50

0.67

0.57

1

0.75

0.75

0.75

4

2

0.92

0.92

0.92

12

3

1.00

0.80

0.89

5

0.83

24

Macro average

0.79

0.78

0.78

24

Weighted average

0.85

0.83

0.84

24

Accuracy

Support 3

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Table 9 Classification report—RF model for PPG signal Precision

Recall

F1 score

0

0.75

0.75

0.75

1

0.75

0.75

0.75

4

2

0.92

1.00

0.96

11

3

1.00

0.80

0.89

5

0.88

24

Macro average

0.85

0.82

0.84

24

Weighted average

0.88

0.88

0.87

24

Accuracy

Support 4

Fig. 11 Learning curve of KNN algorithm for PPG signal

Fig. 12 Learning curve of RF algorithm

4 Conclusion This paper describes a comparative study based on analysis and classification of various physiological signals for identifying state of human emotions accurately as stated in the proposed work. The best average accuracy achieved is EEG-97%, GSR83%, and PPG-83% on the dominance dimension. Compare all sets of results, GSR

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and PPG are more promising for supervised learning, since it can able to detect four types of human emotions satisfactory. Classification performance states that few human emotional states can be identified through brain activities using several sentiments associated with positive, negative, and neutral. Focusing more on parameters and optimizing the model structure along with the analysis of additional physiological signals like EMG, ECG, HRV, and RR will be the future scope of our work.

References 1. Dutta, S., D. Sachikanta, and A. Mitra, eds. 2020. A model of socially connected things for emotion detection. In 2020 International Conference on Computer Science, Engineering and Applications (ICCSEA), 571–580, IEEE, March 13–14. Gunupur, India. 2. Goshvarpour, A., A. Ataollah, and G. Ateke, eds. 2017. An accurate emotion recognition system using ECG and GSR signals and matching pursuit method. Biomedical Journal 40 (6): 355–368. 3. Setyohadi, D., and Budiyanto et al., eds. 2018.Galvanic skin response data classification for emotion detection. International Journal of Electrical and Computer Engineering 2088–8708. 4. Dzedzickis, A., K. Art¯uras, and B. Vytautas, eds. 2020. Sensors 20 (3): 592. 5. Villarejo, M.V., B.G. Zapirain, and A.M. Zorrilla, eds. 2012. A stress sensor based on galvanic skin response (GSR) controlled by ZigBee. Sensors 6075–6101. 6. De˘ger, A., Y. Yaslan, and E.M. Kamasak, eds. 2020. Emotion recognition from multimodal physiological signals for emotion aware healthcare systems. Journal of Medical and Biological Engineering 1–9. 7. Maria, E., M. Ley, and S. Hanke, eds. 2019. Emotion recognition from physiological signal analysis: A review. Electronic Notes in Theoretical Computer Science 35–55. 8. Mouhannad, A., eds. 2020. Emotion recognition involving physiological and speech signals: A comprehensive review. In Recent Advances in Nonlinear Dynamics and Synchronization, 287–302. Springer, Cham. 9. Huong, T.V., Hong, T.K.N., and H.L. Duy, eds. 2019. Emotion recognition based on multimodel: physical—Bio signals and video signal. International Journal of Engineering Research & Technology (IJERT) 8 (10). 10. Granados, S., and Luz, eds. 2018. Using deep convolutional neural network for emotion detection on a physiological signals dataset (AMIGOS). IEEE Access 7: 57–67. 11. Qiu, J.L., W. Liu, and B.L. Lu, eds. 2018. Multi-view emotion recognition using deep canonical correlation analysis. In International Conference on Neural Information Processing. Springer, Cham. 12. Suhaimi, N.S., M. James, and T. Jason, eds. 2020. EEG-based emotion recognition: A state-of-the-art review of current trends and opportunities. Computational Intelligence and Neuroscience. 13. Gu, X., eds. 2021. Eeg-based brain-computer interfaces (bcis): A survey of recent studies on signal sensing technologies and computational intelligence approaches and their applications. IEEE/ACM Transactions on Computational Biology and Bioinformatics. 14. Moraes, J.L., eds. 2018. Advances in photopletysmography signal analysis for biomedical applications. Sensors 18 (6): 1894. 15. Priyadarshini, R., and Gayathri, eds. 2021. Review of PPG signal using machine learning algorithms for blood pressure and glucose estimation. In IOP Conference Series: Materials Science and Engineering, vol. 1084, no. 1.

Wiener and Zagreb Indices for Helm and Web Graph J. Senbagamalar, M. Priyadharshini, P. Rajesh, and Hanaa Hachimi

Abstract The chemical graph is related to the compounds molecular structure, which is similar to the distance and degree-based graphs that are used for molecular design. In this paper we obtained the general results complementing the Helm graph and Web graph for the Wiener Index and the Zagreb indices. Keywords Distance · Connectivity · Degree · Duplication

1 Introduction Consider the graph M has without loops and multiple edges, with carbon atoms, bonds V (M) and E(M), respectively. The bonds incident in a atoms η in a graph M is the neighboring bonds in M, named as deg(η). Generally, shortest bonds between ε and η of graph M is d(ε, η) see [1, 2]. In reference [3] and [4] the path number was introduced by Wiener.  W (M) = nε=1 nη=1 d(vε , vη ). The connectivity indices were enumerated by  Gutman and Trinajstic [5] which are defined as M1 (M) = η∈V (M) d(η)2 .  M2 (M) = εη∈E(M) deg(ε) deg(η). The topological indices (Graph invariants) are computed in a graph for any two pairs of vertices. Models of the relationship between quantitative structure behavior were applied in biological and chemical activities. In the first century, topological indices were based on integer graph features such as topological distances [6, 7]. The concept of line graph has numerous applications in physico chemical alkane molecules. An interesting relation between W (M) J. Senbagamalar (B) · M. Priyadharshini SRM Institute of Science and Technology, Ramapuram, Chennai, India e-mail: [email protected] P. Rajesh Vels Institute of Science and Technology, Chennai, India H. Hachimi Sultan Moulay Slimane University, Beni-Mellal, Morocco e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_9

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and L W (M). In [8, 9] the line graph L(M) is a graph such that the nodes of L(M) are the bonds of M, two nodes of L(M) are neighbors, then their corresponding links in M have a common node. A connected acyclic graph is a tree. [10]. In reference [11, 12] the Inverse or Complement of a graphM, with two nodes are neighbors in M and the same two neighbors should be not a neighbor inM. A graph is called duplication of vertices of M by K 2 denoted by M ⊕ ηK 2 , a graph G is obtained from n copies of K 2 , of which each copy of K 2 corresponds to a unique vertex of M such that each vertex η of M is joined to two vertices of its corresponding copy K 2 . No two atoms intersect each other is a planar graph. Graph Cn ⊕ ηK 2 is an outer planar graph. A Helm graph H2η+1 created in wheel Wη by adding a single bond at each atom on the rim of the wheel Wη . This paper analyses the Wiener index and the Zagreb indices of the complement of Helm graph, web graph, Cε ⊕ ηK 2 and the full binary tree union of the full binary tree.

2 Main Results Theorem 1. For η > 2, the complement of Cε ⊕ ηK 2 are (i) W (Cε ⊕ ηK 2 ) = 2η(η + 1) 2 (ii) M1 (C   ε ⊕ ηK 2 )= η(8η − 32η  + 34) (iii) M2 Cε ⊕ ηK 2 = 4η(η − 2) 2η2 − 8η + 9 Proof For the case η > 2, consider Cε ⊕ ηK 2 . The degree sequence of Cε ⊕ ηK 2 be (2, 4, 2, 4, ..., 4, 2).The graph Cε ⊕ ηK 2 has 2η vertices. The degree sequence of the complement of Cε ⊕ ηK 2 be ((2η − 3), (2η − 5), (2η − 3), ..., (2η − 5)). In Cε ⊕ ηK 2 , we have η nodes have degree (2η − 3) and η nodes have degree (2η − 5).  1   d vε , vη 2 ε=1 η=1 ⎤ ⎡ 1⎣ W (Cε ⊕ ηK 2 ) = ((2η − 3) + 4) + ((2η − 5) + 8)⎦

 

 2  n

n

(i) W (Cε ⊕ ηK 2 ) =

η

W (Cε ⊕ ηK 2 ) = 2η(η + 1).

η

Wiener and Zagreb Indices for Helm and Web Graph



(ii) M1 (M) =

93

d(η)2

η∈V (M)

M1 (Cε ⊕ ηK 2 ) = η(2η − 3)2 + η(2η − 5)2 M1 (Cε ⊕ ηK 2 ) = η(8η2 − 32η + 34) (iii) M2 (M) =



deg(η) deg(ε)

εη∈E(M)

⎧ ⎪   η⎨ M2 Cε ⊕ ηK 2 = (2η − 5) ((2η − 5)(2η − 5)...(2η − 5))

  2⎪ ⎩ (η−3)

⎫ ⎪ ⎬ + (2η − 5) ((2η − 3)(2η − 3)...(2η − 3)) 

⎪ ⎭ (η−2)

+ (2η − 5) ((2η − 3)(2η − 3)...(2η − 3)) 

 (η−2)

⎫ ⎪ ⎬ + (2η − 3) ((2η − 3)(2η − 3)...(2η − 3)) 

⎪ ⎭ (η−1)

  η  M2 Cε ⊕ ηK 2 = 16η3 − 96η2 + 200η − 144   2   M2 Cη ⊕ ηK 2 = 4η(η − 2) 2η2 − 8η + 9 . Theorem 2 For η ≥ 1, the Helm graph H2η+1 are   (i) W H2η+1 = 6η(η − 1) (ii) W (H2η+1 ) = 2η(η + 2) (iii) M1(H2η+1) = η2 + 17η (iv) M1 H2η+1 = 8η3 − 19η2 + 17η (v) M2 (H  2η+1 )= η(4η  + 20)  (vi) M2 H2η+1 = η2 16η3 − 60η2 + 95η − 57 3 (vii) F(H  2η+1 )= η +465η 3 (viii) F H2η+1 = 16η − 60η + 38η2 − η     (ix) The Helm graph and its complement W H2η+1 + W H2η+1 = 2η(4η − 1). Proof Consider the Helm graph H2η+1 has 2η + 1 nodes. In H2η+1 , m nodes have degree 4, η nodes have degree one and a single node has degree η. In the complement of H2η+1 , the degree sequence of η nodes have degree (2η − 1), η nodes have degree (2η − 4) and a single node has degree one.   1 (i) W H2η+1 = {3η + η((3 + 2(η − 3)) + (5 + 3(η − 3))) 2

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+ η((5 + 3(η − 3)) + (8 + 4(η − 3)))} 1 W H2η+1 = {3η + η(5η − 7) + η(7η − 8)} 2   W H2η+1 = 6η(η − 1) 



1 [η(2η − 1 + 2) + η(2η − 4 + 8) + (3η)] 2 W (H2η+1 ) = 2η(η + 2)

(ii) W (H2η+1 ) =

⎛

⎞

⎛

⎞

2 ⎠ + ... + 42 ⎠ + ⎝1 (iii) M1 (H2η+1 ) = ⎝4 2 + 42

+1+ · · · + 1 + η η

M1 (H2η+1 ) = 16η + η + η

η

2

M1 (H2η+1 ) = η2 + 17η (iv) M1 (H2η+1 ) = η(2η − 1)2 + η(2η − 4)2 + η2     M1 (H2η+1 ) = η 4η2 + 1 − 4η + η 4η2 + 16 − 16η + η2   M1 H2η+1 = 8η3 − 19η2 + 17η ⎞

⎛

⎛

⎞

+ ... + 16⎠ (v) M2 (H2η+1 ) = ⎝4η + 4η + ... + 4η⎠ + ⎝16  + 16



 η

⎛

η

⎞

⎠ + ⎝4 + 4 + ... + 4 η

M2 (H2η+1 ) = η(4η + 20)   1 (vi) M2 H2η+1 = η2 (2η − 1) + η(η − 3)(2η − 4)(2η − 4) 2 + η(η − 1)(2η − 4)(2η − 1) + η(η − 1)(2η − 1)(2η − 1)

 + η(η − 1)(2η − 1)(2η − 4) + η2 (2η − 1)   η  M2 H2η+1 = 16η3 − 60η2 + 95η − 57 2 ⎛ ⎞ ⎛ ⎞ 3 ⎠ + ... + 43 ⎠ + ⎝1 + 1 + (vii) F(M2η+1 ) = ⎝4 3 + 43

... + 1 + η η

F(M2η+1 ) = η3 + 65η

η

Wiener and Zagreb Indices for Helm and Web Graph

95

  (viii) F M2η+1 = η(2η − 1)3 + η(2η − 4)3 + η3   F M2η+1 = 16η4 − 60η3 + 38η2 − η. (ix)

    The Helm graph and its complement W M2η+1 + W H2η+1 = 2η(4η − 1).

Theorem 3 The complement of web graph is given by. (i) W (W Bη ) = η2 (9η + 5)   (ii) M1 (W Bη ) = 3η 9η2 − 22η + 15 Proof Consider the web graph W Bη has 3η nodes. The degree sequence of W Bη be (3,3,3,…,3,4,4,…,4,1,1,…,1). In the complement of W Bη , η nodes have degree (3η − 2), η nodes have degree (3η − 4) and η nodes have degree (3η − 5). 1 [η(3η) + η(6 + 3η − 4) + η(8 + 3η − 5)] 2 η W (W Bη ) = (9η + 5) 2

(i) W (W Bη ) =

(ii) M1 (W Bη ) = η(3η − 4)2 + η(3η − 5)2 + η(3η − 2)2   M1 (W Bm ) = 3m 9m 2 − 22m + 15 . Theorem 4 The complement of the full binary tree T ∪ L(T ); η ≥ 3   (i) W T ∪ L(T ) = 12(22η ) + 16(2η ) + 10     (ii) M1 T ∪ L(T ) = 64(23η ) − 272 22η − 18(2η ) − 300. η+1 − 1. The number of Proof The number of nodes  η+1 in the full binary tree T is 2 − 3 vertices. nodes in T ∪ L(T ) is 2 2

      1   (i) W T ∪ L(T ) = 2η 2 2η+1 − 3 + 22η+2 2    + 2η+2 − 2 + 22η+2 − 2η − 2η+3 + 2      + 2η+3 + 2 + 2η − 22 2η+2 + 2   W T ∪ L(T ) = 12(22η ) + 16(2η ) + 10     (ii) M1 T ∪ L(T ) = 64(23η ) + 272 22η − 18(2η ) − 300    2  2 M1 T ∪ L(T ) = 2η 2η+2 − 3 + 2η+2 − 6   2 2 + (2η − 2) 2η+2 − 7 + 2η 2η+2 − 8  2  2 + 2 2η+2 − 9 + (2η − 4) 2η+2 − 10

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    M1 T ∪ L(T ) = 64(23η ) + 272 22η − 18(2η ) − 300.

3 Conclusion Recently, graph theory was used to investigate the role of medications in the spread of infectious diseases. The major objective of connectivity indices is to derive multi linear regression models from molecules, isomers, the study of complexity, alters, medication design and database selection.

References 1. Dankelmann, P., I. Gutman, S. Mukwembi, and H.C. Swart. 2009. On the degree distance of a graph. Discrete Applied Mathematics 157 (1): 2773–2777. 2. Furtual, B., I. Gutman, and M. Dehmer. 2013. On structure sensitivity of degree based topological indices. Applied Mathematics and Computation 219 (1): 8973–8978. 3. Baskar Babujee, J., and J. Senbagamalar. 2012. Wiener Index of Graphs using Degree Sequence. Journal of Applied Mathematical Sciences 6 (88): 4387–4395. 4. Wiener, H. 1947. Structural determination of paraffin boiling Points. Journal of American Chemical Society 69 (1): 17–20. 5. Trinajstic, N. 1992. Chemical Graph theory, vol. II, 1936. CRC Press, Inc. Boca Raton, Florida. 6. Devillers, J., and A.T Balaban (eds.). 1999. Topological indices and related descriptors in QSAR and QSPR. Gordan and Breach, Amsterdam, The Netherlands 7. Kandan, P., E. Chandrasekaran, and M. Priyadharshini. 2018. The Revan weighted szeged index of graphs. Journal of Emerging Technologies and Innovative Research 5 (9): 2–8. 8. Dobrynin, A.A., and L.S. Mel’nikov. 2004. Wiener index for graphs and their line graphs. Diskretn Anal Issled Oper Ser 2, 11 (1): 25–44. 9. Gutman, I., and E. Estrada. 1996. Topological indices based on the line graph of the molecular graph. Journal of Chemical Information and Computer Sciences 36 (1): 541–543. 10. Dobrynin, A.A., R. Entringer, and I. Gutman. 2001. Wiener Index for trees: Theory and applications. Acta Applicandae Mathematicae 66 (1): 211–276. 11. Ivan, G., Lu, J., and M.A. Boutiche. 2016. On distance in complement of graphs. Serial A: Applied Mathematics Information and Mechanics 8 (1): 35–42. 12. Senbagamalar, J., J. Baskar Babujee, and Ivan Gutman. 2014. Wiener index of graph complements. Transactions on Combinatorics 3 (2): 11–15.

Disease Classification Using Particle Swarm Optimization with Fuzzy Neural Network S. Leoni Sharmila and S. Poongothai

Abstract Data mining in health care include analysis, early detection, prevention of diseases, prevention of hospital errors, and for cost savings. The importance of machine learning has drastically increased due to predicting and diagnosing disease at an early stage. Due to slow convergence level and it requires a large amount of calculation for fuzzy neural network algorithm, a particle swarm optimization-based fuzzy neural network is used to solve classification of medical dataset. For this work, three medical datasets such as heart, lymph, and hepatitis are used for prediction of diseases. It is proven that there is a better classification rate while using PSO-based FNN. Keywords Fuzzy logic · Neural network · Hybrid technique · Fuzzy neural network · Particle swarm optimization

1 Introduction Data mining is a method of getting valuable information from massive datasets. It is a methodology aimed to obtain the patterns with minimum user input and effort. This is a method of obtaining useful data from large databases [1]. Fuzzy sets and fuzzy logic, which are often used to express and manage complexity, are the optimal method for data mining. It is one of the appropriate ways of handling incomplete and messy data. Fuzzy set theory was used to implement [2], helps to manage ambiguous results. Fuzzy sets and dynamic logic were necessary to execute the proposed program of experts. Using fuzzy logic, the possibility of any specific case falling into any cluster can be quantified, and decisions can then be made based on the value [3]. Fuzzy modeling is skilled for handling professional available knowledge or experience which can be expressed in a set of linguistic’ IF—THEN fuzzy rules and graded membership functions. Therefore, it is user-friendly and provides a detailed portrayal of information. It was also shown in terms of approximation capabilities [4]. S. L. Sharmila (B) · S. Poongothai Department of Mathematics, SRMIST, Ramapuram, Chennai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_10

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Fuzzy structures can therefore be used to seek a high precision approximation of the function. Another interesting characteristic of the fuzzy model which distinguishes it from the NN model is its ability to manage imprecise details using fuzzy rules generated from human experience. ANN has drawn growing interest and has been active in numerous nonlinear machine applications identification and control problems, [5, 6]. NNs are utilized as one of the most common data modeling algorithms in medicines. NNs were the only classification method earlier. The main purpose of using NNs is to recognize patterns and execute the classification tasks [7]. NNs have added benefit as, unlike basic simulation approaches, they can forecast nonlinear relationships [8]. NNs utilize parallel architectures to perform nonlinear mapping between independent variables and dependent variables. This comprises process units that transmits the flow of data through weighted links. NN model has an appealing features that reside in its capacity to approximate and learn in analyzing medical data [9]. Evolutionary techniques are more capable as well as efficient.

2 Particle Swarm Optimization Particle swarm optimization (PSO) is one of the evolutionary computing method created by Eberhart and Kennedy in the year 1995, based on the communal behavior of birds flocking. This algorithm is a population-based optimization technique, with the system starting with a population of random solutions and updating generations to find optimal answers. It is based on population study in which natural behavior flocks of birds are reproduced in a software program. Individuals are initially formed at random and are known as particles. Each particle has a velocity connected with it. Particles in the search space are flying, and their speed is always changing due to the swarm’s behavior, so they prefer to fly to the best solution in the search space. The characteristics of every particle in a swarm are as follows [10]: Yk : particle’s current location. vk : particle’s current velocity. Pbk : particle’s preferred location. If f is a function of quality, which indicates how close the result is to the best possible one, and n is the present time, then the particle’s best location is updated as follows [10, 11]:  Pbk (n + 1) =

Pbk , if f (yk (n + 1)) ≥ f (Pbk (n)); yk (n + 1), if f (yk (n + 1)) ≥ f (Pbk (n))

 Pb0 (n), . . . , Pbe (n) / f (G b (n))   = min f ( Pb0 (n)), . . . , f (Pbe (n))

G b (n) ∈

(1)



The optimum location in the swarm at time n is computed using Eq. (2)

(2)

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The number of elements in a swarm is represented by e [10]. vkl (n + 1) = wvkl (n) + a1r1l (n)[Pbk (n) − Ykl (n)] + a2 r2l (n)[G bl (n) − Ykl (n)]

(3)

The velocity is determined using Eq. (3) vkl (n)—velocity of kth element in l measurement at time n. Ykl (n)—position of the item kth element in l measurement at time n. Pbk (n)—best local position visited by kth element for the first time. w—weights of initial value of {0, 1}. r1k , r2l —random values in the range (0, 1). a1 , a2 —denotes the acceleration coefficients. K,l—particle position. Finally, kth particle is updated in yk by the subsequent equation: Yk (n + 1) = Yk (n) + vk (n) + vk (n + 1)

(4)

Selection is made by best fitness function for every component, and it is represented as Gbest. Now, the velocity component is computed according to Eq. (3), and updated location is given by Eq. (4) [10]. The following are the steps in the PSO algorithm: (i) (ii) (iii)

(iv) (v)

Set the particles’ velocities and locations in the search space to some random values. Begin calculating the corresponding value of the swarm particles’ fitness function. Equip recent value of the particle’s Pbest with fitness value evaluation. Set current value as new Pbest value and set Pbest if current value is better than Pbest . After that evaluate the fitness value to your prior overall best. If current value is than Gbest , Gbest is reset to the index of the array and value of current particle. Finally, apply these values to swarm particle’s appropriate position and velocity.

3 Fuzzy Min–Max Method The FMM classification system is developed utilizing fuzzy hyper-box sets. Hyperbox is specified solely by its lowest and highest points. With respect to these hyperbox, the membership function is defined as min–max position; this explains the extent to which the pattern fits within the hyper-box. A cubic unit K n is described as the spectrum of membership values here between 0 and 1. To an input model of n-dimensions, a pattern found in the hyper-box has the membership quality of one and are described as follows:

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Fig. 1 Min–max hyper-box K j = {mj , nj } in I 3

   K j = y, m j , n j , f y, m j , n j ∀y ∈ I n

(5)

where mj and nj are the ones relevant min and max points. The three-dimensional box, min and max points are illustrated in Fig. 1. Implementing the above condition of a fuzzy hyper-box set consisting of the cumulative fuzzy set to classify Pth pattern class, Fp, is Fp =



Kj

(6)

j∈P

In which, F is the index set to class p associated hyper-boxes. The majority of the calculations are concerned with position, and fine-tuning of class boundaries is an essential property of this method. The system of learning is, FMM enables the overlapping of same class hyperboxes and prevents overlapping of unusual classes. The membership function of jth hyper-box fj (Ua), 0 ≤ f j (Ua) ≤ 1 computes the degree to which the ath ip pattern V a falls exterior to hyper-box K j . It can be used as a metric to see how much each factor is superior (or lesser) than the peak (or least) value together with the dimension that lies outside the hyper-box’s min–max boundaries. Still, when fj (Ua) reaches 1, it shows that the point in the hyper-box should be more “protected.” The purpose that fulfills all of those parameters is the amount of the average of breach of max point and the average amount of breach of min point. The membership function is: n     1  max 0, 1 − max 0, γ min 1, vai − n ji 2n 1=1    

+ max 0, 1 − max 0, γ min 1, c ji − vai

f j (Va ) =

(7)

  where Va = (va1 , va2 , . . . , van ) ∈ I n is the ath input pattern, m j = w1 , w2 , ...., w jn is the min point for K j , n j = n 1 , n 2 , ...., n jn provides the max point of K j , and γ is the sensitivity variable that standardize time when the membership values decline and gap from Oa to F j rises. FMM layout is a network of three levels, given in Fig. 2. The initial layer is the input layer which is identical to the measurements of protocol template as input nodes were as output level has nodes identical to class numbers. Every nodule comprises a sophisticated chain of hyper-boxes, with links from the input layer to the hidden

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Fig. 2 Three-layer FMM network

layer treated as min–max points. The feature of hidden transfer layer is to find hyperbox membership function that is described in (7). The least and highest points, respectively, are represented as matrices m and n. Binary values contained in matrix U are the relations between secret layer and output layer nodes.

4 Experiments and Results 4.1 Datasets In this experiment, three datasets are been used, namely heart-staglog, lymph, hepatitis, and they are taken from UCI repository. The description of each dataset is given is explained in Table 1. It explains about number of instances and number of variables each set contains. Despite the fact that fuzzy ANNs offer greater advantages for classification, they only take into account quantitative rather than qualitative elements. The evolutionarybased FNN can significantly improve the accuracy of a selection. FNN will not become stranded at the local minimum as a result of this suitable variables (Attributes) Table 1 Dataset description

S. No.

Dataset

No. of instances

No. of variables

1

Heart-staglog

270

14

2

Lymph

148

19

3

Hepatitis

155

20

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Table 2 Reduced attribute description of heart-staglog dataset

S. No.

Variables

Description

1

3

chest

2

7

resting_electrocardiographic_results

3

8

“maximum_heart_rate_achieved”

4

9

“exercise_induced_angina”

5

10

oldpeak

6

12

“number_of_major_vessels”

7

13

“thal”

are reduced using PSO which is very important in biological data mining. Let’s consider each dataset, (i)

Heart-Staglog: Totally, 14 variables were present in this dataset, to find the subset of variables using PSO selection feature. Finally, seven variables were selected which are more useful to predict the disease; the attributes that were chosen are listed in Table 2.

(ii)

Lymph: This dataset consists of 19 variables; initially, using PSO, selection of attributes reduces to 10 as given in Table 3

(iii)

Hepatitis: Consists of 20 variables, using PSO selection, it is reduced to 10 that is shown in Table 4.

The fuzzy neural network receives the input from features extracted by the collection of variables obtained by PSO. Table 3 Reduced attribute description of lymph dataset

S. No.

Variables

Description

1

1

lymphatics

2

2

block_of_affere

3

7

regeneration_of

4

8

early_uptake_in

5

9

lym_nodes_dimin

6

10

lym_nodes_enlar

7

11

changes_in_lym

8

13

changes_in_node

9

15

special_forms

10

18

no_of_nodes_in

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Table 4 Reduced attribute description of hepatitis dataset S. No.

Variables

Description

1

1

Age

2

2

Sex

3

6

Malaise

4

11

Spiders

5

12

Ascites

6

13

Varices

7

14

Bilirubin

8

17

Albumin

9

18

Protime

10

19

Histology

Table 5 Comparison accuracy level of FNN and PSO with FNN Dataset

No. of variables

FNN (%)

Reduced variables using PSO

PSO with FNN (%)

Heart-staglog

14

66

7

84

Lymph

19

58

10

87

Hepatitis

20

74

10

83

4.2 Results After the reduction of variables using PSO, each dataset is processed in fuzzy neural network algorithm. In heart-staglog data, with seven reduced variables, FNN has achieved an accuracy rate of 84%. Lymph dataset with ten reduced variables got 87%, and hepatitis dataset with ten reduced variable got 83% accuracy. A comparative classification rate by FNN and PSO with FNN is illustrated in Table 5

5 Conclusion This work gives a study on fuzzy neural network model for classification of biological data. Also, this gives a relative study of particle swarm optimization-based FNN. Since the convergence level is less in FNN, PSO-based FNN is used for classification. Three biological datasets such as heart-staglog, lymph, and hepatitis are used for this work. From the experimental results, it is noticed that PSO has reduced the variable drastically to fasten the convergence level. Results are shown in . This shows that the hybrid technique could be successfully used to help the diagnosis of medical datasets.

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Further, this work can be extended to other medical dataset, and the experiment can be carried out in other evolutionary methods in neural network along with fuzzy logic.

References 1. Kantardzic, M. 2003. Data mining: Concepts, models, methods, and algorithms. New Jersey: John Wiley. 2. Zadeh, L.A. 1998. Some reflection on soft computing granular computing and their roles in the conception. Design and Utilization of Information/Intelligent System, Soft Computing 2 (117): 23–25. 3. Mali, K., and S. Bhattacharya. 2013. Soft computing on\medical—Data (SCOM) for a countrywide medical system using data mining and cloud computing features, global. Journal of Computer Science and Technology Cloud and Distributed 13: 7–13. 4. Wang, L.X., and J.M. Mendel. 1992. Fuzzy basis function, universal approximation and orthogonal least-squares learning. IEEE Translation of Neural Networks 3: 807–814. 5. Chen, S., S.A. Billings, C.F.N. Cowan, and P.M. Grant. 1990. Non-linear systems identification using radial basis functions. International Journal Systems Science 21: 2513–2539. 6. Chen, S., and S.A. Billings. 1992. Neural networks for nonlinear dynamic system modelling and identification. International Journal of Control 56: 319–346. 7. Dunham, M.H. 2003. Data mining introductory and advanced topics. Upper Saddle River, NJ: Pearson Education Inc. 8. Fathima, A.S., D. Manimegalai, and N. Hundewale. 2011. A review of data mining classification techniques applied for diagnosis and prognosis of the arbovirus-dengue. International Journal of Computer Science Issues 8: 322–328. 9. White, H. 1990. Connectionist nonparametric regression: Multilayer feedforward networks can learn arbitrary mappings. Neural Networks 3: 535–549. 10. Leoni Sharmila, S., C. Dharuman, and Venkatesan, P. 2017. Neuro-fuzzy system with evolutionary computing for classification. International Journal of Pure and Applied Mathematics (IJPAM) 113 (11): 37–45. 11. Engelbrecht, A.P. 2012. Computational intelligence an introduction (2 edn). John Wiley & Sons Ltd, West Sussex, England.

A Comparison of Fuzzy and ACO-Based Fuzzy for Classification of Bio-Medical Database S. Poongothai and S. Leoni Sharmila

Abstract This paper deals with fuzzy evolutionary algorithms which is the fast developing area in the field of machine learning and artificial intelligence. In today’s world, lots and lots of data are available for various real-life problems, where finding the relevant information from huge quantities become a challenge to the human society. This leads the way for data mining which plays an important role to extract the relevant information from vast amount of data. In this paper, deadly diseases such as lung cancer and breast cancer are considered for classification. Also, the concept of ant colony optimization (ACO) is used for variable selection, and fuzzy concept is used for classification. Based on the activities of the real ant systems, ACO is developed by Marco Dorigo in the year 1992, and it is utilized in solving many complex problems of optimization. It is one of the types of evolutionary computations. Its main aim is to identify the shortest distance between the source of food and its nest. The behavior of ants that deposits their incense on the top layer of ground to make a desired path so that other members of the colony should follow the path. Using the concept of fuzzy unordered rule induction algorithm (FURIA), classification has been done, and the results are compared with and without hybridization. Lung cancer dataset consists of 57 variables. Fuzzy gives classification accuracy as 75% with total variables. By ACO, 57 variables are reduced to five variables, and fuzzy gives classification after reduction of variables is 84.37%. Similarly, breast cancer consists of ten variables, and fuzzy classification gives accuracy as 73.07% with total variables. ACO reduced ten variables to three variables, and fuzzy gives classification after reduction of variables is 75.17%. Hence, the hybridization concept consumed less time and also less cost. By finding the relevant variables, identifying the disease at earlier stage becomes easier, and curing the disease becomes faster. Keywords Fuzzy logic · Evolutionary algorithms · Ant colony optimization · Hybridization S. Poongothai (B) · S. L. Sharmila Department of Mathematics, SRMIST, Ramapuram, Chennai, India e-mail: [email protected] S. L. Sharmila e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_11

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1 Introduction The process of finding the relevant information from large set of data is termed as data mining [1]. It includes Web, information repositories, and databases. In this paper, using ant colony optimization, relevant variables are identified and classified using the fuzzy concept for the datasets lung cancer and breast cancer which are taken from UCI machine learning repository. Also, normal classification and hybrid classification are compared in which the hybridization classification shows the improvement in accuracy and consumption of less time. The results shows that hybridization methods yield the fruitful results.

2 Machine Learning Methods Machine learning is the process of creating programs or algorithms that is used to extract relevant information from high-dimensional data automatically. The basic concepts of machine learning are data, a model, and the learning [2]. For the sake of finding useful information from the large set of data, ML needs to design models that creates data similar to the dataset considered. A model is trained to learn the behavior of the dataset if its performance improves for the particular task after the dataset is taken into account [3]. By learning process, it leads to the way to find related useful information and patterns from large dataset by optimizing the model parameters.

2.1 Fuzzy Logic In 1965, Lotfi Zadeh found that unlike machines, manual decision-making requires various intermediate possibilities between YES and NO, such as surely yes, maybe yes, neither yes or no, maybe no, surely no which gave the origin of fuzzy logic [4]. Fuzzy logic is useful for both commercial and practical purposes. Machines and consumer products can be controlled by fuzzy logic. It may accept reasoning which is not exactly right but appropriate.

2.2 Evolutionary Algorithms In 1960, Rechenberg introduced the concept of evolutionary algorithms. EAs are developed based on the Darwinian model [5], and it is a type of stochastic optimization algorithm, using the principle of “the survival of fittest.” Major operators of EAs are selection, crossover, and mutation [6]. Initially, members in population are selected at random using the Roulette wheel process in which each member has

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survival capability for the next generation [7]. New generation is obtained in which some members are undergoing crossover process from the present generation. Mutation is the process of changing the member randomly in the recent population [8]. The term population, group of individuals, is defined as 

x1 , x2 , . . . , x N p



(1)

where x i is the ith member and Np is the total population size. EAs represent the chromosome in the form of strings. Each x comprises a string of parameters x i , called genes, that is, {x1 , x2 , . . . , xn }

(2)

where n is the number of genes totally present in the string of chromosome. Let f (x) be the fitness or objective function. The group of all genotypes with their corresponding fitness values is termed as landscape [9].

2.3 Ant Colony Optimization (ACO) Marco Dorigo introduced the idea of ant colony optimization (ACO) in 1992 [10]. It was also called as ant systems. Based on the activities of the real ant systems, ACO is developed, and it is utilized in solving many complex problems of optimization. It is also a type of evolutionary computations [11, 12]. Consider the set F of n variables, identify the subset S, which comprises of m factors (m < n, S ⊂ F ) to improve classification accuracy [13]. Consider n variables that constitutes the real set, F = { f1 , f2 , . . . , fn }

(3)

Let T i be the trail of incense intensity corresponding to the variable feature f i . For each case j, a list containing the subset of selected function, S j = {s1 , s2 , . . . , sm }

(4)

“Each sample must randomly select a function subset of m variable features in the first iteration. Only, the best k subsets, k < na, will be used to update the pheromone test and affect the next iteration’s function subsets. After first iteration, each sample starts with m–p variables selected at random k-best subsets from previous generation, where p is number ranging from 1 to m − 1.” By continuing this process repeatedly, the variables which are having capability to survive for next generation form the k subsets. Selection (Sel) is measured by the equation

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s Seli j

=

⎧ ⎨ ⎩

sj

(Ti )η (L i )k

sj η k g∈S j (Ti ) (L i )

0, otherwise

⎫ , if i ∈ / Sj ⎬ ⎭

(5)

s

where L i j is considered as the local significant value of the variable f i in the subset s S j . L i j is defined as ⎡ s Li j

= l(C;

f j )⎣

⎤ 2

 − 1⎦ Sj −α D 1+e i

(6)

where s



 H ( f i ) − l( f i , f s ) f i∈S j H ( fi )   γ   l(Ci ( f i , f s )) 1   β × s j  l(Ci , f i ) + l(Ci , f s ) f

Di j = min

i∈S j

and α, β, γ are referred as constants.

3 Motivational Database The database of lung cancer and breast cancer has been taken from UCI [15] for this paper. The dataset of lung cancer has 32 instances with 57 variables [16], whereas the dataset of breast cancer has 10 variables and 286 instances.

4 Methods and Results 4.1 Fuzzy Classification Fuzzy unordered rule induction algorithm (FFURIA) shows its excellence by its expert act in defining and differentiating the class from other classes. Hence, class orders are not relevant [17, 18]. Based on RIPPER algorithm, this algorithm has developed which extends the pruning technology for generating the successor and the revision rule. Classification matrix is formed from the four outcomes. Using metrics like (i) sensitivity (ii) specificity, and (iii) accuracy, classification can be done for the required dataset [19].

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By FURIA, lung cancer and breast cancer with the full set of variables are classified, and they are shown in Figs. 1 and 2, respectively. For Lung cancer, with 32 cases and 57 variables, Accuracy = 75% Sensitivity = 76% Specificity = 79.59% For Breast cancer, with 286 cases and 10 variables, Accuracy = 73.07% Sensitivity = 74.8% Specificity = 61.11%

Fig. 1 Classification of lung cancer dataset by fuzzy

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Fig. 2 Classification of breast cancer dataset by fuzzy

4.2 Variables Reduction by ACO Applying the method of ACO in WEKA tool, for lung cancer, 57 variables are reduced to five variables which is shown in Fig. 3. Also, for breast cancer, 10 variables are reduced to three variables that is shown in Fig. 4.

4.3 Classification of ACO-based Fuzzy After reduction of variables by ACO, the relevant variables act as an input for FURIA and classified. ACO-based fuzzy classifies lung cancer and gives accuracy rate as 84.38%. It is given in Fig. 5. Also, it gives classification accuracy for breast cancer as 75.17%, and it is given in Fig. 6. By summarizing the above results, it is obviously seen that the classification rate has increased with reduced number of attributes. Table 1 shows the classification results of lung cancer and breast cancer by fuzzy and ACO-fuzzy. Figure 7 shows the comparison of classification of datasets using fuzzy and ACO-fuzzy.

A Comparison of Fuzzy and ACO-Based Fuzzy …

Fig. 3 Variable selection of lung cancer by ACO

Fig. 4 Variable selection of breast cancer data by ACO

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Fig. 5 Classification of lung cancer by ACO-based fuzzy

5 Conclusions Overall, it has been shown that instead of usual classifications, hybrid techniques play better role in classifying the dataset. So, hybrid techniques should be adopted to improve the accuracy rate. By reducing the variables of dataset using evolutionary algorithms-inspired ACO, time is getting reduced as well as cost-effective and also yields better classification results. Further, work is needed to extend fuzzification to other machine learning techniques, and also, more work is needed to formulate hybridization models to a high-dimension databases.

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Fig. 6 Classification of breast cancer by ACO-based fuzzy Table 1 Classification of lung cancer and breast cancer by fuzzy and ACO-based fuzzy Dataset

Total variables

Reduced variables by ACO

Fuzzy (%)

ACO-fuzzy (%)

Lung cancer

57

5

75

84.37

Breast cancer

10

3

73.07

75.17

Fig. 7 Comparison of classification of datasets using fuzzy and ACO-fuzzy

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References 1. Tan, P.N., M. Steinbach, and V. Kumar. 2004. Introduction to data mining. New York: AddisonVesley. 2. Cai, D., C. Zhang, and X. He. 2010. Unsupervised feature selection for multi-cluster data. In Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, pp. 333–342. 3. Yang, Y., H.T. Shen, Z. Ma, Z. Huang, and X. Zhou. 1-norm Regularized Discriminative Feature Selection for Unsupervised Learning. IJCAI. 4. Dehghan, M., B. Hashemi, and M. Ghatee. 2006. Computational Methods for Solving Fully Fuzzy Linear Systems, Applied Mathematics and Computation, vol. 179, pp. 328–343. 5. Fraser, A.S. 1958. Monte Carlo analyses of genetic models. Nature 181: 208–209. 6. Back, T., and H.P. Schwefel. 1993. An Overview of Evolutionary Algorithms or Parameter Optimization, Evolutionary Computation, vol. 1, pp. 1–23. 7. Poongothai, S., C. Dharuman, and P. Venkatesan. 2019. A comparative study of hybrid evolutionary based algorithms with machine learning classifiers for the prediction of medical database. Journal of Physics: Conference Series 1377: 012026. https://doi.org/10.1088/17426596/1377/1/012026. 8. Back, T. (1996). Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms. Oxford University Press. 9. Holland, J. 1975. Adaptation in Natural and Artificial Systems. Ann Arbor: The University of Michigan Press. 10. Liu, L., Y. Dai, and J. Gao. 2014. Ant colony optimization algorithm for continuous domains based on position distribution model of ant colony foraging. The Scientific World Journal 428539: 9. 11. Dorigo, M., V. Maniezzo, and A. Colorni. 1996. Ant system: Optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics 26: 29–41. 12. Parpinelli, R.S., H.S. Lopes, and A.A. Freitas. 2002. Data mining with an ant colony optimization algorithm. IEEE Transactions on Evolutionary Computation 6: 321–332. 13. Stutzle, T., and M. Dorigo. 2002. The ant colony optimization metaheuristic: Algorithms, applications, and advances, In Handbook of Metaheuristics, ed. F. Glover, and G. Kochenberger. Kluwer Academic Publishers, Norwell, MA. 14. Jinyan, Li., H. Liu, J.R. Downing, A.E.J. Yeoh, and W. Limsoon 2003. Simple rules underlying gene expression profiles of more than six subtypes of acute lymphoblastic leukemia (ALL) patients. Bioinformatics 19: 71. 15. UCI Repository of Machine Learning Databases. University of California, Irvine. 1998. http:// www.ics.uci.edu/~mlearn/MLRepository.html. 16. Saleem Durai, M.A, and N.C.S.N. Iyengar. 2010. Effective analysis and diagnosis of lung cancer using fuzzy rules. International Journal of Engineering Science and Technology 2: 2102–2108. 17. Hall, P., B.U. Park, and R.J. Samworth. 2008. Choice of neighbor order in nearest-neighbor classification. The Annals of Statistics 36: 2135–2152. 18. Poongothai, S., C. Dharuman, and P. Venkatesan. 2019. A comparative study of hybrid evolutionary based algorithms with machine learning classifiers for the prediction of medical database. Journal of Physics: Conference Series 1377: 1–7. 19. del Jesus, M., F. Hoffmann, L. Navascues, and L. Sanchez. 2004. Induction of fuzzy rule-based classifiers with evolutionary boosting algorithms. IEEE Transactions on Fuzzy Systems 12: 296–308. 20. Stutzle, T., and M. Dorigo. 2002. The ant colony optimization metaheuristic: Algorithms, applications, and advances. In Handbook of Metaheuristics, ed. F. Glover, and G. Kochenberger. Kluwer Academic Publishers, Norwell, MA.

Secret Information Sharing Using Probability and Bilinear Transformation Kala Raja Mohan, Suresh Rasappan, Regan Murugesan, Sathish Kumar Kumaravel, and Ahmed A. Elngar

Abstract Information security is very much important in this Internet world, especially in electronic communications such as system security, smart card, mobile communications. Cryptography is based on transformation of multiple rounds of transformation of messages in the form of plain text as input into encrypted text message. Through suitable mathematical technique, secrecy of the information is maintained. This paper proposes a cryptographic technique using probability and bilinear transformation for encryption and decryption of a message. The algorithm for encryption and decryption is given. The probability concept is employed to secure the key between the communicator and recipient. The methodologies are used to get a safe communication between communicator and recipient. The bilinear transformation gives more secure for the process in key transformation. The bilinear transformation is used to encrypt the message. The inverse bilinear transformation is used to decrypt the message. The example is presented to validate the theory part. Keywords Cryptography · Data encryption · Decryption · Bilinear transformation · Probability

K. R. Mohan (B) · R. Murugesan · S. K. Kumaravel Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamilnadu 600062, India e-mail: [email protected] S. Rasappan Department of Mathematics, University of Technology and Applied Sciences-Ibri, Ibri, Sultanate of Oman 466, 516 A. A. Elngar Faculty of Computers and Artificial Intelligence, Beni-Suef University, Beni-Suef City, Egypt © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_12

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1 Introduction In Universe, computer networks, Internet and mobile communications are more important and unavoidable part of our society, so that information security is obviously required to protect from hackers. One of the widely used approaches for information security is cryptography [1–4]. The mathematics of encryption, plays a major role in many fields. The main goal of cryptography is to ensure the secret communication between two individuals. Encryption is the process of obscuring information to make it unreadable without special knowledge. A new cryptographic technique applying probability and bilinear transformation is carried out. Bilinear transformation find its application in many fields. In cryptography also, it plays a significant role [5–10]. The concept of probability is included in this method, which acts as a secured key between the sender and the receiver. In this paper, Sect. 2 describes the standard definitions applied in this crypto analysis. Section 3 presents the algorithm for encryption. Section 4 demonstrates the algorithm for decryption [11–13]. The coding table applied for this cryptographic analysis is given in Sect. 5. Encryption process explained in Sect. 3 is demonstrated with an example in Sect. 6. Section 7 demonstrates the decryption process given in Sect. 4 with an example. Section 8 is about the conclusion followed by references.

2 Standard Definitions The following standard definitions are applied in the cryptographic analysis in this paper. 2.1 Plain Text: Plain text is the data which can be directly read by any person. It is the information which has to be shared secretly to the receiver. 2.2 Cipher Text: The transformed form of plain text, which can be read only using the key specified by the sender is called the cipher text [14, 15]. 2.3 Cipher: Cipher refers to the algorithm through which the plain text is transformed to cipher text. 2.4 Encryption: Encryption is the process in which the given information is converted into secret message. This hides the original information reaching unauthorized persons, using the secret key. 2.5 Decryption: Decryption is the reverse process of encryption. This is the process in which the encrypted text gets converted into original text with the usage of the secret key. a2 T = 2.6 Bilinear Transformation: Bilinear transformation is given by T = aa13 B+ B+ a4 a1 B+ a2 . This is applied in the encryption process of this paper. a3 B+ a4 2.7 Inverse Bilinear Transformation: Inverse bilinear transformation is given by a4 T a4 T .B = aa23 − . Decryption process of this process uses this inverse B = aa23 − T − a1 T − a1 transformation.

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2.8 Modulo Operator: The modulo operator gives the remainder value when one number is divided by another number. 2.9 Probability: Probability of the event is the chance of the event to happen. Probability =

number of favourable outcomes total number of outcomes

The concept of probability is applied in framing the bilinear transformation in this paper.

3 Collection of Raw Data The step by step procedure to be followed in encryption are as follows. Step 1: Assign numerical values to alphabets, space and full stop. a2 a2 Step 2: Apply Bilinear Transformation T = aa13 B+ .T = aa13 B+ . Here, B+ a4 B+ a4 a1 , a2 , a3 a1 , a2 , a3 and a4 a4 are chosen using probability method. It can be chosen based on the willingness of both the sender and the receiver which can help to maintain secrecy. Step 3: To each numerical value in step 1, find its equivalent T value. Step 4:Multiply the values of T by 100,000. Step 5: To the values in obtained in step 4, find modulo 54 and its equivalent key values K n .K n . Step 6: Cipher text is obtained from the equivalent encrypted codes of values in step 5.

4 Algorithm for Decryption Decryption is the reverse process of encryption. In this stage the cipher text gets converted to plain text. The process of decryption has the following steps to be performed. The step by step procedure to be followed in encryption are as follows. Step 1: Using each values of key K n K n and the corresponding values of K n , K n , are obtained. Step 2: qn = K n ∗ 54 + E n qn = K n ∗ 54 + E n Step 3: Divide each qn qn by 100,000. a4 T a4 T .B = aa23 − . Step 4: Inverse Bilinear Transformation B = aa23 − T − a1 T − a1 Step 5: With the help of the coding table, the plain text is obtained from the values of B.B.

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5 Coding Table Used for Cyptographic Analysis

A

B

C

D

E

F

1

2

S

T

G

H

3

4

5

6

7

8

U

V

W X

Y

Z

I

J

K

Q

R

9

10 11 12 13 14 15 16 17

18

a

b

j

c

L d

M N e

O

f

g

P h

i

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 k

l

m

n

o

p

q

r

s

T

u

v

w

x

y

Z

36

Space Full Stop

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

0

6 Encryption—Worked Example The encryption process applied here is explained using the word ‘Mathematics’. Step 1: To each letter of the word ‘Mathematics’ the corresponding code is obtained using the code table mentioned in Sect. 5. M

a

t

h

e

m

a

t

i

c

s

13

27

46

34

31

39

27

46

35

29

45

0

Step 2: For choosing the values of a1 , a2 , a3 a1 , a2 , a3 and a4 a4 , the method of probability is applied. In this example, the experiment of tossing two coins is considered. The possible outcomes are as follows: S = {H H, T T, H T, T H }S = {H H, T T, H T, T H } The values of a1 , a2 , a3 a1 , a2 , a3 and a4 a4 are chosen as follows: 2 = 0.5 4 1 a2 = the probability of getting 2 head = = 0.25 4 1 a3 = the probability of getting no head = = 0.25 4 3 a4 = the probability of getting atleast 1 head = = 0.75 4 a1 = the probability of getting 1 head =

Thus, T T is obtained as, T =

0.5B + 0.25 0.5B + 0.25 T = 0.25B + 0.75 0.25B + 0.75

Secret Information Sharing Using Probability and Bilinear …

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Step 3: Assuming the values obtained in step 1 as B B, the values of T T are obtained and listed in the following table. B

13

27

46

34

31

39

T

1.6875

1.83333

1.89796

1.86486

1.85294

1.88095

B

27

46

35

29

45

0

T

1.83333

1.89796

1.86842

1.84375

1.89583

0.3333333

Step 4: Each value of T T is multiplied by 100,000 and assigned to E n E n . T

1.6875

1.83333

1.89796

1.86486

1.85294

1.88095

En En

168750

183333

189796

186486

185294

188095

T

1.83333

1.89796

1.86842

1.84375

1.89583

0.333333

En En

183333

189796

186840

184375

189583

33333

Step 5: Now, to each E n E n modulo 54 is obtained and assigned asCn Cn . Also, the corresponding K n K n is obtained. En En

168750

183333

189796

186486

185294

188095

Cn Cn

0

3

40

24

20

13

Kn Kn

3125

3395

3514

3453

3431

3483 33333

En En

183333

189796

186840

184375

189583

Cn Cn

3

40

0

19

43

15

Kn Kn

3395

3514

3460

3414

3510

617

Step 6: With the E n E n values obtained, the corresponding cipher text is obtained using the coding table. 0

3

40

24

20

13

3

40

0

19

43

15

.

C

n

X

T

M

C

n

.

S

q

O

The plain text ‘Mathematics’ is now converted to the cipher text ‘.CnXTMCn.SqO’. The coding table, the key values K n K n and the probability values a1 , a2 , a3 a1 , a2 , a3 and a4 a4 chosen for framing bilinear transformation are shared only between the sender and the receiver.

7 Decryption—Worked Example The decryption process is explained with the same cipher text ‘.CnXTMCn.SqO’ as follows.

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Step 1: Using the coding table, the numerical values corresponding to ‘.CnXTMCn.SqO’ are obtained. .

C

n

X

T

M

C

n

.

S

q

O

0

3

40

24

20

13

3

40

0

19

43

15

Step 2: With the values of Cn Cn and K n K n , the values of E n E n are obtained using the relation E n = K n ∗ 54 + Cn E n = K n ∗ 54 + Cn . Cn Cn

0

3

40

24

20

13

Kn Kn

3125

3395

3514

3453

3431

3483 188095

En En

168750

183333

189796

186486

185294

Cn Cn

3

40

0

19

43

15

Kn Kn

3395

3514

3460

3414

3510

617

En En

183333

189796

186840

184375

189583

33333

Step 3: Divide each value of E n E n by 100,000 which gives the values of T T . En En

168750

183333

189796

186486

185294

188095

TT

1.6875

1.83333

1.89796

1.86486

1.85294

1.88095

En En

183333

189796

186840

184375

189583

33333

TT

1.83333

1.89796

1.86842

1.84375

1.89583

0.333333

Step 4: The inverse bilinear transformation related to T T is B=

0.25 − 0.75T 0.25 − 0.75T B= 0.25T − 0.5 0.25T − 0.5

Using the values of T T , the values of B B are obtained as listed below. TT

1.6875

1.83333

1.89796

1.86486

1.85294

1.88095

BB

13

27

46

34

31

39

TT

1.83333

1.89796

1.86842

1.84375

1.89583

0.3333333

BB

27

46

35

29

45

0

Step 5: With the help of the coding table, the plain text is obtained from the values of B B as follows. 13

27

46

34

31

39

27

46

35

29

45

0

M

a

t

h

e

m

a

t

i

c

s

.

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8 Conclusion A new cryptographic algorithm applying probability and bilinear transformation has been proposed in this paper. The plain text ‘Mathematics’ has been converted to cipher text using the proposed encryption algorithm. Also its reverse process using decryption algorithm has been furnished to get the plain text. This paper proposes a cryptographic technique using probability and bilinear transformation for encryption and decryption of a message. The algorithm for encryption and decryption is expressed in detail. The probability concept is used to secure the key between the communicator and recipient. The advantage of this method is that the process carries out protection at two stages one at choosing probability and the other at finding the key points. The methodology is useful to get a safe communication between communicator and recipient. The bilinear transformation gives more secure for the process in key transformation. The bilinear transformation is used to encrypt the message. The inverse bilinear transformation is used to decrypt the message. The example is presented to validate the theory part.

References 1. Hiwarekar, A.P. 2014. New mathematical modeling for cryptography. Journal of Information Assurance and Security, MIR Lab USA 9: 027–033. 2. Genço˘glu, M.T. 2017. Cryptanalysis of a new method of cryptography using laplace transform hyperbolic functions. Communications in Mathematics and Applications 8 (2): 183–189. 3. Hiwarekar, A.P. 2013. A new method of cryptography using laplace transform of hyperbolic functions. International Journal of Mathematical Archive 4 (2): 208–213. 4. Undegaonkar, Hemant K. 2019. Security in communication by using laplace transform and cryptography. International Journal of Scientific & Technology Research 8 (12): 3207–3209. 5. Sujatha, S. 2013. Application of laplace transforms in cryptography. International Journal of Mathematical Archive 4: 67–71. 6. Jayanthi, C.H., and V. Srinivas. 2019. Mathematical modelling for cryptography using laplace transform. International Journal of Mathematics Trends and Technology 65: 10–15. 7. Nagalakshmi, G., A.C. Sekhar, and D.R. Sankar. 2020. Asymmetric key cryptography using laplace transform. International Journal of Innovative Technology and Exploring Engineering 9: 3083–3087. 8. Dhingra, S., A.A. Savalgi, and S. Jain. 2016. Laplace transformation based cryptographic technique in network security. International Journal of Computer Applications 136 (7): 0975– 8887. 9. Saha, M. 2017. Application of laplace-mellin transform for cryptography. Rai Journal of Technology Research & Innovation 5 (1): 12–17. 10. Sedeeg, A.K.H., M.M. AbdelrahimMahgoub, and M.A. SaifSaeed. 2016. An application of the new integral “Aboodh Transform” in cryptography. Pure and Applied Mathematics Journal 5 (5): 151–154. 11. Abdalla, M., J.H. An, M. Bellare, and C. Namprempre. 2008. From identification to signatures via the Fiat-Shamir transform: Necessary and sufficient conditions for security and forwardsecurity. IEEE Transactions on Information Theory 54 (8): 3631–3646. 12. Aliyu, A.A.M., and A. Olaniyan. 2010. Vigenere cipher: Trends. Review and Possible Modifications. In PiE 101: 1.

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13. Sanchez, J., R. Correa, H. Buena, S. Arias, and H. Gomez. 2016. Encryption techniques: A theoretical overview and future proposals. In 2016 Third International Conference on eDemocracy & eGovernment (ICEDEG), 60–64. IEEE. 14. Diffie, W., P.C. Van Oorschot, and M.J. Wiener. 1992. Authentication and authenticated key exchanges. Designs, Codes and Cryptography 2 (2): 107–125. 15. Chatterjee, D., J. Nath, S. Dasgupta, and A. Nath. 2011. A new symmetric key cryptography algorithm using extended MSA method: DJSA symmetric key algorithm. In Communication Systems and Network Technologies (CSNT), International Conference, 89–94.

Encryption on Graph Networks A. Meenakshi, J. Senbagamalar, and A. Neel Armstrong

Abstract Cryptography is the study of techniques for secure communication and practice. Mathematical theory and computer science practices are the tools of modern cryptography. It has been recognized that encryption and decryption mostly emerge from mathematical disciplines. In this paper, we study a new combinatorial technique for encrypting and decrypting confidential data using a labeled network model and the domination technique. The confidential number is divided in to ‘t’ divisions and are assigned on the efficient domination nodes of constructed graph network using total labeling. Finding an efficient dominating set of the given network helps us to encrypt the confidential number. Keywords Network model · Secret number · Encryption · Decryption · Labeled network

1 Introduction Require confidentiality when sent the data of transactions and security passwords and other related confidential data electronically, in this security, makes the numbers plays important role. We describe a new combinatorial technique for encrypting and decrypting the confidential number utilizing a labeled network and an efficient dominating mechanism in this research. Let G(ν, e) be a simple undirected connected network model with v nodes and e links. The set of nodes is represented by V (G); the set of links by E(G). Readers may refer [1] for further details of graphs. A set Y  ⊆ V  is a dominating set if, |N [y] ∩ Y | ≥ 1, for every vertex y in V −Y. The minimum cardinality of a dominating set in G is called the domination number of G and is denoted by γ (G). Many types of domination parameters like paired, total, equitable paired domination [5] etc.,. In particular, if for every vertex y ∈ V , |N [y] ∩ X | = 1, then the dominating set X is said to be efficient dominating set (abbreviated as EDS) A. Meenakshi (B) · J. Senbagamalar · A. N. Armstrong Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_13

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of G [3]. The shortest distance between any two nodes a and b is represented by d (a, b). Graph terminology and notation follows from [1, 7]. Rosa introduced the graph labeling concept in 1967 [6]. For types of labeling, refer [8]. Encryption by labeling was studied by Baskar Babujee [2]; we continue the studies in [4]. By total labeling, we partitioned the pin number into ‘t’ parts and decrypt the pin number using domination technique.

2 Main Results First, we construct a network model with efficient domination number ‘t’ that has say v vertices. Next, we split the confidential number or pin number into ‘t’ partitions. We label the links and nodes and encrypt the numbers on network model. The aim of this work is to divide the confidential number M into ‘t’ partitions and allocate each part to the efficient dominating node set and the links that incident on them so that ⎤ ⎡ t   ⎣ f 1 (u i ) + f 2 (u i v j )⎦ M= i=1

u i v j ∈E

where f 1 and f 2 are pair labeling on nodes and links, respectively.

2.1 Construction of Graph Network Let H = (V, E ) be a simple graph with |V (H )| = ν and |E(H )| = e. Let X = {x 1 , x 2 , x 3, …, x t } be the set of nodes, which are the members of EDS with degrees say deg(x1 ) = k1 , deg(x2 ) = k2 , deg(x3 ) = k3 , . . . , and deg(xt ) = kt where k1 , k2 , k3 , . . . , kt ≥ 1. Let x11 , x12 , . . . , x1k1 ; x21 , x22 , . . . , x2k2 ; x31 , x32 , . . . , x3k3 ; and xt1 , xt2 , . . . , xtkt be the neighboring nodes of x1 , x2 , x3 , . . . , xt , respectively, d(x1 , x1 j1 )=d(x2 , x1 j2 )=d(x3 , x3 j3 )= . . . =d(xt , xt jt )=1, 1 ≤ j1 ≤ k1 ; 1 ≤ j2 ≤  k k2 ; 1 ≤ j3 ≤ k3 ; …, 1 ≤ jt ≤ kt . Define V (H ) = {x1 , x2 , . . . , xt } ∪ x1 j 1 j11=1 ∪ k  k  x2 j 2 j22=1 ∪ . . . ∪ xt j t jtr=1 and |V (H )| = k1 + k2 + k3 + · · · + kt + t = ν.   Let E 1  = x1 j1 x2 j2 , x2 j2 x3 j3 , . . . , x(t−1) jt−1 xt jt for only one j1 , j2 , . . . , jr where 1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jr ≤ kr . Define Emin ={x1 x1 j1 , x2 x2 j2 , . . ., xt xt jt /1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jt ≤ kt } ∪E1 . Then |Emin |=k1 +k2 +· · ·+kt +t=v(say). Define Emax ={(i, j); 1≤i, j≤k1 +k2 +· · ·+kt +t; i = j}−{x1 x2 , x2 x3 , x3 x4 , . . . , xt x1 j1 , xr x2 j2 , . . . , x2 xt−1 xt , x1 x2 j2 , x1 x3 j 3 , . . . , x1 xt j t , x2 x3 j 3 , . . . , x2 xt j t , . . .,

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x(t−1) j(t−1) /1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jt ≤ kt }. Then

|E max | = β(say) = 21 ν2 + ν − 2t (1 + v) , where ν = k1 +k2 +· · ·+kt +· · ·+t. The cardinality of edge set of G satisfies, α ≤ |E(H )| ≤ β. We now provide the algorithm for constructing graph and label the vertices and edges to encrypt the confidential number M.

2.2 Encryption Algorithm: Encrypting Number M ≡ a(mod t), Where a = 0 Input: M ≥ r is the secret number, M ≡ a(mod t), where a = 0. Output: Encrypted Labeled graph H Begin Step 1:

Step 2:

Step 3:

Split the confidential number M in to r separations say M0 , M1 , M2 , . . . , Mt−1 where M0 ≡ 0(modr ), M1 ≡ 1(modr ), M2 ≡ 2(mod r ), . . . , Mt−1 ≡ (t − 1)(mod t). Fix deg(x1 ) = j1 , deg(x2 ) = j2 , . . . , deg(xt ) = jt , where j1 , j2 , . . . , jt ≥ 1. Let x11 , x12 , . . . , x1k1 ; x21 , x22 , . . . , x2k2 ; x31 , x32 , . . . , x3k3 ; and xt1 , xt2 , . . . , xtkt be the neighboring nodes of x1 , x2 , x3 , . . . , xt , respectively, d(x1 , x1 j1 ) = d(x2 , x1 j2 ) = d(x3 , x3 j3 ) = . . . = d(xt , xt jt ) = 1, 1 ≤ j1 ≤ k1 ; 1 ≤ j2 ≤ k2 ; 1 ≤ j3 ≤ k3 ;, …, 1 ≤ jt ≤ kt .  k k  Define V (H ) = {x1 , x2 , . . . , xr } ∪ x1 j 1 j11=1 ∪ x2 j 2 j22=1 ∪ . . . ∪  k t xt j t jt =1 and |V (H )| = k1 + k2 + k3 + · · · + kt + t = ν. Let   E 1  = x1 j1 x2 j2 , x2 j2 x3 j3 , . . . , x(t−1) jt−1 xt jt for only one j1 , j2 , . . . , jt where 1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jt ≤ kt .

Define Emin ={x1 x1 j1 , x2 x2 j2 , . . ., xr xr jr /1≤ j1 ≤k1 , 1≤ j2 ≤k2 , . . ., 1≤ jt ≤kt }∪E1 . Then, Define Emax = {(i, j); 1 ≤ i, j ≤ k1 + k2 + · · · + kt + t; i = j}−{x1 x2 , x2 x3 , x3 x4 , . . ., xr −1 xr , x1 x2 j2 , x1 x3 j3 , . . ., x1 xt jt , x2 x3 j3 , . . ., x2 xt jt , . . ., xr x1 j1 , xr x2 j2 , . . ., x2 x(t−1) j(t−1) /1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jt ≤ kt }. Then

|E max | = β(say) = 21 ν2 + ν − 2t (1 + v) , where ν = k1 +k2 +· · ·+kt +· · ·+t. The size of H  satisfies, α ≤ |E(H )| ≤ β. 





Step 4: y0 = Mt 0 , y1 = Mt 1 , y2 = Mt 2 , …, yt−1 = Mtt−1 . Step 5: Split y0 = s11 + s12 + · · · + s1k1 ; y1 = s21 + s22 + · · · + s2k2 ; ..., yr −1 = st1 + st2 + · · · + stkt , where k 1 , k 2 , …, k t ≥ 0. Step 6: Let f  : V ; → {(ω1 , t1 ) : ω1 ∈ {0, 1, 2, . . . t − 1}, 1 ≤ r ≤ ν} defined as f (vi ) = [(r − 1) mod t, r ], 1 ≤ r ≤ ν be a bijective function. Let g : E → {(ω2 , t2 ) : ω2 ∈ N  where N = {s11 , s12 , . . . , s1k1 ; s21 , s22 , . . . , s2k2 ;s31 , s32 , . . . , s3k3 ; . . . ; st1 , st2 , . . . , stkt and ν + 1 ≤ t2 ≤ v + e}

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such that g(x1 x1 j 1 ) = s1 j1 ; g(x2 x2 j 2 ) = s2 j2 ; . . . ; g(xt xt j t ) = st jt for 1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jt ≤ kt be an injective function. Step 7:

Define g(x1i x1 j 1 ) = s1 j1 , 1 ≤ i ≤ t, 1 ≤ j1 ≤ k1 ;

g(x2i x2 j 2 ) = s2 j2 , 1 ≤ i ≤ r, 1 ≤ j2 ≤ k2 ; . . ., g(xri xt j t ) = st jt , 1 ≤ i ≤ t, 1 ≤ jt ≤ kt ; and i = j g(x1i x2 j ) = s1i , 1 ≤ i ≤ t, 1 ≤ j ≤ k2 ; g(x2i x3 j ) = s2i , 1 ≤ i ≤ t, 1 ≤ j ≤ k3 ; …; g(x(t−1)i xt j t ) = s(t−1)it , 1 ≤ i ≤ t, 1 ≤ j ≤ kt ; and i = j. End

2.3 Decryption Algorithm Input: Encrypted Labeled graph H Output: M, the Encrypted Number Begin Step 1: Step 2:

Identify the efficient dominating set S = {x1 , x2 , . . . , xt } such that M[x1 ] ∩ M[x2 ] ∩ . . . M[xt ] = ϕ. k1

g(x1 x1i ) M0 = f (x1 ) + t i=1

M1 = f (x2 ) + t Step 3:

M=

t

k2

i=1

g(x2 x2i ), . . . ,Mt−1 = f (xt ) + t

kr

g(xt xti )

i=1

Mi .

i=0

End

2.4 Encrypting Number M  ≡ a (mod t), Where a  = 0 If the confidential number M is not a multiple of ‘t,’ then let M = A + B where

A = t Mt and B = M ( mod t). In this case, add a new pendent vertex t m+1 adjacent to x r such that d (x r , t m+1 ) = 1 and assign g (x 5 t m+1 ) = B.

2.5 Illustration of Encryption and Decryption Algorithm Here, we given one of the example of split the confidential number M into three separations and assign each part in the efficient dominating node set and the links incident on them so that

Encryption on Graph Networks

M=

127 3 

⎡ ⎣ f (u i ) +



⎤ g(u i v j )⎦

u i v j ∈E

i=1

where f and g are pair labeling on nodes and links, respectively.

2.5.1

Encrypting the Number 1156

To encrypt the confidential number 1156 which is not a multiple of 3, let M = A + B where A = 1155, B = 1. Split the number A = 1155 into three partitions say M 0 , M 1 , and M 2 , namely M 0 = 393 ≡ 0 (mod 3), M 1 = 385 ≡ 1 (mod 3), and M 2 = 377≡ 2 (mod 3). Construct the graph G with |V (G)| = 20, |E(G)| = 36 and its efficient dominating set S = { x 1 , x 2 , x 3 } is shown in fig(i). Arbitrarily fix deg(x 1 ) = 6, deg(x 2 ) = 5, deg(x 3 ) = 5. The neighbors of x 1 , x 2 , and

x 3 are ai , bj , and ck where 1 ≤ i ≤ 6; 1 ≤ j ≤ 5 and 1 ≤ k ≤ 5. Fix y0 = M30 = 131, y1 = M31 = 128,

y2 = M32 = 125. Split y0 = 131 = 22 + 22 + 22 + 22 + 22 + 21 (where p1 = 22, p2 = 22, p3 = 22, p4 = 22, p5 = 22 and p6 = 21). Split y1 = 260 = 21 + 33 + 30 + 22 + 22 (where q1 = 21, q2 = 33, q3 = 30, q4 = 22 and q5 = 22). Split y2 = 125 = 21 + 30 + 30 + 22 + 22 (where r1 = 21, r2 = 30, r3 = 30 , r4 = 22 and r5 = 22 ). {(γ , i) : γ  ∈ {0, 1, 2}, 1 ≤ i ≤ 20} defined Let f : V → ≤ 19. Define N = as f (vi ) = [(i − 1) mod 3, i], 1 ≤ i : E → { p1 , p2 , . . . , p6 , q1 , q2 , . . . q5 , r1 , r2 , . . . r4 , r5 }. Let g {(δ, j) : δ ∈ N , 20 ≤ j ≤ 56} such that g(x1 ai ) = pi ; g(x2 b j ) = q j ; g(x3 ck ) = rk ; for 1 ≤ i ≤ 6, 1 ≤ j ≤ 5, 1 ≤ k ≤ 5. g(ai b j ) = pi for 1 ≤ i ≤ 6, 1 ≤ j ≤ 5, g(b j ck ) = q j for 1 ≤ j ≤ 5, 1 ≤ k ≤ 5, g(ck ai ) = rk for 1 ≤ k ≤ 5, 1 ≤ i ≤ 6, g(ai a j ) = pi for 1 ≤ i, j ≤ 6; i = j, g(bi b j ) = qi for 1 ≤ i, j ≤ 5; i = j, g(ci c j ) = ri for 1 ≤ i, j ≤ 5; i = j.

2.5.2 Step1: Step 2:

Decrypting the Number 1156 Identify the efficient dominating set S = {x1 , x2 , x3 } such that N [x1 ] ∩ N [x2 ] ∩ N [x3 ] = ϕ. k1

g(x1 ai ) = 0 + 3(22 + 22 + 22 + 22 + 22 + Find M0 = f (x1 ) + 3 21) = 393. M1 = f (x2 ) + 3 M2 = f (x3 ) + 3

i=1 k2

j=1 k3

k=1

g(x2 b j ) = 1 + 3(21 + 33 + 30 + 22 + 22) = 385. g(x3 ck ) = 2 + 3(21 + 30 + 30 + 22 + 22) = 377.

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A =

2

Mi = 393 + 385 + 377 = 1155.

i=0

M = A’+B’ = 1155+1 = 1156. 2.5.3 Step1: Step 2:

Decrypting the Number 1156 Identify the efficient dominating set S = {x1 , x2 , x3 } such that N [x1 ] ∩ N [x2 ] ∩ N [x3 ] = ϕ. k1

g(x1 ai ) = 0 + 3(22 + 22 + 22 + 22 + 22 + Find M0 = f (x1 ) + 3 21) = 393. M1 = f (x2 ) + 3

i=1 k2

g(x2 b j ) = 1 + 3(21 + 33 + 30 + 22 + 22) =

j=1

385. M2 = f (x3 ) + 3

k3

g(x3 ck ) = 2 + 3(21 + 30 + 30 + 22 + 22) =

k=1

377. Step 3:

A =

2

Mi = 393 + 385 + 377 = 1155.

i=0

M = A’+B’ = 1155+1 = 1156. Labeled graph H is shown in Fig. 1.

3 Conclusion In this paper, we employed total labeling and an efficient domination strategy in networks to encrypt and decrypt the secret number using a combinatorial technique. The values assigned for s1 j1 , s2 j2 , . . . , st jt , 1 ≤ i ≤ r and 1 ≤ j1 ≤ k1 , 1 ≤ j2 ≤ k2 , . . . , 1 ≤ jt ≤ kt while splitting y0 , y1 , y2 , . . . , yt−1 repetition is possible in our method, resulting in a more intricate labeled network. Only by identifying the efficient dominating set can the encryption be broken. This research collaborates the domination and labeling techniques together and which has been used to encrypt secret number.

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Fig. 1 Labeled graph H

References 1. Bondy Murthy, J.A. 1976. Graph Theory with Applications, Elsevier science publishing co, 10017. New York N.Y: Inc. 2. Baskar Babujee, J., and S. Babitha. 2012. Encrypting and decrypting the number using labeled graphs. European Journal of scientific Research 75: 14–24. 3. Benzenken, C., and P.C. Hammer. 1978. Linear separation of dominating sets in graphs. Annals of Discrete Mathematics 3: 1–10. 4. Meenakshi, A., and J. Baskar Babujee. 2016. Encryption through Labeling using Efficient Domination. Asian Journal of Research in social science and Humanities 6 (9): 1967–1974.

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5. Meenakshi, A., and J. Baskar Babujee. 2016. Paired Equitable domination in graphs. International Journal of Pure and Applied Applied Mathematics 109 (7): 75–81. 6. Rosa, A., (1967). On certain valuation of the vertices of graph, theory of graphs, International symposium, July 1966, Gordon and Breach, N.Y. and Dunod Paris, pp.349–355 7. Teresa W. Haynes, Stephen T. Hedetnimi, and Peter J. Slater, 1998. Fundamentals of Domination in Graphs, Marcel Decker. 8. Wallis, W.D., 2001. Magic Graphs, Birkhause.

A Novel Indexing Scheme Over Lattice of Cuboids and Concept Hierarchy in Data Warehouse Saurabh Adhikari, Sourav Saha, Anjan Dutta, Anirban Mitra, and Soumya Sen

Abstract In data warehouse, lattice of cuboids is very important as it represents all the combination of dimensions for that particular business application. The size of the structure is high as for N dimensions, the total number of cuboids is 2N . If the dimensions maintain concept hierarchies, that results in more numbers of cuboids. Hence, the search time is quite high if the data warehouse maintains cuboids in the form of lattice. Here, a secondary index scheme is proposed over lattice of cuboids by analyzing the existing query set. A novel methodology is proposed to rank the dimensions based on the usage of the cuboids and corresponding dimensions. Secondary index is created based on these ranking and that improves the search time significantly. Both the case study and experimental results show the efficacy of the proposed method. Keywords Lattice of cuboids · Concept hierarchy · Indexing · Secondary index

S. Adhikari · S. Saha Swami Vivekananda University, Kolkata, India e-mail: [email protected] S. Saha e-mail: [email protected] A. Dutta Techno International NewTown, Kolkata, India A. Mitra Sister Nivedita University, Kolkata, India S. Sen (B) University of Calcutta, Kolkata, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_14

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1 Introduction Online analytical processing (OLAP) tools [1] help to perform analytical processing on multi-dimensional data. The analyzed result is useful for different types of decision-making majorly used to generate business intelligence [2]. Data warehouse is the most common tool for OLAP. It comes with an integrated architecture and tools for corporate and business organizations to develop decision support system (DSS) that enriches their business decision. A data warehouse is constructed to focus on a certain business fact or subject. The property subject-oriented signifies the main theme of the data warehouse. A data warehouse works with huge amount of heterogeneous data. These huge data are integrated into data warehouse schema through extraction, transformation, and loading (ETL) tools [3]. A data warehouse works with historical data over a long period. In order to perform a better analysis for business, time-variant data are desirable. A data warehouse is a separate data store over the transactional database. No transaction, concurrency control are performed here. Two operations that take place here are: loading of data and access of data. This feature is termed as non-volatile. A data warehouse is organized with fact table from the underlying dimension tables. A dimension can be considered as an entity that represents some useful information for business. Every dimension is represented in dimension table. A fact table consists of the primary key of the associated dimension table (in fact table, the primary keys are represented as foreign key) and one or more measures. The types of the data warehouse schema depend on how a fact table is constructed from dimension tables. In star schema, a fact table is connected to all the dimension tables. In snowflake schema, some of the dimension tables are connected to fact table, and remaining dimension tables present in the schema are having connection with the existing dimension tables. Every data warehouse schema corresponds to a multidimensional data model, and it views the data in the form of cuboid or data cube. A fact table consists of all dimensions and that corresponds to a cuboid. It has been observed that all possible combinations of dimensions are important for business, and these different combinations represent different cuboids. The structure comprising of all cuboids is knowns as lattice of cuboids. A lattice of cuboids is depicted in Fig. 1 with three dimensions: (i) A, (ii) B, and (iii) C. represents a base cuboid of a fact table consisting of three dimensions. Roll-up is applied on base cuboid to produce the 2D cuboids, then 1D cuboids, and finally the apex cuboid. Drill-down is applied to access the higher dimensional cuboid from the lower dimensional cuboids. A data warehouse having n numbers of dimensions forms a lattice of 2n number of cuboids. Hence, the cuboids grow (in terms of numbers) very fast with the addition of dimensions. Moreover, as the data warehouses deal with time-variant data of a long period, the amount of data is humongous. Managing and accessing these data are a time-consuming task. The number of cuboids increases further if one or more dimensions maintain concept hierarchy [4]. A dimension can be represented in multiple abstractions which is referred to as concept hierarchy. For example, time

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0D Cuboid/ Apex Cuboid



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1D Cuboid

Drill-down

Roll-up



2D Cuboid 3D Cuboid / Base Cuboid

Fig. 1 Three-dimensional lattice of cuboids

dimension can be represented in the forms of: (i) day, (ii) month, (iii) quarter, (iv) year, etc. This represents concept hierarchy of time dimension. The number of cuboids in lattice structure with concept hierarchy is given in Eq. 1. Total number of cuboids =

N  (L i + 1)

(1)

i=1

where L i is the number of abstraction for the given dimension i. If a dimension has no concept hierarchy, then L i = 1. For example, in Fig. 1, all the three dimensions A, B, C have no concept hierarchy; hence, L i = 1 for A, B, C. Thus, the lattice of cuboids in Fig. 1 has 2 × 2 × 2 = 8 values. Let us consider dimension A has three abstractions, B has two abstractions, and C has one abstraction (no concept hierarchy). Then, the number of cuboids will be 4 × 3 × 2 = 24. In real-life applications, these cuboids are to be accessed in real time. However, due to the higher numbers of cuboids and huge amount of data, access time is often high. Therefore, it is a challenge to access the required data from lattice of cuboids in real time. In this research work, an indexing scheme is proposed for quick access of data based on the importance of the cuboids as well as the importance of data. The paper is organized in following sections as given below. Section 2 carries out a survey on different searching mechanism on lattice of cuboids and concept hierarchy. Section 3 briefly discusses on indexing scheme. The motivation of this research work is described in Sect. 4 and that is followed by the proposed methodology in Sect. 5. In Sect. 6, a case study is carried out on real-life dataset, and in Sect. 7, complexity of the proposed methodology is computed. Section 8 is the conclusion section.

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2 Related Work In this section, a survey work is performed on different approaches used to faster the lattice of cuboids operations as well as the on the concept hierarchy. Galois connection-based efficient traversals were proposed for lattice of cuboids [5] and concept hierarchy [6]. Two operations named abstraction and concretization were performed in the Galois connection framework to optimize the traversal in the context of roll-up and drill-down. Another approach named parallel graph olap over largescale attributed graphs (Pagrol) [7] was proposed to aggregate attribute graphs at different levels and granularity. It proposed Hyper Graph Cube model for roll-up and drill-down operations that execute parallel algorithms in Map-Reduce and can work in various social networking. In another research work, threshold is imposed on the measures of cuboids to form iceberg cube and search the partitions that are important and also submerge part of the data cube [8]. This approach is also extended for different aggregate functions of data warehouse. A heuristic-based approach [9] was proposed for finding out optimal path between cuboids. The work considers size of each dimension and also the cardinality in the concerned cuboids. Probabilistic data cube [10] was proposed to address the veracity problem of data. These probabilistic cuboids summarized the aggregated values using probability mass functions (PMFS). This work addresses probabilistic cube generation, aggregation of the cuboids, materialization of the cuboids, and query evaluation. The work supported common aggregation functions, full and partial materialization of cuboids, and OLAP operations such as probabilistic slicing and dicing. In another research work [11], efficient query response time over data cube and updation were considered in terms of the recursive construction of the power set of the different dimension of fact table and a prefix tree structure for the efficient storage of cuboids. Storage and faster memory for computation are two important aspects for computing over lattice of cuboids as it manages gigantic data. In [12], the authors considered memory hierarchy to store and process the lattice of cuboids. Based on the availability of space and cost of the different faster memory, optimized path among the cuboids was identified to reduce the access time of query. The work was further extended [13] to consider concept hierarchies over the lattice of cuboids within different memory hierarchies. Some of the indexing schemes were proposed in data warehouse to access the data quickly. Bitmap join indices [14] were used to optimize the join query time between fact and dimension tables. This work reduced candidate attributes numbers to perform bitmap join. In another work, scatter bitmap index [15] was used to reduce query processing time by applying multiple scans on indexing and low-cost Boolean operations. In this survey work, different works are presented to optimize traversal on lattice of cuboids and also some of the indexing strategies on data warehouse. However, none of them addresses indexing scheme on lattice of cuboids for faster accessing.

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3 Indexing Scheme Indices are the auxiliary data structures that allow faster retrieve of tuples that satisfy certain selection criteria without doing an exhaustive search of the whole relation [16]. Indices are categorized into two types: (i) primary index and (ii) secondary index. (i)

Primary index

Primary index is applied on the data arranged in sorted order. It is generally sorted on the primary key (may not be always). Primary index consists of two fields. The first field represents the primary key attribute, and the second field is an address that stores the address of the particular data block. This index file maintains record for each block of the file that represents data. The average number of blocks (NBA ) using primary index is given by: NBA = log2 B + 1

(2)

Primary indexing is of two types—(i) dense index and (ii) sparse index. (ii)

Secondary index

Secondary index [16] is created either on the candidate key or on a non-key attribute having duplicate values. Secondary indexing is independent on the organization of data in the file. User can define multiple secondary indexes, whereas a single primary index is allowed on a table.

4 Objective The main objective of this research work is to expedite the access time of data from lattice of cuboids where the dimensions may have concept hierarchy. Dynamic indexing scheme will be proposed over the lattice of cuboids so that the faster access time could be achieved based on the nature of queries currently executed in the system. In the fact table as the primary key is composite, it is difficult to create primary indexing based on primary key. Moreover, as data warehouse works with huge amount of data, it is very much time-consuming to sort the data in sequence. Hence, primary index cannot be formed with non-primary key also. The alternative option is to create secondary index on different dimensions. A methodology will be proposed here to rank the dimensions dynamically to create the different secondary index as per requirements. Moreover, all the values of the dimensions are not required. Hence, for every selected dimension, those values which are selected will be stored in the index.

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5 Proposed Methodology The methodology is performed in two steps. Dimensions are ranked in the initial step. Based on these, one or more secondary indices are created. After these, in the next step, the important values belonging to these dimensions are identified and then loaded into the index.

5.1 Ranking of Dimensions In this step, a ranking scheme for the dimensions will be proposed. The process will start with the set of queries executed in the system. Recently executed queries are given higher priority over the older queries. The queries fetch the values from the cuboid. Weightage for that cuboid is computed based on its occurrence number in the query. Now, that weightage value is added to the existing weightage value of all the dimensions that are present in that cuboid. Once all the queries are scanned, then the dimensions are ranked in descending order of weightage. This process is formally defined in the following algorithm. Algorithm Cuboid_Dimension_Ranking Step 1: All the dimensions are uniquely numbered starting from 1 and incremented by 1. [This includes all the abstractions of the dimensions (representing concept hierarchy)]. It is denoted as DimI. (I = 1 to M), M = Total numbers of dimensions including concept hierarchy. Step 2: Initialize ∀DimI_Weight = 0 // Weight of all the dimensions are initialized as 0. Step 3: Count the number of queries in the Query Set Q. Step 4: For j = 1 to N //1 is oldest query and N is the latest query Weightage of every query is calculated as W Cub = qj /N // Cub is the cuboid from which the current query j is accessed. Step 5: ∀ dimensions of the Cuboid DimI_Weight = DimI_Weight + W Cub End of Loop (Step 5) End of Loop (Step 4) Step 6: Rank the dimensions on the descending values of DimI_Weight Step 7: End. Using algorithm Cuboid_Dimension_Ranking, different dimensions are ranked. First, secondary index will be created on the highest ranked dimension. If more secondary indices are created, then the new secondary index will be created on the subsequently ranked dimensions.

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5.2 Loading of Values in Secondary Index This is an optional step. If the size of the secondary index grows high, then this step may be executed. Secondary index is created based on the ranking of the dimension weight using the algorithmCuboid_Dimension_Ranking. Once the secondary index (one or more) is created, all the values of the corresponding dimensions are put into the secondary index and that point to all the existing cuboids that contain that particular value. If the size of the secondary index grows too high, then a thresholding mechanism can be used on the values. A threshold can be applied on every value that belongs to the cuboids. If the values accessed are more than the threshold value, then only those values are put into secondary index structure.

6 Case Study We used the same dataset and data warehouse environment as given in [9]. This dataset is available at https://archive.ics.uci.edu/ml/machine-learning-databases/003 52/, and from this, a star schema is developed with three-dimension tables: Country, Stock, and Time. The fact table comprises of the primary key of these tables, namely Country_id, Stock_code, and Time_quarter. Two measures are used, namely Total_Unit_Sold and Total_amount. The dimension Stock has the highest weightage as per the proposed algorithm. Next, higher priority is of the dimension Time_quarter. Two secondary indices are created for this application based on these dimensions. A comparative study of the execution time with index and without index for this application is depicted in Fig. 2.

ExecuƟon Time(Sec)

2.5 2 1.5 1 0.5 0 0

100000

200000

300000

400000

500000

Data Size without indexing

with indexing

Fig. 2 Execution time with indexing and without using indexing

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7 Complexity Analysis The proposed approach significantly improves the searching time over the methods that do not use any indexing scheme. If the total numbers of dimensions is M and the size of data is n, then for the first dimension, the complexity will be O(lgM (n)). If one more secondary index is created, then the complexity will be O(lg M (n)) + O(lg M−1 (n)) Even if more secondary indices are created, the complexity will be in the order of logarithmic series.

8 Conclusion Data warehouse index on the entire structure of lattice of cuboids is never proposed earlier. Moreover, this work considers concept hierarchy on dimensions. Hence, the structure is gigantic. The proposed indexing scheme improves the searching time drastically as it computes in the order of log. In this research, secondary index is applied on the cuboids based on the priority of dimensions. The work can be extended further to analyze the relationship among dimensions and to create the index based on the relationship of dimensions. Different advanced indexing schemes can be applied such as B- tree, B+ tree, and bitmap indexing to improve the performance over lattice of cuboids.

References 1. Zhang, Y., et al. 2019. Main-memory foreign key joins on advanced processors: Design and re-evaluations for OLAP workloads. Distrib Parallel Databases 37: 469–506. 2. R. Ghosh, S. Halder, and S. Sen. 2015. An integrated approach to deploy data warehouse in business intelligence environment. In Proceedings of the IEEE 3rd International Conference on Computer, Communication, Control and Information Technology (C3IT). ISBN: 978-1-47994446-0. 3. Ali, S.M.F., and R. Wrembel. 2017. From conceptual design to performance optimization of ETL workflows: Current state of research and open problems. The VLDB Journal 26 (6): 777–801. 4. Swamy, M.K., and P.K. Reddy. 2020. A model of concept hierarchy-based diverse patterns with applications to recommender system. International Journal of Data Science and Analytics 10 (2): 177–191. 5. Sen, S., N. Chaki, and A. Cortesi. 2009. Optimal space and time complexity analysis on the lattice of cuboids using Galois connections for data warehousing. In 2009 Fourth International Conference on Computer Sciences and Convergence Information Technology, 1271–1275. IEEE.

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6. Sen, S., and N. Chaki. 2011. Efficient traversal in data warehouse based on concept hierarchy using Galois connections. In 2011 Second International Conference on Emerging Applications of Information Technology, 335–339. IEEE. 7. Wang, Z., Q. Fan, H. Wang, K.L. Tan, D. Agrawal, and A. El Abbadi. 2014. Pagrol: Parallel graph olap over large-scale attributed graphs. In 2014 IEEE 30th International Conference on Data Engineering, 496–507. IEEE. 8. Phan-Luong, V. 2017. Searching data cube for submerging and emerging cuboids. In 2017 IEEE 31st International Conference on Advanced Information Networking and Applications (AINA), 586–593. IEEE. 9. S. Roy, S. Sen, and N.C. Debnath. 2018. Optimal query path selection in lattice of cuboids using novel heuristic search algorithm. In 33rd International Conference on Computers and Their Applications (CATA), 134–139. ISBN: 978-1-5108-5867-1. 10. Xie, X., K. Zou, X. Hao, T.B. Pedersen, P. Jin, and W. Yang. 2019. Olap over probabilistic data cubes ii: Parallel materialization and extended aggregates. IEEE Transactions on Knowledge and Data Engineering 32 (10): 1966–1981. 11. Phan-Luong, V. 2016. A data cube representation for efficient querying and updating. In 2016 International Conference on Computational Science and Computational Intelligence (CSCI), 415–420. IEEE. 12. Roy, S., S. Sen, A. Sarkar, N. Chaki, and N.C. Debnath. 2013. Dynamic query path selection from lattice of cuboids using memory hierarchy. In 2013 IEEE Symposium on Computers and Communications (ISCC), 000049–000054. IEEE. 13. Sen, S., S. Roy, A. Sarkar, N. Chaki, and N.C. Debnath. 2014. Dynamic discovery of query path on the lattice of cuboids using hierarchical data granularity and storage hierarchy. Journal of Computational Science 5 (4): 675–683. 14. An, H.G., and Koh, J.J. 2012. A study on the selection of bitmap join index using data mining techniques. In 2012 7th International Forum on Strategic Technology (IFOST), 1–5. IEEE. 15. Weahama, W., S. Vanichayobon, and J. Manfuekphan. 2009. Using data clustering to optimize scatter bitmap index for membership queries. In 2009 International Conference on Computer and Automation Engineering, 174–178. IEEE. 16. Choenni, S., H. Blanken, and T. Chang. 1993. Index selection in relational databases. In Proceedings of ICCI’93: 5th International Conference on Computing and Information, 491–496. IEEE.

Modeling Interactive E-book: Computational Perspective and Design Principles Nguyen Tung Lam, Vu Minh Trang, Nguyen Hoa Huy, and Ton Quang Cuong

Abstract In the current process of digital transformation in education, teaching is increasingly shaped by highly contextualization and personalization. Electronic books (E-books) have created, space, context, opportunities for teaching and learning activities, interactive and adaptive experiences to meet the diverse learning needs of learners. The building and application of interactive multimedia models in interactive E-book design to activate the personalized learning process of learners are an urgent issue in digital teaching today. The study analyzed and evaluated the feasibility of interactive E-books in contextualization and personalization with the aim of ensuring opportunities to access learning, form and develop competencies for current learners. Keywords Personalized teaching · E-books · Capacity · Digital learning

1 Introduction The current trend of digital transformation in the education industry in Vietnam has been bringing educational technology to gradually become popular, diffuse, integrated and deeply penetrated all elements of the teaching process. Therefore, personalized teaching has been more appreciated and interesting than ever. Because, each student is a “special individual” with different intellectual, cognitive, manipulative skills, learning styles in a collective, learning community and has his own learning N. T. Lam (B) · V. M. Trang · T. Q. Cuong VNU-University of Education, Vietnam National University, Hanoi, Vietnam e-mail: [email protected] V. M. Trang e-mail: [email protected] T. Q. Cuong e-mail: [email protected] N. H. Huy VNU-Center for Education Accreditation, Vietnam National University, Hanoi, Vietnam e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_15

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style to practice achieving the overall goal of the training program. The problem for educators is how to “contextualize” the learning process to suit individual learners. Bloom [1] pointed out that every student can reach his full potential if provided and facilitated before, during and after the learning process; teaching is not about comparing one student to another, but about creating the context in which students can achieve the goals of the educational program [2]. In particular, when students learn in the form of “one-on-one” to meet the requirements of style, needs and learning motivation suitable for students combined with regular assessment, providing opportunities feedback and continuous adjustment guidance, they will improve their cognitive processes, thereby enhancing their cognitive processes and their learning outcomes. On the other hand, the teacher himself will adjust the content, teaching methods and apply educational technology to ensure that the individual’s learning process can be done thoroughly and completely [3]. Since then, the application of educational technology will revolutionize education, “reconfigure” the teaching process in the direction of expanding learning space and resources and provide optimized solutions for collaborating and connecting learning activities, enhancing informal and non-formal learning opportunities and focusing more effectively on individual learners. One of the technology solutions that contribute to the creation of innovative educational products, knowledge and skills is the application of interactive E-books which strongly supports and promotes learning activities so that learning becomes a regular, ongoing, anytime, anywhere process toward lifelong learning [4]. In addition, interactive E-books are also predicted to thrive with a wide range of beneficiaries across all learners’ geographic regions, ages and socioeconomic conditions. Therefore, the report will focus on proposing an interactive multimedia model in E-book design used for personalized teaching and outlining the E-book design process with outstanding interactivities and its application in the design of E-books for teaching general chapters on metals, Grade 12 (K-12) chemistry at high schools in Vietnam.

2 Theoretical Frameworks 2.1 The Trend of Technology Integration in Personalized Teaching In the past few years, there has been a strong transformation from the traditional method of test-based education to the orientation of personalized and interactive teaching. Global education is moving toward learner-centered and inclusive learning based on technology (to better meet learners with special needs). Personalize the “adaptive factors” in learners’ actions (if the learner’s activities tend to be type A, then the adaptive activity factors will be B). According to this trend, the personalization process is based on the following bases: actions and behaviors of learners in different learning spaces (e.g., what actions and behaviors learners have

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performed, how long, how efficient to complete a learning task); skills, capacity to implement the learning experience (e.g., what skills the learner has acquired, the level of effort and concentration to solve the problem, the ability to collaborate and create in accomplishing learning goals).

2.2 Integrated Interactive Multimedia Model (INTERACT) See Fig. 1. Model INTERACT-XP To examine the theory of multimedia learning in interactive E-books, the study adopted the interactive multimedia model (INTERACT). This can help provide research information and build learning components required to design and develop interactive E-books from which the design is made into the INTERACT-XP model. E-book is an effective medium to support many different purposes, especially to support personalized learning. When designing an E-book, it is necessary to fully based on the following criteria: learning environment, behavioral activities, cognitive and metacognitive activities, motivation and emotions, learner-teacher variables and mental models of learners. The interaction process is represented by the feedback loops connecting these components (Fig. 2) [6]. Indeed, research suggests that E-book teachers or designers should follow the principles of interactive multimedia learning during E-book design. From the perspective of INTERACT, learning environments that integrate feedback loops that connect learning systems, behavioral processes and cognitive processes can activate students’ learning motivation and engage them in cognitive activity. For example, an E-book can provide utilities such as video, audio, images,

Fig. 1 Integrated model of multimedia interactivity (INTERACT) [5]

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Fig. 2 Modeling impact of personalization on INTERACT

audio recordings and users’ interactivity that improves feedback loops for learning with E-books. Thus, within the personalization component, students can create individual profiles in their learning for their particular requirements, and these profiles are context dependent and integrated with the E-books. For example, the system can provide functionalities such as electronic annotation, bookmarking and tracking of learning progress, supporting individual behavior and cognitive performance through the learning process or design learning forums so that learners can interact and respond to different objects in the community using E-books, including the teachers.

3 Interactive E-books Design to Support Personalized Teaching 3.1 Interactive E-books Design 3.1.1

E-book Design Principles

E-book is a very effective teaching and learning tool. However, to be able to design and use E-books effectively, it is necessary to note the following principles:

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• Integrate many audio-visual multimedia applications to encourage different ways of receiving knowledge. • Make the most of interactive activities on the E-book interface to make them more learner-friendly. • For phrases that are difficult to understand, providing online explanations (inline dictionary) or descriptive illustrations is necessary. • Use multimedia elements flexibly to attract students’ attention but also ensure the requirement of providing scientific knowledge. • Develop tests and assessments in a suitable way, ensuring the appropriate difficulty and acceptable discrimination. 3.1.2

E-book Design Software

Currently, there are many supporting software for building E-books such as Lectora, iBooks Author, Kotobee Authors. The function of these software generally allows the integration and arrangement of documents and images, tables, interactive applications into one book. Interactive applications include photograph galleries, audio files, video files, short and multiple-choice tests. However, in the framework of this report, we choose Kotobee Author software to be the primary tool for the design. Kotobee Author is an EPUB E-book design software that can be read on many different devices, suitable for educational, training or publishing purposes. The software allows us to create E-books with the designer’s own “colors,” logos and brands. Interactive applications are prioritized thoroughly and especially, this software has a very user-friendly interface, can work on both today’s popular operating system platforms, Windows and Mac OS.

3.1.3 Step 1 Step 2 Step 3 Step 4 Step 5

E-book Design Process Analyze learners’ needs (personalization). Build a framework for the basic content of the E-book. Put basic content into E-book using Kotobee Author software, integrating interactive multimedia applications. Implement the test and impact assessment into the analysis in Step 1. Edit, complete and expand the use of E-books.

3.2 Application of Interactive E-book Design in Chemistry Teaching E-book Structure When a user logs in to Kotobee Reader, an E-book library appears on the screen and the system automatically loads the E-book selected by the student (Fig. 3). A

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Fig. 3 Templates of E-book structure design

good point when using the Kotobee platform is that students can choose to use it on computers, phones, iPads or any other portable smart device, while the screen resolution is still of good quality. The table of contents of the E-book is oriented to build according to the textbook content framework for students to follow easily and added some other content such as user manuals, practical content not mentioned in the textbooks, library of general knowledge for revision, test system… (Fig. 3). Moreover, each page of the book is designed with a full range of tools such as information search, images, sounds, labels, notes, zoom in, zoom out, return to the table of contents (Fig. 4) to maximize the reading ability and the use of E-books according to each student’s needs. The E-book developed in this study includes multimedia content such as videos, audio recordings, images, pdf text (Fig. 5) and interactive components such as the user forums of the application Padlet.com, short quiz using Quizi.com app and relying on linker feature for use on E-book (Fig. 5) [7].

Fig. 4 Templates of E-book structure design of chapter metal (chemistry K-12)

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Fig. 5 Templates of E-book multimedia design of chapter metal (chemistry K-12)

Besides, in order to attract students to read and use E-books, we have designed knowledge content pages in the form of infographics and built knowledge content associated with real life such as an experienced activity design of eight metallic elements with an added link for using the Storymap application (https://storymaps. arcgis.com/). E-book tracks students’ learning progress automatically through live tests on Ebook and quiz on Quizi.com, then the results are sent back to teachers so that teachers can be timely informed and direct students to learn. Finally, students can share information and problems in the learning process with the E-book through the link between the E-book and the Padlet (https://padlet.com/) application installed on the teacher’s phone and receive immediate feedback from the teacher. Students can also choose to share the book they are reading with friends or the community through the Kotobee library. Experimental Results To evaluate the level of interactive E-book using in students learning, the survey and feedback analysis had been conducted from 43 students in experimental class 12A6, Kim Lien High School, Hanoi (Table 1). The study result shows that 91.56% of students were excited about learning with E-books, 90.36% of students think that E-books help practice self-study skills and direct activities for students in the process of studying the lesson, and 87.95% of students commented that the content of the E-books is clear and specific, while over 90% of students understood the requirements and managed to analyze the data contained in the E-books. In the process of using E-books, students have learned to summarize key knowledge, draw conclusions from collecting documents and apply self-study knowledge to reach learning goals. This is because the E-book has interactive feedback with students. Teachers interact with students through the E-book, especially the convenient feature that can be used on many smart devices such as laptop, smartphone and IPad.

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Table 1 Summary of student feedback evaluation results when using interactive E-books in learning Contents

Number of students (43)

Percentage%

1. The extension of using your own E-book to accomplish your learning goals Collect and select materials related to the lesson content

36

84.34

Manage to read and use the chapters in the E-book

40

93.98

Manage to analyze data from E-book

39

91.57

Be able to apply the knowledge and documents in the E-book to complete the learning task

39

90.36

2. Comments about the interactive E-book designed by your teachers The content is specific, clear, complete, beautifully designed

37

87.95

Multimedia resources such as videos, images effectively supporting you in acquiring knowledge

40

91.56

The index of subsections in the E-book helps guide students’ learning activities

37

85.54

Support self-study skills, self-discovery of knowledge

36

84.33

Get excited about learning with the teacher’s E-book

40

91.56

4 Conclusion Interactive E-book is a highly interactive medium and learning material that effectively supports teachers and students in the personalized teaching process. Through the process of designing and testing E-books, the survey results show that the application of E-books is completely feasible and brings good results. Students have responded very positively, and teachers realize the need to innovate in teaching, learning and integrating information technology into personalized teaching. It is an important supplement to textbooks and personalized learning.

References 1. Bloom, B.S. 1981. All Our Children Learning. New York: McGraw-Hill. 2. Eisner, E.W., and B.S. Bloom. 2000. The Quarterly Review of Comparative Education, vol. XXX, no. 3. Paris, UNESCO: International Bureau of Education. 3. Bloom, B.S. 1984. The 2 sigma problem: The search for methods of group instruction as effective as one-to-one tutoring. Educational Researcher 13 (6): 4–16. 4. Harjono, A., G. Gunawan, R. Adawiyah, and L. Herayanti. 2020. An interactive e-book for physics to improve students’ conceptual mastery. International Journal of Emerging Technologies in Learning 15(05): 40–49. https://doi.org/10.3991/ijet.v15i05.10967.

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5. Domagk, S., R.N. Schwartz, and J.L. Plass. 2010. Interactivity in multimedia learning: An integrated model. Computers in Human Behavior 26 (5): 1024–1033. 6. Huang, Y.M., T.H. Liang, Y.N. Su, and N.S. Chen. 2012. Empowering personalized learning with an interactive e-book learning system for elementary school students. Educational Technology Research and Development 60: 703–722. 7. Qiu, X., H. Shishido, R. Sakamoto, and I. Kitahara. 2020. Interactive e-book linking text and multi-view video. In 2020 IEEE 9th global conference on consumer electronics, GCCE 2020 9291752, 813–817. https://doi.org/10.1109/GCCE50665.2020.9291752.

Bipolar-Valued Fuzzy Subhemirings of a Hemiring Under Homomorphism N. Kumaran, T. Gunasekar, Naresh Kumar Jothi, and A. Neel Armstrong

Abstract In 1965, Zadeh (Fuzzy Logic Intelligent Systems 14:1x25–129, 2004) introduced the notion of a fuzzy subset of a set, fuzzy sets are a kind of useful mathematical structure to represent a collection of objects whose boundary is vague. Since then, it has become a vigorous area of research in different domains, there have been a number of generalizations of this fundamental concept such as intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, and soft sets. (Gau and Buehrer in IEEE Transactons on Systems, Man and Cybernetics 23:610–614). Lee (Bulletin of the Malaysian Mathematical Sciences Society 2, 32(3):361–373) introduced the notion of bipolar-valued fuzzy sets. Bipolar-valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [−1, 1]. In a bipolar-valued fuzzy set, the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degree [0, 1] indicates that elements somewhat satisfy the property, and the membership degree [−1, 0] indicates that elements somewhat satisfy the implicit counter property. Bipolar-valued fuzzy sets and intuitionistic fuzzy sets look similar each other. However, they are different each other (Lee in Bulletin of the Malaysian Mathematical Sciences Society 2, 32(3):361– 373, Lee in Proceedings of the International Conference on Intelligent Technologies, Bangkok, pp 307–312, 2000) Anitha, Muruganantha Prasad, and Arjunan (Bulletin of Society for Mathematical Services and Standards 2:52–59, 2013) defined as bipolarvalued fuzzy subgroups of a group. We introduce the concept of bipolar-valued fuzzy subhemiring under homomorphism and established some results. Keywords Fuzzy set · Bipolar-valued fuzzy set · Bipolar-valued fuzzy subhemiring

N. Kumaran (B) · T. Gunasekar · N. K. Jothi · A. Neel Armstrong Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology (Deemed To Be University), Avadi, Chennai, Tamilnadu, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_16

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1 Introduction Bipolar-valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from interval [0, 1] to [−1, 1]. Bipolar-valued fuzzy sets have membership degrees that represent the degree of satisfaction to the property corresponding to a fuzzy sets and its counter property. In a bipolar-valued fuzzy set, the membership degree zero means that elements are irrelevant to the corresponding property, the membership degree on [0, 1] indicates the elements somewhat satisfies the property, and the membership degree on [−1, 0] indicates that elements show that satisfy the implicit counter property Definition 1.1 [2] A non-empty set γ together with two binary operations, denoted by and. is a hemiring if (i) (ii) (iii)

(γ , +) is a semigroup and commutative with zero, (γ , .) is a semi-group,    σα˜ + ςβ˜ .wγ = σα˜ .υγ˜ + ςβ˜ .υγ˜ and σα˜ . ςβ˜ + υγ˜ = σα˜ .ςβ˜ + σα˜ .υγ˜ , for all u, ζβ˜ , vγ˜ ∈ γ .

Example 1.1 (Z, +, .) is a hemiring under usual addition and multiplication, where Z is a set of all integers. Definition 1.2 A non-empty subset N of a hemiring (γ , +, .) is called a subhemiring of γ if N is a hemiring with respect to the operations of γ . Example 1.2 (4Z, +, .) is a subhemiring of (Z, +, .), where Z is a set of all integers. Definition 1.3 Let  be a nonempty set. A fuzzy subset K is a function K : rightthreetimes → [O, 1]. Definition 1.4 [2] Let (γ , +, .) be a hemiring. A fuzzy subset K of γ is said to be a fuzzy subhemiring (FSH) of γ if,      (i) K σα˜ + ςβ˜ ≥ min K(σα˜ ), K ςβ˜      (ii) K σα˜ ςβ˜ ≥ min K(σα˜ ), K ςβ˜ for all σα˜ , ςβ˜ in γ .

2 Bipolar-Valued Fuzzy Subhemiring (BVFSHR) Definition 2.1 Let (γ , +, .) be a hemiring. A BVFS K of γ is a bipolar-valued FSH of γ if,      (i) K+ σα˜ + ςβ˜ ≥ min K+ (σα˜ ), K+ ςβ˜ ,    (ii) K+ (σ ς) ≥ min K+ (σα˜ ), K+ ςβ˜ ,

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     K− σα˜ + ςβ˜ ≤ max K− (σα˜ ), K− ςβ˜ ,    K− (σ ς) ≤ max K− (σα˜ ), K− ςβ˜ for all σα˜ and ςβ˜ in γ .

Note Bipolar-valued fuzzy sets K, P, and R mean K = K+ , K− , P = P , P − , and R = R+ , R− , respectively. +

Definition 2.2 Let K be a BVFS of . For λ ∈ [0, 1] and ω ∈ [0, 1], the (λ, ω) level subset of K is K (λ,ω) = {x ∈  : K+ (σ ) ≥ λand K− (σ ) ≤ ω)}. + Definition  2.3 Let K be a BVFS of . For λ ∈ [0, 1], the K -level λ-cut of K is  T K+ , λ = { XP ∈ σ : K+ (σ ) ≥ λ}.

Definition of . For ω ∈ [−1, 0], then K− -level o)-cut of K  Let K be a BVFS  − 2.4 − is N K , 0 ) = {xN ∈ σ : K (σ ) ≤ 0)}. Theorem 2.1 Let K be a BVFSHR of a hemiring H. Then, for λ in [0, 1] and σ ) in [−1, 0], such that λ ≤. K(0) and ( j) ≥ K− (0), K(λ, ω) is a(λ, 0))-level subhemiring of γ , where 0 is a zero in hemiring H.   Proof For all σα˜ , ςβ˜ ∈ K (λ,co) , then K+ (σα˜ ) ≥ λ, K− (σα˜ ) ≤ 0), K+ ςβ˜ ≥ λ, and   K− ςβ˜ ≤ 0). Now,      K+ σα˜ + ςβ˜ ≥ min K+ (σα˜ ), K+ ςβ˜ ≥ min{λ, λ} = λ   = K+ σα˜ + ςβ˜ ≥ λ. And,      K+ σα˜ ςβ˜ ≥ min K+ (σα˜ ), K+ ςβ˜ ≥ min{λ, λ} = λ   = K+ σα˜ ςβ˜ ≥ λ. Also,      K− σα˜ + ςβ˜ ≤ max K− (σα˜ ), K− ςβ˜ ≤ max{0J, 0J} = 0)   = K− σα˜ + ςβ˜ ≤ 0). And,      K− σα˜ ςβ˜ ≤ max K− (σα˜ ), K− ςβ˜ ≤ max{0), 0)} = 0)

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  = K− σα˜ ςβ˜ ≤ 0). Thus, σα˜ + ςβ˜ , σα˜ ς ∈ K (λ,co) . Hence, K (λ0J ) is a(, σ ))-level subhemiring (LSH) of H. Theorem 2.2 Let K be a BVFSHR of a hemiring H. Then, for λ, μ in [0, 1], 0), φ in [−1, 0], λ ≤ K+ (0), μ ≤ K+ (0), 0) ≥ K− (0), φ ≥ K− (0), μ < λ, and / (i J < φ, the tυo(, ( j))-LSH K (λ,cv ) and K (μ∅ ) of K are equal if and only if σ ∈ γ λ > K+ (σα˜ ) > μ and ( j) < K− (σα˜ ) < φ. Proof Assume that K (λ,co) = K (μ,ϕ) . Suppose ∃σα˜ ∈ γ , such that λ > K+ (σα˜ ) > μ and σ J < K− (σα˜ ) < φ. Then, K (λ,co)⊆ K (μ∅) implies σα˜ ∈ K (μ∅) ,’ but not in K (λ,cv ). This is contradiction to K (λ,co) = K (μ∅) . / γ λ > K+ (σα˜ ) > μ and σ J < K− (σα˜ ) < φ. Conversely, if Therefore, σα˜ ∈ / γ such that λ > K+ (σα˜ ) > μ, 0J < K− (σ ) < φ. σα˜ ∈ Theorem 2.3 The homomorphic image of a(, ( j))-LSH of a BVFSHR of γ is a(, ( j))-LSH of a BVFSHR of a hemiring H. Proof Let υ = ϕ(K), where ϕ : γ → γ | and K is a BVFSHR of γ . By the theorem, υ = υ+, υ− is a BVFSHR of H.   Let σα˜ , ςβ˜ ∈ γ , then ϕ(σα˜ ), ϕ ς Sβ˜ ∈ γ |. Let K(λ,00) be a(, σ ))-LSH of K.     That is, K+ (σα˜ ) ≥ λ, K− (σα˜ ) ≤ 0); K+ ςβ˜ ≥ λ, K− ςβ˜ ≤ 0J;         K+ σα˜ + ςβ˜ ≥ λ, K− σα˜ + ςβ˜ ≤ 0), K+ σα˜ ςβ˜ ≥ λ, K− σα˜ ςβ˜ ≤ 0). + + + Now, )) ≥ λ;  υ  (j(σ  α˜ )) ≥ K  (σ  α˜ ) ≥ λ which implies υ (j(σ  α˜ + + + υ ϕ ςβ˜ ≥ K ςβ˜ ≥ λ which implies υ ϕ ςβ˜ ≥ λ.         Then, υ + ϕ(σα˜ ) + ϕ ςβ˜ = υ + ϕ σα˜ + ςβ˜ ≥ K+ σα˜ + ςβ˜ ≥ λ    which implies υ + ϕ(σα˜ ) + ϕ ςβ˜ ≥ λ.         And, υ + ϕ(σα˜ )ϕ ςβ˜ = υ + ϕ σα˜ ςβ˜ ≥ K+ σα˜ ςβ˜ ≥ λ    which implies υ + ϕ(σα˜ )ϕ ςβ˜ ≥ λ.

− − − And, (ϕ(σ α˜ )) ≤ 0);  υ (ϕ(σ  α˜ )) ≤ K (σ  α˜ ) ≤ 0) which implies υ   − − − υ ϕ ςβ˜ ≤ K ςβ˜ ≤ 0) which implies υ ϕ ςβ˜ ≤ 0).         Then, υ − ϕ(σα˜ ) + ϕ ςβ˜ = υ − ϕ σα˜ + ςβ˜ ≤ K− σα˜ + ςβ˜ ≤ 0)    which implies υ − ϕ(σα˜ ) + ϕ ςβ˜ ≤ 0).         And, υ − ϕ(σα˜ )ϕ ςβ˜ = υ − ϕ σα˜ ςβ˜ ≤ K− σα˜ ςβ˜ ≤ 0)    which implies υ − ϕ(σα˜ )ϕ ςβ˜ ≤ 0). Hence, ϕ(K(λ, 0))) is a(λ, 0))-LSH of a BVFSHR υ of H.

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Theorem 2.4 The homomorphic pre-image of a(, ( j))-level subhemiring of a BVFSHR of γ | is a(, ( j))-LSH of a BVFSHR of a hemiring H. Proof Let υ = ϕ(K), where ϕ :  γ → γ |, W is a BVFSHR ofγ |. By the theorem, K is a BVFSHR of γ . Let ϕ(σα˜ ), ϕ ςβ˜ ∈ γ |, but σα˜ , ςβ˜ ∈ γ . Let j(K(λ,0 bea(, ( j))-level subhemiring of W.       That is υ + (ϕ(σα˜ )) ≥ λ, υ − (ϕ(σα˜ )) ≤ 0) ; υ + ϕ ςβ˜

≥ λ, υ − ϕ ςβ˜

≤ 0);

         ≥ λ, υ − ϕ(σα˜ ) + ϕ ςβ˜ ≤ 0J, υ + ϕ(σα˜ )ϕ ςβ˜ ≥ υ + ϕ(σα˜ ) + ϕ ςβ˜    λ, υ − ϕ(σα˜ )ϕ ςβ˜ ≤ 0). Now, K+ (σα˜ ) = υ + (ϕ(σα˜ )) ≥ λ = K+ (σα˜ ) ≥ λ;

       K+ ςβ˜ = υ + ϕ ςβ˜ ≥ λ = K+ ςβ˜ ≥ λ. Then,         K+ σα˜ + ςβ˜ = υ + ϕ σα˜ + ςβ˜ = υ + ϕ(σα˜ ) + ϕ ςβ˜ ≥ λ   = K+ σα˜ + ςβ˜ ≥ λ. And,         K+ σα˜ ςβ˜ = υ + ϕ σα˜ ςβ˜ = υ + ϕ(σα˜ )ϕ ςβ˜ ≥ λ   = K+ σα˜ ςβ˜ ≥ λ. − − − And,  K (σα˜ ) = υ (ϕ(σ α˜ )) ≤ 0) = K  (σ α˜ ) ≤ 0J; − − − K ςβ˜ = υ ϕ ςβ˜ ≤ 0) = K ςβ˜ ≤ 0).         Also, K− σα˜ + ςβ˜ = υ − ϕ σα˜ + ςβ˜ = υ − ϕ(σα˜ ) + ϕ ςβ˜ ≤ 0)   = K− σα˜ + ςβ˜ ≤ 0).         And, K− σα˜ ςβ˜ = υ − ϕ σα˜ ςβ˜ = υ − ϕ(σα˜ )ϕ ςβ˜ ≤ 0)   = K− σα˜ ςβ˜ ≤ 0). Hence, K(λ,00) is a(λ, 0))-level subhemiring of BVFSHR K of H.

Theorem 2.5 The anti-homomorphic image of a(λ, 0))-level subhemiring of a BVFSHR of γ is a(λ, 0))-LSH of a BVFSHR of a hemiring H. Proof Let υ = ϕ(K), where ϕ : γ → γ |, K is a BVFSHR of γ . By the theorem, υ is aBVFSHR of H.   Let σα˜ , ςβ˜ ∈ γ . Then ϕ(σα˜ ), ϕ ςβ˜ ∈ γ |.

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Let K(λ,0)) be a(, ( j))-level subhemiringof K. 

  That is, K+ (σα˜ ) ≥ λ, K− (σα˜ ) ≤ 0); K+ ςβ˜ ≥ λ, K− ςβ˜ ≤ 0).       And, K+ ςβ˜ + σα˜ ≥ λ, K− ςβ˜ + σα˜ ≤ 0J, K+ ςβ˜ σα˜ ≥   K− ςβ˜ σα˜ ≤ 0). Now υ + (ϕ(σα˜ )) ≥ K+ (σα˜ ) ≥ λ = υ + (ϕ(σα˜ )) ≥ λ;

λ and

        υ + ϕ ςβ˜ ≥ K+ ςβ˜ ≥ λ = υ + ϕ ςβ˜ ≥ λ. Also,         ≥ K+ ςβ˜ + σα˜ ≥ λ υ + (+ ϕ(σα˜ ) + ϕ ςβ˜ = υ + ϕ ςβ˜ + σα˜    = υ + (+ ϕ(σα˜ ) + ϕ ςβ˜ ≥ λ. And,      υ + (+ ϕ(σα˜ )ϕ ςβ˜ = υ + (ϕ(ς σ )) ≥ K+ ςβ˜ σα˜ ≥ λ    = υ + (+ ϕ(σα˜ )ϕ ςβ˜ ≥ λ. − − − And, (ϕ(σ α˜ )) ≤ 0);  υ (ϕ(σ  α˜ )) ≤ K (σ  α˜ ) ≤ 0) = υ   υ − ϕ ςβ˜ ≤ K− ςβ˜ ≤ 0) = υ − ϕ ςβ˜ ≤ 0).         Also, υ − ϕ(σα˜ ) + ϕ ςβ˜ = υ − ϕ σα˜ + ςβ˜ ≤ K− ςβ˜ + σα˜ ≤ 0)    = υ − ϕ(σα˜ ) + ϕ ςβ˜ ≤ 0).         And, υ − ϕ(σα˜ )ϕ ςβ˜ = υ − ϕ ςβ˜ σα˜ ≤ K− ςβ˜ σα˜ ≤ 0)    = υ − ϕ(σα˜ )ϕ ςβ˜ ≤ 0). Hence, ϕ(K(λ, 0isa(λ, 0))-level subhemiring of BVFSHR υ of H.

Theorem 2.6 The anti-homomorphic pre-image of a(, σ ) -LSH of a BVFSHR of a hemiring γ | is a(, σ ) -LSH of a BVFSHR of a hemiring H. Proof Let υ = ϕ(K), where ϕ : γ → γ |, υ is a BVFSHR of γ |. By the theorem, K = K+, K−is aBVFSHR of H. Let ϕ(σα˜ ), ϕ ςβ˜ , but σα˜ , ςβ˜ ∈ γ .   + subhemiring Let ϕ K(λ,00) be a(,  σ ))-level    of W. That is, υ (ϕ(σα˜ )) ≥ λ, υ − (ϕ(σα˜ )) ≤ 0); υ + ϕ ςβ˜ ≥ λ, υ − ϕ ςβ˜ ≤ 0);        υ + ϕ ςβ˜ + σα˜ ≥ λ, υ − ϕ ςβ˜ + ϕ(σα˜ ) ≤ 0),         υ + ϕ ςβ˜ ϕ(σα˜ ) ≥ λ, υ − ϕ ςβ˜ ϕ(σα˜ ) ≤ 0). Now, K+ (σα˜ ) = υ + (ϕ(σα˜ )) ≥ λ = K+ (σα˜ ) ≥ λ;

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       And, K+ ςβ˜ = υ + ϕ ςβ˜ ≥ λ = K+ ςβ˜ ≥ λ. Then,          K+ σα˜ + ςβ˜ = υ + ϕ σα˜ + ςβ˜ = υ + ϕ ςβ˜ + j(σα˜ ) ≥ λ   = K+ σα˜ + ςβ˜ ≥ λ. And,          K+ σα˜ ςβ˜ = υ + ϕ σα˜ ςβ˜ = υ + ϕ ςβ˜ ϕ(σα˜ ) ≥ λ   = K+ σα˜ ςβ˜ ≥ λ. − − − And,  K (σα˜ ) = υ (ϕ(σ α˜ )) ≤ 0) = K  (σ α˜ ) ≤ 0J; − − − K ςβ˜ = υ ϕ ςβ˜ ≤ 0J = K ςβ˜ ≤ 0).          Also, K− σα˜ + ςβ˜ = υ − ϕ σα˜ + ςβ˜ = υ − ϕ ςβ˜ + ϕ(σα˜ ) ≤ 0).   = K− σα˜ + ςβ˜ ≤ 0).          And, K− σα˜ ςβ˜ = υ − ϕ σα˜ ςβ˜ = υ − ϕ ςβ˜ ϕ(σα˜ ) ≤ 0).   = K− σα˜ ςβ˜ ≤ 0). Hence, K(λ,0)) is a(λ, 0))-level subhemiring of BVFSHR Kofγ .

Theorem 2.7 Let K be a BVFSHR of H. Then, for σ ) in closed interval-1, O, K− −1 evel o)-cut N K− , 0 ) is a K− − level o)-cut subhemiring of H.   Proof For all σα˜ , ςβ˜ ∈ N (K− , 0 then K− (σα˜ ) ≤ 0), K− ςβ˜ ≤ 0). Now,      K− σα˜ + ςβ˜ ≤ max K− (σα˜ ), K− ςβ˜ ≤ max {( j), ( j)} = ( j).   = K− σα˜ + ςβ˜ ≤ 0).      And, K− σα˜ ςβ˜ ≤ max K− (σα˜ ), K− ςβ˜ ≤ max{0), 0)} = 0)   = K− σα˜ ςβ˜ ≤ 0).   Thus, σα˜ + ςβ˜ , σα˜ ςβ˜ ∈ N K− , 0 .  −  Hence, N K , 0 is a K− -level 0)-cut subhemiring of H.

3 Conclusion and Future Work In this paper, bipolar-valued fuzzy subhemiring is introduced and bipolar-valued fuzzy subhemiring concept of theorem is introduced and proved. BVFNSHR, pseudo

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bipolar-valued fuzzy subhemiring, translations of BVFNSHR, and related concepts in BVFNSHR of a hemiring have been introduced and studied. (λ, ω)-level subset of BVFI of a hemiring and related concept going to define using these definitions some (λ, ω)-level subset theorem in BVFI of a hemiring are to derive. (λ, ω) level subset of BVFSHR K + level λ cut of BVFSHR , K − level ω of BVFSHR an bipolar-valued fuzzy translation of BVFSHR of a hemiring will be defined; using this definition, some (λ, ω) level of subset theorem in BVFSHR of a hemiring will be derived.

References 1. Anitha, M.S., Muruganantha Prasad, and K. Arjunan. 2013. Notes on bipolar-valued fuzzy subgroups of a group. Bulletin of Society for Mathematical Services and Standards 2 (3): 52–59. 2. Anthony, J.M., and H. Sherwood. 1979. Fuzzy groups redefined. Journal of Mathematical Analysis and Applications 69: 124–130. 3. Arsham, B.S. 2009. Bipolar-valued fuzzy BCK/BCI-algebras. World Applied Sciences Journal 7 (11): 1404–1411. 4. Rosenfeld, Azriel. 1971. Fuzzy groups. Journal of Mathematical Analysis and Applications 35: 512–517. 5. Choudhury, F.P., A.B. Chakraborty, and S.S. Khare. 1988. A note on fuzzy subgroups and fuzzy homomorphism. Journal of Mathematical Analysis and Applications 131: 537–553. 6. Gau, W.L., and D.J. Buehrer. 1993. Vague sets. IEEE Transactons on Systems, Man and Cybernetics 23: 610–614. 7. Lee, K.J. 2009. Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras. Bulletin of the Malaysian Mathematics Science and Society 2, 32 (3): 361–373. 8. Lee, K.M. 2000. Bipolar-valued fuzzy sets and their operations. In Proceedings of International Conference on Intelligent Technologies, Bangkok, Thailand, 307–312. 9. Lee, K.M. 2004. Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets and bipolarvalued fuzzy sets. Journal of Fuzzy Logic Intelligent Systems 14 (2): 125–129. 10. Akgul, Mustafa. 1988. Some properties of fuzzy groups. Journal of Mathematical Analysis and Applications 133: 93–100. 11. Samit Kumar Majumder. 2012. Bipolar valued fuzzy sets in -semigroups. Mathematica Aeterna 2 (3): 203–213. 12. Young Bae Jun and Seok Zun Song. 2008. Subalgebras and closed ideals of BCH-algebras based on bipolar-valued fuzzy sets. Scientiae Mathematicae Japonicae Online, 427–437. 13. Zadeh, L.A. 1965. Fuzzy sets. Inform. and Control 8: 338–353. 14. Muhammad Akram. 2013. Bipolar fuzzy graphs with applications. Knowledge-Based Systems 39: 1–8. 15. IsabelleBloch. 2012. Mathematical morphology on bipolar fuzzy sets: General algebraic framework. International Journal of Approximate Reasoning 53 (7): 1031–1060.

Hybrid Phase Synchronization for Generalized Stretch, Twist, Fold Flow Chaotic System of Fractional Order Nagadevi Bala Nagaram, Suresh Rasappan, Regan Murugesan, Kala Raja Mohan, and Hanaa Hachimi

Abstract Secure communication is essential in all fields in our day to life. While the communication, the error occurs due to some circumstance. To rectify that many methodologies are adopted. The hybrid phase synchronization is one of the new techniques to be used in secure communication. This synchronization technique is helpful in such an area. This article is exposed the generalized chaotic system of stretch, twist, fold flow for fractional order. The stability and chaotic nature of the system are examined. This article examined synchronization via a suitable control function. The evaluation process is done to be the consistency of the proposed system. Hybrid phase synchronization is investigated for the indistinguishable chaotic system of stretch, twist and fold flow. The entire synchronization and anti-synchronization is taken to control the system by employing feedback technique. To validate the proposed theory, numerical simulation is presented. The chaotic nature of the system is presented through MATLAB software. Keywords Chaotic theory · Equilibrium point · Hybrid · Synchronization

N. B. Nagaram (B) · R. Murugesan · K. R. Mohan Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamilnadu 600062, India S. Rasappan Department of Mathematics, University of Technology and Applied Sciences-Ibri, Ibri 466, 516, Sultanate of Oman H. Hachimi Secretary General of Sultan Moulay Slimane University, Applied Mathematics & Computer Science, Beni-Mellal, Morocco e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_17

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1 Introduction The chaos theory is high sensitivity dynamical system with its initial state. This was first discovered by Lorentz. After some decades the researchers intensively studied about the chaotic systems and developed. The magnetic field is designed by stretch, twist, fold flow mechanism, an example of fast dynamo action [1]. The fractional order system is a dynamic systems in the field of control theory. It can be modeled as a non-integer order derivative of differential equation [2]. It is important to study the anomalous behavior of dynamic systems in various field like physics, chaotic systems, etc. [3, 4]. The fractional order system is a dynamic systems in the field of control theory [5–10]. It can be modeled as a non-integer order derivative of differential equation [11, 12]. It is important to study the anomalous behavior of dynamic systems in various field like physics, chaotic systems, etc. [13–15]. x(t) ˙ = αz − 8(b + 1)x y y˙ (t) = 11x 2 + 3y 2 + z 2 + βx z − 3c2 z˙ (t) = −αx + 2yz − βx y

(1)

where α, β represent a real value parameters. The nature of the chaotic is determined by the parameters b, c, α and β. This paper is structured as follows, Sect. 2 offers preliminary definition and theorems which support the theory part. The consistency of the proposed system is evaluated in Sect. 3. Using feedback control, the synchronization part is analyzed in Sect. 4. Section 5 also provides numerical simulations to support the theory.

2 Basic Concepts To examine the stability and chaotic nature of the system (1) some preliminary definitions and theorems are presented in this section. Definition 1: Let n = [q] + 1 and D q be the caputo differential operator then fractional derivative be expressed as, D q f (t) =

1 (n − q)



t

(t − τ )−q+n−1 f n (τ ) dτ

0

Definition 2 : For x ∈ n and 0 < q ≤ 1 the system of fractional order be expressed as D q (x) = f (x)

(2)

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At the equilibrium point E has one non-negative real Eigen value then the system (2) is said to be saddle point with index 1. Theorem 1 : A condition which is necessary for system (2) to be chaotic is to have at least one Eigen value should be non-negative real Eigen value. That is,   |Im(λ)| 2 arctan . q> π Re(λ) Theorem 2 : For all the Eigen values of the system (2) should satisfies arg(λ) > qπ , 2 then the equilibrium points are called locally asymptotically stable. Otherwise, if then the system (2) is called unstable. arg(λ) ≤ qπ 2

3 The Mathematical Model The mathematical model of generalized fractional order STF flow is described as, D q x(t) = αz − 8(b + 1)x y D q y(t) = 11x 2 + 3y 2 + z 2 + βx z − 3c2 D q z(t) = −αx + 2yz − βx y

(3)

Description of the system: • Let q be a positive non integer, where 0 < q ≤ 1 and (x, y, z) ∈ 3 . The real parameters b, c, α&β are determined the nature of chaotic.

4 Equilibria Analysis When α ≥ 0 and β ≥ 0 we obtained the equilibrium points as follows, E c1 = (0, c, 0) E c2 = (0, −c, 0) E c3 = (xc3 , yc3 , z c3 ) E c4 = (−xc3 , yc3 , −z c3 ) E c5 = (xc4 , yc4 , z c4 ) E c6 = (−xc4 , yc4 , −z c4 ) where

(4)

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  2 α2 S2  3(c − 1024(b+1) 2 αs sx , zc = xc3 =  , yc3 = βS S2 32(b + 1) 3 4 11 + 16 + 4   2 α2 R2  3(c − 1024(b+1)  2 xc4 =  , S = β − β 2 + 64(b + 1), and βR R2 11 + 16 + 4  R = β + β 2 + 64(b + 1). By the transformation (x, y, z) → (−x, y, −z) the system (4) is invariant and it has rotatixon about Y-axis symmetrically. Thus the system is not conservative for a long time, since the divergence is ∇ · V = −8by. Theorem 3 : If α > 0 , β > 0 , b = 0 and c = 1, then the system’s equilibrium is all on the unit sphere S surface. Proof To solve the following equation αz − 8(b + 1)x y = 0

(5)

11x 2 + 3y 2 + z 2 + βx z − 3c2 = 0

(6)

−αx + 2yz − βx y = 0

(7)

and show that the solution x, y and z satisfy the condition x 2 + y 2 + z 2 = 1 when b = 0, c = 1. From (5) α=

8(b + 1)x y z

(8)

From (6) 11x 2 + 3y 2 + z 2 = 3c2 − βx z

(9)

−αx + 2yz = βx y

(10)

βx z = 2z 2 − 8(b + 1)x 2

(11)

Substituting (8) in (10)

Substituting (11) and (9) we get,

Hybrid Phase Synchronization for Generalized …

11x 2 + 3y 2 + z 2 − 8x 2 − 8bx 2 = 3c2

163

(12)

When b = 0, c = 1 in Eq. (12) we get sphere equation with radius 1. Thus all equilibrium points lies on the surface S. For α = 0.1, β = 1.5, c = 1.7 and b = 0.001 the equilibrium points can be written as, E c1 = (0, 1.7, 0) E c2 = (0, −1.7, 0) E c3 = (0.6514, −0.0207, −1.0819) E c4 = (−0.6514, 0.0207, 1.0819) E c5 = (0.6514, 0.0301, 1.5704) E c6 = (−0.6514, 0.0301, −1.5704)

(13)

The linearized form for the system (4) is obtained by using Jacobian matrix is, ⎡

⎤ −8(b + 1)y −8(b + 1)x α J = ⎣ 22x + βz 6y 2z + βx ⎦ −α − βy 2z − βx 2y

(14)

Theorem 4: The saddle points of the system (3) are E c1 , E c3 and E c4 , each with index 2. Furthermore they are unstable. Proof: The Eigen values at E c1 of Eq. (14) are λ1 λ1 = 10.2, λ2 = 3.3844 and λ3 = −13.59801. By Theorem 1 and Definition 3 we can conclude that E c1 is unstable. Similarly E c3 and E c4 are also have Eigen values with positive real value. Thus E c3 and E c4 are unstable. Theorem 5: The saddle points of the system (3) are E c2 , E c5 and E c6 , each with index 1. Furthermore they are unstable. Proof: The Eigen values at E c2 of Eq. (14) are λ1 = −10.2, λ2 = 13.628 and λ3 = −3.4144. By Theorem 1 and Definition 2 we can conclude that E c2 is unstable. Similarly E c5 and E c6 are also have Eigen values with positive real value. Thus E c5 and E c6 are unstable. Remark. The equilibrium point of saddle points with index 2 are E c1 , E c3 and E c4 and scrolls are generated around on it.

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Hybrid Phase Synchronization

In this section, by using feedback controller to synchronize the system (3) is analyzed. The master system be taken as, D q x1 (t) = αz − 8(b + 1)x1 y1 D q y1 (t) = 11x12 + 3y12 + z 12 + βx1 z 1 − 3c2 D q z 1 (t) = −αx1 + 2y1 z 1 − βx1 y1

(15)

The response system is described by D q x2 (t) = αz − 8(b + 1)x2 y2 + u 1 D q y2 (t) = 11x22 + 3y22 + z 22 + βx2 z 2 − 3c2 + u 2 D q z 2 (t) = −αx2 + 2y2 z 2 − βx2 y2 + u 3

(16)

To synchronize the system (15) and (16) the controller u = (u 1 , u 2 , u 3 )T is introduced in the response system (16). Consider the error states as, e1 = x2 − x1 e2 = y2 + y1 e3 = z 2 − z 1

(17)

The error states e1 and e3 are said to be entirely synchronized and e2 is said to be entirely anti-synchronized. The error dynamical system is studied by, D q e1 (t) = αe3 − 8(b + 1)(x2 y2 − x1 y1 ) + u 1 D q e2 (t) = 11(x22 + x12 ) + 3(y22 + y12 ) + (z 22 + z 12 ) + β(x2 z 2 − x1 z 1 ) − 6c2 + u 2 D q e3 (t) = −αe1 + 2(y2 z 2 − y1 z 1 ) − β(x2 y2 − x1 y1 ) + u 3

(18)

Choose the control functions as, u 1 = 8(b + 1)(x2 y2 − x1 y1 ) + V1 u 2 = −11(x22 + x12 ) − 3(y22 + y12 ) − (z 22 + z 12 ) − β(x2 z 2 − x1 z 1 ) + 6c2 + V2 u 3 = −αe1 + 2(y2 z 2 − y1 z 1 ) − β(x2 y2 − x1 y1 ) + V3

(19)

The linear controller V1 , V2 and V3 are introduced in system (19). Then the system (18) can be written as,

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165

D q e1 (t) = αe3 + V1 D q e2 (t) = V2 D q e3 (t) = −αe1 + V3

(20)

Hence, we obtained the synchronization of two identical system (15) and (16) by feedback control. That is, the error will converge to zero when t → ∞. Let us consider 3×3 positive real value matrix M and also defined the feedback control as, ⎤ V1 

⎣ V2 ⎦ = M e1 e2 e3 V3 ⎡

(21)

Then the system (19) is asymptotically stable.

6 Numerical Simulation ⎛

⎞ −2 0 0 Let M be chosen as ⎝ 0 −3 0 ⎠ and substitute the values V1 , V2 and V3 in (19) 0 0 −4 we get, ⎛

⎞ −2 0 0 J (e1 , e2 , e3 ) = ⎝ 0 −3 0 ⎠ 0 0 −4

(22)

and the characteristic polynomial as, λ3 + 9λ2 + (2b + α 2 )λ + (24 + 3α 2 ) = 0. By taking the parameter value α = 0.1, the Eigen values of the Eq. (22) are λ1 = −4.666717101, λ2,3 = −2.16664 ± 1.04569i. Hence, the error states e1 , e2 and e3 converges to zero when t → ∞. Hence the synchronization and stability is achieved for the system (15) and (16). Figure 1 describes the bounded chaotic nature of the system.

7 Conclusion This article expressed the generalized chaotic system of stretch, twist and fold flow for fractional order. The stability and chaotic nature of the system are examined. For synchronization, a suitable control function is derived. The evaluation process is done through feedback control to the proposed system. Hybrid phase

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Fig. 1 The chaotic nature of the system

synchronization is investigated and achieved the synchronization for the indistinguishable chaotic system of stretch, twist and fold flow. The entire synchronization and anti-synchronization is considered to control the system. In order to validate the proposed theory, numerical simulation is presented. The chaotic nature of the system is presented through MATLAB software.

References 1. Vainshtein, S.I., R.Z. Sagdeev, R. Rosner, and E.J. Kim. 1996. Fractal properties of the stretchtwist-fold magnetic dynamo. Physical Review E 53 (5): 4729. 2. Vainshtein, S.I., R.Z. Sagdeev, and R. Rosner. 1997. Stretch-twist-fold and ABC nonlinear dynamos: Restricted chaos. Physical Review E 56 (2): 1605. 3. Nain, A.K., R.K. Vats, and A. Kumar. 2021. Caputo-Hadamard fractional differential equation with impulsive boundary conditions. Journal of Mathematical Modeling 9 (1): 93–106. 4. Liu, W., and G. Chen. 2003. A new chaotic system and its generation. International Journal of Bifurcation and Chaos 13 (01): 261–267. 5. Abdelouahab, M.S., and N.E. Hamri. 2012. A new chaotic attractor from hybrid optical bistable system. Nonlinear Dynamics 67 (1): 457–463. 6. Wang, Z. 2010. Chaos synchronization of an energy resource system based on linear control. Nonlinear Analysis: Real World Applications 11 (5): 3336–3343. 7. Wu, X., and S. Li. 2012. Dynamics analysis and hybrid function projective synchronization of a new chaotic system. Nonlinear Dynamics 69 (4): 1979–1994.

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8. Muthukumar, P., and P. Balasubramaniam. 2013. Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonlinear Dynamics 74 (4): 1169–1181. 9. Petráš, I. 2006. Method for simulation of the fractional order chaotic systems. Acta Montanistica Slovaca 11 (4): 273–277. 10. Jun-Jie, L., and L. Chong-Xin. 2007. Realization of fractional-order Liu chaotic system by circuit. Chinese Physics 16 (6): 1586. 11. Qiang, H., L. Chong-Xin, S. Lei, and Z. Da-Rui. 2013. A fractional order hyperchaotic system derived from a Liu system and its circuit realization. Chinese Physics B 22 (2): 020502. 12. Park, J.H., and O.M. Kwon. 2003. A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos, Solitons and Fractals 17: 709–716. 13. Lu, J., X. Wu, X. Han, and J. Lu. 2004. Adaptive feedback synchronization of a unified chaotic system. Physics Letters A 329: 327–333. 14. Childress, S., and A.D. Gilbert. 1995. Stretch, Twist, Fold: The Fast Dynamo. Springer Science & Business Media. 15. Nagaram, N.B., S. Rasappan, and N. K. Jothi. 2019. The backstepping control technique to break-up the life cycle of plasmodium parasite. In AIP Conference Proceedings, Tamil Nadu, India. 020017_1–020017_7.

Glucose Distribution and Drug Diffusion Mechanism in the Fuzzy Fluid Connective Tissue in Human Systems: A Mathematical Modelling Approach Sachindra Nath Matia, Animesh Mahata, Shariful Alam, Banamali Roy, and Balaram Manna Abstract The crucial point of this article is to discuss the application of fuzzy differential equation (FDE) on a biological problem. In this paper, a mathematical modelling for the diffusion of glucose and a typical drug molecule in the blood stream as the fluid connective tissue in human system has been designed and studied in a fuzzy and a heterogeneous environment, in order to understand the biological importance of the diffusion kinetics of the model drug molecule and the secreted glucose molecule from the meal to the blood stream for achieving the drug efficacy of treatment and the resulting glucose distribution in the body. Generalised Hukuhara derivative approach has been envisaged, wherein the mechanism of conversion of the mathematical models into system of crisp differential equations is discussed in details. The stability analysis of the same is performed in a fuzzy uncertain environment elaborately. The numerical solutions of the models are calculated and illustrated in an efficient way using MATLAB for understanding the theoretical basis of the study in details. Keywords Glucose diffusion · Drug diffusion · Hukuhara derivative · Fuzzy stability · Crisp equation

S. N. Matia Kukrahati High School (H.S.), Kukrahati, Sutahata, Purba Medinipur, West Bengal 721658, India A. Mahata (B) Mahadevnagar High School, Maheshtala, Kolkata, West Bengal 700141, India e-mail: [email protected] S. Alam Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India B. Roy Department of Mathematics, Bangabasi Evening College, Kolkata, West Bengal 700009, India B. Manna Department of Mathematics, Swami Vivekananda University, Telinipara, Barasat-Barrackpore Rd, Bara Kanthalia, West Bengal 700121, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_18

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1 Introduction According to the biological sciences, the absorption of glucose into the blood stream involves two main stages: (i) diffusion into the blood and (ii) active transport. Even though the modelling of physical phenomena is hypothetical in nature, however, the consequences bring about realistic outcome once analysed and checked observationally. Abd-el-Malek et al. [1] studied the diffusive models for absorption of drugs in membrane. Earlier, Khanday and Rafiq [2] used finite element technique to find the solution of diffusion equation about drug delivery system. Moreover, Khanday and Rafiq [3] proposed a model on TDD system and solve it using Crank–Nicholson and finite element method. Recently, modelling on biology in uncertain environment is the interesting and emerging field. It became recognised that distinct types of problems had been confronted due to the fact researcher first targeted to investigate mathematical system in existence of vagueness [4–7], imprecision and ambiguity in the system. In bio-mathematical model, uncertainty may also arise because of assumption of state variable, coefficients and initial situation. Data are constantly suffering from uncertainty, record nicely now not given, blunder in measurement process and so on. In that state of affairs to tackle the matter that arises from inexact environment fuzzy set theory [8], stochastic method and interval valued function [9–11] are functional tools. We look onto some of the previously published work [12–18] which can be of help to the researchers for understanding the topics of fuzzy boundary value. In this study, we considered fuzzy differential equation for diffusion of drug and glucose in blood stream and discussed solutions of dynamical system analytically and numerically in a fuzzy environment. The remaining part of this article is arranged as in Sect. 2, we consist of some preliminaries, in Sect. 3, we formulate mathematical model for diffusion of glucose and drug in the blood stream in a fuzzy environment, and in last section, i.e. Sect. 4, conclusion of the paper is included.

2 Preliminaries 

Definition1: TFN A triangular fuzzy number (TNF) is an order triplet as N1 = (N11 , N12 , N13 ), and representation of membership function is

ϕ



N1

=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

0, x−N11 , N12 −N11 N13 −x , N13 −N12

0,

x ≤ N11 N11 ≤ x ≤ N12 N12 ≤ x ≤ N13 x ≥ N13

Definition 2: Generalised Differentiability Let ϕ1 : [c, d] → E 1 and s0 ∈ [c, d]. We can say that ϕ1 is Hukuhara differential at s0 if ∃ an elementϕ1 (s0 ) ∈ E 1 such that for eachs > 0 very small, then ∃ ϕ1 (s0 + s)  ϕ1 (s0 ), ϕ1 (s0 )  ϕ1 (s0 − k)

Glucose Distribution and Drug Diffusion Mechanism …

and lim

s→0

ϕ1 (s0 +s)ϕ1 (s0 ) s

= lim

s→0

ϕ1 (s0 )ϕ1 (s0 −k) s

171

= ϕ1 (s0 ) for which the limits exists in

D 1 metric. Definition 3 The generalised Hukuhara difference of two fuzzy numbers p1 , p2 ∈ RF is defined by p1 g p2 ↔ {(i) p1 = p2 ⊕ r or (ii) p1 = p2 ⊕ (−1)r }. Definition 4: gH Derivative The gH derivative of a function (fuzzy valued)ψ1 : (a1 , a2 ) → Rf at t 0 can be defined as ψ1 (t0 ) = lim

h→0

ψ1 (t0 + h)ψ1 (t0 ) h

(1)

If (1) exists and ψ1 (t 0 ) ∈ Rf satisfies (i), we are able to say that ψ1 (t) is gH differentiable at t 0 . We can say that ψ1 (t) is (i) gH differentiable at t 0 if   [ψ1 (t)]a = [ψ11 (t, α), ψ12 (t, α)]

(2)

ψ1 (t) is (ii) gH differentiable at t 0 if   [ψ1 (t)]a = [ψ12 (t, α), ψ11 (t, α)]

(3)

2.1 Characterisation Theorem Let the fuzzy initial value problem be as dv(t) = f (t, v(t)), t ∈ [t0 , T ] dt

(4)

With initial condition v(t0 ) = v0 . f : I × E → E is a continuous fuzzy function, and v0 or the coefficient of the differential equation (DE) (4) or both are fuzzy number. The interval may be like [0, T ] for some T > 0 or I = [0, ∞). Theorem If f : I × E → E is a continuous fuzzy function such that ∃ m > 0 such that ( f (t, u), f (t, v)) ≤ m D(u, v) ∀t ∈ I, y, z ∈ E. Then, (4) has two different solutions, namely (i) gH differentiable solution and (ii) gH differentiable solution on I. Proof See article [18].

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3 Mathematical Model Formulation In the human body, V be the volume of blood, and consider that the initial concentration of glucose in the blood stream is g(0). Let glucose be introduced in the blood stream at a constant amount I. Glucose is also eradicated from the blood stream because of the biological process of human body at a rate proportionate to g(t), so that the continuity principals provides V

dg = I − kg dt

(5)

Now, consider a dose D of a drug is provided to a patient at a consistent intervals of time t each. The drug also evaporates from the system at a rate proportionate to c(t), concentration of drug in the blood stream, then the differential equation given by the continuity principle is V

dc = −kc dt

(6)

3.1 Fuzzy Solution of the Fuzzy Diffusion of Drug in the Blood Stream Model The problem of diffusion of drugs in the blood stream is very complex and interesting. Now, the prediction of drug availability in blood stream after certain period of exposure time may be of great challenge to understand. The amount of drugs in blood stream after some time of exposure depends on two things: (i) the amount of drug given initially (that means the initial condition c(0) = c˜0 ) and (ii) the concentration rate of the drug into the blood (that means k). From the concept of fuzzy differential equation (FDE), we study here only one following case: When the Initial Condition c(0) = c˜0 (That Means the Amount of Drug Given Initially Is a Fuzzy Number): Using the concept of gH derivatives, we converted the FDE into system of ordinary differential equations in two ways. The process is now applied bellow: Case 1: We find solutions of transformed differential equations when concentration c(t) of drug at time t in the blood is (i) gH differentiable, and stability analysis of the transformed model is explored analytically and numerically. (a)

Transformed Model: Differential Eq. (6) is transformed to V dc1dt(t,α) = −kc2 (t, α) V

dc2 (t, α) = −kc1 (t, α) dt

(7)

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Fig. 1 Reflects the fuzzy solutions of the system (7) for α(= 0, 0.5, 1)

(b)

with the initial condition c1 (0,α) = c01 (α) and c2 (0,α) = c02 (α). Exact Solution: Solving differential equations of the system (7), we get the solutions as given bellow: 1 1 k k {c01 (α) − c02 (α)}e V t + {c01 (α) + c02 (α)}e− V t 2 2 1 1 k k c2 (t, α) = {c02 (α) − c01 (α)}e V t + {c01 (α) + c02 (α)}e− V t 2 2 c1 (t, α) =

(c)

(d)

Stability Analysis: E 11 (0, 0) is critical (equilibrium) point  (7). The  of the kmodel 0 −V . The eigen variational matrix at E 11 (0,0) is given by M(E 11 ) = − Vk 0 values of M(E 11 ) are λ1,2 = KV , − KV . Hence, the equilibrium point E 11 (0,0) is saddle point which is unstable. Numerical Analysis: To plot Fig. 1, we consider values as k = 0.9776, V = 1, c 0 = (450, 500, 550), t ∈ [0, 3]. From Fig. 1, it is displayed that c1 (t, α) less than ( 0 : p = p0 m 2 , Q = Q ∞ + (Q w − Q ∞ ) D m  ,     H  = H∞ D m  , at z = 0 + Hw − H∞

(4)

p → 0, Q → Q ∞ , H  → H∞ as z → ∞. p2

where D = e0 . Introducing the following non-dimensional quantities:  2 1/3  p 1/  p 1/ p0 3 3 0 0 P=p 2 m, Z = z 2 ,m = e e e g=

M − M∞ iβ(Mw − M∞ ) , Gr = 1 Mw − M∞ (e p0 ) /3

  ) H  − H∞ iβ(Hw − H∞ , Gc = ,   1 Hw − H∞ (e p0 ) /3  1 e μHo σ B02 e 3 , Sc = , Pr = K = 2 ρ l A p0

H=

(5)

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257

in Eqs. (1)–(4), leads to ∂p ∂2 p −K P = Gr i + Gc H + ∂m ∂ Z2

(6)

∂g 1 ∂2g = ∂m Pr ∂ Z 2

(7)

1 ∂2 H ∂H = ∂m Sc ∂ Z 2

(8)

The initial and boundary conditions in non-dimensional quantities are P = 0, g = 0, H = 0 for all Z,m = 0 m > 0 , P = m 2 g = m , H = mat Z = 0

(9)

P → 0, g → 0, H → 0 at Z → 0.

1.1 Method of Answer The dimensionless governing Eq. (6) to (eight), situation to the corresponding initial and boundary conditions (9) are tackled using Laplace remodel technique, and the solutions are derived as follows:  √  √    2   γ Pr 2 (10) g = m 1 + 2γ Pr erfc γ Pr − 2 √ exp −γ Pr π  √  √     2  γ Sc 2 H = m 1 + 2γ Sc erfc γ Sc − 2 √ (11) exp −γ Sc π

    √  γ 2 + km m   √  exp 2γ km erfc γ + km P =2∗ 4k   √  √  + exp −2γ km erfc γ − km √  √  √  γ m(1 − 4km)   exp −2γ km erfc γ − km + 3 2 8k /

 √   √  γm 2 − exp 2γ km erfc γ + km − √ exp(−(γ + km)) 2k π

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 m   √   √  exp 2γ km erfc γ + km + b∗   2 √  √   + exp −2γ km erfc γ − km √  √  √  γ m  − √ exp −2γ km erfc γ − km 2 k   √  √   − exp 2γ km erfc γ + km    

√ √   2γ Sc 2 2 exp(−γ Sc) − b ∗ m 1 + 2γ Sc erfc(γ Sc) − √ π (12) where a =

k , Pr−1

b=

k , Sc−1

c=

Gr , a 2 (1−Pr)

d=

Gc . b2 (1−Sc)

2 Results and Discussion Figure 1 reveals that the consequences of the magnetic discipline parameter on the rate while K = 3, 4, 5, Gr = Gc = 4, Pr = 6 and m = 0.2. See (Fig. 2). Figure 3 appears that the flow field effects of different thermal Grashof number Gr = 2, 5, mass Grashof number, with the corresponding Prandtl number. Fig. 1 Velocity profiles

An Unstable Flow Past in a Vertical Plate …

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Fig. 2 visually explains that behavior

Fig. 3 Velocity profiles for different Gr, Gc

3 Conclusion Accurate study of the distribution over the reflective and flexible weight distribution within the proximity of the attractive subject is examined. Unlimited viewing conditions are revealed with the standard Laplace remodel design. The effect of speed, temperature, and awareness of various parameters such as Grashof temperature (Gr), Grashof weight number (Gc), and attractive title parameter (K) and time parameter (t) is assumed to increase, K and m. But, the slant is true and has grown into a circle in terms of the attractive title parameter

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References 1. Asimoni, Nor Raihan Mohamad, and Nurul Farahain, Mohammad. 2017. MHD free convective flow past a vertical plate. In 1st International Conference on Applied & Industrial Mathematics and Statistics 2017, vol. 890, 8–10 August 2017. 2. Turkyilmazoglu, Mustafa. 2012. MHD fluid flow and heat transfer due to a stretching rotating disk. International Journal of Thermal Sciences, 51, 195–201. 3. Jiao, Zeren, Trent Parker, Yue Sun, Qingsheng Wang. 2020. Recent application of Computational Fluid Dynamics (CFD) in process safety and loss prevention: A review. Journal of Loss Prevention in the Process Industries, 67, 1042–1052. 4. Shanker, B., and N. Kishan. 1997. ‘ The effects of mass transfer on the MHD flow past an impulsively started infinite vertical plate with variable temperature or constant heat flux. Journal of Energy Heat and Mass Transfer 19: 273–278. 5. Attia, H. A., and N. A. Kotb. 1996. MHD flow between two parallel plates with heat transfer. Acta Mechanica, 117, 215–220. 6. Jha, Basanth Kumar, and Ravindra Prasad. 1990. Free convection and mass transfer effects on the flow Past an accelerated vertical plate with heat sources. Mechanics Research Communications, 17, 143–148. 7. Cramer, K. R., and S. I. Pai. Magneto Fluid Dynamics for Engineers and Applied Physicists. Mc. Graw Hill, New York. 8. Hossain, M. A., and L. K. Shayo. 1986. The skin friction in the unsteady free convection flow past an accelerated plate. Astrophysics and Space Science 125, 315–324. 9. Kafousias, N.G., and A. Raptis. 1981. Mass transfer and free convection effects on the flow past an accelerated vertical infinite plate with variable suction or injection. Revue Roumaine des Sciences Techniques Serie de Mecanique Appliquee 26, 11–22.

Opinion of Faculty About the Effectiveness of Online Class During COVID Pandemic Sathish Kumar Kumaravel, Suresh Rasappan, Regan Murugesan, Kala Raja Mohan, and Vicente García-Díaz

Abstract This paper presents the results of a survey on faculty assessment on online classes, which was conducted in the midst of the epidemic. The analysis is focused on many characteristics of online classrooms, such as teaching efficacy, tool usage, and the quality of assessments delivered through online platforms. The fuzzy models like combined effective time-dependent matrix (CETD), average timedependent data matrix (ATD), and refined time-dependent data matrix (RTD) are applied using the fuzzy matrix theory concepts for the purpose of analysis to bring out the students views about the online classes. The effects of online classes on faculty members in a pandemic are evaluated using the average and standard deviation (SD) of real data matrices. Collect comments by circulating a few questionnaire survey among faculty via primary data collection in order to do the analysis. The results are presented in a graphical format. Keywords ATD · RTD · CETD · Fuzzy Matrix

S. K. Kumaravel (B) · R. Murugesan · K. R. Mohan Department of Mathematics, Vel Tech Rangarajan Dr Sagunthala R & D Institute of Science and Technology, Avadi, Chennai, Tamilnadu 600062, India e-mail: [email protected] S. Rasappan Department of Mathematics, University of Technology and Applied Sciences Ibri, Ibri, Sultanate of Oman V. García-Díaz Department of Computer Science, University of Oviedo, Oviedo, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_27

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1 Introduction Online classes are the lone tool of training during this pandemic circumstance. At the point when the entire world of academics stalled out up during the pandemic circumstance, online classes have given some assistance to defeat the present circumstance. Amidst numerous troubles, guardians make their wards study [1–3]. Thus, it is fundamental to give quality education. Concerning Web classes, faculties face hardships in various aspects. All things considered. In battle faced by the faculty, it is necessary to redress by the resources, for better instructing climate [4–8]. This paper targets drawing out the positive and negatives of online classes handled by faculty. The consequence of this investigation would help in drawing out a productive outcome about online classes. Section 2 describes the methodology of the work. Section 3 explains the type of survey from the data [9–11]. Sections 4 and 5 demonstrate the formation of ATD, RTD, and CETD matrix. Section 6 discussed about the graphical representation, and Sect. 7 gives the conclusion of the study.

2 Preliminaries and Methodology Fuzzy matrix has gained recent attention among many researchers. It has got its applications in numerous ongoing issues [12–15]. Fuzzy matrix investigation of traffic stream, investigation of demonetization, learn about students’ demeanor, COVID-19 pandemic, and study on viable of typing assessment have been performed. The hypothesis applied behind this examination is fuzzy matrix which includes CETD matrix, ATD matrix, and RTD matrix. This paper targets drawing out the investigation on online classes from understudy’s criticism utilizing fuzzy matrix.

2.1 Formation of ATD Matrix Original information gets changed to an original time subordinate information matrix by bringing the lines with the five classes that follow as strongly disagree (R1 ), disagree (R2 ), neutral, agree (R3 ), and strongly agree (R4 ). Furthermore, survey raised to the facilities as response about their viability and hardships of online classes during this pandemic circumstance. Utilizing the raw information network, we make it into the ATD matrix (ci j ) by separating every passage to raw information grid by the value 10 considered as a time span. The standard deviation of the sections happening on each segment of the ATD grid was found. The entries of ATD matrix represent the number which is divided by time period.

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2.2 Structure of RTD Matrix The refined time-dependent (RTD) matrix is given by condition, with parameter υ in [0, 1], the average μ j , and υ j at R1 column of the ATD matrix, (i) if ci j ≤ (μ j − υ(β j )), then eij = −1 (ii) if ci j ∈ (μ j − υ(β j ), μj + υ(β j )), then eij = 0 (iii) if ci j ≥ (μ j − υ(β j )), then eij = 1 The RTD matrix is obtained by the entries belong to the set {-1, 0, 1} by redefine ATD matrix.

2.3 CETD Matrix The CETD matrix is developed from RTD matrix by assuming the various values for α in [0, 1]. The summation of row is attained, and conclusions are derived for CETD matrix. Also, graphical representation of the row sums is given. This investigation depends on the response acquired by faculties for nine inquiries identified with online classes. The reaction to these inquiries is recorded depending on five classifications of answers. The individual responses of 410 faculties of a reputed organization are gathered. From the crude information network of these reactions, ATD and RTD grids are formed. This assistance to plan CETD fuzzy framework which is utilized for this investigation. Obtain some of the various refined respect to time fuzzy matrices for different value of α. The adoption of the RTD fuzzy matrix has the primary goal of reducing the amount of time necessary to deliver matrix. −1, 0 or 1 are the entries for this matrix.

3 Collection of Raw Data This examination depends on faculties’ response that are collected from survey on online classes. In online classes, faculties face both simplicity and trouble in investigation; it is important to overcome the troubles faced by the faculties. The poll depends on the troubles, and simplicity of the understudies has been outlined. An online survey has been conducted of establishment. The questionnaires is framed to check different aspects of studies through online classes. The issues identified are framed from (F1 ) to (F9 ) as below. F1 : Adopting to online learning is not difficult. F2 : Have enough time to prepare the course contents to deliver online live classes.

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F3 : This live class teaching experience will help to develop flipped courses, peerinstruction courses, and MOOCs in future. F4 : Own laptop/tablet is available. F5 : Digital divide among the students is quite low. F6 : E-content design and development tools (open source/subscribed) are readily Accessible. F7 : Online Platform used in online assessment is reliable and does not favor any student. F8 : Only very few students are indulging in malpractices during the online assessment tests. F9 : Overall quality of online assessment and evaluation is satisfactory. The responses of 410 from the faculty based on five classifications of answers are recorded. The response acquired from the following matrix.

G1 G2 G3 G4 G5

F1

F2

F3

1

0

5

8

4

39 196 166

F4

F5

F6

F7

F8

F9

12

4

2

5

20

6

7

20

24

13

23

54

25

36 185

60 218

33 123

153 209

85 242

70 188

92 197

90 232

185

210

222

20

68

124

47

57

4 Formation of ATD Matrix From the raw data, ATD and RTD matrices are calculated. Each entry of the respective matrix is divided by ten to obtain the ATD matrix which is given below

F1 G1 G2 G3 G4 G5

F2

F3

F4

F5

F6

F7

F8

F9

0.1

0

0.5

1.2

0.4

0.2

0.5

2

0.6

0.8

0.4

0.7

2

2.4

1.3

2.3

5.4

2.5

6

7

9.2

9

3.9

3.6

3.3

15.3

8.5

19.6

18.5 21.8 12.3

20.9

24.2

18.8 19.7 23.2

16.6

18.5

2

6.8

12.4

12

22.2

4.7

5.7

Now, to each column, elements of the ATD matrix, average, and standard deviation are obtained which are listed as below. Average and Standard Deviation matrix.

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265

F1

F2

F3

F4

F5

F6

F7

F8

F9

Average

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

Standard deviation

8.239

8.502

7.997

8.053

8.302

8.598

6.716

6.193

8.027

By choosing α = 0.9, the following RTD matrix is obtained.

F1 G1 G2 G3 G4 G5

F2

F3

F4 0

F5 −1

F6 −1

F7 −1

F8 −1

F9

−1

−1

−1

−1

0

−1

−1

0

0

0

0

0

0

0 1

0 1

0 1

0 0

0 1

0 1

0 1

0 1

0 1

1

1

0

1

0

0

0

0

0

5 Formation of ATD Matrix With the motive to form CETD matrix, by varying different values of α, the corresponding values for each response are obtained which are given as follows. The RTD matrix obtained by when α = 0.1, αα = 0.1 is given below.

F1 G1 G2 G3 G4 G5

F2

F3

F4

F5

F6

F7

F8

F9

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

1

0

−1

1

0

1

1

1

1

1

1

1

1

1

1

1

1

1

−1

−1

1

−1

−1

The RTD matrix obtained by when α = 0.3,α = 0.3 is given below.

F1 G1 G2 G3 G4 G5

F2

F3

−1

−1

−1

F4

F5

F6

F7

F8

F9

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

0

−1

1

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

1

1

−1

0

1

−1

−1

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The RTD matrix obtained by when α = 0.3,α = 0.9 is given below. F1

F2

F3

F4

F5

F6

F7

F8

F9

G1

-1

-1

-1

0

-1

-1

-1

-1

1

G2 G3 G4

0 0 1

-1 0 1

-1 0 1

0 0 0

0 0 1

0 0 1

0 0 1

0 0 1

0 0 1

G5

1

1

0

1

0

0

0

0

0

Using the values in above RTD matrix, we obtained CETD matrix given below.

F1 G1

G2 G3 G4 G5

-5 -4 -3 5 5

F2

-5 -5 -3 5 5

F3

-5 -5 -1 5 2

F4

F5

-4 -4 -3 3 5

-5 -3 4 5 -4

F6

-5 -4 0 5 -1

F7

-5 -4 -1 5 3

F8

-5 -2 1 5 -3

F9

-5 -4 0 5 -2

6 Result and Discussion Graphical representation of row sum matrix for α = 0.1, 0.3, 0.5, 0.7, 0.9 has been figured out. Together with this, the graphical representation of combined row sum matrix of all αα values together has also been portrayed. In addition, CETD matrix obtained is also represented graphically. Figures 1, 2, 3, 4, and 5 give the graphical representation with respect to α = 0.1, 0.3, 0.5, 0.7, 0.9. Fig. 6 portrays the graphical representation of combined form of all five α values together. Figure 7 is the graphical representation of CETD matrix. Figures 1 and 2 depict the graphical representation of row sum matrix for the parameter values α = 0.1 and α = 0.3. It shows that the greater number of faculty has Fig. 1 Graphical representation of row sum matrix for α = 0.1

10 5 0 -5 -10

Opinion of Faculty About the Effectiveness of Online Class … Fig. 2 Graphical representation of row sum matrix for α = 0.3

10 5 0 -5 -10

Fig. 3 Graphical representation of row sum matrix for α = 0.7.

10 5 0 -5 -10

Fig. 4 Graphical representation of row sum matrix for α = 0.5

10 5 0 -5 -10

Fig. 5 Graphical representation of row sum matrix for α = 0.9

10 5 0 -5 -10

267

268 Fig. 6 Combined graphical representation of row sum matrix for α = 0.1, 0.3, 0.5, 0.7, 0.9

S. K. Kumaravel et al.

Combined Matrix 10

0

-10

Fig. 7 Graphical representation CETD matrix

10

Combined Matrix

0

-10

given agree comment and then strongly agree about online class for questionnaires. Hence, that the feedback about online classes are in agree level from the responses of faculty. Figures 3 and 4 depict the graphical representation of row sum matrix for the parameter values α = 0.5 and α = 0.7. It shows that more number of faculty were given agree and strongly agree comments as feedback about online class for nine questionnaires apart from the rest of categories. So that the feedbacks about online classes from the faculty sides are in positive responses. Figure 5 depicts the graphical representation of row sum matrix for the parameter value α = 0.9. It shows that a greater number of faculties have been agree comments in as response about online class for the questionnaires apart from the rest of categories. The responses about online classes from the faculties’ sides are in agree level. Figure 6 depicts the combined graphical representation of row sum matrix for all the parameter values such as α = 0.1, 0.3, 0.5, 0.7 and α = 0.9. It shows that a greater number of faculty have been agree comments in order as feedback about online class for questionnaires. The rest of categories such as strongly disagree and disagree comments are placed in the negative values. The responses about online classes from the faculty side are almost in agreeing level.

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7 Conclusion The investigation shows that the faculty responses of neutral and agree have reached about the same value when adopting fuzzy matrix approaches. Based on the five categories of responses to the nine online class questionnaires, a study on the effectiveness and difficulties of online classes among teachers was conducted. The graphical representation of row sum matrices clearly shows the evident for the highest responses. This analysis reveals that more than half of the faculty gave neutral responses, indicating that they are willing to experiment with any mode of study, whether offline or online. Faculty that agreed with a “agree” suggest that they have been comfortable with online classes, i.e., the faculty does not face any difficulties in attending classes via the Internet. Based on the findings, faculty can improve their teaching methods and avoid flaws with regard to faculty who disagree with the online class and who fail to pay attention to learn.

References 1. Rajarajeswari, P., and P. Dhanalakshmi. 2013. Intuitionistic fuzzy soft matrix theory and its application in decision making. International Journal of Engineering Research and Technology 2 (4): 1100–1111. 2. Ahn, J.Y., K.S. Han, S.Y. Oh, and C.D. Lee. 2011. An application of interval-valued intuitionistic fuzzy sets for medical diagnosis of headache. International Journal of Innovative Computing, Information and Control 7 (5): 2755–2762. 3. Bellman, R.E. and L.A. Zadeh. 1970. Decision-making in a fuzzy environment. Management Science, 17(4), B-141. 4. Bezdek, J.C., B. Spillman, and R. Spillman. 1979. Fuzzy relation spaces for group decision theory: An application. Fuzzy Sets and Systems 2 (1): 5–14. 5. Dhingra, A.K., S.S. Rao, and V. Kumar. 1992. Nonlinear membership functions in multiobjective fuzzy optimization of mechanical and structural systems. AIAA Journal 30 (1): 251–260. 6. Kim, Y.H., S.K. Kim, S.Y. Oh, and J.Y. Ahn. 2007. A fuzzy differential diagnosis of headache. Journal of the Korean Data and Information Science Society 18 (2): 429–438. 7. Kokila, R., and C. Vijayalakshmi. 2013. Technique of fuzzy matrix to analyse the knowledge gathering attitude of research scholars. International Journal of Engineering Associates 2 (4): 17–19. 8. Vasantha, W.B., V. Indra, and S. Mandalam. 2000. Applications of fuzzy cognitive maps to determine the maximum utility of a route. Journal of Fuzzy Mathematics 8 (1): 65–78. 9. Zadeh, L.A. 1965. Information and control. Fuzzy Sets 8 (3): 338–353. 10. Cagman, N., and S. Enginoglu. 2012. Fuzzy soft matrix theory and its application in decision making. Iranian Journal of Fuzzy Systems 9 (1): 109–119. 11. Borah, M.J., T.J. Neog, and D.K. Sut. 2012. Fuzzy soft matrix theory and its decision making. International Journal of Modern Engineering Research 2 (2): 121–127. 12. Sut, D.K. 2012. An application of fuzzy soft relation in decision making problems. International Journal of Mathematics Trends and Technology 3 (2): 51–54.

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13. Abdullah, L., and N. Zakaria. 2012. Matrix driven multivariate fuzzy linear regression model in car sales. Journal of Applied Sciences 12 (1): 56–63. 14. Chetia, B., and P.K. Das. 2012. Some results of intuitionistic fuzzy soft matrix theory. Advances in Applied Science Research 3 (1): 412–423. 15. Lin, C.J., and W.W. Wu. 2008. A causal analytical method for group decision-making under fuzzy environment. Expert Systems with Applications 34 (1): 205–213.

A Deterministic Replenishment Policy for Constant Deteriorating Giffen Goods with Time-Dependent Demand Saranya Palanivelu and E. Chandrasekaran

Abstract In this paper, a deterministic inventory model for deteriorating Giffen goods with time-dependent linear demand and constant rate of deterioration is developed for Giffen good inventory system. The exception of law of demand is considered in here. The implementation of this model is illustrated by using some numerical examples. Sensitivity analysis is performed to show the impact on the inventory cost, economic order quantity, and cycle length by the changes in the parameters of the model on the optimum solution. Diagrams are used to demonstrate how the key variables of the inventory model get the changes as a result of variations in the model parameters. By the sensitivity analysis, it is observed that the deterioration and demand play the sensitive role on maintaining the inventory for a period of time. Keywords Giffen goods · Deterioration · Linear demand · Replenishment · Inventory

1 Introduction In many inventory systems, the effect of deterioration is critical. Deterioration is described as deterioration or damage to an item to the point where it can no longer be used for its intended purpose [1]. Food, drugs, pharmaceuticals, and radioactive materials are just a few examples of products that might go bad significantly throughout the regular storage time of the units, and as a result, this loss must be factored into the system analysis. The law of demand is rendered to be broken in the case of a Giffen good. The amount bought is inversely proportional to the price, according to the law of demand. In other words, the greater the price, the lower the amount requested. Because of the distinctive characteristics of a Giffen good, the amount required increases as the price rises. The price of an inferior commodity rises when people S. Palanivelu (B) · E. Chandrasekaran VelTech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamil Nadu 600062, India e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_28

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consume more of it, which is a gross violation of the rule of demand. Giffen goods include coarse grains (barley, maize, bajra, etc.) and cooking kerosene. In the case of rice, the economists discovered considerable evidence of Giffen behavior among Hunan families in China. Household demand for rice was reduced when the price of rice was lowered through the subsidy, whereas raising the price (by eliminating the subsidy) had the reverse effect.

1.1 Giffen Good Definition Giffen goods are low-cost, low-quality consumer goods that people purchase in greater quantities as the price of the item rises, and vice versa. Giffen goods, commonly referred to as inferior goods, are low-cost consumer goods that defy the fundamentals of supply and demand.

1.2 Conditions to Be a Giffen Good There are three fundamental requisites for a good to be termed a Giffen good, as Alfred Marshall said: • It must be an inferior good. When there is a severe lack of money or a budget cutback in a consumer’s family, the good is typically consumed. • There cannot be no close substitutes for the good. Even if there is an alternative, it should be enormously more overpriced than the Giffen good. It is generally accepted that even if the price of the usual equivalent item rises in the case of Giffen goods, demand for the Giffen commodity will stay static. • The product must account for a significant portion of the consumer’s income and usage. Meanwhile, it should not bring to any changes in the use of related common products. Figure 1 shows that the upward sloping demand curve for Giffen products indicates price and quantity demanded has a direct relationship. In general, the point of equilibrium for normal goods is defined as the intersection of the demand and supply curves, which determines the price of a normal good in a sustainable market. However, on Giffen products, the intersection of the demand and supply curves results in a rise in price and demand [2]. Now, in present situation, EOQ models have to integrate with the predictable economic considerations. The decision and strategy makers are under burden to include these added aspects into the decisionmaking process as well as the conservative economics. Supportable improvement is becoming a significant concern for companies in global [3]. Mathematical inventory models with a constant or time-dependent deterioration rate of inventory items were established by Ghare and Schrader [4], Covert and Philip [5], Shah [6], and others. Their models take into account the constant demand for the item. Their models take

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Fig. 1 Demand curve for a Giffen good

Fig. 2 Giffen good inventory level

into account the constant demand for the item. Other scientists followed suit and built inventory models with time-dependent demand rates for deteriorating items. Donaldson [7], Silver [8], Ritchie [9], Goel and Aggarwal [10], Datta and Pal [11], etc., only, a few of them are worth mentioning. Among the most recent works on this topic, Datta and Pal [12] and Pal and Mandal [13] stand out. With square demand, linear storage costs, and bottlenecks, Uthayakumar and Karuppasamy [14] created the pharmaceutical inventory model for the healthcare sector (Fig. 2).

2 Presumptions and Symbols The following presumptions and symbols are used to construct the mathematical model.

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2.1 Presumptions Only, one item is taken into account by the inventory system. A linear time-dependent demand rate is used to represent the demand rate. Holding cost is assumed to be a constant. The rate of replenishment is unrestricted. The lead time is zero. Shortages are not allowed. The rate of deterioration is considered to be constant for unit time at on-hand inventory, and there is no cure or replacement of deteriorated items during the cycle.

2.2 Symbol A—the ordering cost per order. Deterministic demand, D(t) = a + bt, a, b > 0 C—constant deterioration rate. q—the economic order quantity (EOQ). I(t)—the inventory on the level of time t; HC—carrying cost per cycle. DC—total deterioration cost per cycle; TC—total inventory cost per unit time. T —replenishment cycle time.

3 Mathematical Model dI (t) = −(a + bt) − cI (t) dt

(1)

With boundary conditions I (T ) = 0, I (0) = q

(2)

On solving (1), I (t) = C1 e−ct −

  b 1 a + bt − . c c

On using the boundary conditions,     ac − b −ct 1 b I (t) = q + e − a + bt − c2 c c

(3)

A Deterministic Replenishment Policy for Constant …

275

And q=

 1  cT e (ac + bcT − b) − (ac − b) 2 c

(4)

Then now I (t) =

   −ct 1 1  cT b e − + bcT − b) e a + bt − (ac c2 c c

(5)

Holding Cost     q  bT 2 ac − b (ac − b)T −cT − + − HC = ∫ h I (t)dt = h 1 − e c c3 2c c2 0 T

(6)

Deterioration Cost        T bT 2 a−b a − b + bT cT ∫ e − − aT + DC = d q − (a + bt)dt = d c c 2 0 (7) Total Inventory Cost TC = (OC + HC + DC)/T      q bT 2 ac − b (ac − b)T − A + h 1 − e−cT + − c c3 2c c2        bT 2 a − b + bT cT a−b e − − aT + +d c c 2

TC =

1 T



(8)

The goal is to keep minimum of overall inventory costs per unit time. The following are the required and sufficient requirements for cost minimization, d2 TC dTC = 0 and >0 dT dT 2 As a result, we must differentiate Eq. (8) with respect to T in order to satisfy the necessary condition, dTC = 0 expressed the nonlinear equation in T. dT

     b a − b + bT 1 b h 2 1 − ecT − ecT + d ecT (a − b + bT ) + ecT − (a + 2bT ) T c c c

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      q bT 2 ac − b 1 (ac − b)T −cT − + − − 2 A+h 1−e T c c3 2c c2      2   bT a − b + bT cT a−b e − − aT + + d c c 2 =0

(9)

The value of the cycle length T which will give if it satisfies the condition d2 TC >0 dT 2

(10)

Equation (9) is in nonlinear, making it challenging to solve using analytic methods as well as to check in the inequality (10).

4 Numerical Example To point up the model which built on above, few examples are considered with the various parameter values. Example 1 An example is considered based on the following values of parameters: a = 200, b = 0.5, A = 500 per order, h = 2, d = 40 per unit, and c = 0.01 to illustrate the model generated. T = 1.4331 is obtained by substituting and into Eq. (10). (44 day). When this best value of T * is substituted in equations, the minimum total cost per unit time is obtained as TC∗ = 695.3129 and economic order quantity q ∗ = 289.20. Example 2 In this example, the value of parameters are considered as in example 1 except with the deterioration cost, here d = 50 per unit. With the parameter values a = 200, b = 0.5, A = 500 per order, h = 2 and c = 0.01, the model gives the optimum value of cycle time T ∗ = 1.4043. And on substituting the optimal cycle time in the equations, one can obtain the minimum total cost TC∗ = 709.60 and the economic order quantity q ∗ = 283.34. Example 3 Suppose the ordering cost is increased, in that situation to explain about the optimum solution, let the parameter values as a = 200, b = 0.5, A = 1000 per order, h = 2, d = 50 per unit and c = 0.01. By taking those values into the equation, the optimal cycle length T ∗ = 1.9803. By this value, the total cost TC∗ = 1004.98 and the economic order quantity q ∗ = 401.005.

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Table 1 Sensitivity analysis for model parameters Parameter

Variation

T∗

TC∗

q∗

a

180

1.5096

659.85

274.37

190

1.4699

677.81

281.88

200

1.4331

695.31

289.2

210

1.3989

712.38

296.34

b

c

h

220

1.3671

729.05

303.3

0.3

1.4344

694.98

289.27

0.4

1.4338

695.15

289.23

0.5

1.4330

695.31

289.20

0.6

1.4324

695.48

289.16

0.7

1.4317

695.64

289.13

0.005

1.5001

664.99

301.70

0.01

1.4331

695.31

289.20

0.015

1.3741

724.43

278.15

0.02

1.3217

752.43

268.3

0.025

1.2746

779.58

259.45

1

1.8722

531.64

378.86

1.5

1.6092

618.93

325.10

2

1.4331

695.31

289.20

2.5

1.3045

764.07

263.05

3

1.2054

827.12

242.90

5 Sensitivity Analysis Example 1 is subjected to do a sensitivity analysis to observe how the values of T *, TC*, and q* change when other parameters change. Table 1 explains the impact of the change of values in the parameters.

6 Discussion of Results On increasing in the demand, deterioration rate and holding cost, the cycle length will be decreasing; the total inventory cost and economic order quantity are tend to be in increasing manner. If deterioration rate is much more, we cannot keep the much inventory. By observing the numerical examples, sensitivity analysis Table 1 and from the diagrams Figs. 3, 4, 5, and 6, we can make the following deductions which is briefly provided in Table 2.

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Fig. 3 Comparison with the parameter a

Fig. 4 Comparison with the parameter b

Fig. 5 Comparison with the parameter c

• Increase in all of the model parameters will result to the impact of increase in total inventory cost. • It can be noticed that the changes of TC∗ and q ∗ are same order of as changes in the demand shape parameter ‘a’.

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Fig. 6 Comparison with the parameter h

Table 2 Influence on cycle length, total cost, and EOQ through model parameters

Parameter

Influence on cycle length, total cost, and EOQ T∗

TC∗

q∗

↑ in a

↓

↑

↑

↑ in b

↓

↑

↓

↑ in c

↓

↑

↓

↑ in h

↓

↑

↓

• T ∗ , TC∗ , and q ∗ are almost insensitive toward the changes in the scale parameter ‘b’. • T ∗ , TC∗ , and q ∗ are moderately sensitive with the raise of the value of the parameter c. • T *and q*are sensitive to changes in the parameter h; T * and q* fall as the value of holding cost per unit h rises, but TC* rises as ‘h’ rises.

7 Conclusion In this research, a time-dependent demand in increasing pattern and constant deterioration rate is used to build an inventory model for Giffen goods (which will decay). The proposed model’s sensitivity has been applied to all of the system’s parameters. By this developed model for the deteriorated Giffen goods, one can get the optimum cycle length, order quantity, and inventory cost. We can add other parameters such as time-dependent deterioration rate, shortage cost, variable carrying cost, and nonlinear demand to the established model.

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References 1. Manna, S. K., Chaudhuri, K. S. 2006. An EOQ model with ramp type demand rate, time dependent deterioration rate, unit production cost and shortages. European Journal of Operational Research 171, 557–566. 2. Haagsma, R. 2012. Notes on some theories of Giffen Behavior. In New Insights into the Theory of Giffen Goods, 5–19. Springer-Verlag Berlin Heidelberg. ISBN 978-3-642-21776-0. 3. Srivastava, Saurabh, and Harendra Singh. 2017. Deterministic inventory model for items with linear demand, variable deterioration and partial backlogging. International Journal of Inventory Research 4: 333–349. 4. Ghare, P.M., and G.F. Schrader. 1963. A model for exponentially decaying inventories. Journal of Industrial Engineering. 14: 238–243. 5. Covert, R. P., and Philip, G. C. 1973. An EOQ model for items with Weibull distribution deterioration. AIIE Transactions 5, 323–326. 6. Shah, Y. K. 1977. An order level inventory model for a system with constant rate of deterioration. Journal of Operations Research 14(3), 174–184. 7. Misra, R. B. 1975. Optimum production lot.size model for a system with deteriorating inventory. The International Journal of Production Research 13, 495–505. 8. Silver, E. A. 1979. A simple inventory replenishment decision rule for a linear trend in demand. Journal of Operational Research Society 30, 71–75. 9. Ritchie, E. 1984. The EOQ for linear increasing demand, A simple optimum solution. Journal of the Operational Research Society 35: 949–952. 10. Goel, V. P., and S. P. Aggarwal. 1981. Order level inventory system with power demand pattern for deteriorating items. In Proceedings All India Seminar on Operational Research and Decision making, University of Delhi, Delhi-110 007. 11. Datta, T.K., and A.K. Pal. 1988. Order level inventory system with power demand pattern for items with variable rate of deterioration. Indian Journal of Pure and Applied Mathematics 19 (11): 1043–1053. 12. Datta, T.K., and A.K. Pal. 1992. A note on a replenishment policy for an inventory model with linear trend in demand and shortages. Journal of the Operational Research Society 43 (10): 993–100. 13. Pal, A. K., and B. Mandal. 1997. An EOQ model for deteriorating inventory with alternating demand rates, The Korean Journal of Computational & Applied Mathematics 4(2). 14. Uthayakumar, R., and S.K. Karuppasamy. 2016. A pharmaceutical inventory model for healthcare industries with quadratic demand, linear holding cost and shortages. International Journal of Pure Applied Mathematics 106 (8): 73–83.

Randomly Selection of Interior Points in SV Learning Algorithm Uses of Confidence Parameter M. Premalatha and C. Vijayalakshmi

Abstract Machine learning is a one of the subsets of AI. When a data received in the training set, according to the training dataset to build an algorithmic model based on machine learning concept, it is a specific testable prediction with progressive approach of input and output data that design can be applicable for large dataset in the real-time approach. Support vector machine (SVM) design is in large data applications. In sixties, Russia developed the algorithm based on nonlinear SV which is called generalized portrait algorithm. We evaluate the recognition accuracy of classification of the event using portrait algorithm. The data dependent and data independent show the effectiveness of a support vector machine learning algorithm approach of non-parametric methods to tractable for massive datasets in feature of high dimensional. In this approach, the new set of points, generated to the probability distribution PY ((y, f (y))) . Y ((y. f (y))), represented in the example is a support vector, and the probability distribution controls the marginal value and also corresponding weighted vector value. These weighted vectors belong to the training set but not classified with confidence factor. The active SVM learning based on points and their distance between points and hyperplane and newly enters the confidence factor is known as adaptive factor. The confidence factor value is computed from the availability of information around us by using the principle of the k-nearest neighbor, N dimensional data as a input X in to K dimensional space (feature), then K always greater than N through the mapping function φ. SVM algorithm gives both computation efficiency and accuracy in a simple manner. In large date set, we can solve by predictor–corrector method in step by step in this process, the similar dates are combined together and give maximum accuracy and also optimum solution to reduce the training error and test error through probability distribution. Here, SVM M. Premalatha (B) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India e-mail: [email protected] C. Vijayalakshmi Department of Statistics and Applied Mathematics, Central University of Tamil Nadu, Thiruvarur, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_29

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gives the efficient idea based for function estimation that covers both dependent and independent data. Keywords Machine learning · Support vector machine · Confidence factor · Recognition accuracy · Random generation · Hyperplane

1 Introduction Training datasets are labeled as S = ((x1 , y1 ) . . . (xm , ym )), yi belongs to ±1. A function of systems that learning the decision function of the u(x) = sign(w. x + b) a label belongs ±1 to for an unseen example x (previously). We have to find the hyperplane with maximum margin, distances from the closest example x in respective classes of the hyperplane. SVM performs the size of margin bounds on unseen data in the function of hyperplane. SVM design is related to the process of mathematical calculation and memory [1].

2 Support Vector Machine Methodology The nonlinear functions of SVM methods are defined as: ⎛ ⎞ m    y = sign⎝ α j y j k x j .x + b⎠

(1)

j=1

In VC, dimension can be minimizing the bound and misclassification error. A set of data does consist with a new set of points and previous SVs on Training SVM [2–4]. For a separating hyperplane f (x), separate the date in to two classes by passing through the middle of these two classes. An SVM classification gives the function as maximization with relates to w and b:   1 w − α j y j (w.x j + b) + αj 2 j=1 j=1 M

L pri =

M

S = {(x1 , y1 ), (x2 , y2 ), . . . . (x R , y R ). Here, xj (j = 1,…, R) and yj are the corresponding outputs of the given input x j . If xj = [x 1 ,…, x M ]T is the collected vectors from the availability of surroundings, yj is the corresponding class. The input vectors x j (j = 1,…, R) in N dimensionality. There are only two classes, and yi belongs to ±1. This assumption is given by the first step linearly separable of the classes and second step to evaluate the function that divides the data into two classes as linear

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283

Fig. 1 SVM separating hyperplane

(Fig. 1). To solve by using of linear learning of SV, to compute that separates both region belonging to the classes [5]. T To define the hyperplane, M j=1 w j x j +b = 0 where w = [w1 ,…, wM ] represents the weighted vector and b as the bias. The support vector machine method of learning process gives the better separation of the points into two classes. The hyperplane that generates the largest margin between the points in the training set of the class belongs to ±1. max M

w b w

y j (x j .w + b) ≥ M

j = 1, . . . , M

(2)

y j (x j .w + b) ≥ 1 j = 1, . . . , M

(3)

min M wb

The margin distribution gives the value as minimum and functional distribution of the marginal hyperplane (w, b) with relates to set S (training). In this SVM learning method, hyperplane defines as maximum margin hyperplane and also provides the solution for linear classification of training data Eqs. (2, 3). In Lagrangian multipliers, the maximum marginal hyperplane (w*, b*) forms: By applying Kuhn-Tucker models w∗ =

R 

y j λy ∗j X j

j=1

max y j =−1 (w∗ .x j ) + min y j =1 (w∗ .x j ) b∗ = 2

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Table 1 Sample of admissible kernel functions

Kernel functions K (x j , xk ) = K (x j , xk ) =

Type of classifiers

tanh(x Tj xk + c) (x Tj xk + 1)n

Multilayer perceptron Polynomial of degree n

K (x j , xk ) =  −1 1

(x j .xk ) e 2 (x j .xk )T

max W (λ) =

R 

λ

λj −

j=1

Gaussian RBF

R 1  y j yk λ j λk (x j .xk) 2 j,k=1

(4)

subject to R 

yjλj = 0

j,k=1

λ j ≥ 0 , j = 1, 2, 3, . . . , R In Eq. (1), the vectors perform in the input space (x j . x j ). This problem is not explicitly actual mapping function (xi ); By equivalent  l functions of kernel K(x j , x k ) defined as: K (x j , xk ) = (u j .u k ) = u Tj u k = T x j  (xk ), as performing the mapping function (Table 1). Slack variables are introduced in the constraints of the margin [6–8]. To solve the nonzero misclassification function rate of cost is on the training data. Then, the constraints (4) become to get the new solution of misclassification rate. R 

yjλj ≥ 1 − ξj

j=1

subject to

ξj ≥ 0 λj ≥ 0

To minimize the generalization error by using SV learning is as follows min : φ(w, ξ ) = (w.w) + C

R 

ξj

(5)

j=1

The solution obtained in the function is under decision variable of the form: ⎡ ⎤ R  f (x) = sign⎣ y j λ j (x.x j ) + b⎦ (6) j=1

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285

Only Lj coefficients are a small fraction of nonzero element and pairs of x j are known as support vectors in the decision function Eq. (6).

3 Structure of Place Classification In this paper proposed, raw data input is not directly applied into inputs of SVM. We assumed each sensor observation that is x = {n1 ,…, nM } composed the measures functions nj = (x j , d j ) where x j and d j are the bearing measures and range measures. The SVM algorithm has observation xi and its classification ci in each training examples [9]. It is given by E = {(xi , ci ) : ci belongs to Q}

(7)

Q is the set of classes Eq. (7). We assumed the training data class’s examples in advance (Fig. 2). To classify unseen data in the surroundings and learn a classification systems, generalize Eq. (5) from these training examples [10]. Define R as the set of all possible observation, and observation obeys r belongs to R defined as f j (q):R → Q. The methodologies are used to characteristic features from the gathered information of the training phase (Table 2). The following characteristic can be applied: • Measure of length should be average in all the class. • Difference between the lengths and consecutive measures it also average measure only. • A maximum possible value only admits.

Fig. 2 Overall structure of classification system

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Table. 2 Results of training error and test error Method

Training error

Test error

Linear SVM

0.24

0.29

SVM degree 4

0.17

0.24

SVM radial

0.15

0.22

Linear SVM degree4

0.18

0.27

Linear SVM degree4 RK

0.12

0.21

0.6 0.5 0.4 0.3 0.2 0.1 0

Test error Training error Linear SVM SVM SVM Degree 4 Radial

Linear Linear SVM SVM degree4 degree4 RK

Graph 1 Comparison analysis of test and training error

• Differences between the standard deviation and lengths of consecutive measures. • Limited-range measures in SD. • Absolute difference between consecutive measures and given threshold (b) are maximum. By transforming raw data, S in the SVM input becomes: S = {( f (x1 ), c1 ), ( f (x2 ), c2 ) . . . ( f (x R ), c R )} ci is the desired output classes (Graph 1).

4 Adaptive Confidence Factor with Active Learning Algorithm of Support Vector The objective of proposed algorithm, here new set of points, generated to the probability distribution PY ((y, f (y))) . Y ((y. f (y))) representing the example is a support vector. The active SVM learning based on points and their distance between points and hyperplane, and newly enters the confidence factor is known as adaptive factor. The current dataset of support vectors be V = (v1 , v2 ,…, vn ), test set T = (x 1 , x 2 ,…, x m ). For every vi belonging to V can evaluate the k-nearest points in the set, let k j− and k j+ are the points that have labels ±1. To define c (confidence factor):

Randomly Selection of Interior Points in SV Learning …

c=

287

l 2  min(k −j , k +j ) lk j=1

K is an integer k =  min c =

√ l

1 if k +j = k −j 0 if (k +j , k −j ) = 0

The principle case that all the SVs lie closest to the boundaries of the class and the vector set V = (v1 , v2 , …, vn ) is close support vector set involved in the actual set. Thus, the confidence factor c represents measures the nearest degree of V to the SVs involved in actual set. In confidence factor of c value is higher, when the present SV set is close to the actual set of SV’s [11].

5 Confidence Factor SVM is the repetition of the process in order to generate a sequence of probability distribution of outcomes which is familiar to the select examples for the subsequent step. A learning methodology for large margin classifiers in which the process under iteratively proceeds the point nearest to the current hyperplane. The algorithm is to select the new set of data points that divide the hyperplane space into two parts having counterpoise strength at each step, as expected in actual SVs. The class probabilities compute by using of logistic regression, which is used to calculate the error expected after adding a data [12]. This method involved the single point, but the result used in different sizes of batches included to single point.

6 Conclusion A classifier is built for each pair of classes, and the final classifier is the one that controls the most. The proposed method, the new set of points, generated to the probability distribution PY ((y, f (y))) . Y ((y. f (y))) represented in the example is a support Vector. The active SVM learning based on points and their distance between points and hyperplane and newly enters the confidence factor is known as adaptive factor. We used machine learning algorithmic approaches of non-parametric methods to tractable for massive datasets in feature of high dimensional. In this approach, the new set of points, generated to the probability distribution PY ((y, f (y))) . Y ((y. f (y))) represented in the example is a support vector, and the probability distribution controls the marginal value and also corresponding weighted vector value. The active SVM learning based on points and their distance between points and hyperplane and

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newly enters the confidence factor is known as adaptive factor. The process repeat’s until all the data is decomposed to reached optimization value successfully to avoid the error for classification. The confidence factor can be calculated by using the principle of k-nearest neighbor. SVM learning problems are maintaining a good approximation solution in the quick process and minimizing the time limit and also giving great speed. SVM can be easy to reach the scope and to perform numerical calculations easily.

References 1. Cerri, R., and A. de Carvalho. 2010. New top-down methods using SVMs f or hierarchical multi label classification problems. In IJCNN 2010, 1–8. IEEE Computer Society. 2. Campbell, C., and Y. Ying. 2011. Learning with Support Vector Machines. Synthesis Lectures on Artificial Intelligence and Machine Learning 5(1): 1–95. 3. Chen, B., and J. Hu. 2012. Hierarchical multi-label classification based on over-sampling and hierarchy constraint for gene function prediction. IEEE Transactions on Electrical and Electronic Engineering 7: 183–189. 4. Soudry, Daniel, Elad Hoffer, SuriyaGunasekar MorShpigelNacson, and Nathan Srebro. 2018. The implicit bias of gradient descent on separable data. Journal of Machine Learning Research 19 (1): 2822–2878. 5. Hush, Don, Patrick Kelly, Clint Scovel, and Ingo Steinwart. 2006. QP algorithms with guaranteed accuracy and run time for support vector machines. Journal of Machine Learning Research 7: 733–769. 6. García, Fernando Turrado, Luis Javier García Villalba, Javier Portela. 2012. Intelligent system for time series classification using support vector machines applied to supply-chain. Expert Systems with Applications 39: 10590–10599. 7. Talwar, Kunal. 2020. On the error resistance of hinge-loss minimization. Advances in Neural Information Processing Systems 33: 4223–4234. 8. Sur, Pragya, and Emmanuel J. Candès. 2019. A modern maximum-likelihood theory for high dimensional logistic regression. Proceedings of the National Academy of Sciences 116 (29): 14516–14525. 9. Premalatha, M., and C. Vijayalakshmi. 2014. Using optimization methodologies to find the solution of support vector machine with maximum accuracy. Pensee Journal, publication on La Pensee Multidisciplinary Journal, Paris, France, 0031–4773. 10. Wu, Qi, and Rob Law. 2011. The complex fuzzy system forecasting model based on fuzzy SVM with triangular fuzzy number input and output. Expert Systems with Applications 38: 12085–12093. 11. Xu, Yitian, Laisheng Wang, and Ping Zhong. 2012. A rough margin-based ν-twin support vector machine. Neural Computing and Applications 21: 1307–1317. 12. Liu, Yang, and Gareth Pender. 2015. A flood inundation modelling using v-support vector machine regression model. Engineering Applications of Artificial Intelligence 46: 223–231.

Using Convolutional Neural Networks for Fault Analysis and Alleviation in Accelerator Systems Jashanpreet Singh Sraw and M. C. Deepak

Abstract Today, Neural Networks are the basis of breakthroughs in virtually every technical domain. Their application to accelerators has recently resulted in better performance and efficiency in these systems. At the same time, the increasing hardware failures due to the latest semiconductor technology need to be addressed. Since accelerator systems are often used to back time-critical applications such as selfdriving cars or medical diagnosis applications, these hardware failures must be eliminated. Our research evaluates these failures from a systemic point of view. Based on our results, we find critical results for the system reliability enhancement, and we further put forth an efficient method to avoid these failures with minimal hardware overhead. Keywords Fault analysis · Neural networks · Accelerator systems · Deep neural networks · Convolutional neural network

1 Introduction Deep neural networks (DNNs) have been shown to be effective in massive territories like image processing, video processing, and natural language processing over the years [1–3]. The success further provokes the flourish of customized neural network accelerators with massive parallel processing engines which typically offer much higher performance and energy efficiency compared to general-purpose processors (GPPs) [4–8]. While the performance of neural network accelerators has been intensively optimized from various angles like pruning and quantization, the reliability of the accelerators especially the ones that are deployed on FPGAs remains not wellexplored. The shrinking semiconductor technology greatly improves the transistor J. S. Sraw Thapar Institute of Engineering & Technology, Patiala, India e-mail: [email protected] M. C. Deepak (B) PES College of Engineering, Mandya, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_30

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density of chips, but the circuits with smaller feature size are more susceptible to manufacturing defects and abnormal physical processes like thermal stress, electromigration, hot carrier injection, and gate oxide wear-out. Thereby, the persistent (hard) faults become one of the major sources of system unreliability. Despite the fault tolerance of neural network models, permanent faults in neural network accelerators can cause computing errors and considerable wrong predictions [9]. They may even lead to disastrous consequences to some of the mission-critical applications such as self-driving, nuclear power plants, and medical diagnosis, which are sensitive to neural network prediction mistakes. It is seminal to comprehend the impact of errors on neural network accelerators to improve the model functioning. To this end, several research works are published that contain methods to identify errors in neural network accelerators [10–14]. However, they usually experimented with simulators and focused on the influence of hardware faults on the prediction accuracy of the neural network models. For instance, the authors in [10] mainly investigated the data faults of weights, activities, and hidden states, which are stored in on-chip buffers. Then they explored the neural network model resilience with model-wise analysis and layer-wise analysis. The work in [13] targeted at the analysis of software errors in DNN accelerators and explored the error propagation behaviors based on the structure of the neural networks, data types and so on. The simulation-based approaches are fast for prediction accuracy and analyzing neural network computing, but they typically ignore critical controlling details and interfaces of the DNN accelerators to ensure the simulation speed. Nevertheless, hardware faults on these components may have considerable influence on the overall acceleration system other than the prediction accuracy loss. For instance, faults in the DMA module may result in illegal memory accesses and corrupt the system. This must be addressed to guarantee reliable DNN acceleration especially in missioncritical applications. In addition, many DNN accelerators are implemented on FPGAs for more intensive customization and convenient reconfiguration. While the functionality of the DNN accelerators is mapped to the FPGA infrastructures instead of the primitive logic gates, the simulation-based approaches that usually inject errors to the operations used in DNN processing do not apply to the FPGA based DNN acceleration system. Because hardware faults in FPGAs affect the configuration of the devices instead of the accelerator components directly while the actual parts of the accelerators that are influenced depend on the FPGA placing and routing. To gain further insight of faults in DNN accelerators, we conduct the fault analysis on a running ARM-FPGA system where the FPGA has a representative DNN accelerator with 2D systolic array implemented along with hardware fault injection modules and shares the DRAM with the ARM processors. Since the fault distribution models are implemented with software on the ARM processors, they are convenient to change for fault analysis using different models. Aside from that we work with four different models for detecting the others. Li et al. [15] operated a one-dimensional convolutional neural network along the time axis to capture the temporal dependency. Shi et al. [16]’s work focuses on the internal leakage fault diagnosis caused by the wear based on intrinsic mode functions (IMFs). Unlike prior works that focused on prediction accuracy loss analysis, we try to

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analyze the behaviors of the DNN accelerators under hardware faults and investigate the system functionality, fault coverage, input variation. We studied the system stall that dramatically destroys the system functionality in detail and present a simple yet efficient approach to alleviate the problem. The following lines sum up our contributions to the literature: • Our system focuses on fault analysis of neural network accelerators on ARMFPGAs. To make the analysis easier, it provides high-level interfaces for both fault injection and output data comparison of neural network models. • On top of the fault analysis system, we mainly classify and analyze the resulting behaviors of the DNN accelerators from a system point of view. On top of the prediction accuracy, we also study the fault coverage, system functionality, input variation when the accelerators are exposed to persistent faults. • With comprehensive experiments over representative models, we observe that system stalls that can destroy the system functionality of DNN accelerators cannot be ignored. And we further show that errors in data movement instructions are a key reason for the system stalls, which can be addressed with negligible hardware overhead. The remainder of the paper can be described in the following paragraphs. Section 2 sheds light on the introduction to deep learning neural networks and typical DNN accelerators with 2D computing array. In Sect. 3, we describe the proposed framework on ARM-FPGAs and elaborate on the major aspects that we will investigate from a system point of view. Section 4 details the results and evaluation. In Sect. 5, we give a brief review of prior fault-tolerant analysis and design of neural network accelerators. Finally, we conclude this work in Sect. 6.

2 Background The main causes of unreliability are hardware flaws in DNN accelerators. The impact of errors is inextricably linked to the microarchitecture of the DNN accelerators. In this part, we take a standard DNN accelerator with a normal 2D processing element (PE) array as an example and develop its design to assist, understand and study the fault tolerance of the accelerators. Figure 1 depicts a sample DNN accelerator. It uses output stationary data flow to transfer computations like convolution to a 2D computing array. Every PE completes all the processes necessary to produce an output activation. While each PE just has a single multiplier and accumulator, it progressively accumulates all input activations in a filter window. While input activations are organized in a batch and sent to one column of PE every cycle, each PE in the column shares the same input data through broadcasting and it takes each PE multiple cycles to complete the accumulation. During this period, more batched input activations can be read and sent to the next column of PEs along with the movement of the weights. Output activations flow from right to left in column-wise. Eventually, each row of the PEs array produces a

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Fig. 1 Typical DNN accelerator architecture

set of sequential output activations in the same row of one output feature map on the y-axis. The architecture along with the compact data flow achieves high data reuse under limited on-chip buffer bandwidth provision. Both convolution layer and full connection layer can be mapped to the array efficiently. While pooling and other nonlinear activation functions such as sigmoid will be performed right after the computing-intensive layers like convolution layer in a module named XPE such that the data movement between the two layers can be reduced. All the neural network operations can be mapped to the accelerator. To associate the models with various layer combinations and characteristics such as stride size and kernel size, we define a set of instructions to generate appropriate control signals for different neural network operations. Each neural network will be compiled to a series of instructions and executed sequentially. In addition, neural network input features and weights are usually larger than the on-chip buffers and PE array size, so they must be tiled, and the tiles need to be scheduled to obtain efficient execution on the accelerator. To enable fine-grained optimizations, each instruction only handles operations of a single tile. Thus, tiling is performed during model compilation, and it is transparent to the instructions. Table 1 shows the instruction set of the neural network accelerator. It adopts 64-bit fixed length encoding and consists of four types of instructions including parameter setup, calculation, data movement and control. The parameter setup category defines the input/output feature size, kernel size, Q-code, and DMA parameter. Calculation category includes different operations in neural networks such as convolution, full connection, pooling, addition, softmax, dot-accumulation and activation function

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Table 1 Instruction set of the DNN Accelerator Instruction type

Description

Parameter setup

Setup parameters for the computing operations such as the input/output feature size, kernel size, Q-code, and DMA options

Calculation

Performs various neural network operations such as convolution, full connection, pooling, addition, softmax, dot-accumulation and activation function

Data movement

Move a block of data from buffer to buffer, buffer to DRAM and DRAM to buffer

Control

Control the execution of the accelerator such as Jump, Nop and Stop

etc. Data movement category includes three instructions which move a block of data from DRAM to buffer, buffer to DRAM and buffer to buffer, respectively. Finally, the control category includes three instructions which are Jump, Stop and Nop. Jump instruction is mainly used for repeated execution. Stop is used to terminate the execution of the accelerator. Nop is used to resolve the data dependency between sequential instructions. The neural network accelerator architecture is general enough to support various neural network models. In addition, it typically works along with a general-purpose processor and has an AXI slave port that allows configuration and controlling from the attached processor. It assumes the input data, weight and output data are stored in DRAM that can be accessed directly.

3 DNN Acceleration Fault Analysis Platform and Fault Classification 3.1 Fault Analysis Platform We built a fault analysis platform as represented in Fig. 2. It has an unusual neural network accelerator based on FPGA. In addition to this system, we needed a module as present in Fig. 2. The fault injection data path is marked with orange arrows. It is implemented on both the ARM processor and FPGA. The FPGA errors are prevalent in the four different memory types. For FPGA configuration memory, we leverage Xilinx ICAP port [17] that allows user logic to access configuration memory, to inject errors. We select the frames and the number of bits for each bit error injection arbitrarily. It is possible that the error may be present at any location of the FPGA configuration memory. After finding the error location, we read the whole frame out of the configuration memory [18], change the victim bit in the frame, and write it back to the configuration memory [19]. We also produce an injection mask for block RAM and distributed memory as represented in Fig. 3. It essentially consists of a collection of address and mask

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fault injection path ICAP

Fault Model

Result Compare

AXI

ARM Processor

PE Array

DRAM

result comparison path Configuration Memory Err. Mask

In Buffer

Err. Mask

Out Buffer

Err. Mask

Wt. Buffer

Err. Mask

Inst. Buffer

FPGA

Fig. 2 Overview of the fault analysis system

Fig. 3 Error mask for fault injection to on-chip memory

Error Mask

Mask registers

XOR XOR XOR

XOR

din Buffer waddr from processor

dout

= ==

=

raddr

Address registers

registers. The addresses in the registers reflect the positions of the faults to be injected, while the associated masks keep track of the precise error bits.

3.2 System Fault Classification Prior fault analysis works usually focus on computing errors of the neural networks and the incurred prediction accuracy loss. We argue that the consequences of hardware faults on a DNN acceleration system vary and should be classified into more categories. Table 2 shows the proposed classification. From the perspective of a system, the consequences caused by hardware faults roughly include system exception and accuracy degradation. System exception indicates that the neural network execution behaves abnormally, which may be stalled without returning or returns too fast or too slow. Basically, we define them as system stall and abnormal runtime.

Using Convolutional Neural Networks for Fault Analysis … Table 2 Proposed classification

295

2*system exception

system stall

3*Accuracy degradation

L0

abnormal runtime … LK

For the accuracy degradation, we further classified into two cases including system stall and abnormal runtime. I We chose Neural Networks in four different application scenarios and try to analyze the differences of error tolerance in different application scenarios and networks. The four network applications include ResNet network for image classification, YOLO system for target detection, LSTM network for voice classification, and DCGAN network for image generation. We will evaluate their fault tolerance from accuracy and output consistency. Neural Network accelerators injected with hardware errors may produce unexpected conditions. We define the system halt situation, which refers to a serious error in the system, or working improperly. Such as unable to read and write registers, timeout, abnormal short runtime, etc. When the system halts, we need to reset the FPGA and restart the system. System halt situations are considered the result of errors in the evaluation of network accuracy and are listed separately in the output consistency. Network accuracy refers to the overall accuracy of the network when performing corresponding tasks, such as the accuracy of 20,000 image recognition. When there are errors in the operation, the accuracy of the network will show a downward trend. For classification networks including ResNet and LSTM, top-5 accuracy is used to evaluate their accuracy, and for the YOLO system, mAP is adopted. Output consistency is the difference between the result of running with errors injected and the result of normal running. We ran the corresponding data set when no errors were injected into each network at first and defined the results as standard output. The results of Neural Networks prediction injected with errors are divided into two categories: result with deviation and result match. Result with deviation refers to the system working properly with output differently from standard output. Result match means that the system works properly, and the outputs are still standard output. For the result with deviation case, we define its deviation quantitatively and further subdivide the result. Due to the different application functions of each network, the evaluation criteria of deviation are also different. For the YOLO system, the result is the target detection bounding box, and when the detection result does not match the standard output in object type, it is defined as detection result error. When the result target type is consistent, the error is defined as the intersection area of the error output and the standard output divided by the area of the union. Target types do not match for one single level, the two do not overlap with each other for one level, and then each 20% is divided into one level. For ResNet and LSTM, the outputs are top-5 labels. When the error output is not completely consistent with the standard output, the number of elements in the intersection of the two is taken, and divide the levels

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refer to the number. For DCGAN, we used the universal SSIM standard and divided it into six levels according to the actual visual effects: 0 10%, almost impossible to recognize; 10 20%, barely visible; 20 50%, with large deformation or distortion; 50 80% with partial deformation or distortion; 80 90%, small deformation or distortion can be seen; 90 100%, almost no deformation or distortion is visible.

4 Experiment We want to know how persistent faults affect FPGA-based neural network acceleration systems. Particularly, we try to analyze the influence from a system point of view and figure out the underlying reasons for severe system problems such as system stall and dramatic prediction accuracy loss.

4.1 Device and Environment Xilinx Zynq-7000 SoC ZC706 Evaluation Board will be used in the experiments’ hardware implementation. It has appropriate hardware resources and is easy to develop and use. The hardware design and bitstream file compilation were completed using the upper computer with Intel Core i7-6700 processor and 2 × 8 GB DDR4 2400 MHz memory. The system environment used was Ubuntu 16.04 LTS version, and Xilinx Vivid Design Suite and Xilinx Software Development Kits version 2017.4. The hardware resource utilization of the error analysis platform implementation on ZC706 is shown in Table 3. The four models cover a broad range of applications. Yolo represents a typical neural network model for object detection [20], Resent is a widely adopted neural network model for classification [21], LSTM is the mostly used neural network model for audio classification tasks [22], and DCGAN stands for a typical neural network model for generative tasks [23]. Despite the widespread adoption of deep learning neural networks in various applications, it is particularly Table 3 Utilization report of fault analysis platform

Resource

Utilization

Available

Utilization percentage (%)

LUT (centered)

122,618

218,600

56.09

LUTRAM

185

70,400

0.26

FF

84,641

437,200

19.36

BRAM

203

545

37.25

DSP

297

900

33

MMCM

1

8

12.5

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successful in four categories of tasks including object detection, object classification, voice recognition and style transfer. Among a great number of neural network models, Yolo, ResNet, LSTM and DCGAN are four typical neural networks that are comprehensively explored to handle the four computing tasks, respectively.

4.2 Overview We hope to explore the possible consequences of errors, the fault-tolerant ability of different applications and the influence of different error locations of accelerators, which can provide meaningful references for the subsequent network optimization and the fault-tolerant design of accelerators. We conducted five sets of experiments, which explained the influence of single error on different networks, the difference of fault tolerance ability of different applications, the error classification of different applications, the influence of different accelerator units and whether the input data affected the error representation.

4.3 Error Consequences and Coverage First, we conducted single-bit random error injection experiments for different applications, and conducted 20,000 runs for each application, to analyze the proportion of single-bit random error shielded in the system and the possible influence of single-bit random error on the accelerator system. Figure 4 shows the percentage of application errors caused by a single-bit random hardware error. In the experiment, we found that more than 90% of the errors were masked by software or hardware. However, while most errors are masked without impact to the operation, a single-bit error can lead to serious exceptions, including system halt, serious deviation in results, and so on. In addition, the LSTM network has a better fault tolerance performance than other

1 error

60.19%

34.67% 9.61%

7.92%

5.19%

3.99%

2.51%

1.56%

0.87%

9.96%

5.89%

20

9.09%

18.15%

18.33%

29.03%

31.06%

40

38.95%

50.90%

60

4.58%

Error Rate (%)

80

0 YOLO

ResNet

LSTM

DCGAN

Fig. 4 Application error rate caused by single-bit random hardware error

2 errors 4 errors 8 errors 16 errors

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three networks, and the proportion of the influence caused by single error of the other three networks is about 5–7 times that of LSTM. According to our analysis, this is because the LSTM network is smaller than other networks and uses less storage and computing resources.

4.4 Effect of Error Number

Network Accuracy Rate

We completed a single run with errors injected number of powers of 2, with 20,000 runs for each network. In general, the proportion of system halt and result with deviation increases with the number of errors, and the accuracy of application decreases. Figure 6 shows the proportion of system halt and result deviation, and Fig. 5 shows the accuracy of each network. We found that the influence of multiple errors on the accelerator was obvious and far beyond our expectation. The Neural Network accelerator is still vulnerable to errors. By the time we injected 16 errors in a single run, the system halt rate was more than 1%, meaning it took more time to restore the system than to run it. From the perspective of network accuracy, take the YOLO system as an example, its mAP decreases by 8.95% when it runs with 16 errors, meaning the application function of the system is also seriously affected. The result with deviation proportion of different networks appears differently with the increase of error number. When 16 errors are injected, the LSTM network still has a small result deviation proportion due to its small network structure. The proportion of YOLO systems is very high, with about 70% of the results showing errors. ResNet is relatively low, with only about 35% result deviation. The DCGAN network is in between, and about 50% of the results have numerical errors. We believe that the result deviation may be related to the structure of the output. The output of the YOLO system contains more information such as object type, location and size of bounding box, etc., and the output of ResNet is simple sorting, while the simple output is obviously less susceptible to errors.

100

YOLO mAP

90

ResNet top-5 accuracy

80

LSTM top-5 accuracy

70

DCGAN image SSIM

60 50 0

1

2

4

8

16

Errors per prediction Fig. 5 Network accuracy versus error number

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Fig. 6 Abnormal situation proportion versus error number

4.5 Details of Result with Deviation The results with errors are further classified and analyzed. In general, most of the errors are small, with a certain proportion of serious errors and relatively few moderate ones. We believe that most of the errors do not belong to the errors with global influence, but only affect one or several calculations. For example, a single value in the convolution kernel changes. Alternatively, a portion of the error is masked by subsequent calculations such as the max pooling layer. Some errors may have an impact on the control path or the reusable module, resulting in the accumulation of errors throughout the calculation; Or errors that cause serious deviations in the data, such as sign bit upset, can have serious consequences. Figure 7 shows the details of result distribution of results with deviation situations. Specific to each network, about 70% of the YOLO system’s errors belong to the level of bounding box overlap ratio more than 80% (Fig. 8). The total of object type errors and non-overlapping boxes is about 20%. The remaining 10% or so are moderate errors. We think this is caused by the implementation of YOLO. YOLO divides the input images into a series of grid cells, and each grid cell is only responsible for one kind of object. Moreover, there is a binary judgment Pr(Object) whether there is a target or not in the confidence degree, which leads to more object type error cases. The bounding boxes that are given after object

Fig. 7 Distribution of result with deviation situations

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Fig. 8 Proportion of different error location

recognition based on relative wide and high are less sensitive. More than 70% of the errors in the ResNet and LSTM classification networks were small impact results of matching four or five items. We think this is because the output is the softmax layer, which is less affected by the error, and the error is easy to be hidden when sorting, so the result only shows a small altar. However, with the increase of the number of errors, the proportion of serious errors of no matching item in ResNet results increased and exceeded 10%. Serious errors cannot be hidden, and the network’s fault tolerance for multiple errors is relatively limited. The result of DCGAN is about 90% of the results are small deviation levels of more than 90% of SSIM. Only a few of them have a large deviation. Combined with the image, in 90% of the small deviation results compared with the standard output, basically no difference can be directly seen, and only a few large deviation results have serious distortion of the image. Considering that the output is picture information, these small deviations can be ignored without affecting the visual effect, we believe that DCGAN has relatively strong error tolerance. F. Effect of Error Location In this period, we conducted the experiment results of different error locations. We injected a single-bit random error into a designated location, and conducted multiple experiments to observe the performance of the network. By analyzing the influence of error in different locations on the accelerator, it can provide specific methods for the subsequent fault-tolerant design. In general, the system is less affected by configuration memory errors, considering that the error injection of configuration memory is global, whose errors may not affect the system. Errors in configuration memory can cause system halt or result with deviation. Errors in BRAMs used for instruction buffers can cause system halt or result with deviation, while BRAMs used in other type buffers can only cause numerical deviations. We focus on the analysis of the system halt situations caused by error in the instruction buffer, which takes about 20% part of the system halt situations. Compared with the instruction before and after upset, the system halts caused by instruction errors include three situations: instruction type changes, wrong instruction not defined, and the parameter in the operation is abnormal. Different instruction types of the original instruction are considered. Tables 4 and 5 show the detail of instruction error. About 50% of the system halt situations are caused by the error of DMA instructions. The abnormal access address or boundary violation caused by the abnormal parameters of DMA instruction will lead to the system halt. About 30% are caused by AGU instruction errors, which result in abnormal in-chip control flow. The proportions of errors in instruction buffer led to system halt in ResNet, YOLO and LSTM, and reached 1.15%, 2.55% and 4.25%, respectively, seriously affecting the proper application of the system. For the result with deviation case, in the experiment of YOLO,

Using Convolutional Neural Networks for Fault Analysis … Table 4 Instruction change

301

Situation

Percentage (%)

Error instruction undefined (centered)

2.48

Instruction type change

7.45

Abnormal parameter

Table 5 Original instruction type

90.06

Type

Percentage (%)

DMA (centered)

55.90

AGU

32.92

others

11.18

about half of the result deviation situations caused by instruction buffer errors were serious object type errors. In the experiment of ResNet, many mismatches were also caused. The DCGAN network is limited by instruction errors, because the instruction sequence length of this network is very short, only 2% of the instruction buffer is used, while the instruction sequence of other networks uses an instruction buffer of 50%˜80%. Combined with the result with deviation and system halt case, we propose that the instruction buffer needs to be strengthened in the fault-tolerant design. Errors in the buffers used in dataflow, such as weights, data, and bias buffer, do not cause system halt, only may cause result deviations. In general, data and weights are more sensitive than bias. Taking the YOLO system as an example, the proportion of result deviation caused by single error in weight, data and bias buffer is 48.75%, 56.95% and 6.35%, respectively. Horizontal comparison shows that each network has different sensitivity to different buffer errors, as shown in figure.

4.6 Input-Related Error In the above experiments, we used different input data for experiments, and we verified the relationship between errors and input data in this set of experiments. We test different input data using the same error. Whether a hardware error causes an application error exists in two ways, depending on the input data or not. Errors unrelated to input data, fixed to cause system halt or result deviation, or be masked, that is, different input data will not influence the classification of the result. The other part of the errors is input data related, which can be shown as replacing different input data, result match situation and result with deviation situation both appear. In other words, different input data has different sensitivity to an input-related error. We believe that this is caused by structures which are error affected. Input-unrelated errors may affect

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the control flow of the system, the bus, etc. These structures are used in every operation, and the errors will not be masked by subsequent calculations. Input-related errors may affect the relevant data in the dataflow, and some errors may be masked in the subsequent calculation. Due to the large number of experiments, we only observed this phenomenon without further study on its proportion and distribution.

5 Related Work To gain insight of fault tolerance of neural network accelerators, the authors in [13] investigated the influence of data types, values, data reuses, and types of layers on the resilience of DNN accelerators through experiments on a DNN simulator. Error resilience of DNN accelerators with simulation. Different from [13], the works in [11] mainly focus on the difference of fault tolerance using different data types. Some of the researchers try the reliability of the neural network accelerators. While neural networks with large amounts of redundant computing are resilient, reliability of neural network accelerators becomes critical to resilient neural network execution Query ID="Q4" Text="The sentences ‘(1) Reliability, especially... the neural network’ seem to be incomplete. Please check." . (1) Reliability, especially FPGA based reliability problem. (2) The prevalence of deep learning neural networks provokes the development of convolutional neural network (CNN) accelerators for both higher performance and energy efficiency. existing fault-tolerant CNN accelerator works, Data type analysis with simulation, error propagation analysis Retraining to improve fault tolerance and tolerate the computing errors (Change the neural network model) Computing array-based fault model Relax the design constraints and have the accelerator obtain advantageous design trade-off between precision and performance. DAC’18 work Basically, the analysis focuses on the computing of the neural network. Hardware structure is not discussed in detail. Simulation is the major approach. Lack of system analysis on a CPU CNN accelerator. FPGA based analysis is not covered. Prior FPGA based reliability such as soft processors etc. It is not quite relevant. Neural network inherent fault tolerance. Error analysis on a running system remains not explored. FPGA is a widely used hardware platform, and there are many designs and methods for error injection on FPGAs [24–27]. FPGA-based (also known as emulation-based) error analysis is widely used in soft processor sensitivity analysis and other scenarios. The FPGA-based fault injection techniques have good controllability, observability and ideal speed.

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6 Conclusion With the wide application of neural networks in many scenarios with high safety requirements like autonomous driving, the fault-tolerant performance of neural network accelerators has received more attention. In this paper, we designed a fault emulation and analysis system for a neural network accelerator based on SRAMbased FPGA and completed a series of experiments on this basis. We find that the fault tolerance of neural network accelerators is poor, and analyze the misrepresentation of different numbers, different network applications and different error locations. These analyses provide a reference for the following fault-tolerant design.

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Trust-Based Efficient Computational Scheme for MANET in Clustering Environment Joydeep Kundu, Sitikantha Chattopadhyay, Subhra Prokash Dutta, Koushik Mukhopadhyay, and Souvik Pal

Abstract Mobile ad hoc network (MANET) is a self-coordinated, unconstrained, and highly mobile in nature network. In MANET, because of impermanent collaboration among nodes and the absence of enough information between every node ahead of time, it is hard to build up trust. There are so many trust-based mechanisms have been proposed for MANET; yet practically speaking, the greater part of them are wasteful to fulfill the asset usage and reliable trust calculation necessity because of high communication cost and less versatility. In this work, trust evaluation has been done in two parts, i.e., for intra-cluster and intergroup level in the field of specially appointed networks. We evaluate direct and indirect trust values, respectively, for each group of individuals and bunch heads separately. The analysis of the numerical value of trust calculation shows that our proposed plot is productively distinguished the non-helpful conduct of both group individuals and bunch heads at the intra and between bunch levels. Keywords Clustering · Probability density function · Trust computation scheme

1 Introduction MANET has been applied in [1–3] intelligent system, emergency crisis system, traffic problem, etc., for observing the situation of continuous changes. There is temporary system that means centralized server is absent there. These kind of ascribes make it challengeable against any type of new security attacks which is absent in J. Kundu · S. Chattopadhyay (B) · S. P. Dutta Department of CSE, Brainware University, Barasat, West Bengal, India e-mail: [email protected] K. Mukhopadhyay Department of CSE, Brainware University, Barasat, West Bengal, India S. Pal Department of CSE, Global Institute of Management and Technology, Krishnagar, West Bengal, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_31

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a standard wired organization. Trust-based plans use the trust assessment of the centers to perceive the center points with sketchy assumption and consequently secure the transfer of data communication through nodes with good trust value. Trusted instruments are applied to evaluate the trust assessments of each hub within the network either by prompt or roundabout insight for recognizing the malicious hubs. This paper contains the five segments which are as per the following: Section 2 addresses related work; Sect. 3 talks about generally speaking trust computational plan for bunch-based MANET, and the examination of numerical outcome is clarified in Sect. 4. At last, Sect. 5 describes the conclusion of the paper.

2 Related Work Various existing trust-based components for the clustering conditions [4, 5] have been proposed in MANET. Some of the current trust computation mechanisms are communicated beneath. Li and Yung clarified notoriety-based trust the executives system in the field of group-based ad hoc networks [4]. Here, a cluster head support instrument was kept up. A current CH (most raised trust regard) picks its support which has most noteworthy trust regard inside the group. CH invigorates the information to its all connected support hubs. If the trust assessment of the cluster head is less than the support node, it immediately transfers this obligation to the other CH. In this work [5], any candidate node for cluster head share the information with its energy consumption worth and conveys ability to its every neighbor. Collector members compute the whole load of the sender node with the help of trust value regard with received information about that sender. After that receiver member checks, if whole weight is in overabundance of edge regard, it will decide in favor of sender member. Li et al. [4] explained a selection of trust-based group head dependent on insect state system. Cluster head takes a vital role in clustering environment during the time of resource gathering. This work depends on the natural-inspired procedure known as subterranean insect state enhancement. At the point when an insect goes to various ways for looking through food, it spreads pheromone. Different subterranean insects can follow the way detecting this pheromone. In their proposed calculation, each hub keeps a bunch of pheromone follows for all neighbors. The pheromone follows will figure the likelihood of a specific course picked by the insects. Along these lines, the pheromone follow can be taken as the measure of trust.

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3 Proposed Trust Computation Scheme in Clustering Environment 3.1 Trust Score Range The trust relationship of hubs in MANET is tended to as quantitative qualities. This worth can be a genuine number some place in the range between 0 and 100 [4] or a whole number in the scope of 0 and 50 [6]. In this work, we compute the trust score in terms of a whole (unsigned) number in the stretch some place in the scope of 0 and 25. Depiction of trust regards either as a genuine or entire number isn’t basic in standard wired organization, anyway for MANET this issue is crucial in light of limited information move limit, memory, and transmission, gathering energy. Depiction of trust regard as an unsigned entire number between 0 and 30 which needs 5 bits means 0.62 bytes of memory space; thus, it saves 84.5% of whole memory space when appeared differently in relation to believe regards tended to overall number (4 bytes). Subsequently, by and large less number of touch should be conveyed during the trading of trust regards between CHs. As such, use of transmission and get-together force of CMs and CHs is lessened in the organization.

3.2 Some Essential Presumptions of Proposed Scheme • MANET involves enormous number of bunches that are shaped in an open medium. • CMs are isolated into number of bunches by a gathering system proposed in [7]. • The cluster-based strategy utilizes various leveled organizing which takes care of the issue of scalability that exist in MANET. • Gateway nodes (GWs) are dependable to interface nearby clusters and send GW reference points periodically to advise their individual clusters [8, 9]. We realize that there is no infrastructure-based network in MANET, and moreover, we accept that gateway nodes can’t be undermined by an assailant.

3.3 Intra-cluster Trust Computation At this stage, cluster member calculates its neighbors trust value dependent on the basis of either by direct communication or feedback score, respectively. (1)

Direct trust calculation between cluster members:

The trust value (direct mode) between neighbor member nodes within a cluster p and q is represented by the following equation-

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  m p,q (t) 1 . 1−  T p,q (t) = 25. m p,q (t) + n p,q (t) m p,q (t) + 1

(1)

Window time (t) value esteem is taken dependent on network situations. Successful interaction [m p,q (t)] and unsuccessful [n p,q (t)] communication between members are determined by time window (t) which has a few units of times. In this work, time window (t) has three time units. The window adds present insight (data) rather than past, as time passes. We calculate the trust values into two categories inspired by [10–13] such that trusted and non-trusted. In our scheme, we assume 25 is the optimum trust value.  T p,q (t) = (2)

Trusted, m p,q (t) ≥ 12.5 Non - trusted, n(t) < 12.5

(2)

Recommendation trust calculation between cluster head and cluster member:

We expect that memory limit and burned-through energy of group head (CH) are higher than CMs. Cluster head computes the suggestion trust about its CMs inside a group. Let b quantities of CMs present inside a bunch alongside its CH. In this manner, CH intermittently sends the solicitation message to its (b−1) CMs. CH will preserves its trust values in the T ch within a cluster is. ⎤ T1,ch · · · T1,b−1 ⎥ ⎢ .. .. = ⎣ ... ⎦ . . Tb−1,ch · · · Tb−1,b−1 ⎡

T ch

(3)

Based on the concept of reputation rating from updated beta probability density function [4, 6, 14, 15], we compute the cluster head to cluster members R recommendation (feedback) trust (Tch,n (t)) is represented as

R Tch,n (t) = Re p r  , s 

(4)



r −s Rep r  , s  = (E ρ p| r  , s  − (12.5).2 = 25 r −s+2

(5)

where

We present the standing rating for positive (r  ) and negative (s  ) measure of notoriety esteem inside the range [0, 25] as Rep(r  , s  ).

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3.4 Inter-cluster Trust Computation (1)

Direct trust calculation between cluster heads:

The immediate trust of a cluster head say, m toward its neighbor cluster head say, G n through gateway Tch pp,chq (t) is described as:



  G Tch pp,chq (t) = 25 T p,G pq TG pq ,q

(6)

The trust value T p,G pq and TG pq ,q states the direct trust value between cluster head p with gateway node G pq and gateway node G q p with cluster head q represented in Eqs. (7) and (8), respectively.   m p,G pq (t) 1 . 1−  T p,G pq (t) = 25 m p,G pq (t) + n p,G pq (t) m p,G pq (t) + 1   m G pq ,q (t) 1 . 1−  TG pq ,q (t) = 25 m G pq ,q (t) + n G pq ,q (t) m G pq ,q (t) + 1 (2)

(7)

(8)

Recommendation trust computation between Gateway and CHs:

A group of member nodes (cluster) with s number of nodes have (s−1) number of cluster members that implies any remaining individuals barring the CH can be go about as a gateway in a cluster. We accept that there are k quantities of groups present inside MANET. So, the relationship among all the passages with its CHs is represented (Pt ) below ⎤ Tch1 ,g11 · · · Tch1 ,g1(s−1) ⎥ ⎢ .. .. .. Pt = ⎣ ⎦ . . . Tchs ,gs−1 · · · Tchk ,g1(s−1) ⎡

(9)

Here, the overall gateways to cluster head recommended (TGRk,q ,chq ) trust procedure have been done by the mean of measure of notoriety rating input (sum of f and n) and direct value of trust from k no. of contiguous cluster heads toward a cluster head let, j is displayed as TGRk,q ,chq G

G

Rep f  , n  + = 2

k

G pq

T p.q k

i=1

(10)

T p.qpq ≥ 12.5 and T p.qpq < 12.5 negative feedback toward a CH n, respectively.

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4 Analysis of Newly Proposed Trust Computational Mathematical Result 4.1 Assumption of the Proposed Scheme In this work, a cluster member says, q is trustworthy when the value T p,q (t) ≥ 12.5 for a CM p. Simultaneously, a cluster head q is trustworthy when the value ch T p,q (t) ≥ 12.5 to its adjacent CH p. Otherwise, the trust value is non-trusted. Presently, we will demonstrate numerically our recently proposed trust plot against different situations of non-agreeable conduct of CMs or CHs at both intra and bury bunch level individually with the help of [3, 6, 17, 18].

4.2 Theorem A The trust score (direct communication between cluster member to cluster member) scheme [4, 5] at intra-group level is dependable for the beguiling idea of malicious cluster members. In the present circumstance, we consider three situations which are recorded underneath. Proof When successful transaction is more than 1 (m p,q (t) > 1) and unsuccessful transaction is more than successful transaction n p,q (t) > m p,q (t). When successful transaction is more than 1 that means cluster member (CM) let, q successfully communicates with node p within a window time t. So, the ratio of non-cooperative with cooperative interaction is described by say, v (i.e., positive real n (t) value) and v is equal to, mp,q = v. p,q (t) Next condition n p,q (t) > m p,q (t) describes when non-cooperative interaction is much more than cooperative interaction. Now, we can state on the condition given v ≥ 1 and n p,q (t) + m p,q (t) ≥ 1 that states trust value computation is processed by the past correspondence during the last collaboration between two bunch individuals. So,   m p,q (t) 1 . 1−  T p,q (t) = 25. (11) m p,q (t) + n p,q (t) m p,q (t) + 1 After solving the above equation, it has been found,        n p,q (t) + v(n p,q (t)) − v.n p,q (t) v.n p,q (t) 1  = 25. .  1+v v.n p,q (t).n p,q (t) v + n p,q (t) (12)

Trust-Based Efficient Computational Scheme … Fig. 1 Trust value comparison GTMS versus proposed method in good case

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Trust value comparison GTMS vs Proposed method in good case 71.11

Trust Value (e)

80 60 40 20

14.78

0

Series1

GTMS

Proposed Method

14.78

71.11

From the given conditions, m p,q (t) ≥ 1 and n p,q (t) > m p,q (t) , we states that n p,q (t) > m p,q (t) because m p,q (t) ≥ 1. Therefore, n p,q (t) > 1. So, two neighbor CMs are √   said to be trusted if, 1 √n p,q (t)+v(n p,q (t))−√v.n p,q (t) v.n p,q (t) √ 12.5 ≤ 25. 1+v . √ after solving v.n p,q (t).n p,q (t)

v+n p,q (t)

the equation will be given below, √ v + 1(v + 1)  √ < 2, (v + 1) − v

(13)

which is inconceivable in light of the fact that 1+v > 2. So, by utilizing direct CM to CM trust calculation conspire, the given condition n p,q (t) > m p,q (t) suggests that T p,q (t) ≥ 12.5 is unimaginable. Thus, we might reason that when the amount of ineffective collaboration is more than effective exchange; then, at that point, the immediate trust esteem can’t be surpass trust edge (12.5) which is non-trusted and demonstrates the saying 1. So, we have already plot our obtained trust values based on the rate of successful transaction in different situation, i.e., good case, average case, and poor cases. As a result, in all the cases, our proposed scheme is efficiently improve the trust score with compared to GTMS [6] scheme which has shown, i.e., Figures 1, 2, and 3. Here, we assume that when the rate of successful transaction is more than 80%, it is good case; from 50 to 79%, it is average case, and below 49% is treated as poor case.

4.3 Theorem B Recommendation trust calculation value (bunch head to group member) at group head is solid against the bogus suggestion input of getting rowdy group individuals. R Proof Situation: When m > r and Tch,n (t) ≥ 12.5

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Fig. 2 Trust value comparison GTMS versus proposed method in average case

Trust value comparison GTMS vs Proposed method in average case Trust Value (e)

50 40 30 20 10 0

Series1

Fig. 3 Trust value comparison GTMS versus proposed method in poor case

41.67

8.64

GTMS

Proposed Method

8.64

41.67

Trust value comparison GTMS vs Proposed method in poor case 13.33

Trust Value (e)

15 10 5 0 Series1

2.93 GTMS

Proposed Method

2.93

13.33

R Here, s, r, and Tch,n (t) are the negative, positive, and recommended trust score by the group head. On the off chance that we can have the option to demonstrate the logical inconsistency between the given conditions; then, at that point, adage 3 is demonstrated. As indicated by condition (5) R Tch,n (t) = 25

r −s r −s+2

The feedback trust upsides of bunch individuals by group head is trusted if, R Tch,n (t) ≥ 12.5 = 25

r −s 1 r −s ≥ 12.5 = ≥ = r ≥ 3s + 2 (14) r −s+2 r −s+2 2

So, in this, condition repudiates the given suspicion. Henceforth, axiom 3 has been demonstrated.

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4.4 Theorem C Inter-bunch trust calculation (Gateways to group head) is adaptable against the suggestion data of non-agreeable CHs. Proof Likewise, we can have the option to demonstrate the above adage at entomb bunch level as saying 3 was clarified.

5 Conclusion In this work, another productive trust calculation conspire has been proposed for MANET. It further develops asset proficiency and distinguishes the hubs with malignant goal by keeping away from suggestion input trust between neighbor group individuals. In future, there will be an extension to further develop the trust calculation conspire at both the intra-group and entomb bunch level by utilizing some bio-roused strategy like delicate registering, and so forth to acquire more exact and adaptable outcome in the field of specially appointed organization.

References 1. Venkatraman, S., and M. Alazab. 2018. Use of data visualisation for zero-day malware detection. Security and Communication Networks, 2018. 2. Sirajuddin, M., C. Rupa, and A. Prasad. 2018. A trusted model using improved-AODV in MANETS with packet loss reduction mechanism. Advances in Modelling and Analysis B 61 (1): 15–22. 3. Jhajj, H., R. Datla, and N. Wang. 2019. Design and implementation of an efficient mul- tipath AODV routing algorithm for MANETs. In Proceedings of the CCWC, 0527–0531, Las Vegas, NV, USA. 4. Li, Y., P. Yung, Z. Jianpeng. 2010. Trust CH election algorithm based on ant colony systems. In 2010 Third International Joint Conference on Computational Science and Optimization (CSO), vol. 2, 419–422. 5. Bhagyalakshmi and A. K. Dogra. 2018. QAODV: A flood control ad-hoc on demand distance vector routing protocol. In Proceedings of the ICSCCC, 294–299, Jalandhar, India. 6. Shaikh, R.A., H. Jameel, B.J. d’Auriol, H. Lee, and S. Lee. 2009. Group-based trust management scheme for clustered wireless sensor networks. IEEE Transactions on Parallel and Distributed Systems 20 (11): 1698–1712. 7. Kundu, Joydeep, K. Majumder. An efficient trust management scheme for cluster based MANET using Beta Reputation rating. In 2014 International Conference on Signal Propagation and Computer Technology (ICSPCT 2014). 8. Shaikh, R., H. Jameel, d’Auriol, Brian J., Lee, Heejo, Lee, Sungyoung, and Song, YoungJae. Group-based trust management scheme for clustered wireless sensor networks. IEEE Transactions on Parallel and Distributed Systems 20(11). 9. Femila, L., and M. Marsaline Beno. 2019. Optimizing transmission power and energy efficient routing protocol in MANETs. Wireless Personal Communications 106(3), 1041–1056.

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10. Li, Xiaoyong, Feng, Z., and Junping, Du. 2013. LDTS: A lightweight and dependable trust system for clustered wireless sensor networks. IEEE Transactions on Information Forensics and Security 8(6). 11. Sirajuddin, Mohammad, C. Rupa,Celestine Iwendi, and Cresantus Biamba. 2021. TBSMR: A trust-based secure multipath routing protocol for enhancing the QoS of the mobile ad hoc network. 2021, Article ID 5521713. https://doi.org/10.1155/2021/5521713. 12. Iwendi, C., S. Khan, J. H. Anajemba, M. Mittal, M. Alenezi, and M. Alazab. 2020. The use of ensemble models for multiple class and binary class classification for improving intrusion detection systems. Sensors 20(9), Article ID 2559. 13. Olanrewaju, Rashidah F., Burhan U. Islam Khan, Farhat Anwar, Bisma Rasool Pampori, Roohie Naaz Mir. 2020. MANET security appraisal: Challenges, essentials, attacks, countermeasures & future directions. International Journal of Recent Technology and Engineering (IJRTE) 8(6). ISSN: 2277–3878. 14. Enhancement in Manet performance using trust based secured anonymous routing. International Journal of Advanced Science and Technology 29(3), 8578–8591 (2020). 15. Sirajuddin, M. D., C. Rupa, and A. Prasad. 2018. An innovative security model to handle blackhole attack in MANET. In Proceedings of International Conference on Computational Intelligence and Data Engineering, vol. 9, 173–179. 16. Yang, H. 2020. A Study on Improving Secure Routing Performance Using Trust Model in MANET, vol. 2020, Article ID 8819587. https://doi.org/10.1155/2020/8819587. 17. Li, T., J. Ma, and C. Sun. 2019. SRDPV: Secure route discovery and privacy-preserving verification in MANETs. Wireless Networks 25 (4): 1731–1747. 18. Kathiriya, H., A. Pandya, V. Dubay, and A. Bavarva. 2020. State of art: energy efficient protocols for self-powered wireless sensor network in IIoT to support industry 4.0. In Proceedings of the 2020 8th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions) (ICRITO), 1311–1314, Noida, India.

A Comparative Approach for Solving Fuzzy Transportation Problem with Hexagonal Fuzzy Numbers and Neutrosophic Triangular Fuzzy Numbers T. Nagalakshmi, R. Sudharani, and G. Ambika Abstract Nowadays, optimization methods and its techniques are applied for the optimal solution of fuzzy models which is termed as fuzzy optimization. This is done by formulating fuzzy information with the help of its membership functions. Linear programming has wider applications in various fields such as transportation problem, assignment problem, engineering, manufacturing, energy industry, etc. As fuzziness prevails in many daily related activities, it prevails in transportation problems also. In this proposed approach, such a transportation problem involving fuzzy values of cost, supply and demand is considered. The term ‘fuzziness’ was introduced by Zadeh which was developed later to handle MCDM problems by Bellman and Zadeh. In Linear Programming Problems (LPP), transportation problems are specific models that minimize the transportation cost of a commodity from the places of origin such as factories, manufacturing companies etc., to the places of destination such as warehouse and store. In this approach, two general transportation problems with fuzzy parameters in the same interval are considered where their cost, supply and demand parameters are expressed in terms of hexagonal fuzzy numbers (HFNs) and neutrosophic triangular fuzzy numbers (NTFNs), respectively. Prevailing ranking methods are applied to both the HFNs and NTFNs for converting the fuzzy-based transportation problems to crisp-based transportation problems. The problems are then solved by the methods of Vogel’s approximation and MODI for both HFNs and NTFNs. The final solutions are compared. This is illustrated through the numerical examples.

T. Nagalakshmi (B) · G. Ambika Veltech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, Tamilnadu, India e-mail: [email protected] G. Ambika e-mail: [email protected] R. Sudharani Panimalar Engineering College, Chennai, Tamilnadu, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_32

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Keywords Fuzzy transportation problem · Hexagonal fuzzy numbers · Neutrosophic triangular fuzzy numbers · Fuzzy optimization · Defuzzification technique

1 Introduction Many researches have been carried out by various researchers in the field of fuzzy optimization. Liu and Kao [1] proposed an extension principle-based approach for calculating fuzzy objective value of FTPs. Gani and Razak [2] solved a minimizing FTP with two-stage cost involving supplies and demands as TrFNs. Pandian and Natarajan [3] presented the fuzzy zero approach to locate the optimal solution for a FTP with trapezoidal fuzzy numbers. Several ways for rating fuzzy numbers were introduced so that they could be compared more easily. Rajarajeswari and Sudha [4] presented the hexagonal fuzzy number, which has no parameter limits. Dinagar and Kannan [5] proposed a new ranking method which is applied to study the inventory model of fuzzy parameters with allowable shortage. Pavithra and Rosario [6] discussed more problems on FTP using HFNs. Later, a new approach was developed by Maity and Roy [7] in solving a Type-2 FTP. The problem was reduced to three different LPPs, and the solution was obtained using simplex method. Kumar et al. [8] introduced simple method to solve fuzzy Pythagorean transportation problem. A ranking procedure for IFNs with the concept of distance minimizer was introduced by Nishad and Abhishekh [9] to solve a fully FTP. Mathur and Kumar [10] created an inventive approach which optimizes an FTP through generalized trapezoidal fuzzy numbers. Thota and Raja [11] proposed zero average method to solve FTPs where decision maker is not confident about cost, supply and demand of the transportation products. Geetha and Selvakumari [12] found a new method to solve FTP by using pentagonal fuzzy numbers by using range technique. Different authors have approached in several methods to solve these types of problems, viz. [13–19].

2 Terminology 2.1 Fuzzy Set ˜ ‘In a universe of  discourse X,  a fuzzy set A defined by [20] is defined as the set ˜ followed by A = x, μ A˜ (x) : x ∈ X . Here, μ A˜ : X → [0, 1] is a mapping called ˜ the membership degree value of x ∈ X in a fuzzy set A’.

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2.2 Hexagonal Fuzzy Number (HFN) A hexagonal fuzzy number defined by [4] is denoted by Q˜ n = (ρ1 , ρ2 , ρ3 , ρ4 , ρ5 , ρ6 ) where ρ1 ≤ ρ2 ≤ ρ3 ≤ ρ4 ≤ ρ5 ≤ ρ6 are the real numbers that satisfies the relation ρ2 − ρ1 ≤ ρ3 − ρ2 and also ρ5 − ρ4 ≥ ρ6 − ρ5 , and its membership function μ Q˜ n (x) is given by ⎧ 0, x < ρ1 ⎪ ⎪ ⎪ 

⎪ ⎪ ⎪ ⎪ 1 x − ρ1 , ⎪ ρ1 ≤ x ≤ ρ2 ⎪ ⎪ 2 ρ2 − ρ1 ⎪ ⎪

 ⎪ ⎪ ⎪ 1 1 x − ρ2 ⎪ ⎪ + , ρ2 ≤ x ≤ ρ3 ⎪ ⎪ 2 2 ρ3 − ρ2 ⎪ ⎨ 1, ρ3 ≤ x ≤ ρ4 μ Q˜ n (x) = ⎪ 

⎪ ⎪ 1 x − ρ4 ⎪ ⎪ ⎪ , ρ4 ≤ x ≤ ρ5 1− ⎪ ⎪ 2 ρ5 − ρ4 ⎪ ⎪ 

⎪ ⎪ ⎪ 1 ρ6 − x ⎪ ⎪ , ρ5 ≤ x ≤ ρ6 ⎪ ⎪ 2 ρ6 − ρ5 ⎪ ⎪ ⎪ ⎩ 0, x > ρ6

2.3 Neutrosophic Triangular Fuzzy Number (NTFN) A NTFN defined by [21] is denoted by Q˜ N = (σ1 , σ2 , σ3 ; τ1 , τ2 , τ3 ; ϕ1 , ϕ2 , ϕ3 ). Its TMF, IMF and FMF are defined as follows: ⎧ x−σ1 ⎪ , σ1 ≤ x < σ2 ⎪ σ2 −σ1 ⎪ ⎨ 1, x = σ2 TQ˜ n (x) = σ3 −x , ⎪ , σ 2 < x ≤ σ3 ⎪ σ −σ 3 2 ⎪ ⎩ 0, otherwise ⎧ τ2 −x ⎪ , τ ≤ x 0 y Jz1 yk(α) = e(α) =  1−α  α = y Jz1 Jv1 ify < 0    Iz1 I z 1 I v1 I k(α) Iv p(α) = h(α) = (min Jz , Jv , Jv , Jz )1−α (max Jzz1 , JIvv , 1

1

1

1

1

1

Iz Jv1

,

I v1 Jz1

 )α

Where the parametric interval valued function g(α), s(α), r(α), yk(α), p(α), e(α) for constant k and α ∈ [0, 1].

is

3 Model Formulation We consider a time-delayed epidemic mathematical model [19] in which total population (N ) can be split into three compartment, namely S (susceptible), I (infected), R (recovered) individuals. The treatment rate (Holling type III) for recovery of the infected (I) individuals has been imposed in the model as follows β S I (t − τ ) dS(t) = A − δS − dt 1 + al(t − τ ) dI (t) β S I (t − τ ) aI2 = − (δ + d + γ )I − dt 1 + α I (t − τ ) 1 + bI 2

(1)

aI2 dR(t) = + γ I − δR dt 1 + bI 2 The biological parameters A, δ, α, β, γ , a, b, d, and τ represents recruitment constant, normal death rate, protection measures rate (or measure of inhibition effect), transmission rate (or effective contact rate), recovery rate, cure rate, limitation rate in treatment of infected, disease-induced death rate, and time delay defining the latent period of disease, respectively. The system (1) infer that S and I are free from the effect of R, thus the above system in reduced form after incorporating imprecise co-efficient as follows:

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dS(t) βˆ S I (t − τ ) ˆ − = Aˆ − δS dt 1 + αˆ I (t − τ ) ˆ βsl(t − τ) al ˆ 2 dl(t) = − (δˆ + dˆ + γˆ )I − ˆ2 dt 1 + αl(t ˆ − τ) 1 + bl

(2)

d S(t) (β L )(1−ξ ) (β R )ξ S I (t − τ ) = (A L )1−ξ (A R )ξ − (δ L )(1−ξ ) (δ R )ξ S − dt 1 + (α L )1−ξ (α R )ξ I (t − τ ) dI (t) (β L )(1−ξ ) (β R )ξ S I (t − τ ) − ((δ L )1−ξ (δ R )ξ + (d L )1−ξ (d R )ξ = dt 1 + (α L )1−ξ (α R )ξ I (t − τ )   (a L )1−ξ (a R )ξ I 2 + (γ L )1−ξ γ R )ξ I − 1 + (b L )1−ξ (b R )ξ I 2

(3)

With the initial condition is S(σ ) = ∅1 (σ ), I (σ ) = ∅2 (σ ),R(σ ) = ∅3 (σ  )∅i ≥ 3 . Here, 0, σ ∈ [−τ, 0], ∅i (0) > 0(i = 1, 2, 3) Where ,∅i (σ ) ∈ C1 [−τ, 0], R+ 1−ξ ξ ˆ (1−ξ ) ξ ˆ (δ R ) , A = [A L , A R ] = (A L ) (A R ) , δ = [δ L , δ R ] = (δ L ) 1−ξ ξ (1−ξ ) ) (β ) , α ˆ = , α (α R )ξ , dˆ = [d L , d R ] = βˆ = [β L , β R ] = (β = [α ] (α ) L R L R L  1−ξ ξ 1−ξ ξ (d L ) (d R ) , γˆ = γ L , γ R = (γ L ) (γ R ) , aˆ = [a L , a R ] = (a L )1−ξ (a R )ξ , bˆ = [b L , b R ] = (b L )1−ξ (b R )ξ , and C1 represents the continuous function in Banach space.

4 Equilibriums and Stability Analysis 4.1 Disease-Free Equilibrium (DFE) (1−ξ )

ξ

(A R ) L) The DFE point of the system (3) becomes D ( (A , 0). We analyse two parts (δ L )(1−ξ ) (δ R )ξ of the basic reproduction number (i) analysis for R0 = 1 and (ii) analysis for R0 = 1. (1−ξ )

ξ

(A R ) L) Theorem 1: The DFE point D ( (A , 0) becomes locally asymptotically (δ L )(1−ξ ) (δ R )ξ stable (LAS) if R0 < 1 and unstable if R0 > 1 for τ > 0.

Proof: The proof is obvious. (1−ξ )

ξ

(A R ) L) Theorem 2: The DFE D ( (A , 0) of the system (3) changes the stability if (δ L )(1−ξ ) (δ R )ξ R0 = 1 for τ > 0.

Proof: We consider the characteristic equation of (3) and the basic reproduction (1−ξ ) )ξ (A L )(1−ξ ) (A R )ξ where v = ((δ L )1−ξ (δ R )ξ + (d L )1−ξ (d R )ξ + number R0 = (βL ) (δ(β)R(1−ξ ) (δ R )ξ ν L   (γ L )1−ξ γ R )ξ . Now, we analyse the basic reproduction number, R0 = 1.We evaluated system the (3) when R0 = 1 and (β L )(1−ξ ) (β R )ξ = (β L )(1−ξ ) (β R )ξ ∗ = (δ L )1−ξ (δ R )ξ ν has a zero eigenvalue and another eigenvalue is negative. The equi(A L )(1−ξ ) (A R )ξ librium points can not be determined at R0 = 1 , then we use the central manifold

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theorem. Now, we replaced S = x1 and I = x2 in system (3), the system (3) can be written as, dx1 dt

= (A L )1−ξ (A R )ξ − (δ L )(1−ξ ) (δ R )ξ x1 −

(β L )(1−ξ ) (β R )ξ x1 x2 (t−τ ) 1+(α L )1−ξ (α R )ξ x2 (t−τ )

= f (x1 , x2 ) (say)

(β L )(1−ξ ) (β R )ξ x1 x2 (t − τ ) dx2 (a L )1−ξ (a R )ξ x22 = − νx − = g(x1 , x2 ) 2 dt 1 + (b L )1−ξ (b R )ξ x22 1 + (α L )1−ξ (α R )ξ x2 (t − τ ) (4) Let L ∗ be the Jacobi matrix at R0 = 1 and bifurcation parameter (β L )(1−ξ ) (β R )ξ = ∗ (β L )(1−ξ ) (β R )ξ then ∗ (1−ξ ) −a b (β R )ξ (1−ξ ) L∗ = we consider Where a = (δ L )1−ξ (δ R )ξ , b = − (β(δL ) )1−ξ (δ ξ (A L ) L R) 0 0 two eigenvectors u = [u 1 , u 2 ] and w = [w1 , w2 ]t of the corresponding eigenvalue zero.

 −a b w1 ∗ ∗ u L = 0, we get u 1 = 0 and u 2 = 0. Again L w = 0 implies = w2 0 0 (1−ξ )

ξ∗

0 or,−aw1 + bw2 = 0. If w2 = 1,w1 = − (β(δL ))1−ξ (δ(β R)ξ)2 (A L )(1−ξ ) (A R )ξ . Now, we find L R the non-zero partial derivative with function of the system (4) evaluated at R0 = 1 (1−ξ ) ξ (1−ξ ) ξ∗ and (β L ) (β R ) = (β L ) (β R ) are



∂2g ∂ x22



∂2g ∂ x1 ∂ x2

Q



∂2g ∂ x2 ∂ x1

= (β L )(1−ξ ) (β R )ξ

1−ξ

L

∗

Q

(α R ) L) = − 2(α (β L )(1−ξ ) (β R )4+ (A L )(1−ξ ) (A R )ξ , (δ )1−ξ (δ )4 2



R

∂2g ∂ x2 ∂(β L )1−ζ (β R )4+

ξ

(A L ) (A R ) > 0. (δ L )1−ξ (δ R )ξ a1 and b1 are the 1−g

∗

= (β L )(1−ξ ) (β R )ξ ,

 Q

=

bifurcation constant then,

2(β L )1−ξ ! (β R )4+ a1 = − (A L )(1−ξ ) (A R )ξ (δ L )1−ξ (δ R )ξ  (β L )(1−ξ ) (β R )4+ + (δ L )1−ξ (δ R )ξ (α L )1−ξ (α R )ξ < 0 and b1 =

2

i=1 u k wi



∂2g + ∂ xi ∂(β L )(1−g) (β R )b

 Q

  (1−ξ ) (A R )5 L) = = u 2 w2 (A 1−ξ 3 (μ ) (μ ) L

R

(A L )(1−ξ ) (A R )5 (δ L )1−ξ (δ R )3

>

0. From the above analysis, the model system (3) exhibits forward bifurcation if R0 = 1 for τ > 0.

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4.2 Endemic Equilibrium (EE) To check the condition for existence of EE (D ∗ ) = (S ∗ , I ∗ ), we find the value S ∗ , I ∗ from the system (3). As a result, we obtain I ∗ = (A L )(1−ξ ) (A R )ξ −(δ L )1−ξ (δ R )ξ S ∗ . Here, S ∗ is given by the (β L )(1−ξ ) (β R )ξ S ∗ −[(A L )(1−ξ ) (A R )ξ −(δ L )1−ξ (δ R )ξ S ∗ ](α L )1−ξ (α R )ξ following equation, P1 S ∗3 + P2 S ∗2 + P3 S ∗ + P4 = 0 

P1 = [(β L )(1−ξ ) (β R )ξ + (δ L )1−ξ (δ R )ξ .(α L )1−ξ α R )ξ

(5)



[(β L )(1−ξ ) (β R )ξ + 2(β L )(1−ξ ) (β R )ξ .(δ L )1−ξ (δ R )ξ .(α L )1−ξ (α R )ξ + (b L )1−ξ (b R )ξ .{(δ L )1−ξ (δ R )ξ }2 ,   P2 = −[(β L )(1−ξ ) (β R )ξ + b L )1−ξ b R )ξ .{(δ L )1−ξ (δ R )ξ }2 +{(α L )1−ξ (α R )ξ .(δ L )1−ξ (δ R )ξ }

  P3 = [(β L )(1−ξ ) (β R )ξ + (δ L )1−ξ (δ R )ξ .(α L )1−ξ α R )ξ

[{(A L )(1−ξ ) ( A R )ξ } 2 {(α L )1−ξ (α R )ξ } 2 +(b L )1−ξ (b R )ξ .{(α L )1−ξ (α R )ξ } 2 ]+ [{(A L )(1−ξ ) ( A R )ξ .(α L )1−ξ (α R )ξ } +ν][2(A L )(1−ξ ) (A R )ξ {(β L )(1−ξ ) (β R )ξ +  2 (δ L )1−ξ (δ R )ξ .((α L )1−ξ α R )ξ + (b L )1−ξ (b R )ξ   . δ L )1−ξ (δ R )ξ − [(β L )(1−ξ ) (β R )ξ +   (δ L )1−ξ (δ R )ξ .(α L )1−ξ α R )ξ .{(a L )1−ξ (a R )ξ } + [(δ L )1−ξ (δ R )ξ   .(α L )1−ξ (α R )ξ . A L )(1−ξ ) (A R )ξ , P4 = −[{(A L )(1−ξ ) (A R )ξ .(α L )1−ξ (α R )ξ } + ν][{(A L )(1−ξ ) (A R )ξ } 2 {(α L )1−ξ (α R )ξ } 2 +(b L )1−ξ (b R )ξ .{ (α L )1−ξ (α R )ξ } 2 ] + [{(A L )(1−ξ ) (A R )ξ } 2   .(α L )1−ξ (α R )ξ .(a L )1−ξ a R )ξ

By Descartes rule of sign, Eq. (5) has unique real root (> 0) if one condition holds of the following: (i) (ii) (iii)

P1 > 0, P2 < 0, P3 < 0, P4 < 0 P1 > 0, P2 > 0, P3 > 0, P4 < 0 P1 > 0, P2 > 0, P3 < 0, P4 < 0

Putting the value of the S,∗ we get I ∗ , therefore, there is a unique EE (D ∗ (S ∗ , I ∗ )) whether above mention one condition holds. The characteristic equation of the system (3) is evaluated at EE (D ∗ ), and the given transcendental equation, λ2 + g0 λ + h 0 + (g1 λ + h 1 )e−λτ = 0

(6)

Analysis of an Imprecise Delayed SIR Model System …

341

where g0 ,h 0 = (δ L )1−ξ (δ R )ξ v +

2(δ L )1−ξ (δ R )ξ (a L )1−ξ (a R )ξ I ∗ (β L )(1−ξ ) (β R )ξ v I ∗    +    2 1 + (α L )1−ξ (α R )ξ I ∗ 1 + (b L )1−ξ (b R )ξ I ∗2

2(a L )1−ξ (a R )ξ (β L )(1−ξ ) (β R )ξ I ∗2 +    ∗ 2  1 + (b L )1−ξ (b R )ξ I 1 + (α L )1−ξ (α R )ξ I g1 = −

(β L )(1−ξ ) (β R )ξ S φ

ξ

2

L

(β R )ξ S ∗ ξ ∗ R) I } )

(1−ξ )

(δ R ) (β L ) h 1 = − (δL )(1+{(α )1−ξ (α 1−ξ

(1+{(αL )1−ξ (α R )ξ I ∗ })

∗

Theorem 3: For τ = 0, the system (3) at EE (D ∗ ) is LAS if both condition SI ∗ ≤ ∗ ν and SI ∗ ≤ 1 are satisfied where v = ((δ L )1−ξ (δ R )ξ + (d L )1−ξ (d R )ξ + (δ L )1−ξ (δ R)ξ  (γ L )1−ξ γ R )ξ . Proof: At EE (D ∗ ), the transcendental equation for τ = 0 is given by λ2 +g0 λ+h 0 + ∗ ∗ (g1 λ + h 1 ) = 0. It is easy to show that if SI ∗ ≤ (δ )1−ξν (δ )ξ and SI ∗ ≤ 1 are satisfied, L R  (β L )(1−ξ ) (β R )ξ I ∗ 2(a L )1−ξ (a R )ξ I ∗ ξ 1−ξ − then g0 + g1 = [ δ L ) (δ R ) + ν + 2 + (1+{(α L )1−ξ (α R )ξ I ∗ } ) (1+{(b L )1−ξ (b R )ξ I ∗2 } ) (β L )(1−ξ ) (β R )ξ S ∗ (1+{(α L )1−ξ (α R )ξ I ∗ } )2

δ L )1−ξ (δ R )ξ + ν +

+[

2(a L )1−ξ (a R )ξ I ∗  2 (1 + { b L )1−ξ (b R )ξ I ∗2 }

  {(β L )(1−ξ ) (β R )ξ (a L )1−ξ (a R )ξ I ∗2 + ((β L )(1−ξ ) (β R )ξ I ∗ − β L )(1−ξ ) (β R )ξ S ∗ } (1 + {(α L )1−ξ (α R )ξ I ∗ } )2

Similarly, we  can show h 0 + h 1 = (δ L )1−ξ (δ R )ξ v +

>0

2(δ L )1−ξ (δ R )ξ (a L )1−ξ (a R )ξ I +   ∗ 2 1 + (b L )1−ξ (b R )ξ I

2(a L )1−ξ (a R )ξ (β L )(1−ξ ) (β R )ξ I +2 Hence, by the 2    1 + (b L )1−ξ (b R )ξ I ∗ 1 + (α L )1−ξ (α R )ξl  (β L )(1−ξ ) (β R )ξ v I ∗ − (δ L )1−ξ (δ R )ξ (β L )(1−ξ ) (β R )ξ S ∗    + >0 1 + (a L )1−ξ (α R )ξ I 4 Descartes’ rule of sign, the EE (D ∗ ) of system (3) is LAS for τ = 0. 



Theorem 4: The EE (D ∗ ) of the system (3) is LAS for τ > 0 if ∗

1, SI ∗ ≤

ξ 1−ξ 1, (δL ) νs(δ R )

≤

1 2

and

∗

S I∗

≤

ν (δ)1−ξ (δ R )ξ

1 1+{(α L )1−ξ (α R )ξ I ∗ }

≤

satisfied.

Proof: At D ∗ , the character equation at τ > 0 is given by λ2 + g0 λ + h 0 + (g1 λ + h 1 )e−λτ = 0

(7)

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−w2 + h 0 + g1 wsinwτ + h 1 coswτ + i(g1 wcoswτ − h 1 sinwτ + g0 w) = 0

(8)

[Put λ = iω in (7)]. From Eq. (8), we get, g1 w sin wτ + h 1 cos wτ = w2 − g0

(9)

g1 w cos wτ − h 1 sin wτ = −h 0 w

(10)

    w4 + g02 − 2h 0 − g12 w2 + h 20 − h 21 = 0

(11)

From (9) and (10),

  2 2 2 Or, m 21 + Pm 1 + T = 0(putting w = m 1 in (11)) Where P = g0 − 2h 0 − g1 ,  T = h 20 − h 21 . ∗ 1 ≤ 1, SI ∗ 1+{(α L)1−ξ (α R )ξ I ∗ }  then P = g02 − 2h 0 − g12

It is easy to show that if ν

(δ L )1−ξ (δ R )ξ

are satisfied,

 = (δ L )

ξ

2(a L )1−ξ (a R )ξ l ∗

≤ 1, (δL )

1−ξ

νs

(δ R )ξ

≤

1 2

and

(β L )(1−ξ ) (β R )ξ l ∗

+ (δ R ) + v +  ∗  ∗ 2 1 + ((a L )1−ξ (α R )ξl } 1 + ((b L )1−ξ (b R )ξl } ⎡ ⎤ 2(δ L )1−ξ (δ R )ξ (a L )1−ξ (a R )ξ I ∗ 1−ξ ξ  2 ⎢ (δ L ) (δ R ) v +  ⎥ ⎢ ⎥ 1 + (b L )1−ξ (b R )ξ I ∗2 ⎢ ⎥ ⎢ ⎥ (1−ξ ) ξ ∗ (β R ) v I (β L ) ⎢ ⎥ ⎥   + −2⎢ ⎢ 1 + (a L )1−ξ (α R )ξl ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2(a L )1−ξ (a R )ξ (β L )(1−ξ ) (β R )ξ I ∗2 ⎣ +    ⎦ 1−ξ ξ I∗2 2 1−ξ ξ ∗ 1 + {(b L ) (b R ) ) 1 + (α L ) (α R ) I  2 (β L )(1−ξ ) (β R )ξ S φ − −  2 1 + (α L )1−ξ (α R )ξ I ∗ 1−ξ

S∗ I∗

2

≤

Analysis of an Imprecise Delayed SIR Model System …



= (δ L )

1−ξ

(δ R )

 ξ 2

343

2  4 (β L )(1−ξ ) (β R )ξ I ∗2 + v +   4 1 + (b L )1−ξ (b R )ξ I ∗2 2

  2 (δ L )1−ξ (δ R )ξ (β L )(1−ξ ) (β R )ξ I ∗    + 1 + (α L )1−ξ (α R )ξ I ∗ 2   ∗2  (β L )(1−ξ ) (β R )ξ + I − S ∗2   2 1 + (α L )1−ξ (α R )ξ I ∗      2 (α L )1−ξ (α R )ξ I ∗ 2v − (δ L )1−ξ (δ R )ξ > 0 +   2 1 + (b L )1−ξ (b R )ξ I ∗2 Similarly, T = h 20 − h21 ⎤2 1−ξ (δ )ξ (a )1−ξ (a )ξ I ∗ (β L )(1−ξ ) (β R )ξ v I ∗ R L R 1−ξ (δ )ξ v + 2(δ L ) + (δ )    R   2 ⎢ L 1 + (α L )1−ξ (α R )ξ I ∗ ⎥ 1 + (b L )1−ξ (b R )ξ I I ∗2 ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ 1−ξ ξ (1−ξ ) ξ ∗2 I 2(a L ) (a R ) (β L ) (β R ) I ⎦ ⎣ +      2 1 + (b L )1−ξ (b R )ξ I ∗2 1 + (α L )1−ξ (α R )ξ I ∗ 2  2  (β L )(1−ξ ) (β R )ξ (β L )(1−ξ ) (β R )ξ v I ∗ +    1 + (α L )1−ξ (α R )ξ I ∗ (1 + {(α L )1−ξ (α R )ξ I ]∗ }4

  2  v 2 I ∗2 1 + (α L )1−ξ (α R )ξ I ∗ − v 2 I ∗2 + v 2 I ∗2 − ⎡

 2 (δ L )1−ξ (δ R )ξ S ∗2 > 0

Therefore, by the Descartes’ rule of signs, the EE (D ∗ ) of the mathematical model (3) is LAS for τ > 0.

4.3 Hopf Bifurcation If T = h 20 − h 21 in Eq. (11) is negative, then there is a unique root (> 0)ω0 satisfying Eq. (7), i.e., there is one pair of roots (purely imaginary)±iω0 to Eq. (7).  From Eqs. (9) and  (10),τ0 corresponding to ω0 can be written as τ0 = ω02 (h 1 −g0 g1 )−h o h 1 1 .EE (D ∗ ) is stable for τ < τ0 if transversality condition arccos ω0 g12 ω02 +h 21  holds, i.e., d (Reλ)(λ = iω0 ) > 0. Differentiation Eq. (7) with respect to τ , we get dt

dλ = λ(g1 λ + h 1 )e−λτ . {2λ + g0 + g1 e−λτ − (g1 λ + h 1 )τ e−λτ } dτ  dλ −1 2λ+g0 +g1 e−λτ −(g1 λ+h 1 )τ e−λτ 0 1 = = λ(g12λ+g + λ(g1gλ+h − dτ λ(g1 λ+h 1 )e−λτ λ+h 1 )e−λτ 1)

τ 2λ + g0 g1 +  − λ(g1 λ + h 1 ) λ −λ λ2 + g0 λ + h 0

τ λ

= (12)

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d ( dt (Reλ)

 Re

λ

=

2iω0 +g0 −iω0 (−ω02 +ig0 ω0 +h 0 )

+

iω0 )

g1 −g1 ω02 +i h 1 ω0

+

iτ ω0

= 

Re

 dλ −1  (λ dτ

=

iω0 )

    g12 ω0 1 2ω0 ω02 − h 0 + g02 ω0 = ·   − ω0 (g0 ω0 )2 + ω2 − h 0 2 (g0 ω0 )2 + h 24 0   2 ω02 − h 0 + g02 g2 − =   2 (g1 ω0 )2 + h 21 (g0 ω0 )2 + ω02 − h 0

=

(13)

Now, from Eqs. 9 and 10, we have (g1 ω0 )2 + h 21 = (g0 ω0 )2 + (ω02 − h 0 )2 Here, Eq. (9) can be written as   2ω02 + g02 − 2h 0 − g12 d (Reλ)(λ = iω0 ) = dt (g1 ω0 )2 + h 21

(14)

  Under the above condition P = g02 − 2h 0 − g12 > 0, we can easily observe that d (λ = iω0 ) > 0. Therefore, Hopf bifurcation exists at τ = τ0 , ω = ω0 . The dt (Reλ) above analysis we write in a following theorem: Theorem 5: If T = h 20 − h 21 < 0 holds, the EE (D ∗ ) of the mathematical system (3) is asymptotically stable for τ ∈ [0, τ0 ) and its exhibits Hopf bifurcation at τ = τ0 .

5 Numerical Simulation To discuss the imprecise model (3) numerically, first of all, DFE (D) in Table1 is obtained using the model parameter Aˆ = [3, 6], αˆ = [0.51, 0.61], βˆ = [0.011, 0.031], δˆ = [0.1, 0.5], dˆ = [0.51, 0.61], γˆ = [0.22, 0.33], aˆ = [0.031, 0.081], bˆ = [0.021, 0.051] and EE(D ∗ ) is calculated using the epidemiological parameter values Aˆ = [3, 6], αˆ = [0.070, 0.081], βˆ = [0.21, 0.33], μˆ = [0.01, 0.015], dˆ = [0.05, 0.06], γˆ = [0.55, 0.56], aˆ = [0.007, 0.0081], bˆ = [0.05, 0.061]. It is clear from Table-1 that both type of equilibriums increase when the parameter ξ ∈ [0, 1] increases Table 1. Using the interval parameters Aˆ = [3, 6], αˆ = [0.070, 0.081], βˆ = [0.21, 0.33],δˆ = [0.01, 0.015], dˆ = [0.05, 0.06], γˆ = [0.55, 0.56], aˆ = [0.007, 0.0081], bˆ = [0.05, 0.061] of the model (3), we plot Fig. 1 and distinct values of ξ (= 0, 0.5 and 1). Hopf Table1 Values of DFE (D) and EE (D ∗ ) state for ξ = 0, 0.5, 1

Parameter

DFE (D)

EE (D* )

ξ =0

(14.000, 0)

(90.6966, 1.4769)

ξ = 0.5

(23.6643, 0)

(93.6908, 1.9540)

ξ =1

(40, 0)

(97.8268, 2.5642)

Analysis of an Imprecise Delayed SIR Model System …

345

Fig. 1 Bifurcation diagram with respect to τ for ξ = 0, 0.5, 1.

bifurcation diagram is reflected in Fig. 1. In the model system, the EE is stable if τ ∈ [0, τ0 )and unstable if τ > τ0 where τ0 =38.401, 37.81, and 37.31 for ξ = 0, 0.5, and 1, respectively.

6 Conclusion In this paper, we have studied a time-delayed SIR mathematical model in imprecise environment. Holling type-III treatment rate has been imposed into model to reflect the disease dynamics in interval environment. The system is stable at the DFE point when R0 < 1,τ ≥ 0 and the system undergoes unstable when R0 > 1. The imprecise model (3) exhibits transcritical bifurcation at R0 = 1. For τ = 0, the EE (D ∗ ) of ∗ ∗ imprecise epidemic the model (3) is LAS if both SI ∗ ≤ (δ )1−ξν (δ )ξ and SI ∗ ≤ 1 where L R ξ ∈ [0, 1] are satisfied; For τ > 0, the EE (D ∗ ) of imprecise epidemic, the model (3) ξ 1−ξ ∗ ∗ is LAS if 1+{(α )1−ξ1 (α )ξ I ∗ } ≤ 1, SI ∗ ≤ 1, (δL ) νs(δ R ) ≤ 21 , and SI ∗ ≤ (δ )1−ξν (δ )ξ where L R L R ξ ∈ [0, 1] are satisfied. Of late, condition of Hopf bifurcation has been discussed in interval environment. The solution of the model system utilizing MATLAB has been accomplished for the significant numerical result for different values of ξ ∈ [0, 1]. Lastly, we can infer that epidemiological modelling in uncertain environment makes the situation more realistic. So, in future research work, we can introduce modified treatment rate into the model with uncertainty to discuss the disease kinetics.

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Design and Application of Virtual Reality Technology in Digital Pedagogy Tran Doan Vinh

Abstract VR technology is a part of modern life, used in many different areas of life such as entertainment, culture-tourism, education, … The application of VR technology in teaching and learning to adapt to the current educational development is very necessary. In this paper, after highlighting some concepts of two model of bringing VR technology into education and training, characteristics of High School students and their ability to receive information… We introduce the design and application of virtual reality in teaching Informatics at High School on the topic and pedagogical experiment. Keywords Virtual reality model · Teaching with VR · Digital pedagogy · Digital literacy · Competence

1 Introduction Vietnam is in the process of implementing the national renewal process, carrying out industrialization—modernization, realizing the knowledge economy, participating in the process of international integration and globalization, expanding relations with Vietnam with other countries in the world. In that context, Vietnam has had many changes, and the development of science and technology, of information technology, especially VR technology, is one of the fastest, strongest and most obvious changes. VR technology is a part of modern life, used in many different aspects of life such as entertainment, culture-tourism, education, military… The application of VR technology in teaching and learning to adapt to the current educational development is very necessary. Through VR technology, teachers, and students can easily access the rich knowledge of mankind. In 2018, the Vietnamese Ministry of Education and Training issued a General Education Program, including the High School Education Program in Informatics. T. D. Vinh (B) VNU, University of Education, Vietnam National University, Hanoi, Vietnam e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_35

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Informatics Education plays a key role in preparing students for the ability to seek, receive, expand knowledge and be creative in the era of the fourth industrial revolution and globalization. Informatics has a great influence on the way people live, think, and act, and is an effective tool to support turning learning into lifelong self-study. The subject of Informatics helps students to adapt and integrate into the modern society, to form and develop students’ computer competence to study, work, and improve the quality of life, to contribute to the construction career and defend the Fatherland. The content of Informatics course develops three blended knowledge circuits: Digital literacy (DL), information and communication technology (ICT), computer science (CS) and is divided into two phases [2]. • Basic Education Stage: (Primary and Junior High School level) Informatics helps students form and develop the ability to use digital tools, familiarize themselves with, and use the Internet; initially form and develop problem-solving thinking with the support of computers and computer systems; understand and follow the basic principles of information exchange and sharing. • Stage of Career-oriented Education: (High School level) The subject of Informatics has a deep division. Depending on their interests and future career plans, students choose one of two orientations: Applied Informatics and computer science. The two orientations share a number of sub-themes, and each of these has its own sub-themes. Oriented Applied Informatics meets the needs of using computers as a tool of digital technology in life, study, and work, bringing adaptation and service development capabilities in the digital society. Computer science orientation meets the initial purpose of understanding the operating principles of computer systems, developing computer thinking, the ability to explore and discover information systems, and develop applications on computer systems. In addition to the core educational content, students can choose from a number of study topics depending on their interests, needs, and career orientation. Topics in the direction of Applied Informatics aim to enhance applied practice, help students become more proficient in using essential software, and create practical digital products for learning and life. Computer science-oriented topics introduce educational robot control programming, algorithm design techniques, some data structures, and some principles of computer network design. Subject-based teaching is a form of searching for concepts, ideas, knowledge units, lesson content, topics, etc. with interference and mutual similarities, based on relationships theory and practice covered in the subjects or in the modules of that subject (i.e., an integrated path from content from a number of related units, lessons,

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or subjects) make the lesson content in a more meaningful, more realistic topic, so that students can work more on their own to find knowledge and apply it in practice. Subject-based teaching is a combination of traditional and modern teaching models, in which teachers do not teach only by imparting knowledge but mainly guide students to find information, use apply knowledge to solve practical tasks. Based on the above grounds, based on the researches on VR technology [5], the applications of VR in High Schools [5], we propose the topic: Design and application of VR technology in teaching Informatics at High Schools on Topic. If effectively exploiting teaching situations using VR technology, it will actively improve students’ activities, help students understand the nature of some computer concepts, and contribute to improving the quality of teaching and learning in Informatics 11th grade, High School—This scientific hypothesis of the topic. To understand the research problems, we need to answer the following questions: • How can teachers effectively apply VR technology to teaching grade 11 Informatics? • How to build lessons so that teachers and students can effectively use VR technology in teaching and learning? • How will the students’ learning Informatics in grade 11 be improved if VR technology is applied? In this scientific report, we will present the following contents: • Model of bringing VR technology into education and training; • Design and application of VR technology in teaching Informatics in High Schools on topics; • Pedagogical experiment.

2 Model of Bringing VR Technology into Education and Training 2.1 Two Model of Bringing VR Technology into Education and Training • Top-Down Model: According to this model, we need to comprehensively consider the education system (Fig. 1) and then study the overall solution and deploy the application on a large scale. Whether this program has the desired effect also depends on many factors, for example: The teachers have enough knowledge, ability, and enthusiasm to help bring VR technology into the classroom and enjoy adapt the teaching process/content to be more appropriate; schools have the physical resources to initiate and sustain the integration of VR technology into the classroom.

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Fig. 1 Top-down model (Source Author)

•Comprehensive Review of Education •Research the Total Solution •Application Development on a Large Scale

• Bottom-Up Model: According to this model, we, first of all need to find a small solution and then study the whole and deploy the application on a large scale. In parallel with the top-down model, the active bottom-up model (Bottom-up) proved to be more effective in some societies such as the USA. Investment cash flow from organizations/individuals began to pour more into VR projects. The class/school scale VR project flourished. These factors in turn motivate schools to research as well as integrate VR technology into teaching. We can see that this model has the advantage of being able to stimulate social dynamics and make each small part better prepared for the mass application of technology [1]. However, it also means that more students will have the opportunity to be exposed to technology earlier than other places (if there are any project is invested); and some projects with high investment efficiency in the locality are not completely suitable for the purpose of mastication, so it is not possible to optimize investment funds. So which model would be more suitable for bringing VR technology into the classroom? To give the right answer, planners should consider many environmental and human factors.

2.2 Age Characteristics of High School Students and the Ability to Receive Information Our target audience is High School students.

2.2.1

Features of the Intellectual Development of High School Students

The age of High School students is an important period in intellectual development. Because their bodies have been perfected, especially their well-developed nervous system creates conditions for the development of intellectual abilities. Their senses and perception have reached the level of adults. The process of observation is associated with thinking and language. The ability to observe a personal quality also begins to develop in them. Therefore, the application of VR in teaching will make students have more vocabulary, increase their ability to imagine thinking. However, the children’s observations are often scattered, not highly focused on a

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certain task, while observing an object is still general, one-sided, making hasty conclusions without foundation reality. Therefore, the application of VR technology in teaching also increases the ability to feel, concentrate, and improve the ability to observe a specific object. The memory of High School students is also markedly developed. Intentional memory plays a dominant role in intellectual activity. The students knew how to rearrange learning materials in a new order, with a scientific method of memorization. That is, when studying, they already know how to draw the main ideas, mark important passages, the main ideas, make a summary outline, make a comparison table, compare… Thinking activities of High School students develop strongly. They were able to think logically and abstractly more independently and creatively. The highly developed ability to analyze, synthesize, compare and abstract helps children to comprehend all complex and abstract concepts. They like to generalize, like to learn the general laws and principles of everyday phenomena, of the knowledge that must be absorbed… The ability to think development has contributed to the emergence of a new psychological phenomenon that is scientific skepticism. Before a problem, they often ask questions or use antithesis to understand the truth more deeply. In general, the thinking of High School students thrives; intellectual activities are more flexible and sensitive. They have the ability to judge and solve problems very quickly. However, some students still have the disadvantage that they have not fully developed their ability to think independently and have hasty conclusions based on feelings. Therefore, teachers need to guide and help students to think positively and independently to analyze and evaluate things and draw their own conclusions. The development of students’ cognitive ability in teaching is one of the important tasks of teachers.

2.2.2

The Learner’s Receptivity Level

The pyramid shows the level of absorption of learners according to different methods. Inside: Learners will remember 5% of the content when listening to a lecture (traditional method); 10% when learners read books; 20% from audio-visual devices; 30% from simulated devices (similar to simulation methods); 50% from group discussion (similar to participatory methods); 75% from practice, self-experience; 90% through teaching others. Then, the question arises: How should learners learn to get the most effective? We know that books, classroom lectures, videos… are all non-interactive learning methods and as a result, 80–95% of knowledge only remains in learners. The best solution for high efficiency here is that through practice with a short period of time, we should focus our time, energy, and resources on the object. The learning pyramid is represented by Fig. 2. The learning pyramid is the foundation for building learning methods in the classroom. Today, learners increasingly prefer modern learning methods instead of traditional learning methods, which is simply listening to teacher’s lecture. Vietnamese

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Lesson 5% Reading Books - 10% Listen and See 20% Presentation - 30% Discussion Groups - 50% Self-experience - 75% Teach others - 90% Fig. 2 Learning pyramid (Source Author)

students today are still familiar with traditional teaching methods, that is, teachers teach, students listen, and take notes. Therefore, teaching and learning according to modern methods should have the following five processes: • Introduction: By teaching by the teacher, by asking students to read out loud the information in the lesson and learn through audio-visual equipment with vivid images and sounds (20%); • Teaching concepts: After students have read the information, students can be asked to present it from memory; then, the teacher gives examples for students to discuss together to achieve the goal of “group discussion” (50%); • Applying the concept: After grasping the theory, students have to explain right and wrong, explain the knowledge to others (90%); • Apply and Expand: After students can explain to other students, they can practice, make products, and connect with other knowledge.

2.3 Teaching by Topic in High Schools Today The General Education Program in 2018 and the High School Education Program in Informatics in particular are built in an open direction and in diverse educational forms, selecting practical and attractive topics, creating favorable conditions for students to learn and apply the subject not only within that subject but also in other subjects, not only on school grounds but also in off-campus environments, serving vocational education and education STEM [2]. As such, it makes perfect sense to develop lessons based on themes and interact with the new program. The creation of topics helps students easily unify blocks of knowledge, the acquired knowledge is linked with each other according to a certain

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Fig. 3 VR glasses (Source Author)

Table 1 VR glasses

VR glasses

Frequency

Percent

Yes

7

20.00

No

28

80.00

Total

35

100.00

network. In addition, the knowledge is very practical for students to apply in life, hone skills and develop their own capacity. Currently, thematic teaching is chosen by many teachers, even interdisciplinary integration to create a block of knowledge that is easy to understand and apply for students.

2.4 Applying VR Technology Teaching After taking a survey of 35 students from class 11A4 at Tay Ho High School, Hanoi, we have some information on the extent to which VR glasses (Fig. 3). (VR technology) have been used in computer science subjects as shown in Table 1. The table tells us that many students do not use VR glasses before. Through the above survey, it was found that students have not used many VR applications. The best solution is to improve students’ active learning, passion, and discovery of VR technology; Informatics teachers need to design lessons and innovate teaching methods.

3 Design, Application of Virtual Reality in Teaching Informatics High School on Topic In this section, we will present the steps to build a teaching topic, Applying Realistic VR in High School Education? First of all, we would like to present the steps to build a teaching topic.

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3.1 The Process of Developing a Teaching Topic The process of developing a teaching topic is carried out according to the following steps: Step 1: Determine the topic name, implementation duration: (Performing group) • Determine the title of the topic: At the beginning of the school year, the professional group reviews the program content, the current textbooks to adjust and arrange the contents in the textbooks of each subject (can remove old, outdated information… Supplement, update new information). A collection of knowledge units are closely related in theory and practice, thereby structuring and rearranging the teaching content into a teaching topic. • Duration: The number of periods for a topic should have a moderate capacity (about 2–5 periods) for the compilation and implementation to be feasible, ensuring the total number of periods of the program of each subject after editing re-composed with topics that do not exceed or fall short of the time specified in the current program. Step 2: Develop the goals to be achieved by the topic: (The team does it) • Setting goals: The professional group sets up goals on the knowledge and skills standards to be achieved by the topic suitable for their students (based on the knowledge and skills standards). • Teaching methods and techniques: Depending on each topic, physical conditions and students, teachers actively choose appropriate teaching methods and techniques: State the problem, according to the contract, according to the project…; in particular, it is necessary to pay attention to the correct application of the process and steps of active teaching methods and techniques as prescribed. • Organizational form of teaching: Based on the content of the topic, the students, and the actual conditions of the school, the teacher chooses the appropriate teaching method for each period of the topic: teaching whole class, individual, group, outdoor, sightseeing… • Teaching equipment: Exploiting and using to the maximum and effectively the means, equipment and teaching aids, especially the subject classrooms and the school library, avoid the situation of vegetarian teaching, teaching heavy on academic theory, few practical skills, not attached to practice. Step 3: Build a description table: (The team does it) On the basis of the general goal of the topic, the specialized group concretizes the goals.

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Table 2 Analysis of level of awareness Content/Topics/Standards

Recognize

Understand

Apply

Apply highly

Content 1 Content 2

For each content according to the level of awareness in Table 2. Step 4: Compilation of questions/exercises: (Performed by the group) For each developed topic, four levels of requirements (recognition, understand, application, and high application) of each type of question/exercise can be used to test and evaluate capacity and quality of students in teaching. On that basis, compile specific questions/exercises according to the required levels described for use in the process of organizing teaching activities and testing, assessing and practicing on the topic build. Step 5: Develop a plan to implement the topic: (Implementation group) The development of the topic implementation plan is described in Table 3. Step 6: Organize implementation This step of organization and implementation includes: designing the teaching process, organizing the teaching and observing and assigning and drawing lessons learned. Detail: • Designing the teaching process (personally implemented) The topic teaching process is organized into student learning activities that can be done in class and at home; each class period can only perform some activities in the pedagogical process of the method teaching and techniques teaching used. Some of the activities that we need to pay attention to are: Activity 1. Warm-up/unpacking. Detail: 1. Objectives: …; 2. Students’ learning tasks: ...; 3. How to carry out the operation: … Activity 2. Forming new knowledge; Detail: 1. Objectives: …; 2. Students’ learning tasks: …; 3. Operation method: Step 1. Assign tasks; Step 2. Perform assigned tasks; Step 3. Report the results and discuss; Step 4. Evaluate the results. Table 3 Analysis of implementation plan Contents Content 1 Content 2

The form of organization of the teaching process

Duration

Time

Teaching equipment, learning materials

Notes

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Activity 3. Practice. Detail: 1. Objectives: …; 2. Students’ learning tasks: …; 3. How to carry out the operation: … Activity 4. Applying. Detail: 1. Objectives: …; 2. Students’ learning tasks: …; 3. How to carry out the operation: … Activity 5. Extensive exploration. Detail: 1. Objectives: …; 2. Students’ learning tasks: …; 3. How to carry out the operation: … In a topic with many lessons, it can be compiled together, without having to separate each period, without having to repeat common parts (such as: common goals of the topic, standard requirements for knowledge, skills, and attitudes, …). For topics with many lessons, the teacher actively distributes the appropriate time and knowledge according to the students; the recording the number of lessons in the program distribution in the lesson book. • Organizing teaching and attending classes On the basis of the established teaching topics, the specialized team assigns teachers to conduct lessons to observe, analyze, and draw lessons from teaching hours. When attending class, it is necessary to focus on observing students’ learning activities through the implementation of learning tasks with the following requirements: – Transfer of learning tasks: Learning tasks are clear and suitable to students’ abilities, reflected in product requirements that students must complete when performing tasks; the form of assigning tasks is lively, attractive, stimulating the cognitive interest of students; ensure that all students are receptive and ready to perform their duties. – Performing learning tasks: Encouraging students to cooperate with each other when performing learning tasks; timely detect difficulties of students and take appropriate and effective support measures; no student is “forgotten.” – Report results and discussion: Report format suitable for learning content and active teaching techniques are used; encourage students to exchange and discuss with each other about learning content; handle pedagogical situations that arise in a reasonable manner. – Evaluation of the results of the performance of the learning task: Commenting on the process of performing the learning task of the students; analyze, comment, and evaluate the results of the task performance and the discussion ideas of the students; correct the knowledge that students have learned through the activity. • Analyze and learn lessons The teaching process of each subject is designed into student learning activities in the form of successive learning tasks, which can be done in class or at home. Students are active, proactive, and creative in carrying out learning tasks under the guidance of teachers. Analyzing teaching hours from that point of view is to analyze the effectiveness of students’ learning activities and at the same time to evaluate the organization, testing, and orientation of learning activities for students.

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3.2 Realistic Virtual Reality Applying in High School Informatics Education With the strong development of modern technology, the use of virtual reality glasses in education, especially in classes, is very necessary and urgent. Virtual reality glasses are an intermediate device, bringing students into a threedimensional space. In that three-dimensional space, students are like entering another world, a “virtual world.” That world was a completely different space from the current space the student was in. To apply VR in education, we can record actual videos and create videos with various software, such as Windows Movie Maker, ProShow Producer, Imovie, … Essential for each theme are VR glasses, smartphones, VR box. The application of VR in teaching High School themed needs to develop the following steps: (i) (ii) (iii) (iv) (v) (vi)

Step 1: Choose an appropriate theme to use VR; Step 2: Build a lesson plan; Step 3: Choose suitable VR videos (can be obtained on YouTube or designed by yourself or created with specialized software); Step 4: Prepare before class (VR glasses, smartphones…); Step 5: Teaching; Step 6: Learn from experience.

To evaluate the effectiveness of the application of VR in teaching Informatics, we need to learn in the pedagogical experiment.

4 Pedagogical Experiment 4.1 Experimental Purpose Pedagogical experiments were carried out to test the feasibility and effectiveness of using VR technology in teaching the topic “Informatics applications,” testing the correctness of scientific hypotheses.

4.2 Experiment Description (i)

(ii)

Step 1: Choose an appropriate theme to use VR: In this step, we have chosen the topic “Informatics applications”—Informatics grade 11 in High School. Step 2: Build a lesson plan: After choosing a topic, we integrate images, songs, music… as data and proceed to prepare lesson plans.

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(iii)

Step 3: Select suitable VR images and videos (can be obtained on YouTube or designed by yourself or created with specialized software); Step 4: Prepare before class (VR glasses, smartphone, VR box…); Step 5: Teaching: As they conduct the teaching, they realize that their goals have been achieved. Students feel very excited and excited about the class. Step 6: Learn from experience.

(iv) (v)

(vi)

4.3 Results and Processing of Pedagogical Experimental Results After experimenting, we found that the help of VR in teaching the topic “Application of Informatics,” Informatics 11, at Tay Ho High School, Hanoi, has helped students promote visual observation, from that make comments, discover new problems. In addition, students are given the opportunity to handle real-life situations with the application of VR. Students are provided with the most favorable conditions to promote the activeness of students. At the same time, it helps teachers to save time and control the cognitive activities of students, the test, and assessment of students’ learning results also goes smoothly, and at the same time trains students in self-study skills, self-study, conscious access to applied technologies in learning and thinking development. However, the use of VR technology in teaching “Informatics application” is still limited. After conducting experiments, with the combined use of VR glasses and smartphones in teaching “Informatics applications,” students feel excited, and difficulties and obstacles are gradually removed. To find out more about students’ interest in the content “Informatics applications” with the application of VR technology compared to traditional teaching methods, we distributed a questionnaire consisting of five questions and asked students to answer the questions in the questionnaire for two classes: 11A4: 35 students and 11D9: 35 students. And, we got the following results (Table 4). After the implementation, most of the students in the experimental class felt interested in learning the content “Informatics applications” with the support of VR technology and Smartphones. Table 4 Statistics of students’ interest after the experiment Like so much

Like

Normal

Dislike

No students

20

8

7

0

Rate (%)

57.14

22.8

20.06

0.00

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4.4 General Assessment of Pedagogical Experiment From the results obtained when teaching the topic “Informatics applications” with the help of VR technology to improve the quality of Informatics teaching, it is completely scientifically based and feasible. However, the use of VR in teaching “Informatics applications” is still limited [3, 4]. Because some teachers’ knowledge and skills in VR and IT are still limited, not enough to exceed the threshold for passion and creativity, and even avoid. On the other hand, the old teaching method is still a hard way to change. This makes VR in particular and IT in general, even though it has been included in the teaching process, still cannot promote its full positive and effective. In summary, through the analysis of the results obtained, combined with monitoring the learning process of students during the research period and the subject experiment, we have confirmed the correctness of the scientific hypothesis that the topic is proposed established talent. Therefore, applying VR to teaching content “Informatics application” according to the research direction of the topic is completely feasible.

5 Conclusion This report provides some concepts about two model of bringing VR technology into education and training; characteristics of High School students and their ability to receive information; teaching by topic at current High Schools. Furthermore, we also offer applying VR technology in teaching, and design, application of VR technology in teaching Informatics at High Schools on topic. Through the analysis of pedagogical experimental results, we see the feasibility and effectiveness of the application of VR technology in teaching and applying Informatics. Thus, it can be affirmed that: the research purpose has been carried out, the research task has been completed, and the scientific hypothesis is acceptable. Acknowledgements This work has been partly supported by the Vietnam National Education and Science Programs under the project KHGD/16-20DT.042.

References 1. Daniela, L. (2019). Didactics of smart pedagogy. Smart pedagogy for technology enhanced learning. Springer Switzerland AG [ISBN 978-3-030-01550-3]. 2. Ministry of Education and Training. (2018). High school education program—Overall program (Promulgated together with Circular No. 32/2018/TT-BGDDT dated December. Minister of education and training, Hanoi, Vietnam). 3. Pal, S., Cuong Ton Q., Nehru, R. S. S. (Eds.). (2022). Digital education for the 21st century: Technologies and protocols. CRC Press. [ISBN: 9781774630075]

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4. Pal, S., Cuong Ton Q., Nehru, R. S. S. (Eds.). (2020). Digital education pedagogy: Principles and paradigms. CRC Press. [ISBN 9781771888875] 5. Vinh, T. D. (2020). Virtual reality: A study of recent research trends in worldwide aspects and application solutions in the high schools. In 2nd International conference on innovative computing and cutting-edge technologies (ICICCT-2020).

Adjacent Graph for Some Finite Groups R. Kamali and C. J. Chris Lettecia Mary

Abstract In this paper, a new concept Adjacent Graph for Groups is introduced, which is based on the study of algebraic graph theory. Some properties were derived for this adjacent graph for groups. For this derivation, algebraic methods are applied and explained using graphs because the most common algebraic methods include graphical methods. As graph theory is about the relationship between edges and vertices, consequences of adjacent graph for some finite groups were developed. Also, we have established the relation between the non-self-invertible elements of finite group G and the vertices of the adjacent graph AI (G). Also, we discussed that the adjacent graph is not complete, and the adjacent graph is null graph when the number of elements of I is zero. If the elements of I is two, then the adjacent graph is bipartite, planar, 2-colourable, and having a clique number 2. Also, the adjacent graph is 2-regular if the number of elements of G is four. If the number of elements of G is greater than or equal to 3, then the degree of the adjacent graph is greater than or equal to 1, and the adjacent graph is connected. The diameter of adjacent graph AI (G) is always two. We illustrated few sufficient graphs as examples to show our adjacent graph for groups is unique among other familiar graphs. Keywords Cycle · Finite Abelian group · Finite group · Graph · Planar graph AMS Subject Classification 05C38 · 20K01 · 20D05 · 05C25 · 05C10

1 Introduction In the area of mathematics, graph theory [10] is the study of graphs that involves the connection between points and lines as edges and vertices. A graph [5] is a pictorial depiction of a set of objects where pairs of objects are united by links. Some R. Kamali · C. J. Chris Lettecia Mary (B) Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies, Chennai, Tamil Nadu 600117, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_36

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of the applications of graph theory are computer science, information technology, biosciences, mathematics, and linguistics [11]. A group [13] is an assembly of components or objects that are merge together to perform some operations on them. The study of a set of elements in a group is called group theory [7]. An idea of group is derived from abstract algebra [2]. Rings, fields, and vector spaces are some familiar algebraic structures [1] to be considered as groups with some additional operations and axioms. Whenever an object’s property is same, the object can be examined under group theory; being group theory is the study of symmetry [4]. The method to solve Rubik’s cube also acts based on group theory [8]. Algebraic graph theory [6] is a branch of mathematics that involves to find the solutions for algebraic methods [3] by using graph theory concepts. Linear algebra, group theory, and the study of graph invariants are three main divisions of algebraic graph theory [14]. There is a bind between graph theory and group theory [12], which is shown by Arthur Cayley. He was the first to introduce the Cayley graphs to finite groups [9]. The main aim of this paper is to introduce adjacent graph for groups and to discuss some of its properties. Few graphs are demonstrated as an example to show how this adjacent graph is entirely different from the existing graphs.

2 Preliminaries Definition 1 The Eccentricity [8] ecc(v) of a vertex v in graph X is the maximum distance from v to any other vertex u  X. Definition 2 The Diameter [3] of a graph X is denoted by diam(X) and is defined by diam(X) = max {ecc(v), for all v  X}. Definition 3 The Neighborhood [10] of a vertex v in a graph X is the set of all vertices adjacent to v. N(v) = {u  X/uv  E(X)}. Definition 4 An Adjacent Graph AI (G) [6]is defined for the finite group (G, ∗ ) as graph whose set of vertices coincides with G and I = {g  G/g = g−1 } such that two distinct vertices g and h are adjacent if and only if g ∗ h  I or h ∗ g  I. Example 1 See (Fig. 1). Definition 5 The clique of largest possible size is called maximum clique, and the size is known as Clique Number [3].

3 Adjacent Graphs for Groups and Properties Theorem 1 In a finite group G, the adjacent graph AI (G) is null graph if and only if the number of elements of I is zero.

Adjacent Graph for Some Finite Groups

363

1

Fig. 1 Adjacent graph for the group (Z 7 −{0}, .) [Here I = {2, 3, 4, 5}]

2

6 5

3

4

Proof: Assume that the adjacent graph AI (G) is null graph. Then, by the Definition 4 For any g, h  G, g ∗ h  I. But, g = g − 1, h = h − 1, and g = h. Therefore, g ∗ h ∈ / I. Hence, I = ϕ. ⇒ |I| = 0. Therefore, the number of elements of I is zero. Conversely assume that The number of elements of I is zero. (i.e.,) I = ϕ. All the elements of G are self-invertible. (i.e.,) g = g−1 , h = h−1 , and g = h where g, h  G. Then, by Definition 4, the adjacent graph AI (G) is null graph. Example 2 Let us consider a group G = (Z3 −{0},.) ⇒ G = {1, 2}. Here, non-self-invertible element I = ϕ. The adjacent graph of (Z 3 −{0}, .) is the null graph of order two. Theorem 2 The adjacent graph AI (G) is not complete for every finite group G. Proof: Case (i) Let (G,.) be a group. Let g, h be any two vertices of the adjacent graph AI (G). If g. h = 1 and g = h also 1 ∈ / I. Hence, g and h are not adjacent in AI (G). Case (ii) Let (G, + ) be a group. Let i and j be any two vertices of the adjacent graph AI (G). If i + j = 0 and i = j.

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Since 0 ∈ / I, i and j are not adjacent in AI (G), Hence, any adjacent graph is not complete. + Theorem 3 For a group G with |G| ≥ 3 and I ⊂ G with I = ϕ, then the deg (AI (G)) ≥ 1. Proof: Let AI (G) be the adjacent graph of the group G. Given I is a non-empty subset of non-self-invertible elements. Let {e} be the identity element of G, {e} is adjacent to all vertices except the self-invertible elements. Let z be the self-invertible element. There exist a vertex y ∈ I such that y ∗ z  I. Therefore, z is adjacent to y. Hence, the degree of adjacent graph is greater than or equal to 1. Theorem 4 For any finite abelian group G, |G| ≥ 3 and I ⊂ G; I = ϕ, the adjacent graph AI (G) is connected. Proof: Let {e} be the identity element. Obviously, identity element is adjacent to every element in I. Enough to prove: G−{I ∪ {e}} is adjacent to at least one element in I. Consider g ∗ h, g  I, and h  G−{I ∪ {e}}, ∗ is the operation defined on G. (g ∗ h)−1 = h−1 ∗ g−1 . = g-1 ∗ h-1 [∵G is Abelian group]. = g−1 ∗ h = g ∗ h[∵g  I, g = g−1 , h = h−1 ]. Therefore, g ∗ h  I. ⇒ G−{I ∪ {e}} is adjacent to at least one element in I. The graph AI (G) is connected. Corollary 1 Let G be any group with |G| ≥ 3, I ⊂ G; I = ϕ, { e} be the identity element of G, then the elements of G−{I ∪ {e}} are adjacent to each elements of I in the adjacent graph AI (G). Example 3 Consider the Fig. 2, in which G = {1, 2, 3, 4, 5, 6}, I = {2, 3, 4, 5} and e = {1}. Then, I ∪ {e} = {1, 2, 3, 4, 5}. Also, G−{I ∪ {e}} = {6}. Therefore, {6} is adjacent to all elements in I. Theorem 5 The diameter of any adjacent graph is two. Fig. 2 Null graph

1

2

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Proof: Let AI (G) be the adjacent graph of the finite group G, I ⊆ G with I = ϕ and {e} be the identity element. Case (i). To prove: The eccentricity of e is two. Let I’ = G−{I ∪ {e}}. By Definition 3, N(e) = {u ∈ AI (G) such that eu ∈ E(AI (G))}. For any u  I and v  I’, N(e) ∩ I’ = ϕ. But, u is adjacent with both {e} and v. Therefore, d(e, v) = e. Hence, the eccentricity of e is two. Case (ii). To prove: The eccentricity of u is two. By Definition 3, N(u) = {v ∈ AI (G) such that uv ∈ E(AI (G))}. For any u  I and v  I’, u−1 be the inverse of u in AI (G). / N(u). u−1 ∈ But, v and e belongs to N(u). Therefore, d(u, u−1 ) = 2. Hence, the eccentricity of u is two. Case (iii). To prove: The eccentricity of v is two. By Definition 3, N(v) = {u ∈ AI (G) such that uv ∈ E(AI (G))}. For any u  I and v  I’, e ∈ / N(v). But, u is adjacent with both {e} and v. Since I’ contains only the element {v}, Therefore, e−v is the longest path in the graph X and d(e, v) = 2. ⇒ the eccentricity of v is two. Therefore, the diameter of any adjacent graph is two. Theorem 6 The adjacent graph AI (G) is Km,n where m = 2, n = |G|−2 for a finite Abelian group G, I ⊆ G with |I| = 2. Proof: Let V 1 = I and V 2 = G−I be the partitions in AI G, then V(AI (G)) = {V 1 , V 2 }. Case (i) To prove: Any two elements in V 1 are not adjacent in A1 (G). Let a and b be any two elements such that a, b  V 1 . Since |I | = 2, I = {a, b}. a ∗ b = e  V 2.

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Fig. 3 Complete bipartite graph

1

4

2

3

Since e ∈ / V 1, Therefore, any two elements in V 1 are not adjacent in A1 (G). Case (ii) To prove: Any two elements in V 2 are not adjacent in A1 (G). Let c and d be any two elements such that c, d  V 2 . c ∗ d = z  V 2. Since z ∈ / V 1, Therefore, c and d are not adjacent. Case (iii) Let a and b be any two elements such that a, b  V 1 . Let c and d be any two elements such that c, d  V 2 . a ∗ c = x  V 1. a ∗ d = y  V 1. b ∗ c = y  V 1. b ∗ d = x  V 1. a ∗ c = b ∗ d = x  V 1. a ∗ d = b ∗ c = y  V 1. Each and every elements of V1 is adjacent with all the elements of V2 with |V1| = m and |V2| = n. Hence, the adjacent graph AI (G) is K m,n where m = 2, n = |A|−2 (Fig. 3). Example 4 By Theorem 6, V 1 = I = {2, 3} and V 2 = G−I = {1, 4}. Theorem 7 The adjacent graph AI (G) is 2-regular for a group G with 4 elements and I ⊆ G. Proof: Let G = {e, v1 , v2 , v3 }. Let e be the identity element. Let v3 be the self-invertible element. We know that I is non-empty and I = {v1 , v2 }, by Definition 4. As mentioned in Fig. 4, obviously, AI (G) is a cycle C 4. Therefore, AI (G) is 2-regular. Remark If G is a finite abelian group with |I| = 2, then the adjacent graph AI (G) is.

Adjacent Graph for Some Finite Groups Fig. 4 Cycle C 4

367

2

1

3

Fig. 5 Planar graph

4

0

1 (i) (ii) (iii)

2

Planar 2-colourable Having a clique number 2.

Proof: (i) (ii)

(iii)

Let the adjacent graph AI (G) be (p, q) graph. Here, q ≤ 3 p − 6. It satisfies the condition for planarity. Therefore, the adjacent graph AI (G) is planar. Let us consider the Abelian group (Z 5 −{0},.) as shown in Fig. 5. Here, I = {2, 3} and | I | = 2. We know that the cycle graph C n on n vertices is 2-colourable if and only if n is even. Therefore, the adjacent graph AI (G) is 2-colourable. Let us consider the Abelian group (Z 5 −{0},.) as mentioned in Fig. 5.

Here, K2 is the maximum clique size in the adjacent graph. Therefore, by Definition 5, the adjacent graph having a clique number 2.

4 Conclusion In this paper, a graph called adjacent graph for finite groups is introduced. Also, some properties of this adjacent graph were deliberated, and few adjacent graphs are demonstrated as examples.

5 Result It is possible to find some more properties for the adjacent graph.

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References 1. Anderson, D.F., and P.S. Livingston. 2018. The zero-divisor of a commutative ring. International Electronic Journal of Algebra 23: 176–202. 2. Bubboloni, D. 2009. On bipartite divisor graphs for group conjugacy class sizes. Journal of Pure and Applied Algebra. 3. Bertram, E.A., M. Herzong, and A. Mann. 1990. On a graph related to conjugacy classes of groups. Bulletin London Mathematical Society 22: 569–575. 4. Camina, A. R. 2000. Recognizing direct products from their conjugate type vectors. Journal of Algebra. 5. Chartrand, G., Lesniak, L., and P. Zhang. 2010. Graphs and digraphs, 5th edn. Chapman and Hall/CRC. 6. Godsil, Chris, and Gordon F. Royle. 2001. Algebraic graph theory. New York: Springer science + business media. 7. Kurzweil, Hans, and Bernd Stellmacher. 2004. The theory of finite groups—An introduction, Springer, New York. Inc. 8. Meena, S., and A. Ezhil. 2019. Total prime labelling of some graphs. International Journal of Research in Advent Technology. 9. Alfuraidan, Monther R., and Yusuf F. Zakariya. 2017. Adjacent graphs associated with finite groups. Electronic Journal of Graph Theory and Applications 5 (1): 142–154. 10. Narsigh Deo. 2007. Graph Theory with applications to Engineering and Computer Science. 11. Robin J. Wilson. 1996. Introduction to graph theory, 4th edn. 12. Salama, F., H. Rafat, and M. El-Zawy. 2012. General graph and adjacent graph. Applied Mathematics 3: 346–349. 13. Samir Siksek, Introduction to Abstract Algebra. 14. Yusuf F Zakariya. 2016. Graphs from finite groups, Proceedings of September. Annual National Conference.

Stochastic Inventory Model Using Coxian Distribution with Production and Sales D. Kanagajothi and Hanaa Hachimi

Abstract Inventory is a retaining stock of physical items with financial impact that is retained in different forms by an enterprise in its custody, pending, processing, transformation, use or sale at a later date. Many authors examined this model and discussed its probability density function. In this paper, three products are processed in a production inventory system where output is influenced by seasonal demand and sales. The three separate items are manufactured is sequence. The product production times are general distribution, and the demand season starts after a random time whose distribution is Coxian–Erlang. Laplace the production transform, sales time and their means are obtained. The results are illustrated by simulation studies. Keywords Inventory systems · Production · Products · Coxian distribution

1 Introduction In this paper, we consider that the products manufacturing time has a general distribution, and the demand season begins at a random time with Coxian–Erlang distribution. We showed that sales time and their means derived. Bagchi and Hayya [1] obtained when the demand and lead time distributions are normal and a formula for the probability density function of demand during the lead time of the production level and possible lost sales are calculated using the probability density function. The sum of two independent Erlang distributions is determined by Kadri and Smaili [2]. A flow model is made up of machines with Cox-2 distribution processing times and restricted buffer capacity, according to Stefan [13]. Decomposition of a bigger flow line into a group of related two machine lines yields an exact analysis of a two machine subsystem. The investigator got the motivation to introduce D. Kanagajothi (B) Vel Tech Rangarajan Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, India e-mail: [email protected] H. Hachimi Sultan Moulay Slimane University, Beni Mellal, Morocco e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_37

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the concept of Coxian distribution with production and sales of stochastic inventory model. Parvathi et al. and Kanagajothi et al. studied single and three products inventor systems. Vidalis [15] deal has a two-tier supply chain inventory policy. A stochastic (Q, R) inventory system was described David and Robert [2]. Demands are considered to be of the Poisson distribution, while lead times are assumed to be of the Erlang distribution in the scenario. For queuing and reliability studies, the Coxian-2 distribution is extremely useful. It is vital to understand that a Coxian-2 distribution can approximate a general probability distribution by fitting the first three moments. A positive lower bound random variables in third moment fit is exists in positive random variable are given in this note to completed the figure by Van Der Heijden [14]. Sakthi et al. [12] reported the performance measures; they reported the average number in the system average waiting time, probability of loss. The tool provides the steady-state distribution of number of customers. The tool helps to solve the queue with restricted and unrestricted buffer. Ou et al. [8] introduced a method that can be used in different types of stochastic models. The main idea comes from queuing theory. Ming and Zhang [6] studied that Coxian distributions are used to approximate matrix-exponential distributions. They created a method for constructing Coxian representations of matrix-exponential distributions based on the spectral polynomial algorithm. Ntio and Vidalis [7] analyzed the performance of multi-echelon inventory systems. Specifically, two serial inventory systems with two and three stages, respectively, have been researched. The supply networks are described as discrete-state continuous Markov processes. The structures of those systems transition matrices is investigated, and computer procedures are created to construct them for various system parameter values. The Erlang distribution is followed by the replenishment processes. The external demand is distributed using a pure Poisson process, which means that each customer’s request is one unit in size. Finally, the last upstream node is always saturated.

2 Assumption The following are presented of the model. (i)

(ii)

The corporation makes three separate items A, B and C but only one at a time is produced. The products A, B and C are completed in one by one. A, B and C production times are random variables X, Y and Z , respectively. Product A production time X has CDF U X (.) and PDF u X (.); the product B production time Y has CDF UY (.) and PDF u Y (.) and the product C production time Z has CDF U Z (.) and PDF u Z (.). The production time of the sum X + Y + Z of one triplet has CDF U (.) and PDFg(.)u(.). After an exponential random time with probability q, the product season begins or after two exponential random times with probability p = 1 − q where λ is the parameter of the two exponential distributions. In other words, the season stars after Coxian–Erlang time with two phases.

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(iii)

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The time of sales will start; the k numbers of triplets of products are made otherwise when the season of demand begins.

The products are sold in triplets, and a triplet selling time is determined by the random variable with CDF S(.) and PDF s(.). The CDF of selling time of product A and B is respectively S A (.) and S B (.) with PDF s A (.) and s B (.).

3 Analysis Let us consider, in the interval (0, t), the probability of n units which are produced is given by Un (t)−Un+1 (t) for n is greater than equal nto zero, where Un (t) is n fold CDF (X i + Yi + Z i ). When the nth convolution of U (t) and Un (t) is the c.d.f of i=1 triplet is completed at time x < t, then during the period (x, t), there are possibilities of completion or incompletion of products A, B and C as follows. During the period (x, t). (a) (b) (c)

Product A manufacture time is not yet complete or The product A is manufactured, but the product B is not or The products A and B are manufactured but the product C is not.

Their respective probabilities are given below. Let u n (x) be n fold convolution of u(x) with itself where u(x) is the PDF of X + Y + Z and u n (x) is the PDF of n (X i + Yi + Z i ), where U (x) = 1 − U (x). Then the probability of n triplet i=1 units is produced in (0, t) and the n(n + 1)th production of A is not completed before t is Eq. 1 and the production of A is over before t, but production for B is not over before t is the Eq. 2.  P

n 

(X i + Yi + Z i ) < t
0, y(0) = y0 > 0 and finally, the theorem is proved.

(11)

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4 Positivity Theorem 3.2: Each solutions of (10) with initial conditions (11) exists with s, i and y all are strictly greater zero ∀ τ > 0. Proof The R.H.S. of (10) is totally continuous and locally Lipschitzian conditions holds in R3+ and the solution of (s, i, y) of (10) with initial conditions (11) exists and unique on [0, ψ), where 0 < ψ ≤ +∞ [14]. With help of initial conditions (11) from the system (10), we have t

 s(τ) = s(0)exp ∫ (1 − s(μ1 ) − iμ1 )) − θ1 i(μ1 ) − Iλ θ2 y(μ1 ) dμ1 > 0 0

t

i(τ ) = i(0)exp ∫[θ1 s(μ1 ) − θ3 ]dμ1 > 0 and 0

t     θ4 (1 − y(μ1 )) + pIλ θ˜2 s(μ1 ) d μ1 > 0 y(τ ) = y(0) exp 0

This shows that the model system (10) is positively invariant for all τ ≥ 0.

5 Boundedness Lemma 5.1: Assume that the initial condition (11) of Eq. (10) satisfies ϕ1 (θ ) + ϕ2 (θ ) ≥ 1, θ belongs to [−τ , 0]. Then (a): i(t) + s(t) ≥ 1∀t ≥ 0 and so t → +∞, (s(t), i(t), y(t)) ≥→ E1 = (1, 0, 0) or (b): there exists a t > 0 for which i(t) + s(t) < 1∀t > t0 . Again, when if ϕ1 (θ ) + ϕ2 (θ ) < 1, θ ∈ [−τ, 0], then s(t) + i(t) < 1∀t ≥ 0. Proof: We have considered i(t) + s(t) is greater than equal to 1 1∀t ≥ 0. By the first two equations of (4), we have



d (s + i) = −s((s + i) − 1) − θ3 i − Iλ θ2 ys dt Hence, ∀ t ≥ 0, we have that

ds dt

+

di dt

≤ 0. Let

lim s(t) + i(t) = η

t→∞

(12)

(13)

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If η is strictly greater than 1, then with help of Barbalat lemma, we can write 0

d = lim (I (t) + s(t)) = − lim s(t)((s(t) + i(t) − 1) − θ3 i(t) − Iλ θ2 y(t)s(t) = t→∞ dt t→∞

lim s(t)(1 − η) − θ3 i(t) − Iλ θ2 y(t)s(t) ≤ − min((η − 1), θ3 ) lim (s(t) + i(t)) . t→∞

t→∞

= −η min((η − 1), θ3 < 0 Here, a contradiction occurs for which η = 1 i.e. lim (i(t) + s(t)) = 1

t→∞

(14)

Let w(t) = s(t) + i(t) where t ∈ [0, ∞), w(t) is diff. and w’(t) is continuous uniformly for t ∈ (0, +∞). So, Eq. 15 satisfies every properties of the Barbalat lemma. Hence, lim

t→∞

d (s(t) + i(t)) = 0 dt

(15)

First two equation of (10) gives

d (s(t) + i(t)) = − lim s(t)((s(t) + i(t) − 1) − θ3 i(t) − Iλ θ2 y(t)s(t) t→∞ dt

(16)

Then, Eq. (15) implies that, lim

t→∞



d (S(t) + I (t)) = − lim s(t)((s(t) + i(t) − 1) − θ3 i(t) − Iλ θ2 y(t)s(t) (17) t→∞ dt

Hence, Eqs. 16 and 17 hold if lim i(t) = 0 and lim y(t) = 0. From (15) implies t→∞ t→∞ lim s(t) = 1. Hence, the case (a) is complete. Again, if possible, let condition (a) t→∞ is violated. Then, there exists t0 > 0 for which s(t0 ) + i(t0 ) = 1. From Eq. (17), we have  

d = [s(t0 )(1 − (s(t0 ) + i(t0 )) − θ3 i(t0 ) − Iλ θ2 y(t0 )s(t0 ) < 0 (i(t) + s(t)) dt t=t0 This states that a solution with 0 < (s + i) < 1, then it is bounded with 0 < (s + 1) < 1, i.e. i(t) + s(t) less than 1, ∀t > t0 . If ϕ1 (θ ) + ϕ2 (θ ) < 1, θ belongs to [−τ, 0], so with help of the previous result, it states that i(t) + s(t) is strictly less than 1 ∀ t > 0, i.e. (b) is true. Lemma 5.2: There exist always positive solution of the proposed model (10) with 2 0) y(t) < B, ∀ t, where B = (1+θ4 )(1+γ > 0, and for γ0 ≤ θ3 . 4 Proof: By the previous Lemma 5.2, we have for any (ϕ1 ,ϕ2 , ϕ3 ) ∈ C+ such that ϕ1 (θ )+ϕ2 (θ ) ≥ 1 ≥ 1, θ belongs to [−τ, 0], then either t0 > 0 exists with s(t)+i(t) ≤ 1∀ t > t0 , or lim s(t) = 1, lim i(t) = 0. Furthermore, if ϕ1 (θ ) + ϕ2 (θ ) < 1, θ ∈ t→∞

t→∞

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[−τ, 0], then i(t) + s(t) ≤ 1 for all = t > 0. Hence, in anytime, say t ∗ (greater than equal to zero), exist if i(t) is strictly less than 1 and s(t) is strictly less than 1 + ε, for all t > t ∗ . Let M = s + i + y. Applying the time derivative on both sides of M, we have found for t > t ∗ + τ . dM dt

+ γ0 M ≤

(1+θ4 )(1+γ0 )2 4

+ (γ0 − θ3 )i − (s −

˙ + γ0 M ≤ B where B = M B ∀ t. Hence, the proof is completed.

(1+θ4 )(1+γ0 )2 . 4

1+γ0 2 ) 2

− θ4 (y −

θ4 +γ0 2 ) . 2θ4

Therefore,

Hence, there exist B > 0, for which M(t)
θ3 and Iλ θ2 < 1 are holds, respectively. If

∗ 4 (θ1 −θ3 1 4 λ( 2) 4 λ (θ

4 (θ1 −θ3 ) and Iλ θ2 < θ θθ+Pθ 2 ) are satisfied respectively. 1 4 3 Iλ (θ Proof: Using the variational matrix of the system (10), the eigenvalues at the point E2 are 1 − 1λ (θ2 ), −θ3 and − θ4. , so if Iλ (θ˜2 > 1, then the point E2 is locally asymp(1+θ4 ) , then the system goes to totically stable. Similarly, it is clear if Iλ θ˜2 < θθ44+pI λ (θ2 ) lAS behaviour near to the point E4 . Theorem 6.3: The system always exhibits LAS behaviour around the interior steady θ 1− 3 −Iλ (θ2 )y∗ pθ I θ state E ∗ , where s∗ = θ1 , i∗ = θ1 and y∗ = 1 + 3 λ ( 2 ) . θ2

1+θ1

θ1 θ4

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Proof: The chapter equation of the Jacobian matrix at interior steady state E ∗ is λ3 + A1 λ2 + A2 λ + A3 = 0, where the coefficients are as follows: A1 = (s∗ + θ4 y∗ ),  2 A2 = θ4 s∗ y∗ + θ1 (1 + θ1 )s∗ i∗ + p(Iλ θ˜2 s∗ y∗ and A3 = θ1 θ4 (1 + θ1 )s∗ i∗ y∗ . Obviously, A1 > 0, A3 > 0 and A1 A2 −A3 > 0. Hence, by Routh-Hurwitz properties, the system under goes to LAS behaviour around the interior equilibrium point E ∗ .

6.2 Interpretation of Dynamics of the System

We have observed that if Iλ θ2 exceeds the value 1, then the prey species becomes

(1+θ4 ) dies out. Again if Iλ θ2 can be strictly below some upper threshold between θθ4+pI (θ) 4

λ

2

(θ1 −θ3 ) and θ θθ4+pθ  , then the proposed model undergoes to infected prey stable zone of 4 1 3 Iλ (θ2 ) three species. Now, the condition of stability for the steady states E ∗ and E4 may take in the form as below:



(A) The planner steady state E4 exists if Iλ θ2 < 1 and will be stable if Iλ θ2 < √ −θ4 + (θ42 +4pθ4 (1+θ4 )) ≡ P2 . 2p

(B) The interior steady state E ∗ will exist if θ1 > θ3 and stable if Iλ θ2 < √ √ −θ1 θ4 + {(θ1 θ4 )2 +4pθ3 θ4 (θ1 −θ3 )} ≡ P1 . Now, the stability conditions of E4 and 2pθ3

∗ E are drawn graphically in the Iλ θ2 − θ3 − θ4 space to signify the joint effect of the stability. (C) For the conditions θ3 < θ1 ,P1 and P2 can be taken distinct ‘+’ve values. Let P{min} = Min{P1 , P2 } and P{Max} = Max{P1 , P2 }. Now, if Iλ θ2 < P{Max} , then the proposed system undergoes towards the points E ∗ or E4 . This shows P{Max} is the upper bound of predation rate for the system without any risk.

There are three cases occurs in part (C):

Case-I: If Iλ θ2 < P{Min} , then the system may reach to either at E ∗ or at E4 . This case holds if values of parameter are belongs to the region-1, which seen in Fig. 1. For example, let θ1 = 0.1,θ 3 = 0.02,θ4 = 0.08, then p = 0.9,P1 = 0.414 and P2 = 0.269. Now, as Iλ θ2 < P{Min} = Min{P1 , P2 } = 0.269, so the proposed system tends to either E ∗ or E4 . equilibrium points depending up on the initial restrictions.

Figure 1 exhibits the stability condition jointly of E ∗ and E4 in Iλ θ2 − θ3 − θ4

parametric regions. In this figure, the S-surface and T-surface divided the Iλ θ2 in

Iλ θ2 − θ3 − θ4 parametric region. There are four regions such as Region-1, Region2, Region-3 and Region-4. We have seen that the Region-4 of the proposed model is unstable. The dynamical system is stable around E ∗ and E4 in the Region-2 and Region-3, resp.

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Fig. 1 Depicts that there are two surfaces namely S and T surface which divided the four parametric region

Case-II: Whenever P2 = PMax , i.e. P2 > P1 , the system undergoes to infected prey free steady state that is at E4 only if θ2 ∈ [P1 , P2 ]. This assumption holds if the values (parametric) are enters in region-2 of Fig. 1. For example let θ1 = 0.1, θ3 = 0.07, θ4 = 0.09, p = 0.9 then P1 = 0.148 and P2 = 0.284. In this case PMax = 0.284 and so the model undergoes to E4 if θ2 ∈ [P1 , P2 ] = [0.148, 0.284]. Figure 2 have illustrated that the system will go stable behaviour around the interior steady state E ∗ and Fig. 3 shows that the system enters to the stability zone with follows the infected free steady state E4 . Case-III: Whenever P1 = PMax , i.e. P1 > P2 , then the system undergoes to steady state E ∗ if θ2 ∈ [P2 , P1 ]. This situation holds if the values (parameter) are enters to the region-3 of Fig. 1. For example let θ1 = 0.1, θ3 = 0.03, θ4 = 0.07, p = 0.9 then P1 = 0.3157 and P2 = 0.252. In this case PMax = P1 = 0.316 and thus the model system goes to E ∗ if θ2 ∈ [P2 , P1 ] = [0.252, 0.316]. Figure 4 The picture exhibits that the interior steady state of (3.10) is a stable focus for the parametric values r = 0.5, R = 1.5, k1 = 100, k2 = 10, α = 0.04, γ = 0.07, δ1 = 0.06,β = 0.08,z 0 = [0.6, 0.3, 0.4]. Fig. 2 Depicts that the system will go to stable zone around the equilibrium point (E ∗ )

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Fig. 3 Shows that the system is stable near the infected free steady state (E4 )

Fig. 4 Displays a stable focus around the interior steady state (E ∗ )

7 Conclusion

In our proposed system, we have observed that the rate of predation Iλ θ2 gives an

important factor in the disease-selective predation dynamics. Again when Iλ θ2 outgo a certain threshold value (generated by other model values in parametric approach) then there is a chance for the abrogation of the prey individuals. The pessimistic and optimistic viewpoint of a decision making are stated by Il θ2 and

Ir θ2 , respectively. Again, there is a chance to vanish the prey species, and it increased the consuming time for gestation. Then, we have tried to stay the value of

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Iλ θ2 under some threshold value. We have observed in this article when the disease is deadly, i.e. the mortality rate of infected individuals (θ3 ) non decreases then the prey species is at risk. We have investigated for the selective predation, the system undergoes to stable zone of three species below some conditions on the parameters.

References 1. Anderson, Roy M., and R.M. May. 1978. Regulation and stability of hostparasite population interaction-I: regulatory process. J. Animal Ecol. 47: 219–247. 2. Chattopadhyay, J., and O.A. Arino. 1999. Predator–prey model with disease in the prey. Nonlinear Analysis 36: 747–766. 3. Freedman, H.I. 1990. A model of predator–prey dynamics as modified by the action of a parasite. Mathematical Biosciences 99: 143–155. 4. Hadeler, K.P., and H.I. Freedman. 1989. Predator–prey population with parasite infection. Journal of Mathematical Biology 27: 609–631. 5. Kiesecker, J., D.K. Skelly, K.H. Beard, and E. Preisser. 1999. Behavioral reproduction of infection risk. Proceedings of the National Academy of Sciences of the United States of America 96: 9165–9168. 6. Roy, S., and J. Chattopadhyay. 2005. Disease-selective predation may lead to prey extinction. Math. Method Appl. Sci. 28: 1257–1267. 7. Alam, S. 2009. Risk of disease selective predation in an infected prey-predator system. Journal of Biological Systems 17: 111–124. 8. Bassanezi, R.C., L.C. Barros, and A. Tonelli. 2000. Attractors and asymptotic stability for fuzzy dynamical systems. Fuzzy Sets and Systems 113: 473–483. 9. Barros, L.C., R.C. Bassanezi, and P.A. Tonelli. 2000. Fuzzy modelling in population dynamics. Ecological Modelling 128: 27–33. 10. Peixoto, M., L.C. Barros, and R.C. Bassanezi. 2008. Predator–prey fuzzy model. Ecological Modelling 214: 39–44. 11. Guo, M., X. Xu, and R. Li. 2003. Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets and Systems 138: 601–615. 12. Mizukoshi, M.T., L.C. Barros, and R.C. Bassanezi. 2009. Stability of fuzzy dynamic systems. Int. J. Uncertain. Fuzziness Knowl. Syst. 17: 69–84. 13. Tudu, S., N. Mondal, and S. Alam. 2019. Dynamics of prey–predator system in crisp and fuzzy environment with special imprecise growth rate, rate of conversion and mortality rate. In Recent Advances in Intelligent Information Systems and Applied Mathematics, 194–208. Cham: Springer. 14. Tudu, S., N. Mondal, and S. Alam. 2018. Dynamics of the logistic prey predator model in crisp and fuzzy environment. In Mathematical Analysis and Applications in Modelling, 511–523. Singapore: Springer. 15. Hale, K. 1977. Retarded functional differential equations. In Basic Theory of Functional Differential Equations, 36–56. New York: Springer.

An Analysis for Business Development by the Project Management in Moroccan Companies Kamelia Jahnouni and Hanaa Hachimi

Abstract Several interesting studies have been carried out on IT project management and have given rise to important indicators that allow analysis, conclusions and recommendations whose objective is to optimize the success rate of IT projects which generally encounter great difficulties and incidents in terms of time, budget or content. My ambition is to do a similar work but focused on IT projects in Morocco since this work has never been done in a detailed and precise way. The final objective of this study is to reflect the real state of project management as performed on Moroccan companies in order to come out with recommendations that will give companies the opportunity to optimize the control of their small and large projects by being inspired by the meaningful and interesting experience of the various projects carried out in Morocco. Each project is unique, and its progress cannot be generalized to other projects. This is why it is important for me to study the experience of several projects in several sectors and fields of activity and to provide a vision from several angles and by different profiles. In this communication, I will start by briefly explaining the way I chose to develop my study in order to collect real data from the field. Then make a focus on an analysis for business development by the project management in Moroccan companies. I will analyze the interest that Moroccan companies give to project management and see if they always achieve their objectives within a project methodology structure. And finally zoom in on a part of the results concerning the first parameter clearly causing problems on projects which is the level of information and competences of the projects teams. In other words, the importance that Moroccan companies attach to the organization of training in project management for the benefit of its teams in order to ensure the desired level of competence. Keywords Optimization · Project management · Failure · Success · Statistics · Method · Study · Training K. Jahnouni (B) · H. Hachimi Sultan Moulay Slimane University Systems Engineering Laboratory, Beni Mellal, Morocco e-mail: [email protected] H. Hachimi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_39

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1 Introduction My study’s subject is the Moroccan IT project management optimization. But before talking about the optimization process, I had to start by describing Moroccan IT projects management in a qualitative and quantitative way. This was a heavy task for me, since the data is not available and has never been treated in detail in a previous study. These data are found on the ground within each Moroccan company and are difficult to access. This is why I chose to opt for an online survey that respects the confidentiality and anonymity of each company. The objective of this survey is to build a real database to be analyzed in order to bring out any data that would allow describing and optimizing the management of IT projects in Morocco. The survey questions elaboration is the result of several interviews held with executives having a long experience in the management of mainly IT projects and working either in the private or in the public sector. It should be noted that I have been privileged to have built up over the years, during my studies and my professional experience, a large contacts directory working in the most important public and private Moroccan companies and institutions. This has given me access to very important data and an excellent database to analyze and interpret. The objective of my study is the following: • First, to build a database on project management in Morocco. A real, rich and exhaustive database, extracted from the Moroccan field. • Then to analyze this database in order to bring out interesting indicators and significant statistics. • Then to interpret the detailed analysis results. • And finally to propose recommendations and ways in order to optimize IT project management in Morocco. I was able to draw this objective after several years of professional experience during which I was able to see many cases of IT projects pushing me to a deep reflection on the subject. Why do so many IT projects fail, why so many losses and expenses? I am currently in the analysis phase of my database extracted from the Moroccan field. I am beginning my study with an exploratory analysis of the data. This descriptive analysis allows me to represent the observed data in the most accessible form (histogram, diagram) and to simplify the complex reality that is the questionnaire, into a more digestible presentation [1, 2].

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Fig. 1 Awareness of the project management value by Moroccan companies

Fig. 2 Awareness of the project management value by Moroccan IT companies

2 Are Moroccan Companies Aware of the Project Management Interest in Their Business? In my study, I asked Moroccan companies a question to find out if they are aware of the importance of adopting a methodology in managing their projects. Here is the result on Fig. 1. So we can observe that this represents a very optimistic entry into the world of projects since almost all the Moroccan companies questioned know what project management is and are aware of its importance in their business, in particular the IT companies Fig. 2. I will exclude in the rest of my analysis, the 5% on Fig. 1 because my objective is to understand why a company that confirms being aware of the project management interest will not succeed in all its projects. This leads me to go further in my analysis to understand the causes behind it.

3 Do Moroccan Companies Achieve Their Objectives Within a Project Methodology Structure? Another question was asked to these same companies to see if they always apply a project management method to achieve their objectives? Here is a graph summarizing all the answers received Fig. 3. It should be noted that all these companies are aware of the project management value in their business, but they do not actually apply it in their activity. Only 69%

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Fig. 3 Implementation of project management methodologies by Moroccan companies

of these companies are in the norm and the remaining 31% rarely or never apply a method. Projects are considered “a way to sustainability” [3]; this is why it is very important for all companies to give it a lot of importance and make sure to apply the right methodologies which will help managers through each stage of a project. It has several benefits including [4]: • • • • • • • •

More efficiency Improved customer satisfaction Responsible employees Effective risk management A more efficient team Improved reputation More flexibility Better communication.

A method is a guide. It is the fruit of the experience accumulated during the realization of many projects. It provides a framework for the development of the project by relying on the “good practices” thus acquired. There are a lot of methodologies but the manager must choose the right methodology because it is an essential step to a successful project [5, 6]. But several studies have shown that agile projects have higher success rates than traditional projects [6–9]. The figures for the private sector are better than those for the public sector. In the private sector, ¾ of companies apply a method on a regular basis Fig. 4. This includes more than 80% of large and medium-sized enterprises and almost half of small and micro enterprises Fig. 5. In the public sector, on the other hand, only half of the companies apply a method on a regular basis Fig. 6. This percentage remains almost the same when focusing on public companies of the same size (large, medium, small and micro) Fig. 7. Fig. 4 Implementation of project management methodologies by the Moroccan private sector

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Fig. 5 Implementation of project management methodologies by the Moroccan private sector/segmentation by company size

Fig. 6 Implementation of project management methodologies by the Moroccan public sector

Fig. 7 Implementation of project management methodologies by the Moroccan public sector/segmentation by company size

If we zoom in on IT companies whose business is IT projects, only 79% are in the norm and apply a method regularly Fig. 8. This figure is worrying because these companies in particular need to be 100% on the graph as this is the basis of their business. Managing IT projects without a method is a failure guarantee in all its notions! Here is a first cause of failure of Moroccan projects that emerges clearly from this analysis. It should be noted that integrating project management into the daily activity requires know-how on the part of the project participants; otherwise, it will harm the

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Fig. 8 Implementation of project management methodologies by the Moroccan IT companies

activity instead of optimizing it. This is why it is very important for a company to ensure that its employees have all the knowledge that will enable them to manage projects on the right method. This necessarily involves training; this is why it is necessary to plan it throughout the year according to needs and the planning of expected projects. Training does not have to be expensive for the company; it can call on external training experts to train all the teams if the budget allows, otherwise train just one competent person inhouse who will train all the other project managers. Or, at worst, organize workshops between project managers to share knowledge and experience or provide online training. There are several methods for advancing the skills of project managers. Each project has its own characteristics and objectives and requires specific study and special management. This requires the company to provide training for the different project actors beforehand [10]. It would be interesting or even necessary to systematize the fact that before starting any project, the project manager is obliged to organize training for his entire project team to ensure that everyone is on the same level of training and information. Because nowadays, we notice that many project stakeholders do not have a clear vision of the whole project nor of its adopted method; they just know the task they are going to accomplish without knowing its contribution to the project.

4 What Interest Do Moroccan Companies Have in Project Management Training? Do these Moroccan companies organize presentations and training in project management for their employees to raise awareness of the importance of this practice in achieving objectives? Here is a diagram summarizing all the responses received when we asked this question. Figures 9, 10, 11 and 12. In the following, I will look in detail at what these companies do in terms of organizing project management training and presentations for their staff. Companies that never apply a method in their projects almost never organize training for their employees as shown in Fig. 13. This is quite normal and understandable as they do not feel the need to do so, so why spend money on such a non-profitable action for the company.

An Analysis for Business Development by the Project Management … Fig. 9 Organization of project management training by the Moroccan companies

Fig. 10 Organization of project management training by the Moroccan private sector

Fig. 11 Organization of project management training by the Moroccan public sector

Fig. 12 Organization of project management training by the Moroccan IT companies

Fig. 13 Organization of project management training by Moroccan companies never applying project management methods

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Fig. 14 Organization of project management training by Moroccan companies that rarely apply project management methods

Fig. 15 Organization of project management training by Moroccan companies always using project management methods

Companies that rarely apply a method in their projects rarely or never organize training for their staff Fig. 14. Only 17% of these companies are in the norm; 55% rarely train as they rarely apply which we can hardly admit but the 28% who never train is still a worrying figure; entrusting projects to people who are not properly trained in project management remains a huge risk of failure. Here is a very clear reason for project failure that emerges from this indicator. On the other hand, for companies that always apply a method in their projects, 55% organize training regularly, 37% rarely and 8% never as shown in Fig. 15. This is shocking! These companies consider project management as a part of their business and therefore must ensure that their project managers have all the tools to do their job on the right method. More than half of these companies are IT departments whose business necessarily involves project management that is why they all apply project management methods on a regular basis. An imminent effort should be made by these companies because it is catastrophic to bear this apparent big risk without any anticipation or effort to dilute or even eliminate it. It is necessary to ensure maximum competence and know-how of the project teams which are likely to work under a high degree of stress and interpersonal demands that usually diminish their performance [10].

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5 Conclusion Through this analysis focused on the interest that Moroccan companies give to project management in a theoretical and applicative way, in particular in the planning of project management training for the benefit of employees who work daily in this field, we can conclude that: • Almost all companies are aware of the project management value. • But only 2/3 of them apply methods in their project management. • There is a huge lack of organized training, which undoubtedly has an impact on the quality of project management in these companies and consequently on business results. This is clearly one of the causes of IT project failure in Morocco. It must be said that recruiting a competent project manager is not enough; it is necessary to ensure that their skills are regularly adapted to the specific needs of the company’s projects. Having a competent and regularly trained project manager is not sufficient but necessary to minimize the risk of project failure. Moroccan companies working in the field of project management must give more importance to the training of their employees, because this will certainly optimize the success rate of their projects, the quality of deliverables, the performance of employees and the profitability of the business.

References 1. Observatoire des projets, https://blog-gestion-de-projet.com/observatoire-des-projets/ 2. Observatoire des projets stratégiques, https://dantotsupm.com/2012/01/12/observatoire-desprojets-la-reussite-des-projets-peut-elle-etre-programmee/ 3. José Magano, Gilbert Silvius, Cláudia Sousa eSilva, ÂngelaLeite. 2021. The contribution of project management to a more sustainable society: Exploring the perception of project managers. Project Leadership and Society, 2: 100020. 4. Bénéfices d’un management de projet efficace, https://www.planzone.fr/blog/benefices-man agement-projet-efficac 5. Méthode de Gestion de Projet, https://www.wimi-teamwork.com/fr/blog/methodologie-ges tion-projet/ 6. State of Agile Report, https://stateofagile.com/#ufh-c-7027494-state-of-agile 7. Project Management Institute Pulse of the Profession, https://www.pmi.org/learning/thoughtleadership/pulse 8. Ciro Troise, Vincenzo Corvello, Abby Ghobadian, Nicholas O’Regand. 2021. How can SMEs successfully navigate VUCA environment: The role of agility in the digital transformation era. Technological Forecasting and Social Change, 174: 121227. 9. Lester, Eur Ing Albert. 2017. Project management, planning and control managing engineering, construction and manufacturing projects to PMI. APM and BSI Standards. Seventh Edition: Butterworth-Heinemann.

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10. Hanaa Bel Lahmar, Houdaifa Ameziane. 2019. Le contrôle de gestion de projet au Maroc: Quelles spécificités. La Revue Marocaine de Contrôle de Gestion, ISSN: 2028-4098, No 8. 11. Pavez, Ignacio, Hugo Gómez, Lyonel Laulié, and Vicente A. González. 2021. Project team resilience. The effect of group potency and interpersonal trust. International Journal of Project Management 39 (6): 697–708.

Text Region Identification from Natural Scene Images Using Semi-Supervised MSER Method Shiplu Das, Sitikantha Chattopadhyay, Ritesh Prasad, Joydeep Kundu, and Souvik Pal

Abstract In today’s world, visual detection and recognition of text from an image are very demandable due to its application in content-based image retrieval, robotic navigation, and automatic number plate recognition, extracting information from passport or business cards or bank statements, etc. Text detection in natural images has been increasingly popular because text images represent much technical and digital information. Text can be traced in scattered form from any natural scene images, and it is available in different fonts, colors, and shapes. This paper has been classified into five main chapters. The first chapter presents an idea about the domain of text detection, different challenges, and the motivation behind the present work. The second chapter highlights the evolution of different text detection methods and provides a literature survey of the novel approaches utilized in the avenue of text detection. A brief introduction of the datasets has also been included in this paper. The third chapter provides details of the study of text detection using the maximally stable extremal region (MSER) method. The proposed methodology has been mathematically established for the localization of the texts. The fourth chapter highlights the efficiency of the method in localizing texts in natural images. The corresponding challenges related to our work and the advantages of using this method are also described in this paper. The last chapter concludes and discusses relevant future scope in scene text understanding in natural Images. Keywords MSER · Image processing · Text detection

S. Das · S. Chattopadhyay (B) · R. Prasad · J. Kundu Department of CSE, Brainware University, Barasat, West Bengal, India e-mail: [email protected] S. Pal Department of CSE, Global Institute of Management and Technology, Krishnagar, West Bengal, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_40

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1 Introduction Text detection [1] is a challenging task for an uncontrolled image, and it is often a preprocessing process before text classification. We need to classify the areas of text in natural images. There are various applications of text detection like text image search, landmark identification, etc. Text detection is challenging for some images which consists of noise and blur condition. So it is mostly challenging to map the text in natural images. Each of the languages has the stroke with the thickness of each or even letters remaining practically the same, and the height of the characters and height of the letters remain in a particular fashion to some extent. The problem with stroke with transform (SWT) is that it cannot be so useful for all the texts which are not flat to be detected so easily. Moreover, it also is not so good for natural images which are corrupted with noise. So MSER [2] is a method that detects the text region in a natural image. MSER uses texture properties of the image, and it is used to separate the text from the natural image. It is very sensitive to blur, and among all the different identification techniques, it increases robustness. Text detection in natural images is done by locating text in bounding boxes [3]. Text extraction is done by finalizing the scene images in such a manner that all text pixels are foregrounded, and the rest are background. Text region proposal methods give multiple possible text bounding boxes.

2 Mathematically Established for the Localization of the Texts Let us consider a natural scene image I(p), and p ∈  is a real function of a finite set that is  with a topology τ . Elements of  are called pixels that are smallest addressable element of the screen. For simplicity method, let us  = [1, 2, ..., N]n and p ∈  of the image I(p) is the set of pixels that have intensity which is not greater than I(p), so we can write the following form: S( p) = { y ∈  : I(r ) ≤ I( p)}

(1)

We denote by Rg (I) that is the set of all extremal regions of sample image I and extremal region (Rg) be the maximum natural image value attained in the region Rg. I (Rg) = sup I ( p)

(2)

x∈Rg

Now, an extremal region denotes by Rg of a one-dimensional image I(p). It is shown in Eq. 2. Two corresponding extremal regions are represented as Rg + and Rg − .

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Let  > 0. Let Rg + be the smallest extremal region that contains iR, and the stability of an extremal region Rg is the inverse of the relative area of the region Rg when the intensity level is increased . It has intensity which exceeds of at least  the intensity of Rg, i.e., Rg + = argtmin{|Q| : Q ∈ Rg(I ), Q ⊃ Rg, I (Q) ≥ I (Rg) + } Similarly, let Rg − be the most significant extremal region containing Rg which has intensity which is exceeded by at least  by Rg, i.e., Rg − = argtmax{|Q| : Q ∈ Rg(I ), Q ⊂ Rg, I (Q) ≤ I (Rg) − } Formula of area variation: ρ(Rg; ) =

     Rg +  −  Rg −  |Rg|

An extremal region Rg immediately contains another extremal region Q if Rg ⊃ Q and if Rg´ is another extremal region with Rg ⊃ Rg´ ⊃ Q, then Rg´ = Rg, because the base set  is finite.

3 Implementation of Semi-Supervised Technique Semi-supervised learning is the ML technique used to make unlabeled data for training. Typically, the main function of that process is a small amount of labeled data with a sufficient amount of unlabeled data conversion. Most machine learning researchers have found unlabeled data when used in conjunction with a small amount of labeled data. It can produce improvement in learning accuracy. The cost of this process is associated with the labeling process. For this situation, the semi-supervised technique has a great practical value. We have used L that denotes the collection of training images examples which are also labeled, and U refers to the collection of training images examples, but they are unlabeled. We add the examples, which are successfully classified that are used by the MKL classifier [4] and fall outside the decision boundary, i.e., |fc(x)| ≥ 1 and then adding all the examples in U. The observations are precisely the examples of natural images that would not make any changes to the MKL classifier [5] if they were included in the training data. The second alternative process has come in the picture by the observation that gives information from the MKL classifier. So we have used this technique at the time of training the final visual classifier which denotes the sign of the examples set selected from U. This learning technique falls between supervised and unsupervised learning [6]. It can improve the learning accuracy [7]. The cost of this process is associated with the labeling process. It may render a fully labeled training set infeasible. Support vector machine uses unlabeled data with a label from the sign of MKL score in the natural

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images. LSR is least squares regression that is of MKL scores using the visual kernel using KPCA projection [8]. It works well for text regions because the high consistent color and high contrast of text can stabilize the intensity of the images [9].

4 Data Set Collection and Annotation We have collected 2120 RGB images from a natural scene using a MI note 4 mobile camera and created 424 ground truth images randomly. The images are natural images such as the heading of any shop, hospital name, road sign, banner, car number plate, house name, and also other natural images, where the text portions have different orientations and different font styles. To make the data set usable for any researchrelated work, we have annotated those data. We have used some procedures for data set annotation. Step 1. We have manually drawn the region of interest, i.e., text region using MATLAB application, and saved the coordinates (upper left corner x, y position and height, and width) of that rectangle box. All the coordinates are stored in a matrix format. Step 2. We have written all the coordinate values into an XML file. The XML file also contains the respective image name and respective text name.

5 Detect Text Regions Using Region Detection Method In preprocessing stage, we have enhanced the contrast of our data set images using the histogram equalization method. In this paper, we have used the gradient-based MSER method to detect text regions over the image, then convert the image region into binary form, and find the connected component of those regions. We have filtered non-text regions using selected connected components [10,11] and drawn a bounding box on, respectively, connected components, where the bounding box contains text and non-text in both regions [12]. We have cropped those box regions. We have calculated a histogram of oriented features from each box region and classification using semi-supervised SVM. We have applied semi-supervised algorithm to classify text and non-text regions [13]. In Fig. 1, we have collected a sample image, and Fig. 2 represents the localized ground truth images.

6 Experimental Result and Analysis In this paper, we have used our developed data set, some examples of captured images, the images after contrast enhancement, and also text region detected image

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Fig. 1 Collected images

Fig. 2 Localized ground truth images

Fig. 3 Sample images taken from the developed data set image

using MSER. Some original images also display their detected region before filtering non-text region [14]. We have used 240 images for training purposes and 250 images for testing. We have got 81% accuracy in pixel level from the text region applying on 250 test images.

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Figure 3 deals with sample images taken from the developed data set. Figure 4 focuses on the sample images after contrast enhancement, and with the help of semisupervised MSER algorithm, we can detect the region of natural images, so Fig. 5 represents detected regions in the sample images after using semi-supervised MSER, and Fig. 6 represents some detected regions in the sample images after using MSER method. Original images describe some examples of the original image, contrastenhanced image, and also MSER region detected image, and Fig. 7 represents the accuracy of semi-supervised MSER mode. With the help of this method, accuracy of 81% is calculated for semi-supervised support vector machine. Detected region

Fig. 4 Sample images after contrast enhancement

Fig. 5 Detected regions in the sample images after using semi-supervised MSER

Fig. 6 Detected regions in the sample images after using MSER method

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Fig. 7 Accuracy of semi-supervised MSER model

images [15] describe some original images and also display their detected region before filtering non-text regions [16].

7 Conclusion Text detection from scene images is a complex computer vision task that is being studied by many research laboratories and international companies for its importance. It is used in newly developed technologies, such as automated driving and automated locating of information from visual data. Unfortunately, till now no method proposed in literature achieves semi-supervised text detection rates that are even remotely comparable to human observers’ performances to improve our accuracy and also recognize the text regions. In MSER detection technique, we require a range of threshold values. Since the threshold is highly unstable, the watersheds are not frequently changed. Importantly, this process is very fast in practice [17]. Finally, we remark that the maximally stable extremal region can be defined on any natural image with pixel values from an ordered set approach [18]. In future work, we are trying to proceed to a fully automatic projective reconstruction of 3D scene images. Secondly, we have to find properties of robust similarity measurement areas and their selection which is based on statistical properties of the text.

References 1. Zhong, Zhuoyao, Lianwen Jin, and Shuangping Huang. (2017). Deeptext: A new approach for text proposal generation and text detection in natural images. In 2017 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE. 2. Zhang, Xiangnan, Xinbo Gao, and Chunna Tian. 2018. Text detection in natural scene images based on color prior guided MSER. Neurocomputing 307: 61–71. 3. Xie, Enze, et al. (2019). Scene text detection with supervised pyramid context network. In Proceedings of the AAAI conference on artificial intelligence, vol. 33, No. 01.

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4. Deng, Cheng, et al. 2018. Active multi-kernel domain adaptation for hyperspectral image classification. Pattern Recognition 77: 306–315. 5. Gu, Yanfeng, Jocelyn Chanussot, Xiuping Jia, and Jon Atli Benediktsson. (2017). Multiple kernel learning for hyperspectral image classification: A review. IEEE Transactions on Geoscience and Remote Sensing 55(11): 6547–6565. 6. Sarmadi, Hassan, and Alireza Entezami. 2021. Application of supervised learning to validation of damage detection. Archive of Applied Mechanics 91 (1): 393–410. 7. Harris, A.H., A.C. Kuo, T.R. Bowe, L. Manfredi, N.F. Lalani, and N.J. Giori. 2021. Can machine learning methods produce accurate and easy-to-use preoperative prediction models of one-year improvements in pain and functioning after knee arthroplasty? The Journal of Arthroplasty 36 (1): 112–117. 8. Wang, Jiale, Guohui Li, Peng Pan, and Xiaosong Zhao. 2017. Semi-supervised semantic factorization hashing for fast cross-modal retrieval. Multimedia Tools and Applications 76 (19): 20197–20215. 9. Ge, F., L. Fei, J. Zhang, and C. Wang. 2020. The Electrical-triggered high contrast and reversible color-changing Janus fabric based on double side coating. ACS Applied Materials & Interfaces 12 (19): 21854–21862. 10. Wu, H., B. Zou, Y.Q. Zhao, Z. Chen, C. Zhu, and J. Guo. 2016. Natural scene text detection by multi-scale adaptive color clustering and non-text filtering. Neurocomputing 214: 1011–1025. 11. Chakraborty, Neelotpal, Agneet Chatterjee, Pawan Kumar Singh, Ayatullah Faruk Mollah, and Ram Sarkar. 2021. Application of daisy descriptor for language identification in the wild. Multimedia Tools and Applications 80 (1): 323–344. 12. Bhowmik, Showmik, et al. 2018. Text and non-text separation in offline document images: A survey. International Journal on Document Analysis and Recognition (IJDAR) 21 (1): 1–20. 13. Mosquera, H. P., & Genç, Y. (2019, September). Recognition and classifying sales flyers using semi-supervised learning. In 2019 4th International Conference on Computer Science and Engineering (UBMK), 1–6. IEEE. 14. Naiemi, F., V. Ghods, and H. Khalesi. 2020. Scene text detection using enhanced extremal region and convolutional neural network. Multimedia Tools and Applications 79 (37): 27137–27159. 15. Unar, S., X. Wang, C. Zhang, and C. Wang. 2019. Detected text-based image retrieval approach for textual images. IET Image Processing 13 (3): 515–521. 16. He, Tong, Weilin Huang, Yu Qiao, and Jian Yao. (2016). Text-attentional convolutional neural network for scene text detection. IEEE Transactions on Image Processing 25(6): 2529–2541. 17. Arteta, Carlos, et al. (2016). Detecting overlapping instances in microscopy images using extremal region trees. Medical Image Analysis 27: 3–16. 18. Le, Viet Phuong, et al. (2015). Text and non-text segmentation based on connected component features. In 2015 13th international conference on document analysis and recognition (ICDAR). IEEE.

Visualization of Audio Files Using Librosa Shubham Suman, Kshira Sagar Sahoo, Chandramouli Das, N. Z. Jhanjhi, and Ambik Mitra

Abstract The process of pursuing music as a professional field is thought of as a tedious job in remote regions. In spite of various online materials are available which can be easily accessed, still, the approach of gathering knowledge and proper evaluation is dependent on traditional teaching. So, there is an imminent need for a smart system model to design and access the audio data analysis. Librosa is focused on facilitating a feasible solution to this test case where a user can play audio notes on a musical interface which is thereby evaluated by its performance with selected files. Librosa takes into consideration several variables and factors like loudness factor, tempo, and frequency count with a music instrument for validation purposes. The validation process was undertaken by a series of phases like feature normalization, text, and audio pattern match to retrieve an efficient spectrograph of the played audio. Limitations like detection of musical sequence as well as noise onsets were ignored with the use of a pattern matching model. The performance element is based on the musical notes which the user missed out on, and the extraneous audio notes played by the user. Thus, this methodology is based on cutting-edge technology in the domain of audio learning applying the latest technologies. Keywords Audio · Signal processing · Information retrieval · Spectrograph · Visualization

S. Suman (B) · A. Mitra Kalinga Institute of Industrial Technology, Deemed To Be University, Bhubaneswar, Odisha, India e-mail: [email protected] K. S. Sahoo Department of CSE, SRM University, Amaravati, A.P. 522502, India C. Das L&D Associate, HighRadius Corporation, Bhubaneswar, Odisha, India N. Z. Jhanjhi School of Computer Science and Engineering, Taylor’s University, 47500 Subang Jaya, Malaysia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_41

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1 Introduction Machine learning is concerned with computer vision, language processing, and analysis of audio signals [1, 2]. This audio signal processing deals with digital signal processing as well as visualization of audio signals. This field is a growing subdomain in the machine learning domain. Few widely used computational models based on machine learning and virtual assistants such as Alexa, Siri, and Google Home are the massive framework that is used in the extraction of data from audio signals. The submodule of Librosa consists of a wide domain of generally applied functionalities. In general, it is divided into four labels which include audio and time seriesrelated functionality, spectrogram computation, time and frequency transformation, and pitch-based operations. For convenience, all functions within the core submodule are aliased at the top level of the package hierarchy. Audio and time series involve packages like reading audio from disk through audio read package, re-sampling a signal at a suitable frequency, stereo to mono transformation, time-domain-based auto-association with zero-crossing identification [4, 5, 15]. Spectral representations which are the accumulation of power within a series of frequency domains are the root of several analytics methodologies in digital signal processing [16]. The Librosa characteristics package unit applies several spectral representations. The display unit in Librosa gives very realistic and easy-to-use interfaces to visually integrate audio-related signal information. The module display wave-plot integrates the amplitude envelope of an audio signal using Matplotlib unit which is under Librosa. Mono signals are rendered symmetrically in the horizontal axis; stereo signals combine with the left channel. Audio data visualization is about presenting an analysis and interpreting audio signals taken by digital modules with various implementations in the enterprise, health care, productivity, and smart cities. Applications involve customer satisfaction analysis from customer service calls, media data analysis and retrieval, medical aids and caring of the patient, relative technologies for the public with hearing defacement, and audio details for public safety. Now, let us discuss Python language. Python has many advantages such as it is easily adaptable for every operating system, it is innate and friendly to user language, it allows multiple programming patterns such as object-based, vital, practical, and organized, and it involves several science-oriented libraries which are strongly expanding. Some disadvantages of Python language are, in the audio signaled zone, it lacks constant width integer types, but the use of Numpy makes it possible to provide them in code, changeable fixed arguments, unchangeable to present build-in data types and structures, fixed numbered recording types, use in global interpreted lock mechanisms (GLM). This paper depicts the usage of Python library files in audio processing, its proposed model, and its result analysis. In this study, we have discussed the following things:

Visualization of Audio Files Using Librosa

a.

b.

c.

411

Parameters have taken through which visualization of an audio file, as well as observation of changes in multiple audio files, can be done. Then compared with each parameter of the file and analyze the difference. We have used Python library named Librosa version 0.6.3. It can be defined as a package that is structured as a collection of submodules that further contain other functions. Basically, conversion of each track into a spectrograph and use the spectrograph for comparing the audio files. Each audio track may have a different sampling rate, amplitude, etc. [14]. An audio signal is represented as a one-dimensional Numpy submodule that are aliased at the top level of the package hierarchy, array, denoted as y throughout Librosa.

The remaining work is arranged as follows. Section 2 discusses the literature review, Sect. 3 deliberates the basic principles, Sect. 4 application followed by the proposed model. Then Sect. 5 discusses result analysis and Sect. 6 conclusion.

2 Literature Review Several existing works related to audio visualization are being carried out by different researchers. Authors in [1] suggested that at present education on audio-visual works is constantly developing. This continuous development is dependent on the way in which the institution of audio-visual events is made available. Several existing works related to audio visualization are being carried out by different researchers. In this work, the authors also suggested that at present, education on audio-visual works is constantly developing. This continuous development is dependent on the way in which the institution of audio-visual events is made available. Authors in [2] presented an SVM-based technique to categorize various heterogeneous audio signals. An empirical study of audio signals and data was performed by Chu et al. which was based on characterization, and they used a matching pursuit technique to derive efficient time-frequency variables from audio streams [3]. Zhang and Kuo proposed an automated segmentation method for the classification of audiovisual signals on basis of distinguishing features of audio signals [4]. Elaiyaraja and Meenakshi [5] developed an audio detection and categorization model with the use of audio-based attributes as well as a frame-oriented SVM classifier. A systematic review analysis of seven different techniques for visualization of audio-visual data like spectrograms is discussed by Isaacson [6]. Mitroo et al. in [7] presented an automated model using the computerized technique for the synthesis of visualizations of audio music signals in the real-time scenario. Real-time visuals with the help of design metaphors related to the mental visualization of musician’s audio are proposed for training musicians through visual response [8]. In [9], a powerful technology to read and process musical chords is presented using an automated computer-aided tool called computational musicology. It is based on the retrieval of musical data. In

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[10], the authors discussed the role of information retrieval using advanced computational strategies for musical data analysis and visualization. In another work [11], the authors depicted the significance of data visualization in musical information and highlighted it in the context of music annotations. In [12], the authors discussed an attribute checking technique for valuable classification of disorders in diabetes. It came out to be a well-optimized prototype of predictive learning analytics. The authors discussed a method to sample data which is helpful to optimize skewed records in medical samples [13]. This sampling approach proved to be an efficient model for dimensionality reduction.

3 Basic Principles and Applications While hearing various voices, people compare the listening experience, as well as other experiences. Steady beats easily allow people to think of silence and tranquility. With corporal waves of music files, various systems of music control segregate a piece of music into several intervals. Due to this, audio is converted into control signals which are resulted in particular control units after simultaneous processing. In these products, visual effects are changed with music beats, which are often produced or preset in controlling systems. Audio visualization is used in the education system, communication system, medical help, and even military places, theaters, granting ceremonies, TV programs, and some traditional occasions like operas and shows for the symphony. Audio visualization is used in education system, communication system, medical help and even military places, also with various applications in stage performance and theaters, granting ceremonies, TV programs and some traditional occasions like operas and shows for symphony. In stage performance, audio visualization mostly serves to work with style and theme, gathered organizing, other design stuffs like shows and lights, and performers.

4 Proposed Model The sounds are available in many formats which makes it possible for the computer to read and analyze them. Here, .wav format is used. The proposed work shows the process in which the audio gets visualized. At first, the audio files should be downloaded in the form of a .wav file and then we retrieve the audio tracks or files. Then, the files are extracted by giving the same path in the Python code. The compiler extracts the audio tracks and converts them into spectrograms. The spectrograms are analyzed and compared one by one. The spectrograms are formed based on the aspects like tempo, sampling rate, pitch, amplitude, etc. (Fig. 1). In the following, we discuss the processes in audio visualization.

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Fig. 1 Proposed system model for audio visualization

Loading Audio files The audio files are downloaded from a particular Web site. The audio files might be any song or any specific tune. The audio files must be saved with the .wav extension. This format of audio files is preferred as it is an easy process of the file. If the audio file is present in mp4 format, it must be first converted to wav format. Retrieving audio tracks After downloading the audio tracks, these audio tracks are gathered and stored in a particular place in the system. The path where the audio tracks or files are stored should not be too long so as to make the task quick and less time taking. Librosa Librosa is a package of Python used for the analysis of music and audio. It provides the necessities required to create music information systems that are retrieved. Conversion of audio tracks to spectrographs The Python library used to convert audio tracks to spectrographs is Matplotlib. It is a library of Python used for plotting graphs. It also provides an API that is objectoriented for the need of using a generally purposed graphical user interface (GUI) toolkits, e.g., Tkinter, wxPython, Qt, or GTK+ to lodge plots into applications. Comparing the spectrographs The spectrographs of various audio files which are created by using the Python library Matplotlib are now compared based on the following parameters which are pitch,

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Table 1 Attributes for plotting the spectrographs

Attribute

Type

Comments

sr

int

target sampling rate

res_type

string

Re-sample type

Offset

float

start reading after this time (s)

Mono

bool

convert signal to mono

Duration (time)

int

only load up to this much audio (s)

Dtype

float

Data type of Y value

audio tick, velocity, and audio channel. The spectrographs are processed into bar graphs and line graphs so as to make the reading more precise. Few terminologies related to spectrograph. A.

B.

C.

D.

E.

Sampling Sound is a continuous wave. We can digitize sound by breaking the continuous wave into discrete signals. This process is called sampling [15]. Sampling converts a sound wave into a pattern of samples or a distinct-time signal. The load functions load the audio file and convert it into an array of values which represent the amplitude if a sample at a given point of time. Sampling Rate The sampling rate is the number of audio signal per second. Hz or Hertz is the unit of the sampling rate. 20 kHz is the audible range for human beings. .Audio Tick It is an audible art usually caused by editing that is separated in between samples or peaks of an audio piece and cause the speakers to change voltage suddenly. Audio Channel An audio channel or audio piece is an audio signal communicating channel which is in a stored device or mixing solace, used in operations such as multi-tracking of the records and sound reinforcing. Pitch The changing of a frequency is generally referred to as the pitch of a sound. A high-pitched voice is analogous to a high frequency sound wave, and a less pitched voice is analogous to a low frequency sound wave. Table 1 shows the attributes used with the Matplotlib to plot the spectrographs of the audio files.

5 Result and Analysis The sampling rate (SR) is calculated as the number of times a signal is read in a second. Here, the sampling rate used was 22,050 Hz. The re-sampling type is the mode (low quality or high-quality mode) of the graph. The Kaiser_best (mode) was used which is the high-quality mode in which the spectrograph is represented. The

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offset is the time at which the reading of the audio track starts. In the graph, the reading is starting from zero seconds. The duration is for a period of time the reading is taken. In all the graphs, the x-axis represents the time, and the y-axis represents the variations in the parameters which are used to build a spectrograph. Figure 2a shows the graph representation of audio File 1 showing variations in the pitch, velocity, audio channel, audio tick, and index. Figure 2b shows a line graph obtained on the basis of audio File 1. It makes the readings more accurate, and we get the pitch correct at any instant of time. Similarly, Figs. 3 and 4 illustrate the bar graph representation of audio File 2, File 3, and File 4 showing the variations in the pitch, velocity, audio channel, audio tick, and index. In this figure, the variations are observed to be more regular and with a particular interval. The audio tick is observed to be decreasing at every time interval. Table 2 essentially shows the parameters used to build spectrographs. These parameters are used to make the spectrograph better, sharp, and proper (Table 3). Table 4 shows the values of parameters obtained from the bar graph of audio File 2. The pitch is varying at every 5 s instant of time which we can clearly observe from Fig. 3. Here, we can observe that 7 s in the first graph, in the second graph, we notice that it takes 9 s, in the third graph it takes 11 s, and in the fourth graph, it also takes 11 s. DType is the data type for y-axis(amplitude) which was taken ‘float’ to make the spectrograms. After achieving the bar graphs by using the above-listed attributes in the Python code using the Librosa library, it was observed that there was a slight change in the values of the four bar graphs. The variations were more in the second

Fig. 2 a Bar graph for audio file 1. b. Line graph for the audio file 1

Table 2 Parameters used to build a spectrograph

Attribute

Value

Sampling rate

22,050 Hz

Re-sample type

Kaiser_best

Offset

0.0 s

Mono

True

Time duration

2:40 min in first graph and 50 s in second graph

dtype

Float

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Table 3 Result of audio file 1 Index

Audio channel

Velocity (m/s)

Pitch (m)

0

Audio tick 20

1

100

61

1

40

1

100

68

2

60

1

100

69

3

80

1

100

71

4

100

1

100

56

5

120

1

100

72

6

140

1

100

73

7

160

1

100

78

Audio channel

Velocity (m/s)

Pitch (m)

Table 4 Result of audio file 2 Index

Audio tick

0

5

1

100

61

1

10

1

100

68

2

15

1

100

69

3

20

1

100

71

4

25

1

100

56

5

30

1

100

72

6

35

1

100

73

7

40

1

100

78

8

45

1

100

71

9

50

1

100

67

Fig. 3 a Bar graph for audio file 2. b. Line graph for the audio file 2

graph compared to all the four graphs, the second graph depicted slight change in the values in between, whereas in the first graph, the waves appeared to be simple, regular, and values showed changes at constant intervals throughout the graph. In the second graph, the pitch increased before the pitch increased in the first graph. In

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Fig. 4 a Bar graph for audio file 3. b. Line graph for audio file 3

this experiment, the spectrographs were basically converted to bar graphs to make the reading more precise, understandable, and relatable.

6 Conclusion This paper illustrates a summary of the design considerations and functionality of Librosa. The audio-visuals displayed are clear and well distinguished [16]. Four spectrographs of four audio tracks were visualized using the Matplotlib and Librosa library. The spectral lines were sampled and created properly without any error. The spectrographs were converted into bar graphs and line graphs to find the accurate reading. The performance element is based on the musical notes which the user missed out on and the extraneous audio notes played by the user. Thus, this methodology is based on cutting-edge technology in the domain of audio learning applying the latest technologies.

References 1. Walter Arno Wittich, and Charles Francis Schuller. (1957). Audio-visual materials, their nature and use, 510. New Yorks Harper & Brothers. 2. Dhanalakshmi, and S. Palanivel. 2008. Classification of audio signals using SVM and RBFNN. Journal of Expert Systems with Applications, 36: 6069–6075. 3. S. Chu, S. Narayanan, and C.-C.J. Kuo. (2009). Environmental sound recognition with timefrequency audio features. IEEE Transactions on Audio, Speech, and Language Processing, 17(6): 1142–1158. 4. T. Zhang, and C.-C. Kuo. 2011. Audio content analysis for online audiovisual data segmentation. IEEE Transactions on Speech and Audio Processing, 9(4): 441–457. 5. V. Elaiyaraja, and P. M. Sundaram. 2012. Audio classification using support vector machines and independent component analysis. Journal of Computer Applications (JCA), 5(1). 6. E. Isaacson. 2005. What you see is what you get: On visualizing music. In ISMIR 2005.

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7. B. Mitroo, N. Herman, and N. I. Badler. 1979. Movies from music: Visualizing musical compositions. In SIGGRAPH, 218–225. 8. S. Ferguson, A.V. Moere, and D. Cabrera. 2005. Seeing sound: Real-time sound visualisation in visual feedback loops used for training musicians. In Proceedings iV ’05, 97–102. 9. Mishra, S., Dash, A., and Jena, L. 2021. Use of deep learning for disease detection and diagnosis. In Bio-inspired Neurocomputing, 181–201. Springer, Singapore. 10. Amit Singhal. 2001. Modern information retrieval: A brief overview. IEEE Data Engineering Bulletin, 24(4): 35–43. 11. Mark Cartwright, Ayanna Seals, Justin Salamon, Alex Williams, Stefanie Mikloska, Duncan Mac Connell, E. Law, J. Bello, and O. Nov. 2017. Seeing sound: Investigating the effects of visualizations and complexity on crowdsourced audio annotations. Proceedings of the ACM on Human-Computer Interaction, 1(1). 12. Ray, C., Tripathy, H.K., and Mishra, S. 2020. Assessment of autistic disorder using machine learning approach. Journal: Intelligent Computing and Communication Advances in Intelligent Systems and Computing, 209–219. 13. Sahoo, Kshira Sagar, and Deepak Puthal. (2020). SDN-assisted DDoS defense framework for the internet of multimedia things. ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM) 16(3): 1–18. 14. Nithya, S., et al. 2020. SDCF: A software-defined cyber foraging framework for cloudlet environment. IEEE Transactions on Network and Service Management 17 (4): 2423–2435. 15. Madhu, G., et al. 2021. Imperative dynamic routing between capsules network for malaria classification. CMC-Computers Materials & Continua 68 (1): 903–919. 16. Mishra, S., Mallick P.K., Jena L., Chae G.S. 2020. Optimization of skewed data using samplingbased preprocessing approach. Front Public Health 8:274. Published 2020 July 16. https://doi. org/10.3389/fpubh.2020.00274

Proper Lucky Labeling of Jelly Fish Graph, Cocktail Party Graph and Crown Graph T. V. Sateesh Kumar and S. Meenakshi

Abstract The proper labeling is different natural number for adjacent nodes. Lucky labeling is that the total of labels over adjacent nodes within the graph is not same, and if a node is an isolated node, the sum is zero. The proper lucky number is the minimum positive number that labeled the graph. The η p (G) denotes proper lucky number. In this article, we proved that jelly fish graph J (m, n), cocktail party graph C Pk and crown graph Cn∗ are proper lucky graph. Also, we calculated the proper lucky number of the above-mentioned graph. The relation of the proper lucky number and the degrees of the same is derived. Keywords Proper lucky labeling · Jelly fish graph · Cocktail party graph · Crown graph

1 Introduction It was initiated by Rosa [11]. There are more discoveries and innovative research in past decades. Here, we tried to build those discoveries and innovation to the next level, so we tried with the existing discoveries and came up with fascinating results. Gallian [5] provided vibrant details of graph labeling. Karonsik [8] was start off the proper labeling which says about different natural number for adjacent nodes. Lucky labeling was start off by Ahadi [1]; it means that the total of labels over adjacent nodes within the graph is not quite the same, and if a node is an isolated node, the sum is zero. Labeling lucky is related with proper by Akbari [2]. The proper lucky number is the minimum positive number that labeled the graph [10]. The η p (G) denotes proper lucky number [9]. Sriram computed 1—near mean cordial labeling of jelly fish J(m, n) graphs in 2015 [12]. Gregory computed clique partitions of the cocktail party graph [6]. Rokad derived cordial labeling of jelly fish graphs in 2017 [3]. Daoud computed complexity of crown and cocktail party graph [4]. Thirusangu T. V. S. Kumar · S. Meenakshi (B) Department of Maths, VISTAS, Chennai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_42

419

420 Fig. 1 Jelly fish graph J (m, n)

T. V. S. Kumar and S. Meenakshi x

u1 u2 u3

v1 v2

u

v3

v

um

vn

y

computed Zumkeller labeling of jelly fish graph in 2019 [13]. Aleksandar Jurisic computed 1—homogeneous graphs with cocktail party [7].

2 Preliminaries 2.1 Jelly Fish Graph The J (m, n) has m + n + 4 vertices, i.e., V (J (m, n)) = {u, v, x, y} ∪ {u 1 , u 2 , . . . , u m } ∪ {v1 , v2 , . . . , vn } and m + n + 5 edges, i.e., E(J (m, n)) = {(u, x), (v, x), (x, y), (u, y), (v, y)} ∪ {(u, u i ); 1 ≤ i ≤ m} ∪ {(v, vi ); 1 ≤ i ≤ n} refer Fig. 1 [12].

2.2 Cocktail Party Graph The C Pk , where k is the 2n vertices, i.e., V (C Pk ) = {u 1 , u 2 , . . . , u n } {v1 , v2 , . . . ,vn } ∪ and n + n2 edges,  = ∪ E(C Pk ) (u i , u j ); 1 ≤ i ≤ n, 1 ≤ j ≤ n, i = j i.e., (vi , v j ); 1 ≤ i ≤ n, 1 ≤ j ≤ n, i = j ∪ {(u i , vi ); i = j} refer Fig. 2 [4]. Fig. 2 Cocktail party graph C Pk

u1

u2

u3 un

v1 v2

v3

vn

Proper Lucky Labeling of Jelly Fish Graph… Fig. 3 Crown graph Cn∗

421

u1

u2

u3

u4

un

v1

v2

v3

v4

vn

2.3 Crown Graph   A crown graph Cn∗ is obtained from 2n vertices, i.e., V Cn∗ ∗ {u 1 , u 2 , . . . , u n } ∪ {v1 , v2 , . . . , vn } and n(n − 1) edges, i.e., ECn (u i , v j ); 1 ≤ i ≤ n, 1 ≤ j ≤ n, i = j refer Fig. 3 [4].

= =

3 Main Results 3.1 Theorem A J (m, n) is proper lucky with η p (J (m, n)) = 3. Proof. Considering f (J (m, n)) → {1, 2, 3} be defined by. Case (i): m = n = 1. f (x) = 1. f (y) = 2. f (u) = f (v) = 3. f (u_m) = f (v_n) = 1. Now, adjacent labels are not same. Here, we obtain. s(x) = 8. s(y) = 7. s(u) = s(v) = 4. s(u_m) = s(v_n) = 3. Here, total of adjacent vertices are not same. So J (m, n) is proper lucky with η p (J (m, n)) = 3 when n = m = 1. Case (ii): n > 1, m > 1. f (x) = 1. f (y) = 3. f (u) = f (v) = 2. f (u_m) = f (v_n) = 3. Now, adjacent labels are not same. Here, we obtain. s(x) = 7.

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1(7)

3(2)

3(2)

3(2) 3(2) 3(2)

3(2) 2(22)

2(22)

3(2) 3(2)

3(2) 3(2) 3(2)

3(5)

3(2)

Fig. 4 Proper lucky jelly fish graph J (6, 6)

s(y) = 5. s(u) = 4 + 3m. s(v) = 4 + 3n. s(u_m) = s(v_n) = 2. Here, total of adjacent vertices are not same. So J (m, n) is proper lucky with η p (J (m, n)) = 3 when n > 1, m > 1. Hence, from case (i) and (ii), it is obvious the J (m, n) is proper lucky with proper lucky number η p (J (m, n)) = 3.

3.2 Illustration The proper lucky jelly fish graph J(6, 6) with proper lucky number 3 is shown in Fig. 4, i.e., η p (J (6, 6)) = 3.

3.3 Corollary The proper lucky labeling of jelly fish graph is equal to two more than the minimum degree, i.e., η p (J (m, n)) = δ(J (m, n)) + 2.

3.4 Corollary The proper lucky labeling of jelly fish graph is n subtracted from maximum degree and add with 1, i.e., η p (J (m, n)) = (J (m, n)) − n + 1.

Proper Lucky Labeling of Jelly Fish Graph…

423

2(16)

Fig. 5 Proper lucky cocktail party graph C P8

3(14)

1(18)

4(12)

1(18)

4(12) 2(16)

3(14)

3.5 Theorem The cocktail party graph C Pk where k is the 2n vertices is proper lucky graph with proper lucky number η p (C Pk ) = n. Proof. Let f (C Pk ) → {1, 2, 3, . . . , n} be defined by f (u i ) = f (vi ) = i Where 1 ≤ i ≤ n Now, adjacent labels are not same. Here, we obtain ⎧ ⎨

n 2 + (n − 2)i = 1 s(u i ) = s(vi ) = n + (n − 2) − 2(i − 1) even i ⎩ 2 n + (n − 2) − (2i − 2) odd i > 1 2

Here, total of adjacent vertices are not same. So cocktail party graph C Pk is a proper lucky graph with η p (C Pk ) = n.

3.6 Illustration The proper lucky cocktail party graph C P8 with lucky number n is shown in Fig. 5, i.e., η p (CP8 ) = 4.

3.7 Corollary The proper lucky labeling of cocktail party is 2 greater than minimum degree minus n, i.e., η(C Pk ) = 2 + δ(C Pk ) − n.

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Fig. 6 Proper lucky crown graph C5∗

1(14) 2(13) 3(12)

4(11)

5(10)

1(14) 2(13)

4(11)

5(10)

3(12)

3.8 Corollary The proper lucky labeling of cocktail party is 2, greater than maximum degree minus n, i.e., η(C Pk ) = 2 + (C Pk ) − n.

3.9 Theorem   The crown graph Cn∗ is proper lucky graph with proper lucky number η p Cn∗ = n. Proof. Let f (C Pk ) → {1, 2, 3, . . . , n} be defined by f (u i ) = f (vi ) = i Where

1≤i ≤n

Now, adjacent labels are not same. Here, we obtain ⎧ 2 n +n ⎪ ⎨ 2 − 1i = 1 2 n +n s(u i ) = s(vi ) = − 2i = 2 2 ⎪ ⎩ n 2 +n − ii > 2 2 ∗ Here, total of adjacent vertices are  not same. So crown graph Cn is a proper lucky graph with proper lucky number η p Cn∗ = n.

3.10 Illustration   The lucky crown graph C∗5 with lucky number 5 is shown in Fig. 6, i.e., η C5∗ = 5.

Proper Lucky Labeling of Jelly Fish Graph…

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3.11 Corollary     η p Cn∗ = δ Cn∗ + 1

3.12 Corollary     η p Cn∗ =  Cn∗ + 1

4 Conclusion Here, we compute the proper lucky number of the jelly fish J (m, n), cocktail party C Pk and crown graph Cn∗ . Also, we linked it with highest and lowest degree of graph. It determines that proper lucky number can be calculated for J (m, n), C Pk and Cn∗ ; also, it can be called as proper lucky graph. We extend this analysis to a variety of regular graphs and networks.

References 1. Ahadi, A., A. Dehghan, M. Kazemi, and E. Mollaahmadi. 2012. Computation of lucky number of planar graphs is NP-hard. Information Processing Letters 112: 109–112. 2. Akbari, S., M. Ghanbari, R. Manaviyat, and S. Zare. 2013. On the lucky choice number of graphs and combinatorics 29: 157–163. 3. Amit, Rokad, and Kalpesh Patadiya. 2017. Cordial labeling of some graphs. Aryabhatta Journal of Mathematics and Informatics 9: 2394–9309. 4. Daoud, S.N. 2012. Complexity of cocktail party graph and crown graph. American Journal of Applied Sciences 9: 202. 5. Gallian Joseph. 2019. A dynamic survey of graph labeling Electron Journal Combinatorics DS 6. 6. Gregory, David A., Sean McGuinness, and W. Wallis. 1986. Clique partitions of the cocktail party graph. Discrete Mathematics 59: 267–273. 7. Juriši´c, Aleksandar, and Jack Koolen. 2003. 1-homogeneous graphs with cocktail party µgraphs. Journal of Algebraic Combinatorics 18: 79–98. 8. Karo´nski, M., T. Łuczak, and A. Thomason. 2004. Edge weights and vertex colours. Journal of Combinatorial Theory B 91: 151–157. 9. Kumar, TV Sateesh., and S. Meenakshi. 2021. Lucky and proper lucky labeling of quadrilateral snake graphs. IOP Conference Series: Materials Science and Engineering 1085 (1): 012039. https://doi.org/10.1088/1757-899X/1085/1/012039. 10. Sateesh Kumar, T. V., and S. Meenakshi. 2020. Proper lucky labeling of graph. In Proceedings of First International Conference on Mathematical Modeling and Computational Science: ICMMCS, Springer. 11. Rosa A. 1966. On certain valuations of the vertices of a graph. In Theory of Graphs (Rome: International Symposium). 349–355.

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12. Sriram, S., and R. Govindarajan. 2015. 1-near mean cordial labeling of D2 (Pn ), Pn (+) N m (when n is even), jelly fish J (m, n) graphs. International Journal of Mathematics Trends and Technology (IJMTT) 25: 55–58. 13. Sriram, S., R. Govindarajan, and K. Thirusangu. 2019. Zumkeller labeling of jewel graph and jelly fish graph. Journal of Information and Computational Science 9: 725–731.

Topological Indices on Central Graph J. Senbagamalar, A. Meenakshi, A. Kanchana, and Hanaa Hachimi

Abstract The collection of masses of all the atoms in the molecule was referred to as molecular mass. The grouping of all molecules are known as compound. Consider a hydrogen suppressed atoms and double and triple bonds counted as a single bonds. The number of bonds incident on atoms counted as the valency of a vertex. Most of the chemical descriptors can depend on valences. Physical attributes, chemical reactivity, and biological activity can all be predicted by topological indices based on chemical molecular structure. The central graph C(G) is generated by partitioning every bonds exactly once and connecting all the non-adjacent atoms in G. We construct a central graph for the Zagreb index, Gutman index, Forgotten index, Lanzhou index, and degree distance for trees. Keywords Distance · Degree · Graph · Topological index

1 Introduction The graph invariant is correlated with some properties such as refractive index, molar volumes, and physicochemical properties of alkanes. Isomorphic graphs depend on their degree sequence. As seen, topological indices are numerical value. Consider a graph without loops and multiple edges; a powerful correlation between chemical bonds and graph invariants [1]. Consider edges are correlated with bonds and vertices as carbon atoms. A graph invariant is a topological index based on graph isomorphism. The Wiener index was the first distance-based topological index. J. Senbagamalar (B) · A. Meenakshi Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India e-mail: [email protected] A. Kanchana Rajalakshmi Institute of Science and Technology, Chennai, India H. Hachimi Sultan Moulay Slimane University, Beni Mellal, Morocco e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_43

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 Wiener index W (H ) = m 0 for i = 1, 2, 3, 4, 5.

3 Study of Positivity and Boundedness Proposition For all t ≥ 0, the variables are non-negative. for the system (4) is the closed region

The positive invariant (S, E, I, R, V ) ∈ R5 : 0 < N (t) ≤ /μ0 .

=

Proof From the model (4), we have, C

Dtν (S(t)) =  − β S I − μ0 S − δS ≥ −(β I + μ0 + δ)S ⎛ ⇒ S(t) ≥ S(0)exp⎝−

t

⎞ (β I + μ0 + δ)dq ⎠ > 0.

(6)

0 C

Dtν (E(t)) ≥ −(μ0 + μ1 )E ⎛

⇒ E(t) ≥ E(0)exp⎝−

t 0

C

⎞ (μ0 + μ1 )dq ⎠ > 0. ⎛

Dtν (I (t)) ≥ −(μ0 + μ2 )I ⇒ I (t) ≥ I (0) exp⎝− ⎛ C

(μ0 + μ2 )dq ⎠ > 0.

0

⎛ C

⎞

t

Dtν (R(t)) ≥ −μ0 R ⇒ R(t) ≥ R(0) exp⎝−

Dtν (V (t)) ≥ −μ0 V ⇒ V (t) ≥ V (0) exp⎝−

(7)

t 0

t

(8)

⎞ (μ0 )dq ⎠ > 0.

(9)

⎞ (μ0 )dq ⎠ > 0.

0

Also, C Dtν (S + E + I + R + V )(t) =  − μ0 (S + E + I + R + V ). This gives,

(10)

Study of Time-Delayed Fractional Order SEIRV Epidemic Model C

439

Dtν (N (t)) =  − μ0 N .

(11)

lim SupN (t) ≤ /μ0 .

(12)

Thus, t→∞

Thus, the model (4) is bounded above by /μ0 and S, E, I, R, and V are all positive functions, and hence, system is positive invariant (4). The basic reproduction number (0 ) can be obtained from the leading eigen

+ δ) 0 β/(μ 0 value of the matrix FV−1 [15, 16], where, F = and V = 0 0

0 μ0 + μ1 as −μ1 μ0 + μ2 0 = βμ1 /[(μ0 + δ)(μ0 + μ1 )(μ0 + μ2 )].

(13)

4 Stability Analysis The disease-free equilibrium points E 0 and the epidemic equilibrium point E 1 of the system (4) are obtained from the following equations, C

Dtν (S(t)) = C Dtν (E(t)) = C Dtν (I (t)) = C Dtν (R(t)) = C Dtν (V (t)) = 0.

(14)

= This leads to E 0 = (/μ0 + δ, 0, 0, 0, δ/μ0 (μ0 + δ)), E 1 (S ∗ , E ∗ , I ∗ , R ∗ , V ∗ ), S ∗ = (μ0 + μ1 )(μ0 + μ2 )/βμ1 , E ∗ = I ∗ (μ0 + μ2 )/μ1 , I ∗ = [μ1 /(μ0 + μ1 )(μ0 + μ2 )] − (μ0 + δ)/β, and R ∗ = I ∗ μ2 /μ0 , V ∗ = (μ0 + μ1 )(μ0 + μ2 )δ/βμ0 μ1 . The community matrix of the system (4) at E 0 is given by. ⎡

P11 ⎢0 ⎢ ⎢ J0 = P + Qe−λη1 , where P = ⎢ 0 ⎢ ⎣0 0

0 P22 P32 0 0

P13 P23 P33 P43 0

0 0 0 P44 0

⎡ ⎤ Q 11 0 ⎢ ⎥ 0 ⎥ ⎢0 ⎢ ⎥ 0 ⎥, Q = ⎢ 0 ⎢ ⎥ ⎣0 0 ⎦ P55 Q 51

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

⎤ 0 0⎥ ⎥ ⎥ 0 ⎥, ⎥ 0⎦ 0

with P11 = −μ0 , P13 = −β S0 , P22 = −(μ1 + μ0 ), P23 = β S0 , P32 = μ1 , P33 = −(μ0 + μ2 ), P43 = μ2 , P44 = −μ0 , P55 = −μ0 , and Q 11 = −δ, **Q 51 = δ. As a result of LaSalle’s extension to Lyapunov’s principle [17, 18], when 0 < 1, the disease-free equilibrium point of the system (4) is locally as well as globally

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asymptotically stable, and when 0 > 1, it is unstable with η1 = 0. If 0 > 1, the epidemic equilibrium E 1 = (S ∗ , E ∗ , I ∗ , R ∗ , V ∗ ) is locally asymptotically stable when η1 = 0 [19]. Theorem 1 The disease-free point ofthe system (4)  equilibrium   is locally  asymptotically stable when η1 ∈ 0, η1∗ , η1∗ = k −1 cos−1 (ux + vy)/ x 2 + y 2 .   Proof Characteristic equation of J0 is given by determinant P + Qe−λη1 − λI5 = 0. Now, 

 λ5 + P4 λ4 + P3 λ3 + P2 λ2 + P1 λ + P0   + Q 4 λ4 + Q 3 λ3 + Q 2 λ2 + Q 1 λ + Q 0 e−λη1 = 0.

(15)

where P4 = −(P11 + P22 + P33 + P44 + P55 ), P3 = [P44 P55 + (P44 + P55 )(P22 + P33 ) + (P22 P33 − P23 P32 ) P11 (P22 + P33 + P44 + P55 )], P2 = −[P44 P55 (P22 + P33 )] + (P22 P33 − P23 P32 ) + P11 {P44 P55 (P44 + P55 )(P22 + P33 ) + (P22 P33 − P23 P32 )}], P1 = [P44 P55 (P22 P33 − P23 P32 ) + P11 P44 P55 (P22 + P33 ) + (P22 P33 − P23 P32 )], P0 = −P11 P44 P55 (P22 P33 − P23 P32 ), Q4 = −Q11 , Q3 = Q11 (P22 + P33 + P44 + P55 ), Q2 = −Q11 [P44 P55 + (P22 + P33 )(P44 + P55 ) + (P22 P33 − P23 P32 )], Q1 = Q11 [P44 P55 (P22 + P33 ) + (P22 P33 − P23 P32 )], and Q0 = Q11 [P44 P55 (P22 P33 − P23 P32 )]. Let λ = ik be a root of the Eq. (15), then we have, x cos(kη1 ) + y sin(kη1 ) = u, y cos(kη1 ) − x sin(kη1 ) = v,

(16)

when x = Q 2 k 2 − Q 4 k 4 − Q 0 , y = Q 3 k 3 − Q 1 k, u = P4 k 4 − P2 k 2 + P0 , v = k 5 + P3 k 3 − P1 k, this leads to the following equation, x 2 + y 2 = u 2 + v 2 ⇒ k 10 + R4 k 8 + R3 k 6 + R2 k 4 + R1 k 2 + R0 = 0,

(17)

with R4 = 2P3 + P42 − Q 24 , R3 = P32 − 2P1 − 2P4 P2 − Q 23 + 2Q 2 Q 4 , R2 = −2P1 P3 + P22 + 2P0 P4 2Q 1 Q 3 − Q 22 − 2Q 0 Q 4 , R1 = P12 − 2P0 P2 − Q 21 + 2Q 0 Q 2 , and R0 = P02 − Q 20 . Putting k 2 = t in Eq. (17), then we have, t 5 + R4 t 4 + R3 t 3 + R2 t 2 + R1 t + R0 = 0.

(18)

∗ Thus, if t1∗ is a positive root in Eq. (18), then k = t1 is a positive root in Eq. (17). Eliminating sin from the Eq. (16), we obtain η1∗ = (kη ) 1   2  −1 −1 2 k cos (ux + vy)/ x + y .

Study of Time-Delayed Fractional Order SEIRV Epidemic Model

441

     −1 Theorem 2 If η1 ∈ 0, η1∗ , η1∗ = m −1 (ac + bd)/ a 2 + b2 , the epidemic 10 cos equilibrium E 1 = (S ∗ , E ∗ , I ∗ , R ∗ , V ∗ ) of the model (4) is locally asymptotically stable.   Proof Characteristic equation of J0 is given by determinant G + H e−λη1 − λI5 = 0. Which gives, λ5 + A 4 λ4 + A 3 λ3 + A 2 λ2 + A 1 λ + A 0   + B4 λ4 + B3 λ3 + B2 λ2 + B1 λ + B0 e−λη1 = 0.

(19)

where A4 = −(G 11 + G 22 + G 33 + G 44 + G 55 ), A3 = [G 11 (G 22 + G 33 ) + (G 22 G 33 − G 23 G 32 ) + (G 44 + G 55 )(G 22 + G 33 + G 11 ) + G 44 G 55 ], A2 = −[G 44 G 55 (G 22 +G 33 +G 11 ) + (G 55 + G 44 ){G 11 (G 22 +G 33 ) + (G 22 G 33 −G 23 G 32 )} + {G 11 (G 22 G 33 − G 23 G 32 ) + G 13 G 21 G 32 }], A1 = [(G 55 G 44 ){G 11 (G 22 + G 33 ) + (G 22 G 33 −G 23 G 32 )}(G 55 + G 44 ){G 11 (G 22 G 33 −G 23 G 32 ) + G 13 G 21 G 32 }], A0 = −G 55 G 44 {G 11 (G 22 G 33 − G 23 G 32 ), B4 = −H11 , B3 = H11 (G 33 + G 22 + G 44 + G 55 ), B2 = −[H11 G 44 G 55 + H11 (G 33 + G 22 )(G 44 + G 55 )H11 (G 22 G 33 − G 23 G 32 )(G 44 + G 55 ), B1 = [H11 G 44 G 55 ((G 22 + G 33 ) + H11 (G 22 G 33 − G 23 G 32 )(G 44 + G 55 )], and B0 = −H11 (G 22 G 33 − G 23 G 32 )(G 44 G 55 ), ⎡ ⎤ ⎡ ⎤ H11 0 0 0 0 G 11 0 G 13 0 0 ⎢ 0 0 0 0 0 ⎥ ⎢G G G 0 ⎥ ⎢ ⎥ ⎢ 21 22 23 0 ⎥ ⎢ ⎥ ⎢ ⎥ with G = ⎢ 0 G 32 G 33 0 0 ⎥, H = ⎢ 0 0 0 0 0 ⎥, G 11 = ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 ⎦ ⎣ 0 0 0 G 44 0 ⎦ 0 G 55 0 0 0 H55 0 0 0 0 −(β I ∗ + μ0 ), G 13 = −β S ∗ , G 21 = β I ∗ , G 22 = −(μ1 + μ0 ), G 23 = β S ∗ , G 32 = μ1 , G 33 = −(μ0 + μ2 ), G 44 = μ2 , G 44 = −μ0 , G 55 = −μ0 , H11 = −δ, and H55 = δ. If we consider λ = im 1 to be a root of the Eq. (19), we get, a cos(m 1 η1 ) + b sin(m 1 η1 ) = c, b cos(m 1 η1 ) − a sin(m 1 η1 ) = d,

(20)

where a = B2 m 21 − B4 m 41 − B0 , b = B3 m 31 − B1 m 1 , c = A4 m 41 − A2 m 21 + A0 , d = m 51 + A3 m 31 − A1 m 1 , and using a 2 + b2 = c2 + d 2 , we have the following equation, 8 6 4 2 m 10 1 + L 4 m 1 + L 3 m 1 + L 2 m 1 + L 1 m 1 + L 0 = 0,

(21)

with L 4 = 2 A3 + A24 − B42 , L 3 = A23 −2 A1 −2 A4 A2 − B32 +2B2 B4 , L 2 = −2 A1 A3 + A22 +2 A0 A4 +2B1 B3 − B22 +2B0 B4 , L 1 = A21 −2 A0 A2 − B12 −2B0 B2 , L 0 = A20 − B02 . Put m 21 = j in Eq. (21), then we have, j 5 + L 4 j 4 + L 3 j 3 + L 2 j 2 + L 1 j + L 0 = 0.

(22)

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√ Assuming j0 is a positive root of Eq. (22), then m 10 = j0 is a root in Eq. (21). Eliminating sin (m 1 η1 ) from (20) and substituting m 1 = m 10 ,where m 10, is a posi−1 2 2 + bd)/ a . Differencos + b tive root of Eq. (21), we have η1∗ = m −1 (ac 10 tiating Eq. (19) with respect to η1 and simplifying with λ = im 10 , we obtained,  −1  −1 f ( j ) dλ dλ  dη = a 21+b0 2 , and this gives  dη = 0, provided f 1 ( j0 ) = d fd1 (j j) = 1 1 λ=im 10

λ=im 10

0 at j = j0 holds where f 1 ( j) = j 5 + L 4 j 4 + L 3 j 3 + L 2 j 2 + L 1 j + L 0 . Thus, we established the conclusion of Theorem 2 as per the Hopf bifurcation hypothesis [20].

5 Numerical Scheme for the SEIRV Model For fractional order initial value issues, the Adams–Bashforth-Moulton approach is the most commonly used numerical technique. Let us look at the fractional differential equation given below.   r , Dtν Hj (t) = gj t, Hj (t), Hj (t − η1 ) , t ∈ [−η1 , 0], Hjr (0) = Hj0

(23)

r r = 0, 1, 2, . . . , ν, j ∈ N, where Hj0 is the arbitrary real number, ν > 0, and the ν fractional differential operator Dt is identical to the well-known Volterra integral equation in the Caputo sense.

Hj (t) =

ν−1  n=0

r Hj0

1 tn + n! (ν)

t

  (t − u)ν−1 gj u, Hj (u), Hj (u − η1 ) du, j ∈ N. (24)

0

Using this scheme, we explore the numerical solution of our model (4). For algoˆ Then, the corrector and rithm, we set h = T /m, ˆ tn = nh, n = 0, 1, 2, . . . , m. predictor formulae are given by the Eqs. (25) and (26), respectively,  p p p p  Sn+1 = S0 + (h ν / (ν + 2))  − β Sn+1 In+1 − μ0 Sn+1 − δSn+1 + (h ν / (ν + 2))

n 

  α j,n+1  − β S j I j − μ0 S j − δS j ,

j=0

 p p p  E n+1 = E 0 + (h ν / (ν + 2)) β Sn+1 In+1 − (μ0 + μ1 )E n+1 + (h ν / (ν + 2))

n 

  α j,n+1 β S j I j − (μ0 + μ1 )E j ,

j=0

 p p  In+1 = I0 + (h ν / (ν + 2)) μ1 E n+1 − (μ0 + μ2 )In+1

Study of Time-Delayed Fractional Order SEIRV Epidemic Model

+ (h ν / (ν + 2))

n 

443

  α j,n+1 μ1 E j − (μ0 + μ2 )I j ,

j=0

 p p  Rn+1 = R0 + (h ν / (ν + 2)) μ2 In+1 − μ0 Rn+1 + (h ν / (ν + 2))

n 

  α j,n+1 μ2 I j − μ0 R j ,

j=0

 p p  Vn+1 = V0 + (h ν / (ν + 2)) δSn+1 − μ0 Vn+1 + (h ν / (ν + 2))

n 

  α j,n+1 δS j − μ0 V j .

(25)

j=0 p Sn+1

= S0 + (1/ (ν))

n 

 

j,n+1  − β S j I j − μ0 S j − δS j ,

j=0 p

E n+1 = E 0 + (1/ (ν))

n 

 

j,n+1 β S j I j − (μ0 + μ1 )E j ,

j=0 p

In+1 = I0 + (1/ (ν))

n 

 

j,n+1 μ1 E j − (μ0 + μ2 )I j ,

j=0 p

Rn+1 = R0 + (1/ (ν))

n 

 

j,n+1 μ2 I j − μ0 R j ,

j=0 p

Vn+1 = V0 + (1/ (ν))

n 

 

j,n+1 δS j − μ0 V j ,

(26)

j=0

⎧ ν+1 − (n − ν)(n + 1)ν , if j = 0 ⎨n where ν j,n+1 = (n − j + 2)ν+1 + (n − j)ν+1 − 2(n − j + 1)ν+1 , if 0 ≤ j ≤ n, ⎩ 1, if j = 1,   and j,n+1 = (h ν /ν) (n + 1 − j)ν − (n − j)ν ) , 0 ≤ j ≤ n.

6 Numerical Simulation We use MATLAB to analyze the solutions generated by the scheme given in Sect. 5. Table 1 shows the values for the parameters. In the instance of COVID-19, the following are the parameter values in India. The values of parameters in Table 1 are used to plot the figures in Fig. 2a–e. The behavior of susceptible individuals over time for various fractional orders ν is

444 Table 1 Calculation table of values using different parameters

S. Paul et al. Parameter

Value

Reference



0.0182

Estimated

β

0.476

Estimated

μ0

0.0073

Estimated

δ

0.01

Model to fit

μ1

0.071

[21, 22]

μ2

0.286

[21, 22]

0

1.55

Estimated

seen in Fig. 2a. Moreover, the administration of vaccine shows that the number of susceptible individuals is always less than those in the case of without vaccination for different values of ν as expected. Figure 2b indicates the relation between exposed individuals and time for different fractional order ν. For all values of v, the number of exposed individuals grows with time. The behavior of the number of infected individuals with time for different fractional orders ν is seen in Fig. 2c. The number of infected individuals decreases consistently with time for different fractional values of ν which further decreases with the use of vaccines. The behavior of recovered individuals with time is depict in Fig. 2d. The graph shows that for all values of v, the number of recovered individuals grows with time. It may also be deduced that the recovered individuals increase because of the impact of vaccines. Figure 2e depicts the behavior of vaccinated individuals with time for different fractional order ν. Figure 3a–e depicts that the epidemic equilibrium points E 1 of the system (5) is periodically stable.

7 Conclusion The SEIRV model (4) was investigated with a single time delay parameter η1 .The disease-free equilibrium point of the system is locally asymptotically stable when 0 < 1, and unstable when 0 > 1, according to the system’s stability analysis. The endemic equilibrium E 1 = (S ∗ , E ∗ , I ∗ , R ∗ , V ∗ ) is locally asymptotically stable if 0 > 1, when η1 = 0. However, in the presence  of time delay parameter η1 , E 0 and E 1 are asymptotically stable in the interval 0, η1∗ . According to numerical calculations, if η1 > 80, our system displays Hopf bifurcation. Thus, it becomes apparent that beyond the value of η1∗ = 80, the dynamics of the system becomes unstable. It should be remembered that the time delay parameter was included in (4) to support the claim that the afflicted population would take some time to recover. When the time delay owing to the time period required by the susceptible population to recover from the disease reaches a threshold value, the model discussed here experiences a Hopf bifurcation about the endemic equilibrium point.

Study of Time-Delayed Fractional Order SEIRV Epidemic Model

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Fig. 2 Dynamical behavior of all individuals for various values of ν with respect to time (days) with η1 = 2

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Fig. 3 Time series analysis of the model system (5) for η1 = 80, ν = 1 and parameter values as given in Table 1

References 1. Ji, C., D. Jiang, and N. Shi. 2011. Multigroup SIR epidemic model with stochastic perturbation. Physica A: Statistical Mechanics and Its Applications 390 (10): 1747–1762. 2. Wang, L., and R. Xu. 2016. Global stability of an SEIR epidemic model with vaccination. International Journal of Biomathematics 9 (6). 3. Gumel. A., B. Mccluskey, and C. Watmough. 2006. An SVEIR model for assessing potential impact of an imperfect anti-SARS vaccine. Mathematical Biosciences and Engineering 3 (3): 485–512. 4. Ferretti, L., C. Wymant, M. Kendall, et al. 2020. Quantifying SARS-CoV-2 transmission

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Modeling of Teaching–Learning Process of Geometrical LOCI in the Plane with GeoGebra Pham Van Hoang, Ta Duy Phuong, Nguyen Thi Bich Thuy, Tran Le Thuy, Nguyen Thi Trang, and Nguyen Hoang Vu

Abstract GeoGebra is an open-source software that has been widely used for teaching and learning mathematics worldwide. Other than the English interface, GeoGebra also supports Vietnamese, making it suitable for mathematics education in Vietnam, at school level and higher. As GeoGebra can combine calculation and dynamic drawing of geometrical objects, it is quite effective in geometry classes. This article showed how GeoGebra could be used for teaching lessons on geometrical loci (sets of points satisfying certain conditions). In addition to basic loci, we also illustrated the ability of GeoGebra in solving hard loci (in mathematical competitions) and expanding knowledge of geometrical loci for school students and undergraduates. Keywords GeoGebra · Loci · Computer-assisted teaching · Mathematics education

P. Van Hoang University of Education, Vietnam National University Hanoi, Hanoi, Vietnam e-mail: [email protected] T. D. Phuong (B) Institute of Mathematics, Hanoi, Vietnam e-mail: [email protected] N. T. B. Thuy University of Science, Vietnam National University, Hanoi, Vietnam T. Le Thuy University of Education, National University Hanoi, Hanoi, Vietnam e-mail: [email protected] N. T. Trang Hanoi University of Natural Resources and Environment, Hanoi, Vietnam e-mail: [email protected] N. H. Vu Indochina Institute of Biology and Environmental Sciences, Hanoi, Vietnam © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_45

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1 Introduction Problems involving geometrical loci (see, for example, [1]) are usually hard, and hence often appear in mathematical competitions. Generally, students find it hard to visualize how point A depending on point B (according to a certain relation or rule f ) will move when B moves. Effective methods for teaching loci are therefore required in teaching loci. Loci can be considered as geometrical transformations. For example, when point B moves on a line (on a circle), its image (point A) can move accordingly on a circle (on a line or on a circle) through the transformation f. Problems involving loci are closely related to other kinds of problems such as proofs, geometrical constructions, calculations, and especially geometrical transformations. Moreover, problems on loci can be related to Analytic Geometry. As such, studying locus problems can enhance the knowledge of students in Analytic Geometry. GeoGebra (see, for example, [2]) is a software supporting drawing of dynamic images, i.e., a geometrical point can move and make another point move accordingly. GeoGebra is therefore effective in mathematical education (see, [3]), especially geometry, including geometrical traces (locus). Besides guide books on using GeoGebra for geometry (see [4, 5]); academic articles covering specific problems with loci have also been produced, including simple loci for teaching elementary concepts [6], advanced locus problems for enthusiastic students [7], loci involving conics and other analytical curves [8, 9], to undergraduate topics such as complex functions [10]. However, for school teachers (in Vietnam and worldwide) who need information on how to apply GeoGebra to teaching loci in practical classroom situations, particularly with classification of locus problems and how GeoGebra can be used cover each topic, just looking at a couple of related problems may not be adequate. It is also necessary to account for the differences between math curriculum to determine the correct approach. Hence, this paper aimed to introduce how GeoGebra could be used for geometrical loci in the plane in a more systematic way in Vietnamese math classrooms, with a ground-up approach from the basic loci and how they can be visualized using the basic geometrical objects in GeoGebra to solving more complex loci. Topics presented were chosen based on the way geometry was taught in the Vietnamese National Mathematics curriculum, with a variety of examples drawn from mathematic education sources in Vietnam. Aside from familiar or hard loci from mathematical competitions, we also showed how GeoGebra could help school students familiarize with loci uncommon in Vietnamese school mathematics, including ellipse, parabola, hyperbola, cycloid, etc.

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2 Basic Tasks for Constructing Loci in GeoGebra GeoGebra is an open-source software that can be installed on both personal computers and mobile devices and can be used for teaching and learning many different fields of mathematics: Arithmetic, Algebra, Calculus, Statistics, Geometry, Analytic Geometry, etc. (see, for example, [11–14]). Users can switch the language from English to Vietnamese, so it is effective in mathematics education at both school and undergraduate levels in Vietnam. In addition, GeoGebra can simultaneously draw geometrical objects and do automatic calculations (coordinates of points, segment lengths, area, volume, magnitude of angles, etc.), when conditions change, making it convenient for combining geometry and algebraic calculations. The following basic geometrical objects can be drawn using the tools found in the geometrical toolbar of GeoGebra (called basic tasks in later sections): 1. Point 2. Line through two points 3. Ray 4. Line segment with known endpoints 5. Line segments of given length 6. Line parallel to a given line 7. Line perpendicular to a given line 8. Angle bisector

9. Circle with known center through a point 10. Circle with radius R 11. Circle through three points 12. Angle of given size 13. Midpoint of a segment 14. Triangle 15. Regular polygons

Detailed instructions for these tasks can be found in guides for GeoGebra such as [2, 4, 5]. After specifying the relations between geometrical objects, traces can be visualized in GeoGebra using the following steps: 1. 2.

Points can be moved using the tool. Right click on objects whose traces need to be visualized and select “Show trace.”

While loci can be generated using the Locus command in GeoGebra, in our opinion, it is more visually stimulating to use traces as students can see the locus curve being drawn out while the point moves.

3 GeoGebra as a Tool for Observing and Providing Hints for Geometrical Loci 3.1 Basic Loci Problems on loci often lead to basic loci (part of a line or a circle). GeoGebra can be used to observe all basic loci and help solve specific problems. Examples and problems are listed below (GeoGebra files can be found in [14]).

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Fig. 1 Locus of Problem 3.1.1 in GeoGebra

Basic locus 1 ([1], page 51) The locus of a moving point being equidistant to two fixed points is the perpendicular bisector of the segment connecting these two points. Problem 3.1.1 ([1], page 12) In a given circle, find the locus of the midpoint of a moving chord that is always parallel to a another given chord. Solution Draw a circle (basic task 9). Draw chord AB (basic task 4) by selecting the points A and B on the circle (basic task 1). Select point C on the circle (basic task 1). Draw line CD parallel to AB (basic task 6). M is midpoint of segment CD (basic task 13). Move C using the move tool, the trace of M is diameter perpendicular to segment AB (perpendicular bisector of AB, Fig. 1). Proof Since M is the midpoint of CD and OD = OC, hence OM⊥CD. But C DAB so OM⊥AB and MA = MB. So, the locus of M is the perpendicular bisector of AB (basic locus 1). Problem 3.1.2 ([1], Problem 15, page 51) Ox and Oy are two perpendicular given rays. The vertex A of the right angle of triangle ABC is a fixed point inside angle xOy. The two other vertices, B and C of the triangle move along Ox and Oy, respectively. Find the locus of the midpoint M of segment BC. Solution Draw ray Ox and ray Oy perpendicular to Ox (basic task 7). Pick point A inside angle xOy and point B on Oy (basic task 1). Draw segment AB (basic task 4). Draw the line through A and perpendicular to AB (basic task 7), intersecting Ox at C. Draw segment BC (basic task 4). Point M is the midpoint of segment BC (basic task 13). Move B along Oy, point M can be seen moving on the segment DF perpendicular to AO (F on Ox, D on Oy, Fig. 2). Proof Angle BOC is a right angle. M is the midpoint of BC. Hence OM = BM = CM. Angle BAC is a right angle so MB = MC = AM. Hence OM = AM. This means M is on the perpendicular bisector of AO. The locus of M is the segment DF, with D on Oy, F on Ox. DF is the perpendicular bisector of OA.

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Fig. 2 Locus of Problem 3.1.2 in GeoGebra

Analysis Draw and point out that A and O are two fixed points to show that M is equidistant to A and O. Basic locus 2 ([1], page 49) Locus of points at a given distance d from a given line are two lines parallel to the given line and at a distance d from the given line. Problem 3.1.3 ([1], Problem 1, page 13) Prove that the locus of the midpoint of a moving segment, with one endpoint being a given fixed point P and the other endpoint moving on a given line AB, is a line parallel to AB and equidistant to P and line AB. Solution Draw line AB (basic task 2). Select point P not on AB and point C on AB (basic task 1). E is the midpoint of PC (basic task 13). Move C along line AB, the trace of point E is the line through E and parallel to AB (Fig. 3). Proof Draw PH and EK perpendicular AB. We have EK = 21 PH being constant (Fig. 3). When C moves along line AB, E moves on a line parallel to AB, in the same half plane with P and its distance to AB is half the distance from P to AB (basic locus 2). Basic locus 3 ([1], page 56) The locus of a moving point that is equidistant to two given intersecting lines is the bisectors of the inner and outer angles formed by those two lines.

Fig. 3 Locus of Problem 3.1.3 in GeoGebra

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Fig. 4 Locus of Problem 3.1.4 in GeoGebra

Problem 3.1.4 ([1], Problem 9, page 35) A moving segment has one endpoint being the vertex A of the given triangle ABC, and the other endpoint P moving on the segment BC. Find the locus of the center of the circle inscribed in the triangle APC. Solution Draw triangle ABC (basic task 14). Select point P on segment BC (basic task 1). Join A and P (basic task 4). Draw inner bisectors of angles A and P (basic task 8), intersecting at D. Since D is the center of the inscribed circle of triangle APC, D lies on the (fixed) bisector of angle C. When P moves along BC, D moves along segment CG (with G being the center of the inscribed circle of triangle ABC (Fig. 4). Basic locus 4 ([1], page 43) The locus of a moving point being at a constant distance from a given point is the circle whose center is the given point and whose radius is equal to the given distance. Problem 3.1.5 ([1], Problem 12, page 45) Point P moves on the given circle (O, R). Find the locus of the midpoint M of AP, with A being a point inside the circle. Solution Draw circle with center at O (basic task 9). Select point A inside the circle (O) and point P on the circle (basic task 1). M is the midpoint of AP (basic task 13). When P moves on circle (O), point M moves on the circle (C, R2 ), where C is the midpoint of AO (Fig. 5). Proof Draw MC parallel to PO. Since M is the midpoint AP, C is the midpoint of AO. Hence MC = 21 OP = R2 . Hence the locus of M is the circle (C, R2 ). Fig. 5 Locus of Problem 3.1.5 in GeoGebra

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Fig. 6 Locus of Problem 3.1.6 in GeoGebra

Problem 3.1.6 ([6], Problem 13, page 46) A moving segment DE of constant length l is always parallel to the given line a. The endpoint D moves on the given circle (O; R). Find the locus of E. Solution Draw circle (O, R) (basic task 10) and line (a) (basic task 2). Select point D on circle (O, R) (basic task 1). Draw line (g) through D and parallel to (a) (basic task 6). On (g), construct segment DE = l (with l given) (basic task 5). Move D along circle (O, R). The trace of E is the circle (N, R), with N being a −→ − → point on Ox // (a) such that ON = DE (Fig. 6). Proof Through O, draw a line Ox parallel to (a) and select a point on it such that −→ − → ON = DE. Hence NE = OD = R. Since ON = DE = l, N is a fixed point. When D moves along (O), E moves along (N, R) (Fig. 6). Basic locus 5 ([1], page 58) The locus of a moving point that always subtends right angle with two other fixed points is a circle with diameter equal to the distance between the two fixed points. Problem 3.1.7 ([1], Problem 19, page 60) Let AOB be a fixed diameter of a given circle. BC is an arbitrary chord. Lengthen BC to D so that CD = BC. AC and DO intersect at P. Find the locus of P. Solution Draw circle (O) (basic task 9), diameter AB (basic task 2) and ray Bx (basic task 3) intersecting (O) at C. Select D on Bx so that BC = CD (through the circle (C, CB) - not shown here). Draw segments AC and DO (basic task 4). Let P be the intersection of AC and DO (basic task 1). Draw PE parallel to BD (basic task 6). Since P is the centroid of triangle ABC, we have AE: AB = AP: AC = 2: 3. Hence E is a fixed point. Angle ACB is a right angle so P subtends AE at a right angle. Move C along (O) (move tool 1), point P will move along the circle with diameter AE (Fig. 7). Basic locus 6 ([1], page 62) The locus of a point subtending an angle of given magnitude α with two fixed points A and B are two arcs symmetric through AB.

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Fig. 7 Locus of Problem 3.1.7 in GeoGebra

Problem 3.1.8 ([1], Problem 20, page 62) Let AOB be a fixed diameter of a given circle. AC is a moving chord. Lengthen AC to P, so that CP = CB. Find the locus of P. Solution Draw circle (O) (basic task 9). Draw diameter AB (basic task 2) and ray Ax (basic task 3), intersecting (O) at C. Let P be the intersection of Ax and the circle (C, CB), i.e., CP = BC. Move C along (O), P will move on two arcs BN and BM (Fig. 8). Proof Triangle PCB is an isosceles right triangle. Hence the angles APB = CPB = 45°. So, P subtends AB at an angle of 45°, which means P is on one of the two arcs subtending AB at an angle of 45°.

Fig. 8 Locus of Problem 3.1.8 in GeoGebra

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3.2 Using GeoGebra for Solving Complex Loci Problems Based on the basic loci, GeoGebra can also be applied to more complex locus problems with more sophisticated relationships between quantities and geometrical transformations. Problem 3.2.1 Given line xy and a fixed point A outside that line. Point M moves along xy. Point I is a point on segment AM such that AI.AM = k 2 , the given k is greater than 0 and is less than the distance from A to line xy. Construct square AIJK. Find the locus of point J and point K. Observation with GeoGebra Point I can be constructed as the intersection of AM and circle (A, k 2 /AM). Observing traces of vertices I, J, K of the square AIJK in GeoGebra (basic task 15) show that they all move along circles. Solution Draw AH perpendicular to xy (basic task 7). Let B be the point on AH AI AB = AM . So  AIB and with AB.AH = k 2 . Since AI.AM = k 2 = AB.AH, AH  AHM are similar. Hence the angles AIB = AHM = 90°. So, point I is on the circle with diameter AB. Because IJ = AI = AK and angles AJI = 45°, AKI = 45°, K is the image of I through the rotation (A, 90°) or (A, √ −90°) and J is the image of I through the rotation (A, 45°) or (A, −45°) with a 2 scaling. The loci of these points are the image of the circle with diameter AB through these transformations (Fig. 9). Problem 3.2.2 Given segment AC with midpoint K. Two moving points B and D are always symmetric through K. The bisector of angle BCD intersects lines AB, AD at I and J respectively. Point M is the other intersection (aside from A) of circumcircles of triangles ABD and AIJ. Show that M is always on a fixed circle. Observation with GeoGebra Draw segment AC (basic task 4). Construct midpoint K of segment AC (basic task 13). Pick point B on the plane and construct point D symmetric to B through K (basic task 13). Move B in the plane, M will move Fig. 9 Locus of Problem 3.2.1 in GeoGebra

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Fig. 10 Locus of Problem 3.2.2 in GeoGebra

on the circle (K, KA) (Fig. 10). By knowing the result from this hint, students can work backwards to a proof.

3.3 Extension to Analytical Curves GeoGebra can help school students explore geometrical curves not currently in the curriculum. It is also useful for undergraduate students studying Analytical Geometry. Compared to previous sections, these curves require command inputs to declare objects in GeoGebra in addition to mouse operations. Problem 3.3.1 (Ellipse) Find the set of points with the sum of distances to two given points F1 and F2 being a constant equal to 2a (with 2a > F1 F2 ). Solution Select points F1 and F2 on Ox (basic task 1). Declare parameter t (from 0 to 2a). Draw circles (F1 , t) and (F2 , 2a-t), intersecting at B. Trace of B when t changes is an ellipse, Fig. 11a). Problem 3.3.2 (Parabola) Find the sets of points being equidistance to the given point A and line (a). Solution Select point A (basic task 1) and axis Ox as line (a) (basic task 2). Let D a point that DA = DF (DF is perpendicular to (a)). Declare parameter t. Draw circle (A, t) and declare line y = t, intersecting at D. Trace of D when t changes is a parabola with OA as the symmetrical axis (Fig. 11b). The Hyperbola can also be drawn in a similar fashion like the case with the ellipse (Fig. 11c).

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Fig. 11 Conic curve as loci in GeoGebra: a ellipse, b parabola, c hyperbola

Fig. 12 Cycloid and prolate cycloid as loci in GeoGebra

The cycloid can be drawn in GeoGebra using its parametric form or through rotation of point A with the angle of rotation depending on the parameter of time (Fig. 12a). More complex cases such as a prolate cycloid can also be drawn using similar techniques (Fig. 12b).

4 Conclusion As shown through examples and problems above, GeoGebra could be effectively used for teaching loci problems, especially for the math curriculum in Vietnamese schools, with extensions to bridge the gap to undergraduate topics (analytical geometry, ordinary differential equations, etc.). Many problems involving loci with GeoGebra could be found in [14]. Further research on locus problems with GeoGebra would involve using GeoGebra for 3D loci.

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References 1. Chunfeng, Hua. 1973. Locus. Vietnam Education Publishing House, Hanoi, 96 p (in Vietnames). 2. Steve Phelps, An Introduction To GeoGebra. https://www.math.utah.edu/~emina/teaching/527 0s13/Intro_to_Geogebra.pdf 3. Hernández, Alexánder, Josefa Perdomo-Díaz and Matías Camacho-Machín. 2019. Mathematical understanding in problem solving with GeoGebra: a case study in initial teacher education. International Journal of Mathematical Education in Science and Technology, 1–16. 4. Project Maths Development Team. GeoGebra Geometry. https://www.projectmaths.ie/docume nts/PDF/GeoGebraForGeometry.pdf. 5. Gerry Stahl & The VMT Project Team. 2013. The Math Forum at Drexel University, DynamicGeometry Activities with GeoGebra for Virtual Math Teams, http://gerrystahl.net/vmt/activi ties.pdf, 111 p. 6. Chan, Yip-Cheung. 2013. GeoGebra as a tool to explore, conjecture, verify, and prove: The case of a circle. North American GeoGebra Journal 2 (1): 14–18 (ISSN: 2162-3856). 7. Abdilkadir Altınta¸ S. 2021. Congruent Circles On Locus Problems. International Journal of Computer Discovered Mathematics (IJCDM) 6: 92–96. 8. Botana, Francisco, and Zolt’an Kov’acs. 2016. New tools in GeoGebra offering novel opportunities to teach loci and envelopes. CoRR abs/1605.09153, pp. 1–21. 9. Lazarov, Borislav Yordanov and Dimitar Georgiev Dimitrov. 2020. Introducing Conics in 9th Grade: An Experimental Teaching. In Proceedings of the 12th International Conference on Computer Supported Education (CSEDU 2020), Volume 1, 436–441. 10. Breda, Ana Maria d’Azevedo, and José Manuel Dos Santos Dos Santos. 2015. Using GeoGebra to study complex functions. In Proceedings of the 12th International Conference on Technology in Mathematics Teaching, ICTMT 12, 584–586. 11. Hanh, Nguyen Thi Hong, Ta Duy Phuong, Pham Thanh Tam, Nguyen Thi Bich Thuy, Tran Le Thuy, and Nguyen Hoang Vu. 2021. Using GeoGebra software for in geometry hypothesis testing. Journal Mathematics and Youth Magazine, No. 529 (July 2021), 10-15 (in Vietnames). 12. Hanh, Nguyen Thi Hong, Ta Duy Phuong, Nguyen Thi Bich Thuy, and Tran Le Thuy. 2021. Using GeoGebra software in teaching and learning the space Geometry. In Book Education of Things: Digital Pedagogy, 73–102. 13. Hanh, Nguyen Thi Hong, Ta Duy Phuong, Nguyen Thi Bich Thuy, Tran Le Thuy, and Nguyen Hoang Vu. 2021. Use GeoGebra in teaching definite integral. In Proceedings of 2nd International Conference on Innovative Computing and Cutting-edge Technologies (ICCT), 11 and 12 September, 2020, in the Springer Series “Learning and Analytics in Intelligent Systems”, 2021, 327–335. 14. Hanh, Nguyen Thi Hong, Pham Van Hoang, Ta Duy Phuong, Nguyen Thi Bich Thuy, Tran Le Thuy, Nguyen Thi Trang, and Nguyen Hoang Vu. 2021. Using GeoGebra in Teaching and Learning Geometry. In Book (manuscript) with CD-Rom for solving Geometry Problems on GeoGebra, 250 p (in Vietnames). 15. Antohe, Gabriela-Simona. 2009. Modeling a geometric locus problem with GeoGebra, Annals. Computer Science Series, 7th Tome, 2nd Fasc., pp. 105–112. 16. Trojovská, Eva, and P. Trojovský. 2012. On some problems for loci of points via GeoGebra. In Educational Technologies (Edute’12), Athens, World scientific and engineering academy and society, 117–122.

Application of Pentagonal Fuzzy Number in CPM and PERT Network with Algorithm G. Ambika and T. Nagalakshmi

Abstract Projects are made up of interconnected aspects like goal, time, resource, and environment. The successful project completion is dependent on the controlled use of these dimensions and their appropriate scheduling. The project scheduling process entails outlining project activities and estimating the amount of time and resources needed to complete them. Program Evaluation Review Technique (PERT) and critical path method (CPM) processes made many researchers to study the possible ways of finding the criticality paths and activities in network. The advancement of the CPM and PERT towards a probabilistic environment is still a long way off. However, Artificial intelligence approaches such as Genetic algorithm, Dijkstra’s algorithm, and others are utilized for network analysis within the framework of project management. This study is to help the project manager plan schedule for a construction project in order to determine expected completion time. The first portion is an introduction, followed by a survey of the literature for studies pertinent to the topic in Sect. 2. Section 3 presents the topic’s fundamental definitions. An example problem is given in Sect. 4; finally, in Sect. 5, conclusions and discussion are presented. In this research paper, we describe a method for obtaining the earliest and latest time of a critical path using modified Dijkstra’s algorithm. Keywords Pentagonal Fuzzy Number · Critical path · Modified Dijkstra’s algorithm · Ranking technique · Range technique · CPM · PERT

G. Ambika (B) · T. Nagalakshmi Vel Tech Rangarajan Dr. Sagunthala, R&D Institute of Science Technology, Chennai, Tamilnadu, India e-mail: [email protected] T. Nagalakshmi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_46

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1 Introduction In the year of 1965, Zadeh [1] introduced the concept of fuzzy set theory. In today’s highly competitive world, a large number of problems in fuzzy mathematics have been produced. When the activity periods in a project environment are deterministic in nature, many real-life events are changing at a faster rate by utilizing the idea of fuzziness. Many cases have been discussed where the activity times are not deterministic, but here in this case, PERT method is applicable on probabilistic environment. Different methods and various working techniques are applied in project management. Every procedure has its own span of time to complete the task. Gantt chart, network diagram, CPM, and PERT are a few strategies that are commonly used to tackle projects. With the help of program evaluation review technique (PERT) and critical path method (CPM) along with Dijkstra’s algorithm approached with an example to formulate the critical path and project duration. The main objective values are taken in fuzzy number, we are able to rank the fuzzy number to find the best alternative.

2 Literature Review Yager [2] proposed four indices to order fuzzy quantities in [0, 1]. In Zadeh’s possibility theory, Dubois and Prade [3] introduced a complete set of comparison indices. The task of applying the PERT approach to fuzzy parameters is difficult. Backward recursion facts, as reported, are used to compute the sets of potential values for the latest starting times and float activities. Furthermore, different definitions of the fuzzy critical path produce varied estimates of the criticality grade for the same path. According to the extension concept proposed by Chanas and Zielinski [4], fuzzy CPM in the standard criticality notion is treated as a function of network activity time. The most basic form of PERT and CPM focuses on determining the longest timeconsuming path through a network of tasks as a foundation for project planning and control by Davis et al. [5]. Kuchta [6] proposed a method for evaluating a fuzzy approach of assessing the criticality of project activities and the project as a whole. He adopted a decision-maker mind set and a project network structure. Yao and Lin [7] introduced a new approach to fuzzy critical path method. The use of fuzzy variables introduced by Chanas [8]. Lin and Yao [9] introduced a method that combines fuzzy mathematics with statistics to solve practical problems in unknown or vague situations. For the project network problem, Noto and Soto [10] provided the shortest path by an extended Dijkstra’s algorithm. Jassbi and Khan Mohammadi [11] introduced a new approach based on membership functions for estimated durations and the delays in activities.

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Chen [12] proposed a method for doing critical path analysis on a project network with ambiguous activity timings. Selvakumari [13] recommended range technique for solving a fuzzy transportation problem.

3 Preliminaries Definition 3.1 A fuzzy set can be mathematically formed by assigning a value denoting grade of membership to each individual in the universe of discourse. Definition 3.2 A fuzzy number A˜ is a fuzzy set whose membership function μ A˜ (x) satisfies the following conditions. (i) (ii) (iii)

μ A˜ (x) piecewise continuous μ A˜ (x) is convex μ A˜ (x) is normal i.e. μ A˜ (x0 )

  Definition 3.3 A Pentagonal fuzzy number is defined as A˜ = l1 , l2,l3 , l4 , l5 where the middle point l3 has the grade membership function and w1 & w2 are the grades of point l2 , l4 . Note that every PFN is associated with two weights w1 & w2 , l1 ≤ l2 ≤ l3 ≤ l4 ≤ l5 . Case (i): Triangular fuzzy number and trapezoidal fuzzy number are associated with weights if w1 = 0 and w2 = 0, then PFN is reduced to a triangular fuzzy number, i.e. {l2 , l3 , l4 }. Case (ii): w1 = 1 and w2 = 1, then PFN is reduced to a trapezoidal fuzzy number, i.e. {l1 , l2 , l3 , l4 }. ⎧ 0, x < l1 ⎪

⎪ ⎪ ⎪ w x−l1 , ⎪ l1 ≤ x 1 l2 −l1 ⎪ ⎪

⎪ ⎪ x−l ⎪ 2 ⎪ ⎨ 1 − (1 − w1 ) l3 −l2 , l2 ≤ x μ APFN (x) = 1, 

x = l3 ⎪ ⎪ ⎪ l4 −x ⎪ 1 − (1 − w2 ) l4 −l3 , l3 ≤ x ⎪ ⎪ ⎪ ⎪ ⎪ w l5 −x , l4 ≤ x ⎪ ⎪ ⎩ 2 l5 −l4 0, x > l5

≤ l2 ≤ l3 ≤ l4 ≤ l5

  Definition 3.4 A fuzzy number  A P F N = l1 , l2, l3 , l4 , l5 is called a Canonical PFN if it is closed and bounded; PFN and its membership function is strictly increasing on the interval [l2 , l3 ] and strictly decreasing on [l3 , l4 ] (Fig. 1). Definition 3.5 Operations on PFN Consider two    pentagonal fuzzy numbers A˜ = l1 , l2, l3 , l4 , l5 and B˜ = m 1 , m 2, m 3 , m 4 , m 5 . The following properties are discussed under arithmetic operations.

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Fig. 1 PFN

Addition: A˜ + B˜ = (l1 + m 1 , l2 + m 2 , l3 + m 3 , l4 + m 4 , l5 + m 5 ). Subtraction: A˜ − B˜ = (l1 − m 5 , l2 − m 4 , l3 − m 3 , l4 − m 2 , l5 − m 1 ). Multiplication:  AB = (l1 m 1 , l2 m 2 , l3 m 3 , l4 m 4 , l5 m 5 ).

−1 ˜ Inverse of PFN: A = 1/A = 1 l5 , 1 l4 , 1 l3 , 1 l2 , 1 l1 .

A l2 , l3 , l4 , l5  −1 = l 1 Division:  = AB , m5 m4 m3 m2 m1 . B˜ Definition 3.6 Fuzzy Project Network: PERT—program evaluation review technique—is a project management method for estimating how long it will take to complete a project successfully. It is a simple strategy that uses a beta (β) distribution mechanism. The length of an activity distribution is determined using three time estimates.

(i) Optimistic estimate O˜ T : It requires minimum time for completing the activities, and there is 1% for the activity to complete within time.

chance ˜ (ii) Most Likely estimate MT : Assumes some issues might occur and based on how long the task usually takes under normal circumstances. (iii) Pessimistic estimate ( P˜T ): Assuming everything goes wrong. Fuzzy project network is represented by an acyclic directed graph, in which the nodes (vertices) represents events and directed lines (edges) indicate project

˜ ˜ P˜ ;  A, function to be executed on the network. It is denoted by M = V, let V = (V1 , V2 , V3 , V4 , . . . , Vn ) be the set of vertices. Let E i and L i be the earliest and latest times, respectively, for the event i. Let E j and L j be the earliest and latest time for the event j. Definition 3.7 Expected Time: If an activity is reported in large numbers, it will take an average time, and it depends on the assumption that the beta distribution is followed by the operating time. It is provided by the formula. te =

O˜ T + 4 M˜ T + P˜T 6

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Definition 3.8 Variance: Using this formula, we can calculate the variance for the activity.

te =

P˜T − O˜ T 6

2

Definition 3.9 Ranking Technique 1: In this problem, we applied two ranking techniques to find out the completion of the project duration. (i)

Using the ranking algorithm, fuzzy numbers are immediately inserted into the real line. Let A˜ PFN (x) be the generalized pentagonal fuzzy number; by applying the ranking technique, it is calculated as follows:

l + 2l + 3l + 2l + l 1 2 3 4 5 R A˜ PFN (x) = 9

(ii)

Range Technique 2 [13]: The range is described as the difference between the maximum value and minimum value. Range = Max value − Min value

4 Illustration 4.1 Numerical Example To calculate the time of the critical path, we are considering constructional project network in fuzzy environment with the relevant data collected as a pentagonal fuzzy number. Finding forward and backward passes for the earliest and latest time by applying modified Dijkstra’s algorithm (Tables 1, 2 and 3 and Figs. 2 and 3). Table 1 Given data are in pentagonal fuzzy numbers



Activity Optimistic O˜ T Most likely M˜ T



Pessimistic P˜T

1–2

(2, 4, 6, 8, 10)

(3, 6, 9, 12, 15)

(5, 7, 9, 11, 13)

1–6

(1, 3, 5, 7, 9)

(2, 5, 7, 9, 10)

(4, 6, 8, 10, 12)

2–3

(1, 2, 3, 4, 5)

(3, 4, 5, 6, 7)

(8, 10, 11, 12, 13)

2–4

(3, 6, 9, 12, 15)

(5, 8, 10, 12, 13)

(14, 15, 16, 17, 18)

3–5

(3, 4, 5, 6, 7)

(3, 5, 7, 9, 11)

(4, 6, 8, 9, 10)

4–5

(7, 8, 9, 10, 11)

(9, 11, 13, 15, 17)

(13, 14, 15, 16, 17)

5–8

(8, 9, 10, 11, 12)

(10, 12, 14, 15, 16)

(10, 12, 14, 16, 18)

6–7

(4, 8, 12, 14, 16)

(6, 8, 10, 12, 14)

(12, 13, 14, 15, 16)

7–8

(7, 8, 10, 12, 13)

(11, 12, 13, 14, 15)

(14, 15, 17, 18, 19)

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Table 2 Represents PFN to crisp numbers using ranking technique 1



 T )(Most O˜ T (Optimistic) (M P˜T (Pessimistic) te = Activity likely) O˜ T +4 M˜ T + P˜T

σ2 =

P˜T − O˜ T 6

6

1–2

6

9

9

8.5 ~ 9

0.25

1–6

5

7

8

6.8 ~ 7

0.25

2–3

3

5

11

5.67 ~ 6

1.77

2–4

9

10

16

10.8 ~ 11

1.36

3–5

5

7

8

6.8 ~ 7

0.25

4–5

9

13

15

12.60 ~ 13

1

5–8

10

14

14

13

0.44

6–7

12

10

14

11

0.111

7–8

9

13

14

13

1.36

te =

σ2 =

Table 3 Represents PFN to crisp numbers using range technique





O˜ T Optimistic M˜ T Most P˜T Pessimistic Activity

O˜ T +4 M˜ T + P˜T 6

Likely

P˜T − O˜ T 6

1–2

8

12

8

11

0

1–6

8

8

8

8

0

2–3

4

4

5

4

0.028

2–4

12

8

4

8

1.77

3–5

4

8

6

7

0.111

4–5

4

8

4

6.67

0

5–8

4

7

8

7

0.44

6–7

12

8

4

8

1.77

7–8

6

6

5

6

0.028

Fig. 2 Network diagram for ranking technique 1

2

2

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Fig. 3 Network diagram for range technique 2

4.2 Modified Dijkstra’s Algorithm: In this case, we used the modified Dijkstra’s algorithm developed by Ravi Shankar and Sireesha’s [14] in 2010. This approach is used to find a maximal path, from starting vertices to finishing vertices in a weighted network. We started by calculating the critical path of an activity network. We determine the earliest time, the latest time and the project length time duration with three estimates using the modified Dijkstra’s algorithm. Initially, each vertex v, assigned a number between 1 and n. 1.

2.

Vertex v, gets a new temporary label, since it has not yet been permanently labelled, the result of which is provided by max[old label of i(old label of j + Di j)], where the current vertex j was permanently labelled in the previous step, and Dij is the distance between the two nodes i and j and an edge not connected then Di j = ∞. To find out the longest path, acquire the largest value of all temporary labels j and store it in a pre-eminence queue with the connected list, this will become the permanent label for that vertex.

Steps 1 and 2 are performed till the permanent label v = n is obtained. The project completed duration is indicated by this permanent label denoted as C (Tables 4, 5 and 6). The project completed duration by ranking technique 1 is 46 days. The project completed duration by ranking technique 2 is 33 days.

5 Conclusion and Discussion As we have seen from the elucidation mentioned above, both ranking techniques gives the same results in critical path. But the duration of the project completion differs. Small adjustments in ranking techniques to accomplish in less time are obvious. Here, decision-makers plays a vital part to select best alternative. This algorithm has

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Table 4 Forward pass calculation in earliest time Activity/Vertices

Earliest start time

1

2

3

4

5

6

7

8

0

0

0

0

0

0

0

0

0(C)

9

∞

∞

∞

7

∞

∞

E1 = 0

0(C)

9(C)

15

20

∞

7

∞

∞

E2 = 9

0(C)

9(C)

15

20(C)

13

7

∞

∞

E 4 = 20

0(C)

9(C)

15

20(C)

33(C)

7

∞

∞

0(C)

9(C)

15

20(C)

33(C)

7

∞

13

0(C)

9(C)

15

20(C)

33(C)

7

11

46

0(C)

9(C)

15

20(C)

33(C)

7

11

46(C)

E 5 = 33 E 8 = 46

Table 5 Backward pass calculation in latest time Activity/Vertices

Latest time

8

7

6

5

4

3

2

1

46(C)

46

46

46

46

46

46

46

46(C)

33

46

33

46

46

46

46

46(C)

33

22

33 (C)

20

26

46

46

L 5 = 33

46(C)

33

22

33(C)

20(C)

26

46

46

L 4 = 20

46(C)

33

22

33(C)

20(C)

26

9

46

46(C)

33

22

33(C)

20(C)

26

9

46

46(C)

33

22

33(C)

20(C)

26

9(C)

46

46(C)

33

22

33(C)

20(C)

26

9(C)

0(c)

Table 6 Comparison results

L 8 = E 8 = 46

L2 = 9 L1 = 0

Critical path

Ranking technique 1 (days)

Range technique 2 (days)

1–2-3–5-8

35

29

1–2-4–5-8

46

33

1–6-7–8

31

22

been used with optimization approaches, an AI-based approach, and other methods. This improved algorithm for critical path approach to determine earliest event time and latest event time in a project network is presented in this research study.

References 1 Zadeh, L.A. 1965. Fuzzy sets. Information Control 8: 338–353.

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2 Yager, R.R. 1981. A procedure for ordering fuzzy subsets of the unit interval. Information Sciences 24 (2): 143–161. 3 Dubois, Prade. D. 2000. The Handbooks of Fuzzy Sets, vol. 7. Springer. 4 Chanas, S., and P. Zielinski. 2001. Critical path analysis in the network with fuzzy activity times. Fuzzy Sets and Systems 122: 195–204. 5 Davis, Mark M., Nicholas J., Aquilano, and Richard B, Chase. 2003. Fundamental of Operation Management, 4th, McGraw—Hill, pp. 94. 6 Kuchta, D. 2001. Use of fuzzy numbers in project risk (criticality) assessment. International Journal of Project Management 19: 305–310. 7 Yao, J.S., and F.T. Lin. 2000. Fuzzy critical path method based on signed distance ranking of fuzzy numbers. In IEEE Transactions on Systems, Man, and Cybernetics Part A, pp. 3076–3082. 8 Chanas, S., and J. Kamburowski. 1981. The use of fuzzy variables in PERT. Fuzzy Sets and Systems 5: 1–19. 9 Lin, F.T., and J.S. Yao. 2003. Fuzzy critical path method based on signed-distance ranking and statistical confidence-interval estimates. Journal of Super Computing 24 (3): 305–325. 10 Noto, M., and H. Soto. 2000. A method for the shortest path search by extended Dijkstra’s algorithm. In: IEEE International Conference on Systems, Man, and Cybernetics Part B, vol. 3, pp. 2316–2320. 11 Jassbi, J., and Mohammadi S. Khan. A New Approach For Predicting Project Duration Using Beta Shape Membership Function Sand Simulation. 12 Chen, S.M., and T.H. Chang. 2001. Finding multiple possible critical paths using fuzzy PERT. In: IEEE Transactions on Systems Man, and Cybernetics, vol. 31, pp. 930–937. 13 Selva Kumari, K., and S. Santhi. 2018. A pentagonal fuzzy number in solving fuzzy sequencing problem. International Journal of Mathematics and its Applications 6 (2): 207–211. 14 Ravishankar, N., and V. Sireesha. 2010. Using modified Dijkstra’ s algorithm for critical path method in a project network. International Journal Computational Applied Mathematics 5 (2): 217–225. 15. Yager, R.R. 1980. On a general class of fuzzy connectives. Fuzzy Sets and Systems 4 (3): 235–242. 16. Sathya Geetha, S., and K. Selva Kumari. 2020. A new method for solving fuzzy transportation problem using pentagonal fuzzy number. Journal of Critical Reviews 7 (9): 171–174. 17. Nicholson, T.A.J. 2010. Finding the shortest route between the two points in a network. Computer Journal 9 (3): 275–280. 18. Suresh Babu, S., Y.L.P. Thorani, and N. Ravi Shankar. 2012. Ranking generalized fuzzy numbers using centroid of centroids. International Journal of Fuzzy logic Systems 2: 17–32. 19. Adhilakshmi, S., and N. Ravishankar. 2020. Implemented modified Dijkstra’s algorithm to find project completion time. Advances in Mathematics Scientific Journal 9 (12): 10787–10795. 20. Chen, S.H. 1985. Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems 17: 113–129.

Solving Assignment Problem Using Decision Under Uncertainty and Game Theory: A Comparative Approach with Python Coding N. Kalaivani, E. Mona Visalakshidevi, and Mostafa Ezziyyani

Abstract Assignment problem has a vital role in assigning items to the receivers such that the assignment cost is minimized. Game theory is a theoretical framework for conceiving social situations among competing players. Game theory allows the analysis of situation interdependently. Decision-making and problem-solving are used in all management functions. To attain this objective, in this paper, we consider a different approach to solve the assignment problem by using the new proposed methods like game theory, decision under uncertainty, and also some of the other prevailing methods. Symmetric and non-symmetric assignment problems, game theory and decision under uncertainty are solved using these methods. Also, to emphasize the efficiency of these proposed methods, classical solution is compared with the solution obtained from these proposed methods. In this paper, convergence behavior is discussed to increase the convergence rate of game theory and decision under uncertainty. Then, we get the nearer to the optimal solution by new approach. In this paper, symmetric and non-symmetric assignment problems are solved in two different approaches. Game theory and decision under uncertainty are compared in two tables. Also, the results are tested for the optimality using Python programming language. The graphs for the symmetric and non-symmetric problems are plotted using Python programming language. Keywords Assignment problem · Game theory · Decision under uncertainty · New proposed method · Using Python programming language N. Kalaivani (B) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, IndiaChennai e-mail: [email protected] E. M. Visalakshidevi Department of Mathematics, Misrimal Navajee Munoth Jain Engineering College, Chennai, India e-mail: [email protected] M. Ezziyyani Faculty of Sciences and Techniques of Tangier, Department of Computer Science, Abdelmalek Essa â di University, Tétouan, Morocco e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_47

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1 Introduction The assignment problem pacts in assigning [1–5] the various capitals (items) to various happenings (receivers) on a one-to-one basis in such a mode that the consequential efficacy is improved. It is one of the special cases of transportation problems [6–9]. This is especially essential in the theory of decision-making. This has a wide range of application in fields such as economics, finance, regulation, military, insurance, retail marketing, politics, conflict analysis, energy, production planning, telecommunication, psychological phenomena, and economics. In a typical assignment problem [6, 7, 10] the goal is to allocate the variable sources [8] to the ongoing activities in order to achieve the lowest cost or largest total benefits of allocation. Then, we compare the new proposed method with some of the existing methods. Also, the results are tested for the optimality using Python programming language [11].

2 Mathematical Formulation of the Assignment Problem Definition 1 Assignment Problem Assume that there are n jobs to be done and that n people are available to complete them. Let cij be the cost if the ith person is allocated to the jth work. Assume that each person can complete each job at a term with varying degree of efficiency. The problem is to find an assignment [4, 12] (which work should be allocated to which individual on a one-to-one basis) that minimize the overall cost of executing all activities. This type of problem is known as assignment problem [2, 13]. The assignment problem can be expressed as an n × n cost matrix with C real elements as shown in Table 1.

Table 1 Approach of assignment problem

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3 New Technique Used for Solving the Assignment Problem We use a new technique to solve the assignment problem [15] which is related to decision under uncertainty [13] and game theory [2] but differing from both techniques. This novel strategy offers a smooth approach to solve the assignment problem. This technique is also used to solve an issue, and the results are compared to decision under uncertainty and game theory [9]. Here, we ponder the assignment matrix where pij is the charge of passing on ith jobs to jth appliances. Proposed Method: Assignment problem is solved by the method of Hungarian [14]. Definition 2 Decision Choice Criteria/Quantitative Methods for DecisionMaking under Uncertainty/Approaches for Decision under Uncertainty There are several rules and techniques to take decisions under uncertainty situation [6, 11]. Definition 3 Maximax or Minimin (Criterion of Optimism) Working rule: (a) Locating the payoff value that corresponds [10] to each and every course of action. (b) Preference of an alternative, having the best expected payoff value which maximize the profit and minimize the loss. Definition 4 Criterion of Realism (Hurwicz Criterion) According to the Hurwicz Criterion, the measure of the decision-makers confidence as the decision [3, 15] payoff is weighted by the coefficient of optimism defined as ‘a.’ It lies between 0 and 1 (0 < α < 1): (1) If α = 1.0, the decision-maker is completely optimistic. (2) If α = 0.5, the decision-maker is entirely pessimistic.

4 Numerical Comparison of Existing Methods with the New Proposed Method Problem 1 Our aim is solving the following assignment problem using the new proposed method. Contemplate the problem of passing on five jobs to five persons. The task charges are specified as follows:

A person B C D E

1 8 0 3 4 9

job 2 4 9 8 3 5

3 2 5 9 1 8

4 6 5 2 0 9

5 1 4 6 3 5

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Determine the optimum assignment schedule. Solution: The optimum assignment is given by A → 5, B → 1, C → 4, D → 3, E → 2.The optimum assignment cost = (1 + 0 + 2 + 1 + 5) cost units = 9 units of cost. Problem 2 Solving the problem with decision under certainty types: job 12 345 A84 261 person B 0 9 5 5 4 C 38 926 D43 103 E 95 895 Find (i) Laplace criterion, (ii) criterion of optimism, (iii) criterion of pessimism, (iv) minimax regret criterion, and (v) Hurwicz criterion take α = 0.5 Solution: Different type of matrix to solve. ⎞ ⎛ ⎞⎛ Expected payoff 12345 ⎟ ⎟⎜ 4.2 A⎜ ⎟ ⎜ 8 4 2 6 1 ⎟⎜ ⎟ ⎜ ⎟⎜ 4.6 B ⎜ 0 9 5 5 4 ⎟⎜ ⎟ (i) Laplace criterion: ⎜ ⎟ Payoff E = 7.2, Cost ⎟⎜ ⎟ ⎜ ⎜ ⎟ 5.6 C 38926 ⎟ ⎜ ⎟⎜ ∗∗ ⎠ D ⎝ 4 3 1 0 3 ⎠⎝ (2.2) E 95895 (7.2)∗ D = 2.2 (ii) Criterion of optimism ⎛ ⎞⎛ ⎞ 8 84261 ⎜ 0 9 5 5 4 ⎟⎜ 9 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ (a) To find Maximax criterion: ⎜ 3 8 9 2 6 ⎟⎜ 9 ⎟ Maximax value is B = ⎜ ⎟⎜ ⎟ ⎝ 4 3 1 0 3 ⎠⎝ 4 ⎠ 9 95895 9. That is payoff matrix. ⎛ ⎞⎛ ⎞ 1 84261 ⎜ 0 9 5 5 4 ⎟⎜ 0 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ (b) To find Minimin criterion: ⎜ 3 8 9 2 6 ⎟⎜ 2 ⎟ Minimin value is D = ⎜ ⎟⎜ ⎟ ⎝ 4 3 1 0 3 ⎠⎝ 0 ⎠ 5 95895 0. That is cost matrix. (iii)

Criterion of pessimism:

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⎞⎛ ⎞ 1 84261 ⎜ 0 9 5 5 4 ⎟⎜ 0 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ To find maximin criterion-payoff matrix: ⎜ 3 8 9 2 6 ⎟⎜ 2 ⎟ Maximin ⎜ ⎟⎜ ⎟ ⎝ 4 3 1 0 3 ⎠⎝ 0 ⎠ 5 95895 value is E = 5. ⎛ ⎞⎛ ⎞ 8 84261 ⎜ 0 9 5 5 4 ⎟⎜ 9 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ To find Minimax criterion-cost matrix: ⎜ 3 8 9 2 6 ⎟⎜ 9 ⎟ Minimax ⎜ ⎟⎜ ⎟ ⎝ 4 3 1 0 3 ⎠⎝ 4 ⎠ 9 95895 value is D = 4 ⎛

(a)

(b)

(iv)

Minimax regret criterion: (a)

To find maximization problem: ⎛

1 A⎜ ⎜8 ⎜ B ⎜0 ⎜ C ⎜3 ⎜ D⎝4 E 9

(b)

2 4 9 8 3 5

3 2 5 9 1 8

4 6 5 2 0 9

⎞ ⎛ 5 12 1⎟ A⎜ ⎟ ⎜1 5 ⎟ ⎜ 4⎟ B ⎜9 0 ⎟ R.t = ⎜ 6⎟ C ⎜6 1 ⎟ ⎜ ⎠ 3 D⎝5 6 5 E 04

1 A⎜ ⎜8 ⎜ B ⎜0 ⎜ C ⎜3 ⎜ D⎝4 E 9

2 4 9 8 3 5

3 2 5 9 1 8

4 6 5 2 0 9

⎞ ⎛ 5 12 1⎟ A⎜ ⎟ ⎜8 1 ⎟ ⎜ 4⎟ B ⎜0 6 ⎟ R.t = ⎜ 6⎟ C ⎜3 5 ⎟ ⎜ 3⎠ D⎝4 0 5 E 92

Minimum value is D = 4. Hurwicz criterion: (a)

4 3 4 7 9 0

⎞ ⎞⎛ max 5 ⎜ ⎟ 5⎟ ⎟⎜ 7 ⎟ ⎟ ⎟⎜ 2 ⎟⎜ 9 ⎟ ⎟ ⎟⎜ 0 ⎟⎜ 7 ⎟ ⎟ ⎟⎜ 3 ⎠⎝ 9 ⎠ 4 1

Minimum value is E = 4. To find minimization problem: ⎛

(v)

3 7 4 0 8 1

Maximization problem: Payoff matrix

3 1 4 8 0 7

4 6 5 2 0 9

⎞ ⎞⎛ max .value 5 ⎜ ⎟ 8 0⎟ ⎟ ⎟⎜ ⎟ ⎟⎜ 6 3 ⎟⎜ ⎟ ⎟ ⎟⎜ ⎜ ⎟ ⎟ 8 5 ⎟ ⎟⎜ ⎠ 4 2 ⎠⎝ 9 4

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⎛

1 A⎜ ⎜8 ⎜ B ⎜0 ⎜ C ⎜3 ⎜ D⎝4 E 9

(b)

2 4 9 8 3 5

3 2 5 9 1 8

4 6 5 2 0 9

⎛ ⎛ ⎞⎞⎞ ⎞⎛ max min Weighted outcome 5 ⎜ ⎜ ⎜ ⎟⎟⎟ 4.5 1⎟ ⎟⎟⎟ ⎟⎜ 8 ⎜ 1 ⎜ ⎜ ⎜ ⎟⎟⎟ ⎟⎜ 4.5 4 ⎟⎜ 9 ⎜ 0 ⎜ ⎟⎟⎟ ⎜ ⎜ ⎟⎟⎟. ⎟⎜ ⎟⎟⎟ 5.5 6 ⎟⎜ 9 ⎜ 2 ⎜ ⎜ ⎜ ⎟⎟⎟ ⎟⎜ ⎠⎠⎠ 2 3 ⎠⎝ 4 ⎝ 0 ⎝ 9 5 7 5

Max. value is C = 5.5 Minimization problem: Cost Matrix: ⎛ ⎛ ⎛ ⎞⎞⎞ ⎞⎛ 12345 min max Weighted outcome ⎜ ⎜ ⎟⎟⎟ ⎟⎜ A⎜ 4.5 ⎜ 8 4 2 6 1 ⎟⎜ 1 ⎜ 8 ⎜ ⎟⎟⎟ ⎜ ⎜ ⎜ ⎟⎟⎟ ⎟⎜ B ⎜ 0 9 5 5 4 ⎟⎜ 0 ⎜ 9 ⎜ 4.5 ⎟⎟⎟ ⎜ ⎜ ⎜ ⎟⎟⎟. ⎟⎜ ⎟⎟⎟ C ⎜ 3 8 9 2 6 ⎟⎜ 2 ⎜ 9 ⎜ 5.5 ⎜ ⎜ ⎜ ⎟⎟⎟ ⎟⎜ ⎝ ⎝ ⎝ ⎝ ⎠⎠⎠ ⎠ D 43103 0 4 2 E 95895 5 9 7 Min. value is D = 2.

Problem 3 Solving the following problem with game theory using dominance rule.

A person B C D E

1 8 0 3 4 9

job 2 4 9 8 3 5

3 2 5 9 1 8

4 6 5 2 0 9

5 1 4 6 3 5

Solution: To find the row min and col max in the given problem. 1 2 3 4 5 Rowmin ⎞⎛ ⎞ A 84261 1 ⎟⎜ ⎟ B⎜ ⎜ 0 9 5 5 4 ⎟⎜ 0 ⎟  ⎜ ⎟⎜ ⎟ C ⎜ 3 8 9 2 6 ⎟⎜ 2 ⎟ColMax 9 9 9 9 6 ⎜ ⎟⎜ ⎟ D ⎝ 4 3 1 0 3 ⎠⎝ 0 ⎠ E 95895 5 ⎛

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Step 1: The fifth row dominates on the first row, i.e., elements of fifth row are greater⎛ than first ⎞row. So, eliminating the first row, we get the following 12345 ⎜ B ⎜0 9 5 5 4⎟ ⎟ ⎟ ⎜ matrix. C ⎜ 3 8 9 2 6 ⎟. ⎟ ⎜ D⎝4 3 1 0 3⎠ E 95895 Step2: Now the reduced payoff matrix is given as follows: Step 3: Make initial condition C 45 D



26 95



Then, the value of the game is = 5.5. In this paper, a novel recommended strategy for tracking the assignment problem is presented in addition an example employing the suggested approach, and two current ways are reviewed. The optimal solution for the two methods are compared, with the ideal solution of the assignment problem.

5 Problem Based on Non-Symmetric Matrix Problem 4 Finding the solution of the Assignment problem using Hungarian Work\Job 1 2 3 A 9 26 15 method-1 (MINcase) B 13 27 6 . C 35 20 15 D 18 30 20 Solution: The optimum assignment is given by A → 1, B → 3, C → 2, D → 4. The optimum assignment cost = (9 + 6 + 20 + 0) cost units = 35 units of cost. Problem 5 Solving the decision under certainty types: Work\Job 1 2 3 A 9 26 15 B 13 27 6 C 35 20 15 D 18 30 20 Finding (i) Laplace criterion, (ii) criterion of optimism, (iii) criterion of pessimism, (iv) minimax regret criterion, and (v) Hurwicz criterion take α = 0.5. Solution:

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(i)

The succeeding matrix stretches the payoff of unlike tactics (alternativeswork) A, B, C, D,⎛ E counter to conditions (Events-job) 1, 2, 3, 4 ⎞ Work\Job 1 2 3 Expected payoff ⎟ 16.6 A 9 26 15 ⎜ ⎟ ⎜ ⎟ ⎜ (15.3)∗∗ B 13 27 6 ⎜ ⎟ Payoff E = 23.3, Cost D = 15.3 ⎟ ⎜ ⎠ (23.3)∗ C 35 20 15 ⎝ 22.6 D 18 30 20 Criterion of optimism: ⎛ ⎞⎛ ⎞ 9 26 15 26 ⎜ 13 27 6 ⎟⎜ 27 ⎟ ⎟⎜ ⎟ (a) To find maximax criterion: payoff matrix: ⎜ ⎝ 35 20 15 ⎠⎝ 35 ⎠

(ii)

18 30 20 30 ⎞⎛ ⎞ 9 26 15 9 ⎜ 13 27 6 ⎟⎜ 6 ⎟ ⎟⎜ ⎟ To find minimin criterion: cost matrix: ⎜ ⎝ 35 20 15 ⎠⎝ 15 ⎠ Minimin 18 30 20 18 value is B = 6. Maximax value is C = 35.

(b)

(iii)

⎛

Criterion of pessimism:

(a)

⎛

To find maximin criterion-payoff matrix Maximin value is D = 18

(b)

(iv)

⎛

9 26 ⎜ 13 27 ⎜ To find minimax criterion-cost matrix: ⎝ 35 20 18 30 20 value is A = 26

Minimax regret criterion

(a)

⎞⎛ ⎞ 9 26 15 9 ⎜ 13 27 6 ⎟⎜ 6 ⎟ ⎜ ⎟⎜ ⎟ ⎝ 35 20 15 ⎠⎝ 15 ⎠: 18 30 20 18 ⎞⎛ ⎞ 15 26 ⎜ 27 ⎟ 6 ⎟ ⎟⎜ ⎟ Minimax 15 ⎠⎝ 35 ⎠

To ⎛

26 ⎜ 22 ⎜ ⎝0 17

find 4 3 10 0

maximization

problem:

⎛ A 9 B⎜ ⎜ 13 C ⎝ 35 D 18

⎞⎛ ⎞ 5 26 ⎜ 22 ⎟ 14 ⎟ ⎟⎜ ⎟ Minimum value is = 10 5 ⎠⎝ 10 ⎠ 0 17

30

26 27 20 30

⎞ 15 6 ⎟ ⎟ R.t 15 ⎠ 20

=

Solving Assignment Problem Using Decision …

(b)

To ⎛

0 ⎜4 ⎜ ⎝ 26 9 (v)

A B C D

find 9 7 0 10

minimization

479

problem:

⎛ A 9 B⎜ ⎜ 13 C ⎝ 35 D 18

⎞⎛ ⎞ 9 9 ⎜ 7 ⎟ 0 ⎟ ⎟⎜ ⎟: Minimum value is B = 7 9 ⎠⎝ 26 ⎠ 14 14

26 27 20 30

⎞ 15 6 ⎟ ⎟ R.t 15 ⎠ 20

=

Hurwicz criterion: (a) Maxproblem: payoff matrix (b) min. problem: cost matrix

1 9 13 35 18

2 26 27 20 30

⎞ ⎞⎛ ⎞⎛ ⎞⎛ ⎛ max weigh outcome min min weigh outcome 3 max ⎟ ⎟⎜ 9 ⎟⎜ 26 ⎟⎜ 17.5 17.5 15 ⎜ ⎟ ⎟⎜ ⎟⎜ ⎜ 26 ⎟⎜ 9 ⎟ ⎟⎜ ⎟⎜ ⎟⎜ ⎜ 16.5 16.5 6 ⎜ 27 ⎟⎜ 6 ⎟. ⎟⎜ 6 ⎟⎜ 27 ⎟ ⎟⎜ ⎟⎜ ⎟⎜ ⎜ ⎠ ⎠⎝ 15 ⎠⎝ 35 25 25 15 ⎝ 35 ⎠⎝ 15 30 24 18 18 24 20 30

Max. value is C = 25 and min. value is B = 16.5 Problem 6 Solving the Work\Job 1 2 3 A 9 26 15 B 13 27 6 . C 35 20 15 D 18 30 20

problem

game

theory

using

dominance

rule

Solution: To find ⎛the row⎞ min and col max in the given problem. Work\Job 1 2 3 rowmin ⎟ A 9 26 15 ⎜ ⎜9 ⎟ ⎜ ⎟ B 13 27 6 ⎜ 6 ⎟ 35 30 20 . ⎜ ⎟ ⎠ C 35 20 15 ⎝ 15 D 18 30 20 18 The fifth row dominates on the first row, i.e., elements of fifth row are greater than first row. So, eliminating the first row, we get the following matrix. The fifth column is less than or equal to second column, i.e., fifth column dominates over second column. Hence, delete the dominated second column. Now, the reduced payoff matrix is given 1 3 as follows: Make initial condition C 35 15 Then, the value of the game is = 19.5. D 18 20 In this paper, a novel recommended strategy for tracking the assignment problem is presented in addition an example employing the suggested approach and two current

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ways are reviewed and the optimal solution for the two methods are compared, with the ideal solution of the assignment problem. (Tables 2 and 3). Hence, we complete that this new proposed technique is more effective for answering assignment problem.

6 Numerical Simulation—Using Python Coding By the above table, we get the result of Symmetric and non-symmetric problem: The parameter values are taken as 9 costs, 8 cost, 3 costs as assignment problem, (9.4, 6.05, 6), (9, 6, 6), (9, 5, 6), (8, 4, 4), (9, 8, 6), decision under uncertainty, and (5.5, 2, 3)Game theory in graph (i). The parameter values are taken as 35 costs, 12 cost, 6 costs as assignment problem, (38.6, 11.5, 8), (41, 11, 8), (44, 13, 10), (17, 4, 6), (36.5, 12, 8.5), decision under uncertainty and (19.5, 4, 4), Game theory in Graph (ii). In a comparative study, involving several problems, when resolved using ‘Decision under uncertainty and Game Theory,’ the resulted values are depicted in the following line chart in graph (i) and graph (ii) (Fig. 1).

7 Conclusion In this paper, we proved a novel approach to solve the assignment problem. We first explain the newly proposed approach and demonstrated its effect using a numerical sample. The optimum solution is then obtained which is learned from the optimal solution of decision under uncertainty in addition to game theory. As a result, the strategy learned can be used to solve any assignment problem.

3

2

1

S.No

Total : 8 costs

D3352

B→1

Total : 3 Costs

A25

B 41

A → 2,

D → 2,

C 2140

12

C → 1,

B 5624

A → 3,

B → 2,

1234

Total : 9 costs

E →2

D → 3,

C → 4,

B → 1,

A → 5,

Assignment problem

A 3542

E 95895

D43103

C 38926

B 09554

A 84261

12345

Example Criterion of optimism

Payoff: 3.5 Payoff: 5 Cost: 2.5 Cost: 1

Payoff: 4.3 Payoff: 6 Cost: 1.75 Cost: 0

Payoff: 7.2 Payoff: 9 Cost: 2.2 Cost: 0

Laplace Criterion

Decision under uncertainty

Payoff: 5 Cost: 1

Payoff: 5 Cost: 0

Payoff: 5 Cost: 4

Criterion of pessimisim

Table 2 Comparison of optimal values of three methods using symmetric matrix

Payoff: 2 Cost: 2

Payoff: 2 Cost: 2

Payoff: 4 Cost: 4

Minimax Regret criterion

Payoff: 3.5 Cost: 2.5

Payoff: 6 Cost: 2

Payoff: 7 Cost: 2

Hurwicz criterion

V=3

V=2

V = 5.5

Game theory

3

8

9

Optimum

Solving Assignment Problem Using Decision … 481

3

2

1

S. No

A → 9,

B → 6,

w\J 1 2 3

9 26 15

A → 2,

A → 1,

B → 5,

C →0

Total : 6 Costs

1234

B 5645

C 7203

Total : 12

D → 0,

C → 5,

B → 5,

A1734

C 7535

B 8945

A8628

1234

D 18 30 20

Total : 35

D → 0,

35 20 15

C

C → 20,

B 13 27 6

A

Assignment problem

Example

Payoff: 35 Cost: 6

Criterion of optimism

Payoff: 5 Cost: 3

Payoff: 7 Cost: 1

Payoff: 6.5 Payoff: 9 Cost: 5 Cost: 2

Payoff: 23.3 Cost: 15.3

Laplace Criterion

Decision Under Uncertainty

Payoff: 4 Cost: 6

Payoff: 4 Cost: 9

Payoff: 18 Cost: 26

Criterion of Pessimism

Table 3 Comparison of optimal values of three methods using a non-symmetric matrix

Payoff: 2 Cost: 4

Payoff: 3 Cost: 1

Payoff: 10 Cost: 7

Minimax Regret criterion

Payoff: 5 Cost: 3.5

Payoff: 6.5 Cost: 5

V=4

V=4

Payoff: 20 V = 19.5 Cost: 16.5

Hurwicz criterion

Game theory

6

12

35

Optimum

482 N. Kalaivani et al.

483

Y-axis persons

Solving Assignment Problem Using Decision …

Graph(i)

Jobs

Graph(ii)

Fig. 1 Comparison of optimal values in line chart: graph (i) and graph (ii)

References 1. Basirzadeh, Hadi. 2012. One’s assignment method for solving assignment problems. Applied Mathematical Sciences 6 (47): 2345–2355. 2. HumayraDilAfroz, and Dr. Mohammad Anwar Hossen. 2017. New proposed method for solving assignment problem and comparative study with the existing methods. Journal of Mathematics (IOSR-JM), 13 (2), 84–88. 3. Rai, Neha, Khushbu Rai, and A.J. Khan. 2017. New Approach to Solve Assignment Problem. International Journal of Innovative Science and Research Technology 2 (10). 4. Seethalakshmy, A., and N. Srinivasan. 2016. A new methodology for solving a maximization assignment problem. International Journal of Latest Research in Science and Technology 5 (6): 10–13. 5. Shweta Singh, G.C., and Rajesh Shrivastava Dubey. 2012. Comparative analysis of Assignment problem. IOSR Journal of Engineering 2 (8): 1–15. 6. Goel, B.S., and S.K. Mittal. 1982. Operations Research. Fifty. Meerut, India: PragatiPrakashan. 7. Taha, Hamdy A. 2017. Operations Research: An Introduction, 10th ed. New Jersey: Pearson Prentice Hall. 8. Kumar, A. Ramesh., and S. Deepa. 2015. An application of the assignment problems. International Journal of Physical and Social Sciences 5 (5): 183. 9. Mishra, Shraddha. 2017. Solving transportation problem by various methods and their comparison. International Journal of Mathematics Trends and Technology 44 (4): 270–275. 10. Sharma, K. 2009. Operations Research—Theory and Applications, MacMillan India. 11. Votaw, D.F., and A. Orden. 1952. The personel assignment problem, Symposium on Linear Inequalities and Programming, SCOOP 10, USA ir Force, pp. 15512. Rao Sambasiva, S., and Maruthi Srinivas. 2016. An effective algorithm to solve assignment problems: Opportunity cost approach. International Journal of Mathematics and Scientific Computing 6 (1): 48-50. 13. Ramesh Kumar, A., and S. Deepa. 2016. Solving one’s interval linear assignment problem. International Journal of Engineering Research and Application 6 (10), 69–75. 14. Gupta, Prem Kumar, and D.S. Hira. 1999. Operations Research, An Introduction. New Delhi: S. Chand and Co., Ltd. 15. Kuhn, H.W. 1955. The Hungarian method for the assignment problem. Naval Research Logistics, 2 (1–2), 83-97 (Wiley online Library).

Interval-Valued Fuzzy Dynamic Programming Approach of Capital Budgeting Problem M. Ruthara and G. Uthra

Abstract In this research article, an optimal solution to capital budgeting problems in interval-valued fuzzy environment is obtained. The problem is considered as a multi-stage decision problem and hence solved by dynamic programming approach. The revenue generated is represented as IvTrFN. Recursive equations in the forward and backward directions are formulated in interval-valued fuzzy case. Both recursive equations produce the same optimal solution, as shown in numerical example. Keywords Interval-valued Fuzzy Set · Fuzzy Dynamic Programming · Interval-valued Trapezoidal Fuzzy Number

1 Introduction One of the most important mathematical tools for dealing with multi-stage optimisation issues is fuzzy dynamic programming (FDP). A multi-stage decision-making challenge can be broken down into several stages. Interval-valued fuzzy sets (IVFS), a generalization of Zadeh’s [1] fuzzy sets, are now used in a variety of optimization and decision-making issues involving ambiguous parameters. Bellman and Zadeh’s [2] study on decision-making in a fuzzy environment was eye-opening. All human activities include numerous stages of decision-making. DP is a valuable tool for addressing these issues. However, in the analysis of multi-stage decision issues, several elements such as imprecision, ambiguity, and vagueness exist. Because fuzzy notions are used in DP, the result is referred to as fuzzy dynamic programming (FDP). Many authors were drawn to FDP, and as a result, they contributed significantly to the field’s literature [3–5]. Both Kalprzyk [6, 7] and Esogbue et al. [8] examined improvements in FDP theory and applications. In multi-criteria decision analysis, Abo-Sinna et al. [9] and Hussein [10] used a fuzzy dynamic technique. Nagalakshmi and Uthra [11] used the DP technique to find the best answer to a fuzzy M. Ruthara (B) · G. Uthra P.G. and Research Department of Mathematics, Pachaiyappa’s College, Chennai, Tamil Nadu 600030, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_48

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capital budgeting problem. Atanassov [12] developed the concept of intuitionistic fuzzy sets as an extension of Zadeh’s fuzzy sets (IFS). Shu et al. [13] employed IFS in fault tree analysis on printed circuit board assembly. Wei [14, 15] investigated the use of intuitionistic fuzzy numbers in communal decision-making. Interval-valued fuzzy sets were utilized by Zadeh et al. [16] to supplement the GRA technique for MCDM. Deschrijver and Kerre [17] studied binary aggregation operators in fuzzy set theory. In critical path analysis, D. Stephen Dinagar and D. Abirami used L–R type and interval-valued fuzzy numbers [18, 19]. In this paper, a capital budgeting problem in interval-valued fuzzy environment is considered. The problem is taken as a multi-stage decision problem and hence solved by dynamic programming approach. The revenue of the capital budgeting problem is represented as IvTrFN. The recursive equations are formulated for interval-valued fuzzy case. The optimal solution is obtained by both forward and backward recursion.

2 Preliminaries and Definition 2.1 Fuzzy Set Suppose, A a classical set μ A (x) be a function from A →[0, 1]. A μ A (x) is determined by A = fuzzy set  A with membership function   x, μ A (x) : x ∈ A and μ A (x) ∈ [0, 1] .

2.2 Intuitionistic Fuzzy Set Suppose, X the universe of discourse, then an intuitionistic fuzzy set (IFS) A˜ I in X is provided by A˜ I = {(x, μ A (x), ν A (x))/x ∈ X } , where the functions μ A (x) : X → [0, 1] and ν A (x) → [0, 1] determine the degree of membership and non-membership of the element x ∈ X , respectively, and for every x ∈ X, 0 ≤ μ A (x) + ν A (x) ≤ 1.

2.3 Trapezoidal Fuzzy Number A trapezoidal fuzzy number is designated as four tuples A = (a1 , a2 , a3 , a4 ), where a1 , a2 , a3 and a4 are real numbers and a1 ≤ a2 ≤ a3 ≤ a4 with membership function defined as

Interval-Valued Fuzzy Dynamic Programming …

487

0.1

0

11

12

13

14

15

16

17

18

Fig. 1 Interval-valued Trapezoidal Fuzzy Number

⎧ x−a1 ⎪ ⎪ a −a ⎪ ⎨ 2 1 1 μ A (x) = a4 −x ⎪ a −a ⎪ ⎪ ⎩ 4 3 0

for a1 ≤ x ≤ a2 for a2 ≤ x ≤ a3 for a3 ≤ x ≤ a_4 otherwise

2.4 Interval-Valued Trapezoidal Fuzzy Number (IvTrFN) On R, an interval-valued fuzzy number A˜ is equal to

 β A˜ = x, μαA (x), μ A (x) , x ∈ R β

and μαA (x) ≤μ A (x) for  all x ∈ R.    β β β β α ˜β ˜ ˜ Let A = A , A , where A˜ α = a1α , a2α , a3α , a4α and A˜ β = a1 , a2 , a3 , a4 β

β

β

β

are the trapezoidal fuzzy numbers such that a1 ≤ a1α , a2 ≤ a2α , a3α ≤ a3 , a4α ≤ a4 (Fig. 1).

2.5 Properties of (IVTrFN) 1.

  β β β β  Addition of two IVTrFN. A˜ = a1α , a2α , a3α , a4α , a1 , a2 , a3 , a4   β β β β B˜ = b1α , b2α , b3α , b4α b1 , b2 , b3 , b4

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2.

  A˜ + B˜ = a1α + b1α , a2α + b2α , a3α + b3α , a4α + b4α  β β β β β β β β a1 + b1 , a2 + b2 , a3 + b3 , a4 + b4

Subtraction of two IVTrFN.    A˜ − B˜ = a1α − b4α , a2α − b3α , a3α − b2α , a4α − b1α  β β β β β β β β a1 − b4 , a2 − b3 , a3 − b2 , a4 − b1

2.6 Ranking Method of IvTrFN For every IvTrFN   β β β β  A˜ = a1α , a2α , a3α , a4α , a1 , a2 , a3 , a4 , we use the following ranking function    r A˜ = r A =



β

β

β

β

a1α + a2α + a3α + a4α + a1 + a2 + a3 + a4 8

 (1)

3 DP Approach of Capital Budgeting Problem 3.1 Capital Budgeting Problem To maximize investment revenue, the capital budgeting challenge comprises dividing financial resources across different projects. C0 Units of (amount of money) can be invested in N stages, with the entire revenue of each stage exceeding the previous stage’s total revenue. Let pi be the amount of money set aside for step i. Let’s call the alternatives di . Let Mi (di ) be the estimated revenue from the alternative di allocation at stage i. Let f i ( pi ) represent the optimal revenue of states 1, 2, …, i, and let pi represent the specified states. Then, the LPP model is Maximize Z = M1 (d1 ) + M2 (d2 ) + M3 (d3 ) + · · · + M N (d N )

(2)

Subject to the constraints, d1 + d2 + d3 + ... + d N = C0 and di ≥ 0, i = 1, 2, 3, . . . , N

(3)

Interval-Valued Fuzzy Dynamic Programming …

489

The DP model’s primary characteristics are as follows: stages, moods within each stage, and decision choices inside each step are the three sorts of phases. Both forward and backward recursive equations can be used to solve the problem. Forward recursive equation (FRE) computations start at the beginning and end at the end. Backward recursive equation (BRE) computations are similar in that they start at the end and end at the beginning. FRE The DP model, as presented by 1. 2. 3. 4. 5.

Plant i represents stage i where i = 1, 2, …, N. The amount of money allocated to stages 1, 2, …, i is state pi at stage i and pi = 1, 2, . . . , C0 for i = 1, 2, 3, . . . , N − 1 Let di be the alternatives Let Mi (di ) be the anticipated revenue from allocating the option di at step i. Let f i ( pi ) represent the optimal revenue of states 1, 2, …, i, and pi represent the current state. As a result, the FRE is written as f 1 ( p1 ) = Max{M1 (d1 )} where c1 (d1 ) ≤ p1

(4)

f i ( pi ) = Max{Mi (di ) + f i−1 ( pi − ci (di ))}, where ci (di ) ≤ pi and i = 2, 3, . . . , N (5) BRE The DP model is given by 1. 2. 3. 4. 5.

Plant I represents stage I where i = 1, 2, …, N. The amount of money assigned to stages i, i + 1, …, N is represented by state yi at stage i and yi = 0, 1, . . . , C0 for i = 2, 3, . . . , N Let di be the alternatives. Let Mi (di ) be the estimated revenue from the alternative di allocation at stage i. Let f i (yi ) be the optimal revenue of states i, i + 1, . . . , N and yi be the given state. The BRE is thus given as f N (y N ) = Max{M N (d N )} where c N (d N ) ≤ y N

(6)

f i (yi ) = Max{Mi (di ) + f i+1 (yi − ci (d))}, where ci (di ) ≤ yi and i = 1, 2, , . . . , N − 1

(7)

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Interval-Valued Fuzzy Capital Budgeting Problems: In an interval-valued fuzzy environment, the DP is given by Maximize Z = M˜ 1 (d1 ) + M˜ 2 (d2 ) + · · · + M˜ N (d N ) Subject to the constraints, d1 + d2 + d3 + · · · + d N = C˜ and di ≥ 0, i = 1, 2, . . . , N

(8)

(9)

where M˜ i (di ) be the interval-valued fuzzy expected revenue from the allocation of the alternative di at the state i. Let’s call the possibilities di . The three elements of the interval-valued FDP model are the same case followed in the crisp case. As a result, the fuzzy FRE with interval values are presented as

 f 1 ( p1 ) = Max M˜ 1 (d1 ) where c1 (d1 ) ≤ p1

f˜i ( pi ) = Max M˜ i (di ) + f˜i−1 ( pi − ci (di )) , where ci (di ) ≤ pi and i = 2, 3, . . . , N

(10)

(11)

where f˜i ( pi ) is the optimal interval-valued fuzzy revenue at stages 1, 2, . . . , i given pi . Similarly the interval-valued fuzzy BRE are given by

f˜N (y N ) = Max M˜ N (d N ) where c N (d N ) ≤ y N

f˜i (yi ) = Max M˜ i (di ) + f˜i+1 (yi − ci (di )) , where ci (di ) ≤ yi and i = 1, 2, 3, . . . , N − 1

(12)

(13)

where f˜i (yi ) is the interval-valued optimal fuzzy revenue at stages i, i + 1, . . . , N given yi .

4 Illustrative Example A company has set aside $5 million for each  of its three plants. Each plant must submit its ideas, detailing the total cost c j , and total income (M j ) for each. The costs and revenue are represented in the following Table 1. There is a possibility that funds may not be allocated to individual plants, and hence, zero cost proposals can

Interval-Valued Fuzzy Dynamic Programming …

491

Table 1 Cost and revenue representation Proposal

Plant 1

Plant 2

Plant 3

c1

M1

c2

M2

c3

M3

1

0

(3, 5, 1, 4) (2, 4, 3, 6)

0

(4, 8, 2, 6) (3, 6, 4, 9)

0

(4, 6, 2, 5) (3, 5, 4, 7)

2

1

(7, 10, 5, 9) (6, 9, 7, 11)

2

(7, 11, 7, 10) (8, 10, 9, 12)

1

(8, 10, 5, 9) (6, 9, 8, 12)

3

2

(5, 9, 3, 6) (4, 6, 5, 10)

3

(4, 6, 2, 5) (3, 5, 4, 7)

–

–

4

–

–

4

(8, 10, 5, 9) (6, 9, 8, 11)

–

–

be introduced. The objective of the business firm is the maximization of total revenue obtained by allocating 5 million to the three plants. Solution by Interval-valued Fuzzy FRE The components of the IVFDP model are as follows (Table 2): Interval-valued Fuzzy FRE Stage 1: f˜1 ( p1 ) = Max{M(d1 )} where c1 (d1 ) ≤ p1 (Table 3). Table 2 Components of IVFDP model Stages

Stages

Alternative

Plant 1 is the first stage

p1 = Amount of funds earmarked for Stage 1

d1 = 1, 2, 3

Plant 2 is the second stage

p2 = Amounts set aside for Stages 1 and 2

d2 = 1, 2, 3

Plant 3 is the third stage

p = Amounts allotted to Stages 1, 2, and 3

d3 = 1, 2, 3

Table 3 Interval-valued fuzzy FRE (Stage 1) p1

d1 = 1

d1 = 2

d1 = 3

f˜1 ( p1 )

d1∗

0

(3, 5, 1, 4) (2, 4, 3, 6)

–

–

(3, 5, 1, 4) (2, 4, 3, 6)

1

1

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

–

(7, 10, 5, 9) (6, 9, 7, 11)

2

2

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

2

3

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

2

4

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

2

5

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

2

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M. Ruthara and G. Uthra

Table 4 Interval-valued fuzzy FRE (Stage 2) p2

d2 = 1

d2 = 2

d2 = 3

d2 = 4

f˜2 ( p2 )

d2∗

0

(7, 11, 3, 10) (5, 10, 7, 15)

–

–

–

(7, 11, 3, 10) (5, 10, 7, 15)

1

1

(11, 18, 7, 15) (9, 15, 11, 20)

–

–

–

(11, 18, 7, 15) (7, 11, 15, 20)

1

2

(11, 18, 7, 15) (9, 15, 11, 20)

(10, 16, 8, 14) (10, 14, 13, 18)

–

–

(11, 18, 7, 15) (9, 15, 11, 20)

1

3

(11, 18, 7, 15) (9, 15, 11, 20)

(14, 21, 12, 19) (14, 19, 16, 23)

(7, 11, 3, 9) (5, 9, 7, 13)

–

(14, 21, 12, 19) (14, 19, 16, 23)

2

4

(9, 11, 15, 21) (7, 11, 15, 20)

(14, 21, 12, 19) (14, 19, 16, 23)

(11, 16, 7, 14) (9, 14, 11, 18)

(11, 15, 6, 13) (8, 13, 11, 18)

(14, 21, 12, 19) (14, 19, 16, 23)

2

5

(9, 11, 15, 21) (7, 11, 15, 20)

(14, 21, 12, 19) (14, 19, 16, 23)

(11, 16, 7, 14) (9, 14, 11, 18)

(15, 20, 10, 18) (12, 18, 15, 23)

(14, 21, 12, 19) (14, 19, 16, 23)

2

Table 5 Interval-valued fuzzy FRE (Stage 3) p3

d3 = 1

d3 = 2

f˜3 ( p3 )

d3∗

5

(18, 27, 14, 24) (17, 24, 20, 30)

(22, 31, 17, 28) (20, 28, 24, 35)

(22, 31, 17, 28) (20, 28, 24, 35)

2



Stage 2: f˜2 ( p2 ) = Max M˜ 2 (d2 ) +  f 2 ( p2 − c2 (d2 )) where c2 (d2 ) ≤ p2 (Table 4).

Stage 3: f˜3 ( p3 ) = Max M˜ 3 (d3 ) +  f 2 ( p3 − c3 (d3 )) where c3 (d3 ) ≤ p3 (Table 5). The total (optimal) value is (22, 31, 17, 28) (20, 28, 24, 35) and the corresponding optimal solution is given in Table 6.  ∗ ∗ ∗ d1 , d2 , d3 = (2, 2, 2) Applying the ranking of IVTrFN, we have  22 + 31 + 17 + 28 + 20 + 28 + 24 + 35 205 r Aˆ  = = 8 8 = 25.625 units Table 6 Corresponding optimal solution   p3 d3∗ p2 = p3 − c3 d3∗ 5

2

5−2=3

d2∗

  p1 = p2 − c2 d2∗

d1∗

2

4−2=2

2

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493

Table 7 Interval-valued fuzzy BRE (Stage 3) p3

d3 = 1

d3 = 2

d3 = 3

f˜3 ( p3 )

d3∗

0

(5, 9, 3, 6) (4, 6, 5, 10)

–

–

(5, 9, 3, 6) (4, 6, 5, 10)

1

1

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

–

(7, 10, 5, 9) (6, 9, 7, 11)

2

2

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

2

3

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

2

4

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

2

5

(5, 9, 3, 6) (4, 6, 5, 10)

(7, 10, 5, 9) (6, 9, 7, 11)

(3, 5, 1, 4) (2, 4, 3, 6)

(7, 10, 5, 9) (6, 9, 7, 11)

2

Solution by Interval-valued Fuzzy BRE

Stage 3: f˜3 ( p3 ) = Max M˜ 3 (d3 ) + f˜2 ( p3 − c3 (d3 )) where c3 (d3 ) ≤ p3 (Table 7).

Stage 2: f˜2 ( p2 ) = Max M˜ 2 (d2 ) + f˜2 ( p2 − c2 (d2 )) where c2 (d2 ) ≤ p2 (Table 8).

Stage 3: f˜1 ( p1 ) = Max M˜ 1 (d1 ) where c1 (d1 ) ≤ p1 (Table 9). The total (optimal) value is (22, 31, 17, 28) (20, 28, 24, 35) which is same as the one obtained by forward recursion. The corresponding optimal combination of alternatives given in Table 10 is  ∗ ∗ ∗ d1 , d2 , d3 = (2, 2, 2) Table 8 Interval-valued fuzzy BRE (Stage 2) p2

d2 = 1

d2 = 2

d2 = 3

d2 = 4

f˜2 ( p2 )

d2∗

0

(9, 17, 5, 12) (7, 12, 9, 19)

–

–

–

(9, 17, 5, 12) (7, 12, 9, 19)

1

1

(14, 21, 12, 19) – (14, 19, 16, 23)

–

–

(14, 21, 12, 19) 1 (14, 19, 16, 23)

2

(14, 21, 12, 19) (12, 20, 10, 20) – (14, 19, 16, 23) (12, 16, 14, 22)

–

(14, 21, 12, 19) 1 (14, 19, 16, 23)

3

(14, 21, 16, 19) (14, 21, 12, 19) (9, 15, 5, 11) (14, 19, 16, 23) (14, 19, 16, 23) (7, 11, 9, 17)

–

(14, 21, 12, 19) 2 (14, 16, 19, 23)

4

(14, 21, 16, 19) (14, 21, 12, 19) (11, 16, 7, 14) (13, 19, 8, 15) (14, 19, 16, 23) 2 (14, 19, 16, 23) (14, 19, 16, 23) (9, 14, 11, 18) (10, 15, 13, 21) (14, 19, 16, 23)

5

(14, 21, 16, 19) (14, 21, 12, 19) (11, 16, 7, 14) (13, 19, 8, 15) (14, 21, 12, 19) 2 (14, 19, 16, 23) (14, 21, 12, 23) (9, 14, 11, 18) (10, 15, 13, 21) (14, 19, 16, 23)

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Table 9 Interval-valued fuzzy BRE (Stage 1) p1

d1 = 1

d1 = 2

f˜1 ( p1 )

d1∗

5

(18, 27, 14, 24) (17, 24, 20, 30)

(22, 31, 17, 28) (20, 28, 24, 35)

(22, 31, 17, 28) (20, 28, 24, 35)

2

Table 10 Corresponding optimal combination of alternatives     p1 d1∗ p2 = p3 − c3 d3∗ d2∗ p1 = p2 − c2 d2∗ 5

5−2=3

2

2

4−2=2

d1∗ 2

Applying the ranking of IVTrFN, we have  22 + 31 + 17 + 28 + 20 + 28 + 24 + 35 205 = r Aˆ  = 8 8 = 25.625 units

5 Conclusion In the proposed approach, the capital budgeting problem is considered in an intervalvalued fuzzy environment. The solution is obtained by DP approach using forward and backward recursion. This approach would be effective to solve multi-stage decision problems in finance, planning, research and development projects. Future study will focus on employing intuitionistic interval-valued fuzzy sets in multi-stage decision problems with decision points.

References 1. Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8: 338–353. 2. Bellman, R.E., and L.A. Zadeh. 1970. Decision-Making in a fuzzy environment. Management Science 17: B141–B164. 3. Kacprzyk, I., and A.O. Esogbue. 1996. Fuzzy dynamic programming: Main developments and applications. Fuzzy Sets and Systems 81: 31–45. 4. Li, L., and K.K. Lai. 2001. Fuzzy Dynamic programming approach to hybrid multi objective multistage decision making problems. Fuzzy Sets and Systems 117 (1): 13–25. 5. Baldwin, I.F., and B.W. Pilswoth. 1992. Dynamic Programming for fuzzy systems with fuzzy environment. Journal of Mathematical Analysis and Applications 85: 1–23. 6. Kacprzyk, I. 1994. Fuzzy dynamic programming—basic issues. In Fuzzy Optimization: Recent Advances, ed. M. Delgado, et al., 321–331. Heidelberg: Physica. 7. Kacprzyk, I. 1997. Multistage Fuzzy control: A Model-Based Approach to Fuzzy Control and Decision Making. New York, USA: Wiley.

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8. Esogbue, A.O., M. Theologidu, and K. Guo. 1992. On the application of fuzzy sets theory to the optimal control problem arising in water resources systems. Fuzzy Sets and Systems 48: 155–172. 9. Abo-Sinna, M.A., A.H. Amer, and Hendh EL Sayed. 2008. An interactive algorithm for Decomposing: The parametric space in fuzzy multi-obiective dynamic programming problem. In Fuzzy Multi-Criteria Decision Making. Springer Science + Business Media, LLC, New York. 10. Hussein, M.L., and M.A. Abo-Sinna. 1995. A fuzzy dynamic approach to the multi criteria resource allocation problem. In Fuzzy Sets and Systems, pp. 115–124. 11. Nagalakshmi, T., and G. Uthra. 2016. An application of generalized trapezoidal fuzzy numbers in the optimal solution of a fuzzy capital budgeting problem. International Journal of Pure and Applied Mathematics 109 (9): 63–71. 12. Atanassov, K. 1986. Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20: 87–96. 14. Wei, G.W. 2010. Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Journal of Computers 5: 345–351. 13. Shu, M.H., C.H. Cheng, and J.R. Chan. 2006. Using Intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly. Microelectronics Reliability 46: 2139–2148. 15. Wei, G.W. 2010. Some induced arithmetic aggregation operators with intuitionistic fuzzy trapezoidal fuzzy numbers and their application to group decision making. Journal of Computers 5: 345–351. 16. Zadeh, S.F., S.Y. Liu, and R.H. Zhai. 2011. An extended GRA method for MCDM with interval-valued triangular fuzzy assessments and unknown weights. Computers & Industrial Engineering 61: 1336–1341. 17. Deschrijver, G., and E.E. Kerre. 2005. Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems 153: 229–248. 18. Stephen Dinagar, D., and D. Abirami. 2015. On L-R type interval k valued Fuzzy Numbers in Critical path analysis. International Journal of Fuzzy Mathematical Archive 6(1): 77–83. 19. Stephen Dinagar, D., and D. Abirami. 2015. On critical path in project scheduling using topsis ranking of more generalized interval valued fuzzy numbers. Malaya Journal of Mathematics 2485–495

Characterization of Some Standard Graphs Based on the Eccentric Distance Sequence K. Deepika and S. Meenakshi

Abstract For two vertices a, b of a graph G, the distance between these two vertices is the length of the shortest (a, b) path. Based on the distance, many sequences have been studied in graph theory. In continuation with the distance degree sequence of graphs and eccentric sequence of graphs, we have developed a new sequence called the eccentric distance sequence of a graph, denoted as EDS or eds is defined as the sequence representing the number of vertices (denoted as n(v)) henceforth, at a maximum distance of every vertex of G. For any vertex, say u in G, the n(v) at a maximum distance is called the eccentric distance number, denoted as EDN or edn. Listing or enumerating the eccentric distance numbers of every vertex of the graph as a sequence is the eccentric distance sequence of the graph. In other words, the eccentric distance number is computed for every vertex of the graph, and the eccentric distance sequence refers to the eccentric distance numbers of every vertex of the graph represented in a sequence. In this paper, we have given a constructive characterization of some standard graphs based on the given eccentric distance sequence. That is, given an eccentric distance sequence, we have construct and characterize a graph that realizes or accepts the eccentric distance sequence. Keywords Distance of a graph · Eccentric distance sequence of a graph · Cycle graph · Wheel graph · Complete bipartite graph

1 Introduction Many sequences have been a subject of interest for researchers. This paved a way for them to study and develop various sequences like degree sequence, distance degree sequence, eccentric sequence, status sequence, and path degree sequence. [1, 2]. The

K. Deepika (B) · S. Meenakshi Vels Institute of Science Technology and Advanced Studies, Pallavaram, Chennai, India S. Meenakshi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_49

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realization of any sequence for a given graph was a primary question in the study of sequences of graphs [3, 4]. Degree sequences were the first type of sequences to be developed and studied. Erdos and Gallai [1] gave an existential characterization of graphs, whereas the constructive characterization was given by Havel and Hakimi [5, 6]. In this category, eccentric sequences were the first to be conceptualized and researched [2]. Many results in this direction are applied in practical problems [4]. Later, Nandakumar studied the minimal eccentric sequence [7]. The eccentric sequence for digraphs was developed much later in 2008 by Gilbert and Lopez [8]. Sequences based on the distance were studied which paved way for many researchers to develop various sequences to study the properties of graphs. Randic studied these sequences so that chemical isomers can be differentiated using their graphical structure [9]. Path degree sequences are very helpful in describing atomic environments [10, 11]. Based on the study of these sequences, we have defined a term called the eccentric distance sequence denoted as EDS or eds. The eccentric distance sequence of a graph, denoted as EDS or eds, is defined as the sequence representing the n(v) at a maximum distance of every vertex of G. This definition is based on the lines of distance degree sequence and eccentric sequence of graph [10–13]. For a given graph, finding the eccentric distance sequence is quite easy. But, for a given eccentric distance sequence, finding a graph that realizes the given sequence is quite challenging [14, 15]. This paves way to develop the realization or characterization theorems [13]. Since the eccentric distance sequence is a new concept, rudimentary results have been determined now. Hence, in this paper, we have given a constructive characterization of wheel graph Wn , complete bipartite graph K 2,n and cycle graph Cn based on the eccentric distance sequence. In future, various applications can be modeled based on this concept, and further properties can be determined.

2 Preliminaries For better understanding of this paper, we define some necessary terminologies. We consider G as a simple graph without any multiple edges or loops. Definition 2.1 For two vertices a, b of a graph G, the distance between these two nodes is the length of the shortest (a, b) path. Definition 2.2 For any vertex, say u in G, the n(v) at a maximum distance is called the eccentric distance number of the vertex u. The eccentric distance number is denoted as EDN or edn. Definition 2.3 Listing or enumerating the eccentric distance numbers of every vertex of the graph as a sequence is the eccentric distance sequence of the graph. It is denoted as EDS. In other words, the EDN is computed for every vertex of the graph, and the

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Fig. 1 Graph G

EDS refers to the eccentric distance numbers of every vertex of the graph represented in a sequence (Fig. 1). The eccentric distance number or EDN of the vertices {v1 , v2 , v3 , v4 , v5 } is {1, 2, 1, 1, 1}, respectively. So, the eccentric distance sequence or EDS of G is {1, 2, 1, 1, 1}. Definition 2.4 A connected graph with n vertices and whose edges form a cycle of length n is called a cycle graph of order n and is denoted by Cn . Definition 2.5 A wheel graph is a graph consisting of a cycle of order n − 1. These n − 1 vertices are connected to a singleton graph K 1 or a single vertex. Definition 2.6 A complete bipartite graph is a graph where the node set is divided into two independent sets, namely N1 and N2 , and each vertex of N1 is adjacent with every vertex of N2 . Further, no two vertices of N1 or N2 are adjacent.

3 Main Results Theorem 1 Given the eccentric distance sequence or EDS as {2, 2, 2, . . . 2}, the graph that realizes this EDS is the n—odd cycle. Proof Let the EDS be {2, 2, 2, . . . 2}. We are assuming G as a simple connected graph that realizes the given EDS. Let n be the number of nodes of the graph. We first assume n = 3. Then, the given EDS becomes {2, 2, 2}. Let the vertices of graph be {v1 , v2 , v3 }. From our assumption on the EDS {2, 2, 2}, the eccentric distance number on every vertex v1 , v2 , v3 is 2. That is, the n(v) at maximum distance is 2 for all the vertices. There is a possibility of a regular graph to realize the given EDS. We try to verify if the EDS of C3 or cycle graph with 3 vertices that satisfies the EDS {2, 2, 2} based on the Fig. 2. Clearly, Fig. 2 satisfies the EDS {2, 2, 2}. n(v) at maximum distance is 2, and the maximum distance is 1.

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Fig. 2 Cycle C3

Fig. 3 Cycle Cn

So, the result holds for n = 3. Also, the result holds only n = 5, 7, . . . for odd n. From the Fig. 3, for every vertex, of the graph, the eccentric distance number is 2. Also, this result does not hold for n = 4, 6, . . ., that is, for even n. Continuing this way, we infer that the EDN for every vertex of a cycle Cn is 2, and hence, the EDS is {2, 2, 2, . . . 2} where n is odd. The graph that realizes the given EDS {2, 2, 2, …2} is n − odd cycle Cn from Fig. 3. Hence the result. Corollary Given the eccentric distance sequence or EDS as {1, 1, 1, . . . 1}, the graph that realizes this EDS is the n—even cycle Cn . Proof The proof of this result is similar to the above discussed Theorem 3.1. Theorem 2 For a given eccentric distance sequence, {1, 1, 1, . . . n − 1}, the graph that realizes the given EDS is the wheel graph Wn , where n is odd. Proof Let the given EDS be {1, 1, 1, . . . n − 1}. Let G be the graph that realizes the given EDS with n number of vertices. Let n ≥ 5. Suppose n = 5. Then, the EDS is {1, 1, 1, 1, 4}. From Fig. 4, of the 5 vertices say, {v1 , v2 , v3 , v4 , v5 }, one vertex, say v5 , will have the eccentric distance number 4 which means the number of vertices at maximum distance from that vertex is 4. Also, we note that the maximum distance is

Characterization of Some Standard Graphs Based …

501

Fig. 4 Graph G1

1 and not more than 1 as that will in turn result in more than 5 vertices in the graph, contrary to our assumption that graph has 5 vertices. The vertex with EDN 4 must be adjacent to the remaining 4 vertices. Further, for the rest of the vertices {v1 , v2 , v3 , v4 }, the eccentric distance number is 1 for every vertex from the given EDS. If the above graph has to be the final graph, then the eccentric distance number of these four vertices will be 2. This implies the vertices v1 , v2 , v3 , v4 must be a cycle. Then, the graph that realizes the given EDS is a wheel graph W5 which is clear from Fig. 5. Continuing this way, the result can be proved for Wn as in Fig. 6. n − 1 vertices will have the EDN 1, and one vertex will have the EDN n − 1. The result holds only for odd n, that is n = 5, 7, 9, . . . from Fig. 6, the EDN of vertices {v1 , v2 , . . . vn−1 } of graph Wn is 1, and for the vertex vn , it is n − 1. The result does not hold for even n, that is n = 6, 8, 10, . . .. So, the graph that realizes the EDS {1, 1, 1, . . . .n − 1} is the wheel graph Wn where n is odd. Fig. 5 Wheel graph W5

Fig. 6 Wheel graph Wn

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Hence the result. Corollary For a given eccentric distance sequence, {2, 2, 2, . . . 2, n − 1}, the graph that realizes the given EDS is the wheel graph Wn , where n is even. Proof The proof of this result is similar to the above discussed theorem 3.2. The EDN for n − 1 vertices is 2 which means it should be a regular graph. In particular, it must be a cycle graph. The remaining one vertex will have the EDN n − 1. This means that vertex must be adjacent to every vertex of the outer cycle graph forming a wheel graph realizing the given EDS. Theorem 3 For a given eccentric distance sequence, {1, 1, (n − 1), (n − 1), . . . (n − 1)}, the graph that realizes the given EDS is the complete bipartite graph, K 2,n . Proof Let the given EDS be {1, 1, (n − 1), (n − 1), . . . (n − 1)}. Suppose that G is a simple connected graph that realizes the given EDS. By definition, for any vertex u in G, the eccentric distance number or EDN is n(v) at maximum distance. So, for the given EDS, we understand that the EDN is 1 for two vertices, and for the remaining vertices, the EDN is n − 1. Let us now construct and characterize the graph that realizes the given EDS. We are considering the EDS {1, 1, 1, 1} for n = 2. Let there be 4 vertices say v1 , v2 , v3 , v4 . We obtain the following graph. For this graph, EDN(v1 ) = 1 and EDN(v2 ) = 2. The EDN of vertices v3 , v4 is obtained from the Fig. 7, as 1 and 1, respectively. It is clear that the vertices v1 , v2 and v3 , v4 cannot be adjacent. If there are edges between v1 , v2 and v3 , v4 , then it results in a complete graph as in Fig. 9, and the EDN for all the vertices of this complete graph is 3 which is a contradiction to our given statement. So, v1 and v2 cannot be adjacent, but those two vertices are adjacent to v3 , and v4 . Similarly,v3 and v4 cannot be adjacent but are adjacent to v1 and v2 resulting in a complete bipartite graph K 2,2 as in Fig. 8. The same argument can be made to the generalized complete bipartite graph K 2,n . We consider the EDS {1, 1, 2, 2, 2} for n = 3. Let there be 5 vertices say v1 , v2 , v3 , v4 , v5 . By the previous argument, we obtain the graph in Fig. 10. Fig. 7 Graph G1

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Fig. 8 Graph G

Fig. 9 Complete graph

Fig. 10 Complete graph K 2,3

The EDN of v1 and v2 is 1. The EDN of v3 , v4 , v5 is 2. A complete bipartite graph K 2,3 is thus obtained. Continuing this way, we could construct and characterize a complete bipartite graph K 2,n that realizes the EDS {1, 1, (n − 1), (n − 1), . . . (n − 1)}. Hence the result.

4 Conclusion The study of sequences has seen vast development in the recent years. Many distancerelated sequences like distance degree sequence, eccentric sequence finds applications in the field of chemistry to study the chemical and molecular structures. The eccentric distance sequence is a new terminology introduced in this paper where constructive characterization of some standard graphs has been discussed. This can

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be extended to other graphs as well. It is comparatively easier to compute the eccentric distance sequence for a graph based on its definition. But, given an EDS, to find a graph that realizes the EDS is quite challenging. In future, various properties can be characterized and determined. Future Works Construction and characterization of various standard graphs can be determined based on this work as an extension. The concept of degree sequence is used in message transfer that is in encryption and decryption. Similarly, the concept of eccentric distance sequence can be used in encrypting and decrypting a messages so that a safe cryptosystem can be generated. The concept of distance-based sequences are usually applied in molecular chemistry. Since this EDS is also a distance-based sequence, in future, we may even extend the research in applying the concept of EDS in other fields. Also, various properties can be computed and determined. Also, embedding of graphs can be done based on the eccentric distance sequence of graphs.

References 1. Buckley, F., and F. Harary. 1990. Distance in Graphs. Addison-Wesley. 2. Huilgol, M.I. 2011. Distance degree regular graphs and the ireccentric digraphs. International Journal of Mathematical Sciences and Engineering Applications 5 (6): 405–416. 3. Huilgol, M.I., and C. Ramaprakash. 2014. On some edge rotation distance graphs. IOSR Journal of Mathematics 10 (6): 16–25. 4. Huilgol, M.I., M. Rajeshwari, and S. Syed Asif Ulla. 2013. Embedding in distance degree regular and distance degree injective graphs. Malaya Journal of Matematik 4 (1) : 134–141. 5. Bond, J. 1989. Graph theory and its applications. Annals of the New York Academy of Sciences. 6. Bloom, G.S., L.V. Quintas, and J.W. Kennedy. 1981. Distance degree regular graphs. The Theory and Applications of Graphs (Kalamazoo, Mich.,) 95–108. 7. Huilgol, M.I., M. Rajeshwari, and S.S.A. Ulla. 2012. Products of distance degree regular and distance degree injective graphs. Journal of Discrete Mathematical Sciences and Cryptography 15 (4–5): 303–314. 8. Erdös, P., and T. Gallai. 1960. Graphs with prescribed degrees of vertices. Mat. Lapok 11: 264–274. 9. Ostrand, P.A. 1973. Graphs with specified radius and diameter. Discrete mathematics 4 (1): 71–75. 10. Hakimi, S. L. 1962. On realizability of a set of integers as degrees of the vertices of a linear graph. I. Journal of the Society for Industrial and Applied Mathematics 10 (3) : 496–506. 11. Meenakshi, S., K. Deepika, and R. Abdul Saleem. 2019. Eccentric sequence of graphs. International Journal of Recent Technology and Engineering 8(4S5) : 52–54. 12. Meenakshi, S., and K. Deepika 2019. Some results on the eccentric sequence of graphs. International Journal of Recent Technology and Engineering 8(4S5) : 55–57. 13. Huilgol, M.I., M. Rajeshwari, and S. Syed Asif Ulla. 2011. Distance degree regular graphs and their eccentric digraphs. International Journal of Mathematical Sciences and Engineering Applications 5 (6) : 405–416. 14. Akiyama, J., K. Ando, and D. Avis. 1985. Eccentric graphs. Discrete Mathematics 56 (1) : 1–6. 15. Bollobas, B. 1982. Distinguishing vertices of random graphs. Annals of Discrete Mathematics 13: 33–50.

A Smart Personal Assistant for Visually Challenged Sushruta Mishra, Kunal Anand, and N. Z. Jhanjhi

Abstract At present, there are numerous people in the world suffering from poor vision. We cannot even imagine the numbers that we see, 285 million in which 39 million are visually challenged and the other 246 million have poor vision. To help them and make their life a little easier, an IoT-based model is presented in this work. The introduced device provides a walking support system to visually challenged users and help them to be independent. The system allows the user to move freely as they get notified of any obstacle on their way. It is a portable device with GPS and GSM modules helping the family members keep the wearer’s track. Thus, this device serves as a personal assistant to a visually challenged person, assisting the person and protecting them from obstacles in way. Later a proper implementation of the proposed framework is carried out and a comparative analysis with other existing works is undertaken. The outcome is promising and hence can be recommended. Keywords Internet of Things (IoT) · Visually challenged · GSM · Sensors · Actuators

1 Introduction As the world has evolved, many new technologies have found their way into our lives, and they still keep getting better with time [1, 2]. However, there has not too much evolution in technology to help visually challenged people. We have almost forgotten that how much life is complex and challenging for physically challenged S. Mishra (B) · K. Anand School of Computer Engineering, Kalinga Institute of Industrial Technology, Bhubaneswar, Odisha, India e-mail: [email protected] K. Anand e-mail: [email protected] N. Z. Jhanjhi School of Computer Science and Engineering, Taylor’s University, 47500 Subang Jaya, Malaysia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_51

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people, and visually challenged people being one of them. When we look at the graph of physically challenged people worldwide, we see that approximately 39 million people are entirely visually challenged, whereas the other 246 million suffer from the low vision problem [3, 4]. On the other hand, the Internet of Things (IoT) presents a view of a future Internet technology where end-users, computing systems, and everyday life objects maintaining, sensing, and actuating capabilities having economic advantages. This motivates us to make a device that will help visually challenged people navigate around using IoT technology. The objective of this work is to make, blind people’s life a little bit simpler and more accessible so that they do not need to be dependent on anyone for at least physical movement in their own house. So, a device has made that deriving the inspiration from the type of method adopted by Bat to navigate their way while avoiding the obstacle. The module is a portable device innovated with ultrasonic sensors and buzzers for visually challenged people. It has been implemented with GPS and GSM. Here, the device will emit ultrasonic waves. It will start the buzzer or activate the vibrator if it identifies any obstacle in the path to help the wearer know about an obstacle ahead to change their course. In case of any unwanted incident, the GPS and GSM can send their location to their loved one so that they can reach to them come for rescue.

2 Literature Survey Recently the usage of IoT is changing the scenario drastically by introducing an indispensable role in current trends and uses developing technologies. In the year 2012, Chew proposed a system that used the smart white cane [1]. The cane was named Blindspot. The system uses GPS technology, and a proximity sensor called an ultrasonic sensor to challenge people in navigation visually. S. Gangwar in 2013 [2] proposed an intelligent stick with an IR sensor for signaling the visually challenged people of any obstacle found in their way. However, this stick is only capable of detecting the object. Benjamin et al. in 2014 [3] had developed an aid that used a stick with laser sensors. These laser sensors identify the obstacles and indicate the person about the obstruction using a microphone. Central Michigan University, in 2009, Do et al. [4] used an intelligent cane for visually challenged people and used it along with an RFID tag. The information received from the label is given to the challenged people. It uses a proximity sensor named an ultrasonic sensor to detect the objects and alert the person with a speaker’s help. Md. Helmy Abd Wahab and Amirul A. Talibetal in 2013 [5] also used ultrasonic sensors to detect the obstacles and alarm the wearer using a voice message. Alejandro R. Garcia Ramirez and Renato Fonseca Livramento da Silvaetal in 2012 [6] used haptic sensors to detect barriers and alert the wearer with the help of a message. Sushruta et al. [7] discussed the impact of data redundancy and replication on clustered wireless sensor networks’ overall performance. Panda et al. [8] discussed various features and criteria that lead

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to smartphones’ buying patterns; based on those features, a model framework was proposed using the KNN algorithm.

3 System Design The developed system consists of various components that coordinate with one another and help visually challenged people walk with ease and confidence. Arduino IDE is the essential software required for this study. The hardware components are listed in Table 1.

3.1 Arduino Nano Arduino nano is a small and complete board that is just the size of a matchbox and is breadboard-friendly. It consists of a USB mini port that one can use to provide a power supply and upload the code. The microcontroller used in Arduino nano is Atmega328p. The nano consists of a reset button on the chip’s side that can be used to reset the Arduino. It consists of 4 onboard LEDs RX, TX, PWR, and L. Whenever the Arduino receives data from other devices; the RX LED glows. Whenever the data is sent successfully, then TX LED illuminates. As soon as the power supply is provided, the power LED glows. L LED is connected to D13, and thus anything related to D13 makes this LED glow. There is a 14 MHz crystal oscillator that provides heartbeat/clock pulses to the board. There are 14 digital pins and 8 Analog pins in total. The required components are listed in Fig. 1.

3.2 Ultrasonic Sensor HC-SR04 is a proximity sensor that uses sound waves to identify or detect any obstacle/objects [9]. Ultrasonic sound vibrates above the range of human ears, and hence they do not get affected by the vibrations. Ultrasonic sensors do not get involved Table 1 Specification of the components

Sl. No Components

Input voltage Operating voltage (V)

1

Arduino nano

7–12 V

5

2

Ultrasonic sensor

5V

5

3

GPS/GSM module 5–26 V

4

Buzzer

5V

5

5

Vibrating motor

2.5–4 V DC

3

5

508

S. Mishra et al.

Fig. 1 Components required for the work

with color, dust, light, or smoke, and therefore it is suitable for both indoor and outdoor use. It covers the range of 2–400 cm with an accuracy of up to 3 mm. The module incorporates a controller, receiver, and transmitter.

3.3 GPS/GPRS/GSM Module SIM808 is being used in the system. It is an all-in-one module comprising of GPS, GRPS, GSM, and Bluetooth. The module has got provision for two antennas, one for GSM and the other for GPS. TTL pins are extended TX, RX, and GND for Arduino. It tracks the visually challenged person wearing it and sends their location via text message to family members or friends.

3.4 Buzzer A small and practical component helps make the visually challenged person aware of any obstacle on the way by producing a beep sound. It consists of two pins; the

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positive end is connected to any of the 14 digital pins of the Arduino, and the opposing end connects to the ground.

3.5 Flat Vibrating Motor A flat vibrating motor, also known as Coin vibrating motor, indicates the visually challenged person for any obstacle by vibration if they do not want the buzzer that may affect the people around. It is a fully enclosed motor with no movable parts, and its small size and shape make it easy to use. It helps in making the device portable and compact.

4 Proposed Work The device has been designed and configured to help visually challenged people. It has been made to make the lives of those people a bit more comfortable [10–15]. This device is well equipped with an ultrasonic sensor and Arduino board. The sensor will help in detecting any obstacle present on any side. It worked well in most of the circumstances that visually challenged people to come across usually. It can see the block in the direction present with the help of the ultrasonic sensors. It has a userspecified distance that can be increased or decreased as per the user requirement. It also has a vibrator contained in it so that instead of buzzing continuously, it starts vibrating. Thus, only the person wearing the device can feel the vibration, and the people in the surrounding will not get disturbed. The ultrasonic sensor has a range approx. 400 cm. It has 4 pins named as VCC, Trigger (Trig), Echo, and GND. VCC connected to 5 V. Trigger goes to any of the digital pins, for example, D2. Echo goes to another digital pin, say D3, and GND goes to the Arduino ground. This device is tiny and can be kept inside the person’s pocket that makes it easier to carry and protects it from getting lost. This device is very much affordable as it costs significantly less. Overall, this device gives a better result than any of the devices formed for the visually challenged person. The device comes with a GPS and a GSM module that helps to track the person wearing it and sends the location to his/her family members or friends on their mobile phones, which makes it very easy for them to access. Figure 2 illustrates the overall circuit diagram and Fig. 3 shows the working of an ultrasonic sensor. The system consists of an ultrasonic sensor that detects the obstacle under the range and then notifies the person by producing a beep sound or vibration made by the vibrating motor. The ultrasonic sensor consists of two terminals: the transmitter and the other being the receiver. The transmitter transmits sound waves under a given range, and when those waves hit an obstacle, these sound waves are reflected and are received by the receiver. The distance can be measured by travel time and the speed of sound.

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Fig. 2 Circuit diagram for connection between Arduino nano, ultrasonic sensor, and vibrating motor

Fig. 3 Working of an ultrasonic sensor

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Distance = time ∗ speed of sound. As the sound travels the same distance two times hence Distance = (time ∗ speed of sound)/2. The module consists of 4 pins VCC, trigger, Echo, and ground; to produce the ultrasound, we need to make the trigger pin high for a minimum of 10 ms. Then the module will send out eight sonic bursts of ultrasound at 40 kHz. The Echo pin outputs the time in a microsecond. Once the obstacle is identified, the buzzer produces a beep sound indicating the person of the block. The GPS receiver receives the data in an NMEA format. The latitude and longitude coordinates are extracted from it using the Arduino Tiny GPS library. Then, the GSM module sends this information via text message. A switch is provided, which helps the visually challenged ones to send their location via text message. GPS module keeps tracking the wearer’s live location and sends the site to their family members or friends via text message. The sent area can easily be accessed through Google map. The circuit diagram is illustrated in Fig. 4.

Fig. 4 Circuit diagram of SIM 800 module with Arduino

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5 Result Analysis Choosing the correct sensor to detect the obstacle present on the way is very important [11, 17, 18]. So, we tested different kinds of the sensor on other material and objects [16]. The accuracy result analysis is shown in Fig. 5. The ultrasonic sensors show a better result for almost all materials like wood, rubber, plastic, etc. in analyzing the graph. After considering all the criteria regarding the effect of color, dust, light, outdoor use, etc., we concluded that the ultrasonic sensor is a better option than the rest. To transfer the wearer location, we have used the all-in-one GSM GPS module instead of a separate GSM and GPS module that makes the device more compact. The GSM module connects with the GPS satellite, and the satellite then transfers the information through a wireless network. The wireless network then connects to the GPS server, and the information is transferred to their family members in the form of text messages through the GSM. The system model has visualized in Fig. 6. The device helps track the visually challenged person’s real-time location and supports their family members, always remain connected to them to follow them at the time of emergency easily. Many devices and technologies have come in the past decade, but they all had one or another drawback. The present system consists of a device like a white cane which is the traditional method used by visually challenged

Fig. 5 Accuracy graph of different proximity sensors

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Fig. 6 System model for real-time tracking of visually challenged

to move and detect obstacles. However, those have some drawbacks like getting stuck between objects, cracks easily. Another existing system includes a vision or torch device that comes with a deficiency of portability. It is not easy to be carried and requires a lot of prior training before being used. Visually challenged ones also use pet dogs, but people of a different economic class will not afford it. Considering all these drawbacks, the proposed device for visually challenged, tries to overcome these drawbacks. A comparison has made between the existing method and the proposed method. This device is tiny and can be kept inside the person’s pocket, making the machine easy to carry. It made it easier for the person to move. Table 2 shows a comparison analysis of the proposed device with others. Overall, this device gives a better result than any of the devices formed for the visually challenged person. The device comes with a GPS and a GSM module that helps track the wearer’s location and transfers the information to his/her family members or friends on their mobile phones, which makes it very easy for them to access and get Table 2 Comparison between the existing method and the introduced method Sl. No

Method

Probability

Training

Cost

1 2

White cane

Medium

Not required

Low

Vision or torch

Low

Required

Medium

3

Pet dogs

Low

Required

High

4

Proposed model

High

Not required

Low

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updated. It has a better navigation range, with the ultrasonic sensors’ help, as these sensors do not get affected by light, dust, or color that makes it suitable for outdoor use, unlike the IR sensor. It is also cheap and user-friendly, and hence people of any financial class can afford it. It has been tested on some visually challenged people. The result has been primarily positive in many cases, and the people were able to give up the White cane, and some of them were even able to walk freely without the help of anybody from one place to another. This can help to the visually paired people; this can be further scalable solution by considering the secure communication [19–22]. The right scheme for the selected IoT-based application solutions [23].

6 Conclusion The system introduced shows the design and architecture of a new idea of Arduinobased Virtual Eye for a visually challenged person. A cheap, simple, portable, configurable, and simple to handle automated guidance system with many added unique features and improvements are proposed to give constructive assistance and support to the visually challenged person. The system will be effective in detecting the obstacles that visually challenged people encounter on their way. With the suggested architecture, if constructed with utmost precision, the visually challenged will move from one place to another without the other’s help and with even more confidence. This study suggested the architecture and system of one of the latest concepts of the Arduino-based computerized eye for visually challenged people. This is one of the devices that stick to the person; it can be kept in the wearers pocket and has a very slight chance of getting lost as it detects the obstacle and stops the person from colliding or falling over. It comes with two modes named the vibrating mode and the beep mode. The vibrating method consists of a vibrating motor that prevents the device from causing disturbance to others by the buzzer’s noise, making the person wearing it comfortable and the people around them. The GSM and the GPS module make it even more secure by tracking the wearer’s location and making the family updated by the site via sending the information on their phone, making it even more accessible if implemented with better accuracy.

References 1. Selene Chew. 2012. The smart white cane for blind. National University of Singapore (NUS). 2. Shantanu Gangwar. 2013. Smart stick for blind. New Delhi. 3. Benjamin, J. M., N.A. Ali, and A.F. Schepis. A laser cane for the blind. In Proceedings of the San Diego Biomedical Symposium, vol. 12, 53–57. 4. Do Ngoc Hung, Vo Minh-Thanh, Nguyen Minh Triet, Quoc Luong Huy, Viet Trinh Cuong. 2017. Design and implementation of smart cane for visually challenged people. In International Conference on the Development of Biomedical Engineering.

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5. Wahab, Mohd Helmy Abd., and Amirul A. Talibetal. 2013. A review on an obstacle detection in navigation of visually challenged. International Organization of Scientific Research Journal of Engineering (IOSRJEN) 3 (1): 01–06. 6. Alejandro R. Garcia Ramirez and Renato Fonseca Livramento da Silvaetal. (2012). Artificial EYE An innovative idea to help the blind. In Conference Proceeding of the International Journal of Engineering Development and Research (IJEDR), SRM University, Kattankulathur, 205–207. 7. Mishra, S., H.K. Tripathy, and A.R. Panda. 2018. An improved and adaptive attribute selection technique to optimize dengue fever prediction. International Journal of Engineering Technology 7: 480–486. 8. Panda, Amiya, and Sushruta Mishra. 2018. Smartphone purchase prediction with 3-NN classifier. Journal of Advanced Research in Dynamical and Control Systems 10 (14): 674–680. 9. Mishra, S., B.K. Mishra, H.K. Tripathy, and A. Dutta. (2020). Analysis of the role and scope of big data analytics with IoT in health care domain. In Handbook of data science approaches for biomedical engineering, 1–23. Academic Press. 10. Chattopadhyay, A., S. Mishra, and A. González-Briones. (2021). Integration of machine learning and IoT in healthcare domain. In Hybrid Artificial Intelligence and IoT in healthcare, 223–244. Springer, Singapore. 11. Sahoo, Kshira Sagar, and Deepak Puthal. 2020. SDN-Assisted DDoS defense framework for the internet of multimedia things. ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM) 16 (3): 1–18. 12. Rath, M., and S. Mishra. (2020). Security approaches in machine learning for satellite communication. In Machine learning and data mining in aerospace technology, 189–204. Cham: Springer 13. Tripathy, H. K., S. Mishra, H.K. Thakkar, and D. Rai. (2021). CARE: a collision-aware mobile robot navigation in grid environment using improved breadth first search. Computers & Electrical Engineering, 94, 107327. 14. Mishra, Sambit Kumar, et al. 2020. Energy-aware task allocation for multi-cloud networks. IEEE Access 8 (2020): 178825–178834. 15. Rout, Suchismita, et al. 2021. Energy efficiency in software defined networking: a survey. SN Computer Science 2 (4): 1–15. 16. Jena, L., S. Mishra, S. Nayak, P. Ranjan, and M.K. Mishra. (2021). Variable optimization in cervical cancer data using particle swarm optimization. In Advances in electronics, communication and computing, 147–153. Singapore: Springer. 17. Mohapatra, S.K., P. Nayak, S. Mishra, and S.K. Bisoy. (2019). Green computing: a step towards eco-friendly computing. In Emerging trends and applications in cognitive computing, 124–149. IGI Global. 18. Madhu, G., et al. 2021. Imperative dynamic routing between capsules network for malaria classification. CMC-Computers Materials & Continua 68 (1): 903–919. 19. Humayun, M., N.Z. Jhanjhi, and M.Z. Alamri. 2020. Smart secure and energy efficient scheme for e-health applications using IoT: a review. International Journal of Computer Science and Network Security 20 (4): 55–74. 20. Jacob, S., et al. 2021. AI and IoT-enabled smart exoskeleton system for rehabilitation of paralyzed people in connected communities. IEEE Access 9: 80340–80350. https://doi.org/10.1109/ ACCESS.2021.3083093. 21. Ullah, A., M. Azeem, H. Ashraf, A.A. Alaboudi, M. Humayun, and N. Jhanjhi. 2021. Secure healthcare data aggregation and transmission in IoT—a survey. IEEE Access 9: 16849–16865. https://doi.org/10.1109/ACCESS.2021.3052850. 22. Aadil, F., B. Mehmood, N. Ul Hasan, S. Lim, S. Ejaz, and N. Zaman. 2021. Remote health monitoring using IoT-based smart wireless body area network. CMC-Computers Materials & Continua 68 (2): 2499–2513. 23. Saleh, M., N. Jhanjhi, A. Abdullah, and R. Saher. 2021. Proposing encryption selection model for IoT devices based on IoT device design. In 2021 23rd International Conference on Advanced Communication Technology (ICACT), 210–219. https://doi.org/10.23919/ICACT51234.2021. 9370721.

F-Index for Some Class of Graphs R. Krithika

Abstract A graph L consists of set of nodes and lines. A labeled graph is a chemical graph whose nodes represent the molecule of the compound and lines represent the chemical bonds. Chemical graph theory is to come up with helpful graph invariants. Vertex degrees are essential in chemical graphs. The vertex degree of a chemical graph is the number of neighbors of that vertex. The basic concept on vertex degrees is the degree sequence. The F - index of a graph L denoted by F(L) is defined as the sum of cubes of the vertex degrees of the graph. In this paper to construct an exact formulae for forgotten index for few class of graphs like star graph, crown graph, Mycielski construction of path and cycle, cocktail party graph, friendship graph and wheel graph, detour saturated tree for T3 (l1 ), functional graph, and isomorphic graph. Keywords Index · Degree · Vertices · Graph

1 Introduction Molecular graph theory is a part of graph theory that focuses on the conception of a chemical graph, also known as structurally graph, or constitutional graph, which indicates a graph containing vertices and edges expressing atoms and bonds [3]. However, topological principles have been applied in chemistry, and the structure of the well-known Huckel molecular orbitals is governed by the topology of a molecule rather than its geometry [5]. The valence of a vertex in a chemical graph is the degree of that vertex [1]. Because double bonds and lone-pair electrons can be represented by a single edge in graph L. Chemical graph theory is based on the premise that the information represented in chemical graphs is utilized to examine the physico-chemical possessions of molecules. In 2017, Furtula and Gutman established forgotten index F(L) =

 l1 ∈V (L)

d(l1 )3 =

   d(l1 )2 + d(l2 )2 l1 l2 ∈E(L)

R. Krithika (B) Department of Humanities and Sciences, Rajalakshmi Engineering College, Chennai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_52

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d(l1 ) represents the degree of node in L [2, 4]. Nilanjan De et al. characterized the forgotten index and forgotten coindex of graph decorations [7, 8]. Throughout this paper l1 denotes number of vertices in L. In this study, we provide novel methods for estimating the forgotten index [6]. This solution provides a direct way for computing any graph degrees.

2 Main Results Theorem 1 The F - index of  star graph Sl1 −1 = K 1,l1 −1 with l1 vertices and (l1 − 1)  edges for l1 ≥ 3 is F Sl1 −1 = l13 − 3l12 + 4l1 − 2. Proof: The degree sequence of star graph is {1, 1, 1, . . . 1, l1 − 1}. By representation of forgotten index

F(L) =

 l1 ∈V (L)

[d(l1 )]3 =





l1 l2 ∈E(L)

[d(l1 )]2 + [d(l2 )]2



    F - index is F Sl1 −1 = 13 + 13 + · · · + 13 +(l1 − 1)3



(l1 −1)

= (l1 − 1) + (l1 − 1)3   = (l1 − 1) l12 − 2l1 + 2 = l13 − 3l12 + 4l1 − 2. Theorem 2 The F - index of crown graph Sl01 with 2l1 vertices and l1 (l1 − 1) edges   for l1 ≥ 3 is F Sl01 = 2l1 (l1 − 1)3 .   Proof: Let the two vertex sets be u 1 , u 2 , u 3 , . . . , u l1 and v1 , v2 , v3 , . . . , vl1 . Then, the degree sequence of crown graph with 2l1 vertices is {l1 − 1, l1 − 1, l1 − 1, . . . , l1 − 1}. 

 0

⎡

⎤

F - index is F Sl1 = ⎣(l1 − 1)3 + (l1 − 1)3 + · · · + (l1 − 1)3 ⎦



2l1 times

= 2l1 (l1 − 1) . 3

   Theorem 3: The F - index of Mycielski construction of path Pl1 is F ζ Pl1 = l13 + 91l1 − 150 if l1 ≥ 3. Proof:   The degree sequence of Mycielski construction of path Pl1 is denoted by ζ Pl1 . The degree sequence is (2, 4, 4, . . . , 4, 2, 2, 3, 3, . . . 3, 2, l1 ).

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519

⎡ ⎤    F - index is F ζ Pl1 = ⎣23 + 43 + 43 + · · · + 43 + 4 3 +23 ⎦ (l1 −2)

⎡

⎤

+ ⎣23 + 33 + 33 + · · · + 33 + 3 3 +23 ⎦ + l13 . (l1 −2)

= 23 + 43 (l1 − 2) + 23 + 23 + 33 (l1 − 2) + 23 + l13 = 32 + 91(l1 − 2) + l13    F ζ Pl1 = l13 + 91l1 − 150 if l1 ≥ 3.    Theorem 4: The F - index of Mycielski construction of cycle Cl1 is F ζ Cl1 = l13 + 91l1 − 56 if l1 ≥ 3. Proof:   The degree sequence of Mycielski construction of cycle Cl1 be denoted as ζ Cl1 . The degree sequence is (4, 4, . . . , 4, 3, 2, 3, 3, . . . , 3, l1 ). ⎤ ⎡    F - index is F ζ Cl1 = ⎣ 43 + 43 + . . . .  . . . . . . + 43 + 4 3 +33 ⎦ (l1 −1)

⎡

⎤

+ ⎣23 + 33 + 33 + . . . .  . . . . . . + 33 + 3 3 +33 ⎦ (l1 −1)

= 4 (l1 − 1) + 3 + 2 + 33 (l1 − 1) + l13 3

3

3

= l13 + 91(l1 − 1) + 35 = l13 + 91l1 − 56, if l1 ≥ 3. Theorem 5: Let L be a cocktail party graph. The F - index of cocktail party graph for 2l1 vertices is F(L) = 2l1 (2l1 − 2)3 i f l1 ≥ 2. Proof: The degree sequence of cocktail party graph with 2l1 vertices is {2l1 − 2, 2l1 − 2, 2l1 − 2, . . . , 2l1 − 2}. ⎡

⎤

F(L) = ⎣(2l1 − 2)3 + (2l1 − 2)3 + · · · + (2l1 − 2)3 ⎦



2l1 times

= 2l1 (2l1 − 2) ifl1 ≥ 2. 3

    Theorem 6: The F - index of friendship graph Fl1 is F Fl1 = 8l1 l12 + 2 ifl1 ≥ 2.

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Proof: The degree sequence of 2l1 + 1 vertices of friendship graph is (2, 2, . . . , 2, 2l1 ). F - index is F(L) = (2)3 + (2)3 + · · · + (2)3 +(2l1 )3



(2l1 times)

= 2 (2l1 ) + (2l1 )3   = 2l1 8 + 4l12   = 8l1 l12 + 2 i f l1 ≥ 2 3

  Theorem 7: The F - index of wheel graph Wl1 with l1 vertices for l1 ≥ 4 is F Wl1 = l13 − 3l12 + 30l1 − 28. Proof: The degrees of wheel graph with l1 vertices are du 1 = du 2 = du 3 = du 4 = · · · = du l1 −1 = 3 and du l1 = l1 − 1. 

⎡



⎤

3⎦ 3 F - index is F Wl1 = ⎣3 3 + 33 +  · · · + 3 + (l1 − 1) (l1 −1)

= 3 (l1 − 1) + (l1 − 1)3   = (l1 − 1) l12 − 2l1 + 28   F Wl1 = l13 − 3l12 + 30l1 − 28ifl1 ≥ 4. 3

Theorem 8: The F - index of detour saturated tree for T3 (l1 ) for l1 ≥ 2 is F(T3 (l1 )) = 84(2)l1 − 54. Proof The degree sequence of detour saturated {3, 3, 3, 3, . . . , 3, 3, 3, 3, 1, 1, 1, 1, . . . , 1, 1, 1, 1}. ⎡

⎤

tree

for

⎡

T3 (l1 )

is

⎤

⎢ 3 ⎢ 3⎥ 3 3⎥ F - index is F(T3 (l1 )) = ⎣3 3 + 33 +  · · · + 3 ⎦ + ⎣1 + 1 +  · · · + 1 ⎦ (2l1 −2 ×12−2) (3×2l1 )    = 2l1 −2 × 12 − 2 (3)3 + 3 × 2l1 (1)3     = 2l1 × 3 − 2 27 + 3 × 2l1 

= 84(2)l1 − 54ifl1 ≥ 2. Theorem 9: The F - index of functional graph, cycle Cl1 with l1 vertices for l1 ≥ 3 is F(L) = 8l1 . Proof: The degrees of functional graph, cycle Cl1 with l1 vertices are du 1 = du 2 = du 3 = du 4 = · · · = du l1 −1 = du l1 = 2.

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⎡

⎤

F - index is F(L) = ⎣ 23 + 23 + . . . . . . . . . . + 2 3 ⎦   = l 1 23 = 8l1 ifl1 ≥ 3.

l1

Theorem 10: The F - index of two isomorphic graphs is same. Proof: Forgotten index depends only on degree sequence. For any two isomorphic graphs, the degree sequence is similar. The forgotten index of two isomorphic graphs is equal. Illustration 11: Consider the two graphs L 1 andL 2 (Fig. 1.) For L 1 , the degree sequence is {3, 2, 2, 4, 1}. For L 2 , the degree sequence is {3, 2, 2, 4, 1}. L 1 andL 2 are isomorphic graphs. Since the degree sequence of L 1 andL 2 is same, the F - index is equal. Observation 12. By definition, the F - index for few graphs that is acquired effortlessly as follows. (i)

The F - index of bull graph consisting of five nodes and five lines is F(L) = 13 + 33 + 23 + 33 + 13 = 64.

(ii)

The F - index of butterfly graph consisting of five nodes and six lines is

Fig. 1 Isomorphic graphs L 1 andL 2

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F(L) = 23 + 23 + 43 + 23 + 23 = 96. (iii)

The F - index of diamond graph consisting of four nodes and five lines is F(L) = 33 + 23 + 33 + 23 = 70.

(iv)

The F - index of Wagner graph consisting of eight nodes and twelve lines is F(L) = 33 + 33 + 33 + 33 + 33 + 33 + 33 + 33 = 216.

3 Conclusion F - index is one of the evolving concepts in graph theory. The F - index is important concept in mathematical chemistry. The F - index of many graphs is familiar. Results in this paper are based on simple graphs. We generalized accurate formula for some particular graphs for forgotten index. F - index can be extended for some more critical graphs.

References 1. Baskar, Babujee J., and J. Senbagamalar. 2015. On Wiener and terminal of graphs. International Journal of Biomathematics 8 (5): 1550066–1550077. 2. Furtula, B., and I. Gutman. 2015. A forgotten topological index. Journal of Mathematical Chemistry 53 (4): 1184–1190. 3. Harary, F. 1969. Graph theory. Addison—Wesley, Reading, Mass Press. 4. Gutman, I. 2013. Degree–based topological indices. Croatica Chemica Acta 86 (4): 351–361. 5. Hosam Abdo1., Darko Dimitrov., and Ivan Gutman. 2017. On extremal trees with respect to the F-index. Kuwait Journal of Science 44 (3): 1–8. 6. Pattabiraman, K., and T. Suganya. 2020. F-index and its coindex of block-edge transformation graphs. Bulletin of the International Mathematical Virtual Institute 10 (2): 283–290. 7. Nilanjan De1„ Sk. Md., Abu Nayeem., and Anita Pal. 2016. The F-coindex of some graph operations. De et al. Springer Plus 5 (1): 221–234. 8. Nilanjan De., Sk. Md., and N. Nayeem. 2016. F-index of some graph operations. Discrete Math Algorithm Application 8 (2).

Synchronization of a Modified Colpitts Oscillator with Triangular Wave Non-linearity on Graph Suresh Rasappan, K. A. Niranjan Kumar, R. Narmada Devi, and Ahmed J. Obaid

Abstract A graph theory-based chaos synchronization is introduced. Although different chaotic systems have been formulated by earlier researchers, only a few chaotic systems exhibit chaotic behaviour. The synchronization of chaotic systems, which seems addressed in a range of applications, is addressed in this paper on graphs. This work shown that the structure and properties of the solution of graph for the system of differential equation allows to investigate the synchronization. Keywords Chaos · Colpitts oscillator · Synchronization · Triangular wave non-linearity · Bipartite graph

1 Introduction The concept of synchronization in chaos theory [1, 2] is an interesting one also important for many problems in current situations. The estimation of circuits in Colpitts oscillator [3–5] is dealt in terms of graph theory approach. We represent the Colpitts chaotic system from differential equations of Colpitts into graphical concept [6]. In Sect. 2, synchronization of chaotic system on graph [7–10] is established. In Sect. 3, synchronization of modified Colpitts oscillator on graph [11–13] is derived. Finally, in Sect. 4, we conclude that the synchronization approach in both the methods is same [14, 15]. S. Rasappan (B) Department of Mathematics, University of Technology and Applied Sciences - Ibri, 466, Ibri 516, Sultanate of Oman e-mail: [email protected] K. A. N. Kumar · R. N. Devi Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamilnadu 600062, India A. J. Obaid Faculty of Computer Science and Mathematics, University of Kufa, Kufa, Iraq e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_53

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2 Synchronization of Chaotic System on Graph This section deals the mapping of the synchronization in specified linear form. Let us state a finite-oriented bipartite graph is furnished . This indicates we have two non-intersecting non-empty finite sets of items. A = {ai }(i = 1 to n), B = {bi }(i = 1, 2, . . . , m) which are known as the graph’s nodes, and there is a finite set of ordered two vertices (defined as the graph’s edges). Let αik and βik represent the number of edges {ak , bi } and {bi , ak }, respectively. Let γik = βik − αik be assumed. Let x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) associate with vertex bi . The functions f i (t, x) and h i (t, y) defined with node ak and n + 1 variables, respectively. In the functions yk (t) and jk (t), h i and jk are defined by considered yk as unknown. The drive system on  is defined as  dxk = γik f i (t, x) + gk (t), k ∈ Natural number dt i=1 m

(1)

The response system on  is defined as  dyk = γik h i (t, x) + jk (t), k ∈ Natural number dt i=1 m

(2)

When u is control The error dynamics is defined as  dek = γik (h i (t, y) − f i (t, x)) + jk (t) − gk (t) + u dt i=1 m

(3)

is a system of differential equation on . The initial condition for t > 0 is xk (0) = xk0 , yk (0) = yk0 , (k = 1, 2, 3, 4)

(4)

moreover, consider that xk0 ≥ 0 and yk0 ≥ 0, (k = 1, 2, 3, 4)

(5)

For t ≥ 0, the functions f i (t, x) and h i (t, y) are defined in half space. We assume that

Synchronization of a Modified Colpitts Oscillator with Triangular Wave …

f i (t, x) ≥ 0 and h i (t, y) ≥ 0, where x ∈ xk , y ∈ yk

525

(6)

for t > 0 and xk ≥ 0, yk ≥ 0. The functions gk (t), jk (t) are assumed given that t ≥ 0, continuous and non-negative. Theorem 1 The graph  ×   contains nn  A-vertices and mn  + m  n B-vertices. Proof The graph  can be specified as the mapping of linear forms.

:

n 

αik αk →

k=1

n 

βik αk , (i = 1, 2, . . . , m)

(7)

k=1

with non-negative coefficients. The ak (k = 1, 2, 3, 4) is the A-vertices of graph . The numbers αik and βik refer to the edges, respectively. The scheme of the graph will be described as a system of (10) mappings (5). The schemes (10) and (11) of the graph are given, then it will be able to fix the problem. 



 :

n 



αil αl

→

l=1

n 

  βil αl , i = 1, 2, . . . , m 

(8)

l=1

  then by  ×   whose A-vertices of the graph are the pairs akl = ak , al with the scheme : 

 :

n  k=1 n  l=1

αik αkl →

n 

  βik αkl , i = 1, 2, . . . , m; l = 1, 2, . . . , n 

k=1

αil αkl

→

n 

l=1

  βil αkl , i = 1, 2, . . . , m  , k = 1, 2, . . . , n

(9)

Hence the Theorem 1. Let the error system (6) in which ak (k = 1, 2, 3, 4) takes part n  k=1

αik αk →

n 

βik αk , (i = 1, 2, . . . , m)

(10)

k=1

where αik and βik are the coefficients. Let f i (t, x) and h i (t, y) be the rate of change due to time ‘t’ and of xk , yk of ak , (k = 1, 2, 3, 4). Then, the system of equation of (6) corresponding to the system (13) takes the form (6) with gk = 0. Now consider (13) as in a graph’s plan , then Eq. (6) is equations on . Since the rate of change is non-negative, condition (9) is fulfilled. Usually,

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f i (t, x) = ki x1αi1 x2αi2 · · · xnαin

(11)

h i (t, y) = ki y1αi1 y2αi2 · · · ynαin

(12)

where ki is given positive constants which are chosen as the f i (t, x) and h i (t, y). In (14) we considered xlαi l = 1, ylαi l = 1 for ail = 0, xl = yl = 0 (l = 1, 2, . . . , n).

3 Synchronization of Modified Colpitts Oscillator on Graph The depiction of simplified illustrative diagram for modified Colpitts oscillator is undertaken in Fig. 1. In addition to electronic devices, communication systems also have wide usage of the sinusoidal oscillator. The following are the hypotheses for simplifying the extensive simulation of the complete circuit model. • The base-emitter (B-E) driving point (V-I) characteristic of the R E with triangular wave function is     2a −1 2π sin sin I E = f (VB E ) = I S (x3 ) π p     2a −1 2π and I E = f (VB E ) = I S sin sin (x1 ) π p where I S is the emitter current, a is amplitude and p is period of the B-E junction. • The state space is schematically represented in Fig. 1.

Fig. 1 Circuit diagram

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527

Fig. 2 Graph representation of Fig. 1

dVC1 = V0 − VC1 − VC2 + RC I L − RC f (VBE ) dt dVC2 = V0 − VC1 − VC2 − RC I0 + RC I L RC C 2 dt dVC3 = I L − (1 − α) f (VBE ) C3 dt dI L = −Rb I L − VC1 − VC2 − VC3 L dt RC C 1

Figure 2 describes the graph representation of the state space xi , i = 1 to 4. The modified state space representation of proposed system is x˙1 x˙2 x˙3 x˙4

= σ1 (−x1 − x2 ) + x4 − γ φ1 (x3 ) = ε1 σ1 (−x1 − x2 ) + ε1 x4 = ε2 (x4 − (1 − α)γ φ2 (x1 )) = −x1 − x2 − x3 − σ2 x4



 sin−1 sin 2πp (x3 ) , φ2 (x1 ) = where φ1 (x3 ) = 2a π drive system. The response system is y˙1 y˙2 y˙3 y˙4

2a sin−1 π

(13)



 sin 2πp (x1 ) which is

= σ1 (−y1 − y2 ) + y4 − γ φ1 (y3 ) + u 1 = ε1 σ1 (−y1 − y2 ) + ε1 y4 + u 2 = ε2 (y4 − (1 − α)γ φ2 (y1 )) + u 3 = −y1 − y2 − y3 − σ2 y4 + u 4

(14)

The error is defined by ei = yi − xi , i = 1, 2, 3, 4. The unknown parameters are updated by e˙1 e˙2 e˙3 e˙4

= −σ1 e1 − σ1 e2 + e4 − γ φ1 (y3 ) + γ φ1 (x3 ) + u 1 = −ε1 σ1 e1 − ε1 σ1 e2 + ε1 e4 + u 2 = ε2 e4 − ε2 (1 − α)γ (φ2 (y1 ) − φ2 (x1 )) + u 3 = −e1 − e2 − e3 − σ2 e4 + u 4

(15)

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Fig. 3 Graph representation of the synchronization between (13) and (14)

Figure 3 shows the synchronization representation between the states y1 –y4 and x 1 –x 4 . It is not a simple graph. It forms a Pseudo graph. The edges e5 –e16 are dummy edges. From (13), the drive system is defined on  is when k = 1,

dx1 dt

=

4 

γi4 f i (t, xk ) + g1 (t).

i=1

where γ14 f 1 (t, x1 ) = −σ1 X 1 , γ24 f 2 (t, x1 ) = −σ1 X 2 , γ34 f 1 (t, x1 ) = X 4 , γ44 f 4 (t, x1 ) = −γ φ1 (X 3 ), g1 (t) = 0. when k = 2,

dx2 dt

=

3 

γi4 f i (t, x2 ) + g2 (t).

i=1

where γ14 f 1 (t, x2 ) = −ε1 σ1 X 1 , γ24 f 2 (t, x2 ) = −ε1 σ1 X 2 , γ34 f 3 (t, x2 ) = ε1 X 4 , g2 (t) = 0. when k = 3,

dx3 dt

=

2 

γi4 f i (t, x3 ) + g3 (t).

i=1

where γ14 f 1 (t, x3 ) = ε2 X 4 , γ24 f 2 (t, x3 ) = −ε2 (1 − α)γ φ2 (X 1 ), g3 (t) = 0. when k = 4,

dx4 dt

=

2 

γi4 f i (t, x4 ) + g4 (t).

i=1

where γ14 f 1 (t, x4 ) = X 1 , γ24 f 2 (t, x4 ) = −X 2 , γ34 f 3 (t, x4 ) = −X 3 , γ44 f 3 (t, x4 ) = −σ2 X 4 , g4 (t) = 0. The response system is defined on  is 5 1 = i=1 γi4 h i (t, yk ) + j1 (t). when k = 1, dy dt where γ14 h 1 (t, y1 ) = −σ1 y1 , γ24 h 2 (t, y1 ) = −σ1 y2 , γ34 h 1 (t, y1 ) y4 ,γ44 h 5 (t, y1 ) = −γ φ1 (y3 ), γ54 h 5 (t, y1 ) = u 1 , j1 (t) = 0. 3 2 when k = 2, dy = i=1 γi4 h i (t, y2 ) + j2 (t). dt

=

Synchronization of a Modified Colpitts Oscillator with Triangular Wave …

529

where γ14 h 1 (t, y2 ) = −ε1 σ1 y1 , γ24 h 2 (t, y2 ) = −ε1 σ1 y2 , γ34 h 3 (t, y2 ) = −ε1 σ1 y4 ,γ44 h 4 (t, y2 ) = u 2 , j2 (t) = 0. 2 3 = i=1 γi4 h i (t, y3 ) + j3 (t). when k = 3, dy dt where γ14 h 1 (t, y3 ) = ε2 y4 , γ24 h 2 (t, y3 ) = −ε2 (1 − α)γ φ2 (y1 ), γ34 h 3 (t, y3 ) = u 3 , j3 (t) = 0. 5 4 when k = 4, dy = i=1 γi4 h i (t, y4 ) + j4 (t). dt where γ14 f 1 (t, y4 ) = −y1 , γ24 h 2 (t, y4 ) = −y2 , γ34 h 3 (t, y4 ) = −y3 , γ44 h 4 (t, y4 ) = −σ2 y4 , γ54 h 4 (t, x5 ) = u 4 , j4 (t) = 0. k − dxdtk . The error e is defined by dedtk = dy dt Then, the error dynamics is defined on  is 2 when k = 1, dedt1 = i=1 γi1 (h i (t, y1 ) − f i (t, x1 )) + j1 (t) − g1 (t). where γ11 (h 1 (t, y1 ) − f 1 (t, x1 )) = −σ1 e1 , γ21 (h 2 (t, y1 ) − f 2 (t, x1 )) = −σ1 e2 +u 1 . 3 when k = 2, dedt2 = i=1 γi2 (h i (t, y2 ) − f i (t, x2 )) + j2 (t) − g2 (t). where γ12 (h 1 (t, y2 ) − f 1 (t, x2 )) = −ε1 σ1 e1 , γ22 (h 2 (t, y2 ) − f 2 (t, x2 )) = −ε1 σ1 e2 , γ32 (h 3 (t, y2 ) − f 3 (t, x2 )) = ε1 e4 + u 2 . 2 when k = 3, dedt3 = i=1 γi3 (h i (t, y3 ) − f i (t, x3 )) + j3 (t) − g3 (t). where γ13 (h 1 (t, y3 ) − f 1 (t, x3 )) = ε3 e3 , γ23 (h 2 (t, y3 ) − f 2 (t, x3 )) −ε2 (1 − α)γ (φ2 (y1 ) − φ2 (x1 )) + u 3 . 4 when k = 4, dedt4 = i=1 γi4 (h i (t, y4 ) − f i (t, x4 )) + j4 (t) − g4 (t).

=

where γ14 (h 1 (t, y4 ) − f 1 (t, x4 )) = −e1 , γ24 (h 2 (t, y4 ) − f 2 (t, x4 )) = −e2 ,γ34 (h 3 (t, y4 ) − f 3 (t, x4 )) = −e3 , γ44 (h 4 (t, y4 ) − f 4 (t, x4 )) = −σ2 e4 + u 4 . For t > 0, the synchronization between the system is xk (0) = xk0 , yk (0) = yk0 , k = 1, 2, 3, 4 By applying Theorem 1, (13) and (14) are mapped on  ×   . Figure 4 describes the function between  and   . Fig. 4 Function of modified chaotic Colpitts oscillator (13) and (14)

(16)

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4 Positiveness of the Solution This section deals under specific natural conditions on the function f i (t, x) and h i (t, y) for non-negative gk , jk and initial conditions. Moreover, all vertices of the network that have been connected to the initial vertices by path in are strictly positive (6).  is have a path with the initial vertices. Thus, the solution is obtained. If αik > 0(βik > 0), then the vertex ak (bi ) is known as directly preceding to the vertex bi (ak ). It is taken as each B-vertex obtained by one direct A-vertex. It say that the function f i (t, x) and h i (t, y) are subordinate to the vertex ak . If f i (t, x) = 0 for xk = 0 and h i (t, y) = 0, for yk = 0. Condition I: The functions f i (t, x) and h i (t, y) are subordinate to all A-vertices directly preceding to bi vertex. Theorem 2 If xk0 > 0, yk0 > 0, gk (t) ≥ 0, jk (t) ≥ 0 and the condition I is satisfied. Then, Eq. (6) is positive in. xk0 > 0, yk0 > 0(t ≥ 0, k = 1, 2, 3, 4)

(17)

Proof Consider the value given on [0, T ] in (15). The solutions on t-axis and t0 be the point of intersection, so as E s (t) > 0(0 ≤ t < t0 , s = 1, 2, 3, . . . , n). For definiteness, let ek (t0 ) = (yk (t0 ) − xk (t0 )) = 0

(18)

In (6), γik < 0 and αik > 0 which give ak directly precedes bi , and consequently, h i (t, y) − f i (t, x) = 0 when ek = 0. So that h i (t, y) − f i (t, x) = ϕi (t)ek (t). Now define the notation  (19) ϕ(t) = γik ϕi (t) Here, the summation is carried out over all i for which γik < 0, and ψ(t) =



γik (h i (t, y) − f i (t, x)) + ( jk (t) − gk (t))

i

From above equation, sum value is carried out over all i. Obviously ψ(t) ≥ 0 on the interval [0, t0 ]. From (3), it gives ek (t) = ϕ(t)[yk (t) − xk (t)] + ψ(t). so

Synchronization of a Modified Colpitts Oscillator with Triangular Wave …

ek (t) = (yk (0) − xk (0))ea

t 0

ϕ(s)ds

t

+

e

t r

ϕ(s)ds

· ψ(r )dr

531

(20)

0

Equation (17) implies ek (t0 ) > 0. But from (16), which is contradiction. Hence, the theorem proved. Here, an acyclic graph describes which there are no oriented cycles and B-reachable vertex in a graph. Theorem 3 The adequate and required condition that a A-vertex to be a finite index it is should be reachable from A0 . This theorem investigates the positiveness of the solution of the system (3). A0 denotes the set of all those vertices ak for which xk0 > 0, yk0 > 0, so that xk0 = 0 and yk0 = 0 for all A-vertices. Theorem 4 In interval [0, T ], let x(t) and y(t) be a solution of (3), (7) with satisfied condition. Then, xk (t) ≡ 0, yk (t) ≡ 0(0 ≤ t ≤ T )(that is e(t) ≡ 0) of the graph  which is unreachable from A0 . Proof Let X be the collection of unreachable nodes and N be the collection of vertices from X and kbelongs toN , if γik = 0, then f i (t, y(t) − x(t)) = ϕi (t)(yl (t) − xl (t))

(21)

where l ∈ N , ϕi (t) is on [0, T ]. If αik > 0, then f i (t, y − x) = 0 when yk = xk , i.e. l = k in (18). If αik > 0 and βik > 0, which gives the A-vertices directly preceding bi , with atleast 1 unreachable vertex. Thus, Theorem 3 holds. In (3), xk , yk ∀k ∈ N only considered. According to Eq. (18), we obtain a system of linear (or linear) equations. Since xk0 = 0, yk0 = 0(k ∈ N ). Therefore, the system has only trivial solution. Condition II. If xk > 0, yk > 0 at all vertices ak directly preceding to the vertex bi , then f i (t, y − x) > 0 for t ≥ 0, xk > 0, yk > 0 (k = 1, 2, . . . , n). Theorem 5 Let y(t) − x(t) be a solution (3), (5) on [0, T ] with satisfied I and II. Then, yk − xk > 0(0 < t ≤ T ) for all vertices ak of the graph  reachable from A0 . Proof By the definition A0 , xk0 > 0, yk0 > 0 for all A-vertices (ak ) with index 0. By Theorem 2, ek (t) = (yk (0 − xk (0)))e

t 0

ϕ(s)ds

+

t

e

t r

ϕ(s)ds

· ψ(r )dr for t ∈ [0, T ]

0

and moreover, ψ(t) ≥ 0. Thus, ek (t) > 0. Therefore, the Theorem 5 holds with index 0. Suppose xk (t) > 0, yk (t) > 0 for all vertices ak , with an index less than or equal to s.

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Assume a vertex ak with index s + 1. Here needed to find atleast one B-vertex, followed ak with an index equal to s. Since all A-vertices directly preceding bl have an index not exceeding s, at them, the solution is positive. By virtue of condition II, h l (t, y) − fl (t, x) > 0 Thus, ψ(t) > 0 in ek (t) = ϕ(t)[yk (t) − xk (t)] + ψ(t) and on the basis of (17), ek (t) > 0. Therefore, the above result is exist.

5 Conclusion In this paper, the estimation of a modified Colpitts oscillator is analyzed in graph theory concept. The mapping of the synchronization is discussed. Synchronization of modified Colpitts oscillator on graph is investigated. Positiveness and the properties of the solution of graphical representation for system of differential equation are verified. The modified Colpitts oscillator circuit diagram is represented in graph concept. The general concept of synchronization of mapping is discussed for graph theory concept.

References 1. Kennedy, M.P. 1994. Chaos in the Colpitts oscillator. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 41 (11): 771–774. 2. Elwakil, A.S., and M.P. Kennedy. 1999. A family of Colpitts-like chaotic oscillators. Journal of the Franklin Institute 336 (4): 687–700. 3. Vaidyanathan, S., A. Sambas, and S. Zhang. 2019. A new 4-D dynamical system exhibiting chaos with a line of rest points, its synchronizationand circuit model. Archives of Control Sciences 29. 4. Kammogne, S.T., and H.B. Fotsin. 2011. Synchronization of modified Colpitts oscillators with structural perturbations. Physica Scripta 83 (6): 065011. 5. Pecora, L.M., and T.L. Carroll. 1990. Synchronization in chaotic systems. Physical review letters 64 (8): 821. 6. Vol’pert, A.I. 1972. Differential equations on graphs. Mathematics of the USSR-Sbornik 17 (4): 571. 7. Josic, K. 2000. Synchronization of chaotic systems and invariant manifolds. Nonlinearity 13 (4): 1321. 8. Belykh, I., M. Hasler, M. Lauret, and H. Nijmeijer. 2005. Synchronization and graph topology. International Journal of Bifurcation and Chaos 15 (11): 3423–3433. 9. Wu, C.W. 2005. Synchronization in networks of nonlinear dynamical systems coupled via a directed graph. Nonlinearity 18 (3): 1057. 10. Vaidyanathan, S., and S. Rasappan. 2014. Global chaos synchronization of n-scroll Chua circuit and Lur’e system using backstepping control design with recursive feedback. Arabian Journal for Science and Engineering 39 (4): 3351–3364. 11. Dörfler, F., and F. Bullo. 2012. Exploring synchronization in complex oscillator networks. In 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 7157–7170. IEEE.

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12. Belykh, I., V. Belykh, and M. Hasler. 2006. Generalized connection graph method for synchronization in asymmetrical networks. Physica D: Nonlinear Phenomena 224 (1–2): 42–51. 13. Hou, C.L., C.C. Huang, and J.W. Horng. 2007. A criterion of a multi-loop oscillator circuit. Journal of Circuits, Systems, and Computers 16 (01): 105–111. 14. Jost, J., and M.P. Joy. 2001. Spectral properties and synchronization in coupled map lattices. Physical Review E 65 (1): 016201. 15. Ahmadlou, M., and H. Adeli. 2012. Visibility graph similarity: A new measure of generalized synchronization in coupled dynamic systems. Physica D: Nonlinear Phenomena 241 (4): 326– 332.

T-Fuzzy Modular l-Filters in Commutative Lattice Ordered M-Group D. Vidyadevi and S. Meenakshi

Abstract Researchers from several mathematical disciplines have been seeking to extend their discoveries to a broader framework of the fuzzy world since then. Several authors have also made contributions to the topic of fuzzy lattice theory. This paper defines the t-norm that represents a triangular norm that is induced on fuzzy modular lattice ordered filters in a commutative lattice ordered m-group are characterized for all intents and fruitful purposes. Later, the existence of some most important properties is proved. Acceptable examples tabular columns that exhibit the supremum values and infimum values of the graphical rendering of the partially ordered set were also used to clarify the situation. Finally, we have explored that every fuzzy modular l-filter may be transformed into a T-fuzzy modular l-filter in a commutative lattice ordered m-group by establishing its lattice ordered m homomorphism. The existence of the image and pre-image of T-fml-filters in a commutative group was also demonstrated by this article. Keywords l-m homomorphism · t-norm · T-fuzzy modular l-filters (T-fml-filters) · T-fml-filters in G

1 Introduction Since then, researchers in a variety of mathematical disciplines have been attempting to apply their findings to a wider framework of the fuzzy environment. In addition, several authors have contributed to the field of fuzzy lattice theory.

D. Vidyadevi · S. Meenakshi (B) Vel’s Institute of Science, Technology and Advanced Studies, Chennai, India e-mail: [email protected] D. Vidyadevi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_54

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2 Preliminaries Definition 1 [9]. A set G which is not empty is a commutative l-m-group, then a(mp ∧ mq) = (amp) ∧ (amq) ai(mp ∨ mq) = (amp) ∨ (amq)

i. ii.

∀ a, b, p, q ∈ G, m ∈ M. Example 1. Consider G = {0, mp, mq, mr, 1}. Let μ : P → [0, 1] in G. Figure 1 represents the graphical rendering of a poset with an implied upward orientation. Table 1 also shows the supremum values and similarly, infimum values can be illustrated, which corresponds to Fig. 1. μ : P → [0, 1] of G satisfies all the axioms of fml-filters. Example 2. Consider G = {0, mp, mq, mr, 1}. Let μ : P → [0, 1] in G. Figure 2 depicts a poset with an assumed upward inclination in graphical form. μ : P → [0, 1] of G satisfies all the axioms of fml-filters. Example 1.

Fig. 1 Hasse diagram for the given data

Table 1 Cayley’s table: supremum values ∨

μ(0)

μ(mp)

μ(mq)

μ(mr)

μ(1)

μ(0)

μ(0)

μ(1)

μ(1)

μ(1)

μ(1)

μ(mp)

μ(1)

μ(mp)

μ(1)

μ(1)

μ(1)

μ(mq)

μ(1)

μ(1)

μ(mq)

μ(1)

μ(1)

μ(mr)

μ(1)

μ(1)

μ(1)

μ(mr)

μ(1)

μ(1)

μ(1)

μ(1)

μ(1)

μ(1)

μ(1)

T-Fuzzy Modular l-Filters in Commutative Lattice Ordered M-Group

537

Example 2.

Fig. 2 Hasse diagram for the given data

3 T-Fuzzy Modular l-filter in G Definition 3. ‘A fml-filter (G, µ) of G is called T-fml-filter, if   (i) μ(m(pq)) ≥ T μ(mp), μ(mq) (ii) μ(mp ∨ mq) ≥ T [μ(mp), μ(mq) (iii) μ(mp ∧ mq) ≥ T [μ(mp), μ(mq)   (iv) μ(mp ∨ mq) ∧ μ(mp ∨ mr) ≥ T μ(mp), μ(mq) ∧ μ(mp ∨ mr)  (v) 0 < mp < a ⇒ T [μ(a)] ≥ T μ(mp) , for all mp, mq ∈ G’. Example 3. Consider G = {0, mp, mq, 1}. Let μ : P → [0, 1] in G. Figure 3 is a graphical representation of a poset with an assumed upward direction, and Table 3 also illustrates the supremum values that correlate to Fig. 3. Similarly, the infimum values can be illustrated.: P[0, 1] of G satisfies all the axioms of T-fmlfilters. Example 4. Consider G = {0, mp, mq, mr, 1}. Let: P[0, 1] in G. Example 3.

Fig. 3 Hasse diagram for the given data

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Table 3 Cayley’s table: supremum values ∨

(0)

(mp)

(mq)

(1)

(0)

(0)

(1)

(1)

(1)

(mp)

(1)

(mp)

(1)

(1)

(mq)

(1)

(1)

(mq)

(1)

(1)

(1)

(1)

(1)

(1)

Example 4.

Fig. 4 Hasse diagram for the given data

The graphical representation of a poset with an assumed upward orientation further provides the supremum and infimum values for Fig. 4. T-fml-filters axioms are satisfied by: P[0, 1]. Theorem 1. Let f: G1 → G2 be the l-m homomorphism. Let (G1 , μ1 ) and (G1 , μ2 ) be the T-fml-filter of G1 and G2 such that μ2 [ f (mp)] = μ1 (mp) for all mp ∈ G. Proof: Here, f : G1 → G2 be the l-m homomorphism. For any mp1 , mp2 ∈ G. i.

ii. iii.

μ2 [ f ((mp)(mq))] = μ1 ((mp)(mq)) ≥ T [μ2 ( f (mp)), μ2 ( f (mq))]   μ2 f (mp ∧ mq)   ≥ T μ2 ( f (mp)), μ2 ( f (mq))   Similarly, μ2 f (mp ∨ mq)   ≥ T μ2 ( f (mp)), μ2 ( f (mq))

T-Fuzzy Modular l-Filters in Commutative Lattice Ordered M-Group

iv.



μ2 ( f (mp ∨ mq)) ∨ μ2 ( f (mp ∨ mr))

539



  ≥ T μ2 ( f (mp)), [μ2 ( f (mq)) ∧ μ2 ( f (mp ∨ mr) ⇒ Hence, (G2 , μ2 ) is a T-fml-filter of G2 . Theorem 2. Let f: G1 → G2 be the l-m homomorphism. Let (G1 , μ1 ) and of G1 and G2 . Then, the pre-image of f defined by (G1 , μ2 ) be the T-fml-filter  f −1 (μ2 ) (mp) = μ2 f (mp) is a T-fml-filter of G1 . Proof: Let the l-m homomorphism is defined by f : G1 → G2 . For any mp1 , mp2 ∈ G.   f −1 (μ2 ) (mp)(mq) = μ2 ( f ((mp)(mq)))   i. ≥ T μ2 ( f (mp), μ2 ( f (mq)     ≥ T [( f −1 (μ2 ) (mp) , f −1 (μ2 ) (mq)   f −1 (μ2 ) (mp) ∨ (mq) = μ2 ( f ((mp) ∨ (mq)))   ii. ≥ T μ2 ( f (mp), μ2 ( f (mq)     ≥ T [( f −1 (μ2 ) (mp) , f −1 (μ2 ) (mq)   f −1 (μ2 ) (mp) ∧ (mq) iii. Likewise,     ≥ T [( f −1 (μ2 ) (mp) , f −1 (μ2 ) (mq)   iv. f −1 (μ2 ) (mp ∨ mq) ∧ (mp ∨ mq)      ≥ T [ f −1 (μ2 ) (mp) , f −1 (μ2 ) (mq) ∧ f −1 (μ2 )((mp ∨ mr)) ⇒ Thus, the pre-image of f is a T-fml-filter of G1 . Theorem 3 Let (G, μ) be the T-fml-filter. Then, the set: (g/μ) = μ(g) is the T-fml-filter of Gμ .

G μ

→ [0, 1] determined by

Theorem 4 If (G, μ) is a T-fml-filter of G/I, then there exists T-fml-filter φ of G s.t. φ t = I for t = μ(0).

4 Conclusion Attempts have been made by researchers from a range of mathematical fields to extend their discoveries to a comprehensive scope of the fuzzy world. A number of authors have contributed to the field of fuzzy lattice theory. For all intents and

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purposes, this work defines the t-norm, which represents a triangular norm imposed on fuzzy modular lattice ordered filters in a commutative lattice ordered m-group. Later on, the existence of a few key properties is demonstrated. To clarify the situation, acceptable examples tabular columns displaying the supremum values and infimum values of the graphical rendering of the partially ordered set were also utilized. Finally, we have explored that every fuzzy modular l-filter may be transformed into a T-fuzzy modular l-filter in a commutative lattice ordered m-group by establishing its l-m-homomorphism. The existence of the image and pre-image of T-fml-filters in a commutative group was also demonstrated by this article. We can expand this definition to T-fuzzy modular prime filters and T-fuzzy modular quotient filters in the future.

References 1. Birkhoff, G. 1942. Lattice ordered groups. Annals of Mathematics Second Series. 2. Bakhshi, M. 1992. On fuzzy convep lattice ordered subgroups. Fuzzy Sets and System 51: 235–241. 3. Bhakat, S.K., and P. Das. 2013. On the definition of fuzzy groups. Iranian Journal of Fuzzy Systems 10: 159–172. 4. Malik, D.S., and Mordeson. 1992. Extension of fuzzy subrings and fuzzy ideals. Fuzzy Sets and Systems 45: 245–251. 5. Wu, X.-H., and J.-J. Zhou. 2000. Fuzzy discriminant analysis with kernel methods. Pattern Recognition 39 (11): 2236–2239. 6. Rosenfeld, A. 1971. Fuzzy groups. Journal of Mathematical Analysis and Applications 35: 512–517. 7. Li, J.-B., J.-S. Pan, and Z.M. Lu. 2009. Kernel optimization-based discriminant analysis for face recognition. Neural Computing & Application 18 (6): 603–612. 8. Li, J.-B., J.-S. Pan, and S.-C. Chu. 2008. Kernel class-wise locality preserving projection. Information Science 178 (7): 1825–1835. 9. Vidyadevi, D., and S. Meenakshi. 2021. Homomorphism on fuzzy modular l-ideal and the role of fuzzy modular l-filters in commutative l-m group. Advances in Mathematics: Scientific journal 10 (2): 891–899. 10. Ajmal, N., and K.V. Thomas. 1994. The lattice of fuzzy subgroups and fuzzy normal subgroups. Information Science 76: 1–11. 11. Subramanian, S., R Nagarajan, and Chellappa. 2012. Structure properties of M-fuzzy groups. Applied Mathematical Sciences 6 (11): 545–552. 12. Gu, W.P., S.Y. Li, and D.G. Chen. 1994. Fuzzy groups with operators. Fuzzy Sets and System 66: 363–371. 13. Manuel Ojeda-Hernández, Inma P. Cabrera. 2021. Closure systems as a fuzzy extension of meet-subsemilattices. In Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), 40–47. 14. Eleni Vrochidou, Chris Lytridis, Christos Bazinas, George A, Papakostas, Hiroaki Wagatsuma and Vassilis G. Kaburlasos. 2021. Brain signals classification based on fuzzy lattice reasoning. Mathematics 9 (1063): 1–16.

The Minimum Maximal Mean Boundary Dominating Seidel Energy of a Graph A. Meenakshi and M. Bramila

Abstract We execute an idea of minimum maximal mean boundary dominating M seidel energy of a graph (abbreviated as MMMBDSE) denoted  by SE B (Z ). We   execute the MMMBDSE of some standard graphs. Let Z V , E be a simple graph of g points  v of Z , the open neighbourhood of v is the  and d lines. In any point N (v) = u ∈ V  : (u, v) ∈ E  (Z ) . If v is an eccentric point of u, w is a neigh     bourhood of v, and d(u; w) ≤ d(u; v). Z V , E with V  (Z  ) =  v1 , v2 ,. . . vg , for i = j, a point vi is the boundary point of v j if d v j ; vt ≤ d v j ; vi , for all vt ∈ N (vi ). If v is a nearest boundary of u, then point v is known as of u is known boundary neighbour of u. If u ∈ V  , the boundary neighbourhood   as Nb (u) = u ∈ V  : d(u : w) ≤ d(u; v)∀w ∈ N (u) . If a set B ∈ V  (Z) is a boundary dominating set (BDS), then every point V  difference B is adjacent to some point in B. If V  difference B is not a boundary dominating set of Z , then a boundary dominating set B (Z) is known asmaximal boundary dominating set. Also     2 − g − 2b g |B| + g 2 − g |B| + g study the bounds of SE M and then (Z ), B 

  2 − g + gb2 . |B| g + g ≤ ≤ SE M (Z ) B Keywords Minimum · Maximal · Mean · Boundary · Seidel · Energy

1 Introduction If a set B ∈ V  (Z) is a boundary dominating set (BDS), then every point V  difference B is adjacent to some point in B. If V  difference B is not a boundary dominating set A. Meenakshi Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, India M. Bramila (B) Department of Mathematics, DRBCCC Hindu College, Chennai 600072, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_55

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of z, a boundary dominating set B (Z) is known as maximal boundary dominating set [1–3]. Let M be the dominating matrix related to Z . Let μ1 , μ2 . . . μg be the g  characteristic root, then the dominating energy of Z is E D (Z ) = |μi |. The mean g 

i=1

dominating energy of Z is then denoted as = |μi − μi |. i=1   Let Z with V  = v1, v2 , . . . vg and E  . The seidel matrix of Z is g × g matrix denoted as S(Z ) = Si j , where E DM (Z )

⎧ ⎨ −1 if vi v j ∈ E  Si j = 1 if vi v j ∈ / E ⎩ 0 if vi = v j The characteristic equation of S(Z ) is denoted by f g (Z , μ) = det(μI − S(Z )). The seidel characteristic roots of a graph Z are the characteristic roots of S(Z ).The g  seidel energy [14–17] of Z is denoted as S E(Z ) = |μi |. i=1

2 The Minimum Maximal Mean Boundary Dominating Seidel Energy (MMMBDSE) of Graph   Consider Z with V  = v1 , v2 , . . . vg and edge E  . If V  difference B is not a boundary dominating set of Z , then the boundary dominating set B of Z is known as maximal boundary dominating set. The boundary domination number γb (Z ) of Z is the minimum maximal number of a boundary dominating set [4, 5]. Any maximal boundary dominating set with minimum number is known as minimum maximal boundary dominating set (abbreviated as MMBD) [9–12]. Let B be the MMBD  seidel set of Z = (V , E  ), and the minimum maximal boundary set dominating  (abbreviated as MMBDS) matrix of Z is g × g matrix A B (Z ) = ai j , where ⎧ −1 ⎪ ⎪ ⎨ 1 ai j = ⎪1 ⎪ ⎩ 0

if vi v j ∈ E  if vi v j ∈ / E if i = j, vi ∈ B otherwise

Since A B (Z ) is real and symmetric, its characteristic roots b1 , b2 , . . . bg are real and positive. The MMBDSE of Z is denoted as g  SE B (Z ) = |bi |. i=1

The MMMBDSE  of Z is denoted as g [6–8, 13]  SE M B (Z ) = i=1 bi − b , where b is the mean of b1 , b2 , . . . , bg .

The Minimum Maximal Mean Boundary Dominating …

543

3 Minimum Maximal Mean Boundary Dominating Seidel Energy (MMMBDSE) of Some Standard Graph Theorem 3.1 If g ≥   K ≤ M M M B DS E i s SE M g B

3,

then

the

complete

graph

(g−1)(2g−1)+g(g−2)+1 . g

Kg

of

  Proof If g ≥ 3 with point set of the complete graph K g , V  = v1 , v2 , . . . vg .   The MMBDS set is B = v1 , v2 , . . . vg . The MMBDS matrix is ⎤ ⎡ 1 −1 −1 · · · −1 −1 ⎢ −1 1 −1 · · · −1 −1 ⎥ ⎥ ⎢ ⎥ ⎢ −1 −1 1 · · · −1 −1 ⎥ ⎢   ⎢ ⎥ A B K g = ⎢ .. .. .. . . .. .. ⎥ . . . ⎥ ⎢ . . . ⎥ ⎢ ⎢ −1 −1 −1 · · · 1 −1 ⎥ ⎦ ⎣ . −1 −1 −1 .. −1 1 g×g     Then, the characteristic equation is  A B K g − bI  = 0. (b − 2)g−1 (b + (g − 2)) = 0 Then, b = 2; (g − 1) times, b = −(g − 2) (one time). Then b = g1   MMBDS energy = SE B K g = |2|(g − 1) + |−(g − 2)| = 2(g − 1) + g − 2 ≤ 3g − 4. MMMBDS Energy SE M B



Kg



   g−1    1  1    = 2 − g  + −(g − 2) − g  i=1

  (g − 1)(2g − 1) + g(g − 2) + 1 SE M B Kg ≤ g Theorem 3.2 If g ≥ 2, then the MMMBDSE of a complete bipartite graph K g,g is   g+(g+1)(g−2)  2 + 5. K ≤ + 4g SE M g,g B g   set of the complete bipartite graph K g,g . V = Proof If g ≥ 2 with point v1 , v2 , . . . vg , u 1 , u 2 , . . . u g .   MMBDS set is B = v1 , v2 , . . . vg , u 1 .

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The MMBDS matrix is ⎡

1 1 1 .. .

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢   ⎢ ⎢ 1 A B K g,g = ⎢ ⎢ −1 ⎢ ⎢ −1 ⎢ ⎢ −1 ⎢ ⎢ . ⎣ ..

1 1 1 .. .

1 1 1 .. .

··· ··· ··· .. .

1 1 −1 −1 .. .

1 −1 −1 −1 .. .

··· ··· ··· ··· .. .

⎤ 1 −1 −1 −1 · · · −1 1 −1 −1 −1 · · · −1 ⎥ ⎥ 1 −1 −1 −1 · · · −1 ⎥ ⎥ .. .. .. .. . . .. ⎥ . . ⎥ . . . . ⎥ ⎥ 1 −1 −1 −1 · · · −1 ⎥ ⎥ −1 1 1 1 · · · 1 ⎥ ⎥ −1 1 0 1 · · · 1 ⎥ ⎥ −1 1 1 0 · · · 1 ⎥ ⎥ .. .. .. .. . . .. ⎥ . . ⎦ . . . .

−1 −1 −1 · · · −1 1

1

1 ··· 0

g×g

Then, the characteristic equation is      A B K g,g − bI  = 0   b g (b + 1)g−2 b2 − (2g − 1)b − (g + 1) = 0. Then, b = 0; √(g times), b = −1; (g − 2)times. (2g−1)±

4g 2 +5

b= , 2 Mean b = g1 . Then, MMBDS energy is 

SE B K g,g



   (2g − 1) ± 4g 2 + 5    = |0|(g) + |−1|(g − 2) +  .   2    SE B K g,g ≤ (g − 2) + 4g 2 + 5.

MMMBDS energy SE M B



K g,g



       g    g−2    (2g − 1) ± 4g 2 + 5 1   1 1 0 −  + −1 −  +  − . =  g  i=1  g  2 g i=1   g + (g + 1)(g − 2)  SE M + 4g 2 + 5. B K g,g ≤ g

  Theorem 3.3 If g ≥ 2, then the MMMBDSE of star K 1,g−1 is SE M B K 1,g−1 ≤  (g+1)(g−3)+1 + g 2 − 2g + 9. g   Proof K 1,g−1 is star with V  = v1 , v2 , . . . vg .

The Minimum Maximal Mean Boundary Dominating …

545

The MMMBDS set is B = {v1 , v2 }, and v1 is the centre point. ⎡

1 ⎢ −1 ⎢ ⎢   ⎢ −1 A B K 1,g−1 = ⎢ ⎢ .. ⎢ . ⎢ ⎣ −1 −1

−1 1 1 .. . 1 1

⎤ −1 1 ⎥ ⎥ ⎥ 1 ⎥ .. ⎥ ⎥ . ⎥ ⎥ 1 ··· 1 ⎦ 1 · · · 0 g×g

−1 1 0 .. .

··· ··· ··· .. .

Then, the characteristic equation is      A B K 1,g−1 − bI  = 0   b1 (b + 1)g−3 b2 − (g − 1)b − 2 = 0. Then, b = 0; (1 time), b = −1; (g − 3)times. b=

(g − 1) ±



g 2 − 2g + 9 , 2

Mean b = g1 . Then, MMBDS energy is 

SE B K 1,g−1



   (g − 1) ± g 2 − 2g + 9    = |0|(1) + |−1|(g − 3) +  .   2    SE B K 1,g−1 ≤ (g − 3) + g 2 − 2g + 9.

The MMMBDS energy SE M B



K 1,g−1



   g−3         (g − 1) ± g 2 − 2g + 9 1   1 1 −1 −  +  = 0 −  + − .  g g  2 g i=1

  (g + 1)(g − 3) + 1  SE M + g 2 − 2g + 9. B K 1,g−1 ≤ g Bound of MMMBDSE Theorem 4.1 (i)

Let Z be a simple graph of order g. Let b1 , b2 , ..... bg are the characteristic roots of MMBDS matrix AB (Z). Then

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A. Meenakshi and M. Bramil g 

bi = |B|

i=1

Proof . The trace of A B (Z ) g 

μi =

i=1

(ii)

g 

bii = |B|

i=1

If graph or star graph with point set V  =  a graph Z is either complete  v1 , v2 , . . . vg and edge set E and if b1 , b2 , . . . bg are the characteristic roots of minimum maximal boundary dominating seidel matrix A B (Z ), then g 

bi2 ≤ |B| + g 2 − g.

i=1

Proof. The trace of [A B (Z )]2 g  i=1

μi2

=

g  i=1

bi j

g 

g g   2   2 bi j b ji = bi j b ji = (bii ) + (bii )2 + 2

j=1

i= j

i=1

i=1

i< j

    2 g −g 2 2 − d (1) = |B| + 2 d(−1) + 2   2d + g 2 − g − 2d = |B| + 2 2 ≤ |B| + g2 − g.

Theorem 4.2 Let Z be with g points and d lines. If B is the MMBD set with d elements and b1 , b2 , . . . bg are the characteristic roots of MMBDS matrix A B (Z ) with mean b, then 

    |B| + g 2 − g − 2b g |B| + g 2 − g ≤ SE M B (Z ) ≤



  2 g |B| + g 2 − g + gb

Proof By Cauchy–Schwartz inequality,  g 

2 |ai bi |

i=1

=

 g 

ai2

i=1

  Take ai = 1, bi = bi− b.   g g  2  M 2     Then, SE B (Z ) = bi − b 1 i=1

i=1

 g  i=1

 bi2

The Minimum Maximal Mean Boundary Dominating …

547

 g  g    M 2 2 2 SE B (Z ) ≤ g bi + b i=1

i=1

  M  2 2 SE B (Z ) ≤ h |B| + g 2 − g + gb SE M B (Z )



  2 ≤ g |B| + g 2 − g + gb .  g 2    M 2   bi − b SE B (Z ) = i=1



2 SE M B (Z )

g    bi − b2 ≥ i=1



SE M B (Z )

2

g    |bi |2 −−2b |bi | g

≥

i=1

i=1

     M    2 SE B (Z ) ≥ |B| + g 2 − g − 2b + g |B| + g 2 − g  SE M B (Z )

≥

      |B| + g 2 − g − 2b g |B| + g 2 − g .

4 Conclusion We have studied the mean boundary dominating seidel energy and its minimum maximally, and also studied the bounds of a connected graph. In this paper, we have executed an idea of minimum maximal mean boundary dominating seidel energy of a graph and made its justifications. In future works, some specific graphs may be adopted to find the energy of those graphs. The future researchers may plan to find the dominating seidel energy of various graphs.

References 1. Adiga, C., A. Bayad, I. Gutman, and S.A. Srinivas. 2012. The minimum covering energy of a graph. Kragujevac Journal of Science 4 (3): 39–56. 2. Bapat, R.B. 2011. Graphs and matrices. Hindustan Book Agency. 3. Bapat, R.B., and S. Pati. 2011. Energy of a graph is never an odd integer. Bulletin of Kerala Mathematics Association 1: 129–132. 4. Chartrand, G., D. Erwin, G.L. Johns, and P. Zhang. 2004. On boundary vertices in graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 48: 39–53.

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5. Chartrand, G., D. Erwin, G.L. Johns, and P. Zhang. 2003. Boundary vertices in graphs. Discrete Mathematics 263: 25–34. 6. Chakradhara Rao, M.V., B. Satyanarayana, and K.A. Venkatesh. 2017. The minimum mean dominating energy of graphs. International Journal of Computing Algorithm 6 (1): 2278–2397. 7. Chakradhara Rao, M.V., B. Satyanarayana, and K.A. Venkatesh. 2017. The minimum mean boundary dominating energy of a graph. International Journal of Engineering Technology Science and Research 4 (10): 2394–3386. 8. Dinesh, A.C., and Puttaswamy. 2015. The minimum maximal domination energy of a graph. International Journal of Mathematics And its Applications 3 (3A): 31–40. 9. Gutman, I. The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz 103 (10978): 1–22. 10. Gutman, I., X. Li, and J. Zhang. 2009. Graph energy, (Ed-s: M. Dehmer, F. Em-mert), Sterib., Analysis of complex networks, from biology to linguistics, 145–174. Wiley-VCH, Weinheim. 11. Harary, F. 1969. Graph theory. Massachusetts: Addison Wesley. 12. Haynes, T.W., S.T. Hedetniemi, and P.J. Slater. 1997. Fundamentals of domination in graphs. Marcel-Dekker, Inc. 13. Kathiresan, K.M., G. Marimuthu, and M. Sivanandha Saraswathy. 2010. Boundary domination in graphs. Journal of Mathematics 33: 63–70. 14. Koolen, J.H., and V. Moulton. 2001. Maximal energy graphs. Advanced Applied Mathematics 26: 47–52. 15. X. Li, Y. Shi, and Gutman. 2012. Graph energy. New York Heidelberg Dordrecht, London: Springer. 16. Mohammed Alatif, P. 2016. The minimum boundary dominating energy of a graph. International Journal of Mathematics and its Applications 4 (1–B): 37–46. ISSN: 2347–1557. 17. Rajesh Kanna, M.R., R. Jagadeesh, and B.K. Kempegowda. 2016. Minimum dominating seidel energy of a graph. International Journal of Scientic and Engineering Research 7 (5).

GSβ -Compactness of Topological Spaces with Grills N. Kalaivani, K. Fayaz Ur Rahman, and Ahmad J. Obaid

Abstract The present task ambitioned at formation of the extension of topological system applying the idea of grill. The importance of the theory of grill between the topological spaces has been elaborated. It helps us to quantum the materials that were impossible to quantum, and it is used in many sectors such as computer science along with information technology. In the present aspect, we attain modern discovery like GSβ -compact space, GSβ -compact sets relative to space, GSβ -quasi H closed and countably GSβ -compact space. In this article, we are aiming to examine about the interpretation of GSβ -compact space with the help of GSβ -open sets in grill topological space. Succeeding this interpretation, we are examining the theory of GSβ -quasi H closed, and some of their characterizations has been studied. Using the theory of GSβ -compact space, we have intended for extracting the GSβ -compact sets related to space and examining about the main difference between GSβ -compact sets as well as space. And finally, countable GSβ -compact sets have been imported, and some theorems have been examined. In this present article, we have related the idea such as GSβ -compact space along with GSβ -compact relative to space, and many counter examples are inclined. On another hand, we examined the properties

The following notations have been used through this article:GSβ -compact-GSβ -com, GSβ -compact sets-GSβ -com sets, countably GSβ -compact space-coun GSβ -com space, compact-com, topological space-TS, compactness-Cn, neighbourhood-Nbd, topology-Topo, open-Opn, cover-Cvr, grill-Grl, subset-Sbt, closed-Csd, regular-Rgl, continuous-Cont, topology τ = θ, topological- Tpl, Hausdorff space-Haus space, elements-Elts, operator-Opr, space-Spc, relative-Rlt N. Kalaivani (B) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India e-mail: [email protected] K. F. U. Rahman Department of Mathematics, Aalim Muhammed Salegh College of Engineering, Chennai, India e-mail: [email protected] A. J. Obaid Faculty of Computer Science and Mathematics, University of Kufa, xxx, Iraq e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_56

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of the above-mentioned ideas and their relationships to each one and their earlier accompaniments. Keywords GSβ -com · GSβ -com · GSβ -com sets · Coun GSβ -com spaces

1 Introduction Topological developments are directly applied in topical fields just far artificial intelligence, information systems, economics along with data analysis [1–3]. The theory based on a grill has been imported by Choquet [4], with the ambition of hypothesize the ideas based on topological space. In [5, 6], the idea based on grills has been useful tool like nets along with filters, for getting rooted divination into farther studying of some topological properties like compactifications along with extension problems based on different kinds. Many more analysist [7–13] characterized along with established the properties based on generalized open sets in classical topology. A classical prototype for decomposition based on continuity along with semi-continuity was explained in the article of N. Levine [14]. In [15], Roy and Mukherjee imported along with inspected the concept based on topology θG unite into a grill G based on a topological space (X, θ ). Dasan and Thivagar [16] introduced the concept of N -topological space and also established the N -topological open sets. Nassef and Azzam [17, 18] described a new topological operator via grill. Applying the theory of grill many attractive results, properties and classification has been calculated, via [19–23]. Azzam [24] tested as well hypothesized the irresolute along with quasi-irresolute functions also the nano generalized closed sets via grill. Newly, Roy and Mukherjee [25] imported G-Compact along with inspected its relationship with compactness. In the present article, our aim is to instant some concepts based on compactness with grills like GSβ -Compactness, GSβ -Compact sets relative to space along with Countable GSβ -Compactness. Also, some of its inheritance along with classification are attained.

2 Preliminaries Definition 1. A non-empty recollection G of a spc Y which shift topo θ is forenamed as grl on the indicated spc if the succeeding situations are true. (1) (2) (3)

φ∈ /G A ∈ G along with A ⊆ B ⊆ Y ⇒ B ∈ G If A B ∈ G for A, B ⊆ Y, then A ∈ G or B ∈ G.

Definition 2. Let (Y, θ ) be a TS along with G be a grl upon Y. We define a mapping φ : P(Y ) → P(Y ), defined by φG (A, θ ) or φG (A) or simply φ(A) forenamed the opr combined with the grl G along with the topo θ and is defined by φG (A) =

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551

{y ∈ Y : B ∩ V ∈ G, ∀V ∈ θ (x)} where θ (x) erects the compilation based on all opn nbd based on x. We also entitle a mapping θ : P(Y ) → P(Y ) by θ (B) = B ∪ φ(B), for every B ∈ P(X ), which is a Kuratowski closure opr along with consequently induced a topo θG upon Y. Definition 3. Similar to a grl G upon a TS (Y, θ ), there exist a different topo θG = {V ⊆ Y ; ψ(Y |V )|Y |V }, where for any B ⊆ Y, θ (B) = B ∪ φ(B).a   Definition 4. Accredit G be a grl on a TS (Y, θ ). A cvr Vρ : ρ ∈ ω of Y is fore/ G. named as a β-cvr if there occur a finite β-opn sbt ρ0 of ρ so that Y \ ∪ρ∈ω0 Vρ ∈ A cvr which is not a β-cvr of X is forenamed as β-cvr. Definition 5. A sbt B of a space (Y, θ ) is called β-opn if B ⊆ Cl(Int(Cl(B)))   Definition 6. Let (Y, θ ) be a TS. A cvr V = Vρ : ρ ∈ ω of Y is a forenamed as GSβ -opn cvr if every member of V is an GSβ -opn set of Y .

3 GSβ -Com Spaces Definition 7. Let G be a grl on a TS (Y, θ ). Then, (Y, θ ) is forenamed as GSβ -Com space if every GSβ -opn cvr of Y is a β-cvr. Example 1. Let Y = {a, b, c}, θ = {∅, Y, {a}, {c}, {a, c}}, G = {Y, {a}, {a, c}}, then GSβ -Com spc = {∅, {b}, {c}, {a, b}, {b, c}} Remark 1. Every β-Com spc is GSβ -Com spc but the inverse is not true as the following example shows. Example 2. Let Y = {a, b, c}, θ = {∅, Y, {a}, {c}, {a, c}}, G = {Y, {a}, {a, c}}, GSβ -Com spc = {∅, {b}, {c}, {a, b}, {b, c}} and β-Com spc= {∅, {a}, {b, c}, {a, b}}. Here {b}, {c}, ∅ are GSβ -open sets does not belongs to β-Com spc. Remark 2. Each and every opn set in (Y, θ) is GSβ -opn. But the inverse need not be accurate. Example 3. Let Y = {a, b, c}, θ = {φ, {b}, Y }. Apparently, {a, c} is an GSβ -opn set which is not opn set. Remark 3. Each one of the β-Com spc (Y, θ ) is apparently GSβ -Com spc for any grl G on Y.   Proof If Vρ : ρ ∈ ω is any GSβ -opn  cvr of Y of an β-Com spc (Y, θ ), then there / G. Then, exists a finite sub-cvr Vρ : ρ ∈ ω0 of Y. Considering Y \ ∪ρ∈ω0 Vρ = ∅ ∈ (Y, θ ) is GSβ -Com.

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Proposition 1. Let G = P(Y )\φ, then GSβ -Cn of a spc (Y, θ ) deflate to the Cn and β-Cn of (Y, θ ). Proof For proof, fundamentally,  we want to prove that each one of β-com (Y, θ ) is a com spc. Let Vρ : ρ ∈ ω be any GSβ -opn cvr of Y. Considering each one of the opn set is GSβ -opn, then Vρ : ρ ∈ ω is  of Y. Since (Y, θ ) is β  GSβ -opn cvr comp spc, then there occurs a finite sub-cvr Vρ : ρ ∈ ω0 of Y and (Y, θ ) is a com spc. Now let (Y, θ ) be GSβ com, where G = P(Y )\φ. Let Vρ : ρ ∈ ω be any GSβ -opn cvr of Y. Considering (Y, θ ) is GSβ -com space, then there occurs a finite / G. Considering G = P(Y )\φ, then β-opn sbts ω0 of ω such that Y \ ∪ρ∈ω0 Vρ ∈ / G. Then, Vρ : ρ ∈ ω0 is a finite sub-cvr of Y. So (Y, θ) is β-com Y \ ∪ρ∈ω0 Vρ ∈ spc and com spc. Proposition 2. Let G = P(Y )\φ be a grl on a spc (Y, θ ) and the spc (Y, θG ) is GSβ -com. Then, (Y, θ ) is β-com and com (as θ ⊆ θG ) and hence is GSβ -com spa. Proof Let (Y, θG ) be GSβ -com  G = P(Y )\∅. Now we want to prove  spc, where that (Y, θ ) is β-com spc. Let Vρ : ρ ∈ ω be any GSβ -opn cvr of Y. Considering θ ⊆ θG , then Vρ : ρ ∈ ω be θG GSβ -opn cvr of Y. Considering (Y, θG ) is GSβ -com / G. Considering spc, then there occurs a finite sbt ω0 of ω such  that Y \ ∪ρ∈ω  0 Vρ ∈ / G.Then, Vρ : ρ ∈ ω0 is a finite sub-cvr of Y. G = P(Y )\φ, then Y \ ∪ρ∈ω0 Vρ ∈ So (Y, θ ) is β-com spc and consequently com spc. Remark 4. Each one of GSβ -com spc (Y, θ ) is apparently a β-com spc for any grl G on Y. Proof Let (Y, θ ) be a grl G upon  a TS. We need to show that every  GSβ -com  spc is a β-com spc. Let Vρ : ρ ∈ ω be any GSβ -opn cvr of Y . A cvr Vρ : ρ ∈ ω of Y is / G. forenamed to be β-cvr if there occurs a β-opn sbts ω0 of ω so that Y \ ∪ρ∈ω0 Vρ ∈ Consequently, (Y, θ ) is a β-com on grl TS. Theorem 1. Assume that G be a grl upon a TS (Y, θ ). Formerly, (Y, θG ) is a GSβ -com spc if (Y, θ ) is a GSβ -com spc. (Where θG is discrete topo upon Y ).   Proof Accredit Vρ : ρ ∈ ω be a primary θg GSβ -opn cvr of Y. Formerly by characterization of a base for θG , we get for each ρ ∈ ω, then Uρ = Vρ \Hρ , where / G. Formerly, Vρ : ρ ∈ ω is a θ -opn cvr Vρ ∈ θ and Hρ ∈  of Y. Considering each one of the opn set is β-opn, formerly Vρ : ρ ∈ ω is a θ GSβ -opn sbt cvr of Y. Considering (Y, θ ) is a GSβ -com, then there occurs a finite β-opn  V \H / G. Formerly, Y \ ∪ V = Y \ ∪ ω0 of ω so that Y \ ∪ρ∈ω0 Vρ ∈ ρ∈ω ρ∈ω ρ 0 0  / G. ⊆ Y (∪ρ∈ω0 Vρ \ ∪ρ∈ω0 V ) ∪ ∪ρ∈ω0 Hρ ∈ So Y, θg is a GSβ -com spc. Remark 5. Let G be a grl on a TS (Y, θ ). Then, (Y, θG ) is a β-com spc if (Y, θ ) is a GSβ -com spc.

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Proof Let (Y, θ, G) bea grl TS. We  need to prove that each one of the β-opn cvr is a GSβ -com spc. Let Vρ : ρ ∈ ω be any GSβ -opn cvr of Y , then there occurs a / G. Consequently, each one of the finite β-opn sbts ω0 of ω such that Y \ ∪ρ∈ω0 Vρ ∈ opn cvr is an opn sbt of Y. Consequently, each one of the β-com spc is a GSβ -com spc on a grl TS.

4 GSβ -Quasi H Csd Spc and GSβ -Rgl Spc Definition 8. A TS (Y, θ ) is forenamed as GSβ -quasi H csd (GSβ -QHC in short) if each one of the GSβ -opn cvr V of Y, there prevails a finite sub-compilation V0 of V so that X = ∪{cl(v) : v ∈ V0 }. A Haus GSβ -quasi H csd spc is forenamed as a GSβ H csdspc. Proposition 3. Accredit G be a grl on a TS (Y, θ ), so that θ \φ ⊆ G. If (Y, θ ) is a GSβ -com spc, then (Y, θ ) is a GSβ -quasi H csd.   Proof Assume that Vρ : ρ ∈ ω be any GSβ -opn cvr of Y. Considering (Y, θ ) is a / G. GSβ -com spc, then there occurs a finite β-opn sbt ω0 of ω such that Y \∪ρ∈ω0 Vρ ∈ Then, Int(Y \∪ρ ∈ ω0 Vρ ) = ∅. Or then, Int(Y \∪ρ ∈ ω0 Vρ) ∈ θ \φ ⊆ G and c / G,a conflict. It follows that Y \cl Y \ ∪ρ∈ω0 Vρ = ∅. consequently Y \ ∪ρ∈ω0 Vρ ∈ Consequently, X = ∪ρ∈ω0 cl Vρ and (Y, θ ) is a GSβ -quasi H csd spc. Proposition 4. Let (Y, θ ) be an GSβ -quasi H csd spc, where θ is a discrete topo on Y. Then, (Y, θ ) is a GSβ ρ-com spc, where G ρ is the grl given by G ρ = {B ⊆ Y : int cl(B) = ∅}   Proof Assume that (Y, θ ) be a GSβ -quasi H csd spc. Let V = Vρ : ρ ∈ ω be any GSβ -opn cvr  of Y. Then, thereis a finite sub-compilation V0 = Vρ : ρ ∈ ω0 of V so / G ρ . Or else, that X = ∪ cl Vβ : Vρ ∈ V  0 . Then, Y \ ∪ρ∈ω0 V ∈  Y \ ∪ρ∈ω0 Vρ ∈ G ρ and consequently, Y \Cl Int ∪ρ∈ω0 Vρ = ∅. Thus, Y \ ∪ρ∈ω0 cl Vρ = ∅ is a conflict. Consequently, (Y, θ ) is a GSβ ρ-com spc. Definition 9. Accredit (Y, θ ) be a TS along with G be a grl upon Y. Formerly, the spc Y is forenamed as G Sβ -Rgl if for each GSβ -csd set F in Y with y ∈ / F, there / G. occurs different GSβ -opn sets U along with V so that y ∈ U along with F\V ∈ Proposition 5. Accredit G be a grl upon a Haus spc (Y, θ ).If (Y, θ ) is a GSβ -com spc formerly it is a GSβ -Rgl spc. Proof Accredit F be any GSβ -csd sbt of Y and y ∈ / F. Considering Y to be a Haus spc, formerly every y ∈ F, there occurs two different opn sets Ux and Vy so that   x ∈ Ux and y ∈ Vy . Thus, Vy : y ∈ F ∪{Y \F} is an GSβ -opn cvr of Y. Considering -com, formerly there occurs finitely many points y1 , y2 , y3 , . . . , yn ∈ F (Y, θ ) is GSβ   n n / G. Accredit G = Y \ i=1 so that Y \ clVyi along with i=1 Vyi ∪ (Y \F) ∈

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n H = i=1 Vyi . Formerly, G along withH are different non-empty    GSβ -opn sets in n n / G. Vyi = Y \ V Y such that y ∈ G and F\H = F ∪ Y \ i=1 i=1 yi ∪ (Y \F) ∈ So (Y, θ ) is a GSβ -Rgl spc. Corollary 1. Let G be a grl on a Haus space (Y, θ ) such that θ \∅ ⊆ G. If (Y, θ ) is a GSβ -com spc, then it is a GSβ -csd set along with a GSβ -Rgl spc. Proof Validation trails directly from proposition 4 and 5. Theorem 2. Accredit A be a β-opn sub spc for a TS (Y, θ ). If Y has an opn β-cvr formerly, there is a β-cvr of Y, which repose of elts of A. Proof Accredit B be the compilation of all opn β-cvrs of Y. We know that B  is a partially ordered set. Formerly by the given statement B is a non-empty set. Let Q β be a linearly ordered sbts of B. Formerly Uβ Q β is a covering of Y. We defend that it is an opn β-cvr n of Y. For, if not, formerly there occurs F1 , F2 , F3 , · · · , Fn ∈ Uβ Q β so Fi ∈ / G. Now, there occurs a Q β0 ∈ B such that F1 , F2 , F, · · · , Fn ∈ that Y \ i=1 / B, a conflict. Consequently, by Zorn’s lemma, B consists of Q β0 . Thus, Q β0 ∈ a largest elt P. So if H is an opn set along with H ∈ / P, formerly there occurs / finitely many F1 , F, F3 , · · · , Fn ∈ P so that Y \(H ∪ F1 ∪ F2 ∪ F3 ∪ · · · ∪ Fn ) ∈ G. Hence, it is apparent that the group about opn sets which do not reside to P form a filter. To replete the proof, it is enough to show that A ∩ P is a β-cvr of Y. Let y ∈ Y. Since A is an β opn sub base for (Y, θ ), then there exists H1 , H2 , . . . , Hn ⊆ F. It / P for every follows that there occurs an Hi such that Hi ∈ P. For diversely, if Hi ∈ n Hi ∈ τ \P. Thus, F ∈ θ\P along with F ∈ / P, a i = 1, 2, · · · , n, formerly i=1 conflict. So x ∈ H ∈ A ∩ P along with therefore A ∩ P is a β-cvr of Y.

5 GSβ -Com Sets Relative to Space Definition 10. Let G be a grl on a TS (Y, θ ),a sbt B of aspc (Y, θ ) is forenamed as a GSβ -cm rlt to Y if for each one of the cvr Vρ : ρ ∈ ω of B by GSβ -opn sets of / G. Y, in view there occurs a finite sbt ω0 of ω such that B\ ∪ρ∈ω0 Vρ ∈ Example 4. Let Y = {1, 2, 3} and topo = {∅, Y, {1}, {2}, {1, 2}}, Grl TS G = {Y, {2}, {2, 3}} so GSβ -com rlt toY = {∅, {1}, {3}, {1, 3}}. Remark 6. Every β-Com spc implies GSβ -com rlt to Y but the inverse is not true as following example shows. Example 5. Let Y = {1, , 2, 3} and topo = {∅, Y, {1}, {2}, {1, 2}}, Grl TS G = {Y, {2}, {2, 3}} so GSβ -com rlt toY = {∅, {1}, {3}, {1, 3}}, β-Com spc= {∅, {1}, {1, 3}}. Here, GSβ -opn set {3} ∈ / β-Com spc. Theorem 3. The succeeding conditions are same for a sbt B of Y.

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(1) (2) Proof (1)

(2)

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B is GSβ -comrlt to Y, where the grl G = P(Y )\φ, B is β-comrlt to Y.   ⇒ (2) Let Vρ : ρ ∈ ω be a cvr of B by GSβ -opn set of Y. Considering B is GSβ -comrlt to Y, then there occur a finite sbt ω0 of ω such that B\∪ρ∈ω0 Vρ ∈ / G. Since G = P(Y )\∅, then B\ ∪ρ∈ω0 Vρ = ∅. Then, B = ∪ρ∈ω0 Vρ , thus B is β-comrlt toY.  ⇒ (1) Let Vρ : ρ ∈ ω be a cvr of B by GSβ -opn set of Y.  B  Considering of : ρ ∈ ω is a β-com spc rlt to Y, then there occurs a finite sub-cvr V ρ 0  / G, thus B is a GSβ -com B i.e.B ⊆ ∪ρ∈ω0 Vρ . Then, B\ ∪ρ∈ω0 Vρ = ∅ ∈ spcrlt to Y.

Proposition 6. Accredit G be a grl on a TS (Y, θ ), if Bi , i = 1, 2 are GSβ -com sbtsrlt to a spc (Y, θ ), formerly B1 ∪ B2 is GSβ -com rlt to Y.   Proof Accredit Vρ : ρ ∈ ω be a cvr of B1 B by GSβ -opn sets of Y. Formerly it is an GSβ -opn cvr of Bi for Bi , i = 1, 2. Since Bi is GSβ -com rlt to Y, formerly there / G and there occurs a finite sbt occurs a finite sbt ω1 of ω such that B1 \ ∪ρ∈ω1 Vρ ∈ / G. Considering (B1 \ ∪ρ∈ω1 Vρ ) ∪ (B2 \ ∪ρ∈ω2 Vρ ) ⊇ ω2 of ω so that B2 \ ∪ρ∈ω2 Vρ ∈ / G. Consequently, there occurs a finite sbt ω1 ∪ ω2 of ∧ (B1 ∪ B2 ) ∪ρ∈ω1 ∪ω2 Vρ ∈ / G, thus B1 ∪ B2 is a GSβ -com spc rlt to Y. such that (B1 ∪ B2 )\ ∪ρ∈ω1 ∪ω2 Vρ ∈ Theorem 4. Accredit (Y, θ ) be a spc with a grl G on Y.B is a GSβ -com spc rlt to Y , formerly (B, θ \B) is a G\B-com spc.   Proof Accredit { Vρ∩ B : ρ ∈ω} be θ/B-opn cvr of B, where Vρ ∈ θ for each one of ρ ∈ ω. Now, Vρ : ρ ∈ ω is a cvr of B by GSβ -opn sbts of Y. Considering B is a GSβ -com spc relative to Y , then there occurs a finite sbt ω0 of ω such that / B ∩ G. Considering B\ ∪ρ∈ω0 / G. Thus, B ∩ B\ ∪ρ∈ω0 Vρ  ∈ B\ ∪ρ∈ω0 Vρ ∈ Vρ ∩ B = B\ ∪ρ∈ω0 Vρ = B ∈ / B ∩ G, then there occurs a finite ∩ B\ ∪ V ρ∈ω ρ 0   / G\B. Thus, (B, θ \B) is a G\B-com spc. sbt ω0 of ω so that B\ ∪ρ∈ω0 Vρ ∩ B ∈ Example 6. The inverse of Theorem 4 is not true. Indeed, let Y = {a, b, c} and θ = {Y, φ, {b}}. The set U = {b, c} is a GSβ -opn set in θ. Let B = {a, b} and θ \B = {A, φ}. Evidently, U ∪ B = {b} is not an opnset in θ \B.

6 Coun GSβ -Com Spaces Definition 11. Let G be a grl on a TS (Y, θ ), a spc (Y, θ ) is forenamed as coun GSβ -com spc if for each one of the coun GSβ -opn cvr {Vs : s ∈ S} of Y, there occurs / G, where N denotes the set of positive a finite sbt S0 of S so that Y \ ∪V ∈S0 Vn ∈ integers.

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Example 7. Let Y = {x, y, z}, Topo θ = {∅, Y, {x}, {z}, {x, z}}, Grl topo G = {x}, {x, y}, Y } and Coun GSβ -com= {∅, {x}, {y}, {x, z}, {y, z}} Remark 7. Every β-Comp spc is Coun GSβ -Comp spc but the inverse is not true as the following example shows. Example 8. Let Y = {x, y, z}, Topo θ = {∅, Y, {x}, {z}, {x, z}}, Grl topo G = {x}, {x, y}, Y } and Coun GSβ -com= {∅, {x}, {y}, {x, z}, {y, z}} and β-Com spc = {∅, {x}, {y}, {x, z}}. Here, GSβ opn set {y, z} ∈ / β-Com spc. Proposition 7. Let G be a grl on TS (Y, θ ), if the spc (Y, θ ) is a coun GSβ -com spc, then for any coun family { f s : s ∈ S} of β-csd sets of Y such that ∪{ f s : s ∈ S} = ∅, / G. there occurs a finite sbts S0 of S such that ∪{ f s : s ∈ S0 } ∈ Proof Let { f s : s ∈ S} be a coun family of GSβ -csd sets of Y so that ∪{ f s : s ∈ S} = ∅. Then, {Y \ f s : s ∈ S} is a coun GSβ -opn cvr of Y . Then, there occurs a finite sbt S0 of S so that Y \ ∪s∈S0 (Y \ f s ) ∈ / G. This implies that ∪s∈S0 [Y (Y \ f s )] ∈ / G. Proposition 8. If (Y, θ ) is coun GSβ -com, G and G are two grls on Y so that G ⊇ G

formerly (Y, θ ) is acoun GSβ -com spc.

Proof Accredit {Vs : s ∈ S} be a coun GSβ -opn cvr of Y . Considering (Y, θ ) is a / G. coun GSβ -com spc, formerly there occurs a finite sbt S0 of S so that Y \ ∪s∈S0 Vs ∈

/ G . Consequently, (Y, θ ) is a coun GSβ -com For G ⊇ G , thus we get Y \ ∪s∈S0 Vs ∈ spc. Theorem 5. If G = P(Y )\φ is the grl on the spc (Y, θ ), formerly the succeeding conditions are same. (1) (2)

The spc (Y, θ ) is a coun β-com spc. The spc (Y, θ ) is a coun GSβ -com spc.

Proof (1)

(2)

⇒ (2) Let {Vs : s ∈ S} be a coun GSβ -opn cvr of Y. By (1), there occurs a / G. Thus, (Y, θ ) is a finite sub-cvr {Vs : s ∈ S} of Y . Then, Y \ ∪s∈S0 Vn = φ ∈ coun GSβ -com spc. ⇒ (1) Let {Vs : s ∈ S} be a coun GSβ -opn cvr of Y. By (2), there occurs a finite β-opn sbt S0 of S so that Y \ ∪s∈S0 Vs ∈ / G. Considering G = P(Y )\φ, then Y \ ∪s∈S0 Vs = φ. This entails that {Vs : s ∈ S0 } is a finite sub-cvr of Y. Thus, (Y, θ ) is a coun β-com spc.

Proposition 9. If (Y, θ ) is a coun β-com spc, then (Y, θ ) is a coun GSβ -com spc, where G is a grl on Y. Proof Let (Y, θ ) is a coun β-com spc, we need to show that there occurs a finite sbt / G. Let {Vs : s ∈ S} be a coun GSβ -opn cvr of Y . This S0 of S so that Y \ ∪U ∈S0 Vs ∈ cvr {Vs : s ∈ S} is a coun β-com spc, this opn cvr has a finite sbt ω1 of ω such that X = ∪ρ∈ω1 Vρ ∈ / G. This proves that (Y, θ ) is coun GSβ -com.

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Definition 12. A space (Y, θ ) is called a GSβ -Lindelöf spc if and only if each one of the GSβ -opn cvr of Y has a coun sub-cvr. Theorem 6. If (Y, θ ) is a coun GSβ -com spc and a GSβ -Lindelöf spc, then (Y, θ ) is GSβ -com, where G is a grl on (Y, θ ).   Proof Let Vρ : ρ ∈ ω be an GSβ -opn cvr of Y. Considering (Y, θ ) is a GSβ Lindelöf spc, there occurs a coun β-opn sbt ω1 of ω so that X = ∪ρ∈ω1 Vρ . But (Y, θ, G) is a coun GSβ -com spc and consequently, there occurs a finite sbt of ω0 of / G. Thus, (Y, θ ) is a GSβ -com spc. ω so that Y \ ∪ρ∈ω0 Vρ ∈ Theorem 7. Let f : (A, θ, G) → (B, σ ) be a β-irresolute surjection. If (A, θ, G) is a coun GSβ -com spc, then (B, σ, f (g)) is a coun GSβ f (g)-com spc. Proof Let {Cs : s ∈ S}be a coun GSβ -opn cvr of B. Considering f is a β-irresolute, then f −1 (Cs ) : s ∈ S is a coun GSβ -opn cvr of A, and consequently, there occurs a finite β-opn sbts S0 ofS so that A\ ∪s∈S0 f −1 (Cs ) ∈ / G. Considering f is onto, we / f (G). Thus, (B, σ, f (G)) is a coun have B\ ∪s∈S0 Cs = f A\ ∪s∈S0 f −1 (Cs ) ∈ GSβ f (g)-com spc. Theorem 8. Let f : (A, θ, G) → (B, σ ) be a GSβ -cont surjection, if (A, θ, G) is a coun GSβ -com spc, then (B, σ, f (G)) is a coun f (G)-com spc. Proof Let {Cs : s ∈ S} be a coun GSβ -opn cvr of B. Then, {B\Cs : s∈ S} is a coun csd sbts of B. Considering f is GSβ -cont, then f −1 (B\Cs ) : s ∈ S is a coun  GSβ -csd sbts of A. Then, A\ f −1 (B\Cs ) : s ∈ S is a coun GS  β -opn sbts of A. Considering A\ f −1 (B\Cs ) = f −1 (Cs ), then f −1 (Cs ) : s ∈ S be a coun GSβ opn cvr of A. Considering (A, θ, G) is a coun GSβ -com spc, there occurs a finite / f (G), β-pn sbts S0 of S such that A\Us∈S0 f −1 (Cs ) ∈ / G. Thus, f (A\Us∈S0 Cn ∈ / f (G), and consequently, (B, σ, f (G)) is f (G)-com spc. this gives B\Us∈S0 Cn ∈ Theorem 9. If f : (A, θ ) → (B, σ, G) is a pre-GSβ -opn bijection and (B, σ, G) is coun GSβ -com, then (A, θ ) is coun GSβ f −1 (g)-com. Proof Considering f is pre-GSβ -opn. Then, f (F) is GSβ -opn in (B, σ, G) for each one of the GSβ -opn set F in (Y, θ ). Considering f is bijection, then f −1 : (B, σ, G) → (A, θ ) occurs and an β-irresolute surjection. Therefore, the proof follows from the before theorem 8. The diagrammatic representation of a various spaces is given in Fig. 1.

7 Future Work Using the idea of GSβ -opn set, we have defined GSβ -Compact space, GSβ -Compact set relative to space along with countable GSβ -Compact space. The idea based on GSβ -Compact space on grill topological space can be extended to other analysis areas such as nano topology, fuzzy topology, intuitionistic fuzzy topology, neutrosophic theory, fuzzy filter, digital topology and so on.

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Fig. 1 Diagrammatic representation of relationship amongst various spaces

8 Conclusion A set is countable collection based on elements, without integrity of form. At the same time, some kinds of algebraic actions are applied on this set, indefinitely the elements of this set are correlated into a whole, so ultimately, it grows into a space. Using the theory based on β-opn sets, we have already generated a new series of open set called GSβ -opn set. In this present article, we have introduced the theory of GSβ Compact space along with some of their properties along with their characterization has been discussed. With the basics of the above concepts: β-opn subspc, GSβ -opn set, coun β-com, GSβ -Lindelöf space, β-irresolute surjection, GSβ -cont surjection, pre-GSβ -opn bijection, we have introduced Sβ -Rgl, GSβ -QHC, GSβ -Compact sets related to space along with countable GSβ -Compact space. Using the above concepts, we have linked the above ideas using theorems, propositions and counter examples.

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7. Dontchev, J. 1995. On generalizing semi-preopen sets. Mem. Fac. Sci. Kochi Univ. Ser. A, (Math.) 16: 35–48. 8. Al-Omari, A., and T. Noiri. 2011. Decomposition of continuity via grilles. Jordan Journal of Mathematics and Statistics 4 (1): 33–46. 9. Janaki, C., and I. Arockiarani. 2011. γ-open sets and decomposition of continuity Via grills. International Journal of Mathematical Archive 2 (7): 1087–1093. 10. Nj˙astad, O. 1965. On some classes of nearly open sets. Pacific Journal of Mathematics, 15 (3): 961–970. 11. Karthikeyan, N., and N. Rajesh. 2015. Faint continuous via topological grills. International Journal of mathematical archive 6: 24–28. 12. Reilly, I.L., and M.K. Vamanamurthy. 1985. On α-continuity in topological spaces. Acta Mathematica Hungarica, 45 (1–2): 27–32. 13. Aluja, J.G., and A.M.G. Lafuente. 2012. Towards an advanced modelling of complex economic phenomena. Studies in Fuzziness and Soft Computing 276. 14. Levine, N. 1963. Semi-open sets and semi-continuity in topological spaces. The American Mathematical Monthly 70 (1): 36–41. 15. Roy, B., and M.N. Mukherjee. 2007. On a typical topology induced by a grill. Soochow Journal of Mathematics 33 (4): 771. 16. Dasan, M.A., and M.L. Thivagar. 2021. New classes of grill N-topological sets and functions. Applied Sciences 23: 30–41. 17. Nasef, A.A., and A. Azzam. 2016. Some topological operators via grills. Journal of Linear and Topological Algebra (JLTA) 5 (03): 199–204. 18. Nasef, A.A., and A.F.A. Azzam. 2017. On class of function via grill. Journal of New Theory (17): 18–25. 19. Ghosh, M.K. 2017. Separation axioms and graphs of functions in nano topological spaces via nano β-open sets. Annals of Pure and Applied Mathematics 14 (2): 213–223. 20. Mashhour, A.S. 1982. On precontinuous and weak precontinuous mappings. In Proceedings of the Mathematical and Physical Society of Egypt 53, 47–53. 21. Mashhour, A.S., I.A. Hasanein, S.N. El-Deeb. 1983. α-continuous and α-open mappings. Acta Mathematica Hungarica 41 (3–4), 213–218. 22. Zhao, Y., Y. Yao, and F. Luo. 2007. Data analysis based on discernibility and indiscernibility. Information Sciences 177 (22): 4959–4976. 23. Mondal, D., and M.N. Mukherjee. 2012. On a class of sets via grill: A decomposition of continuity. An. St. Univ. Ovidius Constanta 20 (1): 307–316. 24. Azzam, A.A. 2016. Irresolute and quasi-irresolute functions via grill. Kasmera Journal 44 (7). 25. Roy, B., and M.N. Mukherjee. 2007. On a type of compactness via grills. Matematiˇckivesnik 59 (3): 113–120.

Study of Prey-Predator Model Formulation and Stability Analysis Balaram Manna, Subrata Paul, Ani mesh Mahata, Supriya Mukherjee, and Banamali Roy

Abstract One of the essential aspects in ecology is prey-predator communication. In this paper’s tritrophic phase food chain concept, ecological variables are considered as parametric-functional interval numbers. In comparison with the cost of intermediate predator fear, we provide a three-species food chain model in which prey growth is delayed. The positivity and boundedness of solutions, as well as the existence criterion and local stability analysis of non-negative equilibrium points, were investigated. Using an appropriate Lyapunov function, we also explain the global simulation study of such equilibrium position. It is been discovered that raising the amount of anxiety causes the mechanism to achieve a stable position by switching dynamics many times. In this paper, we looked at how varying levels on cooperation rate impact the system’s dynamics. We present numerical representations of our analytical findings. Lastly, the study concludes with a conclusion that supports all of the analytical conclusions through comprehensive numerical simulations. Keywords Ecosystem · Food chain · Imprecise environment · Stability · Numerical simulation

B. Manna Department of Mathematics, Swami Vivekananda University, Barasat—Barrackpore Rd, Telinipara, Bara Kanthalia, West Bengal 700121, India S. Paul (B) Department of Mathematics, Arambagh Govt. Polytechnic, Arambagh, West Bengal, India e-mail: [email protected] A. Mahata Mahadevnagar High School, Maheshtala, Kolkata, West Bengal 700141, India S. Mukherjee Department of Mathematics, Gurudas College, Kolkata, West Bengal 700054, India B. Roy Department of Mathematics, Bangabasi Evening College, Kolkata, West Bengal 700009, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9_57

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1 Introduction 1.1 Mathematical Modeling Mathematical modeling is a crucial and required technique for addressing a wide range of real-world issues. In several fields of science and technology, researchers have given close attention to modeling on various types of phenomena. Various functions in theoretical ecology and ecological modeling have added up interfaces and interactions between biotic and abiotic components, where many key parameters have been used. Researchers were being provided information to represent real-life occurrences by constructing various types of mathematical models in an imperfect environment during the previous few decades. The dynamical models of such interacting species are provided by Lotka-Volterra systems of the equations, and there are three types of model equations dealing with interactions [1–8]. Three models are (i) predator–prey model, (ii) competition model, and (iii) mutualistic model. The characteristics of the single spices population in which like organisms associate to include population growth and reproductions, regulation and equilibrium, and fluctuation and periodicity [9–18]. Such types of phenomena are the very special topic of study of the population ecologist, as the effect of the chemical and physical factor is the domain of the physiological ecologists. In reality, any ecosystem consists of several species which are interrelated among them. Actually, multi-species population models to understand the nature and diversity of natural ecosystem. So there are different types of interactions between two different species living in the same ecosystem.

1.2 Novelties of the Work In this section, the notions and concept are serve with as follows. (i)

(ii) (iii)

In such a prey-predator system with a system of linear system of equations and a fuzzy environment, the parameters are represented as fuzzy interval numbers. The prey-predator system is explored in several scenarios in the context of fuzzy variables. The findings have confirmed the imprecise solution’s viewpoint, which has been visually and quantitatively connected.

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2 Preliminaries Interval-valued function: Let the interval [c, d] with α > 0, β > 0. A function Hn (x, γ ) = (cn (x))1−β (dn (x))β for 0 ≤ β ≤ 1 can be used to depict the interval [c, d]. Some arithmetic operations can be defined as Let P (x) = [g1 (x), gu (x)] = (g1 (x))1−β (gu (x))β and Q (x) = [h l (x), h u (x)] = (h l (x))1−β (h u (x))β be two interval-valued functions so that g1 (x) > 0, h 1 (x) > 0 · ∀ · x. 



3 Model Formulation The following assumption is taken to develop the model system: u(t) v(t) w(t) r k: a1 : b1 a2 m1 b2 m2 γ1

density of prey, density of intermediate predator, density of top predator, inherent prey population growth rate, environment’s sustaining capacity, according to Holling Type-I functional activation, the intermediate predator’s predation rate, according to the rule of mass action, the top predator’s predation rate, coefficient of intermediate predator’s energy conversion, intermediate predator population mortality rate, coefficient of top predator’s energy conversion, death rate of top predator, inter-specific predation rate top predator.

The given system is 1−β β r r  u du 1−β β 1−β β = L Ru 1− − a1R a1L uv − b1R b1L uw, dt 1 + ρv k dv 1−β β 1−β β = a2L a2R uv − m 1R m 1L v, dt dw 1−β β 1−β β 1−β β = b2L b2R uw − m 2R m 2L w − γ1R γ1L w2 . dt

3.1 Positivity and Boundedness of Proposed Model Theorem 1 The model system’s solutions are all positive. Proof From the model system (1), we have

(1)

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du = uϕ(u, v, w)dt with 

 1−β β rL r R  u 1−β β 1−β β 1− − a1R a1L v − b1R b1L w . ϕ(u, v, w) = 1 + ρv k

(2)

Taking integration, then t

∫ ϕ(u,v,w)dt

u(t) = u(0) e 0

> 0∀t

(3)

t

Similarly, v(t) = v(0)e

1−β β m 1R m 1L ].

∫ η(u,v,w)dt 0

1−β β

> 0 ∀t. Where η(u, v, w) = [a2L a2R u −

t

And w(t) = w(0)e Where

∫ ζ (u,v,w)dt 0

> 0 ∀t.

  1−β β 1−β β 1−β β ζ (u, v, w) = b2L b2R u − m 2R m 2L − γ1R γ1L w .

(4)

As a result, all of the system’s solutions are positive. Theorem 2 The model system (1) has bounded solutions. Proof Let the function

1−β β

Z =u+

1−β β

a1R a1L 1−β β

a2L a2R

v+

b1R b1L 1−β β

b2L b2R

w.

(5)

Differentiating with respect to time on the above function, we have 1−β β

1−β β

du a a1L dv b1R b1L dw dZ = + 1R + 1−β , 1−β β β dt dt a2L a2R dt b2L b2R dt 1−β β r r  u 1−β β 1−β β = L Ru 1− − a1R a1L uv − b1R b1L uw 1 + ρv k 1−β β

+ a1R a1L uv − 1−β β

+ b1R b1L uw −

1−β β

1−β

β

a1R a1L m 1R m 1L 1−β β

a2L a2R 1−β β

1−β

β

b1R b1L m 2R m 2L 1−β β

b2L b2R

v 1−β β

w−

1−β

β

b1R b1L γ1R γ1L 1−β β

a2L b2R

w2 ,

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=

565

1−β β 1−β β 1−β β u  a1R a1L m 1R m 1L rL r R  u 1− − v 1−β β 1 + ρv k a2L a2R 1−β β

+

1−β β

a1R a1L b1R b1L 1−β β

1−β β

1−β

β

uw −

b1R b1L m 2R m 2L

w2 +

rL r R k. (1 + ρv)

1−β β

1−β β

w−

1−β

1−β β

a a b2L b2R b2L b2R 2L 2R1−β β

1−β β 1−β β a a1L b b1L r r ≤ −α u + 1R v + 1R w − L R (u − k)2 1−β β 1−β β (1 + ρv)k a2L a2R b2L b2R 1−β β

−

1−β

β

b1R b1L γ1R γ1L 1−β β

a2L b2R

β

b1R b1L γ1R γ1L

w2 ,

1−β β

 1−β β 1−β β 1−β β 1−β β 1−β β +α L α R z ≤ y L y R . where α = min r L r R , m 1R m 1L , γ1R γ1L . Therefore, dZ dt 1−β β yR

Where y L

0 < Z (t) < < Z (t)
ρv12 + v1 −

1−β β rR 1−β β a1R a1L

rL



1−β

β

m m 1L 1 . 1 − 1R 1−β β a2L a2R k

= 0.

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(d)

To obtain interior equilibrium E ∗ (u ∗ , v∗ , w∗ ), we solve the following system of equations 1−β β u rL r R  1−β β 1−β β 1− − a1R a1L v − b1R b1L w = 0, 1 + ρv k 1−β β

β

1−β

a2L a2R u − m 1R m 1L = 0, 1−β β

1−β

β

1−β

β

b2L b2R u − m 2R m 2L − γ1R γ1L w = 0. After solving the above equations, we get  u∗ =

1−β

β

m 1R m 1L ∗ 1−β β , w a2L a2R

=

1−β

β

m 2R m 2L 1−β β γ1R γ1L

1−β β

1−β

β

b2L b2R m 1R m 1L 1−β β 1−β β a2L a2R m 2R m 2L

(6)

 − 1 . Hence u ∗ and w∗ is positive if

1−β β

1−β

β

1−β β

1−β

β

b2L b2R m 1R m 1L a2L a2R m 2R m 2L

> 1.

(7)

And v∗ is to be obtained from the following equation 1−β β u rL rR  1−β β 1−β β 1− − a1R a1L v − b1R b1L w = 0. 1 + ρv k

(8)

Equation (8) has a positive root for    2  u ∗  1−β β  1−β β 1−β β 1−β β 1−β β a1R a1L ρ . a1R a1L + b1R b1L ρw∗ > 4 b1R b1L w∗ − r L r R 1 − k

4.2 Local Stability Analysis The Jacobian matrix of the model (1) at (u, v, w) can be represent as ⎡ ⎤ A B C 1−β β  rL r R 1−β β 1−β β J (u, v, w) = ⎣ D E O ⎦ where A = 1+ρv 1 − 2u − a1R a1L v − b1R b1L w, k F O G   1−β β r L r R u (1− uk )ρ 1−β β 1−β β 1−β β B = − − a1R a1L u, C = −b1R b1L u, D = a2L a2R v, E = (1+ρv)2 1−β β

1−β

β

1−β β

1−β β

1−β

β

1−β

β

a2L a2R u − m 1R m 1L , F = b2R b2L w, G = b1R b1L u − m 2R m 2L − 2γ1R γ1L w. Theorem 3 The model system (1) always exhibits unstable behavior at E 0 (0, 0, 0). 1−β β rR,

Proof The Eigen values are given by λ1 = r L 1−β β −m 2R m 2L . Clearly, λ1 > 0, λi < 0 for i = 2, 3.

1−β

β

λ2 = −m 1R m 1L , λ3 =

Study of Prey-Predator Model Formulation and Stability Analysis

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Hence, the system (1) is always unstable. Theorem 4 The system (1) is locally stable at E 1 (k, 0, 0) if k > 1−β β rR.

Proof The Eigen value is λ1 = −r L

1−β

β

1−β

β

m 2R m 2L +m 1R m 1L . 1−β β 1−β β a2L a2R +b2R b2L

To solve the equation,

  1−β β 1−β β 1−β β 1−β β λ2 + m 2R m 2L + m 1R m 1L − a2L a2R k − b2R b2L k λ    1−β β 1−β β 1−β β 1−β β + m 1R m 1L − a2L a2R k m 2R m 2L − b2R b2L k = 0,

(9)

we find another two Eigen values. 1−β β 1−β β of negative roots of Eq. (9) is m 2R m 2L + m 1R m 1L <  Therefore, the condition  1−β β 1−β β m m +m 1R m 1L 1−β β 1−β β a2L a2R + b2R b2L k. So the system (1) is locally stable, if k > 2R1−β 2L . β 1−β β a2L a2R +b2R b2L

Theorem 5 The model  system (1) is  locally stable at E 2 (u 1 , v1 , 0) if  1−β β 1−β β 1−β β 1−β β b1R b1L .m 1R m 1L > a2L a2R .m 2R m 2L . Proof The matrix (J ) at E 2 (u 1 , v1 , 0) can be written as

One of the Eigen values of Jacobian matrix is λ1 = and other Eigen values are obtain from the equation



1−β β

1−β

β

b1R b1L .m 1R m 1L 1−β β a2L a2R

 1−β β − m 2R m 2L ,



 1−β β  rL rR 2u 1 1−β β λ −λ 1− − a1R a1L v1 1 + ρv1 k ⎛  1−β β  ⎞    r L r R u 1 1 − uk1 ρ 1−β β 1−β β ⎠ a =0 +⎝ + a a u a v 1 1 1R 1L 2L 2R (1 + ρv1 )2 2

Theorem 6 The model (1) is locally asymptotically stable at E ∗ (u ∗ , v∗ , w∗ ) if Ai > 0, for i = 1, 2, 3 and A1 A2 > A3 . Proof The characteristic equation at E ∗ (u ∗ , v∗ , w∗ ) is given by

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λ3 + A1 λ2 + A2 λ + A3 = 0.

(10)

where 1−β β

rL r R u∗ 1−β β + γ1R γ1L w∗ , (1 + ρv∗ )k  1−β β   u∗ ∗ 1−β β ∗ ∗ 1−β β r L r R u 1 − k ρ 1−β β ∗ A3 = a2L a2R v w γ1R γ1L + a1R a1L u , (1 + ρv∗ )2 A1 =

1−β β

r L r R u ∗ 1−β β ∗ 1−β β 1−β β .γ γ w + b1R b1L u ∗ w∗ .b2R b2L (1 + ρv∗ )k 1R 1L  1−β β   u∗ ∗ 1−β β ∗ r L r R u 1 − k ρ 1−β β ∗ + a2L a2R v + a1R a1L u (1 + ρv∗ )2

A2 =

The roots of (10) are negative or have a negative real component, according to Rowth-Hurwitz criterion if A1 > 0, A3 > 0 and A1 A2 − A3 > 0. Theorem 7 The system (1) is globally asymptotically stable at E ∗ (u ∗ , v∗ , w∗ , if 

 ∗ v−v 

1−β β

rL rR 1−β β − a2L a2R (1 + ρv)(1 + ρv∗)



 1−β β  r L r R ρ(uv∗ − vu ∗ ) 1−β β − w − w∗ b2L b2R ≥ 0. + k(1 + ρv)(1 + ρv∗) Proof Let the function L at E ∗ (u ∗ , v∗ , w∗ ):   u    v    u − u ∗ − u ∗ log ∗ + v − v∗ − v∗ log ∗ u  v   w ∗ ∗ + w − w − w log ∗ w

L=

(11)

Taking derivative of Eq. (11) with respect to time is dL u − u ∗ du v − v∗ dv w − w∗ dw = + + . dt u dt v dt w dt Substituting the values of

du dv , dt dt

and

dw dt

from (1) into (12), we have

  1−β β dL u − u∗ rL r R  u 1−β β 1−β β − a1R a1L uv − b1R b1L uw = u 1− dt u 1 + ρv k

(12)

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569

 v − v∗  1−β β 1−β β a2L a2R uv − m 1R m 1L v v  w − w∗  1−β β 1−β β 1−β β b2L b2R uw − m 2R m 2L w − γ1R γ1L w2 , + w

 1−β β 1−β β = u − u ∗ b2L b2R u − m 2R m 2L  1−β β  rL r R  u 1−β β 1−β β 1−β β 1− − a1R a1L v − b1R b1L w − γ1R γ1L w 1 + ρv k 

 1−β β 1−β β + v − v∗ a2L a2R u − m 1R m 1L 

 1−β β 1−β β 1−β β + w − w∗ b2L b2R u − m 2R m 2L − γ1R γ1L w , +

1−β β 2 r L r R (u − u ∗ )2 1−β β − γ1R γ1L w − w∗ ∗) k(1 + ρv)(1 + ρv   1−β β

  r r 1−β β L R − u − u ∗ v − v∗ − a2L a2R (1 + ρv)(1 + ρv∗)    r L1−β r Rβ ρ(uv∗ − vu ∗ ) 

∗ ∗ 1−β β − w − w b2L b2R . − u−u k(1 + ρv)(1 + ρv∗)

=−

dL ≤ 0, if So dt   1−β β ∗ ∗ r r R ρ(uv −vu ) ∗ 1−β β ≥ 0. + Lk(1+ρv)(1+ρv ∗) − (w − w )b2L b2R

(v − v∗ )



1−β β

rL r R (1+ρv)(1+ρv∗)

1−β β

− a2L a2R



5 Numerical Simulation We use rigorous numerical study to assess and confirm the analytical conclusions of our model system in this section. To graphically forecast the model’s solution, we utilized the mathematical approaches MATLAB (2018). Case-I Analyze the equilibrium at E 1 (k, 0, 0). We analyze the system (1) with the model parameter values from Table 1 and m

1−β

β

m +m

1−β

m

β

1R 1L as given in taking β = 0.7, 1.0, which fulfills the condition k > 2R1−β 2L β 1−β β a2L a2R +b2R b2L Theorem 4. Table 2 depicts the Eigen values at E 1 (k, 0, 0). The system’s time series solution is shown in Fig. 1a, b for β = 0.7, 1.0.

Case-II Study the nature of the system (1) at E 2 (u 1 , v1 , 0). We analyze the system (1) with the model parameter values from Table 1 and taking β = 0.7, 1.0. Table 3 displays the Eigen values. One of the Eigen values is found to be negative, while the others are conjugate complexes with negative real parts. As a result, the planar equilibrium point E 2 (u 1 , v1 , 0) is always stable, confirming our analytical conclusions (see Theorem 5). Figure 2a, b depicts the

570 Table. 1 In Cases I, II, and III, it illustrates the parametric values that were used to investigate the system

B. Manna et al. Parameters

Values (for axial)

Values (for planar)

Values (for interior)

r

[2.01, 2.05]

[1.5, 2.8]

[2.01, 3.1]

a1

[0.51, 0.63]

[0.0020, 0.0026]

[0.05, 0.06]

a2

[0.00075, 0.00084]

[0.010, 0.015]

[0.6, 0.7]

b1

[0.00047, 0.00055]

[0.002, 0.007]

[0.002, 0.003]

b2

[0.0064, 0.0074]

[0.020, 0.028]

[0.05, 0.06]

m1

[0.5, 0.7]

[0.70, 0.77]

[0.5, 0.9]

m2

[1.30, 1.70]

[0.50, 0.55]

[0.01, 0.02]

γ

[1.7, 1.8]

[2.05, 3.06]

[0.7, 0.8]

ρ

0.5

0.5

0.5

k

90

90

90

Table. 2 Depicts the nature at E 1 (k, 0, 0) for β = 0.7, 1.0 β Equilibrium point

Eigen values of the matrix

Nature

0.7 (90, 0, 0)

(−2.0100, −0.4799, −0.7716)

Stable

1.0 (90, 0, 0)

(−2.0500, −0.4244, −1.0341)

Stable

Fig. 1 Nature at E 1 (k, 0, 0) is depicted for β = 0.7, 1.0 Table. 3 Shows the nature at E 2 (u 1 , v1 , 0) for β = 0.7, 1.0 β Equilibrium point

Eigen values of the matrix

Nature

0.7 (54.23, 25.65, 0)

(−0.3575, −0.045 ± 0.287567i)

Stable

1.0 (46.67, 35.73, 0)

(−0.3973, −0.0385 ± 0.309867i)

Stable

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Fig. 2 Depicts the time series plot of the system (1) for β = 0.7, 1.0. The planar equilibrium point E 2 (u 1 , v1 , 0) is stable supported by these graphs

model system’s time series plot across the time range [0, 800], demonstrating the stability at E 2 (u 1 , v1 , 0) for β = 0.7, 1.0. Case-III Explore the effects of E ∗ (u ∗ , v∗ , w∗ ). We analyze the system (1) with the model parameter values from Table 1 and taking β = 0.7, 1.0 that satisfy the requirement specified in Theorem 6. Table 4

Fig. 3 indicates the time series plot of the system (1) for various values of parameter ‘β” in the time range [0, 500]. These graphs show that the interior equilibrium point E ∗ (u ∗ , v ∗ , w∗ ) is stable. Table. 4 The nature at E ∗ (u ∗ , v ∗ , w∗ ) for β = 0.7, 1.0 is depicted β Equilibrium point

Eigen values of the matrix

Nature

0.7 (0.892, 9.13, 0.048)

(−0.035, −0.00244 ± 0.7415i)

Stable

1.0 (0.71, 10.12, 0.534)

(−0.416, −0.0019 ± 1.79678i)

Stable

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displays the Eigen values. As a result, all the Eigen values are negative component, indicating that equilibrium point E ∗ (u ∗ , v∗ , w∗ ) is stable.

6 Conclusion In this study, we look at a prey-predator model in which prey populations get infected as a result of intermediate predators’ fear. Fear generated by intermediate predators is thought to alter prey species’ reproduction and growth rates, thus we included a function, called the fear function, in the development term of our system. In an imprecise scenario, we created the model system and generated all of the system’s stable states points. All of these system equilibrium points have been provided in the supporting environment, along with feasibility studies and stability evaluations. To justify and confirm the model’s analytical conclusions, careful numerical simulations were performed. In nature, foraging behavior is frequent, and the best foraging technique plays an essential role in prey-predator engagements. Prey populations exhibit anti-predator reactions such as habitat switching, reduced feeding time, and heightened vigilance as a result of predator fear. The research would also allow experimental lists to carry out a number of precise planning while developing interspecies relationship-based initiatives in nature, paving the way for the preservation of several important species that are endangered or vulnerable, as well as the natural environment overall.

References 1. Liu, M., and C. Bai. 2016. Analysis of a stochastic tri-trophic food-chain model with harvesting. Journal of Mathematical Biology 73 (3): 597–625. 2. Roy, J., and S. Alam. 2019. Dynamics of an autonomous food chain model and existence of global attractor of the associated non-autonomous system. International Journal of Biomathematics. 12 (8): 1–23. 3. Paul, S., S.P. Mondal, and P. Bhattacharya. 2016. Numerical solution of Lotka Volterra prey predator model by using Runge–Kutta–Fehlberg method and Laplace Adomian decomposition method. Alexandria Engineering Journal. 55 (1): 613–617. 4. Alidousti, J., and M.M. Ghahfarokhi. 2019. Dynamical behavior of a fractional three-species food chain model. Nonlinear Dynamics. 95 (3): 1841–1858. 5. Barbier, M., and M. Loreau. 2019. Pyramids and cascades: a synthesis of food chain functioning and stability. Ecology Letters 22 (2): 405–419. 6. Matia, S.N., and S. Alam. 2013. Prey–predator dynamics under herd behavior of prey. Universal Journal of Applied Mathematics 1 (4): 251–257. 7. Banerjee, C., and P. Das. 2018. Impulsive effect on Tri-Trophic food chain model with mixed functional responses under seasonal perturbations. Differential Equations and Dynamical Systems. 26 (1–3): 157–176. 8. Paul, S., S.P. Mondal, and P. Bhattacharya. 2017. Discussion on fuzzy quota harvesting model in fuzzy environment: fuzzy differential equation approach. Modeling Earth Systems and Environment 3: 3067–3090.

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9. Roy, J., and S. Alam. 2019. Dynamics of an autonomous food chain model and existence of global attractor of the associated non-autonomous system. International Journal of Biomathematics 12 (8): 1–23. 10. Erbe, L.H., V.S.H. Rao, and H.I. Freedman. 1986. Three species food chain models with mutual interference and time delays. Mathematical Biosciences 80: 57–80. 11. Pal, D., and G.S. Mahapatra. Dynamics behaviour or a predator-prey system of combined harvesting with interval-valued parameters. Nonlinear Dynamics. 12. Ricker, W.E. 1948. Method of Estimating Vital Statistics of Fish Populations. Indiana Univ. Publ. Sci. Ser. 13. Matia, S.N., and S. Alam. 2013. Prey-Predator dynamics under herd behavior of prey. Universal Journal of Applied Mathematics 1 (4): 251–257. 14. Alam, S. 2009. Risk of disease-selective prediction in an infected prey predator system. Journal of Biological Systems 17 (01): 111–124. 15. Xiao, Q., B. Dai, and L. Wang. 2015. Analysis of a competition fishery model with intervalvalued parameters: extinction, coexistence, bionomic equilibria and optimal harvesting policy. Nonlinear Dynamics 80 (3): 1631–1642. 16. Alidousti, J., and M.M. Ghahfarokhi. 2019. Dynamical bevabior of a fractional three species food chain model. Nonlinear Dynamics 95 (3): 1841–1858. 17. Roy, J., S. Alam. 2020. Study on autonomous and non autonomous version of a food chain model with intraspecific competition in top predator. Mathematical Methods in the Applied Sciences 43 (6): 3167–3184. 18. Kar, T.K., and H. Matsuda. 2007. Sustainable management of fishery with a strong Allee effect. Trends in Applied Sciences Research 2 (4): 271–283.

Author Index

A Abhishek Dhar, 191 Ahmed A. Elngar, 115 Ahmed J. Obaid, 213, 523, 549 Akanksha Kumari Gupta, 79 Ambika, G., 315, 461 Ambik Mitra, 409 Anandan, R., 201 Animesh Mahata, 169, 335, 379, 435, 561 Anirban Mitra, 131, 231 Anjan Dutta, 131 Ashish Acharya, 335 Ashok Kumar Shaw, 221

B Balaram Manna, 169, 561 Balvinder Singh Gill, 49 Banamali Roy, 169, 335, 435, 561 Bidisha Paul, 191 Bishal Chakraborty, 191 Bramila, M., 541

C Chandramouli Das, 409 Chandrasekaran, E., 1, 271 Chris Lettecia Mary, C. J., 361

D Dac-Nhuong Le, 325 Deepak, M. C., 289 Deepika, K., 497 Doanh-Ngan-Mac Do, 241

Dora Pravina, C. T., 573

F Fayaz Ur Rahman, K., 549

G Gitosree Khan, 79 Gunasekar, T., 151

H Ha Huy Cuong Nguyen, 11 Hanaa Hachimi, 1, 91, 159, 255, 369, 391, 427 Ha Thuy Mai, 241

J Jashanpreet Singh Sraw, 289 Jhanjhi, N. Z., 409, 505 Joydeep Kundu, 305, 401

K Kalaivani, N., 1, 471, 549 Kala Raja Mohan, 115, 159, 261 Kamali, R., 361 Kamelia Jahnouni, 391 Kanagajothi, D., 369 Kanchana, A., 427 Komala Durga, B., 35 Koushik Mukhopadhyay, 305 Krithika, R., 517

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 S.-L. Peng et al. (eds.), Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing 1422, https://doi.org/10.1007/978-981-19-0182-9

575

576 Kshira Sagar Sahoo, 409 Kumaran, N., 151 Kunal Anand, 505 L Leoni Sharmila, S., 97, 105 M Mainak Chakraborty, 221, 435 Manikandan, A., 201 Md Jakir Hossain Molla, 59, 179 Meenakshi, A., 123, 427, 541 Meenakshi, S., 419, 497, 535 Mona Visalakshidevi, E., 471 Mostafa Ezziyyani, 471 Moumita Chatterjee, 231 Muthumari, G., 69 N Nagadevi Bala Nagaram, 159, 325 Nagalakshmi, T., 315, 461 Narayan Mondal, 379 Naresh Kumar Jothi, 151 Narmada Devi, R., 69, 213, 523 Neel Armstrong, A., 123, 151, 255 Nguyen Hoa Huy, 141 Nguyen Hoang Vu, 449 Nguyen Thi Bich Thuy, 449 Nguyen Thi Trang, 449 Nguyen Tung Lam, 141 Niranjan Kumar, K. A., 523 Niranjan, S. P., 35 Norazaliza Mohd Jamil, 49 P Parveen Ahmed Alam, 59 Perumal, S., 25 Poongothai, S., 97, 105 Pham Hoang Van, 449 Premalatha, M., 281 Priyadharshini, M., 91 Pugalarasu Rajan, 11 Punyasha Chatterjee, 179 R Rajesh, P., 91 Rahul Kar, 221 Regan Murugesan, 115, 159, 261, 325 Ritesh Prasad, 401 Runa Ganguli, 179 Ruthara, M., 485

Author Index S Sachindra Nath Matia, 169, 379 Saptarshi Haldar, 79 Saranya Palanivelu, 271 Sateesh Kumar, T. V., 419 Sathish Kumar Kumaravel, 115, 261, 325 Saurabh Adhikari, 59, 79, 131, 179, 191, 231 Senbagamalar, J., 91, 123, 427 Shankar Kr. Shaw, 79 Shariful Alam, 169, 335, 379 Shiplu Das, 401 Shubham Suman, 409 Sitikantha Chattopadhyay, 305, 401 Sk Md Obaidullah, 59 Smriti Ghosh, 221, 335 Sonal Aggarwal, 79 Soumya Sen, 59, 79, 131, 179, 231 Sourav Saha, 59, 131, 179, 191 Souvik Pal, 305, 401 Sreelakshmi, S., 573 Subhendu Maji, 221, 379 Subhra Prokash Dutta, 305 Subrata Paul, 435, 561 Sudharani, R., 315 Sudipta Roy, 231 Suman Bhattacharya, 231 Supriya Mukherjee, 435, 561 Suresh Rasappan, 11, 69, 115, 159, 213, 261, 325, 523 Sushruta Mishra, 505

T Ta Duy Phuong, 449 Thangaraj, M., 35 Ton Quang Cuong, 141 Tran Doan Vinh, 347 Tran Thuy Le, 449 Trung Tran, 241

U Uthra, G., 485

V Vicente García-Díaz, 261 Vidyadevi, D., 535 Vijayalakshmi, C., 281 Vijayalakshmi, G. M., 11 Vijayarangam, J., 25 Viswanath, J., 25, 573 Vu Minh Trang, 141