Advancements of Grey Systems Theory in Economics and Social Sciences 9811999317, 9789811999314

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Series on Grey System

Camelia Delcea Liviu-Adrian Cotfas

Advancements of Grey Systems Theory in Economics and Social Sciences

Series on Grey System Series Editors Sifeng Liu, Institute of Grey Systems Studies, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China Yingjie Yang, Center for Computational Intelligence, De Montfort University, Leicester, UK Jeffrey Yi-Lin Forrest, Department of Mathematics, Slippery Rock University, PA, PA, USA

This series aims to publish books on grey system and various applications in the fields of natural sciences, social sciences and engineering. This series is devoted to the international advancement of the theory and application of grey system. It seeks to foster professional exchanges between scientists and practitioners who are interested in the models, methods and applications of grey system. Through the pioneering work completed over 40 years, grey data analysis methods have become powerful tools in addressing system with poor information. Books published with this series will explore the models and applications of grey system, in order to tackle poor information more effectively and efficiently. The series aims to provide state-of-the-art information and case studies on new developments and trends in grey system research and its potential application to solve practical problems. Coverage includes, but is not limited to: • • • • • • • • • •

Foundations of grey systems theory Grey sequence operators Grey relational analysis models Grey clustering evaluations models Techniques for grey system forecasting Grey models for decision-making Combined grey models Grey input-output models Techniques for grey control Various applications of grey system models in the fields of natural sciences, social sciences and engineering.

Camelia Delcea · Liviu-Adrian Cotfas

Advancements of Grey Systems Theory in Economics and Social Sciences

Camelia Delcea Department of Economic Informatics and Cybernetics Bucharest University of Economic Studies Bucharest, Romania

Liviu-Adrian Cotfas Department of Economic Informatics and Cybernetics Bucharest University of Economic Studies Bucharest, Romania

ISSN 2731-4936 ISSN 2731-4944 (electronic) Series on Grey System ISBN 978-981-19-9931-4 ISBN 978-981-19-9932-1 (eBook) https://doi.org/10.1007/978-981-19-9932-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1 State of the Art in Grey Systems Research in Economics and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dataset Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dataset Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Papers’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 8 9 10 13 18 36 39 40

2 Grey Numbers for Sentiment Analysis and Natural Language Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sentiment Analysis Lexicons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arithmetic of Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Grey Numbers Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval Grey Numbers Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sentiment Analysis Using Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 46 49 49 50 51 53 56 73 82 83

3 Supplier Selection Using Grey Systems Theory . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State of the Art in Supplier Selection Through Grey Systems Theory Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 88 v

vi

Contents

Bibliometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey Supplier Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of Grey Numbers to Supplier Selection . . . . . . . . . . . . . . . . . . Xie and Xin Approach to Supplier Selection Based on Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical Application on Supplier Selection . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .(. . . . . . . . .). . . . . . . Annex A: The Code for Determining the Probability P ⊗a ≤ ⊗b . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review on Airplane Boarding . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review on Grey Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey Clustering and Whitenization Weight Functions . . . . . . . . . . . . . . . . . Assumptions and Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions on Social Distance Measures . . . . . . . . . . . . . . . . . . . . . . . Assumptions on Variations in Back-to-Front Boarding Method . . . . . . Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Agent-Based Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation Results for the Considered Metrics . . . . . . . . . . . . . . . . . . . . Grey Clustering of the Variations in Back-to-Front Boarding Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Public Opinion Assessment Through Grey Relational Analysis Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Argument for the Public Opinion Assessment in the COVID-19 Vaccination Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review on COVID-19 Vaccination . . . . . . . . . . . . . . . . . . . . . Literature Review on Grey Relational Analysis . . . . . . . . . . . . . . . . . . . . Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dataset Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classifier Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stance Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey Relational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . News Incidence Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dataset Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classifier Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stance Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . News Incidence on People’s Opinion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 95 104 114 119 134 134 135 139 139 141 141 144 145 150 153 154 158 162 164 164 169 173 174 179 179 179 181 181 182 184 184 184 186 186 188 189 190 193 194

Contents

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Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6 Grey Systems Theory Approach to Linear Programming . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief Presentation of Linear Programming Basic Elements . . . . . . . . . . . . Forms of the Linear Programming Problem . . . . . . . . . . . . . . . . . . . . . . . Feasible Solutions, Feasible Region and Optimal Solutions . . . . . . . . . Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review on Grey Linear Programming . . . . . . . . . . . . . . . . . . . . . Theoretical Approach to Grey Linear Programming . . . . . . . . . . . . . . . . . . Whitening Parameters Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liu and Lin’s Prediction Type Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lin and Liu’s Positioned Solution Model . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 204 206 208 215 218 221 221 237 242 249 250

7 Complex Projects Management with Interval Grey Numbers . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief Presentation of Projects Management . . . . . . . . . . . . . . . . . . . . . . . . . Activities and the Precedence Relationships . . . . . . . . . . . . . . . . . . . . . . Network Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gantt Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review on Grey Systems Theory in Projects Management . . . . Theoretical and Practical Approach to Project Scheduling with Grey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Approach to Project Scheduling with Grey Numbers . . . . Practical Approach to Project Scheduling with Grey Numbers . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 253 254 254 256 263 264

8 Hybrid Approaches Featuring Grey Systems Theory . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey-Fuzzy Approaches in Economics and Social Sciences . . . . . . . . . . . Dataset Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Papers’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey-Neural Networks Approaches in Economics and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dataset Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Papers’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

281 281 282 282 284 286 287 294

265 265 274 278 279

295 295 298 298 300

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Mixed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey-Genetic Algorithms Approaches in Economics and Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dataset Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Papers’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grey-Rough Sets Approaches in Economics and Social Sciences . . . . . . . Dataset Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Papers’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 308 308 311 313 315 319 320 320 322 324 324 329 331 332

Chapter 1

State of the Art in Grey Systems Research in Economics and Social Sciences

Introduction The beginning of the grey systems theory dates back to March 1982 when Professor Deng J.L. published the work “Control Problems of Grey Systems” in the “Systems & Control Letters” journal [1]. In a 7-page work, Deng [1] discusses the basic concepts of “grey systems” and “grey matrix”, along with more in-depth elements such as the controllability and the stability of a grey systems. Through the use of a comprehensive language, Deng [1] describes in simple terms the idea behind a grey system. The author defines a grey system as being “a system containing known and unknown information” and states that the name of the “grey systems” derives from the concept of the “black box” (Fig. 1.1). According to the black-box concept, the objects were called “black” if the information contained within them was completely unknown. Contrary, when analyzing an object and knowing everything about it, the object was called “white”. The grey systems theory comes in and fills the gap between the black and white systems by nuancing the degree of knowledge related to a system. According to the grey systems theory, there might be situations in which one knows only partially the information related to a system, leaving an entire part of unknown information. In consequence, the information about a system might be considered a mix of known and unknown information, therefore, a mix of white and black information, namely a grey information (Fig. 1.2). Grey information could be further seen though the means of limited data, grey numbers or grey structure when referring to a particular system [2]. Considering the scientific literature developed in the 40 years passed since the term of grey system has been used by Deng [1], there have been a lot of contributions in the area of studying and developing the new models associated with the grey systems theory. The current scientific literature associated with the grey systems theory has enlarged and enriched the global knowledge related to the uncertainty systems, along with other system theories such as fuzzy sets [3] and rough sets [4].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_1

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1 State of the Art in Grey Systems Research in Economics and Social …

Fig. 1.1 The concept of “black box”

Input

Output

Black Box

Fig. 1.2 The concept of “grey system”

Input

Output

Grey system Grey variables

Grey variables

Information partially known and partially unknown

In a speech generically titled “Why the world is grey?”, given in 2011 at a conference dedicated to grey systems theory Professor Andrew A.M., CEO of WOSC (World Organization of Systems and Cybernetics) points to some aspects of grey systems theory. The speaker starts his argument by considering the observation made by the WOSC President, Vallee R., that “we live in a grey world”. Thus, Andrew [5] shows that many of the mathematical methods and models used in analyzes are not as robust as they are supposed to be. An example in this direction is given by the assumption that the methods are based on the fact that data are normally distributed or that they follow a Gaussian distribution. In fact, the speaker argues that these distributions are quite rare in practical applications [5]. It is true that these drawbacks can be “dealt with” in real-life applications by using other methods developed over time. In fact, the purpose of the remark made by Andrew is to pay more attention to the sources of uncertainty, which can be found sometimes at the very basic levels of the analysis [6]. Other two very interesting points completed the speech given by Andrew [6], namely that we are not living in a “black world” (or, at least, not directly in a black world), and that we are not living in a “white world”. Having all these said, the speaker concludes that we are living in a grey world. The existence of a grey world, characterized by limited data or poor information, has attracted scientists from all around the world to embrace, study, apply and contribute to the development of the new theory: the grey systems theory [7]. In the 40 years of development, the grey systems theory has attracted a series of prominent scholars such as, but not limited to: Professor Liu S.F., Professor Yang Y., Professor Xie N.M., Professor Lin Y., Professor Javed S.A., Professor Hipel K., Professor Salmeron J.L., Professor Mi C., Professor Yuan C., Professor Javanmardi E., Professor Mierzwiak R., Professor Khuman A. [8–21], who have enriched the

Introduction

3

knowledge related to grey systems theory and have contributed to its continuous development. In an editorial written by Professor Liu S.F. at the anniversary of the 40 years of grey systems theory, a timeline of the most prominent events that have marked the history of the grey systems theory in these years is provided [22]. A series of associations and societies have promoted over the years the grey systems theory: International Association of Grey Systems and Decision Sciences (IAGSUA), Grey Systems Society of China (GSSC), Grey Systems Society of Pakistan (GSSP), Polish Scientific Society of Grey Systems (PSGS), Chinese Grey System Association (CGSA), Grey Systems Committee (IEEE Systems, Man, and Cybernetics Society), Centre for Computational Intelligence (De Montfort University), etc. [23–27]. The associations have been actively involved in organizing conferences, roundtables, meetings, events, journals, all of them with focus on the development of the grey systems theory. The fields of applicability for the grey systems theory are vast and they include, but they are not limited to: agriculture [28], forestry [29], electricity [30], manufacturing [31], online social networks [32], healthcare [33, 34], aviation [35], traffic safety [36, 37], tourism [38], telecommunications [39], exploring human cognitive capacity [40], human preference[41], criminality assessment [42], pandemics [43], natural resources management [44], waste management [45], portfolio analysis [46], enterprises competences [47], etc. Also, the rapid development of grey systems theory and the interest manifested in the use of this new theory into practical applications by researchers from all around the world has highlighted the need from the scholars involved in its development to provide explanations regarding different elements associated with the theory. As a result, in the scientific literature, one can find various papers containing in the title the “explanation of terms of” syntagm followed by the subject of the explanatory paper: • • • • • •

“Explanation of terms of grey models for decision-making” [48], “Explanation of terms of grey numbers and its operations” [49], “Explanation of terms of grey forecasting models” [50], “Explanation of terms of grey incidence analysis models” [51], “Explanation of terms of grey clustering evaluation models” [12], “Explanation of terms of concepts and fundamental principles of grey systems” [52], • “Explanation of terms of sequence operators and grey data mining” [53]. Nevertheless, the interest in the grey systems theory has brought the need for a specific software, capable of applying the main methods included in the theory on various datasets. As a result, the Grey Systems Modeling Software [54] has been developed and can be used successfully for solving grey-oriented practical applications. Over the years, the state of the art and the advancements made by the grey systems theory have been summarized in various review papers. If interested, the reader can refer to Liu et al. [55, 56], Prakash et al. [57], Javanmardi et al. [58], Delcea [59, 60],

4

1 State of the Art in Grey Systems Research in Economics and Social …

Li and Lin [61], Hu and Liu [62], Li et al. [63], Liu et al. [64], etc. A bibliometric analysis for the 40 years of grey systems theory in engineering is provided by Tao et al. [65]. Also, we invite the reader to consider the biographical material published in 2018 by Professor Liu S.F., entitled “The Father of Grey Systems”, in which the author talks about Professor Deng J.L. In a simple, but beautiful phrase, Professor Liu S.F. characterizes the “Grey Systems Theory” book written by Professor Deng J.L. as “the light in my life’s journey” [66]. Knowing the research activity of Professor Liu S.F. and his achievements in the area of grey systems theory, the authors of the present book are most grateful that we had the opportunity to work and get inspired by the work made by Professor Liu S.F. Thank you, Professor! In this context, given the vast fields of applicability of the grey systems theory, the present book aims to bring some light over the applications that have been built using the grey systems theory in the area of economics and social sciences. Particularly, the objective of the present state of the art chapter is to provide an extensive bibliometric analysis of the existing literature in the area of grey systems theory in the economics and social sciences area. A series of papers have been extracted and analyzed using specific research tools and software, as discussed in the following. As mentioned above, given the various applications that can be built using the grey systems theory, the present chapter aims at bringing some light into the use of the grey systems theory into the economics and social sciences area, leaving to the reader the task of exploring more in depth the sub-areas included here, depending on the reader’s interests.

Materials and Methods Web of Science platform (WoS platform) [67] has been used for extracting the papers referring to the use of grey systems theory into the economics and social science field. The data collection has been limited to the papers written until the end of 2021. The needed steps to extract the papers are listed in Table 1.1. The exploration steps 1–3 are represented by a search action performed on the WoS platform through the use of a list of keywords that include the “grey” or “gray” words such as “grey system”, “grey numbers”, “grey cluster”, “grey control”, “grey decision”, “grey incidence”, “grey model”, “grey theory”, “grey sequence”, “grey prediction”, “gray system”, “gray numbers”, “gray cluster”, “gray control”, “gray decision”, “gray incidence”, “gray model”, “gray theory”, “gray sequence”, “gray prediction” on the papers’ title, abstract and keywords. This approach has been chosen as a simple search using only the “grey” or “gray” keywords would have resulted in a large number of papers, most of them unconnected to the grey systems theory. This situation occurs due to the common use of the word “grey” in various aspects studied in the scientific papers. As a result, 2427 papers have been identifying as containing

Questions on Web of Science

Title

Abstract

Exploration steps

1

2

Table 1.1 Data assortment steps

(continued)

4791

(((((((((AB = (“grey system”)) OR AB #3 = (“grey numbers”)) OR AB = (“grey cluster”)) OR AB = (“grey control”)) OR AB = (“grey decision”)) OR AB = (“grey incidence”)) OR AB = (“grey model”)) OR AB = (“grey theory”)) OR AB = (“grey sequence”)) OR AB = (“grey prediction”)

Contains one of the grey systems specific keywordsa

348

(((((((((TI = (“gray system”)) OR TI = #2 (“gray numbers”)) OR TI = (“gray cluster”)) OR TI = (“gray control”)) OR TI = (“gray decision”)) OR TI = (“gray incidence”)) OR TI = (“gray model”)) OR TI = (“gray theory”)) OR TI = (“gray sequence”)) OR TI = (“gray prediction”)

Count

Contains one of the grey systems specific keywordsb

Query number 2079

Query (((((((((TI = (“grey system”)) OR TI = #1 (“grey numbers”)) OR TI = (“grey cluster”)) OR TI = (“grey control”)) OR TI = (“grey decision”)) OR TI = (“grey incidence”)) OR TI = (“grey model”)) OR TI = (“grey theory”)) OR TI = (“grey sequence”)) OR TI = (“grey prediction”)

Description Contains one of the grey systems specific keywordsa

Materials and Methods 5

Questions on Web of Science

Keywords

Title/abstract/keywords

Exploration steps

3

4

Table 1.1 (continued)

(continued)

7162

587

(((((((((AK = (“gray system”)) OR AK #6 = (“gray numbers”)) OR AK = (“gray cluster”)) OR AK = (“gray control”)) OR AK = (“gray decision”)) OR AK = (“gray incidence”)) OR AK = (“gray model”)) OR AK = (“gray theory”)) OR AK = (“gray sequence”)) OR AK = (“gray prediction”)

Contains one of the grey systems specific keywordsb

#7

3079

(((((((((AK = (“grey system”)) OR AK #5 = (“grey numbers”)) OR AK = (“grey cluster”)) OR AK = (“grey control”)) OR AK = (“grey decision”)) OR AK = (“grey incidence”)) OR AK = (“grey model”)) OR AK = (“grey theory”)) OR AK = (“grey sequence”)) OR AK = (“grey prediction”)

Contains one of the grey systems specific keywordsa

#1 OR #2 OR #3 OR #4 OR #5 OR #6

1371

Count

(((((((((AB = (“gray system”)) OR AB #4 = (“gray numbers”)) OR AB = (“gray cluster”)) OR AB = (“gray control”)) OR AB = (“gray decision”)) OR AB = (“gray incidence”)) OR AB = (“gray model”)) OR AB = (“gray theory”)) OR AB = (“gray sequence”)) OR AB = (“gray prediction”)

Contains one of the grey systems specific keywordsb

Query number

Query

Description

6 1 State of the Art in Grey Systems Research in Economics and Social …

b

a

Limit to Economics and Social Sciences

#9 #10

(#8) NOT PY = (2022) (#9) AND DT = (Article) #11

#8

(#7) AND LA = (English)

Manually selected papers

Query number

Query

869

3326

6743

7019

Count

“Grey system”, “grey numbers”, “grey cluster”, “grey control”, “grey decision”, “grey incidence”, “grey model”, “grey theory”, “grey sequence”, “grey prediction” “Gray system”, “gray numbers”, “gray cluster”, “gray control”, “gray decision”, “gray incidence”, “gray model”, “gray theory”, “gray sequence”, “gray prediction”

Selecting the papers in Economics and Social Sciences

8

Limit to Article

Exclude 2022

Year published

Document type

6

7

Contains one of the grey systems specific keywordsa

Language

Description

Questions on Web of Science

Exploration steps

5

Table 1.1 (continued)

Materials and Methods 7

8

1 State of the Art in Grey Systems Research in Economics and Social …

the searched keywords in the title, 6162 papers have contained the selected keywords in the abstract and 3666 papers in the keywords—Table 1.1. As most of the papers contained the selected keywords in more than one of the three sections of the paper (title, abstract or keywords), a merging action has been done in step 4, resulting a total number of papers on grey systems equal to 7162. From these papers, 7019 papers have been retained in the analysis based on the language criterion, namely these papers have been written in English (please see step 5). This exclusion criterion—through which the papers written in other languages except for English have been eliminated—has been used as the preponderance of the readership and researchers can readily understand English. As the search for the papers has been done at the beginning of 2022, the papers written in 2022 have been excluded from the analysis as the number of papers extracted for this year would have been incomplete when compared to the one of the previous years, not being relevant in the analysis. Please see step 6 for the criterion marking the exclusion of year 2022. The number of papers remaining in the analysis after applying this criterion has been equal to 6743 papers. Document type has been another exclusion criterion. In the analysis only the papers marked as articles have been retained. This action conducted to a reduction of the number of papers to 3326 (step 7). For the 3326 papers, another exclusion criterion has been applied (step 8). This last criterion refers to the selection of the papers that refer only to the economics and social sciences fields. As the action couldn’t have been performed using the filtering options offered by the WoS platform, the selection of the papers has been made manually by reading the papers’ titles, their abstracts and keywords. The result of this action has been represented by retaining into analysis a number of 869 papers, which are discussed and analyzed in the following. The data has been analyzed through bibliometrics, content, visualization and citation analysis. A series of visualizations have been conducted through the use of Biblioshiny software developed by Aria and Cuccurullo [68] from the Federico II University and University of Campania Luigi Vanvitelli. This software has been selected as it provides a series of interesting graphics which put in evidence the interconnections among the considered elements. Nevertheless, the software has been chosen recently in many research studies implying authors networks and papers analysis.

Dataset Analysis The extracted papers in the area of economics and social sciences using grey systems theory are discussed in the following in terms of authors, sources and citations.

Dataset Analysis

9

Dataset Overview The characteristics of the dataset comprising the 869 papers are detailed in Tables 1.2, 1.3, 1.4 and 1.5 by providing the values for a set of indicators. The timespan for the published papers is 1987–2021, while the journals in which the papers have been published is equal to 286 sources—Table 1.2. Most of the papers included in the dataset are recent papers, as it results from the small value recorded for the average years from publication indicator, which is equal to 5.62 years. Based on the data provided in Table 1.2 it can be stated that the papers published have attained an increased attention from the scientific community as the average number of citations per documents is equal to 23.87. As a result, an increased value is also obtained for the average citations per year per document, namely 3.53. In terms of references, there has been recorded a number of 24,409 references in the selected titles, conducting to an average number of references per document of 28.09. Regarding the number of keywords, a total number of 2366 keywords have been reported, with an average number of keywords per document of 2.72—Table 1.3. Nevertheless, the number of keywords plus, namely the index terms automatically generated from the titles of cited articles, has been equal to 1238, with an average of 1.42 keywords per document. Table 1.2 Main information about data

Table 1.3 Document contents

Table 1.4 Authors

Indicator

Value

Timespan

1987:2021

Sources (journals, books, etc.)

286

Documents

869

Average years from publication

5.62

Average citations per documents

23.87

Average citations per year per doc

3.53

References

24,409

Indicator

Value

Keywords plus (ID)

1238

Author’s keywords (DE)

2366

Indicator

Value

Authors

1791

Author appearances

2729

Authors of single-authored documents

68

Authors of multi-authored documents

1723

10 Table 1.5 Authors collaboration

1 State of the Art in Grey Systems Research in Economics and Social … Indicator

Value

Single-authored documents

97

Documents per author

0.485

Authors per document

2.06

Co-authors per documents

3.14

Collaboration index

2.23

In terms of authors, a total number of 1791 researchers have been uniquely identified, while the number of their appearances is equal to 2729—Table 1.4. As the number of authors of single-authored documents is low compared to the number of extracted documents (68 vs. 869), it can be stated that the collaboration network across the authors of the papers written using grey systems theory elements in the field of economic and social sciences is expected to be dense. This assumption is supported by the high number of authors of multi-authored documents compared to the total number of authors (1723 vs. 1791)—Table 1.4. The number of single-authored documents is 97 as it results from Table 1.5. Comparing this value with the value of the authors of single-authored documents indicator (97 vs. 68) it can be stated that the authors who have published papers as single authors have done it for an average of approximatively 1.43 documents. As the number of authors is higher than the number of selected papers, the value of the documents per author indicator is 0.485, with an average of 2.06 authors per paper. Retaining only the papers with more than two authors, it has been observed that the number of co-authors per documents is 3.14. The collaboration index is 2.23, being placed in a normal range for the papers written in the economics and social sciences area. The annual scientific production has recorded an increase in the number of papers—Fig. 1.3, reaching 165 documents in 2021, with an annual growth rate of 21.7%. The evolution of the average article citations value has been between 1.7 and 5.9 starting from 2000, which indicates a good visibility of the papers written in the area of economics and social sciences when using grey systems theory as one of the modeling or analyzing techniques. Smaller values of up to 2.5 have been recorded between 1987 and 1999 (Fig. 1.4).

Sources Considering the most relevant journals in which the selected papers have been published, it can be observed that Journal of Grey System holds the first place, with 86 papers, followed closely by the Grey Systems—Theory and Applications journal, with 44 papers—please see Fig. 1.5. It should be mentioned that the two journals are the main journals indexed in the ISI WoS collection which feature grey

Dataset Analysis

Fig. 1.3 Annual scientific production evolution

Fig. 1.4 Annual average article citations per year evolution

11

12

1 State of the Art in Grey Systems Research in Economics and Social …

systems theory as their main method when dealing with issues from various fields. Taken together, the two journals have been the home of approximatively 14.96% of the papers written in the economics and social sciences area using grey systems theory. Other relevant sources that are connected to the economics and social sciences fields are: Energy—with 44 papers, Mathematical Problems in Engineering and Sustainability—each with 29 papers, Expert Systems with Applications and Journal of Cleaner Production—each with 26 papers and Kybernetes—23 papers. Figure 1.5 presents the top-20 most relevant journals with respect to the number of publications. The importance of the mentioned sources is further underlines through the use of Bradford’s law. According to Nash-Stewart et al. [69], the Bradford’s law claims that there are a few productive periodicals (noted as zone 1) which represents the journals that are most frequently cited in the scientific literature, a larger body of moderate producers (zone 2) which have an average amount of citations, and an even larger body of constantly diminishing productivity sources (zone 3) which are seldom cited. Applied on the selected paper, the Bradford’s law includes in zone 1 a number of 8 sources (Journal of Grey System, Energy, Grey Systems—Theory and Application, Mathematical Problems in Engineering, Sustainability, Expert Systems with Applications, Journal of Cleaner Production and Kybernetes)—please see Fig. 1.6. Even more, the impact of the journals has been analyzed through the use of the H-index indicator. As known, the H-index is a measure of the number of papers published in that journal that have been cited H times each.

Fig. 1.5 Top-20 most relevant journals

Dataset Analysis

13

Fig. 1.6 Bradford’s law on source clustering

In the case of the selected journals, Energy scores the highest value for the Hindex, having 29 papers that with 29 or more citations in the area of grey systems theory used in economics and social sciences—please see Fig. 1.7. As expected, the journals marked as belonging to zone 1 when using the Bradford’s law are among the top-journals when considering the H-index, scoring values between 8 and 29, except for the Mathematical Problems in Engineering with a H-index of 5. Regarding the journals’ growth by considering the number of published papers in the mentioned area, the data in Fig. 1.8 has been obtained for the top-5 journals. As expected, the journal with the highest growth in The Journal of Grey System, followed by Energy and Grey Systems—Theory and Application.

Authors The authors with the most prominent activity based on the number of published documents in the area of economics and social sciences when grey systems theory is used are highlighted in Fig. 1.9. The author with the highest number of documents is Professor Liu S.F., counting 50 documents, representing 5.75% of the total number of papers extracted. Authors with more than 10 papers published are: Wang Z.X. (23 papers), Hu Y.C. (20 papers), Wu L.F. (20 papers), Wang C.N. (18 papers), Fang Z.G. (17 papers), Dang Y.G. (13 papers), Javed S.A. (13 papers), Ma X. (13 papers), Wang Y. (13 papers), Xie N.M. (13 papers), Ding S. (12 papers), Yang Y.J. (11 papers)—Fig. 1.9.

14

1 State of the Art in Grey Systems Research in Economics and Social …

Fig. 1.7 Journals’ impact based on H-index

Fig. 1.8 Journals’ growth (cumulative) based on the number of papers

Dataset Analysis

15

Fig. 1.9 Top-35 authors based on number of documents

Fig. 1.10 Top-35 authors production over time

As for the date of the most recent papers written by the top-35 most prominent authors in term of number of documents, it can be observed from Fig. 1.10 that the articles have been written mostly starting with 2015.

16

1 State of the Art in Grey Systems Research in Economics and Social …

Fig. 1.11 Top-20 most relevant affiliations

The top-20 affiliations of the authors of the selected papers are listed in Fig. 1.11. As expected, the university placed first is the Nanjing University of Aeronautics and Astronautics, the home of the grey systems theory. Other relevant universities follow, as presented in Fig. 1.11. In terms of the corresponding author’s country, China is leading the top-20 hierarchy by far having 629 papers form the total of 869 papers (representing 72.38% of the entire dataset). Also, China is the leader on the values recorded for both the MCP (Multiple Country Publications—an inter-country collaboration index, with a value of 90) and SCP (Single Country Publications—an intra-country collaboration index, with a value of 539)—please see Fig. 1.12. The second, third, fourth and fifth places are held by Turkey (36 articles, MCP equal to 6 and SCP equal to 30), Iran (32 papers, MCP: 5, SCP: 27), India (31 papers, MCP: 9, SCP: 22) and USA (18 papers, MCP: 14, SCP: 4)—Fig. 1.12. Through a world-wide visualization map, the contributions from each country can easily be observed. Figure 1.13 presents the scientific contribution based on the country of each author on the considered papers. The colors in the figure range from dark blue, indicating a high contribution as in the case of China, to grey, indicating no articles published in the economics and social sciences with grey systems theory. Following the trend imposed by the scientific production, it can be observed that the country with the highest number of citations for the papers written by its researchers is China, counting for 12,678 citations—Fig. 1.14. The second country is Turkey with 1738 citations, while the third place is occupied by USA with 1308 citations—Fig. 1.14.

Dataset Analysis

Fig. 1.12 Top-20 most relevant corresponding author’s country

Fig. 1.13 Scientific production based on country

17

18

1 State of the Art in Grey Systems Research in Economics and Social …

Fig. 1.14 Top-20 countries with the most citations

In terms of average citations per paper, China recorded a score of 20.16, lower than in the case of Turkey (48.28 average article citations) and USA (72.67 average article citations). As for the collaborations’ network, the country collaboration map is depicted in Fig. 1.15. China is the country with the highest number of collaborations, counting for 28 other countries, with a total of 131 collaborations. Among the countries to whom the authors from China have collaborated, one can name USA—30 collaborations, United Kingdom—20 collaborations and Pakistan—9 collaborations. Other countries to whom the authors from China have collaborated are presented in Fig. 1.16.

Papers’ Analysis The analysis of the selected papers has been done firstly by considering the top10 most cited papers. The top-10 most cited papers have been characterized in the following by presenting several elements related to the number of authors, provenience country of the authors, number of citations, number of citations per year, etc. Additionally a brief summary has been provided for each paper in order to better understand the main areas of grey systems that have been used for shaping the analysis and models in the top-10 most cited papers. Lastly, the entire dataset of papers has been considered for extracting the most frequent words, for creating the word

Dataset Analysis

19

Fig. 1.15 Country collaboration map

clouds, for observing the word dynamics and trend topics. All these elements are presented in the following.

Top-10 Most Cited Papers Overview Referring to the selected papers, the top-10 most cited documents are listed in Table 1.6. A series of general information regarding the papers are provided, such as the name of the first author, the publication year, the journal in which the paper has been published, the digital object identifier (DOI), the number of authors, the country of the first author. Also, some indicators are provided, such as the total citations (TC). Besides the TC, the total citations per year (TCY) is determined by dividing the number of citations to the number of years since the paper has been published. Also, the Normalized TC (NTC) is provided. NTC is an indicator that provide equal credit of citations to all the authors of the paper that cites a document. The article with the higher number of citations is written by Kayacan et al. [70]. The paper counts 513 citations in the 13 years since its publication, and an TCY equal to 39.46. The paper with the second-high number of citations belongs to Bai et al. [71], with a TCY of 38.92. The reminder of the papers included in the top-10 citations papers have also recorded an increased number of citations—above 200, with an TCY above 16.81 and with an NTC above 3.58. Considering these numbers and the limited number of papers extracted through the mentioned keywords (869 papers), it can be stated that the impact of the top-10 most cited papers in high.

20

1 State of the Art in Grey Systems Research in Economics and Social …

Fig. 1.16 Collaboration network for the authors from China

The average number of authors of the 10-most cited papers is 2.6 authors, most of the papers have 2, 3, or 4 authors and only one paper have a single author. As for the countries of the authors, based on the information listed in Table 1.6, one can say that the results have been expected, as it matches the information provided in Fig. 1.14.

Top-10 Most Cited Papers Review In the following, a brief review of the papers included in the top-10 most cited papers is provided, along with a summary of the works. The top-cited paper of Kayacan et al. [70] deals with the use of the grey systems theory in the time series prediction. The authors present the needed information for creating a grey model of first order with one variable, also known in the scientific literature as GM(1,1). Furthermore, the authors discuss other types of models adapted from the scientific literature by including components taken from the grey systems theory, such as the grey Verhulst model (GVM). As for increasing the accuracy of the

2

4

2

Bai C, 2010, International Journal of Production Economics, [71]

Pao HT, 2011, Energy, [72]

Hashemi SH, 2015, Journal of Production Economics, [73]

Akay D, 2007, Energy, [74]

Li GD, 2007, 3 Mathematical and Computer Modelling, [75]

2

3

4

5

6

Japan

Turkey

Iran, USA, Germany

Taiwan

China, USA

Turkey, Canada

Kayacan E, 2010, 3 Expert Systems with Applications, [70]

1

2

Country/countries

Paper (first author, Number of authors year, journal, reference)

Table No.

Table 1.6 Top-10 most global cited documents

https://doi.org/10. 1016/j.mcm.2006. 11.021

https://doi.org/10. 1016/j.energy. 2006.11.014

https://doi.org/10. 1016/j.ijpe.2014. 09.027

https://doi.org/10. 1016/j.energy. 2011.01.032

https://doi.org/10. 1016/j.ijpe.2009. 11.023

https://doi.org/10. 1016/j.eswa.2009. 07.064

Digital object identifier (DOI)

269

297

302

317

506

513

Total citations (TC)

16.81

18.56

37.75

26.41

38.92

39.46

Total citations per year (TCY)

3.96

4.37

7.91

7.80

7.19

7.29

(continued)

Normalized TC (NTC)

Dataset Analysis 21

Pao HT, 2012, Energy, [77]

Kumar U, 2010, Energy, [78]

Xia XQ, 2015, 3 Journal of Cleaner Production, [79]

8

9

10

3

China, Denmark

China, Denmark

Taiwan, China

Taiwan

Tseng ML, 2009, 1 Expert Systems with Applications, [76]

7

3

Country/countries

Paper (first author, Number of authors year, journal, reference)

Table No.

Table 1.6 (continued)

https://doi.org/10. 1016/j.jclepro. 2014.09.044

https://doi.org/10. 1016/j.energy. 2009.12.021

https://doi.org/10. 1016/j.energy. 2012.01.037

https://doi.org/10. 1016/j.eswa.2008. 09.011

Digital object identifier (DOI)

221

252

259

262

Total citations (TC)

27.62

19.38

23.54

18.71

Total citations per year (TCY)

5.79

3.58

6.90

5.24

Normalized TC (NTC)

22 1 State of the Art in Grey Systems Research in Economics and Social …

Dataset Analysis

23

considered grey models, the authors have used Fourier series [70]. Four models are described in the paper, namely the modified GM(1,1) model using modeling errors and Fourier series (EFGM), modified GM(1,1) model at time domain using Fourier series (TFGM), the modified grey Verhulst model using modeling errors and Fourier series (EFGVM) and the modified grey Verhulst model at time domain using Fourier series (TFGVM). Having the six models (GM(1,1), GVM, EFGM, TFGM, EFGVM, and TFGVM), the authors have applied them on real high noisy data represented by the US dollar to Euro parity extracted for a period of time of 3 years (01.01.2005– 30.12.2007). By comparing the performance of the six models, the authors have concluded that the GM(1,1) model is able to provide accurate predictions when the data series are monotonous, while for highly noisy data, the EFGM and TFGM provide better results. The paper with the second-high number of citations, written by Bai and Sarkis [71] focuses on the supplier selection problem and aims at expanding the approach proposed in the scientific literature, based on grey systems theory and rough sets, by providing additional levels through the consideration of sustainability attributes. The authors state that, form a sustainable perspective, factors such as environmental factors, economic/business factors and social factors and their sub-factors should be considered. From a methodological point of view, the interval grey numbers are used for calculating the values of the considered factors. Mathematical elements are presented in the paper, which highlight the specific operations that can be made on grey numbers, such as addition, subtraction, multiplication and division. Considering synthetic data, the authors provide an example on how to integrate the additional sustainability attribute into the supplier selection problem. Three scenarios are set up, by varying the economic/business attributes and the decision environment, which are compared with a baseline scenario. Using a sensitivity analysis, the authors have shown that the final decision in the supplier selection problem may be sensitive to the attributes used in the evaluation process [71]. Pao and Tsai [72] uses the GM(1,1) model for forecasting the evolution of CO2 emissions, consumption, and GDP (Gross Domestic Product) in the case of Brazil on the purpose of determining the long-run equilibrium between the three variables. On this purpose, data regarding the evolution of the three indicators is extracted for a 27-year period, between 1980 and 2007. Three GM(1,1) models have been used for performing the forecast. Depending on the number of years considered in the dataset, the three GM(1,1) models have been featuring a 6-year period (GP-6), a 5-year period (GP-5) and a 4-year period (GP-4). Three indicators are used for evaluating the out-of-the-sample forecast capability of the considered models, namely the RMSE (root mean square error), MAE (mean absolute error) and MAPE (mean absolute percentage error). Even more, the results obtained through the forecast using the grey models are compared with Box-Jenkins autoregressive integrating moving average model (ARIMA). The results have shown that both the optimal grey model and the ARIMA model have a strong forecasting performance, as both of them registered a value for MAPE of less than 3%. In terms of economic results, the authors state that between the energy consumptions, emissions and income, there is a strong causality. In this context, the authors advise the government to adopt a dual strategy

24

1 State of the Art in Grey Systems Research in Economics and Social …

on increasing investment in energy infrastructure, while reducing the waste of energy. The paper has attained a high attention form the research community as it results from the increased number of citations, 317 citations, placing it on the third place on the top-10 most cited papers in the selected dataset. Grey relational analysis has been included in the process of the evaluation of the suppliers’ performance by Hashemi et al. [73]. The authors aim to include additional criteria in the supplier selection process, by taking into account the environmental issues. Traditionally, according to the authors, economic criteria have been used in the supplier selection process. A series of economic criteria have been extracted by the authors from the scientific literature, referring, but not being limited, to cost, quality, delivery, technology, flexibility, culture, innovativeness and relationship. Additionally, the authors have extracted a list of environmental criteria based on the scientific literature, such as pollution production, pollution control, resource consumption, ecodesign, environmental management system, green image, green competencies, green product, staff environmental training and management commitment. An improved grey relational analysis is presented in the paper, which combined with the advantages offered by the analytical network process, would ensure a supplier selection taking into account the environmental sustainability. Along with the theoretical approach, Hashemi et al. [73] provide a case study on data taken from a manufacturing company. Various scenarios are envisioned by the authors, showing the need for considering the environmental conditions when selecting the suppliers. Grey prediction model has been used by Akay and Atak [74] for forecasting the electricity demand in Turkey. Starting from the description of a classical grey model, namely GM(1,1), the authors argue that instead of using the entire dataset for prediction, as required by the GM(1,1) model, it is recommended that, in the case of chaotic data, to use only recent data. As a result, the authors have employed a modified GM(1,1) model, called GPRM (Grey Prediction with Rolling Mechanism). The model is applied on a dataset extracted for Turkey and its results are compared to the model used by the Turkish Ministry of Energy and Natural Resources, called MAED (Model of Analysis of the Energy Demand). The two models, GPRM and MAED, have been compared in terms of error on both the total electricity consumption and industrial energy consumption. Based on the analysis, it has been determined that the GPRM performs better than MAED on the two considered datasets. As a result, the authors state that GPRM can be successfully used for future electricity projections [74]. The paper attracted a number of 297 citations, placing it on the fifth place on the top-10 most cited papers. The paper written by Li et al. [75] received a number of citations equal to 269 and holds the sixth place on the top-10 most cited papers from the selected dataset. Once more, the supplier selection has been addressed through the means offered by the grey systems theory. In this paper, Li et al. [75] use the grey numbers to express the weights and the rankings of the attributes for all the possible alternatives. Information related to the definition of the grey numbers is given in the paper, along with the basic grey mathematics needed to compute different operations such as addition, subtraction, multiplication and division. The authors provide also a numerical example on how the proposed approach can be applied to the supplier selection problem. Synthetic

Dataset Analysis

25

data is used for numerical example. No comparison with a classical approach to the supplier problem is provided. The authors conclude that the experimental results prove the reliability of the proposed approach. Tseng [76] built a grey-fuzzy DEMATEL approach to decision-making problem in the case of the service quality expectations. The paper attracted the interest of the research community, which has been concretized through the high number of citations, 262 citations, placing it on the seventh place on the top-10 most cited papers. Grey numbers have been included in the proposed approach and have been used for ranking the criteria in customer expectations. In order to prove the applicability of the proposed approach in practice, the authors have used a questionnaire and a series of questions have been addressed regarding elements related to customer service quality expectations such as empathy, reliability, assurance, responsiveness, and tangible aspects. Two elements belonging to the empathy category, namely the fact that the employees are courteous, polite, and respectful and the fact that the employees are trustworthy, believable and honest, have shown to be the most important criterion, respectively the most influencing criterion for the respondents. The author state that even though the methodology can be used in practice, if the study involves large samples, the methodology might become too complicated to apply [76]. Considering the same variables as in Pao and Tsai [72], namely the CO2 emissions, energy consumption and economic growth, Pao et al. [77] perform a similar analysis on the purpose of analyzing the connection among the selected variables. China has been chosen as the source of the data for this study. The authors have used a nonlinear grey Bernoulli model (NGBM) for modeling the connection among the three variables. The results of the considered model have been compared in terms of RMSE, MAE and MAPE with the GM(1,1) and ARIMA models. For The GM(1,1), the authors have considered, as in [72], three variants of the model: GM-4, GM-5 and GM-6. Based on the forecasted data and the values obtained for MAPE, the authors state that the NGBM model can be safely used for future projections. As for the improvement suggestions for the China economy, the authors suggest that it should adopt the duals strategy as proposed in [72]. The paper gained attention from the research community and received a number of 259 citations. GM(1,1) model with rolling mechanism has been used by Kumar and Jain [78] for forecasting the coal and electricity consumption in India. The paper attracted 252 citations, placing it on the ninth place in the top-10 most cited papers. The choice for the GM(1,1) model with rolling mechanism for the two data series has been argued by the authors based on the growth pattern of the historical data which suggested a steady exponential growth. Additionally, the authors have used the GreyMarkov model for the forecasting of the crude-petroleum consumption in India. The choice for the Grey-Markov model has been made by the authors after observing the rising trend of the data series, with great fluctuations throughout the data series. This particularity of the data series is said that will affect the forecasting capability of a classical GM(1,1) model, making the Grey-Markov model more suitable for this type of analysis. Additionally, a singular spectrum analysis (SSA) has been performed by the authors for forecasting the conventional energy consumption in India. The three models have been evaluated by means of MAPE, which has proven their applicability

26

1 State of the Art in Grey Systems Research in Economics and Social …

to the selected data sets. The authors concluded that for short periods of time, the results obtained through the use of the three models are getting very close to the real values [78]. Xia et al. [79] applied grey numbers and DEMATEL for analyzing the internal barriers encountered by the remanufacturers from the automotive parts industry. Based on the scientific literature, a series of possible barriers have been highlighted which come from different areas such as tangible resources, intangible resources and capabilities. Using a questionnaire containing the common barriers and the expertise of four experts and a grey DEMATEL approach, the authors have shown that the lack of funds for research in new technologies represents a high influencing barrier in the process of remanufacturing implementation [79]. Other barriers have been highlighted as important such as the low value of the profit obtained by the remanufacturing firms and the insufficient number of recycled used engines. The paper succeeds in gathering a number of 221 citations in approximatively 7 years from publishing, placing it on the tenth place in the top-10 classification of the selected papers based on the number of citations. Table 1.7 summarizes some of the main elements included in the top-10 most global cited documents, providing information related to the elements considered from the grey systems theory, the type of data used in the analysis, the main purpose of the paper, the presence of a theoretical and/or practical application of the methodology and the presence/absence of a hybrid approach, along with the paper’s name and a brief description based on the name of the first author, publication year and publishing journal. Based on the information in Table 1.7 it can be observed that five of the ten selected papers have used grey prediction models for forecasting various datasets (Refs. [70, 72, 74, 77, 78]). Also, the grey numbers have held an important place on the selected papers as four of the ten papers have used them when dealing with qualitative data (Refs. [71, 75, 76, 79]). Based on the type of data, it can be observed that for the papers using grey numbers, questionnaire has been the main source of data, and, when lacking them, synthetic data has been used to prove the approach’s usefulness on real world situations. Even though commonly known and used in the papers dealing with grey systems theory, the grey relational analysis has only be used by a single paper included in the top-10 cited papers (Ref. [73]). As a result of these considerations, in the following, an analysis related to the most frequent words presented in the papers included in the dataset is provided.

Words Analysis An analysis regarding the most frequent words used in the selected papers has been performed in the following on the purpose of identifying the methods used by the authors in their papers or the area of applicability of their researches. The search has been made in keywords plus, authors’ keywords, titles and abstracts.

Paper (first author, year, journal, reference)

Kayacan E, 2010, Expert Systems with Applications, [70]

Bai C, 2010, International Journal of Production Economics, [71]

Pao HT, 2011, Energy, [72]

Hashemi SH, 2015, Journal of Production Economics, [73]

No.

1

2

3

4

Grey prediction models

Grey numbers

Grey prediction models

Grey systems theory main elements

Pollutant emissions, energy consumption, output data

Synthetic data

US dollar to Euro parity dataset

Data

An integrated green Grey relational Manufacturing supplier selection analysis company data approach with analytic network process and improved Grey relational analysis

Modeling and forecasting the CO2 emissions, energy consumption, and economic growth in Brazil

Integrating sustainability into supplier selection with grey system and rough set methodologies

Grey system theory-based models in time series prediction

Title

Table 1.7 Brief summary of the content of top-10 most global cited documents Theoretical/practical approach

To take into account the environmental issues when the selection of the suppliers is performed

To model, forecast, and analyze the connection between the output, energy consumption and CO2 emissions

To expand the supplier selection methodology by adding sustainability attributes

Both

Both

Both

To compare the Both efficiency of the grey models and modified grey models using Fourier series

Purpose

(continued)

Analytic network process (ANP)

–

Rough set theory (RST)

–

Hybrid approach/theories considered

Dataset Analysis 27

A grey-based decision-making approach to the supplier selection problem

A causal and effect Grey numbers decision making model of service quality expectation using grey-fuzzy DEMATEL approach

Li GD, 2007, Mathematical and Computer Modelling, [75]

Tseng ML, 2009, Expert Systems with Applications, [76]

Pao HT, 2012, Energy, [77]

6

7

8

Forecasting of CO2 Grey prediction emissions, energy models consumption and economic growth in China using an improved grey model

Grey numbers

Grey prediction with Grey rolling mechanism prediction for electricity demand models forecasting of Turkey

Akay D, 2007, Energy, [74]

Grey systems theory main elements

5

Title

Paper (first author, year, journal, reference)

No.

Table 1.7 (continued)

Pollutant emissions, energy consumption, output data

Questionnaire

Synthetic data

Electricity consumption data

Data

Both

Theoretical/practical approach

To model, forecast, and analyze the connection between the output, energy consumption and CO2 emissions

Both

To rank the criteria in Both customer expectations

To select the best Both supplied in a supplier selection problem

To forecast the electricity demand

Purpose

–

Fuzzy

–

–

(continued)

Hybrid approach/theories considered

28 1 State of the Art in Grey Systems Research in Economics and Social …

Paper (first author, year, journal, reference)

Kumar U, 2010, Energy, [78]

Xia XQ, 2015, Journal of Cleaner Production, [79]

No.

9

10

Table 1.7 (continued)

Analyzing internal barriers for automotive parts remanufacturers in China using grey-DEMATEL approach

Time series models (Grey-Markov, Grey Model with rolling mechanism and singular spectrum analysis) to forecast energy consumption in India

Title

Grey numbers

Grey prediction models

Grey systems theory main elements

Questionnaire

Conventional energy data: crude-petroleum consumption, coal, electricity consumption, natural gas consumption

Data

To analyze the internal barriers met by remanufacturers from the automotive parts industry and evaluate the causal barriers encountered by the remanufacturers

To forecast the consumption of conventional energy

Purpose

Both

Both

Theoretical/practical approach

–

–

Hybrid approach/theories considered

Dataset Analysis 29

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1 State of the Art in Grey Systems Research in Economics and Social …

Regarding the keywords plus, it has been observed that the top-10 keywords refer to “model”—148 appearances, “prediction”—74, “performance”—66, “demand”— 63, “china” and “energy consumption”—60 appearances each, “optimization”—58, “management”—57, “system”—47, and “algorithm”—46. Based on the identified top-10 keywords plus, it can be observed that most of the belong to the economics and social sciences domain—Table 1.8. The co-occurrence network for the top-50 words included in keywords plus divides the words into two main clusters as represented in Fig. 1.17. It can be observed that words such as “model”, “performance”, “management”, “system”, “selection”, “systems”, “impact”, “efficiency”, “industry”, “decision-making”, “framework”, “criteria”, “relational analysis”, “design”, “quality”, “barriers”, “sustainability”, “ahp”, “grey”, “topsis”, “information”, “supplier selection”, “dea” are listed as belonging to cluster 1. On the other hand, cluster 2 contains keywords such as: “prediction”, “demand”, “china”, ‘energy-consumption”, “optimization”, “algorithm”, “electricity consumption”, “consumption”, “prediction model”, “co2 emissions”, “economicgrowth”, “gm(1,1)”, “arima”, “neural-networks”, “time-series”, “network”, “neuralnetwork”, “output”, “growth”, “regression”, “time”, “turkey”, “grey model”, “emissions”, “forecasting-model”, “system model”, “bernoulli model”. Based on the keywords included in each cluster, it can be said that the papers included in cluster 1 are more oriented to optimization and decision making, while the papers included in cluster 2 are dedicated to practical applications of grey systems in time-series forecasting. On the other hand, when determining the top-10 most frequent words listed by the authors as keywords of their papers, it can be observed that most of them belong to the grey systems theory as presented in Table 1.9. From the words encountered in Table 1.9 only “forecasting”—with 65 appearances and “dematel”—with 21 appearances are not grey systems theory specific words. A mixed situation between the two can be encountered in the case of the words extracted based on titles of the papers and abstracts. Considering the top10 words encountered in the titles of the scientific papers, it can be observed that Table 1.8 Top-10 most frequent words in keywords plus

Words

Occurrences

Model

148

Prediction

74

Performance

66

Demand

63

China

60

Energy-consumption

60

Optimization

58

Management

57

System

47

Algorithm

46

Dataset Analysis

31

Fig. 1.17 Clusters made by top-50 words based on keywords plus

Table 1.9 Top-10 most frequent words in authors’ keywords

Words

Occurrences

Grey theory

78

Forecasting

65

Grey system theory

63

Grey model

53

Grey prediction

44

Grey system

38

Grey relational analysis

28

Grey prediction model

24

Dematel

21

Grey incidence analysis

20

the following words appear most frequently: “grey”—492, “model”—384, “forecasting”—229, “based”—164, “prediction”—147, “analysis”—133, “consumption”—106, “china”—104, “application”—102, and “energy”—90. As for the case of the top-10 most frequent words in the abstract, the following words have been extracted: “model”—2473, “grey”—1921, “prediction”—819, “results”—800, “data”—774, “method”—710, “forecasting”—687, “study”—670, “proposed”— 669, and “energy”—649. Figure 1.18 depicts the top-50 words based on the four categories (keywords plus, authors’ keywords, titles and abstracts).

32

1 State of the Art in Grey Systems Research in Economics and Social …

(a) Top-50 words based on keywords plus

(c) Top-50 words based on title

(b) Top-50 words based on authors’ keywords

(d) Top-50 words based on abstract

Fig. 1.18 Top-50 words based on keywords plus (a), authors’ keywords (b), title (c), abstract (d)

The top-50 most frequent words extracted from abstracts are clustered using the co-occurrence network analysis in Bibiometrix [68]. As a result, two clusters are formed as presented in Fig. 1.19. Cluster 1 contains words such as: “model”, “prediction”, “results”, “data”, “forecasting”, energy”, “models”, “consumption”, “development”, “gm”, “time”, “china”, “forecast”, “accuracy”, “demand”, “economic”, “future”, “predict”, “algorithm”, “china’s”, “electricity”, “growth”, “gray”. Cluster 2 comprises words such as: “grey”, method”, “paper”, “study”, “proposed”, “system”, “based”, “analysis”, “theory”, “performance”, “evaluation”, “industry”, “information”, “research”, “factors”, “approach”, “decision”, “methods”, “supply”, “applied”, “process”, “quality”, “management”, “purpose”, “decision_making”, “finally”, “criteria”. Based on the listed words, it can be observed that cluster 1 refers mostly to papers based on the grey forecasting model, while cluster 2 is dedicated more to decision making process using grey methods (Fig. 1.19). Based on the bigrams and trigrams analysis applied to titles and abstracts, the top10 bigrams and top-10 trigrams have been extracted as it results from Tables 1.10 and 1.11. As expected even from the analysis of the top-10 most cited papers (presented in the previous sub-section), “grey model” bigram holds the first place in the hierarchy obtained for both abstracts and titles—Table 1.10. Also, “grey prediction” and “energy consumption” are the second and the third most common encountered bigrams—Table 1.10. This outcome has been expected as, even in the case of the

Dataset Analysis

33

Fig. 1.19 Clusters made by top-50 words based on abstract Table 1.10 Top-10 most frequent bigrams in abstracts and titles

Bigrams in abstracts

Occurrences

Bigrams in titles

Occurrences 109

Grey model

352

Grey model

Energy consumption

242

Grey prediction

56

Grey prediction

206

Energy consumption

48

Prediction model

203

Prediction model

43

Grey system

173

Grey system

32

Proposed model

155

Grey theory

28

System theory 146

System theory

25

gm model

137

Neural network

24

Supply chain

120

Grey incidence

23

Grey theory

116

Supply chain

23

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1 State of the Art in Grey Systems Research in Economics and Social …

Table 1.11 Top-10 most frequent trigrams in abstracts and titles

Trigrams in abstracts

Occurrences

Trigrams in titles

Occurrences

Grey system theory

126

Grey prediction model

28

Grey prediction 110 model

Grey system theory

23

Grey relational analysis

65

Grey relational analysis

16

Grey model gm

60

Grey incidence analysis

13

Particle swarm optimization

43

Grey prediction models

11

Grey forecasting model

42

Natural gas consumption

11

Natural gas consumption

42

Grey Bernoulli model

10

Grey prediction models

41

Grey forecasting model

10

Grey incidence analysis

35

Discrete grey model

8

Discrete grey model

31

Fractional grey model

8

top-10 most cited papers, there have been an increased numbers of papers dealing with the energy consumption and its forecasting using the grey prediction models. The analysis of the trigrams in the titles and abstracts show that the top-3 trigrams are the same, but in a slightly different order, in Table 1.11: “grey systems theory”, “grey prediction model” and “grey relational analysis”. One element to be noticed from the analysis of the most frequent 10-unigrams, 10-bigrams and 10-trigrams is the lack of the occurrence of the “grey numbers”, which was expected to be encountered in this list given the high frequency of the use of the grey numbers in the top-10 most cited documents. As the trigrams have been almost the same in abstract and titles, a dynamic of the top-10 trigrams has been extracted from the abstracts as presented in Fig. 1.20. From the top-10 most frequent trigrams, it can be observed that the first identified trigram is “grey prediction model”, appearing first in the extracted data in 1992 and having two articles mentioning it. The most frequent trigram, namely “grey system theory”

Dataset Analysis

35

Fig. 1.20 Top-10 trigrams cumulative dynamics in abstracts

has been identified first in 1996 in one paper. The “grey forecasting model” appears for the first time in three papers written in 2001. Using the trend topics option offered by Bibliometrix [68], a trend topics analysis has been performed on the trigrams that belong to the abstracts, by setting the minimum word frequency to 25 and the number of words per year to 10. As a result, only the continuous period of time in which the trigrams appeared more than 10 times a year, with a minimum total frequency of 25, have been retained in the visualization presented in Fig. 1.21. From Fig. 1.21 it can be observed that the most frequent trigram, “grey systems theory” appears more than 10 times a year only between 2014 and 2020, while “grey prediction model” appears between 2017 and 2021. Other trigrams that have an endyear equal to 2021 are: “natural gas consumption” – with a total of 42 appearances, “grey prediction models”—41 appearances, and “discrete grey model”—31 appearances. Considering the resulting trigrams for the periods ending in 2021, it can be observed the increased interest in the scientific community for the grey models, as part of the grey systems theory.

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1 State of the Art in Grey Systems Research in Economics and Social …

Fig. 1.21 Trend topics for the trigrams in abstracts under specific parameters

Mixed Analysis Considering elements from the above discussions, and using three-fields plots, a mixed analysis is performed for better highlighting the connection between different categories, such as authors, affiliations, countries, keywords, abstracts. First, an analysis regarding the correspondence between the top-20 elements in countries, authors and journals has been conducted—please see Fig. 1.22. As expected, the country of the most prominent authors is China, the leading authors is Professor Liu S.F., and the journal with the highest number of papers is Journal of Grey System. One thing to notice is that the considered authors have selected different journals as the home for their researches, not focusing exclusively on a particular journal. Also, there are a series of authors mentioned in Fig. 1.22, which have had affiliations from multiple countries, underlying the existence of international collaborations in the field of grey systems with applications in economics and social sciences. Another analysis has been made between the affiliations of the authors, the authors and the keywords of the published papers—please see Fig. 1.23. From Fig. 1.23 it can be observed the multitude of keywords used by the most prominent authors, most of them containing the combination between words “grey” and key elements from the grey systems theory. Nanjing University of Aeronautics and Astronautics was the leading university, followed by other renowned universities in the area of grey systems. As in the previous

Dataset Analysis

Fig. 1.22 Three-fields plot: countries (left), authors (middle), journals (right)

Fig. 1.23 Three-fields plot: affiliations (left), authors (middle), keywords (right)

37

38

1 State of the Art in Grey Systems Research in Economics and Social …

analysis, it can be observed that not only some of the authors have been affiliated over the time to universities from foreign countries (please see the left side of the graphic in Fig. 1.22) as a result of international collaborations, but also to different universities from their home country (please consider the left side from Fig. 1.23), showing the strength of national collaborations in this area. Comparing the top-20 elements in keywords, journals and keywords plus, the network in Fig. 1.24 has been obtained. Analyzing the connection between the three elements, it can be stated that there are journals, such as Journal of Grey System, which succeed in publishing papers featuring almost the entire range of keywords from the grey systems theory area. On the other hand, there are other journals, such as Energy, which gather grey systems theory elements from a specific area, such as the prediction using the grey models. As it can be observed from Fig. 1.24, in the case of Energy journal, the main keywords are: “grey prediction model”, “grey model”, “forecasting”, “grey prediction model”. Nevertheless, there are journals, such as Journal of Mathematics and Complexity, which do not feature any of the keywords listed in the left side of Fig. 1.24. As for the keywords plus, it can be observed that they are spread among the journals listed. Grey Systems Theory and Application journal succeeds in having all the keywords plus listed in the right side of Fig. 1.24, while journals such as Energy and Journal of Cleaner production, gather 18 of the 20 keywords plus. Specifically,

Fig. 1.24 Three-fields plot: keywords (left), journal (middle), keywords plus (right)

Concluding Remarks

39

Energy is not connected with “impact” and “management”, while the Journal of Cleaner Production is not connected with “algorithm” and “efficiency”.

Concluding Remarks The advancements made in the grey systems theory since its founding in 1982, along with the number of published papers and received citations has shown the interest manifested by the research community from all around the world. In this chapter, the focus has been made on the articles written in the area of economics and social sciences and which have been published between 1982 and 2021 and indexed in ISI Web of Science database (WoS platform). Through the use of bibliometrics, the evolution of the number of papers have been analyzed from multiple points of view, highlighting the main authors contributing to the field, the countries of their residence, the universities they are affiliated to, the network of international collaborations, the main methods from the grey systems theory featured in the selected papers, the main words used in title, abstract, keywords, etc. Additionally, a review of the top-10 most cited papers has been provided for a better overview of the applications of grey systems theory in the economic and social sciences field. Considering the analysis performed in this chapter, the contribution of the grey systems theory to solving the problems associated with the economic and social sciences field has been shaped. The analysis of the selected papers has some limitations which derives from the selection of the papers discussed. First, the papers included in this chapter have been extracted using the WoS platform, and, therefore, the work is restricted to the papers published in this database. If some other databases would have been used (e.g. Scopus), the number and content of the paper would have been different. Second, the selected papers refer only to papers that have been marked in the WoS platform as “article”. Again, if other categories would have been considered, the number of selected papers would have been greater. Third, a series of keywords have been used for searching the papers and the selected dataset reflects the search made by the selected keywords. Considering other keywords, the dataset created based on the new set of keywords would have been different. Last, as the words associated with the economics and social sciences fields are not easy to find in order to provide the exclusion from the analysis of the papers in other research fields, the search has been conducted manually by the authors by reading the titles, abstracts and keywords. This action cannot be seen as a limitation in a direct sense, but rather as an action which required a lot of time and attention. As the number of papers written in the field of economic and social sciences through the use of grey systems theory increases in the years to follow, it would be hard to

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1 State of the Art in Grey Systems Research in Economics and Social …

select the papers manually and the interested parties in conducting similar analysis should consider a different approach. Nevertheless, as future research one can try to extend the analysis by including some more papers by adding other keywords that have been neglected in this work and which have emerged over the time. Another possible direction to follow is to extend the dataset by including the papers presented at the conferences or by considering some other relevant databases (such as Scopus). Furthermore, the analysis can be done by isolating the papers featuring a specific method from the grey systems theory (e.g. grey numbers, grey relational analysis, grey clustering) or papers featuring a specific economic area (e.g. supplier selection, portfolio analysis, resources allocation, etc.). A part of these analyses is discussed in the following chapters of the present book, with the remark that the analyses that can be performed are not limited to the ones included in this book. We invite the reader to explore more the world of grey systems, both through the applications made in specific fields and through the advancements made in the field from a methodological point of view.

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34. Javed, S.A., Liu, S.: Evaluation of outpatient satisfaction and service quality of Pakistani healthcare projects: application of a novel synthetic grey incidence analysis model. Grey Syst. Theory Appl. 8, 462–480 (2018). https://doi.org/10.1108/GS-04-2018-0018 35. Zhu, Y., Wang, R., Hipel, K.W.: Grey relational evaluation of innovation competency in an aviation industry cluster. Grey Syst. Theory Appl. 2, 272–283 (2012). https://doi.org/10.1108/ 20439371211260234 36. René, S.H., Becker, U., Manz, H.: Grey systems theory time series prediction applied to road traffic safety in Germany. IFAC-Pap. 49, 231–236 (2016). https://doi.org/10.1016/j.ifacol.2016. 07.039 37. Xu, X., Chen, B., Gan, F.: Traffic safety evaluations based on grey systems theory and neural network. In: 2009 WRI World Congress on Computer Science and Information Engineering, pp. 603–607. IEEE, Los Angeles, CA, USA (2009). https://doi.org/10.1109/CSIE.2009.888 38. Yoga, I., Yudiarta, I.G.A.: Grey forecasting of inbound tourism to Bali and financial loses from the COVID-19. Int. J. Grey Syst. 1, 48–57 (2021). https://doi.org/10.52812/ijgs.17 39. Muhammad Muneeb, F., Karbassi Yazdi, A., Wanke, P., Yiyin, C., Chughtai, M.: Critical success factors for sustainable entrepreneurship in Pakistani Telecommunications industry: a hybrid grey systems theory/ best-worst method approach. Manag. Decis. 58, 2565–2591 (2020). https://doi.org/10.1108/MD-08-2019-1133 40. Javanmardi, E., Liu, S.: Exploring the human cognitive capacity in understanding systems: a grey systems theory perspective. Found. Sci. 25, 803–825 (2020). https://doi.org/10.1007/s10 699-019-09618-3 41. Delgado, A.: Why do any secondary students prefer the mathematics? A response using grey systems. In: 2017 International Symposium on Engineering Accreditation (ICACIT), pp. 1–4 (2017). https://doi.org/10.1109/ICACIT.2017.8358082 42. Delgado, A.: Citizen criminality assessment in Lima city using the grey clustering method. In: 2017 IEEE XXIV International Conference on Electronics, Electrical Engineering and Computing (INTERCON), pp. 1–4. IEEE, Cusco, Peru (2017). https://doi.org/10.1109/INT ERCON.2017.8079662 43. Ngo, H.A., Hoang, T.N., Dik, M.: Introduction to the grey systems theory and its application in mathematical modeling and pandemic prediction of Covid-19. In: Agarwal, P., Nieto, J.J., Ruzhansky, M., Torres, D.F.M. (eds.) Analysis of Infectious Disease Problems (Covid-19) and Their Global Impact, pp. 191–218. Springer Singapore, Singapore (2021). https://doi.org/10. 1007/978-981-16-2450-6_10 44. Szpak, D., Tchórzewska-Cie´slak, B.: The use of grey systems theory to analyze the water supply systems safety. Water Resour. Manag. 33, 4141–4155 (2019). https://doi.org/10.1007/ s11269-019-02348-y 45. Grey systems theory for solid waste management. In: Sustainable Solid Waste Management, pp. 829–847. Wiley, Hoboken, NJ, USA (2015). https://doi.org/10.1002/9781119035848.ch23 46. Nowak, M., Mierzwiak, R., Wojciechowski, H., Delcea, C.: Grey portfolio analysis method. Grey Syst. Theory Appl. ahead-of-print (2020). https://doi.org/10.1108/GS-11-2019-0049 47. Wi˛ecek-Janka, E., Majchrzak, J., Wyrwicka, M., Weber, G.W.: Application of grey clusters in the development of a synthetic model of the goals of Polish family enterprises’ successors. Grey Syst. Theory Appl. 11, 63–79 (2020). https://doi.org/10.1108/GS-12-2019-0062 48. Liu, S., Fang, Z., Xie, N., Yang, Y.: Explanation of terms of grey models for decision-making. Grey Syst. Theory Appl. 8, 382–387 (2018). https://doi.org/10.1108/GS-10-2018-081 49. Liu, S., Rui, H., Fang, Z., Yang, Y., Forrest, J.: Explanation of terms of grey numbers and its operations. Grey Syst. Theory Appl. 6, 436–441 (2016). https://doi.org/10.1108/GS-09-20160031 50. Liu, S., Yang, Y.: Explanation of terms of grey forecasting models. Grey Syst. Theory Appl. 7, 123–128 (2017). https://doi.org/10.1108/GS-11-2016-0047 51. Liu, S., Zhang, H., Yang, Y.: Explanation of terms of grey incidence analysis models. Grey Syst. Theory Appl. 7, 136–142 (2017). https://doi.org/10.1108/GS-11-2016-0045 52. Liu, S., Yang, Y., Forrest, J., Rui, H.: Explanation of terms of concepts and fundamental principles of grey systems. Grey Syst. Theory Appl. 6, 429–435 (2016). https://doi.org/10. 1108/GS-09-2016-0030

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Chapter 2

Grey Numbers for Sentiment Analysis and Natural Language Processing

Introduction Sentiment analysis deals with the identification and classification of the sentiments or opinions expressed in a given context [1]. It is an important and constantly growing area of the Natural Language Processing (NLP) field. In the context of the explosive role of social media in our everyday life, users are generating more and more data by posting their opinions on popular platforms such as Facebook and Twitter. The sentiment analysis conducted by considering the messages written by the users on social media platforms might come with a series of difficulties, mostly due to the ambiguousness that might accompany some of the words and phrases, the misspellings, and the slag words. The process of extracting the sentiments associated with a certain message might be challenging even due to the fact that, sometimes, it is hard for the computer to understand human humor. There are cases in which, by applying a different verbal tone to a phrase, two completely different outcomes could be understood. In these cases, it might be hard for a computer to understand the thin line between the two sentiments. Among the tasks associated with the sentiment analysis, the polarity detection is one of the most important one, aiming to determine whether a message or a text has a positive, negative or neutral connotation [2]. Polarity detection requires a multidisciplinary approach, as elements from psychology, linguistic and artificial intelligence need to be combined to ensure the reliability of the obtained results. The approaches used for polarity detection can be classified in lexicon-based methods [3, 4], machine learning methods [1] and hybrid methods that combine lexicons and machine learning techniques [5]. Over time, various sentiment lexicons have been created for helping researchers and other interested parties in better determining the sentiments associated with a message or a text. The sentiment lexicons contain words and sequences of words (called tokens) pre-classified in positive, negative and neutral categories. Furthermore, some of the lexicons provide even a numerical polarity for each token in order to better convey the perception of the given token. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_2

45

46

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

The construction of the lexicons includes a wide range of approaches, ranging from the ones in which the association of the tokens in a given category is made in a pure automatic manner to the one in which the association is done by independent human raters. Each of the approaches provides advantages and disadvantages related to the time needed to build such a lexicon, the number of words and expressions included in the lexicon and the accuracy of the polarity associated to each token. While the automatic generated lexicons are faster to implement, they come with disadvantages related to the noise in data, which can affect the overall sentiment evaluation. On the other hand, when independent human raters are used for creating a lexicon, the noise in data is reduced, but the effort to create such a lexicon might be huge. As a result, in the present chapter, we aim to combine various-existing lexicons to create a lexicon that is both more accurate and comprehensive. To this extent, several lexicons are considered as presented in the following. The grey arithmetic on intervals is used with the newly created lexicon for improving the results of the sentiment analysis. The chapter includes a short description of some of the lexicons in the scientific literature, accompanied by a presentation of grey numbers and of the common arithmetic operations between grey numbers. A practical application is provided in order to prove the applicability of the proposed approach.

Sentiment Analysis Lexicons As various sentiment analysis lexicons have been proposed over time, in this section we will provide a brief list of the some of the well-known lexicons and their characteristics [6–8]. Table 2.1 presents the list of the selected lexicons.

Grey Numbers At the center of the grey systems theory, we can find the concept of grey numbers. Their main characteristic is that, although the exact value for a grey number is not known, they provide nevertheless information regarding an interval or a set of values in which this value occurs [14]. Thanks to this characteristic, grey numbers are able to better represent the uncertainty ever-present in a variety of phenomena. Most often grey numbers are represented using the notation “⊗” [15, 16]. They can be classified as either discrete or continuous. Discrete grey numbers take only a finite or countable number of potential values. For example, the grey number representing the possible polarities of a tweet can have the values −1 (negative), 0 (neutral) and 1 (positive). On the contrary, a continuous grey number can potentially take any value inside an interval, between a lower bound a and an upper bound a, with a < a. In this case the grey number can be written as shown in (2.1).

Grey Numbers

47

Table 2.1 Lexicons for sentiment analysis Lexicon

References

Procedure

Notes

WordNet

Fellbaum [9]

Manual

Database of English words linked by their semantic relationships

ANEW (Affective Norms Bradley and Lang [10] Manual for English Words)

The lexicon contains 1034 weighted unigram terms divided into three categories

SentiWordNet

Baccianella et al. [11]

Semi-automatic Extension of WordNet, having 147,306 synsets annotated into three categories

SenticNet

Cambria et al. [12]

Semi-automatic Contains 14,244 common sense concepts

VADER

Hutto and Gilbert [4]

Manual

Contains 7517 English tokens, including emoticons, acronyms, and commonly used slangs. Words are rated on a scale from extreme negative (−4) to extreme positive (+4)

Maxdiff-Twitter

Kiritchenko et al. [7]

Manually

Contains 1515 tokens, including English words, emoticons, sentiment-related acronyms, and initialisms (e.g., LOL) and commonly used slang (e.g., nah). The intensity range varies between −1, associated to extremely negative, to +1, associated to extremely positive

Sentiment140

Mohammad et al. [13] Automatic

Includes 62,468 unigrams, extracted from the tweets containing emoticons. Words intensity varies between minus infinity to plus infinity. Except for some cases, the words have a polarity between −5 and +5

48

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

[ ] ⊗a ∈ a, a

(2.1)

More specifically, for an interval grey number ⊗, one can define a value-covered set, noted with D, which is a continuous set with both upper and lower boundaries [17]. In this context, the true value of a grey number ⊗ is noted with d ∗ , with the property that ∀⊗ => d ∗ ∈ D. In regard to the elements mentioned above, some remarks can be made [18]: 1. The grey number ⊗ is a real number for which one does not know its determinate value 2. The value d ∗ is a real number, representing the connotation of the grey number ⊗ 3. It is known that d ∗ is covered in the value-covered set D, but one cannot know the determinate value 4. Starting from the hypothesis that the true value of d ∗ is unknown, when conducting basic mathematical operations with grey numbers, one should try to keep in the analysis as much information as possible, as any arbitrary discarded information can conduct to an erroneous value of d ∗ , losing the true value of d ∗ . As defined before, the set D can be either [17]: • The aggregation of a set of continuous intervals • A set of discrete values • Both continuous intervals and discrete sets, namely a mixed grey number valuecovered set as depicted in Fig. 2.1. D1

D2

Dn

(a)

... _

a_ 1 a 1

_

_

a_ 2 a 2

a_ n a n

(b)

d1 D1

D2

d2

...

dn

Dn

(c)

... _

a_ 1 a 1

_

a_ 2 a 2

_

a_ n a n

d1

d2

...

dn

Fig. 2.1 Example of possible forms for the value-covered set D: continuous grey number valuecovered set (a), discrete grey number value-covered set (b), mixed grey value-covered set (c)

Arithmetic of Grey Numbers

49

In the particular case in which a is equal to −∞, the grey number ⊗a ∈ [−∞, a]] [ only has an upper bound, while if a is equal to ∞ the grey number ⊗a ∈ a, ∞ only has a lower bound. We should note that in the case in which a is equal to a, the grey number ⊗a is equivalent to a regular number.

Arithmetic of Grey Numbers Given the special characteristics of grey numbers, a specific mathematic has been defined, which together with the grey numbers theory form the very base of the grey systems theory [18]. As shown in the following, the arithmetic operations closely resemble the ones used for intervals.

Basic Operations [ ] [ ] Having the grey numbers ⊗a ∈ a, a and ⊗b ∈ b, b , with a < a and b < b, and ×. If ⊗c = ⊗a × ⊗b then ⊗c will also be an operation between ⊗a and ⊗[b denoted ] an interval grey number, ⊗c = c, c , for which c < c. The arithmetical operations between grey numbers are defined as follows [18]. The addition operation, or the sum between ⊗a and ⊗b is defined as shown in (2.2) [16, 18, 19]. [ ] ⊗a + ⊗b = a + b, a + b

(2.2)

The difference, or the subtraction operation between ⊗a and ⊗b is defined as shown in (2.3) [16, 18, 19]. ] [ ⊗a − ⊗b = ⊗a + (−⊗b ) = a − b, a − b

(2.3)

The product, or the multiplication operation between ⊗a and ⊗b is defined as shown in (2.4) [16, 18, 19]. }] } { [ { ⊗a × ⊗b = min ab, ab, ab, ab , max ab, ab, ab, ab

(2.4)

[ ] The reciprocal, or the inverse of ⊗a = a, a , where a < a and a × a > 0, is denoted by ⊗a−1 and is defined as shown in (2.5) [16, 18, 19]. / ] [ / ⊗−1 a = 1 a, 1 a

(2.5)

The division operation between ⊗a and ⊗b , where b × b > 0 is defined as shown in (2.6) [16, 18, 19].

50

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

[ { }] } { / a a a a a a a a , , , , ⊗a ⊗b = ⊗a × ⊗−1 = min , max , , b b b b b b b b b

(2.6)

Finally, the grey scalar product between a real number k and the grey number ⊗a is defined as shown in (2.7) [16, 18, 19]. ] [ k ∗ ⊗a ∈ ka, ka

(2.7)

Examples of Basic Operations Let us consider some examples of grey numbers and to apply the above basic operations in order to perform addition, subtraction, multiplication and division. For this, let consider two independent numbers: ⊗a = [1, 5] and ⊗b = [7, 8] In the case of addition, one gets: ⊗c = ⊗a + ⊗b = [1, 5] + [7, 8] = [1 + 7, 5 + 8] = [8, 15] While in the case of multiplication, one has: ⊗d = ⊗a × ⊗b = [1, 5] × [7, 8] = [min(1 × 7, 1 × 8, 5 × 7, 5 × 8), max(1 × 7, 1 × 8, 5 × 7, 5 × 8)] = [7, 40] However, the situation becomes slightly different (sometimes confusing) in the case of subtraction as the result of the subtraction for the considered example is: ⊗c − ⊗a = [8, 15] − [1, 5] = [8 − 5, 15 − 1] = [3, 14] Which is not equal to ⊗b = [7, 8] and therefore: ⊗c − ⊗a /= ⊗b . A similar situation occurs in the case of the division operation: [ { } { }] / / 7 7 40 40 7 7 40 40 , max , , , ⊗d ⊗a = [7, 40] [1, 5] = min , , , 1 5 1 5 1 5 1 5 = [1.4, 40] /= ⊗b

Comparison of Grey Numbers

51

As unusual as the result of the two operations (subtraction and division) might appear, it is acknowledged in the scientific literature where it is mentioned that the addition and the subtraction operations are not invertible operations and that not even the multiplication and the division are not invertible operations in interval numbers [17]. This situation appears even in the case of interval fuzzy numbers [17].

Comparison of Grey Numbers The rules for comparing both discrete and interval grey numbers have been discussed by [18], by taking into account the probability interval that can be attributed to each number. In the following, the comparison of the two types of grey numbers, namely discrete and interval numbers, is briefly discussed. In the considered situations, one assumes that the grey numbers to be compared are independent. In the case in which the numbers are not independent, further investigation should be made in order to revisit the operations through which the numbers have become dependent and should be restored to the original composition of independent grey numbers. Otherwise, where possible, the grey numbers should be expressed as joint information in order to ensure their independence. ] [ Let us consider two independent grey numbers noted as ⊗a ∈ a, a and ⊗b ∈ ] [ b, b , with a < a and b < b. We will try to address the comparison issue first through a graphical approach and then by using the idea included by the probability density function. For the moment, we shall refer to the comparison of the two independent grey numbers by not taking into account their nature (discrete or continuous) in order to better explain how the comparison can be seen on a general case. Later, we will move to discussing specifically each of the two situations (discrete and continuous grey numbers). Graphically, in order to compare two grey numbers, one can consider a twodimensional x O y system, in which on one of the axes, O x, we will consider the probability value range of grey number ⊗a , while on the other axes, O y, the probability value range of grey number ⊗b . The probability value that the grey number ⊗a is equal to the probability value of the grey number ⊗b can be found on the straight line where x = y (please see Fig. 2.2). The other two cases, in which the probability value of one of the numbers to be greater than the probability value of the other are on one side and the other of this straight line—e.g. when x > y the probable value of the grey number ⊗a is greater than the probable value of the grey number ⊗b , while when x < y the probable value of the grey number ⊗a is lower than the probable value of the grey number ⊗b [18]. Considering the probability density functions of the two grey numbers, one can write:

52

2 Grey Numbers for Sentiment Analysis and Natural Language Processing y

x=y

xy

x

O

Fig. 2.2 Example on comparing two grey numbers

{+∞ f (x)d x = 1

(2.8)

f (y)dy = 1

(2.9)

−∞

{+∞ −∞

where: f (x) is the probability density function of ⊗a and f (y) is the probability density function of ⊗b . Consequently, the joint probability function of the two grey numbers, ⊗a and ⊗b , noted as f (x, y), is [18]: {+∞ {+∞ f (x, y)d xd y = 1

(2.10)

−∞ −∞

Therefore [18]: { { p(⊗a > ⊗b ) =

f (x, y)d xd y x>y

Three results are possible depending on the value of p(⊗a > ⊗b ) [18]: • When p(⊗a > ⊗b ) = 1 then ⊗a > ⊗b

(2.11)

Comparison of Grey Numbers

53

• When p(⊗a > ⊗b ) = p, with 0 < p < 1 then ⊗a > p ⊗b —in this case, by > p it has been noted that the probable value of ⊗a is greater than the probable value of ⊗b with the probability p • When p(⊗a > ⊗b ) = 0 then ⊗a < ⊗b .

Discrete Grey Numbers Comparison In order to discuss the discrete grey numbers comparison, we will consider in the following two discrete grey numbers ⊗a and ⊗b described as follows: { / / / } ⊗a = da1 pa1 , da2 pa2 , . . . , dan pan and / / } { / ⊗a = db1 pb1 , db2 pb2 , . . . , dbn pbn where: • pai is the probability of ⊗a at the point dai • pbi is the probability of ⊗b at the point dbi And: n ∑ i=1 m ∑

pai = 1 pbj = 1

j=1

If one considers the graphical approach as mentioned above and keeps the same notations, using the bidimensional space x O y, with O x representing the probability value range of the discrete grey number ⊗a and O y representing the probability value range of the discrete grey number ⊗b , the same results can be obtained, namely [18]: 1. The straight line x = y represents the case in which the probable value of ⊗a is equal to the probable value of ⊗b 2. While when x > y the probable value of the grey number ⊗a is greater than the probable value of the grey number ⊗b , and 3. When x < y the probable value of the grey number ⊗a is lower than the probable value of the grey number ⊗b . Figure 2.3 provides a graphical representation on the comparison of two discrete grey numbers. Choosing two random values from the value-covered sets of the two discrete grey numbers, noted through dai —the random value in the value covered set of ⊗a ,

54

2 Grey Numbers for Sentiment Analysis and Natural Language Processing y

x=y

d bm ...

xy

d b1

O

d a1

...

d a2 d a3

d ai

...

d an

x

Fig. 2.3 Example on comparing two discrete grey numbers

and dbj —the random value in the value covered set of ⊗b , one can determine the following probability [18]: ⎧ dai < dbj ) ⎨ 0, ( p dai > dbj = 0.5 pai pbj , dai = dbj ⎩ pai , pbj dai > dbj Which conducts to [18]: p(⊗a > ⊗b ) =

m n ∑ ∑ ( ) p dai > dbj i=1 j=1

Which is equivalent to: p(⊗a > ⊗b ) =

n m ∑ ∑

p(dai > dbj )

j=1 i=1

Therefore, three cases are possible [18]: • If p(⊗a > ⊗b ) = 1 then ⊗a > ⊗b • When p(⊗a > ⊗b ) = p, with 0 < p < 1 then ⊗a > p ⊗b —as mentioned above, in this case, by > p it has been noted that the probable value of ⊗a is grater than the probable value of ⊗b with the probability p • If p(⊗a > ⊗b ) = 0 then ⊗a < ⊗b .

Comparison of Grey Numbers

55

For better understanding the discrete grey numbers comparison, we will consider in the following a numerical example. Let us consider two discrete grey numbers having the following form: / / / / } { / ⊗a = da1 pa1 , da2 pa2 , da3 pa3 , da4 pa4 , da5 pa5 { / / / / / } = 5 0.1, 6 0.2, 7 0.4, 7.5 0.2, 8 0.1 and / / / } { / ⊗b = db1 pb1 , db2 pb2 , db3 pb3 , db4 pb4 { / / / / } = 6.5 0.2, 7 0.4, 7.4 0.3, 8.2 0.1 For comparing ⊗a with ⊗b , we have to analyze the probability that dai , with i = 1, 5, is greater than dbj , with j = 1, 4. First, it can be observed that the requirements related to the sum of probabilities being equal to 1 are fulfilled: 5 ∑

pai = 0.1 + 0.2 + 0.4 + 0.2 + 0.1 = 1

i=1 4 ∑

pbj = 0.2 + 0.4 + 0.3 + 0.1 = 1

j=1

Next, the probabilities of each dai being greater than each dbj are determined based on the Eq. (2.11): p(da1 > db1 ) = p(5 > 6.5) = 0 p(da1 > db2 ) = p(5 > 7) = 0 p(da1 > db3 ) = p(5 > 7.4) = 0 p(da1 > db4 ) = p(5 > 8.2) = 0 p(da2 > db1 ) = p(6 > 6.5) = 0 p(da2 > db2 ) = p(da2 > db3 ) = p(da2 > db4 ) = p(da3 > db1 ) = p(da3 > db2 ) =

p(6 > 7) = 0 p(6 > 7.4) = 0 p(6 > 8.2) = 0 p(7 > 6.5) = pa3 pb1 = 0.4 ∗ 0.2 = 0.08 p(7 > 7) = 0.5 pa3 pb2 = 0.5 ∗ 0.4 ∗ 0.4 = 0.08

p(da3 > db3 ) = p(7 > 7.4) = 0 p(da3 > db4 ) = p(7 > 8.2) = 0 p(da4 > db1 ) = p(7.5 > 6.5) = pa4 pb1 = 0.2 ∗ 0.2 = 0.04 p(da4 > db2 ) = p(7.5 > 7) = pa4 pb2 = 0.2 ∗ 0.4 = 0.08

56

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

p(da4 > db3 ) = p(7.5 > 7.4) = pa4 pb3 = 0.2 ∗ 0.3 = 0.06 p(da4 > db4 ) = p(7.5 > 8.2) = 0 p(da5 > db1 ) = p(da5 > db2 ) = p(da5 > db3 ) = p(da5 > db4 ) =

p(8 > 6.5) = pa5 pb1 = 0.1 ∗ 0.2 = 0.02 p(8 > 7) = pa5 pb2 = 0.1 ∗ 0.4 = 0.04 p(8 > 7.4) = pa5 pb3 = 0.1 ∗ 0.3 = 0.03 p(8 > 8.2) = 0

Thus, according to (2.12), we can get the probability of ⊗a to be greater than ⊗b : p(⊗a > ⊗b ) =

5 ∑ 4 ∑ ) ( p dai > dbj i=1 j=1

= 0.08 + 0.08 + 0.04 + 0.08 + 0.06 + 0.02 + 0.04 + 0.03 = 0.430 Therefore, one can affirm that ⊗a is greater than ⊗b with the probability of 0.430. This affirmation is equivalent to saying that ⊗b is greater than ⊗a with the probability of 1 − 0.430 = 0.570.

Interval Grey Numbers Comparison In the case of interval grey numbers comparison, it should be stated that one of the conditions for making the comparison is represented by the independence of the compared interval grey numbers. As mentioned above, if this condition is violated, few steps should be considered in order to eliminate the dependence between the compared numbers prior to comparison. Let us consider two interval grey numbers and we will note them through ⊗a and ⊗b , with: ] [ ⊗a ∈ a, a and ] [ ⊗b ∈ b, b , with a < a and b < b. For each of the two considered interval grey numbers, one can define a probability density function noted as f (·) [18]:

Comparison of Grey Numbers

57

{a f (x)d x = 1 a

{b f (y)dy = 1 b

where through f (x) we have noted the probability density function for ⊗a , while through f (y) we have noted the probability density function for ⊗b . By using a graphical representation in a bi-dimensional ) area ( can ) space, ( )a rectangle ) ( ( be encountered when representing the four points: a, b , a, b , a, b and a, b , as presented in Fig. 2.4. Using the idea of joint probability, Xie and Liu [18] define the probability that ⊗a is greater than ⊗b , noted as p(⊗a > ⊗b ), as being: { { p(⊗a > ⊗b ) = { {

D1

f (x, y)d xd y

D1 +D2

(2.12)

f (x, y)d xd y

where through D1 it has been noted the area positioned in the right-side of the x = y line, while through D2 it has been noted the area positioned in the left-side of the x = y line, as depicted in Fig. 2.4. Further, let us define [18]: y

xy

O

_a

_ a

x

Fig. 2.4 Example grey numbers when considering the area resulted by ) ( on comparing ) two( interval ) ( ) ( the points a, b , a, b , a, b and a, b

58

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

{

{ f (x, y)d xd y = σ

(2.13)

D1 +D2

( ) ( ) ( ) ( ) Considering the position of the points a, b , a, b , a, b and a, b , Xie and Liu [18] noted that there are six possible situations in which one can be when comparing two interval grey numbers: • • • • • •

Situation 1: b < b < a < a Situation 2: a < a < b < b Situation 3: a < b < a < b Situation 4: b < a < b < a Situation 5: b < a < a < b Situation 6: a < b < b < a.

which are discussed in the following using a one-dimensional and a bi-dimensional graphical approach along with a mathematical expression which provides the calculus of the p(⊗a > ⊗b ). The simplest situations in which one needs to compare interval grey numbers and with the most intuitive results are the ones in which the two interval grey numbers are not overlapping, namely Situation 1 and Situation 2. • Situation 1: b < b < a < a The succession b < b < a < a can be shortly rewritten as b < a, keeping in mind that b < b and that a < a. In order to better visualize this situation a one-dimensional graphical approach can be used as presented in Fig. 2.5. As it can be observed from the Fig. 2.5, in this case ⊗a is greater than ⊗b , thus the probability that ⊗a is greater than ⊗b is equal to 1. The probability that ⊗a is greater than ⊗b , p(⊗a > ⊗b ), can be determined mathematically by solving the following [18]: ¨ p(⊗a > ⊗b ) =

/ f (x, y)d xd y σ = 1

(2.14)

D1

The result of (2.14) is equal to 1 and can also be easily observed by using the bi-dimensional graphical representation as in Fig. 2.6.

b_

_ b

_a

Fig. 2.5 One-dimensional representation of Situation 1

_ a

x

Comparison of Grey Numbers

59

y

xy

b_ O

_a

_ a

x

_ b

x

Fig. 2.6 Bi-dimensional representation of Situation 1

_ a

_a

b_

Fig. 2.7 One-dimensional representation of Situation 2

• Situation 2: a < a < b < b Even in this case, the succession a < a < b < b can be shortly rewritten as b > a, knowing that b < b and that a < a. From the one-dimension graphical representation in Fig. 2.7 it can be observed that the probability that ⊗a is greater than ⊗b is equal to 0. The same result is reached if one solves the following mathematical equation [18]: ¨ p(⊗a > ⊗b ) =

/ f (x, y)d xd y σ = 0

(2.15)

D1

) ( Or by observing the position of the rectangle formed by the four points a, b , ) ( ) ) ( ( a, b , a, b and a, b in Fig. 2.8. • Situation 3: a < b < a < b In this situation, the two interval grey numbers are partly overlapping as it results from the one-dimensional graphical representation in Fig. 2.9. In order to determine the probability that ⊗a is greater than ⊗b , one should solve [18]:

60

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

_ b

xy

O

_ a

_a

x

Fig. 2.8 Bi-dimensional representation of Situation 2

_ b_ a

_a

_ b

x

Fig. 2.9 One-dimensional representation of Situation 3

¨ p(⊗a > ⊗b ) =

/

{a { y

f (x, y)d xd y σ = D1

b

/ f (x, y)d xd y σ

(2.16)

b

Graphically, the two areas positioned in the left-side and right-side of the x = y line are as in Fig. 2.10. • Situation 4: b < a < b < a Even in the case of Situation 4, the two interval grey numbers are overlapping as depicted in Fig. 2.11. In this case, the probability that ⊗a is greater than ⊗b is determined as [18]: ¨ ¨ / / f (x, y)d xd y σ = 1 − f (x, y)d xd y σ p(⊗a > ⊗b ) = D1

D2

Comparison of Grey Numbers

61 xy

O

_ a

_a

x

Fig. 2.10 Bi-dimensional representation of Situation 3

_ a_ b

b_

_ a

x

Fig. 2.11 One-dimensional representation of Situation 4

{b {b =1− a

/ f (x, y)d xd y σ

(2.17)

y

In a bi-dimensional space, Situation 4 is as presented in Fig. 2.12. • Situation 5: b < a < a < b Given the succession of the values for a, a, b and b, Situation 5 characterizes a case in which the interval grey number ⊗b include the interval grey number ⊗a , as depicted in Fig. 2.13. For determining the probability that ⊗a is greater than ⊗b the following equations should be solved as determined by Xie and Liu [18]: ¨ p(⊗a > ⊗b ) = D1

/ f (x, y)d xd y σ

62

2 Grey Numbers for Sentiment Analysis and Natural Language Processing xy

O

_ a

_a

x

Fig. 2.12 Bi-dimensional representation of Situation 4

b_

_ a

a_

_ b

x

Fig. 2.13 One-dimensional representation of Situation 5

⎤ ⎡ / {a {a {a { y ⎥ ⎢ f (x, y)d xd y + f (x, y)d xd y ⎦ σ =⎣ a

a

b

(2.18)

a

This situation is presented using the bi-dimensional space in Fig. 2.14. • Situation 6: a < b < b < a Even Situation 6 is an example of overlapping situation, but in this case the interval grey number ⊗a include the interval grey number ⊗b —Fig. 2.15. The probability that ⊗a is greater than ⊗b is determined as suggested by Xie and Liu [18]: ¨ / f (x, y)d xd y σ p(⊗a > ⊗b ) = D1

Comparison of Grey Numbers

63 xy

b_

O

_ a

_a

x

Fig. 2.14 Bi-dimensional representation of Situation 5

a_

b_

_ a

_ b

x

Fig. 2.15 One-dimensional representation of Situation 6

⎤ ⎡ / {a { y {a {a ⎥ ⎢ f (x, y)d xd y + f (x, y)d xd y ⎦ σ =⎣ a

a

b

(2.19)

a

Using the bi-dimensional representation, a situation similar to the one presented in Fig. 2.16 can be encountered. In the particular case in which the probabilities of any two values in both the covered-sets of the intervals grey numbers to be compared are equal, the probability function, f (x, y) = 1, and the probability that ⊗a is greater than ⊗b can be determined as [17, 18]: p(⊗a > ⊗b ) =

A1 A1 + A2

(2.20)

) ( ) ( where A1 is the area of the rectangle characterized by the points a, b , a, b , ) ( ) ( a, b , a, b and located in the right-side of the x = y line, and A2 is the area of

64

2 Grey Numbers for Sentiment Analysis and Natural Language Processing xy

b_

O

_ a

_a

x

Fig. 2.16 Bi-dimensional representation of Situation 6

( ) ( ) ) ( ) ( the rectangle characterized by the points a, b , a, b , a, b , a, b and located in the left-side of the x = y line, as presented in Fig. 2.17. Considering the situations mentioned above and the particular case in which the probabilities of any two values in both the covered-sets of the intervals grey numbers to be compared are equal, one can state that the probability that ⊗a is greater than ⊗b is determined as proposed in [17, 18]: y

x=y xy

O

_a

_ a

Fig. 2.17 Area-approach to comparing two interval grey numbers

x

Comparison of Grey Numbers

65

⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ (a−b)2 ⎪ ⎪ ⎨ 2(a−a )(b−b) 2 p(⊗a > ⊗b ) = 1 − (b−a ) ⎪ 2(a−a )(b−b) ⎪ ⎪ ⎪ ⎪ (a+a−2b) ⎪ ⎪ ⎪ 2(b−b) ⎪ ⎪ ⎪ ⎩ (2a−b−b) 2(a−a )

a>b b>a a b.up: print("a.low > b.up") return 1 elif b.low > a.up: print("b.low > a.up") return 0 elif a.low < b.low < a.up < b.up: print("a.low < b.low < a.up < b.up") return (a.up-b.low)**2 / (2 * (a.up - a.low) * (b.up b.low)) elif b.low < a.low < b.up < a.up: print("b.low < a.low < b.up < a.up") return 1 - (b.up - a.low)**2 / (2 * (a.up - a.low) * (b.up - b.low)) elif b.low < a.low < a.up < b.up: print("b.low < a.low < a.up < b.up") return (a.low + a.up - 2 * b.low)

/ (2 * (b.up -

b.low)) elif a.low < b.low < b.up < a.up: print("a.low < b.low < a.up < b.up") return (2 * a.up - b.up - b.low) / (2 * (a.up a.low)) else: raise Exception("Sorry, this case is not supported")

values associated with the tokens. As highlighted in Table 2.1 sentiment lexicons can be constructed using both automatic and manual approaches. While automatic approaches can have a very good coverage, containing large numbers of words and expressions, they are also commonly very noisy, which can negatively affect the performance of the sentiment classification algorithms [3]. On the other hand, manually crafting highly accurate lexicons with the help of independent human raters can be a tedious task, due to which, most manually created lexicons have limited coverage. In this context, an approach based on grey numbers theory is proposed in Cotfas et al. [3], which investigates how existing lexicons can be combined in order to improve both coverage and accuracy. The approach heavily relies on the introduction

Sentiment Analysis Using Grey Numbers

75

Table 2.4 Comparison between the number of positive and negative tokens Negative Maxdiff VADER Sentiment140 (unigrams)

726

Neutral 14

Positive 775

Total 1515

4173

–

3344

7517

24,154

–

38,311

62,465

of grey sentiment lexicons. Compared to traditional sentiment lexicons that assign a single numeric value to each token, a grey sentiment lexicon characterizes the affective value of the tokens using grey numbers [3]. The lexicons considered in the study are Maxdiff-Twitter, VADER and Sentiment140. Since word polarities can vary significantly across domains, the three lexicons have been chosen from the same domain, namely microblogging. As shown in Table 2.4, the number of tokens and the ratio between the number of negative and positive tokens varies greatly. The Maxdiff lexicon [7] contains 1515 tokens and has been created manually through crowdsourcing using the Maxdiff approach. The polarity values of the tokens vary between [−1], for extremely negative, to [+1] for extremely positive. On the other hand, the VADER1 lexicon [4] contains approximately 7500 tokens and has been manually created with the help of 10 independent human raters, that have received training in order to accurately evaluate the tokens on a scale from [−4], corresponding to extremely negative, up to [+4], corresponding to extremely positive. The resulting lexicon includes only the tokens for which the mean sentiment rating was non-zero and the standard deviation of the ratings was below 2.5. Finally, the Sentiment140 lexicon [13] has been created in an automatic manner, starting from the emoticons contained in tweets. While the intensity of the words varies between minus infinity and plus infinity, save for a few exceptions, all the tokens have a polarity between [−5] and [5]. Out of the three lexicons, only Maxdiff includes a limited number of neutral terms. The following steps have been taken by Cotfas et al. [3] in order to create the grey sentiment lexicons by combining several classical ones. First, the values in all the classic lexicons are normalized in the interval [−1, 1]. Afterwards the grey sentiment value for each token is determined as the interval between the minimum and the maximum values with which the token appears in the considered classical lexicons (Table 2.5). Four lexicons have been constructed in total by Cotfas et al. [3], three of them combining two by two the classical lexicons and a fourth one combining all three classical lexicons. In order to validate the benefits of grey sentiment lexicons, in comparison with the classical ones, in the following the base-line grey sentiment analysis algorithm proposed in [3] has been applied. The algorithm follows the approach proposed by Hutto and Gilbert [4], but without implementing the heuristics, such as the ones related to negations and booster words. While implementing also

1

https://github.com/cjhutto/vaderSentiment.

76

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

Table 2.5 Grey sentiment values for several affective words Maxdiff-Twitter

VADER

Sentiment140

Grey sentiment

Horrible

−0.89

−0.625

−0.3902

[−0.8900, −0.3902]

Bad

−0.5

−0.625

−0.2594

[−0.6250, −0.2594]

Relaxed

0.688

0.55

0.2464

[0.2464, 0.6880]

Happy

0.734

0.675

0.2392

[0.2392, 0.7340]

Perfect

0.766

0.675

0.1958

[0.1958, 0.7660]

Best

0.812

0.8

0.1574

[0.1574, 0.8000]

the heuristics would have resulted in a better classification performance, the baseline algorithm can nevertheless provide a relatively fair comparison of the analyzed lexicons. During the first step of the algorithm, the tweet is divided into separate units, called tokens, which correspond to the words, mentions and hashtags in the tweet. All the resulting tokens are then converted to lowercase, in order to facilitate the identification of the tokens that are also present in the lexicon. Afterwards, the grey polarity scores for these tokens are added using the addition operation for grey numbers in order to compute the overall grey sentiment score of the tweet. As mentioned by Cotfas et al. [3], the grey scalar product could be used to increase the importance of tokens following a booster word, such as “extremely”. In the following, the approach will be evaluated using the STS-Gold dataset [23] that contains 2034 tweets, out of which 632 are classified as positive tweets and 1402 are marked as negative tweets. The evaluation has been performed in terms of precision, recall and accuracy, computed from the True Positives (TP), True Negatives (TN), False Positives (FP) and the False Negatives (FN). The TP represents the number of real positive tweets classified as positive, while the TN contains the number of negative tweets correctly classified as negative. On the other hand, FP represents the number of real negative tweets classified incorrectly as positives, while FN is the number of real positive tweets incorrectly classified as negative. The accuracy is computed as the ratio between the correctly classified tweets (TP + TN) and the total number of tweets (TP + TN + FP + FN). The precision metric quantifies the ratio between the correctly classified positive tweets (TP) and the overall number of tweets that have been classified as positive (TP + FP). The recall on the other hand, conveys the ratio between the correctly classified positive tweets and the total number of positive tweets. The performance achieved by the base-line sentiment analysis algorithm using the classical lexicons has been evaluated first. As it can be noticed in Table 2.6, the best results in terms of precision (75%) and accuracy (85%) have been achieved using the Sentiment140 lexicon. The best results in terms of recall (79%) have been obtained in the case of the Maxdiff lexicon. The worst values in terms of precision and recall have been encountered in the case of the VADER lexicon, while the accuracy was equally bad in comparison to the Maxdiff lexicon. Arguably, since the automatically created Sentiment140 lexicon contains far more tokens than the other two manually

Sentiment Analysis Using Grey Numbers Table 2.6 Classification performance using the initial lexicons

77 Maxdiff

VADER

Sentiment140

True positives

498

300

487

True negatives

457

620

1238

False positives

945

782

164

False negatives

134

252

145

Precision (%)

35

28

75

Recall (%)

79

54

77

Accuracy (%)

47

47

85

created lexicons, it has a better coverage of the words that appear in tweets and can provide a better classification. Four grey sentiment lexicons have been created next that contain all the terms in the classical lexicons from which they have been created. As it can be noticed from Table 2.7, the grey sentiment lexicon created by keeping all the terms from VADER and Sentiment140 provides good results when considering all the metrics. The accuracy for this lexicon (70%) is the best among the four considered grey sentiment lexicons and is only exceed by the one of the Sentiment140 lexicon (75%). The same holds true for accuracy (84%), which is the best among the considered lexicons, except for the Sentiment140 lexicon (85%). However, the recall of the lexicon (82%) is better than any of the ones achieved by the classical lexicons. The best recall value (88%) has been achieved by the grey sentiment lexicon created from all the terms in the classical lexicons, but this lexicon provides relatively low values for the precision and recall metrics. Additionally, it can be noticed that the lexicon created by combining all the tokens in the Maxdiff and VADER lexicons provides better performance in terms of precision, recall and accuracy than any of the two initial lexicons. The case in which only the common terms in the initial lexicons are kept in the grey lexicon has also been analyzed, as highlighted in Table 2.8. It can be noticed that the results in terms of precision, recall and accuracy of all the lexicons in Table 2.8 is worse than those of the corresponding lexicons created by keeping all the terms from Table 2.7. Arguably, since these lexicons are created by keeping only the common tokens, they are smaller and thus have a reduced coverage of the words that appear in the analyzed tweets. In the following, the grey lexicon created by combining all the terms in the VADER and Sentiment 140 lexicon, that provides good result for all the considered metrics, will be used to determine the polarity of several tweets. For the tweet with the id 1264910848545300492 “I love Ice Cream Sandwich:) #Android #Google #Samsung” the overall polarity can be computed as shown in Table 2.9. The tokens that have been identified in the grey lexicon are “love” with grey sentiment polarity [0.2166, 0.8] and “:)” with a grey sentiment polarity of [0.5, 0.5]. Using the addition operation for grey numbers, the overall polarity of the tweet 2

http://twitter.com/51783905/status/126491084854530049.

58 87

39

81

54

Recall (%)

Accuracy (%) 77

83

Precision (%)

392

817

117

False positives

549 1010

515

585

True positives

True negatives

False negatives

Maxdiff ∪ Sentiment140

Maxdiff ∪ VADER

Table 2.7 Evaluation of the grey sentiment lexicons that contain all the tokens

84

82

70

111

224

1178

521

VADER ∪ Sentiment140

77

88

59

74

385

1017

558

Maxdiff ∪ VADER ∪ Sentiment140

78 2 Grey Numbers for Sentiment Analysis and Natural Language Processing

39 78

25

50

37

Precision (%)

Recall (%)

Accuracy (%)

769

55

140

966

314

False positives

492 633

318

436

True positives

True pegatives

False negatives

Maxdiff ∩ Sentiment140

Maxdiff ∩ VADER

5

5

2

600

1326

76

32

VADER ∩ Sentiment140

Table 2.8 Evaluation of the grey sentiment lexicons that contain only the common tokens

41

51

26

311

895

507

321

Maxdiff ∩ VADER ∩ Sentiment140

Sentiment Analysis Using Grey Numbers 79

80

2 Grey Numbers for Sentiment Analysis and Natural Language Processing

Table 2.9 Computing the polarity for the tweet with the id 126491084854530049

Token

Operation

Grey sentiment polarity

Love

+

[0.2166, 0.8]

Ice

+

[0.067, 0.067]

Cream

+

[0.0574, 0.0574]

Sandwich

+

[−0.0026, −0.0026]

:)

+

[0.5, 0.5]

#android

+

[0.058, 0.058]

#google

+

[−0.0008, −0.0008]

=

[0.8956, 1.479]

I

#samsung Overall polarity

is [0.8956, 1.479]. Given the fact that the polarity is greater than 0, the tweet has a positive connotation. In the case of this tweet we can notice also a shortcoming of the base-line sentiment algorithm, which treats “Ice Cream Sandwich” as independent words and not as a group of words referring to a version of the Android operating system. However, this does not impact the results of the classification process, as even if we would disregard the values associated with the tokens “ice”, “cream” and “sandwich”, the tweet would still be recognized as positive, with an overall sentiment polarity of [0.7738, 1.3572]. In the case of the tweet with the id 126519329025040384,3 “Ice Cream Sandwich to stop carriers bullying smartphone users #google #android http://t.co/BZNy74Nn”. Several tokens have been found in the grey sentiment lexicon, most of them having a negative signification associated. Among them, we can recognize “bullying”, which is profoundly negative, with a grey sentiment polarity of [−0.725, −0.2022]. The overall polarity of the tweet is [−1.0956, −0.3456], as shown in Table 2.10, which indicates a negative tweet. The tweet with the id 1262923355406991364 has the content “@Apple iTunes is the worst program ever. For such a great phone, you make some awful software.”. Many of the tokens in the tweet are found in the lexicon, some with a positive polarity and others with a negative one, such as “worst” and “awful”. The overall grey polarity of the tweet is [−1.0634, 0.0172], as shown in Table 2.11, which captures the fact that even though the tweet is essentially a negative one, it also mentions some positive aspects. For the tweet with the id 125840039031738368,5 having the content “@apple your simply the best.”, the overall grey polarity is positive one, as highlighted in Table 2.12, given the fact that all the matched tokens have a positive connotation.

3

http://twitter.com/94469491/status/126519329025040384. http://twitter.com/18679448/status/126292335540699136. 5 http://twitter.com/286714503/status/125840039031738368. 4

Sentiment Analysis Using Grey Numbers Table 2.10 Computing the polarity for the tweet with the id 126519329025040384

81

Token

Operation

Grey sentiment polarity

Ice

+

[0.067, 0.067]

Cream

+

[0.0574, 0.0574]

Sandwich

+

[−0.0026, −0.0026]

To

+

[−0.033, −0.033]

Stop

+

[−0.3, −0.0728]

Carriers

+

[−0.2226, −0.2226]

Bullying

+

[−0.725, −0.2022]

Smartphone

+

[0.0298, 0.0298]

Users

+

[−0.0238, −0.0238]

#google

+

[−0.0008, −0.0008]

#android

+

[0.058, 0.058]

=

[−1.0956, −0.3456]

Operation

Grey sentiment polarity

+

[−0.023, −0.023]

Worst

+

[−0.775, −0.375]

Program

+

[0.0192, 0.0192]

Ever

+

[−0.0024, −0.0024]

+

[0.0054, 0.0054]

Great

+

[0.2354, 0.775]

Phone

+

[−0.1542, −0.1542]

You

+

[0.1352, 0.1352]

Make

+

[0.014, 0.014]

Some

+

[0.077, 0.077]

Awful

+

[−0.5, −0.359]

Software

+

[−0.095, −0.095]

Overall polarity

=

[−1.0634, 0.0172]

http://t.co/BZNy74Nn Overall polarity Table 2.11 Computing the polarity for the tweet with the id 126292335540699136

Token @apple iTunes Is The

For Such A

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2 Grey Numbers for Sentiment Analysis and Natural Language Processing

Table 2.12 Computing the polarity for the tweet with the id 125840039031738368

Token

Operation

Grey sentiment polarity

Your

+

[0.161, 0.161]

Simply

+

[0.1316, 0.1316]

Best

+

[0.1574, 0.8]

Overall polarity

=

[0.45, 1.0926]

@apple

The

Fig. 2.38 Comparison between the polarity of tweet1 and that of tweet2

Given the fact the overall polarity of the tweets is expressed using grey numbers, one can compare them. This aspect might be useful in real-life applications, as it might offer a better overall view of the people’s opinion regarding certain topics. The comparison approach will be showcased using the tweets in Tables 2.9 and 2.12. As the tweet in Table 2.9 (noted tweet1) has a polarity score of [0.8956, 1.479], while the one in Table 2.12 (noted tweet2) has an overall polarity of [0.45, 1.0926], the probability that the positivity of tweet1 is greater than that of tweet2 is approximately 0.95, as shown in Fig. 2.38.

Concluding Remarks While the creation of grey sentiment lexicons has been evaluated by Cotfas et al. [3] using the Sanders6 dataset, in the present chapter we have chosen to perform the evaluation using the STS-Gold dataset [23]. Even in the case of this dataset it has been shown that the grey lexicons can exceed the performance of the initial lexicons. The grey lexicons have afterwards been used in the context of a base-line sentiment analysis algorithm to perform polarity analysis, during which the tokens in the analyzed text are compared with the ones in the lexicon. The values in the lexicon for the tokens that appear in the text are added using the grey addition operation. The algorithm can be improved by implementing heuristics, such as the ones proposed in the VADER algorithm. Booster words can be taken into account and their impact can be quantified using the grey scalar product operation. Finally, the comparison

6

http://www.sananalytics.com/lab/twitter-sentiment.

References

83

between grey numbers has been used to evaluate whether a tweet has a more positive connotation than another one.

References 1. Neethu, M.S., Rajasree, R.: Sentiment analysis in twitter using machine learning techniques. In: 2013 Fourth International Conference on Computing, Communications and Networking Technologies (ICCCNT), pp. 1–5. IEEE, Tiruchengode (2013). https://doi.org/10.1109/ICC CNT.2013.6726818 2. Ghiassi, M., Skinner, J., Zimbra, D.: Twitter brand sentiment analysis: a hybrid system using ngram analysis and dynamic artificial neural network. Expert Syst. Appl. 40, 6266–6282 (2013). https://doi.org/10.1016/j.eswa.2013.05.057 3. Cotfas, L.-A., Delcea, C., Roxin, I.: Grey sentiment analysis using multiple lexicons. In: Proceedings of the 15th International Conference on Conference on Informatics in Economy (IE 2016), pp. 428–433. Bucharest University of Economic Studies Press, Cluj-Napoca, Romania (2016) 4. Hutto, C.J., Gilbert, E.: VADER: A Parsimonious Rule-Based Model for Sentiment Analysis of Social Media Text (2014) 5. Aloufi, S., Saddik, A.E.: Sentiment identification in football-specific tweets. IEEE Access. 6, 78609–78621 (2018). https://doi.org/10.1109/ACCESS.2018.2885117 6. Dey, A., Jenamani, M., Thakkar, J.J.: Senti-N-Gram: an n-gram lexicon for sentiment analysis. Expert Syst. Appl. 103, 92–105 (2018). https://doi.org/10.1016/j.eswa.2018.03.004 7. Kiritchenko, S., Zhu, X., Mohammad, S.M.: Sentiment Analysis of Short Informal Texts. JAIR 50, 723–762 (2014). https://doi.org/10.1613/jair.4272 8. Cotfas, L.-A., Delcea, C., Roxin, I., Paun, R.: Twitter ontology-driven sentiment analysis. In: New Trends in Intelligent Information and Database Systems, pp. 131–139. Springer International Publishing (2015) 9. Fellbaum, C.: Towards a representation of idioms in WordNet. https://reader.elsevier.com/ reader/sd/pii/S095741741830143X?token=8A0D7A6667D2C6001ECCF61478F91E4259A2 7CE9378223C585D614B0BDC7F63E078BC7CB57AA072AEDB60C8BA6AD839A&ori ginRegion=eu-west-1&originCreation=20221201111409. Last accessed 12 Jan 2022. https:// doi.org/10.1016/j.eswa.2018.03.004 10. Bradley, M., Lang, P.: Affective Norms for English Words (ANEW): Instruction Manual and Affective Ratings. OSF, University of Flordida (1999) 11. Baccianella, S., Esuli, A., Sebastiani, F.: SentiWordNet 3.0: an enhanced lexical resource for sentiment analysis and opinion mining. In: LREC, pp. 2200–2204 (2010) 12. Cambria, E., Havasi, C., Hussain, A.: SenticNet 2: a semantic and affective resource for opinion mining and sentiment analysis. In: Proceedings of the 25th International Florida Artificial Intelligence Research Society Conference, FLAIRS-25, pp. 202–207 (2012) 13. Mohammad, S., Kiritchenko, S., Zhu, X.: NRC-Canada: building the state-of-the-art in sentiment analysis of tweets. In: Second Joint Conference on Lexical and Computational Semantics (*SEM), Volume 2: Proceedings of the Seventh International Workshop on Semantic Evaluation (SemEval 2013), pp. 321–327. Association for Computational Linguistics, Atlanta, Georgia, USA (2013) 14. Liu, S., Yang, Y., Forrest, J.: Grey numbers and their operations. In: Liu, S., Yang, Y., and Forrest, J. (eds.) Grey Data Analysis : Methods, Models and Applications, pp. 29–43. Springer, Singapore (2017). https://doi.org/10.1007/978-981-10-1841-1_3 15. Liu, S., Lin, Y.: Grey Systems. Springer, Berlin Heidelberg, Berlin, Heidelberg (2011) 16. Liu, S., Forrest, J.Y.-L.: Advances in Grey Systems Research. Springer Science & Business Media (2010)

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17. Xie, N., Xin, J.: Interval grey numbers based multi-attribute decision making method for supplier selection. Kybernetes 43, 1064–1078 (2014). https://doi.org/10.1108/K-01-2014-0010 18. Xie, N., Liu, S.: Novel methods on comparing grey numbers. Appl. Math. Model. 34, 415–423 (2010). https://doi.org/10.1016/j.apm.2009.05.001 19. Liu, S., Lin, Y.: Grey Information: Theory and Practical Applications. Springer, London (2006) 20. Delcea, C.: Grey systems theory in economics—bibliometric analysis and applications’ overview. Grey Syst. Theory Appl. 5, 244–262 (2015). https://doi.org/10.1108/GS-03-20150005 21. Delcea, C.: Grey systems theory in economics—a historical applications review. Grey Syst. Theory Appl. 5, 263–276 (2015). https://doi.org/10.1108/GS-05-2015-0018 22. Mukhtar, N., Khan, M.A., Chiragh, N.: Lexicon-based approach outperforms supervised machine learning approach for Urdu sentiment analysis in multiple domains. Telematics Inform. 35, 2173–2183 (2018). https://doi.org/10.1016/j.tele.2018.08.003 23. Saif, H., Fernandez, M., Alani, H.: Evaluation Datasets for Twitter Sentiment Analysis. A Survey and a New Dataset, the STS-Gold (2013)

Chapter 3

Supplier Selection Using Grey Systems Theory

Introduction Supplier selection is a multi-attribute decision-making (MADM) problem largely discussed in the scientific literature. Basically, it deals with selecting the most suitable alternative from a finite number of possible suppliers (feasible alternatives) based on the values recorded for each considered attribute with respect to every alternative, having the evaluations provided by a set of experts. Over the years, the supplier selection problem has evolved from a commoditybased operational decision approach—which considers mainly aspects related to economic issues, described through a series of strategic performance measures (such as cost, quality, time, flexibility, innovativeness, etc.) or organizational factors (such as culture, technology, relationship, etc.) [1, 2], to a sustainable supplier selection problem [3]. As the economic environment is in a continuous development in connection with the new trends and needs provided by the consumers, the classical economic and cost-based supplier selection has been left behind step-by-step, leaving the place to a more complex supplier selection problem, which is based on a system of sustainable performance attributes [4, 5]. Various approaches can be encountered in the scientific literature from a green, environmental or ecological aspect [3]. As a result, the scientific literature acknowledges the problem of green supplier selection [6–9], environmental supplier selection [10, 11] or ecological supplier selection [12]. Now-a-days, in the light of the sustainability and social aspects, the sustainable supplier selection problem is highly titrated. In this problem, elements such as pollution, company’s green image, resources consumption hold the spot [13, 14]. Over the time, a series of methodologies have been used for solving the supplier selection process, such as, but not limited to [15–25]: • Multi-Criteria Decision-Making (MCDM) techniques: – Analytic Network Process (ANP), – Analytic Hierarchy Process (AHP), © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_3

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– Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), – Decision-Making Trial and Evaluation Laboratory (DEMATEL), – Preference Ranking Organization Methods for Enrichment Evaluations (PROMETHEE), – Elimination and Choice Expressing Reality (ELECTRE), – Multicriteria Optimization and Compromise Solution (VIKOR), – Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP), – Simple Additive Weighting (SAW), – Complex Proportional Assessment of Alternatives (COPRAS), – Weighted Aggregated Sum-Product Assessment (WASPAS), – Multi-Objective Optimization on the basis of Ratio Analysis (MOORA), – Multi-objective optimization by Ratio Analysis plus the full Multiplicative Form (MULTIMOORA), – Step-wise Weight Assessment Ratio Analysis (SWARA), – Additive-Veto Method (AVM), – Combined Compromise Solution Method (CoCoSo), – Best Worst Method (BWM), – Qualitative Flexible Multiple Criteria Method (QUALIFLEX). • Mathematical Programming (MP) techniques: – – – – – – – – – – – – –

Data Envelopment Analysis (DEA), Linear Programming (LP), Nonlinear Programming (NLP), Stochastic Programming (SP), Multi-Objective Programming (MOP), Goal-Programming (GP), Mixed Integer Linear Programming (MILP), Mixed Integer Nonlinear Programming (MINLP), Stochastic Mixed Integer Linear Programming (MILP + SP), Stochastic Mixed Integer Nonlinear Programming (MINLP + SP), Multiobjective Mixed Integer Linear Programming (MILP + MOP), Multiobjective Mixed Integer Nonlinear Programming (MINLP + MOP), Multiobjective Stochastic Mixed Integer Programming (MIP + MOP + SP).

• Artificial Intelligence (AI) techniques: – – – – – – – –

Grey Systems Theory (GST), Genetic Algorithm (GA), Neural Networks (NN), Rough Set Theory (RST), Fuzzy Set Theory (FST), Decision tree (DT), Bayesian Networks (BN), Particle Swarm Optimization (PSO),

Introduction

– – – – – –

87

Ant Colony Algorithm (ACA), Case-Based Reasoning (CBR), Support Vector Machine (SVM), Dempster Shafer Theory of Evidence (DST), Association Rule (AR), K-means clustering.

• Strategic Planning (SP) techniques: – Strengths–Weaknesses–Opportunities–Threatens (SWOT), • Hybrid Approaches (HA)—by combining techniques from the above-mentioned categories. Considering the papers published between 2008 and 2012, Chai et al. [21] provided a systematic review of 123 papers on supplier selection. Most of the techniques mentioned in the above list have been identified in the selected data sample. Counting for the techniques used in the selected papers, the authors have shown that AHP was the most frequently techniques used in supplier selection (24.39%), followed by LP (15.44%) and TOPSIS (14.63%). The considered MCM techniques (AHP, ANP, ELECTRE, PROMETHEE, TOPSIS, VIKOR, DEMATEL, and SMART) accounted for 61.79%, the MP techniques (DEA, LP, NLP, MOP, GP, SP) for 48.78%, while the AI techniques (GST, GA, NN, RST, BN, DT, CBR, PSO, SVM, AR, ACA, DST) for 28.46%—the sum of the percentages exceeded 100% due to the hybrid approaches which have used a combination of the above-mentioned techniques [21]. From the AI category, the most-employed technique has been GA (6.50%), followed by GST (4.88%) [21]. As for the papers written between 2013 and 2018, Chai and Ngai [25] conducted another study and revealed that the most-employed technique has been LP (19 papers), followed by AHP (13 papers) and SP (11 papers). The use of the GST has been identified in 3 papers in the above-mentioned period [25]. Additionally, new techniques have been identified in the 2013–2018 period [25], compared to 2008– 2012 period [21], such as: all the mixed programming techniques (MILP, MINLP, MILP + SP, MINLP + SP, MILP + MOP, MINLP + MOP, MIP + MOP + SP), QUALIFLEX and K-means clustering. Nevertheless, each of the mentioned methodologies come with both advantages and disadvantages. For a review of the techniques used in supplier selection, the reader can refer, but not limit to: Dutta et al. [26], de Boer et al. [27], Chai et al. [21], Mardani et al. [28], Simic et al. [29], Chai and Ngai [25], Zimmer et al. [30]. Regarding the supplier selection problem, one of the main drawbacks mentioned in the scientific literature is related to the requirement made by some of the methodologies to have non-interrelated criteria (as in the case of AHP), while other cannot be used as single methodologies, but rather in conjunction with other methodologies (situation known as a hybrid approach) [15]. As Hekmat et al. [15] mentioned, besides the disadvantage related to the existence of non-interrelated criteria, some methodologies suffer from other disadvantages,

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such as the inability to provide a complete ranking of suppliers (sometimes called decision-making units or DMU) or the incapacity of dealing with many inputs and outputs. On the other hand, the entire supplier selection process is subject to a complex and uncertain situation which derives from the subjectivity nature of the evaluation made by the experts over the considered attributes. As a result, it might be sometimes hard to express evaluations using precise numbers and, therefore, an approximate form might be more adequate considering the experts’ different knowledge and perception on the investigated attributes [31]. As Aslani et al. [32] pointed out, in real cases, a degree of uncertainty in the decision process is inevitable. Therefore, the use of grey systems theory has become a viable alternative to the classical methods for solving the supplier selection process (AHP, ANP, and DEMATEL) [32]. In the present chapter we aim at providing a comprehensive literature review on the papers written on the supplier selection theme where grey systems theory elements have been used for solving the problem. A series of attributes have been extracted from the scientific literature and presented, along with some key elements of the papers using a grey approach to the supplier selection problem. A brief bibliometric analysis is provided for better shaping the main characteristics of the papers written in the field of grey supplier selection. Additionally, a method for determining the solution for the supplier selection problem, which features elements taken from grey systems theory is presented from a theoretical point of view. A numerical application accompanies the theory for a better understanding on the needed steps to be consider. As the supplier selection problem is a common problem to be solved in the economic sciences, we hope that you will find the reading interesting and fruitful.

State of the Art in Supplier Selection Through Grey Systems Theory Approach In order to provide an overview on the scientific papers written in the area of grey supplier selection, a filter has been made on the paper manually selected in Chap. 1 by using the succession of words “supplier selection”. As a result of the search step, a number of 24 papers have extracted and discussed in the following. First, a short bibliometric analysis is provided on the purpose of familiarizing the reader with the characteristics of the papers to be discussed, along with identifying the main authors in the field, the journals in which they have published the papers related to grey supplier selection, the affiliation of the authors, their home country, and the most frequent words used in their works. Second, a review of the 24 selected papers is conducted. Besides discussing the specific aspects considered by the authors in the papers, a summary of the elements from the grey systems theory used in their researches is provided, along with a

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89

summary of the types of evaluation criteria considered, the number of suppliers, attributes and experts used within each case study.

Bibliometric Analysis The selected dataset on supplier selection problem using the grey systems theory has the characteristics depicted in Table 3.1. As it can be seen, the 24 papers have been written in the 2007–2021 period. As the basis of the grey systems theory have been put in 1987, it can be observed that the first paper that has used the grey systems theory in the supplier section has been published 20 years later, in 2007. The journals in which the papers dealing with the supplier selection with grey systems theory have been 19, while the number of cited references has been 1018. Regarding the average citations per documents a value of 75.75 has been recorded, higher than the one recorded for the dataset consisting in general economics and social sciences field discussed in Chap. 1 (equal to 23.87). Also, the average citations per year per document is higher in the dataset comprising the papers on the supplier selection (8.34) when compared to the general dataset from Chap. 1 (namely 3.53), showing the increasing interest of the scientific community in the papers written on this subject. Considering the authors of the 24 papers, it has been observed that the authors have been in number of 63, appearing 71 times in the selected papers—Table 3.2. Only one document has been written by a single author, the rest of the 23 papers being the result of a collaboration. Table 3.1 Main information about data

Table 3.2 Authors

Indicator

Value

Timespan

2007:2021

Sources (Journals, Books, etc.)

19

Documents

24

Average years from publication

6.62

Average citations per documents

75.75

Average citations per year per doc

8.34

References

1018

Indicator

Value

Authors

63

Author appearances

71

Authors of single-authored documents

1

Authors of multi-authored documents

62

90 Table 3.3 Authors collaboration

3 Supplier Selection Using Grey Systems Theory Indicator

Value

Single-authored documents

1

Documents per author

0.381

Authors per document

2.62

Co-authors per documents

2.96

Collaboration Index

2.7

The average number of documents per author has been 0.381, while the number of authors per document being 2.62—Table 3.3. Considering in the analysis only the papers with 2 or more authors, the co-authors per document indicator has a value of 2.96. As for the collaboration index, the recorded value of 2.7 is showing a slightly increased collaboration activity than in the case in which the entire dataset on economics and socials sciences is considered and where the value of this indicator has been equal to 2.23 (see Chap. 1). In terms of annual scientific production, one can observe a series of peaks, the period 2014–2019, being the most productive from the number of published papers point of view—Fig. 3.1. The annual growth rate reported by Biblioshiny was 11.61%, smaller than the one recorded for the entire dataset (of 21.7%). The journals in which the 24 selected papers have been published are depicted in Fig. 3.2 along with the number of published papers. As it can be observed, the International Journal of Production Economics has published 3 papers (12.50%),

Fig. 3.1 Annual scientific production evolution

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91

Fig. 3.2 Journals

while International Journal of Advanced Manufacturing Technology has published 2 papers (8.33%). Both journals focusing on grey systems theory, namely Journal of Grey System, and Grey Systems—Theory and Application have published 2 papers each. The 4-mentioned journals are included in zone 1 according to Bradford’s law— Fig. 3.3. Other journals in which 1 paper has been published are: Annals of Operations Research, Complexity, Expert Systems with Applications, Industrial Management & Data Systems, International Journal of Information Technology & Decision Making, International journal of Production Research, International Journal of Systems Science, Operations & Logistics, Journal of Civil Engineering and Management, Journal of Cleaner Production, Journal of Enterprise Information Management, Journal of Intelligent & Fuzzy Systems, Kybernetes, Mathematical and Computer Modelling, Sustainable Development, Technological and Economic Development of Economy—Fig. 3.2. In terms of authors, there are 8 authors which have co-authored 2 papers: Xie NM, Hashemi SH, Karimi A, Li GD, Nagai M, Sarkis J, Tavana M, and Yamaguchi D. As it can be observed from Figs. 3.4 and 3.5 of the 8 authors have written papers in the area of supplier selection with grey systems theory in successive years, while for the rest of 3 papers, the time interval between the successive papers has been of 4 to 8 years. The most relevant universities (for which the number of papers has been greater or equal to 2 papers) are depicted in Fig. 3.5. The country of the corresponding author is presented in Fig. 3.6. As in the case of the general set discussed in Chap. 1, it can be observed that the country with the most

92

Fig. 3.3 Bradford’s law on source clustering

Fig. 3.4 Top-8 authors production over time

3 Supplier Selection Using Grey Systems Theory

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93

Fig. 3.5 Most relevant affiliations

contributions in the field of supplier selection is China, with 8 documents—Fig. 3.6. The second place is occupied by USA with 5 documents, while India, Iran and Japan are sharing the third place, each of them with 2 documents. Fourth place is divided between Brazil, Denmark, Korea, Turkey and United Kingdom with 1 document.

Fig. 3.6 The corresponding author’s country

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Fig. 3.7 Number of citations per paper

In terms of MCP (Multiple Country Publications—an inter-country collaboration index) and SCP (Single Country Publications—an intra-country collaboration index) it can be observed that both China and USA have inter-country and intra-country collaborations in the selected papers, while Denmark and United kingdom have only inter-country collaborations. The rest of the countries (India, Iran, Japan, Brazil, Korea and Turkey) present only intra-country collaborations—Fig. 3.6. Among the 24 considered papers, the most cited paper has been written by Bai et al. [3], counting for 506 citations out of 1818 total citations, representing a percentage of 27.83%—Fig. 3.7. In fact, the paper written by Bai et al. [3] has taken even the second place in the general ranking based on the number of citations discussed in Chap. 1. The second, third and fourth place belong to Hashemi et al. [33], Li et al. [34] and Su et al. [35], with 302, 269 and respectively 162 citations—Fig. 3.7. As half of the 24 papers have more than 20 citations, one can state again the importance given by the scientific community to the papers written in the area of supplier selection using grey systems theory. Considering the author’s keywords, as expected, the most-used keyword is “supplier selection”, appearing 13 times, followed by “grey theory”—6 times, and “grey relational analysis”—4 times. These keywords appear also in the co-occurrence network based on author’s keywords as depicted in Fig. 3.8, along with “grey number”, “fuzzy set” and “multi-criteria decision-making”. Other author’s keywords include the techniques used additionally to the grey systems theory for providing a better approach to the supplier selection problem, such as “dematel”, “best–worst method”, “fuzzy set”, “mcdm”, “principal component

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Fig. 3.8 Co-occurrence network based on author’s keywords

Fig. 3.9 Top-50 words based on author’s keywords

analysis”, “topsis”, “analytic network process”, “cocoso”—Fig. 3.9. As a result, in the section dedicated to the review of the 24 selected papers, we expect to encounter these techniques when addressing the supplier selection problem. Using a three-fields plot, in Fig. 3.10 the connection between the authors’ affiliations, their names and the keywords they have used in the scientific papers is presented. Once more, it can be observed that among the extracted keywords, except for the ones related to the grey systems theory, the remainder of the keywords are mainly referring to other methods and techniques that can be used in the economic analysis.

Grey Supplier Selection A review of the 24 selected papers is provided in the following. The papers have been divided into two categories: paper which have used only elements from grey systems

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Fig. 3.10 Three-fields plot: affiliations (left), authors (middle), keywords (right)

theory (grey numbers, grey relational analysis) to address the supplier selection problem, and papers which have used a mixed approach, combining the advantages of grey systems theory with other methods and techniques.

Supplier Selection with Grey Numbers And/Or Grey Relational Analysis Abdulshahed et al. [36] have used grey numbers in order to take into account the incompleteness and imprecision of human judgements in the case of supplier problem applied for an steelmaking company in Libya. The authors have considered two benefit criteria (quality and logistic service) and two cost criteria (direct costs and delivery lead time) for shaping the supplier selection problem. The data has been extracted using questionnaires, which have been administered to the company’s managers and experts. The number of respondents has been equal to 4, while the number of considered suppliers has been equal to 6. The importance and the performance scales have considered 7 attributes, ranging from very low/very poor to very high/very good. The application has been solved using Excel, which proves the easiness in considering the approach described by the authors for this type of problems. Considering the suppliers order, the authors have stated that the direct costs and the quality of the material are the most crucial criteria for the case of the steelmaking company [36].

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A similar approach is used by Li et al. [37]. Considering the advantages offered by the grey numbers and calculus of the possibility degree, Li et al. [37] propose a grey-based approach to supplier selection problem, in which rough sets are included. An ideal supplier solution is provided along the problem solving. The performance of the considered suppliers is then compared with the ideal supplier in order to rank them. Through the use of synthetic data, a numerical example is provided for a better understanding of the needed steps of the approach. Overall, even though the number of steps is numerous (namely 8 steps), the method is simple, easy to understand and apply. The approach is further used by Thakur and Anbanandam [38] on 6 economic criteria selected for the case of banking industry and 4 suppliers. The authors suggest a possible development of the used method by considering the industries in which one cannot rely on a single supplier. Xie and Xin [31] uses grey numbers to address the supplier selection problem. The authors present in a very comprehensive manner the needed steps to be considered for solving this type of problem. Each step is described in depth, using a rigorous mathematical approach. The paper is accompanied by a synthetic data example which brings light into the actual use of the grey systems theory for solving the supplier selection problem. As the approach proposed by Xie and Xin [31] considers an unform distribution of the interval grey numbers, the authors suggest that a possible extension of their current work would be in focusing on other distribution forms for the implied data. Memon et al. [16] considers the grey numbers for expressing the experts’ view on the selection criteria and for attributing weights to each objective. A whitenization procedure with equal-weight mean is applied for obtaining the value of the grey parameters. The supplier selection problem is solved afterwards through the use of LINGO solver. A numerical example is provided with synthetic data, featuring 4 decision-makers, 3 suppliers and 7 criteria. Diba and Xie [39] propose a new approach when dealing with the supplier selection problem. The authors have employed the Deng’s grey relational analysis (GRA) model, absolute GRA model (ADGRA) and a novel second synthetic GRA (SSGRA) model, along with the conservative (maximin) criteria and Hurwicz criteria for determining the best solutions in the supplier selection problem. The attributes taken into account have targeted both economic and socio-environmental aspects. The application of the proposed approach has been made on a Milk company. As a result, the 5 attributes have been chosen to properly reflect the needs of the selected company, namely: cost, logistics and quantity, technology, environmental management, standard quality and management commitment as a part of environmental aspect [39]. Different from the classical approaches regarding the supplier selection when grey systems theory is used, in the proposed approach, for each supplier, the attributes (criteria) have been ranked from best to worst based on the grey coefficients. These coefficients have been determined through the application of the three degrees of relational analysis. As the authors noted, there couldn’t have been found a consensus in the order of the criteria in the case of each supplier. For example, environmental management has been found as a dominant factor for 4 of the 5 suppliers when Deng’s grey relational analysis model has been used, while when the absolute GRA

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model has been used, the cost has been marked as the dominant factor in 3 of the 5 considered suppliers. As a result, the authors found necessarily to use additionally the conservative criteria and the Hurwicz criteria for determining the best supplier. Last, the results of the considered supplier selection problem are presented in terms of the considered criteria (namely cost, logistic and quality, technology, environmental management, standard quality, management commitment). For each of the six criteria, a ranking of the suppliers has been provided. As one might not necessarily be interested in the ranking order on each of the considered criteria, the authors have provided an overall evaluation in which the supplier have been ranked in accordance with the form of the results usually obtained in the supplier selection problem. Hashemi et al. [40] propose an improved grey relational analysis which is employed in the steps needed for solving the supplier selection problem. The application featured in the paper is focusing on green supplier selection. As a result, the attributes of the suppliers are in the area of carbon governance, carbon policy, measures of carbon management, supplier collaboration and carbon accounting and inventory. The authors mentioned that, as the proposed approach uses elements from grey systems theory for dealing with the uncertain decision-makers judgements, it is capable to deal with vagueness and uncertainty in decision-making. As drawbacks to the method proposed in [40], the authors mentioned that in the situations in which the criteria are internally dependent, a ANP or a DEMATEL approach are needed in addition to the proposed method [40]. As a result, in a new paper, Hashemi et al. [33] combine ANP with the improved grey relational analysis for solving the supplier selection problem. The results are discussed in the next section dedicated to mixed grey approaches. Jiefang [41] proposes a grey correlation model for addressing the supplier selection problem, in which the preference of the decision-maker is measured for any two considered suppliers. As a result, the author defines different degrees of preference for decision-maker, ranging between strong and little preference when making the comparison between the suppliers. Based on them, a maximum cross correlation matrix is determined, with the role in suppliers’ ranking. The method is easy to apply in the supplier selection process. With all these, the method only considers the preference manifested by a single decision-maker, additional adjustments being needed for the case in which multiple decision-makers are considered. Quan et al. [8] uses a weighted grey relational model in the supplier selection process. The authors describe in detail the steps needed to be followed. A numerical application featuring 8 criteria and 7 suppliers is provided by the authors. The approach does not consider the presence of experts as the authors believe that it is better to use data collected by the company rather than to use the scores provided by experts. This approach is, in the opinion of the authors, more authentic and effective. As a result of the application of the proposed method based on grey theory, the ranking of the suppliers is determined.

State of the Art in Supplier Selection Through Grey Systems Theory …

99

Supplier Selection with Mixed Grey Approaches Mixed Grey Approaches have been used in the scientific literature for dealing with the situations in which the classical approaches suffer from a series of disadvantages, either related to the limited capacity of working with interrelated criteria or providing a complete ranking of the suppliers. As the grey systems theory methods have proven their applicability on limited datasets, there have been cases in which the mixed approaches with grey systems theory have been preferred as they do not require a specific distribution of the involved data. In terms of approaches to the supplier selection problem, two possible types of results have been envisioned: the case in which the solution of the supplier selection is a ranking of the suppliers starting from the most-desired supplier to the least-desired supplier, or the case in which the criteria (attributes) used for choosing among the suppliers are ranked from the most-desired to the least-desired criterion. In the following, the papers belonging to each of the two possible results are discussed.

Supplier Ranking Aslani et al. [32] propose three methods for solving the supplier selection problem by considering the results obtained in terms of methodology in the scientific literature. As a result, the authors combine the advantages offered by the grey systems theory with other approaches from the field, such as the best–worst method (BMW), the weighted aggregated sum product assessment (WASPAS), and the technique for order of preference by similarity to ideal solution (TOPSIS). Along with the description of the steps needed to perform the analysis of the suppliers based on the proposed approaches, the authors provide an application which features the existence of 3 types of selection criteria: economic (7 indicators), social (5 indicators) and environmental (5 indicators). The evaluation of the 30 suppliers is made by 3 evaluators. The results of the application of grey WASPAS and grey TOPSIS methods are evaluated through a sensitivity analysis. Based on this analysis, the authors mention that the sequence of the suppliers depends on the selected scenarios. Also, the authors state that the supplier selection becomes more and more a challenging problem for the decisionmakers as, beside the cost elements one should consider, in the now-a-days complex economic environment, one should also consider the environmental and social factors when selecting the supplier. Bai and Sarkis [3] start from the idea of an existing rough set environment and uses grey numbers to address the suppler selection problem. The authors consider 9 attributes, 3 from each of the economic, environmental and social categories for selecting the supplier. A series of examples of indicators from each of the abovementioned 3 categories are presented based on the scientific literature. The needed steps, presented theoretically in the paper, are then explicated through a numerical example. In conclusions, the authors state that the final decision may be sensitive to the attributes used in the evaluation process.

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Considering a slightly different approach than the one provided in Bai and Sarkis [3], Bai and Sarkis [42], combines the advantages of grey numbers with the technique for order of preference by similarity to ideal solution (TOPSIS) and run the supplier selection proposed approach on the same set as the authors did in [3]. In terms of top-ranked supplier, the authors mention that the same supplier has been selected as in Bai and Sarkis [3]. Even more, the authors have showed that in both approaches the top-6 suppliers have been the same, while only two have been transposed in the ranking comparison [42]. Nevertheless, the two approaches are different. As the authors have mentioned, the grey-based TOPSIS approach is less complex from a mathematical point of view, being, at the same time more intuitive than the grey rough set approach. The suitable supplier has been determined by Li et al. [34] through the use of grey systems theory in conjunction with rough set theory. The authors have justified the need for combining the two theories by mentioning the decreasing of the unnecessary information and compensate for the insufficiency of decision making. As in Li et al. [37], in the present approach, the authors have determined the profile of the ideal supplier. Based on the values of the attributes for the ideal supplier, the ranking of the considered suppliers has been determined. A synthetic data example is provided by the authors for a better understanding of the proposed approach. The attributes for ranking the suppliers have been chosen from the economic area. Celebi and Selvi [43] provides a fuzzy-grey approach to the supplier selection problem. Even though the paper claims that grey numbers are used in conjunction with fuzzy sets, the analysis is mostly conducted by the means of grey numbers. The fuzzy sets are used on a limited portion, namely the one dealing with converting the linguistic expressions to fuzzy numbers. The linguistic performance scale has a small granularity, containing 9 possible answers, ranging from very poor to very good. The authors provide a case study made for a Turkish firm in the Wagon industry. A number of 4 criteria are used for the evaluation, namely delivery, quality, flexibility and service, all of them in the area of economic criteria. The number of suppliers has been high (equal to 15), while the number of evaluators has been equal to 4. When comparing the order of the suppliers to the one obtained through a fuzzy selection method, the authors conclude that the top-5 suppliers are in the same order in both approaches [43]. In Hashemi et al. [33], the authors start from the observation made in their previously published paper (Hashemi et al. [40]) regarding the need to add ANP in the supplier selection method and mix the two approaches. Taking the case study of an automotive parts company, the authors applied the new method on 11 scenarios in which the decision criteria are either restricted to only economic criteria, only environmental criteria or both [33]. Among the advantages of the new proposed method, one can name the fact that the addition of the ANP considers more the internal relationships among the green supplier evaluation criteria, an important aspect that should be considered by the supply chain decision makers. The authors also pointed out that despite the fact that the traditional criteria (economic criteria) might be more transparent and easier to evaluate, the decision-makers should try to give specific attention to the driving criteria in order to increase the performance of their companies.

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Starting from the disadvantages brought by the classical methods for dealing with the supplier selection—such as the inability to provide a complete ranking of suppliers, the incapacity of working with too many inputs and outputs, or the lacking of handling the interrelated criteria—Hekmat et al. [15] proposed a Grey Principal Component Analysis-Data Envelopment Analysis (GPCA-DEA) methodology for the supplier selection. The addition of the principal component analysis to DEA has been justified by the drawbacks DEA possesses, such as not being designed to deal with interrelated criteria, not being able to provide a complete ranking of suppliers in most of the cases, and the decrease in the discriminant power of DEA as the number of inputs or outputs increases [15]. Furthermore, the principal component analysis can solve the mentioned drawbacks of DEA, but it is still requiring a normally distributed data in order to provide accurate results. Therefore, the authors decided to add grey relational analysis for dealing with the lack of data sources which might determine the data insufficiency and can affect the determination of the data distribution [15]. The case study is made on the payment subsystem of the banking industry and the results are compared with classical MADM methods (SAW, TOPSIS, VIKOR), proving its reliability. As drawbacks, the GPCA-DEA method does not consider the case in which the preference weights of the criteria are non-identical. Sadeghieh et al. [44] addresses the part supplier selection problem by proposing a method which combines the advantages of grey numbers, goal-programming and genetic algorithms. In terms of methodology, after gathering the values for the criteria to be used, the geometrical mean for grey numbers is employed for determining the most important objectives for the evaluation process. Then, a mathematical goalprogramming problem is set up and evaluated through the means of genetic algorithm. The results have shown the suitable combination of part suppliers. A method based on grey numbers and DEMATEL has been proposed by Su et al. [35]. A questionnaire containing 22 criteria structured in 4 main aspects has been completed and used for selecting the best supplier from a list of 4 possible suppliers for an electronic manufacturing focal firm. A hybrid approach based on grey relational analysis and additive-veto model has been proposed by Garcez et al. [45]. The particularity of the proposed approach consists in the fact that the supplier problem aimed to be solved involves the existence of only one decision-maker. In this context, the proposed method contains a preliminary phase in which the decision maker is characterized. The additiveveto option incorporated in the method ensures the vetoing of the alternatives. This action is important as if offers to the decision-maker the option to make trade-offs among criteria, allowing the decision-maker to compensate a low performance of the supplier in one attribute to a high performance in another attribute [45]. According to the authors, the incorporation in the grey relational analysis of the additiveveto model helps the decision-maker to provide more reasonable judgements when ranking suppliers. In terms of attributes, the authors consider 6 economic criteria. The method is compared by the authors with a simpler approach, which do not consider the additive-veto option, proving its advantages.

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Pitchipoo et al. [46] used grey relational analysis (GRA) along with entropy measurement (EM) and principal component analysis (PCA) for testing the solutions offered by three models: GRA, GRA-Entropy and GRA-PCA in the supplier selection problem. Through a numerical application which considered the existence of 5 suppliers and 4 economic criteria, the authors have shown that the 3 considered models provide comparative results. Even more, the sensitive analysis conducted by the authors has been pointing out that the distinguishing coefficients have a direct impact on the performance of the considered models. Starting from the classical approach in which the supplier selection is determined based on grey decision matrix, Golmohammadi and Mellat-Parast [47] proposed a novel approach based on grey relational analysis, fuzzy set theory and a pairwise comparison for suppliers based on certain criteria. The results of the model are compared with the classical model results by considering the information provided by a company in the automotive industry. The authors state that the advantages of the proposed approach derives from the fact that the method address the imprecision in judgment for weight assignment and preference aggregation of decision makers [47]. On the other hand, one of the main drawbacks of the proposed method is that due to the dependence of each alternative on the other alternatives, the ranking of alternatives is highly sensitive on the changes in the number of alternatives, either by adding or deleting alternatives. Yazdani et al. [48] extends the combined compromise solution method (CoCoSo) by including grey numbers in order to measure the performance of the suppliers in a construction company. The importance of the supplier criteria is provided through the use of two weighting methods, namely DEMATEL and BWM. The results of the proposed approach are compared with a grey complex proportional assessment method (COPRAS). The comparison results have shown consistency in the bestplaced supplier in both the considered methods.

Criteria Ranking Considering a particular type of supplier, namely the one dealing with logistic outsourcing, also known as third-party logistics supplier (3PL or TPL), Govindan et al. [49] propose a grey DEMATEL approach to ranking the criteria needed for selecting a 3PL. In the proposed approach DEMATEL is used for creating a causal diagram of interdependent criteria, while grey systems theory is needed to handle the human judgements. A number of 12 criteria are considered and the opinions of 4 experts are used for determining the final hierarchy in each of the 5 scenarios considered by the authors. By applying the methodology on a 3PL provider selection, the authors have found that the most important criteria for the selection of 3PL are on-time delivery performance, technological capability and financial stability.

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Jiang et al. [6] propose a mixed approach based on a combined DEMATEL-based ANP (known as DANP) and grey relational analysis for determining an importancepreference matrix for the attributes to be considered in the supplier selection decisionmaking process. Grey relational analysis has been used for generating the direct influence matrix from the Delphi questionnaire. This addition has been made in order to avoid the situation in which the respondents completed directly the influence matrix. A number of 8 attributes have been considered, 4 of them from the economic area (cost, quality, technology, and delivery time) and 4 from the environmental area (environmental management system, ecological design, pollution control, and management commitment). The result of the importance performance analysis of the considered criteria has shown that the criteria with the highest importance ranking and performance value are the technology and the environmental management system, while the criteria with the lowest importance ranking and performance value are cost, management commitment, and ecological design.

Discussion Considered the 24 selected papers, it has been observed that most of the studies have used either the grey numbers (15 papers) or the grey relational analysis (11 papers) for addressing the supplier selection problem. Grey numbers have been used in a hybrid approach in 10 papers, while the grey relational analysis has been used in hybrid approach in 6 papers. Regarding the methods with which the hybridization has been made, they have been from the area of multi-criteria decision making (14 papers), artificial intelligence (6 papers), mathematical programming (2 papers). Among the methods from the multi-criteria decision making, the following have been used along with the grey systems theory approaches: DEMATEL (4 papers), TOPSIS (2 papers), BWM (2 papers), ANP (2 papers), WASPAS (1 paper), AVM (1 paper), EM (1 paper), CoCoSo (1 paper). The most used artificial intelligence methods have been RST and PCS (2 papers each), followed by FST and GA (1 paper each). Last, from the mathematical programming, only DEA and GP have been used (in 1 paper each). Figure 3.11 present the distribution of the techniques used along with the GST in the selected papers. Additionally, a summary on the approaches used, the type of evaluation criteria used, a brief description of the data used in the numerical application, the number of criteria used, number of experts considered and number of suppliers to be ranked are presented in Table 3.4. As the evaluation criteria belonging to each of the three considered categories (economic, social and environmental) have been different in the considered studies, a review of them is provided in Table 3.5. Along with a brief explanation, each criterion has been classified as either a “cost” or a “benefit” based on the definition provided to each criterion. Last, the references considering the mentioned criterion have been listed with the observation that in some of the cases, the name of the

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MCDM

MP

PCA

GA

FST

RST

GP

DEA

CoCoSo

EM

AVM

WASPAS

ANP

BWM

TOPSIS

DEMATEL

No. of papers

Hybridization techniques 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

AI

Techniques

Fig. 3.11 Hybridization techniques used along with GST

criterion has not been exactly the one listed in Table 3.5, but the reference has been kept as it was reflecting a similar/the same idea. Based on the review papers, some brief remarks can be made: • the final decision may be sensitive to the attributes used in the evaluation process, • in some cases, the final ranking might be sensitive to the addition/deletion of the suppliers, • the sequence of the suppliers depends on the selected scenarios, • the ranking result is sensitive to the weights used in evaluation (e.g. equal of non-equal weights).

Application of Grey Numbers to Supplier Selection An application for the supplier selection is provided in the following. The theoretical background is in line with the work of Xie and Xin [31]. Also, the works of Thakur and Anbanandam [38] and Abdulshahed et al. [36] have been considered when defining the elements presented in this section. The notations have been kept as in Xie and Xin [31].

✔

✔ ✔ – ✔ –

–

✔

Aslani et al. [32] 2020 Grey numbers, ✔ TOPSIS, WASPAS, BWM ✔ ✔ ✔ ✔

✔

✔

Bai and Sarkis [42]

Celebi and Selvi 2014 Grey numbers, [43] Fuzzy sets

2021 Grey relational ✔ analysis, Grey numbers, additive-veto model

Bai and Sarkis [3]

Diba and Xie [39]

Garcez et al. [45]

Golmohammadi 2012 Grey relational and analysis Mellat-Parast [47]

Govindan et al. [49]

2

3

4

5

6

7

8

9

2015 Grey numbers, DEMATEL

2019 Grey relational analysis

2018 Grey numbers, TOPSIS

2010 Grey numbers, Rough sets

–

✔

2017 Grey numbers

Abdulshahed et al. [36]

–

–

–

✔

–

✔

✔

✔

–

Economic Social Environmental

Evaluation criteria

1

Approach

Year

No Author(s)

–

–

–

✔

–

✔

✔

✔

–

Third-party logistics provider

Company in automotive industry

Synthetic data illustration

Milk company

Wagon company

Synthetic data illustration

Synthetic data illustration

Electrified pickup trucks

Iron and steel company

Sustainable Application

Table 3.4 Review on grey approaches, evaluation criteria and application to supplier selection

12

6

6

6

4

9

9

17

4

4

2

1

4

4

4

4

3

4

(continued)

1

4

15

5

15

13

13

30

6

Number of Number Number attributes/criteria of of experts suppliers

Application of Grey Numbers to Supplier Selection 105

2015 Grey relational analysis, ANP

Hashemi et al. [33]

Hekmat et al. [15]

Jiang et al. [6]

Jiefang [41]

Li et al. [37]

Li et al. [34]

Memon et al. [16]

11

12

13

14

15

16

17

2015 Grey numbers

–

–

✔

✔

–

✔

–

–

✔

2008 Grey numbers, grey ✔ relational analysis, rough sets

–

✔

–

–

–

✔

–

–

–

✔

–

✔

✔

Economic Social Environmental

Evaluation criteria

–

2007 Grey numbers

2013 Grey relational analysis

2018 Grey relational analysis, DEMATEL, ANP

2021 Grey relational analysis, principal component analysis, DEA

2013 Grey relational analysis

Hashemi et al. [40]

10

Approach

Year

No Author(s)

Table 3.4 (continued)

–

–

–

–

–

–

–

–

Synthetic data illustration

Synthetic data illustration

Synthetic data illustration

Synthetic data illustration

Automotive industry

Payment industry

Automobile parts company

Synthetic data illustration

Sustainable Application

7

4

4

5

12

7

6

5

4

4

4

1

8

–

4

4

(continued)

3

7

6

6

1

5

5

5

Number of Number Number attributes/criteria of of experts suppliers

106 3 Supplier Selection Using Grey Systems Theory

–

✔

Yazdani et al. [48]

24

–

✔

Xie and Xin [31] 2014 Grey numbers

23

–

✔

Thakur and Anbanandam [38]

22

✔

✔

Su et al. [35]

21

–

2012 Grey numbers, ✔ goal-programming, genetic algorithm

Sadeghieh et al. [44]

20

–

✔

Quan et al. [8]

19

2019 Grey numbers, combined compromise solution method, BWM, DEMATEL

2015 Grey numbers

2016 Grey numbers, DEMATEL

2018 Grey relational analysis

Pitchipoo et al. [46]

✔

–

–

✔

–

✔

–

Economic Social Environmental

Evaluation criteria

–

Approach

2014 Grey relational ✔ analysis, entropy measurement, principal component analysis

Year

18

No Author(s)

Table 3.4 (continued)

–

–

–

✔

–

–

–

7

8

4

Construction company

Synthetic data illustration

Banking industry

7

5

6

7

4

5

20

4

–

1

10

7

4

4

22

7

5

Number of Number Number attributes/criteria of of experts suppliers

Electronic 22 manufacturing focal firm

Process plant–coffee maker

Chemical processing industry

Medium-sized electroplating industry

Sustainable Application

Application of Grey Numbers to Supplier Selection 107

Description The quality of the materials/products determined based on the number of rejected items in each batch or by the rates of defection in final products

Different types of cost: direct cost of the material/items, production cost, ordering cost, logistic cost

Evaluation criterion

Quality

Cost

Type of criterion

Economic

Table 3.5 Review on evaluation criteria

Abdulshahed et al. [36]; Aslani et al. [32]; Bai and Sarkis [3], [42]; Diba and Xie [39]; Garcez et al. [45]; Golmohammadi and Mellat-Parast [47]; Govindan et al. [49]; Hashemi et al. [33]; Hekmat et al. [15]; Jiang et al. [6]; Li et al. [34]; Pitchipoo et al. [46]; Quan et al. [8]; Sadeghieh et al. [44]; Thakur and Anbanandam [38]; Xie and Xin [31]; Yazdani et al. [48]

✔

(continued)

Abdulshahed et al. [36]; Aslani et al. [32]; Bai and Sarkis [3]; Golmohammadi and Mellat-Parast [47]; Bai and Sarkis [42]; Garcez et al. [45]; Govindan et al. [49]; Hashemi et al. [33]; Hekmat et al. [15]; Jiang et al. [6]; Li et al. [34]; Memon et al. [16]; Quan et al. [8]; Sadeghieh et al. [44]; Thakur and Anbanandam [38]; Xie and Xin [31]

References

✔

Benefit

Type of criterion Cost

108 3 Supplier Selection Using Grey Systems Theory

Type of criterion

Table 3.5 (continued) Description The amount of time passed between the moment an order has been placed and the moment it has been received

Evaluation of the logistic service quality, including the transportation time

Supplier’s ability to respond to the fluctuations in the demand or to handle the eventual disruptions

Evaluation criterion

Delivery lead time

Logistic service

Flexibility

✔

Cost

Aslani et al. [32]; Bai and Sarkis [3], [42]; Celebi and Selvi [43]; Golmohammadi and Mellat-Parast [47]; Govindan et al. [49]; Hashemi et al. [33]; Jiang et al. [6]; Thakur and Anbanandam [38]; Yazdani et al. [48]

✔

(continued)

Abdulshahed et al. [36]; Celebi and Selvi [43]; Diba and Xie [39]; Garcez et al. [45]; Hekmat et al. [15]; Li et al. [34]; Memon et al. [16]; Xie and Xin [31]

Abdulshahed et al. [36]; Aslani et al. [32]; Bai and Sarkis [3], [42]; Celebi and Selvi [43]; Garcez et al. [45]; Golmohammadi and Mellat-Parast [47]; Govindan et al. [49]; Hashemi et al. [33]; Hekmat et al. [15]; Jiang et al. [6]; Jiefang [41]; Li et al. [34]; Memon et al. [16]; Pitchipoo et al. [46]; Quan et al. [8]; Sadeghieh et al. [44]; Thakur and Anbanandam [38]; Xie and Xin [31]; Yazdani et al. [48]

References

✔

Benefit

Type of criterion

Application of Grey Numbers to Supplier Selection 109

Type of criterion

Table 3.5 (continued)

Aslani et al. [32]; Govindan et al. [49] Bai and Sarkis [3], [42]; Garcez et al. [45]; Hashemi et al. [33]; Xie and Xin [31] Bai and Sarkis [3], [42]; Hashemi et al. [33]; Jiang et al. [6]

✔ ✔

✔

Represents the public image of the supplier Measures the new launch of products or the use of new technologies Reflects the management attitude, the strategic fit, the compatibility of the top management, and the compatibility among levels and functions. It also incorporates the feeling of trust

Brand

Innovativeness

Culture

(continued)

Aslani et al. [32]; Govindan et al. [49]; Yazdani et al. [48]

✔

Reflects the position of the supplier in the market by making a comparison with other similar suppliers (competitors)

References

Financial position

Benefit Aslani et al. [32]; Bai and Sarkis [3], [42]; Diba and Xie [39]; Golmohammadi and Mellat-Parast [47]; Govindan et al. [49]; Hashemi et al. [33]; Hekmat et al. [15]; Jiang et al. [6]; Quan et al. [8]; Thakur and Anbanandam [38]

Cost

Type of criterion ✔

Description

Production capability and Measures the supplier capability in facilities terms of machinery and infrastructure capabilities

Evaluation criterion

110 3 Supplier Selection Using Grey Systems Theory

Social

Type of criterion

Table 3.5 (continued)

Reflects supplier’s ability to create a friendly and motivational work environment, and a good work atmosphere Comprises elements related to work conditions (e.g. equity labour sources, employee contracts, flexible working schedules, diversity, discrimination, employment compensations, career and development, job opportunities, research and development, etc.)

Employment practices

Evaluates the long-term relationship, the relationship closeness, reputation from the point of view of integrity, communication openness

Relationship

Supportive activities

Description

Evaluation criterion Cost

Aslani et al. [32]

Aslani et al. [32]; Bai and Sarkis [3], [42]; Govindan et al. [49]; Su et al. [35]

✔

✔

(continued)

Bai and Sarkis [3], [42]; Hashemi et al. [33]; Hekmat et al. [15]; Thakur and Anbanandam [38]

References

✔

Benefit

Type of criterion

Application of Grey Numbers to Supplier Selection 111

Environmental

Type of criterion

Table 3.5 (continued)

Measures the supplier’s ability to create financial prosperity and social equity in the community Reflects the interest of the supplier to guarantee safety at workplace for its employees (considers the health and safety incidents, and the health and safety practices)

Development commitment

Health and safety management system

Evaluates the degree to which the supplier contributes to the deterioration of the environment through pollutants, wastewater, solid waste, air emissions, hazardous materials, etc. which are the result of its everyday activity

Evaluates the supplier’s effects on the community from a multiple perspective: economic welfare, economic growth, health, education, housing, service infrastructure, security, cultural properties, social cohesion, social pathologies, supporting community projects, grants and donations, etc

Local community influence

Pollution

Description

Evaluation criterion

✔

Cost

Aslani et al. [32]

Aslani et al. [32]; Bai and Sarkis [3], [42]

✔

✔

(continued)

Aslani et al. [32]; Bai and Sarkis [3], [42]; Diba and Xie [39]; Hashemi et al. [33]; Jiang et al. [6]; Memon et al. [16]; Quan et al. [8]; Su et al. [35]; Yazdani et al. [48]

Aslani et al. [32]; Bai and Sarkis [3], [42]

References

✔

Benefit

Type of criterion

112 3 Supplier Selection Using Grey Systems Theory

Type of criterion

Table 3.5 (continued)

Reflects the degree to which the supplier respects certain regulations (such as ISO-14001 certificates)

Indicates the public opinion regarding the activity of the suppliers when considering the environment Evaluates the resources consumption. ✔ One can refer to energy, water, raw material, etc Assesses the supplier’s capability of designing products with low impact on the environment as a result of considering the actions of reuse and recycle or as a result of consuming as little energy and material as possible

Environmental management system

Green image

Resource consumption

Eco-design

Cost

✔

Aslani et al. [32]; Hashemi et al. [33]

✔

Aslani et al. [32]; Hashemi et al. [33]; Jiang et al. [6]

Aslani et al. [32]; Bai and Sarkis [3], [42]; Hashemi et al. [33]; Memon et al. [16]

Aslani et al. [32]; Bai and Sarkis [3], [42]; Diba and Xie [39]; Hashemi et al. [33]; Jiang et al. [6]; Memon et al. [16]; Su et al. [35]; Yazdani et al. [48]

References

✔

Benefit

Type of criterion

Description

Evaluation criterion

Application of Grey Numbers to Supplier Selection 113

114

3 Supplier Selection Using Grey Systems Theory

Xie and Xin Approach to Supplier Selection Based on Grey Numbers Let us consider a supplier selection problem defined by the following variables: • k decision-makers (DM), with l = 1, 2, . . . , k, • m possible suppliers (S), with i = 1, 2, . . . m, • n attributes of the suppliers (A), with j = 1, 2, . . . , n. Let us define the following sets: • S = {S1 , S2 , . . . , Sm }—the discrete set of m possible suppliers, • A = {A1 , A2 , . . . , An }—the discrete set of n attributes, • ⊗w = {⊗w1 , ⊗w2 , . . . , ⊗wn }—the set of attribute weights. It should be noted that the attribute weights are viewed as linguistic variables, expressed as grey numbers. The steps need for solving the supplies selection problem are summarized in Fig. 3.12 and fully described in the following: Step 1: Determine the grey rating value matrix for each DM In this step, each DM evaluates the attributes of each supplier and gives a linguistic value for each evaluation. As a result, for each DM a matrix containing the evaluation of each supplier for each attribute is created, having the following form [31]: ⎡

⊗l11 ⊗l12 ⎢ ⊗l ⊗l ⎢ 21 22 Dl = ⎢ . .. ⎣ .. . l ⊗m1 ⊗lm2

⎤ · · · ⊗l1n · · · ⊗l2n ⎥ ⎥ . . .. ⎥ . . ⎦ · · · ⊗lmn

where: [ ] ⊗li j = a li j , a li j (i = 1, 2, . . . m; j = 1, 2, . . . , n; l = 1, 2, . . . , k) representing the rating of the lth DM about the ith supplier for the jth attribute. Therefore, the general structure of the supplier selection problem appears as in Table 3.6. Step 2: Calculate the synthesized rating value for each attribute for each supplier Based on the scale for the attribute ratings presented in Table 3.7, the synthesized rating value for each attribute and for each supplier is determined using the following formula [31]:

Application of Grey Numbers to Supplier Selection Fig. 3.12 Steps needed for solving the supplier selection problem

115

Step 1: Determine the grey rating value matrix for each DM

Step 2: Calculate the synthesized rating value for each attribute for each supplier

Step 3: Create the synthesized grey decision matrix (D)

Step 4: Establish the normalized grey decision matrix (D*)

Step 5: Define the mean attribute weight based on the DMs’ attribute weights

Step 6: Create the weighted normalized grey decision matrix (WD)

Step 7: Describe the ideal supplier attribute sequence

Step 8: Determine the synthesized possibility degree between each supplier and the ideal one

Step 9: Rank the suppliers and make decision

Table 3.6 General structure of the supplier selection problem Supplier

Decision maker k A1

A2

···

An

···

⊗k12 ⊗k22

···

⊗k1n

···

⊗k11 ⊗k21

···

⊗k2n

.. .

..

.. .

.. .

..

.. .

⊗1mn

···

⊗km1

⊗km2

···

···

An

··· ···

⊗11n ⊗12n

.. .

..

⊗1m2

···

A1

A2

S2

⊗111 ⊗121

⊗112 ⊗122

.. .

.. .

Sm

⊗1m1

S1

··· ···

Decision maker 1

.

.

.

⊗kmn

116

3 Supplier Selection Using Grey Systems Theory

Table 3.7 The scale for attribute ratings (⊗)

⊗i j =

Scale

⊗

Very low (VL)

[0.00, 3.00]

Low (L)

[3.00, 5.00]

Medium (M)

[5.00, 7.00]

High (H)

[7.00, 9.00]

Very high (VH)

[9.00, 10.00]

] ) [ 1( 1 ⊗i j + ⊗i2j + . . . + ⊗ikj = a i j , a i j (i = 1, 2, . . . m; j = 1, 2, . . . , n) k

where: ai j =

1∑ l a k l=1 i j

ai j =

1∑ l a k l=1 i j

k

k

The synthesized rating value is determined by dividing the value of the sum of the attribute ratings to the number of decision makers (k) as the decision makers are seen as having equal rights. Step 3: Create the synthesized grey decision matrix (D) The values calculated in the previous step are put into the D matrix, defined as follows [31]: ⎡

⊗11 ⊗12 ⎢ ⊗21 ⊗22 ⎢ D=⎢ . .. ⎣ .. . ⊗m1 ⊗m2

⎤ · · · ⊗1n · · · ⊗2n ⎥ ⎥ . . .. ⎥ . . ⎦ · · · ⊗mn

where: [ ] ⊗i j = a i j , a i j Step 4: Establish the normalized grey decision matrix (D*) The normalized grey decision matrix (D*) is defined as follows [31]:

Application of Grey Numbers to Supplier Selection

117

⎡

⊗∗11 ⊗∗12 ⎢ ⊗∗ ⊗∗ ⎢ 21 22 D∗ = ⎢ . .. ⎣ .. . ⊗∗m1 ⊗∗m2

⎤ · · · ⊗∗1n · · · ⊗∗2n ⎥ ⎥ . . .. ⎥ . . ⎦ · · · ⊗∗mn

where: [ ] ⊗i∗j = a i∗j , a i∗j The elements of the (D*) matrix are determined based on the type of attribute one is considering [31]: [ ] • For a cost attribute, ⊗i∗j = a i∗j , a i∗j is calculated as: [ ⊗i∗j

=

a min a min j j , ai j ai j

]

where: { } a min = min a i j j 1≤i≤m

[ ] • For a benefit attribute, ⊗i∗j = a i∗j , a i∗j is calculated as: [ ⊗i∗j

=

ai j

ai j max , max aj aj

]

where: { } a max = max a i j j 1≤i≤m

Step 5: Define the mean attribute weight based on the DMs’ attribute weights Based on the values in Table 3.8 the evaluations of the decision makers with regard to the attribute weights are determined as [31]: ⊗w j =

) [ ] 1( 1 ⊗w j + ⊗w2j + . . . + ⊗wkj = w j , w j ( j = 1, 2, . . . , n) k

where: 1∑ l w k l=1 j k

wj =

118

3 Supplier Selection Using Grey Systems Theory

Table 3.8 The scale for the attribute weights (⊗w)

Scale

⊗w

Very low (VL)

[0.00, 0.20]

Low (L)

[0.20, 0.40]

Medium (M)

[0.40, 0.60]

High (H)

[0.60, 0.80]

Very high (VH)

[0.80, 1.00]

And 1∑ l wj = w k l=1 j k

Step 6: Create the weighted normalized grey decision matrix (WD) The weighted normalized grey decision matrix (W D) is determined by considering the aggregated attributes’ weights. The form of the matrix is [31]: ⎡

w⊗11 w⊗12 ⎢ w⊗21 w⊗22 ⎢ WD = ⎢ . .. ⎣ .. . w⊗m1 w⊗m2

⎤ · · · w⊗1n · · · w⊗2n ⎥ ⎥ .. ⎥ .. . . ⎦ · · · w⊗mn

where: [ ] w⊗i j = bi j , bi j And w⊗i j = ⊗i∗j × ⊗w j Step 7: Describe the ideal supplier attribute sequence The ideal supplier is identified based on the values of a set of attributes. For a set{ of suppliers defined as S = }{S1 , S2 , . . . , Sm }, the ideal supplier is noted max max and it is determined as [31]: as S max = ⊗wmax 1 , ⊗w2 , . . . , ⊗wn Smax ⎫ ⎧ ⎪ [max{0.116, 0.150, 0.281, 0.116, 0.173, 0.116}, max{0.167, 0.259, 0.950, 0.167, 0.316, 0.167}], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ [max{0.131, 0.150, 0.245, 0.146, 0.300, 0.193}, max{0.300, 0.240, 0.514, 0.240, 0.800, 0.360}], ⎪ = [max{0.263, 0.132, 0.108, 0.140, 0.108, 0.210}, max{0.900, 0.225, 0.158, 0.246, 0.158, 0.491}], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [max{0.155, 0.545, 0.545, 0.058, 0.467, 0.583}, max{0.450, 0.900, 0.875, 0.350, 0.800, 0.900}], ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ [max{0.278, 0.417, 0.153, 0.417, 0.306, 0.306}, max{0.545, 0.700, 0.389, 0.700, 0.583, 0.583}]

Application of Grey Numbers to Supplier Selection

119

Step 8: Determine the synthesized possibility degree between each supplier and the ideal one The synthesized possibility degree between each supplier and the ideal one is determined by comparing the set of attributes. As the attributes are defined as grey numbers, the grey numbers comparison is used. Each supplier from the S = {S1 , S2 , . . . , Sm } set is compared with the ideal supplier, S max . The possibility degree is determined as [31]: P(Si ≤ S max ) =

n ) 1∑ ( P ⊗wi j ≤ ⊗max j n j=1

Step 9: Rank the suppliers and make decision Based on the values determined at the previous step, the set of suppliers {S1 , S2 , . . . , Sm } is ranked. A supplier Si is considered to have a higher ranking if the value for the P(Si ≤ S max ) is smaller. Contrary, when the value of P(Si ≤ S max ) is higher, the ranking order of Si is worse [31]. After the ranking of the suppliers is made, the best supplier is selected.

Practical Application on Supplier Selection A company producing clothes and distributing them through a chain of shops opened in Bucharest is interested in selecting the best supplier from a possible set of 6 suppliers. The criteria (attributes) taken into account for deciding on the best supplier are quality, cost, delivery lead time, environmental management system and pollution. In order to make the decision, 4 persons with knowledge in the area have been appointed as experts and have been asked to evaluate the attributes using linguistic variables. Grey systems theory through the use of the grey numbers and the methodology presented above [31] is employed for determining the best supplier in this case. In the following, the steps presented from a theoretical point of view are now depicted using the information extracted for the case of the clothes company. The starting information regarding the supplier selection is: • 4 decision-makers (DM), with l = 1, 2, . . . , 4, • 6 possible suppliers (S), with i = 1, 2, . . . 6, • 5 attributes of the suppliers (A), with j = 1, 2, . . . , 5. Along with following sets: • S = {S1 , S2 , . . . , S6 }—the discrete set of 6 possible suppliers, • A = {A1 , A2 , . . . , A5 }—the discrete set of 5 attributes, • ⊗w = {⊗w1 , ⊗w2 , . . . , ⊗w5 }—the set of attribute weights.

120

3 Supplier Selection Using Grey Systems Theory

Step 1: Determine the grey rating value matrix for each DM Based on each of the 4 DMs evaluation regarding the 5 attributes, the information in Table 3.9 is extracted. Step 2: Calculate the synthesized rating value for each attribute for each supplier The scale for the attribute ratings presented in Table 3.7 is used for determining the synthesized rating value for each attribute and for each supplier. • For supplier S1 the synthesized rating value for each attribute is determined as follows: ) 1( 1 ⊗ + ⊗211 + ⊗311 + ⊗411 4 11 1 = ([9.00, 10.00] + [7.00, 9.00] + [9.00, 10.00] + [9.00, 10.00]) 4 1 = [34.00, 39.00] = [8.50, 9.75] 4

⊗11 =

) 1( 1 ⊗12 + ⊗212 + ⊗312 + ⊗412 4 1 = ([7.00, 9.00] + [5.00, 7.00] + [5.00, 7.00] + [7.00, 9.00]) 4 1 = [24.00, 32.00] = [6.00, 8.00] 4

⊗12 =

) 1( 1 ⊗ + ⊗213 + ⊗313 + ⊗413 4 13 1 = ([3.00, 5.00] + [0.00, 3.00] + [3.00, 5.00] + [0.00, 3.00]) 4 1 = [6.00, 16.00] = [1.50, 4.00] 4

⊗13 =

) 1( 1 ⊗14 + ⊗214 + ⊗314 + ⊗414 4 1 = ([0.00, 3.00] + [3.00, 5.00] + [0.00, 3.00] + [5.00, 7.00]) 4 1 = [8.00, 18.00] = [2.00, 4.50] 4

⊗14 =

) 1( 1 ⊗15 + ⊗215 + ⊗315 + ⊗415 4 1 = ([5.00, 7.00] + [5.00, 7.00] + [7.00, 9.00] + [3.00, 5.00]) 4

⊗15 =

A1

VH

H

L

VH

F

H

Supplier

S1

S2

S3

S4

S5

S6

L

L

VH

L

VH

H

A2

VL

VH

F

VH

F

L

A3

Decision maker 1

A4

H

F

VL

H

H

VL

A5

F

L

H

L

H

F

VH

F

H

VL

F

H

A1

F

VL

H

L

H

F

A2

L

VH

F

VH

F

VL

A3

Decision maker 2

VH

H

VL

F

H

L

A4

Table 3.9 The decision makers ratings for each supplier on each attribute A5

H

F

VH

VL

VH

F

VH

F

VH

VL

F

VH

A1

H

L

H

L

F

F

A2

F

H

H

VH

H

L

A3

Decision maker 3 A4

F

H

VL

VH

H

VL

A5

F

H

F

L

F

H

VH

L

VH

L

F

VH

A1

F

L

H

F

VH

H

A2

L

VH

F

H

H

VL

A3

Decision maker 4 A4

VH

F

L

H

H

F

A5

F

H

VH

F

VH

L

Application of Grey Numbers to Supplier Selection 121

122

3 Supplier Selection Using Grey Systems Theory

=

1 [20.00, 28.00] = [5.00, 7.00] 4

• For supplier S2 , only the calculus for the first and the last values (⊗21 and ⊗25 ) are provided. We invite the reader to calculate the rest of them. ) 1( 1 ⊗21 + ⊗221 + ⊗321 + ⊗421 4 1 = ([7.00, 9.00] + [5.00, 7.00] + [5.00, 7.00] + [5.00, 7.00]) 4 1 = [20.00, 28.00] = [5.50, 7.50] 4

⊗21 =

.. . ) 1( 1 ⊗25 + ⊗225 + ⊗325 + ⊗425 4 1 = ([7.00, 9.00] + [9.00, 10.00] + [5.00, 7.00] + [9.00, 10.00]) 4 1 = [30.00, 36.00] = [7.50, 9.00] 4

⊗25 =

• For supplier S3 : ) 1( 1 ⊗31 + ⊗231 + ⊗331 + ⊗431 4 1 = ([3.00, 5.00] + [0.00, 3.00] + [0.00, 3.00] + [3.00, 5.00]) 4 1 = [6.00, 16.00] = [1.50, 4.00] 4

⊗31 =

.. . ) 1( 1 ⊗35 + ⊗235 + ⊗335 + ⊗435 4 1 = ([3.00, 5.00] + [0.00, 3.00] + [3.00, 5.00] + [5.00, 7.00]) 4 1 = [11.00, 20.00] = [2.75, 5.00] 4

⊗35 =

• For supplier S4 :

Application of Grey Numbers to Supplier Selection

) 1( 1 ⊗41 + ⊗241 + ⊗341 + ⊗441 4 1 = ([9.00, 10.00] + [7.00, 9.00] + [9.00, 10.00] + [9.00, 10.00]) 4 1 = [34.00, 39.00] = [8.50, 9.75] 4

⊗41 =

.. . ) 1( 1 ⊗45 + ⊗245 + ⊗345 + ⊗445 4 1 = ([7.00, 9.00] + [9.00, 10.00] + [5.00, 7.00] + [9.00, 10.00]) 4 1 = [30.00, 36.00] = [7.50, 9.00] 4

⊗45 =

• For supplier S5 : ) 1( 1 ⊗ + ⊗251 + ⊗351 + ⊗451 4 51 1 = ([5.00, 7.00] + [5.00, 7.00] + [5.00, 7.00] + [3.00, 5.00]) 4 1 = [18.00, 26.00] = [4.50, 6.50] 4

⊗51 =

.. . ) 1( 1 ⊗ + ⊗255 + ⊗355 + ⊗455 4 55 1 = ([3.00, 5.00] + [5.00, 7.00] + [7.00, 9.00] + [7.00, 9.00]) 4 1 = [22.00, 30.00] = [5.50, 7.50] 4

⊗55 =

• For supplier S6 : ) 1( 1 ⊗61 + ⊗261 + ⊗361 + ⊗461 4 1 = ([7.00, 9.00] + [9.00, 10.00] + [9.00, 10.00] + [9.00, 10.00]) 4 1 = [34.00, 39.00] = [8.50, 9.75] 4

⊗61 =

123

124

3 Supplier Selection Using Grey Systems Theory

.. . ) 1( 1 ⊗65 + ⊗265 + ⊗365 + ⊗465 4 1 = ([5.00, 7.00] + [7.00, 9.00] + [5.00, 7.00] + [5.00, 7.00]) 4 1 = [22.00, 30.00] = [5.50, 7.50] 4

⊗65 =

Step 3: Create the synthesized grey decision matrix (D) Gathering all the values calculated in the previous step for each attribute in the case of each supplier, the synthesized grey decision matrix in Table 3.10 is obtained. Step 4: Establish the normalized grey decision matrix (D*) In the case of the normalized grey decision matrix the type of attribute is of utter importance, namely the fact that an attribute is a cost or a benefit attribute. In our case, the attributes A1 , A2 , A3 representing the quality, cost and delivery lead time are cost attribute as a smaller value is better, while the attributes A4 , A5 representing the environmental management system and pollution are benefit attributes, a higher value being better. Table 3.10 Synthesized grey rating value matrix Supplier

A1

A2

A3

A4

A5

S1

[8.50, 9.75]

[6.00, 8.00]

[1.50, 4.00]

[2.00, 4.50]

[5.00, 7.00]

S2

[5.50, 7.50]

[7.50, 9.00]

[6.00, 8.00]

[7.00, 9.00]

[7.50, 9.00]

S3

[1.50, 4.00]

[3.50, 5.50]

[8.50, 9.75]

[7.00, 8.75]

[2.75, 5.00]

S4

[8.50, 9.75]

[7.50, 9.25]

[5.50, 7.50]

[0.75, 3.50]

[7.50, 9.00]

S5

[4.50, 6.50]

[2.25, 4.50]

[8.50, 9.75]

[6.00, 8.00]

[5.50, 7.50]

S6

[8.50, 9.75]

[5.00, 7.00]

[2.75, 5.00]

[7.50, 9.00]

[5.50, 7.50]

A4

A5

Table 3.11 Normalized grey decision matrix Supplier

A1

A2

A3

S1

[0.154, 0.176]

[0.281, 0.375]

[0.375, 1.000]

[0.222, 0.500]

[0.556, 0.778]

S2

[0.200, 0.273]

[0.250, 0.300]

[0.188, 0.250]

[0.778, 1.000]

[0.833, 1.000]

S3

[0.375, 1.000]

[0.409, 0.643]

[0.154, 0.176]

[0.778, 0.972]

[0.306, 0.556]

S4

[0.154, 0.176]

[0.243, 0.300]

[0.200, 0.273]

[0.083, 0.389]

[0.833, 1.000]

S5

[0.231, 0.333]

[0.500, 1.000]

[0.154, 0.176]

[0.667, 0.889]

[0.611, 0.833]

S6

[0.154, 0.176]

[0.321, 0.450]

[0.300, 0.545]

[0.833, 1.000]

[0.611, 0.833]

Application of Grey Numbers to Supplier Selection

125

Therefore, the values of the normalized grey matrix will be determined different in the case of A1 , A2 , A3 than in the case of A4 , A5 —as presented in the previous section. Below, we detail the calculus for the values of the normalized grey matrix elements in the case of attribute A1 and we provide only partially the calculus for the values of the normalized grey matrix elements in the case of attribute A2 − A5 , inviting the reader to determine the values not detailed in the following Table 3.11. • For attribute A1 : { } a1min = min a i1 = min{8.50, 5.50, 1.50, 8.50, 4.50, 8.50} = 1.50 1≤i≤6

⊗∗21 ⊗∗31 ⊗∗41 ⊗∗51 ⊗∗61

[

] ] [ a1min a1min 1.50 1.50 , , = [0.154, 0.176] = a 11 a 11 9.75 8.50 [ min min ] [ ] a1 a1 1.50 1.50 = , , = [0.200, 0.273] = a 21 a 21 7.50 5.50 [ min min ] [ ] a a 1.50 1.50 = 1 , 1 , = [0.375, 1.000] = a 31 a 31 4.00 1.50 [ min min ] [ ] a1 a1 1.50 1.50 = , , = [0.154, 0.176] = a 41 a 41 9.75 8.50 [ min min ] [ ] a a 1.50 1.50 = 1 , 1 , = [0.231, 0.333] = a 51 a 51 6.50 4.50 [ min min ] [ ] a1 a1 1.50 1.50 = , , = [0.154, 0.176] = a 61 a 61 9.75 8.50

⊗∗11 =

• For attribute A2 : { } a2min = min a i2 = min{6.00, 7.50, 3.50, 7.50, 2.25, 5.00} = 2.25 1≤i≤6

⊗∗12

[

] ] [ a2min a2min 2.25 2.25 = , , = [0.281, 0.375] = a 12 a 12 8.00 6.00 .. .

⊗∗62 • For attribute A3 :

[

] ] [ a2min a2min 2.25 2.25 = , , = [0.321, 0.450] = a 62 a 62 7.00 5.00

126

3 Supplier Selection Using Grey Systems Theory

{ } a3min = min a i3 = min{1.50, 6.00, 8.50, 5.50, 8.50, 2.75} = 1.50 1≤i≤6

[

] ] [ a3min a3min 1.50 1.50 = , , = [0.375, 1.000] = a 13 a 13 4.00 1.50

⊗∗13

.. . [

] ] [ a3min a3min 1.50 1.50 = , , = [0.300, 0.545] = a 63 a 63 5.00 2.75

⊗∗63 • For attribute A4 :

a4max = max {a i4 } = max {4.50, 9.00, 8.75, 3.50, 8.00, 9.00} = 9.00 1≤i≤6

⊗∗14

[

] ] [ a 14 a 14 2.00 4.50 = max , max = , = [0.222, 0.500] a4 a4 9.00 9.00 .. .

⊗∗64 =

[

] ] [ a 64 a 64 7.50 9.00 , , = [0.833, 1.000] = a4max a4max 9.00 9.00

• For attribute A5 : a5max = max {a i5 } = max{7.00, 9.00, 5.00, 9.00, 7.50, 7.50} = 9.00 1≤i≤6

⊗∗15 =

[

] ] [ a 15 a 15 5.00 7.00 , , = [0.556, 0.778] = a5max a5max 9.00 9.00 .. .

⊗∗65 =

[

] ] [ a 65 a 65 5.50 7.50 , , = [0.611, 0.833] = a5max a5max 9.00 9.00

Step 5: Define the mean attribute weight based on the DMs’ attribute weights In order to complete this step, the 4 DMs have been asked to evaluate the importance of each attribute using linguistic values, which are further evaluated using the values in Table 3.8. The evaluations of each DM are presented in Table 3.12 along with the weights of attributes determined as:

Application of Grey Numbers to Supplier Selection

127

) 1( 1 ⊗w1 + ⊗w12 + ⊗w13 + ⊗w14 4 1 = ([0.80, 1.00] + [0.80, 1.00] + [0.80, 1.00] + [0.60, 0.80]) 4 1 = [3.00, 3.80] = [0.75, 0.95] 4

⊗w1 =

) 1( 1 ⊗w2 + ⊗w22 + ⊗w23 + ⊗w24 4 1 = ([0.60, 0.80] + [0.40, 0.60] + [0.60, 0.80] + [0.80, 1.00]) 4 1 = [2.40, 3.20] = [0.60, 0.80] 4

⊗w2 =

) 1( 1 ⊗w3 + ⊗w32 + ⊗w33 + ⊗w34 4 1 = ([0.60, 0.80] + [0.60, 0.80] + [0.80, 1.00] + [0.80, 1.00]) 4 1 = [2.80, 3.60] = [0.70, 0.90] 4

⊗w3 =

) 1( 1 ⊗w4 + ⊗w42 + ⊗w43 + ⊗w44 4 1 = ([0.80, 1.00] + [0.60, 0.80] + [0.60, 0.80] + [0.80, 1.00]) 4 1 = [2.80, 3.60] = [0.70, 0.90] 4

⊗w4 =

) 1( 1 ⊗w5 + ⊗w52 + ⊗w53 + ⊗w54 4 1 = ([0.60, 0.80] + [0.40, 0.60] + [0.40, 0.60] + [0.60, 0.80]) 4 1 = [2.00, 2.80] = [0.50, 0.70] 4

⊗w5 =

Table 3.12 The DMs evaluations of the attributes and the weights of attributes (⊗w) Attribute Decision maker Decision maker Decision maker Decision maker ⊗w 1 2 3 4 A1

VH

VH

VH

H

[0.75, 0.95]

A2

H

M

H

VH

[0.60, 0.80]

A3

H

H

VH

VH

[0.70, 0.90]

A4

VH

H

H

VH

[0.70, 0.90]

A5

H

M

M

H

[0.50, 0.70]

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3 Supplier Selection Using Grey Systems Theory

Step 6: Create the weighted normalized grey decision matrix (WD) The weighted normalized grey decision matrix (W D) is determined by considering the aggregated attributes’ weights provided in Table 3.12. The elements of the matrix are determined as presented in the following: • For supplier S1 : w⊗11 = ⊗∗11 × ⊗w1 = [0.154, 0.176] ∗ [0.75, 0.95] = [0.116, 0.167] w⊗12 = ⊗∗12 × ⊗w2 = [0.218, 0.375] ∗ [0.60, 0.80] = [0.131, 0.300] w⊗13 = ⊗∗13 × ⊗w3 = [0.375, 1.000] ∗ [0.70, 0.90] = [0.263, 0.900] w⊗14 = ⊗∗14 × ⊗w4 = [0.222, 0.500] ∗ [0.70, 0.90] = [0.155, 0.450] w⊗15 = ⊗∗15 × ⊗w5 = [0.556, 0.778] ∗ [0.50, 0.70] = [0.278, 0.545] • For supplier S2 : w⊗21 = ⊗∗21 × ⊗w1 = [0.200, 0.273] ∗ [0.75, 0.95] = [0.150, 0.259] w⊗22 = ⊗∗22 × ⊗w2 = [0.250, 0.300] ∗ [0.60, 0.80] = [0.150, 0.240] w⊗23 = ⊗∗23 × ⊗w3 = [0.188, 0.250] ∗ [0.70, 0.90] = [0.132, 0.225] w⊗24 = ⊗∗24 × ⊗w4 = [0.778, 1.000] ∗ [0.70, 0.90] = [0.545, 0.900] w⊗25 = ⊗∗25 × ⊗w5 = [0.833, 1.000] ∗ [0.50, 0.70] = [0.417, 0.700] • For supplier S3 : w⊗31 = ⊗∗31 × ⊗w1 = [0.375, 1.000] ∗ [0.75, 0.95] = [0.281, 0.950] w⊗32 = ⊗∗32 × ⊗w2 = [0.409, 0.643] ∗ [0.60, 0.80] = [0.245, 0.514] w⊗33 = ⊗∗33 × ⊗w3 = [0.154, 0.176] ∗ [0.70, 0.90] = [0.108, 0.158]

Application of Grey Numbers to Supplier Selection

w⊗34 = ⊗∗34 × ⊗w4 = [0.778, 0.972] ∗ [0.70, 0.90] = [0.545, 0.875] w⊗35 = ⊗∗35 × ⊗w5 = [0.306, 0.556] ∗ [0.50, 0.70] = [0.153, 0.389] • For supplier S4 : w⊗41 = ⊗∗41 × ⊗w1 = [0.154, 0.176] ∗ [0.75, 0.95] = [0.116, 0.167] w⊗42 = ⊗∗42 × ⊗w2 = [0.243, 0.300] ∗ [0.60, 0.80] = [0.146, 0.240] w⊗43 = ⊗∗43 × ⊗w3 = [0.200, 0.273] ∗ [0.70, 0.90] = [0.140, 0.246] w⊗44 = ⊗∗44 × ⊗w4 = [0.083, 0.389] ∗ [0.70, 0.90] = [0.058, 0.350] w⊗45 = ⊗∗45 × ⊗w5 = [0.833, 1.000] ∗ [0.50, 0.70] = [0.417, 0.700] • For supplier S5 : w⊗51 = ⊗∗51 × ⊗w1 = [0.231, 0.333] ∗ [0.75, 0.95] = [0.173, 0.316] w⊗52 = ⊗∗52 × ⊗w2 = [0.500, 1.000] ∗ [0.60, 0.80] = [0.300, 0.800] w⊗53 = ⊗∗53 × ⊗w3 = [0.154, 0.176] ∗ [0.70, 0.90] = [0.108, 0.158] w⊗54 = ⊗∗54 × ⊗w4 = [0.667, 0.889] ∗ [0.70, 0.90] = [0.467, 0.800] w⊗55 = ⊗∗55 × ⊗w5 = [0.611, 0.833] ∗ [0.50, 0.70] = [0.306, 0.583] • For supplier S6 : w⊗61 = ⊗∗61 × ⊗w1 = [0.154, 0.176] ∗ [0.75, 0.95] = [0.116, 0.167] w⊗62 = ⊗∗62 × ⊗w2 = [0.321, 0.450] ∗ [0.60, 0.80] = [0.193, 0.360] w⊗63 = ⊗∗63 × ⊗w3 = [0.300, 0.545] ∗ [0.70, 0.90] = [0.210, 0.491]

129

130

3 Supplier Selection Using Grey Systems Theory

Table 3.13 Weighted grey normalized decision matrix Supplier

A1

A2

A3

A4

A5

S1

[0.116, 0.167]

[0.131, 0.300]

[0.263, 0.900]

[0.155, 0.450]

[0.278, 0.545]

S2

[0.150, 0.259]

[0.150, 0.240]

[0.132, 0.225]

[0.545, 0.900]

[0.417, 0.700]

S3

[0.281, 0.950]

[0.245, 0.514]

[0.108, 0.158]

[0.545, 0.875]

[0.153, 0.389]

S4

[0.116, 0.167]

[0.146, 0.240]

[0.140, 0.246]

[0.058, 0.350]

[0.417, 0.700]

S5

[0.173, 0.316]

[0.300, 0.800]

[0.108, 0.158]

[0.467, 0.800]

[0.306, 0.583]

S6

[0.116, 0.167]

[0.193, 0.360]

[0.210, 0.491]

[0.583, 0.900]

[0.306, 0.583]

w⊗64 = ⊗∗64 × ⊗w4 = [0.833, 1.000] ∗ [0.70, 0.90] = [0.583, 0.900] w⊗65 = ⊗∗65 × ⊗w5 = [0.611, 0.833] ∗ [0.50, 0.70] = [0.306, 0.583] The elements of the weighted normalized grey decision matrix (W D) are presented in Table 3.13. Step 7: Describe the ideal supplier attribute sequence The ideal supplier is noted as: } { max max max max S max = ⊗wmax 1 , ⊗w2 , ⊗w3 , ⊗w4 , ⊗w5 and it is determined as: Smax ⎫ ⎧ ⎪ [max{0.116, 0.150, 0.281, 0.116, 0.173, 0.116}, max{0.167, 0.259, 0.950, 0.167, 0.316, 0.167}], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ [max{0.131, 0.150, 0.245, 0.146, 0.300, 0.193}, max{0.300, 0.240, 0.514, 0.240, 0.800, 0.360}], ⎪ = [max{0.263, 0.132, 0.108, 0.140, 0.108, 0.210}, max{0.900, 0.225, 0.158, 0.246, 0.158, 0.491}], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [max{0.155, 0.545, 0.545, 0.058, 0.467, 0.583}, max{0.450, 0.900, 0.875, 0.350, 0.800, 0.900}], ⎪ ⎪ ⎪ ⎭ ⎩ [max{0.278, 0.417, 0.153, 0.417, 0.306, 0.306}, max{0.545, 0.700, 0.389, 0.700, 0.583, 0.583}]

Smax = {[0.281, 0.950], [0.300, 0.800], [0.263, 0.900], [0.583, 0.900], [0.417, 0.700]}

Step 8: Determine the synthesized possibility degree between each supplier and the ideal one By comparing the set of attributes between each supplier and the ideal one the synthesized possibility degree is determined in each case: The possibility degree is: ) 1∑ ( P ⊗w1 j ≤ ⊗max j 5 j=1 5

P(S1 ≤ S max ) =

Application of Grey Numbers to Supplier Selection

131

) ( ) ( ) 1[ ( P ⊗w11 ≤ ⊗max + P ⊗w12 ≤ ⊗max + P ⊗w13 ≤ ⊗max 1 2 3 5 ( ) ( )] +P ⊗w14 ≤ ⊗max + P ⊗w15 ≤ ⊗max 4 5 1 = [P([0.166, 0.167] ≤ [0.281, 0.950]) 5 + P([0.131, 0.300] ≤ [0.300, 0.800])

=

+ P([0.263, 0.900] ≤ [0.263, 0.900]) + P([0.155, 0.450] ≤ [0.583, 0.900]) +P([0.278, 0.545] ≤ [0.417, 0.700])] P(S1 ≤ S max ) =

1 (1.000 + 1.000 + 0.500 + 1.000 + 0.891) = 0.878 5

) 1∑ ( P ⊗w2 j ≤ ⊗max j 5 j=1 5

P(S2 ≤ S max ) =

) ( ) ( ) 1[ ( P ⊗w21 ≤ ⊗max + P ⊗w22 ≤ ⊗max + P ⊗w23 ≤ ⊗max 1 2 3 5 ( ) ( )] +P ⊗w24 ≤ ⊗max + P ⊗w25 ≤ ⊗max 4 5 1 = [P([0.150, 0.259] ≤ [0.281, 0.950]) 5 + P([0.150, 0.240] ≤ [0.300, 0.800]) =

+ P([0.132, 0.225] ≤ [0.263, 0.900]) + P([0.545, 0.900] ≤ [0.583, 0.900]) +P([0.417, 0.700] ≤ [0.417, 0.700])] P(S2 ≤ S max ) =

1 (1.000 + 1.000 + 1.000 + 0.553 + 0.500) = 0.811 5

) 1∑ ( P ⊗w3 j ≤ ⊗max j 5 j=1 5

P(S3 ≤ S max ) =

) ( ) ( ) 1[ ( P ⊗w31 ≤ ⊗max + P ⊗w32 ≤ ⊗max + P ⊗w33 ≤ ⊗max 1 2 3 5 ( ) ( )] +P ⊗w34 ≤ ⊗max + P ⊗w35 ≤ ⊗max 4 5 1 = [P([0.281, 0.950] ≤ [0.281, 0.950]) 5 + P([0.245, 0.514] ≤ [0.300, 0.800]) =

+ P([0.108, 0.158] ≤ [0.263, 0.900]) + P([0.545, 0.875] ≤ [0.583, 0.900]) +P([0.153, 0.389] ≤ [0.417, 0.700])]

132

3 Supplier Selection Using Grey Systems Theory

P(S3 ≤ S max ) =

1 (0.500 + 0.829 + 1.000 + 0.592 + 1.000) = 0.784 5

) 1∑ ( P ⊗w4 j ≤ ⊗max j 5 j=1 5

P(S4 ≤ S max ) =

) ( ) ( ) 1[ ( P ⊗w41 ≤ ⊗max + P ⊗w42 ≤ ⊗max + P ⊗w43 ≤ ⊗max 1 2 3 5 ( ) ( )] +P ⊗w44 ≤ ⊗max + P ⊗w45 ≤ ⊗max 4 5 1 = [P([0.116, 0.167] ≤ [0.281, 0.950]) 5 + P([0.146, 0.240] ≤ [0.300, 0.800]) =

+ P([0.140, 0.246] ≤ [0.263, 0.900]) + P([0.058, 0.350] ≤ [0.583, 0.900]) +P([0.417, 0.700] ≤ [0.417, 0.700])] P(S4 ≤ S max ) =

1 (1.000 + 1.000 + 1.000 + 1.000 + 0.500) = 0.900 5

) 1∑ ( P ⊗w5 j ≤ ⊗max j 5 j=1 5

P(S5 ≤ S max ) =

) ( ) 1[ ( P ⊗w51 ≤ ⊗max + P ⊗w52 ≤ ⊗max 1 2 5 ( ) ( ) ( )] +P ⊗w53 ≤ ⊗max + P ⊗w54 ≤ ⊗max + P ⊗w55 ≤ ⊗max 3 4 5 1 = [P([0.173, 0.316] ≤ [0.281, 0.950]) 5 + P([0.300, 0.800] ≤ [0.300, 0.800]) + P([0.108, 0.158] ≤ [0.263, 0.900]) =

+ P([0.467, 0.800] ≤ [0.583, 0.900]) +P([0.306, 0.583] ≤ [0.417, 0.700])] P(S5 ≤ S max ) =

1 (0.993 + 0.500 + 1.000 + 0.776 + 0.824) = 0.819 5

) 1∑ ( P ⊗w6 j ≤ ⊗max j 5 j=1 5

P(S6 ≤ S max ) =

) ( ) ( ) 1[ ( P ⊗w61 ≤ ⊗max + P ⊗w62 ≤ ⊗max + P ⊗w63 ≤ ⊗max 1 2 3 5 ( ) ( )] +P ⊗w64 ≤ ⊗max + P ⊗w65 ≤ ⊗max 4 5

=

Application of Grey Numbers to Supplier Selection Table 3.14 The synthesized possibility degrees

133

The synthesized possibility degree P(Si ≤ S max )

Value

P(S1 ≤ S max )

0.878

P(S2 ≤

S max )

0.811

P(S3 ≤ S max )

0.784

P(S4 ≤ S max )

0.900

P(S5 ≤

S max )

0.819

P(S6 ≤ S max )

0.831

1 [P([0.116, 0.167] ≤ [0.281, 0.950]) 5 + P([0.193, 0.360] ≤ [0.300, 0.800])

=

+ P([0.210, 0.491] ≤ [0.263, 0.900]) + P([0.583, 0.900] ≤ [0.583, 0.900]) +P([0.306, 0.583] ≤ [0.417, 0.700])] P(S6 ≤ S max ) =

1 (1.000 + 0.978 + 0.854 + 0.500 + 0.824) = 0.831 5

In order to easily determine the probabilities when comparing two interval grey numbers, a Jupyter Notebook written in Python can be used as provided at the following link: https://github.com/liviucotfas/grey-systems-book/blob/main/ grey-numbers-comaprison-smaller-or-equal.ipynb. The source code for determining ( ) the probability P ⊗a ≤ ⊗b is listed in the Annex A, provided at the end of this chapter. The code can be run in the Chap. 2 where the code for ) ( as presented determining the probability P ⊗a > ⊗b was provided. The synthesized possibility degrees between each supplier and the ideal one are summarized in Table 3.14. Step 9: Rank the suppliers and make decision Based on the values in Table 3.14, the following ranking is obtained: S3 > S2 > S5 > S6 > S1 > S4 Therefore, supplier S3 is the best choice from the set of possible suppliers. Some other good choices are represented by suppliers S2 and S5 , which occupies the second and the third position. Supplier S4 is the worst choice in the set of suppliers.

134

3 Supplier Selection Using Grey Systems Theory

Concluding Remarks The supplier selection problem, largely discussed in the scientific literature and solved through the means of different methods and techniques, has been discussed in this chapter from the point of view of the grey systems theory. Starting from the papers published in the scientific literature, a bibliometric analysis has been provided along with a literature review of the selected papers. Based on the literature review, a series of approaches have been observed, all of them either combining elements from the grey systems theory and other theories from the field or using solely the advantages brought by the grey systems theory. Considering the approaches featuring finding a solution for the supplier selection problem only through the means of the grey systems theory, one of the approaches from the scientific literature proposed by Xie and Xin [31] is presented from a theoretical point of view. The approach has been applied in the case of a clothing company which aimed to select the best supplier from a set of 6 possible suppliers. The solution of the application has been presented in detail for facilitating the reader to reproducibility of the work under similar conditions. Last, we invite the reader to explore the other approaches to supplier selection by reading and applying the methods proposed by the authors mentioned along the chapter. Nevertheless, we hope that the reader will be interested to contribute further to the development of the models for addressing the supplier selection and that grey systems theory will be among the considered techniques.

Annex A: The ) Code for Determining the Probability ( P ⊗a ≤ ⊗b class GreyNumber(object): def __init__ (self, low, up): self.low = low self.up = up def probability_smaller_or_equal(a1 : GreyNumber, a2 : GreyNumber) -> float: ”’returns the probability that a1 ≤ a2”’ if a1.low > a2.up: print("a1.low > a2.up") return 0 elif a1.low < a2.low float: # 1. Whitenization weight function of lower measure if self.turning_points[0] == "-" and self.turning_points[1] == "-" and self.turning_points[2] != "" and self.turning_points[3] != "-": return self.type1(value) # 2. Whitenization weight function of upper measure elif self.turning_points[0] != "-" and self.turning_points[2] == "-" and self.turning_points[3] == "": return self.type2(value) # 3. Whitenization weight function of moderate measure elif self.turning_points[0] != "-" and self.turning_points[1] != "-" and self.turning_points[2] == "" and self.turning_points[3] != "-": return self.type3(value) # 4. Typical whitenization weight function elif self.turning_points[0] != "-" and self.turning_points[1] != "-" and self.turning_points[2] != "" and self.turning_points[3] != "-": return self.type4(value)

(continued)

152

4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

Table 4.3 (continued) else: raise ValueError() def type1(self, value): if value < 0.0 or value > float(self.turning_points[3]): return 0.0 elif value < float(self.turning_points[2]) and value >= 0.0: return 1.0 elif value >= float(self.turning_points[2]) and value < float(self.turning_points[3]): return (float(self.turning_points[3]) - value) / (float(self.turning_points[3]) float(self.turning_points[2])); else: return 0 def type2(self, value): if value < float(self.turning_points[0]): return 0.0 elif value >= float(self.turning_points[0]) and value < float(self.turning_points[1]): return (value - float(self.turning_points[0])) / (float(self.turning_points[1]) float(self.turning_points[0])); elif value >= float(self.turning_points[1]): return 1.0 else: return 0 def type3(self, value): if value < float(self.turning_points[0]) or value > float(self.turning_points[3]): return 0 elif value >= float(self.turning_points[0]) and value < float(self.turning_points[1]): return (value - float(self.turning_points[0])) /

(continued)

Assumptions and Metrics

153

Table 4.3 (continued) (float(self.turning_points[1]) float(self.turning_points[0])) elif value >= float(self.turning_points[1]) and value

float(self.turning_points[3]): return 0 elif value >= float(self.turning_points[0]) and value = float(self.turning_points[1]) and value < float(self.turning_points[2]): return 1 elif value >= float(self.turning_points[2]) and value max: max = coefficents[i] index = i cluster.append(index) # Create a copy of the original objects. df_results = df_objects.copy() # Add the cluster column df_results["cluster"] = cluster

Fig. 4.9 Clustering coefficients for the first 7 objects Table 4.8 Code for displaying the clustering coefficients clustering_coefficients df_clustering_coefficients = pd.DataFrame(clustering_coefficients) df_clustering_coefficients

Assumptions and Metrics

157

Fig. 4.10 Clusters assigned to the first 7 objects Table 4.9 Code for displaying the objects with the assigned cluster df_results

Fig. 4.11 The objects belonging to each cluster Table 4.10 Code for displaying the objects belonging to each cluster for i in range(no_of_functions): cluster_index = i+1 object_indexs = str(list(df_results[df_results['cluster'] == i].index + 1)) # +1 to have the indexes starting from 1

Aisle

3

3

3

2

1

3

3

3

2

1

3

3

3

2

1

3

3

3

2

1

3 3 3 3

3

3

3

3

3 3

Fig. 4.12 The first variation in the Back-to-front boarding method

3

3

Empty seat

3

3

3

3 3

3

3 3

3

3 3 3 3

3

3 3 3

3

3 3 3 3

3

3rd group

3

3 3 3

3

3

3

3 3

3

group

3 3

2

nd

3

3

3

3

3 3 3

3

3 3

3

1 group

2

3 3

3

3

3

3

3

3

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

3

9

3

8

3

7

3

3

3

3

3

3

3

6

4

2 st

5

3

1 1

Front door

3

print(str(cluster_index) + ": "+object_indexs)

1 1

1

1

1

1 1

1

1

1 1

group

3

2

2

3rd group

Empty seat

1 1

1 1

1

1

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1

9

1

8

1

7

1

6 nd

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

1

1

5

1st group

1 1

1

1 1

1

1

1

1

4

1

3

1

2

1

1

1 1

1

2

1

1 1 1

2 2

3 3

2

3

1 1

Front door

1

4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering 3

158

Aisle

3rd group

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

2

1

2

2

2

2

2 2

2

Empty seat

1

2 2 2 2

2 2 2 2

2

2 2 2

2 2 2

2

2

3

2nd group

2 2 2

2

3

2 2

2

2

3

4

3

3 3 3 3

3

1st group

3

3

3

3

3

3

3 3

3

3 3 3 3

3 3 3 3

2

3

3

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

3

9

3

8

3

7

3

6

3

5

1 1

Front door

3

Fig. 4.13 The last variation in the Back-to-front boarding method

Aisle

Fig. 4.14 The middle variation in the Back-to-front boarding method g1 g2

Front door

1

2

3

Seat

4

5

6

7

Empty seat

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Aisle

Fig. 4.15 The general scheme for the variations in the Back-to-front boarding method

the number of seat-rows (24 in our case) the seat-rows allocated to the first and the second group, g1 and g2. Few theoretical implications can be mentioned for g1 and g2 [13]: • g1 > 0, g2 > 0, integers • g1 ≤ 22 for a 24-seat-row airplane configuration • g2 ≤ 23—g1 for a 24-seat-row airplane configuration

Metrics Five metrics are used for analyzing the considered variations in Back-to-front boarding methods, two of them being dedicated to the risk associated with the

Assumptions and Metrics

159

COVID-19 transmission, one related to the cost of boarding and two to the customers’ satisfaction while boarding. The metrics are in line with other works from the field [13, 42, 58, 61].

Risk Associated with the Aisle Seats The Risk Associated with the Aisle Seats metric (AisleSeats Risk) measures the risk to which the passengers having an aisle seat are exposed during the pandemic. The metric measures the risk for all the passengers with aisle seats who have already taken their seats while some other passengers are standing on the aisle as they walk to their assigned seat or store the luggage in the overhead compartment. The formula for this metric is [62]: AisleSeats Risk =





⎛ ⎝ RowT ime pr ∗



⎞ AisleSeat p r ⎠

p < p

p r ≤RowSeat p

where p = passenger advancing towards his/her seat r = row index RowSeat p = row in which passenger p has a seat RowTimepr = time that passenger p spends in row r p’ = passenger boarding before passenger p. AisleSeat p  r =



1 i f passenger p  has an aisle seat in row r 0 other wise

Risk Associated with the Window Seats The Risk Associated with the Window Seats metric (W indowSeats Risk) measures the extent to which the passengers with window seats who have already occupied their seats can contact the disease as there are passengers on the aisle who are walking to their assigned seats or are storing the luggage in the overhead compartment and who can be infected and transmit the virus. The risk is determined in the same manner as the Risk Associated with the Aisle Seats [62]: W indowSeats Risk =





p r ≤RowSeat p

⎛ ⎝ RowT ime pr ∗

p < p

⎞ W indowSeat p r ⎠

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

where p = passenger advancing towards his/her seat. r = row index. RowSeat p = row in which passenger p has a seat. RowTimepr = time that passenger p spends in row r. p’ = passenger boarding before passenger p 

W indowSeat p r =



1 i f passenger p  has an aisle seat in row r 0 other wise

As the window seats are located at a certain distance from the aisle—in comparison with the aisle seats that are located near the aisle—a higher importance will be given to the AisleSeats Risk. In both cases, lower values are preferred for the metrics.

Boarding Time Boarding time (Boar dingT ime) is a cost metric and measures the time passed between the start of the boarding process marked by the arrival of the first passenger inside the aircraft and the moment in which the last passenger sits on his/her assigned seat. The metric is expressed in time units. A lower value for this matric is preferred.

Customers Satisfaction Metric Related to Type-3 Seat Interferences Let us first discuss the idea of seat interferences, the types of seat interferences and why are we particularly interested in the type-3 seat interference. A seat interference occurs in the situation in which a passenger with either middle or window seat arrive to his/her allocated seat after the passenger/passengers with aisle or middle or aisle and middle seats have already taken their assigned seats. As the distance between the seat-rows in the airplane is reduced, it is not possible for the passengers with window or middle seats to occupy their assigned seats without asking the already-seated passengers in the same row to clear their path. This action takes time and might prolong the overall boarding time as during this action the aisle is blocked, and no other passenger located in the back of this point can advance down the aisle to his/her allocated seat. The connection between the seat interferences and the overall boarding time has been discussed in the scientific literature, showing that the occurrence of seat interferences prolongs the overall boarding time [63]. Four types of seat interferences can be encountered in airplane boarding, noted as type-1, type-2, type-3 and type-4 seat interferences—Fig. 4.16. Depending on the number and position of the already seated passengers, any of the four types can occur during a boarding process. Taking a close look on the schemes defining the seat interferences from Fig. 4.16, it can be observed that type-1, type-2 and type-4 involve the presence of a seated passenger on the middle seat. As in the case of our approach, the middle seat is left empty, none of the three types of seat interferences can occur.

Assumptions and Metrics

161 Type-2

Type-1

Front door

1

2 Seat

3

4

5

6

7

8

9

Type-3

Type-4

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Aisle

Fig. 4.16 The four types of seat interferences

Thus, the only type of interference that can be encountered for the considered situation is the type-3 seat interference. As presented in Fig. 4.16, type-3 seat interference occurs in the situation in which a passenger having a window seat on one side of the row arrives to his/her seat after the passenger with aisle seat located in the same side of the aisle have already taken his/her seat. In this situation, the passenger with aisle seat needs to clear the path for the passenger with window seat, action that blocks the aisle for approximatively 10 s (ranging between 9 and 13 s) as measured by Schultz [64] in field trials. The occurrence of type-3 seat interference reduces the customer satisfaction. Therefore, the Customers Satisfaction Metric Related to Type-3 Seat Interferences (AvgTimesType3SeatInt) is designed to catch this decrease in customer satisfaction. The following formula is used [13]: AvgT imesT ype3Seat I nt =



pas Sit T ype3 /|P|

Sit T ype3

where pas = number of passengers affected by each type-3 seat interference. |P| = total number of passengers to board. SitType3 = situations in which a type-3 interference occurs.

Customers Satisfaction Metric Related to Aisle Interferences A similar situation that contributes to the decrease in customers’ satisfaction is represented by the occurrence of aisle interferences. These interferences appear when a passenger is placing his/her luggage in the overhead compartment and blocks the aisle for several moments of time, depending on the type of luggage loaded and the availability of the overhead compartment (which depends on the luggage already stored by the other passengers who have arrived in the airplane prior to the arrival of the current passenger). The time needed for storing the luggage in the overhead compartment (T stor e) has been determined in this work using the formula suggested by [65] and previously used in [40, 66]:

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

T stor e = (N bin L arg e + 0.5 N binSmall + N passenger Large + 0.5 N passenger Small) ∗ (N passenger Large + 0.5 N Passenger Small)/2) ∗ T r ow where Tstore is the time to store the luggage NbinLarge is the number of large bags in the bin prior to the passenger’s arrival NbinSmall is the number of small bags in the bin prior to the passenger’s arrival NpassengerLarge is the number of large bags carried by the passenger NpassengerSmall is the number of small bags carried by the passenger Trow is the time for a passenger to walk from one row to the next (when not delayed by another passenger in front) Furthermore, the Customers Satisfaction Metric Related to Aisle Interferences (AvgT imes AisleI nt)) is used as suggested in [13]: AvgT imes AisleI nt =



paa Sit Aisle /|P|

Sit Aisle

where paa = number of passengers affected by each aisle seat interference. |P| = total number of passengers to board. SitAisle = situations in which an aisle interference occurs. All the metrics discussed in this section have been implemented in the agent-based model created in NetLogo and have been measured through simulations.

Agent-Based Model Implementation An agent-based model in NetLogo has been created for representing the airplane boarding process when variations in Back-to-front boarding method is used. NetLogo has been chosen as the modeling environment as it offers a series of advantages when compared to other software designed for agent-based modeling and simulation [67–70]. Among the advantages, one can name the presence of the graphical interface where the process of airplane boarding can be observed in real-time. Also, the commands used for writing the code are simple and easy to understand even for persons that are not familiar with programming. Another advantage relies on the fact that NetLogo provides a platform where examples containing models from different fields are provided to the interested parties, along with a good documentation for commands and code writing [71]. In recent times, NetLogo has gained an increased attention due its efficacy in modeling situations in which human behavior is involved. Among some of the most

Agent-Based Model Implementation

163

well-known applications, the case studies in the area of evacuation and airplane boarding can be named [38, 72–74]. Various types of agents can be defined in NetLogo, which ease the modeling process. For the airplane boarding model, the turtles and patches agents have been used. First, the patches agents have been utilized for representing the environment. Different colors have been given to the patches in order to represent the aisle (dark blue), the empty seats (grey) and the occupied seats (light grey)—Fig. 4.17. Each patch has been assumed to be equivalent to 0.4 m × 0.4 m in real life [28, 64, 75]. Second, the turtle agents have been set to have similar characteristics with the passengers boarding in an airplane. The turtle agents are represented using human shapes in the agent-based model. The agents have speed, which depends on the quantity of luggage brought inside the airplane and it is influenced by the speed of the persons walking down the aisle in front of each agent. The speed can range between 0 and 1 patch/tick. The time unit in NetLogo is named “tick” and it has been determined to be equal to 1.2 s in the case of airplane boarding [35, 64, 76]. The speed can be equal to 0 parches/tick if the agent has reached its allocated seat or if it is involved in an aisle or seat interference. Each agent has associated a row and a seat through the agent-seat-row and agent-seat-column variables. As a social distance of 1 m is considered while walking down the aisle, the comfort-distance variable retains the space needed for each agent in order to preserve the mentioned social distance. If an agent has luggage, the time needed to store it in the overhead compartment is determined based on the quantity of luggage the agent has and based on the overhead bin occupancy as mentioned in the previous section through the use of Tstore variable. More details regarding the ranges of the variables implied in the agent-based model can be found in Delcea et al. [60]. As a result, the agent-based model interface is as presented in Fig. 4.18.

Unavailable seat patch

Aisle patch

Available seat patch

Fig. 4.17 Example of various patches in the agent-based model

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

Fig. 4.18 The agent-based model interface

Results Starting from the agent-based model, a simple code used in Python has been adapted from Delcea et al. [13] for running the 253 variations in Back-to-front boarding method, as presented in Table 4.11. For each situation, 1000 runs have been performed, conducting to a total number of 253,000 simulations. The average values of the simulations are presented and discussed in the following.

Simulation Results for the Considered Metrics For each of the considered metrics, the performance of the variations in Back-to-front boarding method is discussed. Table 4.11 Code for computing the boarding metrics Compute-boarding-metrics in: number of simulation runs noSimulations out: boarding metrics for all group configurations m for g1 = 1 to 22 do for g2 = 1 to 23—g1 do m[g1][g2] ← ABMSimulation(g1, g2, noSimulations) end for end for return m

Results

165

Fig. 4.19 The performance of the variations in Back-to-front in terms of AisleSeatsRisk

Risk Associated with the Aisle Seats The Risk Associated with the Aisle Seats metric (AisleSeats Risk) ranges between 2308.01 s and 4659.32 s. Figure 4.19 presents the performance of the 253 variations in Back-to-front boarding method using colors. As a lower value for the AisleSeats Risk is preferred, the points marked in red presents the best/good performance, while the points colored in dark-blue/pink the lower performance. The best-performing variations in the Back-to-front boarding method are the ones characterized by a value of g1 and g2 ranging between 7 and 9—Fig. 4.19. On the other side, the worst-performing variation in the Back-to-front boarding method are the ones for which we have s small value for g1 and g2 (e.g., g1 = 1, g2 = 1, resulting a value for g3 = 22) or a small value for g1 and a large value for g2 (e.g., g1 = 1, g2 = 22, resulting a value for g3 = 1) or a large value for g1 and a small value for g2 (e.g., g1 = 22, g2 = 1, resulting a value for g3 = 1). Considering all these situations, it can be stated that the smallest values for the AisleSeats Risk are obtained in the variations in the Back-to-front boarding method that are as close as possible to the situation in which the three groups are equal (e.g., g1 = g2 = g3 = 8), while the worst-performing variations in Back-to-front boarding method are the ones characterized by a large difference among the three groups (e.g., two of the groups being made by only one row, and another group having almost 22 seat-rows).

Risk Associated with the Window Seats The Risk Associated with the Window Seats metric (W indowSeats Risk) ranges between 2079.80 and 4424.06 s—Fig. 4.20.

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

Fig. 4.20 The performance of the variations in Back-to-front in terms of WindowSeatsRisk

It can be observed that in this case, as in the case of the AisleSeats Risk, the best-performing variations in the Back-to-front boarding method are the ones having values for the g1, g2 and g3 as close as possible to the value of 8. The same observation can be made for the case of the worst-performing boarding methods: even in the case of W indowSeats Risk these methods are characterized by a large gap between the length of two of the groups and the third group (g1 = 1, g2 = 1, g3 = 22; or g1 = 1, g2 = 1, g3 = 22; or g1 = 1, g2 = 1, g3 = 22).

Boarding Time The values obtained for the Boarding time (Boar dingT ime) range between 823.29 s and 912.69 s. The difference between the worst and the best variation in Back-to-front boarding method is of 89.4 s. Considering the average cost of $53.5 per min [14, 15], a cost reduction of $79.72 per boarding process is determined in the case in which the best-performing variation in used instead of the worst-performing variation. This value, even though seems low when compared to the other costs the airlines have, should be regarded in the context in which the number of flights for 2021 has been of almost 22.2 million [77]. Contrary to the previous two metrics, it can be observed from Fig. 4.21 that the best performing methods are in this case the ones characterized by a g1 = 22 and a g2 = 1 or a g1 ranging between 1 and 3 and a g2 ranging between 19 and 21. Therefore, in this case, the best performing methods in terms of Boar dingT ime are the ones for which the first two groups (g1 or g2) have very small for one of the indicator and very large for the other one. Thus, the best-performing methods have very small values for the third group (g3 ranging between 1 and 4). As for the worst-performing metrics, it may be observed from Fig. 4.21 that they are characterized by similar values of the three groups (g1, g2 and g3 ranging between 7 and 9).

Results

167

Fig. 4.21 The performance of the variations in Back-to-front in terms of BoardingTime

Customers Satisfaction Metric Related to Type-3 Seat Interferences The Customers Satisfaction Metric Related to Type-3 Seat Interferences (AvgTimesType3SeatInt) takes values between 11.76 and 18.10 times—Fig. 4.22. A value equal to 11.76 times shows that, on average, while boarding, each passenger is affected approximatively 11.76 times by the occurrence of a type-3 seat interference as he/she advances to his/her assigned seat. The value increases with almost 50% for the case in which the worst-variations in the Back-to-front boarding methods are used. Considering the configurations for which the best and the worst values are obtained in terms of AvgTimesType3SeatInt, it can be observed that the worst-performing variations are the ones that have a g1 and a g2 ranging between 4 and 7—thus, Fig. 4.22 The performance of the variations in Back-to-front in terms of AvgTimesType3SeatInt

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

relatively smaller then-equal boarding groups are preferred for g1 and g2, while g3 is taking values between 10 and 16 seat-rows. As for the case of the best-performing boarding methods, it can be noted that these are characterized by either two small boarding groups (each one of them having one seat-row) and a large boarding group (comprising of 22 seat-rows), or a situation in which the first group is very small (1–3 seat-rows), and the second group is large (having between 18 and 22 seat-rows).

Customers Satisfaction Metric Related to Aisle Interferences The Customers Satisfaction Metric Related to Aisle Interferences (AvgT imes AisleI nt) takes values between 3.23 and 6.03 times—Fig. 4.23. A value equal to 3.23 times shows that, on average, while boarding, each passenger is affected approximatively 3.23 times by the occurrence of an aisle interference as he/she advances to his/her assigned seat. The value almost doubles for the case in which the worst-variations in the Back-to-front boarding methods are used. Considering the configurations for which the best and the worst values are obtained in terms of AvgT imes AisleI nt, it can be observed that the worst-performing variations are the ones that have a g1 and a g2 ranging between 5 and 8—thus, relatively smaller then-equal boarding groups are preferred for g1 and g2, while g3 is taking values between 8 and 14 seat-rows. In the case of the best-performing boarding methods, it can be noted that these are characterized by two small boarding groups (each one of them having one seat-row) and a large boarding group (comprising of 22 seat-rows). Fig. 4.23 The performance of the variations in Back-to-front in terms of AvgTimesAisletInt

Results

169

Grey Clustering of the Variations in Back-to-Front Boarding Method Based on the individual values obtained for the variations in the Back-to-front boarding method, the whitenization weigh functions have been built on the purpose of dividing the 253 variations into three clusters based on their performance. The whitenization weigh functions for the five considered performance metrics are presented in Table 4.12. As lower values are better for all the indicators, cluster 1 will contain the desirable (best) variations in Back-to-front boarding method, cluster 2 will contain the variations with medium performance and cluster 3 the variations that should be avoided to be used. In the following, we aim at determining the variations in Back-to-front boarding method in each of the three considered clusters based on some objectives, such as: • • • •

Equal Importance to Risk, Cost and Customer Satisfaction Moderate Importance to Risk and Cost High Importance to Risk and Moderate to Cost Exclusive Importance to Cost Additional objectives can be formulated in accordance with the policy of each airline. We invite the reader to explore furthermore some other objectives that can be formulated by the airlines and the variations in Back-to-front boarding method the airlines should be used for each new formulated objective. Table 4.12 The whitenization weight functions for the five metrics Metric

Index

Cluster

f jk (•)

AisleSeats Risk

j=1

k=1

f 11 [−, −, 2778, 3249]

k=2

f 12 [2778, 3249, 3719, 4189]

k=3

f 13 [3719, 4189, −, −]

k=1

f 21 [−, −, 2549, 3018]

k=2

f 22 [2549, 3018, 3486, 3955]

k=3

f 23 [3486, 3955, −, −]

k=1

f 31 [−, −, 841, 859]

k=2

f 32 [841, 859, 877, 895]

k=3

f 33 [877, 8950, −, −]

k=1

f 41 [−, −, 13, 14]

k=2

f 42 [13, 14, 16, 17]

k=3

f 43 [16, 17, −, −]

k=1

f 51 [−, −, 3.8, 4.4]

k=2

f 52 [3.8, 4.4, 4.9, 5.5]

k=3

f 53 [4.9, 5.5, −, −]

W indowSeats Risk

Boar dingT ime

AvgTimesType3SeatInt

AvgT imes AisleI nt

j=2

j=3

j=4

j=5

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

Fig. 4.24 Cluster performance of the variations in Back-to-front in terms of equal importance to risk, cost and customer satisfaction

Clusters for Equal Importance to Risk, Cost and Customer Satisfaction For this objective, the risk, cost and customer satisfaction have been assumed to contribute equally to the cluster formation. As a result, the following values have been used for the weights: η1 = 0.1675, η2 = 0.1675, η3 = 0.33, η4 = 0.1675, η5 = 0.1675. The clustering results are depicted in Fig. 4.24. The variations in the Back-to-front boarding method belonging to cluster 1 are depicted in red in Fig. 4.24 and listed in Table 4.13. Based on the characteristics of the variations in Back-to-front boarding method included in cluster 1 it can be observed that they feature a disproportion among the seat-rows included in the three boarding groups: there are usually small values for two of the boarding groups (ranging between 1 and 4 seat-rows) and high values for the other group (ranging between 18 and 22 seat-rows)—Table 4.13. Cluster 3 retains the variations in Back-to-front boarding method that have g1 between and g2 as presented in blue in Fig. 4.24. As a rule, the size of the three groups is equal in the methods included in cluster 3 or have variations of up to 6 seat-rows around the value of 8 seat-rows. The remainder of the variations, which are depicted in green in Fig. 4.24 belong the cluster 2, which provides a medium performance according to this objective.

Clusters for Moderate Importance to Risk and to Cost When moderate importance is given to risk and cost, the results presented in Fig. 4.25 are obtained. The considered values for the weights are: η1 = 0.175, η2 = 0.175, η3 = 0.35, η4 = 0.15, η5 = 0.15.

Results Table 4.13 Variations included in cluster 1 when equal importance is given to risk, cost and customer satisfaction

171 Cluster Cluster 1

g1

g2

g3

1

1

22

1

2

21

1

19

4

1

20

3

1

21

2

1

22

1

2

1

21

2

18

4

2

19

3

2

20

2

2

21

1

20

3

1

21

1

2

21

2

1

22

1

1

Fig. 4.25 Cluster performance of the variations in Back-to-front in terms of moderate importance to risk and cost

As it can be observed from Fig. 4.25, the variations in the Back-to-front boarding method included in cluster 1 are numerous in comparison with the ones in cluster 3. The best-performing methods, marked in red in Fig. 4.25, are having values for g1 and g2 ranging between 1 and 15, with the observation that these methods do not have simultaneously small values in both g1 and g2, but rather a small g1 and a high g2, or a high g1 and a small g2, or high values for both g1 and g2. In this context, the values for g3 range between 1 and 11.

172

4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

Fig. 4.26 Cluster performance of the variations in Back-to-front in terms of high importance to risk and moderate importance to cost

Looking at the shape of cluster 1 in Fig. 4.25, it can be seen that it follows the shape of the best-performing variations when only risk indicators are considered but being more ample as a result of the cost indicator influence.

Clusters for High Importance to Risk and Moderate Importance to Cost A situation that implies a high importance to risk and a moderate importance to cost has conducted to the clusters presented in Fig. 4.26. The following values have been considered for the weights: η1 = 0.3, η2 = 0.3, η3 = 0.2, η4 = 0.1, η5 = 0.1. Due to the increased importance given to risk—when compared to the previous situation—it can be observed that cluster 1 has almost the same shape, but a smaller number of variations in Back-to-front boarding method included in this cluster. This situation occurs due to the higher values for the weights in the case of the risk indicators, which conducts to a restrained red area in Fig. 4.26. The values for g1 between 2 and 14, while the ones for g2 range between 1 and 15. As in the previous case, the variations in the Back-to-front boarding method included in cluster 1 have either a small g1 and a high g2, or a high g1 and a small g2, or high values for both g1 and g2. Due to an increase of the range for the values of g1—when compared to the previous situation—a decrease in the range of g3 is observed.

Clusters for Exclusive Importance to Cost In the case in which the airline is particularly interested in having low costs associate with the airplane boarding process, the variations in the Back-to-front boarding method depicted in red in Fig. 4.27 can be considered. For determining them, the

Concluding Remarks

173

Fig. 4.27 Cluster performance of the variations in Back-to-front in terms of exclusive importance to cost

following weights have been used: η1 = 0, η2 = 0, η3 = 1, η4 = 0, η5 = 0. This situation might occur in the non-pandemic times, when the number of flights operated by a company has highly increased and the airline is no longer interested in addressing pandemic risks nor elements related to customer satisfaction. The best-performing variations in Back-to-front boarding method are listed in Table 4.14. From the values of the three indicators it can be observed that these variations have either small values for g1 accompanied by high values for g2, or high values for g1 accompanied by small values for g2. As a result, the values for g3 are small in all the variations in Back-to-front boarding method included in cluster 1—Table 4.14. Comparing the results obtained in this chapter for an airplane with 24 seat-rows to the ones obtained in Delcea et al. [13] for an airplane with 30 seat-rows, it can be noticed that the observations made for the number of seat-rows relative to the number of the seat-rows of the airplane remains broadly the same. As a result, when an airline uses a different type of airplane in terms of number of seat-rows and a Backto-front approach for passengers boarding, it can consider the above results when substantiating its decision to a particular variation of the Back-to-front boarding method.

Concluding Remarks In this chapter a part of the airplane boarding problem has been discussed in terms of using the variations in Back-to-front boarding method. Inspired from the recent pandemic conditions, the application developed in this chapter puts together the cost generated by the airplane boarding process with the risk associated for the

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4 Risk Assessment and Transport Cost Reduction Based on Grey Clustering

Table 4.14 Variations included in cluster 1 when exclusive importance is given to cost

Cluster Cluster 1

g1

g2

g3

1

19

4

1

20

3

1

21

2

1

22

1

2

18

4

2

19

3

2

20

2

2

21

1

3

18

3

3

19

2

3

20

1

4

18

2

4

19

1

5

17

2

20

3

1

21

1

2

21

2

1

22

1

1

passengers in times of pandemics, adding also an element of customer satisfaction when a particular boarding method is used. Through the use of agent-based modeling and grey clustering, the variations in Back-to-front boarding method have been segregated into three clusters based on the objectives airline might have at a particular moment of time. Based on each objective, the best-performing variations in Back-to-front boarding method have been discussed. Changing the objectives and applying the same approach, the airlines can better decide upon which variation in Back-to-front boarding method best fits their interests.

References 1. Salari, M., Milne, R.J., Delcea, C., Kattan, L., Cotfas, L.-A.: Social distancing in airplane seat assignments. J. Air Transp. Manag. 89, 101915 (2020). https://doi.org/10.1016/j.jairtraman. 2020.101915 2. Taylor, S.: Air Transport 2035: Four Possible Post-COVID-19 Scenarios for Aviation, https:// apex.aero/2020/05/04/air-transport-2035-webinar-results. Last accessed 31 May 2020 3. Vandycke, N.: Transport and COVID-19: short-term chaos could bring long-term transformation, https://blogs.worldbank.org/transport/transport-and-covid-19-short-term-chaos-couldbring-long-term-transformation. Last accessed 31 May 2020

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66. Milne, R.J., Delcea, C., Cotfas, L.-A.: Airplane boarding methods that reduce risk from COVID19. Saf. Sci. 134, 105061 (2021). https://doi.org/10.1016/j.ssci.2020.105061 67. Wilensky, U., Rand, W.: An Introduction to Agent-Based Modeling: Modeling Natural, Social, and Engineered Complex Systems with NetLogo. The MIT Press, Cambridge, Massachusetts (2015) 68. Stonedahl, F., Wilensky, U.: Finding Forms of Flocking: Evolutionary Search in ABM Parameter-Spaces. In: Bosse, T., Geller, A., and Jonker, C.M. (eds.) Multi-Agent-Based Simulation XI, pp. 61–75. Springer, Berlin Heidelberg (2011). https://doi.org/10.1007/978-3-64218345-4_5 69. Ponsiglione, C., Roma, V., Zampella, F., Zollo, G.: The Fairness/Efficiency Issue Explored Through El Farol Bar Model. In: Gil-Aluja, J., Terceño-Gómez, A., Ferrer-Comalat, J.C., Merigó-Lindahl, J.M., and Linares-Mustarós, S. (eds.) Scientific Methods for the Treatment of Uncertainty in Social Sciences, pp. 309–327. Springer International Publishing, Cham (2015). https://doi.org/10.1007/978-3-319-19704-3_26 70. Cotfas, L.-A., Delcea, C., Mancini, S., Ponsiglione, C., Vitiello, L.: An agent-based model for cruise ship evacuation considering the presence of smart technologies on board. Expert Syst. Appl. 214, 119124 (2023). https://doi.org/10.1016/j.eswa.2022.119124 71. Delcea, C., Bradea, I.A.: Economic Cybernetics. An Equation-Based Modeling and AgentBased Modeling Approach. EdituraUniversitara (2017) 72. Delcea, C., Cotfas, L.-A.: Increasing awareness in classroom evacuation situations using agentbased modeling. Phys. Stat. Mech. Its Appl. (2019). https://doi.org/10.1016/j.physa.2019. 04.137 73. Delcea, C., Cotfas, L., Craciun, L., Molanescu, A.G.: Establishing the proper seating arrangement in elevated lecture halls for a faster evacuation process. IEEE Access. 7, 48500–48513 (2019). https://doi.org/10.1109/ACCESS.2019.2909637 74. Delcea, C., Cotfas, L.-A., Craciun, L., Molanescu, A.G.: An agent-based modeling approach to collaborative classrooms evacuation process. Saf. Sci. 121, 414–429 (2020). https://doi.org/ 10.1016/j.ssci.2019.09.026 75. Alizadeh, R.: A dynamic cellular automaton model for evacuation process with obstacles. Saf. Sci. 49, 315–323 (2011). https://doi.org/10.1016/j.ssci.2010.09.006 76. Schultz, M.: Fast aircraft turnaround enabled by reliable passenger boarding. Aerospace 5, 8 (2018). https://doi.org/10.3390/aerospace5010008 77. Jay, A.: Number of Flights Worldwide in 2022/2023: Passenger Traffic, Behaviors, and Revenue, https://financesonline.com/number-of-flights-worldwide/. Last accessed 25 Nov 2022

Chapter 5

Public Opinion Assessment Through Grey Relational Analysis Approach

Introduction This chapter is dedicated to the application of the grey relational analysis—one of the basic elements of the grey systems theory—in the public opinion mining. Due to the complex economic and social situation the entire mankind has passed in the last few years since the occurrence of the COVID-19 pandemics, the chapter features an application on the public opinion assessment in connection with the COVID-19 vaccination process. As the application of the grey relational analysis into the field of public opinion assessment represents only one of the possible cases in which the grey relational analysis can be applied in the field of economic and social sciences, the chapter dedicates some space to the discussion related to the different types of grey relational analysis, the difference between correlation and causation and to the papers that have used grey relational analysis in economic and social sciences field. Also, different types of grey relational analysis are mentioned in the section dedicated to the literature review, along with some relevant references. As in this chapter only the traditional type of grey relational analysis is performed [1, 2], we invite the reader, if interested, to read about the remainder types of the grey relational analysis by following the references provided in the literature review section (and not only) and to apply them in practice.

Argument for the Public Opinion Assessment in the COVID-19 Vaccination Context The widespread of the coronavirus disease has determined the governments worldwide to take unprecedented measures in order to limit its impact. In this context, the need for a vaccine was of utter importance and, on November 9, 2020, the first vaccine with a more than 90% effective rate was created. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_5

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Considering the situation faced by people all around the world at the time the vaccine was announced, characterized mostly by lockdowns and mask wearing in public spaces, posting their reaction on social media platforms has been a natural way of sharing their thoughts to a large audience, interested in the news related to the new vaccine. This situation has not been new as social media platforms have been, even in the past, the meeting point and the mediating channel between the individual and the rest of the world, where different aspects related to public interest have been extensively discussed [3, 4]. Among the platforms used by people to express their opinions, Twitter has gained a particular attention due to the characteristics of the messages (called tweets) that can be posted here which are limited to 280 characters, forcing their content contributors to sharply express their opinions over a given issue [5]. As a result, over time, various studies have been conducted in the scientific literature with the purpose of analyzing the people’s opinions related to different aspects, such as, but not limited to: vaccination [5], infectious diseases [3, 6], healthcare [7], evaluating companies’ services [8] and reputation [9], forecasting the prices of cryptocurrencies [10], natural disasters and social movements [11], refugee crisis [12] and political attitudes [13]. In this context, the present chapter aims to analyze the incidence of the news posted in tweets on the general public sentiment in connection with the COVID-19 vaccination. On this purpose, two main periods have been considered: the first one represented by the 2-week period following the announcement that a vaccine against COVID-19 is available—representing November 9, 2020—November 22, 2020; and a second one represented by the 2-week period following the start of the vaccination process in the UK—namely December 8, 2020—December 21, 2020. A total number of 3,764,303 tweets have been extracted and through a “cleaning” process, a number of 1,002,706 unique tweets have been retained for analysis. Based on a machine learning approach, the tweets have been divided into three main categories, named neutral for tweets conveying news, in favor for the pro-vaccination tweets and against for the tweets presenting an anti-vaccination message. The evolution of the number of tweets in each category has been observed in the two periods of time considered and the incidence of the number of tweet news on the number of in favor and against tweets has been determined using grey systems theory. The decision to consider two periods of time has been determined by the need of observing if there have been any changes in the incidence of the number of news on the number of in favor and against tweets between the moment when the efficacy of the first vaccine has been announced and the moment when the vaccination campaign has actually started in UK. The contribution of the chapter is four-folded: a COVID-19 vaccination dataset has been collected; the best performing classifier for stance detection in case of COVID-19 vaccination process has been determined; the evolution of the number of tweets in each of the considered categories has been observed; the incidence of the number of news on the number of in favor and against tweets has been determined. The chapter is organized as follows: the next section provides a brief literature review of the papers dedicated to COVID-19 vaccination and of the applications developed in the research literature on the basis of grey relational analysis. Then, the

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methodology used for extracting and analyzing the tweets, which combines stance detection and grey relational analysis is presented. Last, an analysis is conducted on the selected dataset and the results are interpreted in accordance with the values obtained from the grey relational analysis.

Literature Review In the following, a brief literature review is provided on both COVID-19 vaccination and grey relational analysis. From the literature review it can be observed that the use of a large dataset—as the one extracted from Twitter—can be beneficial in better shaping the opinion of a large amount of population worldwide rather than using questionnaires applied to small datasets. Additional, by conducting a grey relational analysis, the relation between the number of tweets classified as news and the tweets classified as in favor and against COVID-19 vaccination can be better observed.

Literature Review on COVID-19 Vaccination The occurrence of the novel coronavirus and the development of vaccines to fight against the spread of the disease has determined the division of the opinions people have regarding the COVID-19 vaccination between three main groups: in favor, against and neutral to vaccination. Considering the papers written in the scientific literature, it can be observed that the most frequently employed technique used for analyzing the people’s opinion regarding COVID-19 vaccination has been represented by questionnaires analysis. As a result, different percentages related to COVID-19 vaccination have been recorded all around the world. In the following, some of the results of the questionnaire analysis related to COVID-19 are briefly mentioned. The reader is invited to consider the mentioned works in order to find out more details related to the reluctance of COVID-19 vaccination, in accordance with the questions used in the listed studies. In a study conducted on a large sample of UK respondents, Paul et al. [14] concluded that 14% of the respondents are unwilling to vaccinate, while 22.5% are not sure about the decision to make. Soares et al. [15] on Portugal respondents reported a 65% vaccine acceptance rate, while Biasio et al. [16] on Italian respondents determined a 91% vaccine acceptance rate. High COVID-19 vaccine acceptance rates have been reported by Detoc et al. [17] on French population—75%, Liu et al. [18] on Chinese population—82.25%, Reiter et al. [19] on US population—69%, Borriello et al. [20] on Australian population—86%, while Alfageeh et al. [21] on Saudi population reported a lower vaccine acceptance rate of only 48%. As the use of questionnaires for analyzing the people’s opinions can have some drawbacks, mostly related to the sample selection and representatively, some other studies regarding COVID-19 vaccination have focused on large datasets extracted

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from Twitter messages. Praveen et al. [22] determined a 35% in favor rate for the COVID-19 vaccination and a 16.65% against rate for the Indian tweets, while Cotfas et al. [23] reported a 17% against rate for English tweets. Both studies have referred to the periods around the first announcement that a vaccine against COVID-19 is available. Later on, Cotfas et al. [24] investigated the hesitancy for COVID-19 vaccination in the month following the start of the vaccination process in UK, highlighting that the sources for hesitancy have been related to mistrust, scam, hiding relevant information, vaccine inefficiency, side effects, missing the feeling of freedom, existence if alternatives. In a new analysis made for the one month period after the third booster dose arrived for the COVID-19 vaccination, Delcea et al. [25] concluded that the tweets in the against category ranged between 4.89% in the entire dataset (the dataset containing all the tweets, including the retweets) and 7.44% in the cleaned dataset (the dataset which does not contain the retweets). Going further with the analysis, the authors have studies the hesitancy reasons and they observed that the number of reasons has been reduced compared to Cotfas et al. [24]. The hesitancy reasons identified for the third booster dose have been only related to side effects, existence of alternatives, hiding relevant information, mistrust and scam. In this context, to the best of our knowledge, the present chapter is the first to analyze the evolution of COVID-19 vaccination opinions and to discuss their numerical evolution in accordance with the types of identified tweets through the use of grey systems theory methods.

Literature Review on Grey Relational Analysis The research around the grey relational analysis has started with the work made by Professor Deng JL, who proposed the first grey relational analysis in 1984, two years later since he put the basics of the grey systems theory [26]. Over time, however, the interest in grey relational analysis, also referred to as “grey incidence analysis” or “grey correlation analysis” [27] has been steadily rising, which has led to numerous methods of calculating the grey incidence grades. Some of these methods are [28, 29]: the degree of incidence of type T [30], the degree of grey incidence taking into account entropy [31], the degree of local grey incidence [32], the degree of Euclidean grey incidence [33], the degree of incidence of type B [34], the degree of incidence of type C [35], the degree of incidence of the slope [36], the degree of absolute grey incidence [1], the degree of relative grey incidence [1], the degree of synthetic grey incidence [1], improved degree of grey T-type incidence [37], the degree of grey incidence based on similarity [38], the degree of grey periodic incidence [39]. The research on improving the exiting grey relational analysis (grey incidence analysis) has continued over the years. For example, Xie and Liu [40] constructed a novel grey relational model based on grey number sequences, while Li et al. [41] proposed a new grey relational model based on discrete Fourier transformations. Javed and Liu [42] proposed a bidirectional absolute grey relational analysis starting

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from the classical absolute grey relational analysis, but which has the capacity to better handle uncertain systems represented by uncertain data. Also, in a recent work, Professor Liu SF has proposed a series of negative grey relational models which can be used for measuring the relationship between reverse sequences. As a result, the negative grey absolute relational analysis model, the negative grey relative relational analysis model, the negative grey comprehensive relational analysis model and the negative Deng’s grey relational analysis model are proposed and discussed along with the definition of reverse sequence and the negative grey similarity relational analysis model [43]. By retaining the advantages brought by Deng’s grey relational analysis of the bidirectional absolute grey relational model, Javed et al. [44] uses a new grey relational analysis, called the second synthetic grey relational analysis model and proves its practical applicability. In a keynote speech at The 2022 International Congress on Grey Systems and Uncertainty Analysis, Professor Javed SA discussed the difference among the between causation and correlation and stated that the grey relational analysis is a measure of correlation and not of distance [26]. Furthermore, the speaker presented a dynamic grey relational analysis [26] in which the data normalization step, so needed in the classical Deng’s grey relational analysis, is included in the model, making it easy to be applied in practice. The author applied the proposed model to the multi-sourcing and supplier classification. Regarding the practical applications in which the grey relational analysis has been used, it can be observed from the start that there are plenty of applications which benefit from this type of analysis. In the following, we will highlight some of the applications in which the grey relational analysis has been successfully used, noting that the selection made represents only a small part of the scientific literature which accompanies the grey relational analysis. Li and Li [45] applied the grey relational analysis in the factors determining China’s technological innovation ability. The authors considered four factors which can influence China’s technological innovation ability, namely research and development expenditure, research and development personnel, university student number and public library number and concluded that the results obtained through applying grey relational analysis are in line with the expectations [45]. In the area of social networks, Guo and Zhang [46] have used the grey relational analysis for discovering social communities by considering the propagation affinity of the messages among members. The authors highlight the possibility of using the proposed approach for discovering communities in many social networks, which could bring benefits for the interested parties in sending advertisements or identify valuable customers [46]. The connection between the companies characteristics and their performance is addressed through the means of grey relational analysis by Delcea et al. [47]. In the area of firm’s bankruptcy and the occurrence of bankruptcy syndrome, Scarlat and Delcea [48] proposed an approach based on grey relational analysis for shaping the relationship among early-warning variables. Focusing on the qualitative characteristics and the companies’ performance, Delcea et al. [49] measures the strength of the relationship between the two elements through the means of grey relational analysis,

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concluding that a continuous observation of the quality characteristics can be helpful in evaluating companies’ performance. The attitudes of patients in relation with the medical services are analyzed by Zhang et al. [50]. The data regarding the attitudes of 600 patients are extracted through the use of questionnaires and the grey relational analysis is used to explore the differences in attitudes in terms of the cost of the medical services. The authors concluded that the proposed approach can be useful for determining the existing problems of the medical services in terms of price. To the mentioned applications, one can add the papers dedicated to supplier selection problem, as presented in Chap. 3 of this book. To the supplier selection papers discussed in Chap. 3, one can add the newly papers written by Asgharnezhad and Darestani [51], Javed [26], Ghosh et al. [52], Afrasiabi et al. [53]. A summary of the grey relational analysis in the field of marine economics and management is provided by Xuemei et al. [54].

Methodology The main steps needed for detecting the news incidence on people’s opinion related to COVID-19 vaccination are presented in Fig. 5.1 and are detailed in the following sub-sections.

Dataset Collection In order to extract the English tweets related to vaccination in the two considered periods, November 9, 2020–November 22, 2020 and December 8, 2020–December 21, 2020, keywords related to both vaccination (vaccine, vaccination, vaccinate, vaccinating, vaccinated) and COVID-19 (covid19, covid-19, coronavirus, corona outbreak, coronavirus pandemic, Wuhan virus, 2019nCoV ) have been used [23–25].

Classifier Selection From the tweets dataset extracted by using the above-mentioned keywords, another dataset (called cleaned) has been extracted by eliminating the duplicates as suggested by Aloufi and Saddik [55] and D’Andrea et al. [5], having the property of containing only unique tweets. The annotated dataset is afterwards constructed by randomly extracting a number of 0.28% tweets from the cleaned dataset and annotating them in accordance with three categories: neutral—the tweets containing news related to COVID-19 vaccination; in favor—tweets containing pro-vaccination statements and against—tweets

Methodology

185

Fig. 5.1 News incidence detection steps

containing anti-vaccination statements. This set will be further used for determining the best-performance classifier. Pre-processing is a crucial step, through which, for the annotated tweets, a series of elements (such as user mentions, email addresses and urls) are eliminated while others (such as minor spelling mistakes, elongated words) are corrected [5, 56]. For performing this step, the “re” python modules has been used along with the ekphrasis library and the Natural Language Toolkit (NLTK) library [23, 57, 58]. The reduction of the weights associated with the most frequent words has been conducted, as suggested in the scientific literature, through Term Frequency—Inverse Document Frequency (TF-IDF) [23, 24]. In order to determine the best-performing classifier, a series of classical machine learning and deep leaning classifiers have been considered. The analyzed classical machine learning algorithms have been: Multinomial Naive Bayes (MNB) [59, 60], Random Forest (RF) [61, 62] and Support Vector Machine (SVM) [63, 64].

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The considered deep learning classifiers use the transformer architecture: Bidirectional Encoder Representations from Transformers (BERT) [65], Robustly Optimized BERT Pretraining Approach (RoBERTa) [66], ALBERT (A Lite BERT) [67] and XLNet [68]. The following indicators have been used for evaluating their performances, keeping in mind that higher values for the indicators are preferred [5]: Accuracy =

TP + TN TP + TN + FP + FN

Precision = Recall = F − score = 2 ·

TP TP + FP

TP TP + FN

Precision · Recall Precision + Recall

where: TP is the number of real positive tweets classified as positive; TN is the number of negative tweets correctly classified as negative; FP represents the number of real negative tweets classified incorrectly classified as positives; and FN is the number of real positive tweets incorrectly classified as negative.

Stance Detection A similar pre-processing step as in the case of the annotated dataset is conducted for the cleaned dataset and the best classifier determined in the previous step is used for dividing the tweets into the three considered categories (in favor, neutral or against). As we are interested in the evolution of the number of tweets in each category for each day of the two considered periods, after the classification, a visual representation of the variation in the number of tweets is provided.

Grey Relational Analysis The grey relational analysis is performed two times, for each of the considered periods. The number of tweets in the three categories for each day is considered when conducting the grey relational analysis. The theoretical approach needed for applying the grey relational analysis is describe next. As in other chapters of this book, we have tried to keep the notations as close as possible to the ones provided in the references. The synthetic degree is determined as follows, based on the absolute and the relative degree, as suggested in the scientific literature [69, 70]:

Methodology

187

• The Absolute Degree Considering two sequences of data with non-zero initial values and with the same length, data X0 and Xj , j = 1…n, with t = time period and n = variables [69, 70]:   X0 = x1,0 , x2,0 , . . . , xt,0   Xj = x1,j , x2,j , . . . , xt,j The zero-start points’ images are:    0 0  0 Xj0 = x1,j − x1,j , x2,j − x1,j , . . . , xt,j − x1,j = x1,j , x2,j , . . . , xt,j The absolute degree is:   1 + |s0 | + sj      ε0j = 1 + |s0 | + sj  + s0 − sj    with |s0 | and sj  computed as follows [69, 70]:   t−1   1  0 0  |s0 | =  xk,0 + xt,0   2  k=2  t−1      1 0 0  sj  =  xk,j + xt,j   2  k=2

• The Relative Degree Considering two sequences of data with non-zero initial values and with the same length, X0 and Xj , j = 1…n, with t = time period and n = variables [70] as presented in (5) and (6), the initial values images of X0 and Xj are: x1,0 x2,0 xt,0 , ,..., x1,0 x1,0 x1,0    xt,j x1,j x2,j     , ,..., Xj = x1,j , x2,j , . . . , xt,j = x1,j x1,j x1,j

     = X0 = x1,0 , x2,0 , . . . , xt,0



The zero-start points’ images calculated based on (11) and (12) for X0 and Xj are:   0 0         0 = x1,0 , x2,0 , . . . , xt,0 X00 = x1,0 − x1,0 , x2,0 − x1,0 , . . . , xt,0 − x1,0

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          0 0 0 = x1,j Xj0 = x1,j − x1,j , x2,j − x1,j , . . . , xt,j − x1,j , x2,j , . . . , xt,j The relative degree is computed as [69, 70]:     1 + s0  + sj   r0j =       1 + s0  + sj  + s0 − sj   t−1       1  0 0 s  =  x x +  0 k,0  2 t,0  k=2

      with s0  and sj :  t−1     1 0     0 xk,j + xt,j  sj  =   2  k=2

• The Synthetic Degree The synthetic degree is based on both the absolute and the relative degrees [69, 70]: ρ0j = θ ε0j + (1 − θ )r0j with j = 2, …, n, θ ∈ [0, 1] and 0 < ρ0j ≤ 1. Grey relational analysis is increasingly used in economic and social sciences due to the fact that it focuses on the closeness of relations between factors, being based on the similarity level of the geometrical patterns of sequence curves [69]. The size of the degree of incidence is directly proportional to the degree of similarity between the considered variables’ curves.

News Incidence Detection A dataset containing 3,764,303 tweets has been extracted based on the keywords mentioned above. From this dataset, it has been determined that the number of unique tweets (cleaned dataset) has been of 1,002,706, from which 328,834 have been posted in the period November 9, 2020–November 22, 2020, and 673,872 in the period December 8, 2020–December 21, 2020.

News Incidence Detection

189

Dataset Characteristics The evolution of the number of tweets in the cleaned datasets in the two periods analyzed is depicted in Figs. 5.2 and 5.3. As it can be observed by comparing the evolution of the number of tweets in the considered periods, twice as many tweets have been published in the second period compared to the first period. Regarding the first period analyzed, depicted in Fig. 5.2, it can be observed that on the day the announcement regarding the creation of a vaccine against COVID19 has been made—November 9, 2020, the number of tweets has reached a peak, counting for 56.768 tweets. After this date, the daily number of tweets has presented a decreasing trend, with a significant peak on November 16, 2020—the day in which it has been publicly announced that Moderna’s COVID-19 vaccine shows 94.5% efficiency in clinical trials—counting for 38.072 tweets. Two other small peaks are recorded on November 18, 2020—when the news regarding Sinovac’s COVID-19 vaccine induces a quick immune response has been released to the press (28.148 tweets), and on November 20, 2020—when Pfizer’s announcement regarding COVID-19 vaccine emergency authorization has been made (21.036 tweets). Based on Fig. 5.2, it can be stated that the number of tweets posted in the analyzed period has presented peaks in accordance with the major events reported in the social media. As for the second period, presented in Fig. 5.3, it can be observed that the start of the vaccination campaign in UK on December 8, 2020, has determined a high number of tweets (71.010 tweets). Along the analyzed period, three other peaks have

NUMBER OF TWEETS

60,000

TWEETS IN THE CLEANED DATASET

50,000 40,000 30,000 20,000 10,000 0

DATE

Tweets Fig. 5.2 Evolution of the number of tweets in cleaned dataset for November 9, 2020–November 22, 2020

190

5 Public Opinion Assessment Through Grey Relational Analysis Approach TWEETS IN THE CLEANED DATASETS 80,000

NUMBER OF TWEETS

70,000 60,000 50,000 40,000 30,000 20,000 10,000 0

DATE

Tweets Fig. 5.3 Evolution of the number of tweets in cleaned dataset for December 8, 2020–December 21, 2020

been recorded on December 14, 2020 (65.271 tweets), December 18, 2020 (57.189) and December 21, 2020 (42.015 tweets). Considering the events mentioned in the press in the three periods, it has been observed that they correspond to the start of the COVID-19 vaccination with Pfizer in US, the plan of Food and Drug Administration to approve the Moderna vaccine and the event during which the US President-elect Joe Biden got the vaccine. Table 5.1 presents a selection of randomly extracted tweets from both periods, classified as in favor, neutral or against COVID-19 vaccination.

Classifier Selection The selection of the best-performing classifier has been made based on the values obtained for the metrics accuracy, precision, recall and F-score. Tables 5.2, 5.3, 5.4 and 5.5 present the results obtained for both the classical machine learning (MNB, RF, SVM) and deep learning algorithms (BERT-cased, BERT-uncased, RoBERTa, ALBERT, XLNet). According to the accuracy indicator, the best-performing classifier is RoBERTa, with an accuracy of 81.84%, followed by the other deep learning classifiers analyzed in this chapter: XLNet—79.64%, ALBERT—79.56%, BERT-cased—79.12% and BERT-uncased—78.68%. Among the classical machine learning, SVM has succeed in achieving the highest accuracy value (Table 5.2).

News Incidence Detection

191

Table 5.1 Randomly extracted tweets Stance

Tweet

Against

I will not take the #vaccine, as I am not in a #COVID19 high risk category and believe that the vaccine needs more rigorous testing to assure that people with existing antibodies and certain allergies (including to polyethylene glycol) will not suffer adverse events. #FactsNotFear https://t.co/HtVQzvC5Bt There will be so many side effects from the vaccinations that there won’t be time to put any restrictions in place. Bell’s Palsy has already been identified as Pfizer vaccination side effect in multiple people. And people are just going to say oh sure inject me with your poison https://t.co/21pcMiQL5X There might be adverse side effects vaccine https://t.co/HyGAuT1w9r

no one wants to be the test monkii for a new

Miss me with that vaccine. I ain’t taking that poison. https://t.co/guIdXX2XUe Neutral

Allina Health seeks volunteers for COVID-19 vaccine trial https://t.co/tzjISOcl84 via @KARE11 Kids are participating in COVID-19 vaccine trials. Here’s what their parents think https://t.co/6i4gMVwzQl Healthcare workers, uniformed personnel, as well as the poor will be among those to be prioritized in getting the COVID-19 vaccine once it is available, Health Secretary Francisco Duque III said. | @DYGalvezINQ https://t.co/Oz0IdPR4KR Australia begins production of Oxford-developed COVID-19 vaccine https://t.co/ tM6UFLV9iM

In favor

This thing isn’t going away anytime soon, I hope a vaccine will be available to everyone who needs it. https://t.co/wmeY7KUQVl Covid vaccine: First vaccine offers 90% protection - this is amazing news like 2021 is going to be better #COVID19 https://t.co/SNVHGnfIe7

looks

I am going to take this vaccine https://t.co/aZFewOK3x0 Normally it takes about 12 years to develop a vacines. But a COVID-19 vaccines is ready in a less than a year. FDA, EMA and co still need to review data from the clinical trials. Most exciting news i have heard so far Table 5.2 Classifiers performance in terms of accuracy

Classifier

Accuracy (%)

MNB

74.31

RF

71.97

SVM

76.76

BERT-cased

79.12

BERT-uncased

78.68

RoBERTa

81.84

ALBERT

79.56

XLNet

79.64

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5 Public Opinion Assessment Through Grey Relational Analysis Approach

Table 5.3 Classifiers performance in terms of precision

Classifier MNB

Table 5.4 Classifiers performance in terms of recall

Table 5.5 Classifiers performance in terms of F-score

Class In favor (%)

Neutral (%)

Against (%)

76.82

78.22

69.87

RF

75.04

69.69

72.23

SVM

74.88

81.45

75.11

BERT-cased

77.42

82.71

78.77 80.48

BERT-uncased

76.65

80.61

RoBERTa

82.51

84.19

80.19

ALBERT

81.21

80.79

77.02

XLNet

76.94

84.04

78.89

Classifier

Class Neutral (%)

Against (%)

In favor (%) MNB

66.23

72.09

84.63

RF

61.48

75.77

78.67 84.73

SVM

73.16

72.41

BERT-cased

79.67

76.60

81.09

BERT-uncased

79.41

78.16

78.08

RoBERTa

81.82

79.52

83.88

ALBERT

76.77

79.72

82.07

XLNet

82.17

74.34

80.67

Classifier

Class In favor

Neutral

Against

MNB

71.1

74.95

76.47

RF

67.5

72.57

75.25

SVM

73.92

76.58

79.53

BERT-cased

78.22

79.42

79.40

BERT-uncased

77.82

79.00

78.86

RoBERTa

82.01

81.50

81.75

ALBERT

78.84

80.21

79.44

XLNet

79.25

78.52

79.55

In terms of precision, RoBERTa has obtained the best values for the classification of the in favor and neutral tweets, while for the against tweets the performance of RoBERTa has been close to the BERT-uncased, performing with only 0.29% worse—Table 5.3.

News Incidence Detection

193

According to the recall indicator, the best-performing classifier has been XLNet for the in favor tweets—82.17%, ALBERT for the neutral tweets—79.72% and SVM for the against tweets—83.88%—Table 5.4. Among all classifiers, RoBERTa has succeed to have second-best performing values for the in favor and neutral tweets, while for the against tweets it has obtained a better performance than all the other deep learning algorithms considered. For the F-score, RoBERTa has provided the highest value for all the three classes considered—Table 5.5. The other deep learning algorithms have provided, in general, better results than the classical machine learning algorithms. As a result of the values obtained for the four performance indicators considered, it has been determined that RoBERTa provides better results in most of the cases and has been used in the following for performing the stance detection on the cleaned dataset.

Stance Detection The stance evolution of the tweets in the two considered periods is depicted Figs. 5.4 and 5.5. In the case of the tweets posted between November 9, 2020–November 22, 2020 it has been observed that most of the tweets have been in the neutral category (219.583 tweets, representing 66.78%), followed by in favor tweets (76.875 tweets, 23.38%) and against tweets (32.376 tweets, 9.84%). Considering the daily number

40,000

STANCE EVOLUTION

35,000 30,000 25,000 20,000 15,000 10,000 5,000 0

in favor

neutral

against

Fig. 5.4 Stance evolution of tweets in cleaned dataset for November 9, 2020–November 22, 2020

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5 Public Opinion Assessment Through Grey Relational Analysis Approach

STANCE EVOLUTION 60,000 50,000 40,000 30,000 20,000 10,000 0

in favor

neutral

against

Fig. 5.5 Stance evolution of tweets in cleaned dataset for December 8, 2020–December 21, 2020

of published tweets it can be observed that in all the days considered in the analysis, the number of tweets in the neutral category has had the highest value—Fig. 5.4, with the highest number of neutral tweets, 34.394, being posted on November 9, 2020. As for the second period considered, it has been observed that the highest number of tweets continues to belong to the neutral category (504.019 tweets, 74.79%), followed by in favor tweets (124.205, 18.43%) and against tweets (45.648 tweets, 6.77%). Even in this case, it can be observed that the daily number of tweets has been higher for the neutral tweets in all the days of the considered period—Fig. 5.5. Even more, for the second period, it can be observed that there are two peaks in which the number of neutral tweets has been the highest, namely December 8, 2020—with 50.087 tweets, and December 14, 2020—with 47.879 tweets. As mentioned above the two peeks correspond to two major events represented by start of the COVID-19 vaccination with Pfizer in US and the plan of Food and Drug Administration to approve Moderna vaccine.

News Incidence on People’s Opinion For the grey relational analysis, the Grey Modeling Software 6.0 [71] has been used. The number of tweets recorded in each category, in favor, neutral and against, has been utilized for determining the synthetic degree for both periods.

Concluding Remarks

195

In the first period following the announcement of the creation of a COVID-19 vaccine (November 9, 2020–November 22, 2020) it has been determined that the degree of grey incidence between the number of daily news posted on Twitter and the number of positive tweets has been 0.8612, while between the number of daily news and the number of negative tweets has been slightly smaller, a value of 0.7725 being determined. For the second period, representing the two-weeks after the start of the vaccination process in UK (December 8, 2020–December 21, 2020), it has been determined that the synthetic degree between the number of daily news and the number of in favor tweets has been of 0.78685, while the one between the number of daily news and the number of against tweets has been of 0.7086. Considering the four values calculated for the synthetic degree, it can be stated that there is a powerful relationship between the number of tweets containing news related to the COVID-19 vaccination and the number of tweets containing in favor or against messages regarding COVID-19 vaccination, highlighting once more that the social media reacts in real time to the news posted on these networks. As a result, if interested, one can develop a monitoring system, which can extract in real time the people’s opinion over a given issue, which can help the interested parties in developing or adjusting the discourse over a given matter.

Concluding Remarks In this chapter, the incidence of COVID-19 vaccination news on the people’s opinion regarding vaccination has been analyzed during two periods of time, each of them equal to two-weeks. The periods have been chosen starting from two major events related to COVID-19 vaccination, namely the first announcement that an effective vaccine is available and the start of the vaccination campaign. A series of tweets have been extracted using specific COVID-19 and vaccination keywords. Classical machine learning and deep learning algorithms have been tested in terms of accuracy, precision, recall and f-score in order to determine the best performing stance classifier. As a result, RoBERTa has been chosen to be used for dividing the tweets into three categories: in favor, neutral and against vaccination. While the neutral tweets contain news or pieces of information, the in favor and against tweets have reflected the people’s opinion regarding the COVID-19 vaccination. Based on the evolution of the number of tweets in the three categories during the two periods, a grey relational analysis has been conducted and it has been determined that for both periods, there is a powerful relationship between the number of tweets containing news related to the COVID-19 vaccination and the number of tweets containing in favor or against messages regarding COVID-19 vaccination—grey synthetics degrees between 0.7086 and 0.8612, higher in the case of the relationship between news and in favor tweets. The high values obtained for the synthetic degree underlines, once more, that the social media reacts in real time to the news posted on the social networks and a proper social networks monitoring can be useful in

196

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determining in real time the people opinion over a given issue. This observation can be useful for the interested parties in creating awareness campaigns for different situations that might occur and might affect the general public. The analysis conducted in this chapter has some limitations related to the periods considered in the analysis, the tweets selection process—highly dependent on the used keywords, the selected deep learning classifier and the language of the extracted tweets. The research can be further extended by prolonging the considered periods or by including other languages into the analysis.

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33. Zhao, L.Y., Wei, S.Y., Mei, Z.X.: Grey euclid relation grade. J. Guan Xi Univ. 1, 10–13 (1998) 34. Wang, Q.Y.: The grey relational analysis of B-Model. J. Huazhong Univ. Sci. Technol. 2, 77–82 (1999) 35. Wang, Q.Y., Zhao, X.H.: The grey relational analysis of C-model. J. Huazhong Univ. Sci. Technol. 2, 75–77 (1999) 36. Dang, Y.G., Liu, Si.: Improvement on Degree of Grey Slope Incidence. Eng. Sci., 23–26 (2004) 37. Sung, Y.G., Dang, Y.G.: Improvement of grey T correlation degree. Syst. Eng. Theory Pract. 4, 135–139 (2008) 38. Cui, J.A.: A kind of new grey similarity incidence and application. Stat. Decis.-Mak. 1, 14–16 (2008) 39. Shi, X.H., Liu, S.F., Fang, Z.G., Zhang: The Model of Grey Periodic Incidence and Their Rehabilitation. Chin. J. Manag. Sci. 4, 131–136 (2008) 40. Xie, N., Liu, S.: A novel grey relational model based on grey number sequences. Grey Syst. Theory Appl. 1, 117–128 (2011). https://doi.org/10.1108/20439371111163747 41. Li, X., Zhang, Y., Yin, K.: A new grey relational model based on discrete Fourier transform and its application on Chinese marine economic. Mar. Econ. Manag. 1, 79–100 (2018). https:// doi.org/10.1108/MAEM-07-2018-004 42. Javed, S.A., Liu, S.: Bidirectional absolute GRA/GIA model for uncertain systems: application in project management. IEEE Access. 7, 60885–60896 (2019). https://doi.org/10.1109/ACC ESS.2019.2904632 43. Liu, S.: Negative grey relational model and measurement of the reverse incentive effect of fields medal. Grey Syst. Theory Appl. ahead-of-print, (2021). https://doi.org/10.1108/GS-102021-0148 44. Javed, S., Khan, A., Dong, W., Raza, A., Liu, S.: Systems evaluation through new grey relational analysis approach: an application on thermal conductivity—petrophysical parameters’ relationships. Processes. 7, 348 (2019). https://doi.org/10.3390/pr7060348 45. Li, L., Li, X.: Analysis on the related factors of China’s technological innovation ability using greyness relational degree. Grey Syst. Theory Appl. 12, 651–671 (2021). https://doi.org/10. 1108/GS-06-2021-0089 46. Guo, K., Zhang, Q.: Detecting communities in social networks by local affinity propagation with grey relational analysis. Grey Syst. Theory Appl. 5, 31–40 (2015). https://doi.org/10.1108/ GS-11-2014-0039 47. Delcea, C., Scarlat, E., M˘ar˘acine, V.: Grey relational analysis between firm’s current situation and its possible causes: a bankruptcy syndrome approach. Grey Syst. Theory Appl. 2, 229–239 (2012). https://doi.org/10.1108/20439371211260199 48. Scarlat, E., Delcea, C.: Complete analysis of bankruptcy syndrome using grey systems theory. Grey Syst. Theory Appl. 1, 19–32 (2011). https://doi.org/10.1108/20439371111106704 49. Delcea, C., Scarlat, E., Cotfas, L.: Companies’ quality characteristics vs their performance: a grey relational analysis—evidence from Romania. Grey Syst. Theory Appl. 3, 129–141 (2013). https://doi.org/10.1108/GS-09-2012-0038 50. Zhang, C., Duan, L., Liu, H., Zhang, Y., Yin, L., Sun, Q., Lu, Q.: Analysis of patients’ attitudes towards medical service prices in different regions based on grey relational theory. Grey Syst. Theory Appl. 9, 143–154 (2019). https://doi.org/10.1108/GS-09-2018-0042 51. Asgharnezhad, A., Avakh Darestani, S.: A green supplier selection framework in polyethylene industry. Manag. Res. Rev. 45, 1572–1591 (2022). https://doi.org/10.1108/MRR-01-2021-0010 52. Ghosh, S., Mandal, M.C., Ray, A.: Green supply chain management framework for supplier selection: an integrated multi-criteria decision-making approach. Int. J. Manag. Sci. Eng. Manag. 17, 205–219 (2022). https://doi.org/10.1080/17509653.2021.1997661 53. Afrasiabi, A., Tavana, M., Di Caprio, D.: An extended hybrid fuzzy multi-criteria decision model for sustainable and resilient supplier selection. Environ. Sci. Pollut. Res. 29, 37291– 37314 (2022). https://doi.org/10.1007/s11356-021-17851-2 54. Xuemei, L., Cao, Y., Wang, J., Dang, Y., Kedong, Y.: A summary of grey forecasting and relational models and its applications in marine economics and management. Mar. Econ. Manag. 2, 87–113 (2019). https://doi.org/10.1108/MAEM-04-2019-0002

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Chapter 6

Grey Systems Theory Approach to Linear Programming

Introduction Programming problems belong to the wider category of decision-making problems and refer to the situation in which, given a set of constraints, one can achieve the best solution (i.e. optimal) for the situation under investigation [1]. As Hillier and Lieberman [2] noted, the development of the linear programming has been one of the most scientific advances made in the twentieth century, which has helped the companies to save money. As a result, it has been faster adopted in various economic fields and has attained the interest from both practitioners and researchers. The issues discussed under the programming umbrella are various. Some simple examples refer to the cases in which a firm needs to determine the quantity to be produced from several goods under the limited resources constraints [3]. In this case, the “programming” term do not refer to computer programming, but rather it is assimilated to “planning” term, while “linear” describes the functions characterizing the mathematical model which describes the problem to be solved [2]. In the case of linear programming, both the objective function and the constraint conditions are linear functions. In the case in which any of these functions are altered, being described as nonlinear functions, we are facing another type of programming problem, namely the nonlinear programming problem. As a matter of fact, in the case in which the result of the problem to be solved is either to take a certain action or rather not to take it, namely the result is either a “yes” or a “no” action, then we are dealing with a 0–1 programming problem [1]. Given the numerous practical applications associated with the linear programming problems, in this chapter we will only focus to this particular part of the operations research area. As there are many aspects to be discussed in connection with the linear programming problem (e.g. Hillier and Lieberman [2] granted seven chapters of their operations research book to linear programming problem), in this chapter only the basic information are given to the reader. If interested, the reader can further consider

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_6

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thr works written by Dantzig and Thapa [4], Hillier and Lieberman [2], Taha [5], Vanderbei [6] and not only. With the occurrence of the grey systems theory and the new type of understanding the numbers, as described by the grey numbers, the linear programming problems become even harder to be solved [1]. As a result, in this chapter we will discuss some possible means to address the linear programming problems in the case in which some/all the involved input variables are grey numbers. To this aim, in the following, some basic information related to the form of the linear programming problems and means to solving it are presented. A literature review considering the papers written on linear programming when grey approach is considered are discussed in order to better identify the possible means to address the problem. As a result of the literature review, some particular cases are discussed from a theoretical point of view and some numerical examples are provided for a better understanding on the possible means to find a solution to the problem under investigation.

Brief Presentation of Linear Programming Basic Elements From a theoretical point of view, a linear programming problem can be written as a system having the following general form: max(min) f (x1 , x2 , . . . , xn ) = c1 x1 + c2 x2 + . . . + cn xn ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ a 11 x 1 + a12 x 2 + . . . + a1n x n ≤ (=, ≥)b1 ⎪ ⎪ ⎨ a21 x1 + a22 x2 + . . . + a2n xn ≤ (=, ≥)b2 ⎪... ⎪ ⎪ ⎪ ⎪ am1 x1 + am2 x2 + . . . + amn xn ≤ (=, ≥)bm ⎪ ⎪ ⎩ x1 ≥ 0; x2 ≥ 0; . . . ; xn ≥ 0 where • x1 , x2 , . . . , xn are the unknown variables to be determined through the problem solving—in most of the cases one can identify them as quantities to be produced from a set of products P1 , P2 , . . . , Pn . • f (·) is the objective function, which might be either maximized, max f (·), or minimized, min f (·)—in most of the maximization problems, the function refers to the overall profit/output of the firm producing the products, while in the case of the minimization problems, the function refers to the overall cost implied by the production of the x1 , x2 , . . . , xn products. • c1 , c2 , . . . , cn are the coefficients of the objective function - in the maximization problems, the c1 , c2 , . . . , cn coefficients are the profit/output contribution of each

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type of product to the profit/output, while in the minimization problems, the c1 , c2 , . . . , cn coefficients refer to the costs involved by the production of each type of product. • a11 , a12 , . . . , amn are the technological coefficients—usually indicating the resource consumption per unit of product. • b1 , b2 , . . . , bm are the limited quantities of resources to be used in the production process. Considering the above linear programming problem, one can easily identify its three main parts: • The linear objective function: f (x1 , x2 , . . . , xn ) = c1 x1 + c2 x2 + . . . + cn xn • The constraints given by the limited amount of resources: ⎧ a11 x1 + a12 x2 + . . . + a1n xn ≤ (=, ≥)b1 ⎪ ⎪ ⎨ a21 x1 + a22 x2 + . . . + a2n xn ≤ (=, ≥)b2 ⎪... ⎪ ⎩ am1 x1 + am2 x2 + . . . + amn xn ≤ (=, ≥)bm • The sign conditions imposed to the variables: x1 ≥ 0; x2 ≥ 0; . . . ; xn ≥ 0 The linear programming problem can be briefly written as: max(min) f (x1 , x2 , . . . , xn ) = ⎧ ⎪ Subject to: ⎪ n ⎪ ⎪ ⎪ ⎨  j=1 ai j x j ≤ bi , i ∈ I1 n j=1 ai j x j ≥ bi , i ∈ I2  ⎪ n ⎪ ⎪ ai j x j = bi , i ∈ I3 ⎪ ⎪ ⎩ j=1 x1 ≥ 0; x2 ≥ 0; . . . ; xn ≥ 0

n 

cjxj

j=1

where I1 ∪ I2 ∪ I3 = {1, 2, . . . , m}, Il ∩ Ik = φ, l = k

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Forms of the Linear Programming Problem In order to study a linear optimization problem, the standard form is often considered [2]. One can identify a standard form for the maximization problem: n max f (x1 , x2 , . . . , xn ) = cjxj j=1 ⎧ to: ⎨ Subject n a x ≤ bi , i = 1, m i ⎩ j=1 j j x j ≥ 0, j = 1, n And a standard form for the minimization problem: n min f (x1 , x2 , . . . , xn ) = cjxj j=1 ⎧ to: ⎨ Subject n a x ≥ bi , i = 1, m ⎩ j=1 i j j x j ≥ 0, j = 1, n where the involved vectors have the same explanation as provided above. The above-listed forms are equivalent to [2]: max f (x1 , x2 , . . . , xn ) = c T x ⎧ ⎨ Subject to: Ax ≤ b ⎩ x≥0 and to: min f (x1 , x2 , . . . , xn ) = c T x ⎧ ⎨ Subject to: Ax ≥ b ⎩ x≥0 where

⎛

⎜ ⎜ • x =⎜ ⎜ ⎜ ⎝

x1 . . . xn

⎞ ⎟ ⎟ ⎟ ⎟ ∈ Rn n-vector representing the objective function variables ⎟ ⎠

Brief Presentation of Linear Programming Basic Elements

⎛

c1 . . . cn

⎜ ⎜ • c=⎜ ⎜ ⎜ ⎝ ⎛

205

⎞ ⎟ ⎟ ⎟ ⎟ ∈ Rn n-vector representing the objective function coefficients ⎟ ⎠

⎞ b1 ⎜ . ⎟ ⎜ ⎟ ⎟ • b=⎜ ⎜ . ⎟ ∈ Rm m-vector representing the right-hand side of the constraints ⎜ ⎟ ⎝ . ⎠ b  m • A = ai j ∈ Mm,n (R) the matrix of the constraints coefficients, having the following form: ⎡ ⎢ ⎢ A=⎢ ⎣

a11 a12 · · · a1n a21 a22 · · · a2n ··· ··· ... ... am1 am2 . . . amn

⎤ ⎥ ⎥ ⎥ ⎦

Any linear programming problem can be written in both maximization and minimization form by using the following transformation [3]: max f (x) = −min(− f (x)); n j=1

ai j x j ≤ bi ⇔ −

n j=1

ai j x j ≥ −bi ;

 n n ai j x j ≤ bi j=1 j=1 ai j x j ≥ bi n  ai j x j = bi ⇔ or j=1 − j=1 ai j x j ≤ −bi − nj=1 ai j x j ≥ −bi  xj ≤ 0 ⇒ x j ≥ 0; x j = −x j

n



x j ∈ R, x j = x j − x j ; x j ≥ 0; x j ≥ 0 Another form of the linear programming problem is the augmented form, as defined by Taha [5], also known as the equation form [7]: max(min) f (x1 , x2 , . . . , xn ) = c T x ⎧ ⎨ Subject to: Ax = b ⎩ x≥0

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where b ≥ 0. The particularity of the augmented form is represented by the fact that it has all equality type constrains and nonnegative right-hand side terms and all nonnegative variables.

Feasible Solutions, Feasible Region and Optimal Solutions According to Hillier and Lieberman [2], in linear programming one has a different understanding of the term “solution”, which do not refer strictly to the final answer to a problem, but rather to any specification of values for the decision variables x1 , x2 , · · · .xn . As a result, a series of different types of solutions can be identified for a linear programming problem and, in order to distinguish among them, usually an adjective is attached to the “solution” term. Therefore, a “feasible solution” is a solution for which all the constraints are satisfied, including the sign conditions imposed to the variables. Contrary, by using the “infeasible solution” term one refers to the case in which at least one constraint, including the sign conditions imposed to the variable, is violated [2]. The reunion of all the feasible solutions constitutes the “feasible region”. In the case in which one deals with a two-variable linear programming problem, the feasible region, FP , can be represented in a two-dimensional space. Figure 6.1 provides an example of a feasible region represented in a twodimensional space, found at the intersection of five hyperplanes which derives from the constraints imposed to the variables and two hyperplanes which reflects the nonnegativity conditions imposed to the variables. The feasible region, FP , consists of the points withing the pictured boundaries, along with the corner points (A, B, C, D, E) and all the points positioned on the exterior segments of the region ([A, B], [B, C], [C, D], [D, E] and [E, A]). There are two situations in which a feasible region can be found [2, 3]: • when FP = φ—case in which the linear programming problem in infeasible, and • when FP = φ—case in which the linear programming problem can be bounded (Fig. 6.1) or unbounded (Fig. 6.2). A series of characteristics of the feasible region can be stated. As these characteristics exceeds the purpose of the discussion from this chapter, we invite the reader, if interested, to discover them by considering books dedicated to operations research— linear programming, such as, but not limited to: Dantzig and Thapa [4], Hillier and Lieberman [2], Taha [5], etc. An “optimal solution” is a feasible solution which brings the most desired outcome when considering the objective function [2]. By “the most desired outcome” we refer to either the maximum value (e.g. the largest value) of the objective function if the linear programming problem is a maximization problem, or the minimum value (e.g. the smallest value) of the objective function if the linear programming problem is a minimization problem [2].

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Fig. 6.1 Example of the representation of the feasible region in a two-dimensional space D E C

Y-Axis

Fp B

A

X-Axis

Y-Axis

Fig. 6.2 Example of the representation of the feasible region in a two-dimensional space when the linear programming problem is unbounded

Fp M

N

X-Axis

In most of the cases, the linear programming problems have a unique optimal solution. However, there are cases in which multiple optimal solutions can be encountered for a linear programming problem. This often happens when the objective function has the same slope as one of the constraints’ lines. For example, this situation is easily observed in a two-dimensional space where the line of the objective function is parallel with the line with one of the constraints. We can refer to each of the existing multiple optimal solutions through the “alternate basic solution” syntagm. The characteristics of this infinite number of solutions is that the

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optimal value of the objective function is the same for all of them, providing either the maximum or the minimum of the objective function, depending on the type of problem (maximization/minimization). Nevertheless, there are cases in which no optimal solution can be found for the linear programming problem. Two main situations can lead to this outcome: either the problem has no feasible solutions or the objective is unbounded [2]. An important result in determining the optimal solution of the linear programming problem is given by the Dantzig’s corner point theorem, which states that if the feasible region, FP , is nonempty and bounded, and the objective function is bounded on the FP , then at least one optimal solution is located at an extreme (corner) point of FP [3]. The extreme (corner) points of the FP are usually referred as “corner-point feasible solutions” or “vertices” [2]. In the case in which the decision variables are more than two, a graphical solution of the linear programming problem is hard to be found and Simplex algorithm is used instead. In this chapter, finding a solution through the Simplex algorithm is not presented as it exceeds the purpose of the discussion. If interested, the reader can find more information related to the components of the Simplex tableau, finding the optimal solution using Simplex, the duality in linear programming, solving linear programming problems using adequate software packages by considering the classical books of operations research or specialized articles dedicated to the linear programming problem.

Numerical Example In the following, we will consider a simple linear programming problem consisting in two unknown variables and three technological constraints that will be solved on the purpose of pointing out the theoretical elements exposed above. We are not going to give an economic text to the problem as the example is only a theoretical one. If interested, the reader can find economic texts that might accompany the linear programming problems in the specific textbooks and papers dedicated to operations research or grey linear programming. Thus, let us consider the following linear programming problem: max f (x1 , x2 ) = 4x1 + 5x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ x1 + x2 ≥ 10 ⎪ ⎪ ⎨ 0.5x1 + x2 ≤ 20 ⎪ 3x1 + x2 ≤ 45 ⎪ ⎪ ⎪ ⎪ 3x1 + x2 ≥ 20 ⎪ ⎪ ⎩ x1 ≥ 0; x2 ≥ 0

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Fig. 6.3 The first quadrant

First, as the sign conditions are indicating that the unknown variables, x1 , x2 , are greater or equal to zero, we know that the solutions should be in the first quadrant. Using GeoGebra Geometry [8], we will depict in the following from a graphical point of view, the feasible region, the feasible solution and the optimal solution of the proposed linear programming problem. First, let us depict the first quadrant in which we will find the feasible solutions as a result of the sign conditions. For this, please consider Fig. 6.3. Next, we will consider each equation once a time and draw them. For each of the equations, the line corresponding to the equation will divide the space into two half-planes and, each time, only one of the two half-planes will be considered for the position of the feasible solutions. Let us start with the first equation: x1 + x2 ≥ 10 For this equation, the line L 1 : x1 + x2 = 10 Divides the space into two half-planes. First, let us draw L 1 by arbitrary selecting two points of this line: x1 = 0 ⇒ x2 = 10

A(0, 10)

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Fig. 6.4 The semi-plane delimited by the first constraint and the sign conditions

x2 = 0 ⇒ x1 = 10

B(10, 0)

Based on the two points, the line L 1 is draw as in Fig. 6.4 (in red) and it delimits the two half-planes: x1 + x2 > 10 and x1 + x2 < 10. As we are interested in finding all the points for which x1 + x2 ≥ 10, we can simply choose arbitrary a point on one-side or the other of the x1 + x2 = 10 line and check the value of true for the x1 + x2 ≥ 10 by replacing the values of x1 and x2 with the coordinates of the arbitrary selected point. Less calculus is made if we choose the O(0, 0) point. For this point we have 0 + 0 ≥ 10 which is false, indicating the fact that we have to select the half-plane that do not contains the O(0, 0) point. In the particular case in which the line that delimits two half-planes contains the O(0, 0) point, that point won’t be used, we will pick another point that is not on the line for making the comparison. In our case, the selected half-plane is marked in light red (being placed in the right-side of the L 1 line). By making the intersection between this half-plane and the one resulting from the sign conditions (depicted in Fig. 6.3), the new plane containing the solutions that satisfy the sign conditions and the first constraint is represented in Fig. 6.4. Let us continue with the second constraint: 0.5x1 + x2 ≤ 20 The line, L 2 , has the equation:

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L 2 : 0.5x1 + x2 = 20 The points intersecting the axes are: x1 = 0 ⇒ x2 = 20

C(0, 20)

x2 = 0 ⇒ x1 = 40

D(40, 0)

Figure 6.5 depicts the L 2 line in dark green and the half-plane for which 0.5x1 + x2 ≤ 20 in green. The intersection between the first two constraints and the sign conditions is represented by the ABC D polygon—see Fig. 6.5. We will proceed in the same manner for the third constraint: 3x1 + x2 ≤ 45 The L 3 line has the equation: L 3 : 3x1 + x2 = 45 The intersection of L 3 line with the axes is given by the points: x1 = 0 ⇒ x2 = 45

E(0, 45)

Fig. 6.5 The polygon delimited by the first two constraints and the sign conditions

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Fig. 6.6 The polygon delimited by the first three constraints and the sign conditions

x2 = 0 ⇒ x1 = 15

F(15, 0)

Considering the third constraint, the half-plane delimited by the L 3 line that respects the third constraint, is depicted with yellow in Fig. 6.6. Now, the polygon in which all the points respect the first three constraints, and the sign conditions is the AB F GC polygon—as represented in Fig. 6.6. While the coordinates of the points A, B, F, C are known as they have been determined by intersecting the considered lines with the axes, the coordinated for G can be determined by solving the following system: L 2 ∩ L 3 = {G} 

0.5x1 + x2 = 20 3x1 + x2 = 45

Therefore, we have G(10, 15). Last, we need to consider the fourth constraint: 3x1 + x2 ≥ 20 The L 4 line has the equation: L 4 : 3x1 + x2 = 20

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Fig. 6.7 The polygon delimited by all the constraints and the sign conditions

The intersection of L 4 line with the axes is given by the points: x1 = 0 ⇒ x2 = 20 x2 = 0 ⇒ x1 ∼ 6.67

C(0, 20) H (6.67, 0)

The L 4 line (depicted in purple) divides the plane into two half-planes as in Fig. 6.7. The selected half-plane with respect to L 4 line is the one positioned in the right-hand side of the line, also depicted in light purple. Considering the intersection of all the half-planes determined by the constraints and by the sign conditions, the C G F B I polygon is obtained. The coordinates for the new point, I, are obtained as follows: L 1 ∩ L 4 = {I} 

x1 + x2 = 10 3x1 + x2 = 20

Therefore: I (5, 5). As a result, the feasible region, FP , is the CGFBI polygon—represented though a chessboard pattern in Fig. 6.8.

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Fp

Fig. 6.8 The feasible region—CGFBI polygon

As the feasible region, FP , is nonempty and bounded, while the objective function is bounded on the FP , then we can use the result of the theorem stating that at least one optimal solution is located at an extreme (corner) point of FP [3]. Therefore, we only need to determine the value of the function in the five corners of the CGFBI polygon and to choose the point/points for which this function is maximum: f (C) = 4x1 (C) + 5x2 (C) = 4∗0 + 5∗20 = 100 u f (G) = 4x1 (G) + 5x2 (G) = 4∗10 + 5∗15 = 115 u f (F) = 4x1 (F) + 5x2 (F) = 4∗15 + 5∗0 = 60 u f (B) = 4x1 (B) + 5x2 (B) = 4∗10 + 5∗0 = 40 u f (I ) = 4x1 (I ) + 5x2 (I ) = 4∗5 + 5∗5 = 45 u We have to choose the maximum value. Therefore:   f ∗ = f x ∗ = 115 u x1∗ = 10 u x2∗ = 15 u With these theoretical aspects, we will discuss in the following how the linear programming problem can be addressed when grey numbers are implied.

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Grey Linear Programming As Voskoglou [9] mentioned, grey linear programming problem differs from the classical linear programming due to the fact that the coefficients of the objective function and/or the technological coefficients and right-side-terms of the constraints are represented by grey numbers expressed as intervals. This situation that involves the presence of the grey numbers within the linear programming problem instead of real numbers as they were in the classical linear programming problem might occur due to various reasons, mostly related to the fact that in some situations the necessary data cannot be easily or precisely determined and, therefore, it is estimated resulting in the presence of grey interval numbers. Other situations that might lead to the use of the grey numbers within the linear programming problem might be related to the existence of a variety of changing factors for whom it is hard to determine their evolution function and, as a precaution measure, one can easier estimate an interval in which the values might range [1, 9]. According to Darvishi et al. [10] the inherent uncertainty one encounters in reallife problems, have stressed the need for developing and using new tools more adequate to solving these types of problems. As a result, fuzzy linear programming and grey linear programming have been developed in the area of linear programming. One of the main features of the grey linear programming leans on the fact that it is appropriate for dealing with inaccurate conditions [10]. The general form of a maximization grey linear programming problem is [1, 10, 11]: n c j (⊗)x j max ⊗ f (x1 , x2 , . . . , xn ) = j=1 ⎧ to: ⎨ Subject n a (⊗)x j ≤ bi (⊗), i = 1, m ⎩ j=1 i j x j ≥ 0, j = 1, n While the general form of a minimization grey linear programming problem is: n c j (⊗)x j min ⊗ f (x1 , x2 , . . . , xn ) = j=1 ⎧ to: ⎨ Subject n j=1 ai j (⊗)x j ≥ bi (⊗), i = 1, m ⎩ x j ≥ 0, j = 1, n With • x j ∈ Rx j ∈ R   • c j (⊗) ∈ c j , c j , c j ≥ 0, c j ≤ c j , j = 1, 2, . . . , n   • bi (⊗) ∈ bi , bi , bi ≥ 0, bi ≤ bi , i = 1, 2, . . . , m

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  • ai j (⊗) ∈ a i j , a i j , a i j ≥ 0, a i j ≤ a i j , i = 1, 2, . . . , m; j = 1, 2, . . . , n A more convenient manner of writing the grey linear programming problems is by using the matrix notation. Therefore, for the general form of a maximization grey linear programming problem, one can write [1, 10, 11]: max ⊗ f (x1 , x2 , . . . , xn ) = c(⊗)T x ⎧ ⎨ Subject to: A(⊗)x ≤ b(⊗) ⎩ x≥0 and for a general form of a minimization grey linear programming problem: min ⊗ f (x1 , x2 , . . . , xn ) = c(⊗)T x ⎧ ⎨ Subject to: A(⊗)x ≥ b(⊗) ⎩ x≥0 where

⎛

⎜ ⎜ • x =⎜ ⎜ ⎜ ⎝

x1 . . . xn

⎞ ⎟ ⎟ ⎟ ⎟ is a n-vector representing the objective function variables ⎟ ⎠ ⎛

⎜ ⎜ • c(⊗) = ⎜ ⎜ ⎜ ⎝

c1 (⊗) . . . cn (⊗)

⎞ ⎟ ⎟ ⎟ ⎟ is a grey n-vector representing the objective function ⎟ ⎠

  coefficients with c j (⊗) ∈ c j , c j , c j ≥ 0, c j ≤ c j , j = 1, 2, . . . , n,c j , c j ∈ R ⎞ ⎛ b1 (⊗) ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ • b(⊗) = ⎜ . ⎟ is a grey m-vector representing the right-hand side of the ⎟ ⎜ ⎠ ⎝ . bm (⊗)   constraints with bi (⊗) ∈ bi , bi , bi ≥ 0, bi ≤ bi , i = 1, 2, . . . , m, b j , b j ∈ R   • A(⊗) = ai j (⊗) is the grey matrix of the constraints coefficients, having the following form:

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⎡

⎤ a11 (⊗) a12 (⊗) . . . a1n (⊗) ⎢ a21 (⊗) a22 (⊗) . . . a2n (⊗) ⎥ ⎥ A=⎢ ⎣... ⎦ ... ... ... am1 (⊗) am2 (⊗) . . . amn (⊗)   with ai j (⊗) ∈ a i j , a i j , a i j ≥ 0, a i j ≤ a i j , i = 1, 2, . . . , m; j = 1, 2, . . . , n, ai j , ai j ∈ R As an observation, in this case, even the vector of the objective function variables, x, is a grey vector as well [1, 10]: ⎛ ⎜ ⎜ ⎜ x(⊗) = ⎜ ⎜ ⎝

x1 (⊗) . . . xn (⊗)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

  x j (⊗) ∈ x j , x j , x j ≥ 0, x j ≤ x j , j = 1, 2, . . . , n;x j , x j ∈ R This is a true statement as, it is known that any real number can be transformed into an interval grey number, which has the same value for the lower bound, q , as for the upper bound, q, as presented below [10]:   q(⊗) ∈ q , q ; q = q This type of number is also known as a white number, but in the theory of the grey systems, it is treated as a special case of grey number [1]. The inverse procedure, of moving from an interval grey number to the appropriate crisp value can be made by using a whitening function as defined in the theory of grey systems and presented in the following [1, 12]. Let us consider a grey number, r (⊗), defined as:     r (⊗) ∈ r , r = t ∈ r /r ≤ t ≤ r where t is the information and r and r are the lower and the upper bounds of the information [1, 10, 12]. ˜ r , can be The whitenization value of the interval grey number r (⊗), noted ⊗ determined as: ˜ r = αr + (1 − α)r ⊗

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with α ∈ [0, 1]. This procedure through which one transforms an interval grey number into a crisp value by using the above equation is known as “equal weight whitenization” [1]. An equal weigh whitenization for which α = 0.5 is called an “equal weight mean whitenization” [1]. If the reader is interested in finding more on the whitenization of the grey numbers, he/she should consider [1] for further information. Various forms of grey linear programming problems can be encountered. Few examples provided for the maximization linear programming problem include [10]: • Grey linear programming problem with grey cost: n max ⊗ f (x1 , x2 , . . . , xn ) = c j (⊗)x j j=1 ⎧ to: ⎨ Subject n a x ≤ bi , i = 1, m ⎩ j=1 i j j x j ≥ 0, j = 1, n • Grey linear programming problem with grey right-hand: n max f (x1 , x2 , . . . , xn ) = cjxj j=1 ⎧ to: ⎨ Subject n a x ≤ bi (⊗), i = 1, m ⎩ j=1 i j j x j ≥ 0, j = 1, n • Grey linear programming problem with grey technological coefficients: n max f (x1 , x2 , . . . , xn ) = cjxj j=1 ⎧ to: ⎨ Subject n a (⊗)x j ≤ bi , i = 1, m ⎩ j=1 i j x j ≥ 0, j = 1, n Furthermore, other variants of the grey linear programing problems can be encountered by combining the three examples provided above [10]. The various types of grey linear programming problems have been discussed in the scientific literature and various solutions for addressing them have been proposed accordingly. Some of them are discussed in the next section.

Literature Review on Grey Linear Programming A series of approaches have been proposed in the scientific literature for addressing the grey linear programming problem.

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Before discussing the solving methods for the grey programming problems, as Darvishi et al. [13] stated in their paper, one should note the difference between grey programming and interval programming. This difference represents the base for properly solving the grey programming problem. According to the authors, the difference consists in the fact that while the first use the idea of a grey parameter (namely a parameter that belongs to an interval), the latter strictly refers to an interval parameter (namely a set in the form of an interval). Considering the literature, Darvishi et al. [13] identified eight methods for solving the grey linear programming problems: • • • • • • • •

Best-worst method Confidence degree solution Whitening parameters model Prediction type Positioned solution Genetic algorithm Covered solution Multi-objective model.

In the following, we will briefly discuss the main idea behind each of the approach, highlighting some information related to the expected type of solutions that can be achieved when using a certain approach and the conditions needed to be fulfilled for applying each method. The main idea behind the Best–worst method resides on the division of the grey linear programming problem into two sub-models which are solved sequentially [13]. An example in this category can constitute the paper of Huang et al. [14], who develop a model based on chance-constrains for solving the grey linear programming problem and uses it for solid waste management planning. Recently, Mahmoudi et al. [15] proposed a grey best–worst method for multiple experts multiple criteria decision making under uncertainty. The approach behind the Confidence degree solution model is based on fuzzy theory applied in an iterative manner. One can refer to the paper written by Yaguang [16]. One of the easiest models for solving the grey linear programming problems consists in the Whitening parameters model. Considering the models proposed in the scientific literature (e.g. Voskoglou [9] and Ardabili [17]), the main idea behind finding a solution is the whitenization of the grey parameters. Depending on the case, the solution can be a crisp value but, if needed, it can be transformed to a grey solution. One of the main advantages of this type of model is that it is easy to understand and to apply in a given situation. As for the drawbacks, it might happen that some of the obtained solutions to range outside of the feasible region. For this reason, it is indicated to check the feasibility of the obtained solutions prior to declare them feasible solutions. Another class in the grey liner programming problem is represented by the Prediction type model. This model applies to a special class of grey linear programming problems, namely the ones in which the current and/or future value of a parameter

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is unknown, but the historical values related to that variable are known. In order to determine the needed value of the parameter, a grey prediction model can be used— in most of the cases the model used being GM(1,1). Examples on how this approach can be used in theoretical and practical situations have been provided by Liu and Lin [1], Yuan et al. [18], Darvishi and Babaei [12], etc. In the case of the Positioned solution method, the grey linear programming problem is seen as a set of ordinary linear programming problems. If one would have an infinite power of calculus, one could have considered all these problems and solve them as basic linear programming problems. As this action is not feasible in the case of a complex grey linear programming problem, one can be interested in solving few linear programming problems and determine a solution based on the obtained results. Therefore, the linear programming problem with grey parameters can be of three main types: ideal, critical or θ -positioned programming. Liu et al. [19] and Liu and Lin [1] discussed these concepts and presented means to address them. A similar approach is used by Huang and Moore [20] for the practical case of water resource planning and decision making under uncertainty. Another practical application refer to animal diet, presented by Salookolaei et al. [21]. Genetic algorithm has been used to solve the grey linear programming problem by Bi et al. [22]. Referring to the work of Bi et al. [22], Darvishi et al. [13] stated that the result clearly proofs the efficiency of the proposed approach. The Covered solution approach to the grey linear programming problem resides on the use of Simplex algorithm and on the covered operation of the inverse grey matrix and it has been proposed by Li et al. [23]. One of the main advantages of the proposed model consists in the fact that the proposed covered solution is suitable for any norm. On the other hand, the main disadvantage is due to the high amount of calculus and the failure to provide a solution in some of the cases. The Multi-objective model addresses the grey linear programming problems by transforming their objective function and constraints through the means of the grey numbers arithmetic operation in order to solve them. Depending on the type of grey numbers involved in the objective function, namely the existence of positive bounds, the existence of negative bounds, the presence of grey numbers that contain different sign and zero in their intervals, a series of steps are needed to be taken. One of the models belonging to this class have been developed by Hajiagha et al. [24]. Later on, Mahmoudi et al. [25] criticized the shortcoming in the model proposed by Hajiagha et al. [24], while in Mahmoudi et al. [26] the authors proposed a novel model for addressing the multi-objective linear programming with interval coefficients through a grey-fuzzy approach. Other approaches of grey in the area of programming problems refers, but are not limited to: grey integer programming [27], sensitivity analysis of grey linear programming problems [10], applying duality results to solve the linear programming problem with grey parameters [28, 29], primal simplex algorithm for solving linear programming problems with grey cost coefficients [30], a dual simplex method for grey linear programming problems based on duality results [31], using the grey linear programming algorithm for calculating and distributing the benefits of the players

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in a n-person grey game [32], grey multi-objective linear model to determine the critical path for a project [33], etc. Nevertheless, a good starting point for a complete overview on the papers written in the area of grey linear programming problem is made by Darvishi et al. [13] and we advise the reader to consider this paper if interested in a general coverage of the grey linear programming issue.

Theoretical Approach to Grey Linear Programming A selection of the approaches to grey linear programming proposed in the scientific literature are presented in the following on the purpose of providing to the reader a set of examples solvable using the grey systems theory. The theoretical approach presented in this chapter belong to the cited authors, while the numerical examples for illustrating the practical approaches are developed within the framework of this chapter. As only a selection of theoretical approaches is provided, we invite the reader to consider more approaches by going through the papers dedicated to this topic. In this endeavor, we suggest the review paper of Darvishi et al. [13].

Whitening Parameters Models A series of whitening parameter models have been used in the scientific literature for studying the grey linear programming problem [13, 20]. According to these models, a needed step in solving the grey linear programming problem is the one represented by the whitenization of the implied grey numbers. Let us see in the following the rules accompanying the models proposed by Voskoglou [9] and Ardabili et al. [17].

Voskoglou’s Whitening Parameters Model The steps proposed by Voskoglou [9] for solving the grey linear programming problem are: Step 1: Whitenization of the involved grey numbers—depending on the type of the grey linear programming problem (e.g. with grey cost values, with grey right-hand values, with grey technological coefficients, or a combination of these situations), ˜ r value) through the whitenization the grey numbers are whitenized (finding the ⊗ method as presented in [1] and briefly synthetized in this chapter; Step 2: Determining the solution of the linear programming problem with whitenizated values—the resulting linear programming problem after the whitenization process is a standard linear programming problem, which can be solved by following

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the classical theory offered by the operations research theory. To this extent, graphical method presented previously in this chapter can be used in the case of a linear programming problem with two variables, or, in the case of multiple variables, Simplex method can be successfully applied. Step 3: Conversion of the optimal solution to grey variables by applying the desired rank of greyness (not mandatory)—as the Voskoglou [9] mentioned, this is not a mandatory step, but it might be useful in the case in which the decision maker seeks for a grey result rather than a crisp result. This might happen situations.  in vague  The rank of greyness (noted RoG), for a grey number, r (⊗) ∈ r , r , is determined as [9]: RoG r (⊗) = r − r Note that in the case of a white number, the value of the RoG is zero, while in the case of a black number the value of RoG is +∞ [9]. For better understanding the use of rank of greyness and the whitenization value of a grey number, let us consider a simple example in which the following information is known: RoG r(⊗) = 6 ˜ r = 10 ⊗ α = 0.5 And we want to find the grey number, r (⊗), then we can use the underlying equations: RoG r (⊗) = r − r ˜ r = αr + (1 − α)r ⊗ From: RoG r (⊗) = r − r and RoG r(⊗) = 6 it results that: r − r = 6. ˜ r = 10 and α = 0.5, we have that: r − r = 20. ˜ r = αr + (1 − α)r , ⊗ From: ⊗ Next, from: r − r = 6 and r − r = 20, we have that: r = 13 and r = 7. Therefore: r (⊗) ∈ [7, 13]. Now, let us consider a grey linear programming problem and let us apply the Voskoglou’s Whitening Parameters Model for solving it, following the steps presented above:

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max ⊗ f (x1 , x2 ) = ⊗ [9.6, 10.4] ⊗ x1 + ⊗[6.9, 7.1] ⊗ x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ ⊗[1.8, 2.2] ⊗ x1 + ⊗[0.7, 1.3] ⊗ x2 ≤ ⊗[7.7, 8.3] ⎪ ⎪ ⎨ ⊗[1.7, 2.3] ⊗ x1 + ⊗[4.6, 5.4] ⊗ x2 ≤ ⊗[19.7, 20.3] ⎪ ⊗[2.9, 3.1] ⊗ x1 + ⊗[0.6, 1.4] ⊗ x2 ≥ ⊗[2.8, 3.2] ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 Step 1: Whitenization of the involved grey numbers By applying “equal weight mean whitenization” for all the known grey numbers in the grey linear programming problem, the following linear programming problem is obtained: max f (x1 , x2 ) = 10x1 + 7x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 2x1 + x2 ≤ 8 ⎪ ⎪ ⎨ 2x1 + 5x2 ≤ 20 ⎪ 3x 1 + x2 ≥ 3 ⎪ ⎪ ⎪ ⎪ x 1 ≥0 ⎪ ⎪ ⎩ x2 ≥ 0 Step 2: Determining the solution of the linear programming problem with whitenizated values As the linear programming problem consists of two variables to be determined, we can easily solve it using the graphical method and corner’s theorem: Therefore, we have the lines associated with the equations and their intersection points with the axes: L 1 : 2x1 + x2 = 8 The intersection of L 1 line with the axes is given by the points: x1 = 0 ⇒ x2 = 8

A(0, 8)

x2 = 0 ⇒ x1 = 4

B(4, 0)

L 2 : 2x1 + 5x2 = 20 The intersection of L 2 line with the axes is given by the points: x1 = 0 ⇒ x2 = 4

D(0, 4)

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x2 = 0 ⇒ x1 = 10

C(10, 0)

L 3 : 3x1 + x2 = 3 The intersection of L 3 line with the axes is given by the points: x1 = 0 ⇒ x2 = 3

F(0, 3)

x2 = 0 ⇒ x1 = 1

G(1, 0)

And the intersection point between L 1 and L 2 : point E(2.5, 3). The selected half-planes obtained by considering all the constrains and the sign conditions are presented using colors in Fig. 6.9. The feasible region is represented by the DEBGF polygon—marked with a chessboard pattern in Fig. 6.9. The optimal solution can be found among the corner points: f (D) = 10x1 (D) + 3x2 (D) = 10∗0 + 7∗4 = 28 u f (E) = 10x1 (E) + 3x2 (E) = 10∗2.5 + 7∗3 = 46 u f (B) = 10x1 (B) + 3x2 (B) = 10∗4 + 7∗0 = 40 u f (G) = 10x1 (G) + 3x2 (G) = 10∗1 + 7∗0 = 10 u f (F) = 10x1 (F) + 3x2 (F) = 10∗0 + 7∗3 = 21 u

Fp

Fig. 6.9 The feasible region—DEBGF polygon

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We have to choose the maximum value. Therefore:   f ∗ = f x ∗ = 46 u x1∗ = 2.5 u x2∗ = 3 u Step 3: Conversion of the optimal solution to grey variables by applying the desired rank of greyness We will consider the same value of the RoG is determined as in Voskoglou [9], namely RoG = 1. As a result, the grey values associated with the unknown variables are: x1 ∈ [2, 3] x2 ∈ [2.5, 3.5] Considering the extreme values of the two grey variables, namely the points K (2, 2.5), L(2, 3.5), M(3, 2.5) and N (3, 3.5), it can be observed from the Fig. 6.10 that some of them are positioned (as expected) outside the feasible solutions region of the linear programming problem. Let us check to which extent the four points mentioned above can be feasible solutions for the grey linear programming problem, by replacing them in the constraints of the problem (we do not need to check for the sign conditions as it can be easily observed that they are fulfilled for all the four points): • Point K (2, 2.5) : Based on the coordinates of point K , the three constrains become equivalent to: ⎧ ⎨ ⊗[1.8, 2.2]∗2 + ⊗[0.7, 1.3]∗2.5 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗2 + ⊗[4.6, 5.4]∗2.5 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗2 + ⊗[0.6, 1.4]∗2.5 ≥ ⊗ [2.8, 3.2] Therefore: ⎧ ⎨ ⊗[3.6, 4.4] + ⊗[1.75, 3.25] ≤ ⊗[7.7, 8.3] ⊗[3.4, 4.6] + ⊗[11.5, 13.5] ≤ ⊗[19.7, 20.3] ⎩ ⊗[5.8, 6.2] + ⊗[1.5, 3.5] ≥ ⊗ [2.8, 3.2] The three comparisons needed are as follows: ⎧ ⎨ ⊗[5.35, 7.65] ≤ ⊗[7.7, 8.3] ⊗[14.9, 18.1] ≤ ⊗[19.7, 20.3] ⎩ ⊗[7.3, 9.7] ≥ ⊗ [2.8, 3.2]

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Fig. 6.10 Possible extreme values for the variables when RoG equals 1

The comparison among the grey numbers can be done by following the theoretical approach presented in Chap. 2. As an alternative, the following two links can be used for making the comparison: – For smaller or equal comparison: https://github.com/liviucotfas/grey-systemsbook/blob/main/grey-numbers-comaprison-smaller-or-equal.ipynb – For greater or equal comparison: https://github.com/liviucotfas/grey-systemsbook/blob/main/grey-numbers-comparison-greater.ipynb ⎧ ⎨ p(⊗[5.35, 7.65] ≤ ⊗[7.7, 8.3]) = 1 p(⊗[14.9, 18.1] ≤ ⊗[19.7, 20.3]) = 1 ⎩ p(⊗[7.3, 9.7] ≥ ⊗ [2.8, 3.2]) = 1 As all the calculated probabilities are equal to 1, it can be stated that the solution determined starting from the optimal solution of the linear programming problem and applying the RoG of 1 is a feasible solution. We will check in the same manner the remainder of the points. • Point L(2, 3.5): By replacing the coordinated of point L within the constraints, we have:

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⎧ ⎨ ⊗[1.8, 2.2]∗2 + ⊗[0.7, 1.3]∗3.5 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗2 + ⊗[4.6, 5.4]∗3.5 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗2 + ⊗[0.6, 1.4]∗3.5 ≥ ⊗ [2.8, 3.2] Which conducts to: ⎧ ⎨ ⊗[3.6, 4.4] + ⊗[2.45, 4.55] ≤ ⊗[7.7, 8.3] ⊗[3.4, 4.6] + ⊗[16.1, 18.9] ≤ ⊗[19.7, 20.3] ⎩ ⊗[5.8, 6.2] + ⊗[2.1, 4.9] ≥ ⊗ [2.8, 3.2] The comparison of the following grey numbers should be performed: ⎧ ⎨ ⊗[6.05, 8.95] ≤ ⊗[7.7, 8.3] ⊗[19.5, 23.5] ≤ ⊗[19.7, 20.3] ⎩ ⊗[7.9, 11.1] ≥ ⊗ [2.8, 3.2] The probability associated with each of the constrains is: ⎧ ⎨ p(⊗[6.05, 8.95] ≤ ⊗[7.7, 8.3]) = 0.672 p(⊗[19.5, 23.5] ≤ ⊗[19.7, 20.3]) = 0.125 ⎩ p(⊗[7.9, 11.1] ≥ ⊗ [2.8, 3.2]) = 1 Given the probability obtained for each of the three comparisons, it can be observed that there might be cases in which, based on the values of the grey numbers associated with the first two constraints, the solution L(2, 3.5) will be outside the feasible region. Therefore, we can state that L(2, 3.5) is a feasible solution only with a probability of 0.084 (the value of the cumulative probability has been obtained by multiplying the individual probabilities as the three constraints have been considered independent). As a result, we are expecting more that the solution is unfeasible than to be feasible. • Point M(3, 2.5): We make the calculus by considering the coordinate of point M : ⎧ ⎨ ⊗[1.8, 2.2]∗3 + ⊗[0.7, 1.3]∗2.5 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗3 + ⊗[4.6, 5.4]∗2.5 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗3 + ⊗[0.6, 1.4]∗2.5 ≥ ⊗ [2.8, 3.2] Thus: ⎧ ⎨ ⊗[5.4, 6.6] + ⊗[1.75, 3.25] ≤ ⊗[7.7, 8.3] ⊗[5.1, 6.9] + ⊗[11.5, 13.5] ≤ ⊗[19.7, 20.3] ⎩ ⊗[8.7, 9.3] + ⊗[1.5, 3.5] ≥ ⊗[2.8, 3.2] The comparison of grey numbers:

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⎧ ⎨ ⊗[7.15, 9.85] ≤ ⊗[7.7, 8.3] ⊗[16.6, 20.4] ≤ ⊗[19.7, 20.3] ⎩ ⊗[10.2, 12.8] ≥ ⊗ [2.8, 3.2] The result of the comparison for the grey numbers: ⎧ ⎨ p(⊗[7.15, 9.85] ≤ ⊗[7.7, 8.3]) = 0.314 p(⊗[16.6, 20.4] ≤ ⊗[19.7, 20.3]) = 0.984 ⎩ p(⊗[10.2, 12.8] ≥ ⊗ [2.8, 3.2]) = 1 The cumulative probability of the three constraints is 0.309. Even in this case, the obtained cumulative probability is quite small, which questions the feasibility of point M(3, 2.5). Comparing the result in terms of probability with point L(2, 3.5), one can state that we are expecting with a higher probability that point M(3, 2.5) is a feasible one than point L(2, 3.5). • Point N (3, 3.5): For point N the constraints become: ⎧ ⎨ ⊗[1.8, 2.2]∗3 + ⊗[0.7, 1.3]∗3.5 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗3 + ⊗[4.6, 5.4]∗3.5 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗3 + ⊗[0.6, 1.4]∗3.5 ≥ ⊗ [2.8, 3.2] Equivalent to: ⎧ ⎨ ⊗[5.4, 6.6] + ⊗[2.45, 4.55] ≤ ⊗[7.7, 8.3] ⊗[5.1, 6.9] + ⊗[16.1, 18.9] ≤ ⊗[19.7, 20.3] ⎩ ⊗[8.7, 9.3] + ⊗[2.1, 4.9] ≥ ⊗[2.8, 3.2] Therefore, we have to compare: ⎧ ⎨ ⊗[7.85, 11.15] ≤ ⊗[7.7, 8.3] ⊗[21.2, 25.8] ≤ ⊗[19.7, 20.3] ⎩ ⊗[10.8, 14.2] ≥ ⊗[2.8, 3.2] The probability associated with each of the constrains is: ⎧ ⎨ p(⊗[7.85, 11.15] ≤ ⊗[7.7, 8.3]) = 0.051 p(⊗[21.2, 25.8] ≤ ⊗[19.7, 20.3]) = 0 ⎩ p(⊗[10.8, 14.2] ≥ ⊗[2.8, 3.2]) = 1 In this case, the value of the cumulative probability is 0 as the value of the probability p(⊗[21.2, 25.8] ≤ ⊗[19.7, 20.3]) is 0. As a result, one can state for sure that the point N (3, 3.5) is not a feasible solution.

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As a result, the presence of a vague expression of the solution does not guarantee that all the values are within the feasible region—this situation is pointed out even by Voskoglou [9], representing one of the main drawbacks of the proposed approach. The situation can be solved partly by checking each time the solution by considering the imposed constraints and sign conditions (as presented above). Darvishi et al. [13] pointed out, the choice of a smaller value for the RoG might conduct to a more credible solution. With all these, even by proceeding on this manner, the occurrence of the situations as the one presented above is not completely excluded. Saffar Ardabili et al.’s Whitening Parameters Model The steps proposed by Ardabili et al. [17] for solving the grey linear programming problem are: Step 1: Whitenization of the involved grey numbers by considering the center of the grey numbers—the center of a grey number, r (⊗) ∈ [r , r ], noted cr (⊗) , is defined as: cr (⊗) =

r +r 2

Step 2: Determining the solution of the linear programming problem with whitenizated values in order to obtain the center of the grey decision variable—the resulting standard linear programming problem can be solved by either the graphical method (in the case of a linear programming problem with two variables) or by using the Simplex method (in the case of multiple variables). The values for the cx(⊗) and c f (⊗) are determined. Step 3: Writing the linear programming problem with grey information by using the width of the grey numbers in the initial grey linear programming problem—a new linear programming problem is set up by considering the width of the grey numbers from the initial grey linear programming problem. The width of a grey number, r (⊗) ∈ [r , r ], noted wr(⊗) , is defined as: wr(⊗) =

r −r 2

Step 4: Solving the linear programming problem containing the width values in order to obtain the width of the grey decision variable—the resulting linear programming problem is solved either by using the graphical method or the Simplex algorithm and the solution is listed accordingly. The values for wx(⊗) and w f (⊗) are determined. Step 5: Determining the solution of the grey linear programming problem by applying the following transformations:     x(⊗) ∈ x, x = cx(⊗) − wx(⊗) , cx(⊗) + wx(⊗)

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    f ∈ f , f = c f (⊗) − w f (⊗) , c f (⊗) + w f (⊗) Next, we will use the same grey linear programming problem as in the case in which we have applied the Voskoglou’s whitening parameters model on which we will apply Saffar Ardabili et al.’s whitening parameters model. The grey linear programming problem reads as: max ⊗ f (x1 , x2 ) = ⊗[9.6, 10.4] ⊗ x1 + ⊗[6.9, 7.1] ⊗ x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊗[1.8, 2.2] ⊗ x1 + ⊗[0.7, 1.3] ⊗ x2 ≤ ⊗[7.7, 8.3] ⎨ ⊗[1.7, 2.3] ⊗ x1 + ⊗[4.6, 5.4] ⊗ x2 ≤ ⊗[19.7, 20.3] ⎪ ⊗[2.9, 3.1] ⊗ x1 + ⊗[0.6, 1.4] ⊗ x2 ≥ ⊗[2.8, 3.2] ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 Step 1: Whitenization of the involved grey numbers by considering the center of the grey numbers By calculating the center of the grey numbers and by including it in the grey linear programming problem instead of the grey numbers, the same linear programming problem as in the case of Voskoglou’s whitening parameters model is obtained: max f (x1 , x2 ) = 10x1 + 7x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 2x1 + x2 ≤ 8 ⎪ ⎪ ⎨ 2x1 + 5x2 ≤ 20 ⎪ 3x 1 + x2 ≥ 3 ⎪ ⎪ ⎪ ⎪ x 1 ≥0 ⎪ ⎪ ⎩ x2 ≥ 0 Step 2: Determining the solution of the linear programming problem with whitenizated values in order to obtain the center of the grey decision variable This step is similar to Step 2 in Voskoglou’s whitening parameters model. Considering the notations in Saffar Ardabili et al.’s whitening parameters model, one determined that: cx1 (⊗) = 2.5 u cx2 (⊗) = 3 u

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c f (⊗) = 46 u Step 3: Writing the linear programming problem with grey information by using the width of the grey numbers in the initial grey linear programming problem By calculating the width of the grey numbers and by including it in the grey linear programming problem instead of the grey numbers, the following linear programming problem is obtained: max f (x1 , x2 ) = 0.4x1 + 0.2x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.2x1 + 0.3x2 ≤ 0.3 ⎨ 0.3x1 + 0.4x2 ≤ 0.3 ⎪ 0.1x 1 + 0.4x 2 ≥ 0.2 ⎪ ⎪ ⎪ ⎪ ≥ 0 x 1 ⎪ ⎪ ⎩ x2 ≥ 0 Step 4: Solving the linear programming problem containing the width values in order to obtain the width of the grey decision variable The lines associated with the equations and their intersection points with the axes are: L 1 : 2x1 + 3x2 = 3 The intersection of L 1 line with the axes is given by the points: x1 = 0 ⇒ x2 = 1 x2 = 0 ⇒ x1 = 1.5

A(0, 1) B(1.5, 0)

L 2 : 3x1 + 4x2 = 3 The intersection of L 2 line with the axes is given by the points: x1 = 0 ⇒ x2 = 0.75 C(0, 0.75) x2 = 0 ⇒ x1 = 1

D(1, 0)

L 3 : x1 + 4x2 = 2

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Fp

Fig. 6.11 The feasible region—CGE triangle

The intersection of L 3 line with the axes is given by the points: x1 = 0 ⇒ x2 = 0.5 x2 = 0 ⇒ x1 = 2

E(0, 0.5) F(2, 0)

And the intersection between L 2 and L 3 we find point G(0.5, 0.375), while at the intersection point between L 1 and L 3 we find point H (1.2, 0.2). All the points mentioned above, along with their corresponding lines and selected half-planes are represented in Fig. 6.11. The feasible region is represented by the C G E triangle marked with chessboard pattern in Fig. 6.11. The optimal solution can be found among the corner points of the feasible region: f (C) = 0.4x1 (C) + 0.2x2 (C) = 0.4∗0 + 0.2∗0.75 = 0.15 u f (G) = 0.4x1 (G) + 0.2x2 (G) = 0.4∗0.5 + 0.2∗0.375 = 0.275 u f (E) = 0.4x1 (E) + 0.2x2 (E) = 0.4∗0 + 0.2∗0.5 = 0.10 u We have to choose the maximum value. Therefore:   f ∗ = f x ∗ = 0.275 u x1∗ = 0.5 u x2∗ = 0.375 u The values for wx(⊗) and w f (⊗) are:

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wx1 (⊗) = 0.5 u wx2 (⊗) = 0.375 u w f (⊗) = 0.275 u

Step 5: Determining the solution of the grey linear programming problem by applying the following transformations     x1 (⊗) ∈ x1 , x1 = cx1 (⊗) − wx1 (⊗) , cx1 (⊗) + wx1 (⊗) = [2.5 − 0.5, 2.5 + 0.5]     x2 (⊗) ∈ x2 , x2 = cx2 (⊗) − wx2 (⊗) , cx2 (⊗) + wx2 (⊗) = [3 − 0.375, 3 + 0.375]   f ∈ f , f ]=[c f (⊗) − w f (⊗) , c f (⊗) + w f (⊗) ]=[46 − 0.275, 46 + 0.275 Thus, we have:   x1 (⊗) ∈ x1 , x1 = [2, 3]   x2 (⊗) ∈ x2 , x2 = [2.625, 3.375]   f ∈ f , f = [45.725, 46.275] Let us check even in this case the feasibility of the extreme points in the obtained solution. First, we have graphically identified them on a graphic containing the feasible region of the linear programming problem as presented in Fig. 6.12. The points are: P(2, 2.625), Q(2, 3.375), R(3, 2.625) and S(3, 3.375). It can be observed that we do not need to check for the sign conditions as it can be easily observed that they are fulfilled for all the four points. • Point P(2, 2.625) : Based on the coordinates of point P, the three constrains become equivalent to: ⎧ ⎨ ⊗[1.8, 2.2]∗2 + ⊗[0.7, 1.3]∗2.625 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗2 + ⊗[4.6, 5.4]∗2.625 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗2 + ⊗[0.6, 1.4]∗2.625 ≥ ⊗[2.8, 3.2] Therefore:

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Fig. 6.12 Possible values for the variables after applying Saffar Ardabili et al.’s whitening parameters model

⎧ ⎨ ⊗[3.6, 4.4] + ⊗[1.8375, 3.4125] ≤ ⊗[7.7, 8.3] ⊗[3.4, 4.6] + ⊗[12.075, 14.175] ≤ ⊗[19.7, 20.3] ⎩ ⊗[5.8, 6.2] + ⊗[1.575, 3.675] ≥ ⊗[2.8, 3.2] The three comparisons needed are as follows: ⎧ ⎨ ⊗[5.4375, 7.8125] ≤ ⊗[7.7, 8.3] ⊗[15.475, 18.775] ≤ ⊗[19.7, 20.3] ⎩ ⊗[7.375, 9.875] ≥ ⊗[2.8, 3.2] By comparing the grey numbers, we have that: ⎧ ⎨ p(⊗[5.4375, 7.8125] ≤ ⊗[7.7, 8.3]) = 0.995 p(⊗[15.475, 18.775] ≤ ⊗[19.7, 20.3]) = 1 ⎩ p(⊗[7.375, 9.875] ≥ ⊗ [2.8, 3.2]) = 1

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In this case, the cumulated probability is 0.995, so we expect that point P(2, 2.625) to be in the feasible region, even though we cannot make this affirmation with certainty (it is not a strong affirmation). • Point Q(2, 3.375): By replacing the coordinated of point L within the constraints, we have: ⎧ ⎨ ⊗[1.8, 2.2]∗2 + ⊗[0.7, 1.3]∗3.375 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗2 + ⊗[4.6, 5.4]∗3.375 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗2 + ⊗[0.6, 1.4]∗3.375 ≥ ⊗[2.8, 3.2] Which conducts to: ⎧ ⎨ ⊗[3.6, 4.4] + ⊗[2.3625, 4.8375] ≤ ⊗[7.7, 8.3] ⊗[3.4, 4.6] + ⊗[15.525, 18.225] ≤ ⊗[19.7, 20.3] ⎩ ⊗[5.8, 6.2] + ⊗[2.025, 4.725] ≥ ⊗[2.8, 3.2] The comparison of the following grey numbers should be performed: ⎧ ⎨ ⊗[5.9625, 8.7875] ≤ ⊗[7.7, 8.3] ⊗[18.925, 22.825] ≤ ⊗[19.7, 20.3] ⎩ ⊗[7.825, 10.925] ≥ ⊗[2.8, 3.2] The probability associated with each of the constrains is: ⎧ ⎨ p(⊗[5.9625, 8.7875] ≤ ⊗[7.7, 8.3]) = 0.721 p(⊗[18.925, 22.825] ≤ ⊗[19.7, 20.3]) = 0.275 ⎩ p(⊗[7.825, 10.925] ≥ ⊗[2.8, 3.2]) = 1 The cumulative probability is equal to 0.198 in this case. As a result, we are expecting more that the solution is unfeasible than feasible. • Point R(3, 2.625): We make the calculus by considering the coordinate of point M : ⎧ ⎨ ⊗[1.8, 2.2]∗3 + ⊗[0.7, 1.3]∗2.625 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗3 + ⊗[4.6, 5.4]∗2.625 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗3 + ⊗[0.6, 1.4]∗2.625 ≥ ⊗[2.8, 3.2] Thus: ⎧ ⎨ ⊗[5.4, 6.6] + ⊗[1.8375, 3.4125] ≤ ⊗[7.7, 8.3] ⊗[5.1, 6.9] + ⊗[12.075, 14.175] ≤ ⊗[19.7, 20.3] ⎩ ⊗[8.7, 9.3] + ⊗[1.575, 3.675] ≥ ⊗[2.8, 3.2] The comparison of grey numbers:

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⎧ ⎨ ⊗[7.2375, 10.0125] ≤ ⊗[7.7, 8.3] ⊗[17.175, 21.075] ≤ ⊗[19.7, 20.3] ⎩ ⊗[10.275, 12.975] ≥ ⊗[2.8, 3.2] The result of the comparison for the grey numbers: ⎧ ⎨ p(⊗[7.2375, 10.0125] ≤ ⊗[7.7, 8.3]) = 0.274 p(⊗[17.175, 21.075] ≤ ⊗[19.7, 20.3]) = 0.724 ⎩ p(⊗[10.275, 12.975] ≥ ⊗[2.8, 3.2]) = 1 The cumulative probability of the three constraints is 0.198, similar to the one obtained for the point Q(2, 3.375). Even in this case, given the low result for the cumulative probability, we are expecting more that the solution is unfeasible than feasible. • Point S(3, 3.375): For point N the constraints become: ⎧ ⎨ ⊗[1.8, 2.2]∗3 + ⊗[0.7, 1.3]∗3.375 ≤ ⊗[7.7, 8.3] ⊗[1.7, 2.3]∗3 + ⊗[4.6, 5.4]∗3.375 ≤ ⊗[19.7, 20.3] ⎩ ⊗[2.9, 3.1]∗3 + ⊗[0.6, 1.4]∗3.375 ≥ ⊗[2.8, 3.2] Equivalent to: ⎧ ⎨ ⊗[5.4, 6.6] + ⊗[2.3625, 4.8375] ≤ ⊗[7.7, 8.3] ⊗[5.1, 6.9] + ⊗[15.525, 18.225] ≤ ⊗[19.7, 20.3] ⎩ ⊗[8.7, 9.3] + ⊗[2.025, 4.725] ≥ ⊗[2.8, 3.2] Therefore, we have to compare: ⎧ ⎨ ⊗[7.7635, 11.4375] ≤ ⊗[7.7, 8.3] ⊗[20.625, 25.125] ≤ ⊗[19.7, 20.3] ⎩ ⊗[10.725, 14.025] ≥ ⊗[2.8, 3.2] The probability associated with each of the constrains is: ⎧ ⎨ p(⊗[7.7635, 11.4375] ≤ ⊗[7.7, 8.3]) = 0.065 p(⊗[20.625, 25.125] ≤ ⊗[19.7, 20.3]) = 0 ⎩ p(⊗[10.725, 14.025] ≥ ⊗[2.8, 3.2]) = 1 Even in this case we have obtained a value for one of the probabilities equal to 0, which makes that the cumulative probability to be 0 as well. As a result, it can be stated that for sure that the point S(3, 3.375) is not a feasible solution. Comparing the solutions obtained through the two whitening parameters models and the associated probabilities for the four extreme points, it can be

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stated that the credibility of the vague expression of the grey linear programming solution should be checked with respect to the constraints before accepting it as a feasible solution.

Liu and Lin’s Prediction Type Model The grey linear programming of prediction type implies the existence of the historical data and their evolution which can be used for forecasting the actual and future data. In this section, we will present the grey linear programming of prediction type proposed by Liu and Lin [1]. First, some hypothesis about the form of the grey linear programming model of prediction type should be stated. These hypotheses refer to the existence of one or more right-hand values which are not known at the moment at which the grey linear programming model is aimed to be solved, but some historical data with regard to these variables is known. We will preserve the notations made above for the grey linear programming model. Therefore, the maximization grey linear programming problem has the following form: max ⊗ f (x1 , x2 , . . . , xn ) = c(⊗)T x ⎧ ⎨ Subject to: A(⊗)x ≤ bˆ ⎩ x≥0 where, different from the elements stated above, in this case, the right-hand vector b contains the predicted values for the current year—these values are not known at the moment in which the grey linear programming is aimed to be solved, but the historical values for are known for a given unit of time, noted with s [1]: 

bi (⊗) = (bi (1), bi (2), . . . , bi (s)) The grey linear programming problem is called in this case “a linear programming problem of grey prediction type” [1]. The steps needed to solve the problem are as follows: Step 1: Whitenization of the involved grey numbers—the grey numbers provided in the grey linear programming problem are whitenized. To this, one can use the mean whitenization process. Step 2: Use the GM(1,1) model for determining the needed values for the righthand vector based on historical data—during this step, where historical data is provided, the values for the interested moments of time (e.g. s + 1, s + 2, . . . , s + k, etc.) are determined and their values are included in the linear programming problem of grey prediction type.

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Step 3: Determine the solution of the linear programming problem of grey prediction type by solving it according to the known methods for solving the linear programming problem. Let us consider a grey linear programming problem similar to the one considered in the previous examples: max ⊗ f (x1 , x2 ) = ⊗ [9.6, 10.4] ⊗ x1 + ⊗[6.9, 7.1] ⊗ x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ˆ ⎪ ⎪ ⎪ ⊗[1.8, 2.2] ⊗ x1 + ⊗[0.7, 1.3] ⊗ x2 ≤ b1 (5 + k) ⎨ ⊗[1.7, 2.3] ⊗ x1 + ⊗[4.6, 5.4] ⊗ x2 ≤ ⊗[19.7, 20.3] ⎪ ⊗[2.9, 3.1] ⊗ x1 + ⊗[0.6, 1.4] ⊗ x2 ≥ ⊗[2.8, 3.2] ⎪ ⎪ ⎪ ⎪ x ≥ 0 ⎪ 1 ⎪ ⎩ x2 ≥ 0 with bˆ1 (5 + k), with k = 1, . . . , 5 is unknown and can be determined based on the historical data: b1 (1) = 7.2; b1 (2) = 7.45; b1 (3) = 7.6; b1 (4) = 7.8; b1 (5) = 7.85 Now, let us apply Liu and Lin’s prediction type model: Step 1: Whitenization of the involved grey numbers As a result of the mean whitenization process the following linear programming problem is determined: max f (x1 , x2 ) = 10x1 + 7x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 2x1 + x2 ≤ bˆ1 (5 + k) ⎪ ⎪ ⎨ 2x1 + 5x2 ≤ 20 ⎪ 3x1 + x2 ≥ 3 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 Step 2: Use the GM(1,1) model for determining the needed values for the right-hand vector based on historical data The GM(1,1) model as presented in scientific literature will be applied. If interested, the reader can consider the work of Liu et al. [34] for the needed theoretical approach related to the GM(1,1). For determining the values of bˆ1 (5 + k) for k = 1, . . . , 5, one can use the Grey System Theory Modeling Software 6.0 as presented in Liu and Lin [35]. The solution screen is presented in Fig. 6.13, having as input variables the values of b1 (1) = 7.2; b1 (2) = 7.45; b1 (3) = 7.6; b1 (4) = 7.8; b1 (5) = 7.85.

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Fig. 6.13 GM(1,1) using Grey System Theory Modeling Software 6.0

After running the GM(1,1) model with Grey System Theory Modeling Software 6.0, for which it has been set a precision of results of 1, and the steps needed for forecast equal to 5, the results are listed in the bottom-right side of the screen. From the Grey System Theory Modeling Software 6.0, one can read the following predicted values: b1 (6) = 8.0; b1 (7) = 8.2; b1 (8) = 8.3; b1 (9) = 8.5; b1 (10) = 8.6. The steps made by the Grey System Theory Modeling Software 6.0 in order to determine these values are listed in the left-side of the software’s screen and presented in Table 6.1. Step 3: Determine the solution of the linear programming problem of grey prediction type by solving it according to the known methods for solving the linear programming problem The linear programming problem, (LPP1 ), for b1 (6) = 8.0 is written as:

240 Table 6.1 The steps used by Grey System Theory Modeling Software 6.0 for forecasting the b1 values using GM(1,1) model

6 Grey Systems Theory Approach to Linear Programming Steps – –––––––––––––––––Start–––––––––––––––––– (1) Initialization of raw sequence 7.2,7.45,7.6,7.8,7.85 (2) 1-AGO of sequence 7.2,14.7,22.3,30.1,37.9, (3) Proximate average generation of 1-AGO 11.0,18.5,26.2,34.0 (4) Compute development coefficient a and grey action quantity b a = 0.0 b = 7.3 (5) Compute simulated values 7.2, 7.57.67.77.9 (6) Compute residual errors 0.0, 0.0, 0.0, 0.1, 0.0, (7) Compute relative errors 0.0%, 0.2%, 0.0%, 0.7%, 0.4%, (8) Compute the average relative error 0.4%, (9) Compute the sum of squares of residual error 0.0 – –––––––––––––––––End––––––––––––––––––

max f (x1 , x2 ) = 10x1 + 7x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 2x1 + x2 ≤ 8 ⎪ ⎪ ⎨ 2x1 + 5x2 ≤ 20 (LPP1 ) ⎪ 3x 1 + x2 ≥ 3 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 As it can be observed, we have previously solved this problem, so in the following we are only providing the results for the optimal solution:   ∗ f LPP = f x ∗ = 46 u 1 x1∗ = 2.5 u x2∗ = 3 u For the second predicted value, b1 (7) = 8.2, the linear programming problem (LPP2 ) is written as:

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max f (x1 , x2 ) = 10x1 + 7x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 2x1 + x2 ≤ 8.2 ⎪ ⎪ ⎨ 2x1 + 5x2 ≤ 20 (LPP2 ) ⎪ 3x1 + x2 ≥ 3 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 Let us solve it by using the graphical approach. As only the first constraint has been changed (as a result of the change in the value of the right-hand value), we will change only the line associated with this constraint. As a result, the equation of line L 1 has the following form: L 1 : 2x1 + x2 = 8.2 The intersection of L 1 line with the axes is given by the points: x1 = 0 ⇒ x2 = 8.2

A (0, 8.2)

x2 = 0 ⇒ x1 = 4.1

B (4.1, 0)

The feasible region after including this change is as in Fig. 6.14. The intersection between L 1 and L 2 is point H (2.625, 2.95).

Fp

Fig. 6.14 The linear programming feasible region, when b1 (7) = 8.2

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The optimal solution for the linear programming problem (LPP2 ) is determined using the corner’s method: f (D)  +  3x2 (D)  =  10*0 + 7*4 = 28 u  = 10x1 (D) f  H  = 10x1 H  + 3x2 H = 10*2.625 + 7*2.95 = 46.9 u f B = 10x1 B + 3x2 B = 10*4.1 + 7*0 = 41 u f (G) = 10x1 (G) + 3x2 (G) = 10*1 + 7*0 = 10 u f (F) = 10x1 (F) + 3x2 (F) = 10*0 + 7*3 = 21 u Therefore:   ∗ f LPP = f x ∗ = 46.9 u 2 x1∗ = 2.625 u x2∗ = 2.95 u As the value of b1 (7) = 8.2 has been greater than the value of b1 (6) = 8.0, it was expected that the value of the objective function to be greater in the case of the linear programming problem (LPP2 ) than in the case of the linear programming problem (LPP1 ). For determining the optimal solution of the remainder of linear programming problems for which b1 (8) = 8.3; b1 (9) = 8.5; b1 (10) = 8.6, we will proceed in the same manner. We invite the reader to determine the optimal value of the three remaining linear programming problems and to compare the results with the ones obtained for the case of (LPP1 ) and (LPP2 ). As expected, the optimal value of the function should be greater to the one obtained for (LPP2 ).

Lin and Liu’s Positioned Solution Model Liu and Lin [1] presented their approach to the linear programming with grey parameters problem by considering it as being composed of several problems of linear programming. To this extent, the authors defined several types of grey linear programming models such as the ideal, the critical and the positioned model of the grey programming with grey parameters. With regard to the solutions of these problems, Darvishi et al. [13] stated that in this approach, one can obtain satisfactory solutions which can be quickly implemented in practical applications rather than the optimal solutions. Before presenting the different types of linear programming with grey parameters problems as described by Liu and Lin [1], several notations should be made as discussed in the following. Let us consider the positioned coefficients of the objective function, denoted as ρ j ∈ [0, 1], , with j = 1, . . . , n; the positioned coefficients of the right-hand vector, denoted as βi ∈ [0, 1], with i = 1, . . . , m; and the positioned coefficients

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of the technological constraints, denoted as δi j ∈ [0, 1], with i = 1, . . . , m and j = 1, . . . , n, which can be used for determining the white values of the grey parameters within the grey linear programming problem [1, 13]:   ˜ cj = ρ j c j + 1 − ρ j c j ⊗   ˜ bi = β j bi + 1 − β j bi ⊗   ˜ ai j = δi j ai j + 1 − δi j ai j ⊗ The linear programming having the parameters the whitenized values of the initial grey parameters, written as:  T ˜ x max ⊗ f (x1 , x2 , . . . , xn ) = c ⊗ ⎧ ⎨ Subject   to:   ˜ x ≤b ⊗ ˜ A ⊗ ⎩ x≥0 it is called a positioned program of the linear programming problem [1, 13]. Furthermore, let us assume that [1, 13]: ρ j = ρ, βi = β, δi j = δ, ∀i = 1, 2, . . . , m, j = 1, 2, . . . , n Considering the values ρ, β, δ as defined above, the corresponding positioned program of the linear programming problem is called a (ρ, β, δ)-positioned programming, noted as LP (ρ, β, δ) [1, 13]. In connection with this program, the following versions of the LP (ρ, β, δ) can be identified [1, 13]: • ideal model of the linear programming with grey parameters—also noted as LP (1, 1, 0), for which the optimal value of the objective function is noted as max f ; • critical model of the linear programming with grey parameters—also noted as LP (0, 0, 1), for which the optimal value of the objective function is noted as max f ; • θ-positioned model of the linear programming with grey parameters—also noted as LP (θ, θ, θ ), for which the optimal value of the objective function is noted as max f θ . A particular case of this situation is the one in which the θ = 0.5, LP (0.5, 0.5, 0.5) being known as the mean whitenization programming. According to Liu and Lin [1], this type of program is the most typical form of linear programming problem with grey parameters. As Liu and Lin [1] noted, between the solutions of the three types of linear programming problems with grey parameters, the following relationship can be identified: max f ≤ max f (ρ, β, δ) ≤ max f

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Particularly, it can be stated that [1, 13]: max f ≤ max f θ ≤ max f Lastly, as Liu and Lin [1] noted, a satisfying degree of positioned programming LP (ρ, β, δ) can be determined through applying the following formula:  max f 1 1− μ(ρ, β, δ) = 2 max f (ρ, β, δ)

+

1 max f (ρ, β, δ) 2 max f

The value of formula is put in connection with a grey target [μ0 , 1] = D. If μ(ρ, β, δ) ∈ D, then the corresponding solution is called a pleased solution of the linear programming problem with grey parameters [1]. Now, let us solve the same problem we have considered for the other models presented in this chapter using Lin and Liu’s positioned solution model. The grey linear program to be solved is: max ⊗ f (x1 , x2 ) = ⊗[9.6, 10.4] ⊗ x1 + ⊗[6.9, 7.1] ⊗ x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊗[1.8, 2.2] ⊗ x1 + ⊗[0.7, 1.3] ⊗ x2 ≤ ⊗[7.7, 8.3] ⎨ ⊗[1.7, 2.3] ⊗ x1 + ⊗[4.6, 5.4] ⊗ x2 ≤ ⊗[19.7, 20.3] ⎪ ⊗[2.9, 3.1] ⊗ x1 + ⊗[0.6, 1.4] ⊗ x2 ≥ ⊗[2.8, 3.2] ⎪ ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎩ x2 ≥ 0 The ideal model of the linear programming with grey parameters has the following form: , x2 ) = 10.4x1 + 7.1x2 max f (x1⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 1.8x 1 + 0.7x 2 ≤ 8.3 ⎪ ⎪ ⎨ 1.7x1 + 4.6x2 ≤ 20.3 LP(1, 1, 0) ⎪ 2.9x1 + 0.6x2 ≥ 3.2 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 The form of the LP (1, 1, 0) has been determined by considering ρ = 1, β = 1, δ = 0. Thus, we have considered the upper values for the objective function coefficients and for the right-hand values, and the lower values for the matrix of the technological coefficients. Using GeoGebra [8] we have represented the lines corresponding to the constrains:

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Fp

Fig. 6.15 The LP (1, 1, 0) feasible region

L 1 : 1.8x1 + 0.7x2 = 8.3 L 2 : 1.7x1 + 4.6x2 = 20.3 L 3 : 2.9x1 + 0.6x2 = 3.2 and we have considered the half-planes which respects the constraints and the sign conditions. At the intersection of the five half-planes (three half-planes from the constrains and two half-planes given by the sign conditions), the feasible region, represented by the ABCD quadrilateral, has been obtained as presented in Fig. 6.15. The coordinates of the A, B, C, D points have been determined with a precision of two decimal places using GeoGebra [8]: A(3.38, 3.16), B(0.21, 4.34), C(4.61, 0), D(1.1, 0). The value of LP (1, 1, 0), noted as max f is determined by choosing the maximum value between the values obtained for the f function among the four corners of the feasible region: ¯f (A) = 10.4 x1 (A) + 7.1 x2 (A) = 10.4* 3.38 + 7.1*3.16 = 57.59 u ¯f (B) = 10.4 x1 (B) + 7.1 x2 (B) = 10.4*0.21 + 7.1*4.34 = 33 u ¯f (C) = 10.4 x1 (C) + 7.1 x2 (C) = 10.4* 4.61 + 7.1*0 = 47.94 u ¯f (D) = 10.4 x1 (D) + 7.1 x2 (D) = 10.4* 1.1 + 7.1* 0 = 11.44 u Therefore:

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max ¯f = 57.59 u obtained for point A(3.38, 3.16). The critical model of the linear programming with grey parameters has the following form: max f (x⎧ 1 , x 2 ) = 9.6x 1 + 6.9x 2 ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ + 1.3x2 ≤ 7.7 2.2x 1 ⎪ ⎪ ⎨ 2.3x1 + 5.4x2 ≤ 19.7 LP(0, 0, 1) ⎪ 3.1x1 + 1.4x2 ≥ 2.8 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 The form of the LP (0.0, 1) has been determined by considering ρ = 0, β = 0, δ = 1. In other words, we have considered the lower values for the objective function coefficients and for the right-hand values, and the upper values for the matrix of the technological coefficients. We have used again GeoGebra [8] for representing the lines corresponding to the constrains and for determining the coordinates of the feasible region: L 1 : 2.2x1 + 1.3x2 = 7.7 L 2 : 2.3x1 + 5.4x2 = 19.7 L 3 : 3.1x1 + 1.4x2 = 2.8 With respect to the half-planes determined by the constraints and sign conditions, the feasible region, A B C D E polygon, depicted in Fig. 6.16 has been determined. The corners of the feasible region have the following coordinates rounded to two digits: A (1.8, 2.88), B (3.5, 0), C (0.9, 0), D (0, 2), E (0, 3.65). The values for the optimum function of the LP (0.0, 1), noted max f , can be found among the corners of the feasible region. The value of the objective function in the corners is calculated and the maximum value is selected:       f A = 9.6 x1 A + 6.9 x2 A = 9.6*1.8 + 6.9*2.88 = 37.15 u −       f B = 9.6 x1 B + 6.9 x2 B = 9.6*3.5 + 6.9*0 = 33.6 u −       f C = 9.6 x1 C + 6.9 x2 C = 9.6*0.9 + 6.9*0 = 8.64 u −       f D = 9.6 x1 D + 6.9 x2 D = 9.6*0 + 6.9*2 = 13.8 −       f E = 9.6 x1 E + 6.9 x2 E = 9.6*0 + 6.9* 3.65 = 25.19 u −

Therefore:

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Fp

Fig. 6.16 The LP (0, 0, 1) feasible region

max f = 37.15 u −

obtained for point A (1.8, 2.88). Now, let us determine the optimal value for the θ -positioned model of the linear programming with grey parameters—also noted as LP (θ, θ, θ ). The optimal value of the objective function is noted as max f θ . Let us consider, for example that θ = 0.7. The coefficients of the objective function are determined as follows: ˜ c1 = θ c1 + (1 − θ )c1 = 0.7∗10.4 + 0.3∗9.6 = 10.16 ⊗ ˜ c2 = θ c2 + (1 − θ )c2 = 0.7∗7.1 + 0.3∗6.9 = 7.04 ⊗ We will proceed in the same manner for the right-side terms and for the elements of matrix A. The linear programming problem to be solved reads: max f 0.7 (x1 , x2 ) = 10.16x1 + 7.04x2 ⎧ ⎪ Subject to: ⎪ ⎪ ⎪ ⎪ 2.08x1 + 1.12x2 ≤ 8.12 ⎪ ⎪ ⎨ 2.12x1 + 5.16x2 ≤ 20.12 LP(0.7, 0.7, 0.7) ⎪ 3.04x1 + 1.16x2 ≥ 3.08 ⎪ ⎪ ⎪ ⎪ x1 ≥ 0 ⎪ ⎪ ⎩ x2 ≥ 0 The lines associated with the three constraints are:

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6 Grey Systems Theory Approach to Linear Programming

Fp

Fig. 6.17 The LP (0.7, 0.7, 0.7) feasible region

L 1 : 2.08x1 + 1.12x2 = 8.12 L 2 : 2.12x1 + 5.16x2 = 20.12 L 3 : 3.04x1 + 1.16x2 = 3.08 The lines corresponding to the constrains have been drawn using GeoGebra [8]— please see Fig. 6.17. The feasible region is given by the polygon A B C D E —Fig. 6.17, while the coordinates of the corner points determined with two-digit precision are:

A (2.32, 2.95), B (3.9, 0), C (1.01, 0), D (0, 2.66), E (0, 3.9) The values for the optimum function of the LP (0.7.0.7, 0.7), noted max f 0.7 , can be found among the corners of the feasible region:       f 0.7  A  = 10.16 x1  A  + 7.04 x2  A  = 10.16*2.32 + 7.04* 2.95 = 44.34 u f 0.7  B  = 10.16 x1  B  + 7.04 x2  B  = 10.16*3.9 + 7.04*0 = 39.62 u f 0.7 C  = 10.16 x1 C  + 7.04 x2 C  = 10.16*1.01 + 7.04*0 = 10.26 u f 0.7  D  = 10.16 x1 D  + 7.04 x2 D  = 10.16*0 + 7.04*2.66 = 18.73 u f 0.7 E = 10.16 x1 E + 7.04 x2 E = 10.16*0 + 7.04* 3.9 = 27.46 u Therefore: max f 0.7 = 44.34 u

Concluding Remarks

249

obtained for point A (2.32, 2.95). The first observation is that one considers the values determined for LP(1, 1, 0), L P(0, 0, 1) and LP(0.7, 0.7, 0.7), it can be observed that the proposition stated by Liu and Lin [1] remains true, namely: max f ≤ max f (ρ, β, δ) ≤ max f In our case: 37.15u ≤ 44.34u ≤ 57.59u37.15u ≤ 44.34u ≤ 57.59u. Another observation is related to the value of the function when θ = 0.5, LP(0.5, 0.5, 0.5), which is known and equal to: max f 0.5 = 46u—being determined through the Voskoglou’s whitening parameters model. Even for this case, we have that: max f ≤ max f (ρ, β, δ) ≤ max f as 37.15u ≤ 46u ≤ 57.59u. Now, considering a value for the grey target, e.g. μ0 = 0.4, we have to determine a satisfying degree of 0.7-positioned programming model by applying the formula given by Liu and Lin [1]:  max f 1 1− μ= 2 max f 0.7

+

1 max f 0.7 2 max f

By replacing the values of the functions with the values determined above, we have that:  1 37.15 1 44.34 + ∗ = 0.47 μ(0.7, 0.7, 0.7) = ∗ 1 − 2 44.34 2 57.59 As D = [μ0 , 1] = [0.4, 1] and μ(0.7, 0.7, 0.7) ∈ D (as 0.47 ∈ [0.4, 1]), the solution obtained for LP(0.7, 0.7, 0.7) is called a pleased solution of the linear programming problem with grey parameters. Based on the calculus presented above, it can be stated that in the case of this model, the computation quantity is quite high. Additionally, Darvishi et al. [13] stated that, even in the case one uses this model for solving the grey linear programming problem, there are many points of the solution which are outside of the feasible region.

Concluding Remarks Grey linear programming continues to be a developing field in which grey systems theory can be applied. Over the years, a series of steps have been taken to solve various linear programming problems. As the advancements in the field can be observed with the time passing, there is still place for improvement and advancement. In this chapter, we aimed to introduce the reader into the linear programming problem and to show the differences between the classical linear programming problem and the grey linear programming problem.

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A brief literature review has been presented on the purpose of increasing the reader’s interest in the methods developed for determining the solutions of the grey linear programming problems. Among the various means to address the grey linear programming problem, we have discussed in depth four models, pointing out their main steps, characteristics, advantages and disadvantages. A numerical example has been considered and has been used for solving the grey linear programming problem through the four selected models. The results containing the obtained solution in each of the four cases can be used by the reader as a starting point in understanding the applicability of the selected models. If interested, from here, the reader can try to develop his/her own models for addressing the grey linear programming problem or to bring improvements to the models from the scientific literature which have addressed this problem. We hope that you have found the chapter interesting and worth reading.

References 1. Liu, S., Lin, Y.: Grey Information: Theory and Practical Applications. Springer, London (2006) 2. Hillier, F.S., Lieberman, G.J.: Introduction to Operations Research. McGraw-Hill, New York, NY (2015) 3. Gramatovici, S., Delcea, C.: A Brief Introduction to Operations Research | Editura ASE. ASE Printing House, Bucharest (2021) 4. Dantzig, G.B., Thapa, M.N.: Linear Programming. Springer, New York (1997) 5. Taha, H.A.: Operations Research: An Introduction. Pearson, Boston (2016) 6. Vanderbei, R.J.: Linear Programming: Foundations and Extensions. Springer, New York (2013) 7. Cooke, W.P.: Quantitative Methods for Management Decisions. McGraw-Hill, New York (1985) 8. GeoGebra. https://www.geogebra.org/. Last accessed 20 Oct 2022 9. Gr. Voskoglou, M.: Solving linear programming problems with grey data. Orient. J. Phys. Sci. 3, 17–23 (2018). https://doi.org/10.13005/OJPS03.01.04 10. Darvishi, D., Pourofoghi, F., Forrest, J.Y.-L.: Sensitivity analysis of grey linear programming for optimisation problems. Oper. Res. Decisions 31, 35–52 (2021). https://doi.org/10.37190/ ord210402 11. Chen, Z., Chen, Q., Chen, W., Wang, Y.: Grey linear programming. Kybernetes 33, 238–246 (2004). https://doi.org/10.1108/03684920410514166 12. Darvishi, D., Babaei, P.: Grey prediction in linear programming problems. Int. J. Applied Oper. Res. Open Access J. 9, 11–18 (2019) 13. Darvishi, D., Liu, S., Yi-Lin Forrest, J.: Grey linear programming: a survey on solving approaches and applications. GS. 11, 110–135 (2020). https://doi.org/10.1108/GS-04-20200043 14. Huang, G.H., Baetz, B.W., Patry, G.G.: Grey chance-constrained programming: application to regional solid waste management planning. In: Hipel, K.W., Fang, L. (eds.) Stochastic and Statistical Methods in Hydrology and Environmental Engineering, pp. 267–280. Springer Netherlands, Dordrecht (1994). https://doi.org/10.1007/978-94-017-3081-5_20 15. Mahmoudi, A., Mi, X., Liao, H., Feylizadeh, M.R., Turskis, Z.: Grey best-worst method for multiple experts multiple criteria decision making under uncertainty. Informatica 31(2), 331– 357 (2020). https://doi.org/10.15388/20-INFOR409 16. Yaguang, Y.: Fuzzy method for solving gray parameter linear programming. Cybern. Syst. 19, 199–213 (1988). https://doi.org/10.1080/01969728808902164

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17. Ardabili, J.S., Salokolaei, D.D., Ofoghi, F.P.: Application of center and width concepts to solving grey linear programming. Int. J. Appl. Comput. Math. 6, 49 (2020). https://doi.org/10. 1007/s40819-020-0800-2 18. Yuan, C., Yang, Y., Liu, S., Fang, Z.: An investigation into the relationship between China’s economic development and carbon dioxide emissions. Clim. Dev. 9, 66–79 (2017). https://doi. org/10.1080/17565529.2015.1067182 19. Liu, S., Dang, Y., Forrest, J.: On positioned solution of linear programming with grey parameters. In: 2009 IEEE International Conference on Systems, Man and Cybernetics, pp. 751–756. IEEE, San Antonio, TX, USA (2009). https://doi.org/10.1109/ICSMC.2009.5346825 20. Huang, G., Moore, R.D.: Grey linear programming, its solving approach, and its application. Int. J. Syst. Sci. 24, 159–172 (1993). https://doi.org/10.1080/00207729308949477 21. Salookolaei, D.D., Liu, S., Nasseri, S.H.: A new approach in animal diet using grey system theory. Grey Syst. Theor. Appl. 8, 167–180 (2018). https://doi.org/10.1108/GS-11-2017-0040 22. Bi, Y.M., Li, J.W., Li, G.M.: Genocop algorithm for solving gray linear programming problem. Syst. Eng. Theor. Pract. 20(2), 79–83 (2000). https://doi.org/10.12011/1000-6788(2000)2-79 23. Li, Q.-X., Liu, S., Wang, N.-A.: Covered solution for a grey linear program based on a general formula for the inverse of a grey matrix. Grey Syst. Theor. Appl. 4, 72–94 (2014). https://doi. org/10.1108/GS-10-2013-0023 24. Hajiagha, S.H.R., Akrami, H., Hashemi, S.S.: A multi-objective programming approach to solve grey linear programming. Grey Syst. Theor. Appl. 2, 259–271 (2012). https://doi.org/10. 1108/20439371211260225 25. Mahmoudi, A., Feylizadeh, M.R., Darvishi, D.: A note on “a multi-objective programming approach to solve grey linear programming.” Grey Syst. Theor. Appl. 8, 35–45 (2018). https:// doi.org/10.1108/GS-08-2017-0027 26. Mahmoudi, A., Feylizadeh, M.R., Darvishi, D., Liu, S.: Grey-fuzzy solution for multi-objective linear programming with interval coefficients. GS. 8, 312–327 (2018). https://doi.org/10.1108/ GS-01-2018-0007 27. Huang, G.H., Baetz, B.W., Patry, G.G.: Grey integer programming: an application to waste management planning under uncertainty. Eur. J. Oper. Res. 83, 594–620 (1995). https://doi. org/10.1016/0377-2217(94)00093-R 28. Pourofoghi, F., Salokolaei, D.D.: Applying duality results to solve the linear programming problems with grey parameters. Control Optim. Appl. Math. 5, 15–28 (2020). https://doi.org/ 10.30473/coam.2021.56072.1152 29. Nasseri, S.H., Darvishi, D.: Duality results on grey linear programming problems. J. Grey Syst. 30, 127–143 (2018) 30. Nasseri, S.H., Yazdani, A., Salokolaei, D.D.: A primal simplex algorithm for solving linear programming problem with grey cost coefficients. J. New Res. Math. 1(4), 115–136 (2016). https://doi.org/10.13140/RG.2.1.2645.8002 31. Salookolaei, D.D., Nasseri, S.H.: A dual simplex method for grey linear programming problems based on duality results. GS. 10, 145–157 (2020). https://doi.org/10.1108/GS-10-2019-0044 32. Kose, E., Forrest, J.Y.-L.: N-person grey game. Kybernetes 44, 271–282 (2015). https://doi. org/10.1108/K-04-2014-0073 33. Mahdiraji, H.A., Hajiagha, S.H.R., Hashemi, S.S., Zavadskas, E.K.: A grey multi-objective linear model to find critical path of a project by using time, cost, quality and risk parameters. E+M. 19, 49–61 (2016). https://doi.org/10.15240/tul/001/2016-1-004 34. Liu, S., Yang, Y., Forrest, J.: Grey Data Analysis : Methods, Models And Applications. Springer, Singapore (2017) 35. Liu, S., Forrest, J.Y.-L.: Grey Systems: Theory and Applications. Springer-Verlag, Berlin Heidelberg (2011). https://doi.org/10.1007/978-3-642-16158-2

Chapter 7

Complex Projects Management with Interval Grey Numbers

Introduction The projects management deals with the process of leading a team to achieve all the goals with a project, while considering the various constraints that may arise, related to funds, time and/or resources. The term “complex” refers, as in other cases, to the idea of “many” [1]. Having many activities within the project, many interdependencies among activities, many resources that can be used for project’s fulfillment, many activities that, at a given moment of time, are competing for the same limited resources, etc., makes the projects’ management a complex activity which involves a lot of effort on behalf of the project manager for properly addressing them and fulfilling the project within the estimated amount of time. Adding uncertainty and limited information to the problem, makes the projects’ management a difficult and complex task. As the presence of uncertainty and limited information creates a suitable ground for the use of grey systems theory, we will discuss along this chapter how the characteristic elements of grey systems theory can help in dealing with the complex projects management. As the grey numbers are the core of the grey systems theory, even in this case, the approach to the projects management will be done through the use of this special type of numbers and their mathematic. First, a basic presentation of the project management is made from the perspective of the operations research, having the purpose of familiarizing the reader with the terms used and with the current approach in determining the time needed for completing a complex project. Two approaches related to the use of networks in representing the activities within a project and their interconnections are detailed in connection with the state of the art in the field. For a better understanding, the theoretical presentation is accompanied by a practical numerical example. After that, a literature review on the papers discussing the problem of project management and project scheduling from the grey systems theory is provided, highlighting some of the most prominent works from the field. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_7

253

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7 Complex Projects Management with Interval Grey Numbers

Based on the literature review, three approaches for dealing with the complex projects’ management which use elements from the grey systems theory are presented and discussed from a theoretical point of view. The approaches are accompanied by a numerical example, which will offer the reader the possibility to observe how one can use them on practical applications. The concluding remarks are ending the chapter.

Brief Presentation of Projects Management A project can be defined as a set of interrelated activities, characterized by a series of properties, including the consumption of time and resources [2, 3]. The division of the project among the activities and the establishment of their interconnections is a difficult task and should be conducted properly in order to ensure the feasibility of the project completion. As the project is a temporary endeavor and it contrasts with the classical business-as-usual activities, it might require an increased attention on behalf of the project manager. The techniques designed for easily representing the projects’ activities have served in assisting the project managers to plan and coordinate the project and to have a realistic overview in time regarding the project completion. As the networks are an appropriate mean for visualizing the flow of activities, they have been used for representing the activities among the project, their relationships and interdependencies [2]. As a result, two network-oriented operations research techniques have been developed over time and have been used for projects’ representation: PERT (program evaluation and review technique) and CPM (critical path method) [2]. Later on, the two techniques have merged by keeping the best features of both of them in the so-called PERT/CPM approach [2]. In the following, some basic information related to the activities within a project, their relationships, the rules accompanying the network representation for a complex project are provided on the purpose of introducing the reader to projects management.

Activities and the Precedence Relationships An activity can be seen as the smallest portion of a project, which allows the project manager to plan, track or control the project for ensuring its on-time completion. The activities are scheduled phases in the project and each of them is characterized by a beginning and by an end. The break-down of a project into individual activities depends on the project manager’s desired level of details. Based on the experience of the project manager, it might happen that the project break-down process to produce various results when different project managers are involved.

Brief Presentation of Projects Management

255

As a project is characterized by a set of activities, precedence relationships arise among them. The presence of a precedence relationship among two activities indicates the fact that one of the activities cannot start unless the other one is finished. In most of the cases, one can define the relationships between the activities by indicating the immediate predecessor for each activity. An immediate (or a direct) predecessor of an activity is an activity that should have been finished immediately before the other activity should start. There are cases in which an activity can have more than one predecessor activities. Also, there are cases in which one can know instead of the immediate predecessors of the activities within a project, the immediate successors, even though these applications are rarely discussed in the literature. The idea behind the immediate successor is similar to the one of immediate predecessor, with the exception that, in this case, one knows for each activity the activity that should start next instead of knowing the activity that has just been finished. Let us consider a project for the release of a new computer software within a company, characterized by the activities described in Table 7.1. Even though we know that the activities implied are various based on a series of elements one might consider, and, in real-life, the number of activities can be higher, we will keep the number of activities to a lower limit as the example is intended to be used for explaining the terms associated with the project management. The project presented in Table 7.1, for which the immediate predecessors are known, is similar to the one presented in Table 7.2, where the immediate successors are provided. Let us see in the following how the representation of a project succession of activities can be done. Table 7.1 Activities for the release of a new computer software—with immediate predecessors Activity code

Activity description

Immediate predecessors

A

Define the specifications

–

B

Software design

A

C

Establishing the hardware requirements

A

D

User-interface design

A

E

Software development—stage 1

B, D

F

Testing software development—stage 1

E

G

Creating a draft for user manual—version 1

C, D

H

Software development—stage 2

E

I

Testing software development—stage 2

F, H

J

Creating a draft for user manual—version 2

F, G

K

Creating the final version of the user manual

I, J

L

Customer-acceptance of the software

I

M

Releasing the software and its user manual

K, L

256

7 Complex Projects Management with Interval Grey Numbers

Table 7.2 Activities for the release of a new computer software—with immediate successors Activity code

Activity description

Immediate successors

A

Define the specifications

B, C, D

B

Software design

E

C

Establishing the hardware requirements

G

D

User-interface design

E, G

E

Software development—stage 1

H, F

F

Testing software development—stage 1

I, J

G

Creating a draft for user manual—version 1

J

H

Software development—stage 2

I

I

Testing software development—stage 2

K, L

J

Creating a draft for user manual—version 2

K

K

Creating the final version of the user manual

M

L

Customer-acceptance of the software

M

M

Releasing the software and its user manual

–

Network Representation For a better view on the succession of the activities with a project, a network representation can be used. There are two alternative types of network representations, each one with its own advantages and disadvantages [2]: • Activity-on-Arc (noted AoA)—in this approach, each activity is represented by an arc, while the nodes are used to separate an activity from each of its immediate predecessors. • Activity-on-Node (noted AoN)—in this representation, a node is associated for each activity, while the arcs are used for depicting the precedence relationship among activities. Let us discuss them in the following.

Activity-on-Arc Representation The Activity-on-Arc representation assumes for each activity an arc that points in the direction of the progress of the project, being flanked by two distinct nodes (vertices). Figure 7.1 presents a standard representation of an activity within two nodes, noted with i and j. In Fig. 7.1: A is the name (acronym) of the activity to be represented, d (A) is the duration of activity A expressed in time units (t.u.), EET (i) is the Earliest Event Time for node i, while LET (i) is the Latest Event Time for node i. While the duration of the activity is given as determined before the network representation, the

Brief Presentation of Projects Management Fig. 7.1 The representation of an activity within AoA

Fig. 7.2 Addition of a dummy activity within AoA

257

EET(i)

i

i

LET(i)

EET(j)

A

j

d(A)

EET(i)

A

LET(i)

d(A)

LET(j)

EET(j)

j

LET(j)

f B d(B)

EET(k)

k

LET(k)

EET (i) and LET (i) are determined within the network representation process, as we will discuss later in this chapter. As a rule, two or more activities can have the same starting node, but cannot have the same ending node. Therefore, if needed, dummy activities, with no time associated can be created to avoid the situations in which two or more activities with the same starting node, would have had the same ending node. Usually, the dummy activities are represented with an interrupted arrow. Figure 7.2 presents the addition of dummy activity f with no time associated and of node k in order to avoid the situation in which activities A and B, which have the same starting node, would have had the same ending node. For each activity A, delimited by node i as start node and j as end node, the index for the nodes will be chosen so that i < j. Next, let us associate durations to the activities listed in Table 7.1—as presented in Table 7.3—and let us draw the network diagram in the AoA representation. The duration of each activity is represented in time units (t.u.). Following the rules presented above, the AoA network representation is provided in Fig. 7.3. As it can be observed, a number of five fictive activities have been added to avoid the situation in which two activities with the same initial node would have had the same final node (noted with f1 , . . . , f5 ). In order to compute the EET and LET values for each node, the following rules should be used: • Forward step: the network is traversed from left to right and the values for the EET are determined and filled in the top-right side of each node. The value for the EET of the first node is equal to zero, all the other values for the remainder of the nodes are computed using the formula:

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7 Complex Projects Management with Interval Grey Numbers

Table 7.3 Project activities, immediate predecessors, and duration

B 3

Activity code

Immediate predecessors

Duration (t.u.)

A

–

4

B

A

3

C

A

1

D

A

2

E

B, D

7

F

E

1

G

C, D

1

H

E

5

I

F, H

2

J

F, G

2

K

I, J

1

L

I

2

M

K, L

1

E

4

7

6

H

f2 1

A 4

2

D 2

1

3

5

2

L

10

2

f4 F 1

f5

7

12

M 1

13

f3

f1 C

I

8

5

G 1

9

J 2

11

K 1

Fig. 7.3 AoA representation for the computer software releasing project

EET (j) = EET (i) + d (A) where j is the index of the node for which the EET is determined, i is the index of the previous node, and d (A) is the duration of activity A, with A being te activity that takes place between node i and j, with i < j. In the case in which node j is the end node for two or more activities, the above formula is changed by applying maximum to the right-handed term: EET (j) = max(EET (i) + d (A), EET(k) + d (B), . . . , EET (l) + d (M )) where i, k, . . . , l are the indexes of the previous nodes, A, B, . . . , M are the activities taking place between i, k, . . . , l nodes and j node, while d (A), d (B), . . . , d (M ) are the duration of the A, B, . . . , M activities. • Backward step: During the backward step, the network is traversed from right to left and the values for the LET are filled-in in the bottom-right side of each node. The LET value of the final node is equal to its EET value. The LET values for all

Brief Presentation of Projects Management B 3

4

7

E

7

7

6

259

14

H

14

5

8

f2 1

0 0

A 4

2

4

D

4

2

3

1

5

I

19

2

10

21

L

21

2

f4

6

F

7

1

7

f5

15

12

19

23 23

M 1

13

24 24

f3

f1 C

19

6

G

19

1

9

15

J

20

2

11

21

K

22

1

Fig. 7.4 AoA representation for the computer software releasing project with EET and LET

the other nodes are determined based on the formula: LET (i) = LET (j) − d (A) where i is the index of the node for which the LET is determined, j is the index of the previous node, and d (A) is the duration of activity A, with A being the activity that takes place between node i and j, with i < j. In the case in which node i is the start node for two or more activities, the above formula is changed by applying minimum to the right-handed term: LET (i) = min(LET (j) + d (A), LET (m) + d (P), . . . , LET (s) + d (W )) where j, m, . . . , s are the indexes of the previous nodes, A, P, . . . , W are the activities taking place between the i node and the j, m, . . . , s nodes, while d (A), d (P), . . . , d (W ) are the duration of the A, P, . . . , W activities. Next, let us see how the EET and LET are filled-in in the case of the software development project presented in Table 7.3. After applying the above rules, the network diagram is as presented in Fig. 7.4. As it can be observed from Fig. 7.4, the completion time for the project is 24 t.u. if no changes appear in the duration of the implied activities. Furthermore, for each activity, A, represented in the network diagram between the start node i and the end node j, four terms can be determined [3]: • • • •

Earliest Starting Time: EST (A) = EET (i); Latest Finishing Time: LFT (A) = LET (j); Earliest Finishing Time: EFT (A) = EET (i) + d (A); Latest Starting Time: LST (A) = LET (j) − d (A).

Relatively to these four terms, activity A could take place any time between EST (A) and LFT (A). Let us determine the four times for the activities of the project depicted in Fig. 7.4. The values are provided in Table 7.4. In connection with these terms, for each activity, A, one can define the total float of activity A, noted (TF(A)), as the difference between LST (A) and EST (A), or as the difference between LFT (A) and EFT (A) [3, 4]. The total float indicator

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7 Complex Projects Management with Interval Grey Numbers

Table 7.4 EST, LST, EFT and LST of the software project activities (in t.u.) EST

LFT

EFT

LST

A

0

4

4

0

B

4

7

7

4

C

4

19

5

18

D

4

7

6

5

E

7

14

14

7

F

14

19

15

18

G

6

20

7

19

H

14

19

19

14

I

19

21

21

19

J

15

22

17

20

K

21

23

22

22

L

21

23

23

21

M

23

24

24

23

Activity

shows to which extent an activity can be delayed without delaying the entire project. In connection with the value of the total float (which can be zero or higher), the activities of a project can be divided between critical activities (for which the total float is zero) and non-critical activities (with total float different than zero). Let us determine the total float for the software project activities. The results are presented in Table 7.5. In the example provided in Fig. 7.4 it can be observed that we have seven critical activities (A, B, E, H , I , L, M ) and six non-critical activities (C, D, G, F, J , K). Table 7.5 TF of the software project activities (in t.u.)

Activity A

TF 0

B

0

C

14

D

1

E

0

F

4

G

13

H

0

I

0

J

5

K

1

L

0

M

0

Brief Presentation of Projects Management B 3

4

7

E

7

7

6

261

14

H

14

5

8

f2 1

0 0

A 4

2

4

D

4

2

3

1

5

I

19

2

10

21

L

21

2

f4

6

F

7

1

7

f5

15

12

19

23 23

M 1

13

24 24

f3

f1 C

19

6

G

19

1

9

15

J

20

2

11

21

K

22

1

Fig. 7.5 Critical path for the software development project

The succession of the critical activities within a network representation gives the critical path. In a network representation, there can be one or more critical paths. The length of the critical path is the sum of the duration of the activities within the critical path. Among all the possible paths in the network, the critical path is the one with the longest length. In the case of the software project development example, one can identify only one critical path, represented by the succession of activities: A → B → E → H → I → L → M . The critical path associated with our example is depicted using red and high-weight arrows in Fig. 7.5. The length of the critical path is equal to 24 t.u.

Activity-on-Node Representation According to Hillier and Lieberman [2], the Activity-on-Node representation is considerably easier to construct than the Activity-on-Arc representation, being, in the same time, easier to understand for the persons not familiar with project management and easier to revise when needed. The rules accompanying the AoN representation are also very simple: for each activity we will associate a node represented by a rectangle divided into six small rectangles as presented in Fig. 7.6. The six elements to be included in each node are: the code of the activity, the duration of the activity, the Earliest Starting Time (EST ), Earliest Finishing Time (EFT ), Latest Starting Time (LST ), and Latest Finishing Time (LFT ). As a rule, any AoN network has only one start node and only one finish node. Therefore, if the project has two or more activities that are starting at time zero, a dummy node representing a dummy activity should be added to the network, prior to the activities starting at zero. The duration of the dummy activity is zero and it does not affect the overall project duration. The same observation is made in the case in which the project has two or more ending activities: one should add a dummy finish activity with a duration of zero. The computations steps in the case of the AoN representation are [3, 5]: • Forward step: the network is traversed from the left to the right and the Earliest Starting Time (EST ) and the Earliest Finishing Time (EFT ) are calculated. For

262

7 Complex Projects Management with Interval Grey Numbers Activity code

Fig. 7.6 A node in AoN representation

EST(A)

A

EFT(A)

LST(A)

d(A)

LFT(A)

Activity duration

the first activity, the EST is zero. The EFT for an activity, A, is determined as the sum between EST and the duration of the activity, d (A): EFT (A) = EST (A) + d (A) The EST of any other activity in the network (other than the first one) is determined by choosing the maximum value of the EFT of the direct predecessor activities of the current activity for which the EST is determined. • Backward step: the network is traversed from the left to the right and the Latest Starting Time (LST ) and the Latest Finishing Time (LFT ) are calculated. For the finish node (the node that has no other predecessor activity), the value of the LFT is equal to its EFT . The LST of any activity, A, is determined as: LST (A) = LFT (A) − d (A) The LFT of any other activity than the last one is determined by choosing the minimum value of the LST of the direct successors activities. Having these guidelines for determining the values of each node in the network diagram, let us apply them to the software development project discussed above. The resulting network diagram is depicted in Fig. 7.7. 4

B

7

7

E

14

14 H

19

19

I

21

21

L

23

4

3

7

7

7

14

14

5

19

19

2

21

21

2

23

14

F

15

23 M 24

18

1

19

23

0

A

4

4

D

6

0

4

4

5

2

7

4

C

5

18

1

19

6

G

7

15

J

17

21

K

22

19

1

20

20

2

22

22

1

23

Fig. 7.7 AoN representation of the software development project

1

24

Brief Presentation of Projects Management

263

As expected, the same values as in the case of AoA representation are determined for the four indicators of activity times. The critical path, made by the critical activities, is depicted in Fig. 7.7 by highlighting in light-red the implied critical activities.

Gantt Diagram For a visual display of the activities within a project against time, a Gantt diagram can be used. Figure 7.8 presents the Gantt diagram for the software development project discussed above.

Activity

M L K J

Y-Axis

I H G F E D C B A

0

4 5 6 7

14 15 17 X-Axis

Fig. 7.8 Gantt diagram for the software development project

19

21 22 23 24

Time

264

7 Complex Projects Management with Interval Grey Numbers

Literature Review on Grey Systems Theory in Projects Management Projects management is another area in which grey systems theory has been applied. As in many other economic and social sciences areas, even in project management, one can encounter the uncertainty associated with different events or measures. In the case of the project management, a source of uncertainty is related to the determination of the duration of the involved activities. As a result, the researchers from the field have tried to capture this uncertainty through the use of grey numbers. Xie [6] integrates interval grey numbers into project scheduling problem and proposes a method through which the starting/ending times of the activities can be determined. Through a network representation approach, the author determines the critical activities of the project, the critical path/paths and the project completion time. The accent in the approach is put on the difference between independent and non-independent duration of the activities, which alters the calculus of the latest staring times for the implied activities. A similar approach, in which a difference is made between the independent and non-independent duration of the activities is proposed by Chen et al. [7]. The authors are using grey interval numbers for the durations of the activities and a network representation for the succession of the projects’ activities. The result provides the main elements need for conducting a project management activity under grey durations associated with the activities. Peng et al. [8] uses the whitenization of the grey numbers for addressing the project scheduling problem with interval grey numbers for the duration of the activities. The authors propose a series of steps needed to be considered for determining the critical path, keeping in mind that the durations of the activities might be sometimes overestimated by the project managers. Mahdiraji et al. [9] consider the case in which not only the completion time is important in project scheduling, but also some other aspects related to cost, quality and risk. As a result, the authors have proposed a grey multi-objective linear model for determining the critical path of a project. The proposed approach can also serve for the situations in which only one objective is considered (e.g. the projects’ completion time). A numerical application is provided by the authors to highlight the applicability of the proposed approach. A linear model with interval grey numbers is proposed by Huang et al. [10] for the project scheduling. The authors consider both the time and cost associated with the implied activities and make a difference between the crash and the normal time/cost. The proposed approach is used by the authors for the planning a project in the construction domain. Another application in the construction field is proposed by Trang and Hai [11]. The authors propose a grey wolf optimizer for project scheduling and use a minimization linear model for reducing the project duration. Other approaches are proposed by Song and Yan [12], Zhao et al. [13], and Zhongmin and Xizu [14].

Theoretical and Practical Approach to Project Scheduling with Grey …

265

Theoretical and Practical Approach to Project Scheduling with Grey Numbers During this section, we have selected three approaches used for project scheduling when the duration of the activities is represented through the use of the grey numbers. The approaches are discussed from a theoretical point of view with reference to the papers in which these approaches have been proposed. A numerical example accompanies the theoretical approach in order to better present how the solution for these types of problems can be reached through the use of grey numbers.

Theoretical Approach to Project Scheduling with Grey Numbers The theoretical approach to project scheduling presented in closely following the proposed approaches of the selected authors. We have tried to keep, as much as possible, the same notations used by the authors. At the same time, we have tried to be consequent with the notations used in the previous chapters of this book, so slightly differences might be noted regarding the notations made by the mentioned authors.

Xie’s Project Scheduling Model The approach proposed by Xie [6] follows the basic information related to the interval grey numbers, while putting an accent on the idea of independent interval grey numbers, which are the key in solving the project scheduling problem with grey numbers. Before giving the rules for determining the times included in the network representation, few definitions and elements should be established in connection with the independent interval grey numbers as presented by Xie [6]. First, two interval grey numbers are considered to be independent if the changes in one of the numbers is not followed by the changes in the other one. Depending on the type of interval grey numbers, namely independent or non-independent, different linear operations can take place.

Linear Operations of Independent Interval Grey Numbers Let us consider a group of independent interval grey numbers, noted ⊗1 , ⊗2 , . . . , ⊗n . In connection with these numbers and keeping the notations from Chap. 2, we define the true value of a grey number ⊗, noted with d ∗ , with the property that ∀⊗ => d ∗ ∈ D, where D is the value-covered set of ⊗.

266

7 Complex Projects Management with Interval Grey Numbers

For the set defined above, one will have Di value-covered sets and di∗ true values of the ⊗i independent interval grey numbers, with i = 1, . . . , n. Considering that each ⊗i has the following form ⊗i = [ai , bi ] for any independent interval grey numbers, with i = 1, . . . , n, the general linear operation result ⊗ can be defined as [6]: ⊗ = c1 ⊗1 +c2 ⊗2 + . . . + cn ⊗n where ⊗ = [a, b]. Further on, as the values of the ⊗1 , ⊗2 , . . . , ⊗n cannot be obtained, the variables x1 , x2 , . . . , xn are used for representing the ⊗1 , ⊗2 , . . . , ⊗n numbers. In this situation, the values covered sets can be seen as a range among which lies every interval grey number [6]. Therefore, the values of a and b can be determined by calculating the solution of two linear models [6]: a = min f (x1 , x2 , . . . , xn ,◦ ) ⎧ ⎪ Subjectto : ⎪ ⎪ ⎪ ⎪ ⎨ a1 ≤ x1 ≤ b1 a2 ≤ x2 ≤ b2 ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎩ an ≤ xn ≤ bn and: b = max f (x1 , x2 , . . . , xn ,◦ ) ⎧ ⎪ Subjectto : ⎪ ⎪ ⎪ ⎪ ⎨ a1 ≤ x1 ≤ b1 a2 ≤ x2 ≤ b2 ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎩ an ≤ xn ≤ bn where ◦ is used for defining any type of operation (addition or subtraction).

Linear Operations of Non-Independent Interval Grey Numbers In this case, let us consider that the ⊗1 , ⊗2 , . . . , ⊗n are independent interval grey numbers, while ⊗A and ⊗B are two non-independent interval grey numbers defined as [6]: ⊗A = cA1 ⊗1 +cA2 ⊗2 + . . . + cAn ⊗n ⊗B = cB1 ⊗1 +cB2 ⊗2 + . . . + cBn ⊗n

Theoretical and Practical Approach to Project Scheduling with Grey …

267

The general linear operation result, ⊗, can be determined as [6]: ) ) ( ◦ ( ◦ ◦ ⊗ = ⊗◦A ⊗B = (cA1 cB1 ) ⊗1 + cA2 cB2 ⊗2 + . . . + cAn cBn ⊗n where ◦ is used for defining any type of operation (addition or subtraction) and ⊗ = [a, b]. Even in this case, a and b can be determined by calculating the solution of two linear models as presented above [6].

Xie’s Proposed Approach to Project Scheduling Similar to the general theory associated to projects management in the case in which grey numbers are used for describing the duration of the activities, a series of time values will be determined. As these values are also grey numbers, the following notations will be used for a generic activity noted with A: • • • • •

Duration of activity A: ⊗d(A) Earliest Starting Time: ⊗EST (A) Latest Finishing Time: ⊗LFT (A) Earliest Finishing Time: ⊗EFT (A) Latest Starting Time: ⊗LST (A). Additional two notations are needed [6]:

• The upper boundary of an interval grey number ⊗: Upper (⊗) • The lower boundary of an interval grey number ⊗: Lower (⊗). As in the case of the general project schedule problem, the first step in determining the duration of the project and the starting and finishing times for all the activities is represented by the drawing of the network diagram—as shown above, any of the AoA or AoN representations can be used. As the AoN representation has the advantage of providing all the four times for each activity that we are interested in, namely ⊗EST , ⊗EFT , ⊗LST and ⊗LFT , being at the same time easier to use and understand, we will describe the theoretical and practical approach to project scheduling by using this approach. In completing the values for the ⊗EST , ⊗EFT , ⊗LST and ⊗LFT in the network diagram, the forward and backward steps will be followed: • Forward step: the network is traversed from the left to the right and the Earliest Starting Time (⊗EST ) and the Earliest Finishing Time (⊗EFT ) are calculated. For the first activity, the ⊗EST is zero. As in this case, we work with interval grey numbers, the ⊗EST of the fist activity will be [0, 0]. The ⊗EFT for an activity, A, is determined as the sum between ⊗EST and the duration of the activity, ⊗d (A): ⊗EFT (A) = ⊗EST (A) + ⊗d (A)

268

7 Complex Projects Management with Interval Grey Numbers

The ⊗EST of any other activity in the network (other than the first one) is determined by choosing the maximum value of the ⊗EFT of the direct predecessor activities of the current activity for which the ⊗EST is determined. As in this case interval grey numbers are involved in the comparison, one should evaluate the probability that the ⊗EFT of an activity is greater than the ⊗EFT of the other predecessors activities. For conducting such a comparison, the theoretical approach to grey numbers comparison presented in Chap. 2 should be considered. Alternatively, the theoretical approach presented by Xie and Liu [15] can be used. One might find the following link useful for directly comparing two interval grey numbers: https://github.com/liviucotfas/grey-systems-book/blob/main/ grey-numbers-comparison-greater.ipynb. Let us consider two predecessor activities, noted with A and B, for the current activity, noted with C, for which we want to determine ⊗EST , noted ⊗EST (C). Knowing the ⊗EFT (A) and ⊗EFT (B), we need to compare the two grey numbers. The result of the comparison is a probability that ⊗EFT (A) is greater than ⊗EFT (B), noted p(⊗EFT (A) > ⊗EFT (B)). As Xie [6] mentioned, if the p(⊗EFT (A) > ⊗EFT (B)) > 0.5, then we can say that activity A is superior to activity B, and therefore, ⊗EST (C) = ⊗EFT (A). Contrary, if p(⊗EFT (B) > ⊗EFT (A)) > 0.5, ⊗EST (C) = ⊗EFT (B). In the case in which the number of predecessor activities is greater than two, we will proceed in the same manner, comparing the ⊗EFT of all the predecessor activities two by two and, based on the obtained result, deciding the superior activity, which will give the value of the ⊗EST for the current activity. • Backward step: the network is traversed from the left to the right and the Latest Starting Time (⊗LST ) and the Latest Finishing Time (⊗LFT ) are calculated. For the finish node (the node that has no other predecessor activity), the value of the ⊗LFT is equal to its ⊗EFT . The ⊗LST of any activity, A, is determined by taking into account if the duration of the activity is independent or non-independent with the ⊗LFT of the activity. Therefore, following the Xie’s [6] approach, we need to determine: ⊗LST (A) { Upper(⊗LFT (A)) − ⊗d (A), if ⊗ d (A)is independent with ⊗ LFT (A) = ⊗LFT (A) − ⊗d (A), if ⊗ d (A)is non−independent with ⊗ LFT (A) The ⊗LFT of any other activity than the last one is determined by choosing the minimum value of the ⊗LST of the direct successors activities. As in the previous case when the network was traversed through the forward step, even in the case of the backward step, the choose of the minimum is made by comparing interval grey numbers. As a result, the comparison will be characterized by a probability that one ⊗LST to be greater than the other ⊗LST . The value of 0.5 will be used as a threshold for the probability even in this case.

Theoretical and Practical Approach to Project Scheduling with Grey …

269

Let us consider again two activities, D and E, which are the successors of another activity, F, for which we want to determine the ⊗LFT (F). The values for ⊗LST (D) and ⊗LST (E) are known. First, we need to compare ⊗LST (D) and ⊗LST (E). If the probability p(⊗LST (D) > ⊗LST (E)) < 0.5, then the activity D is superior to activity E, and its ⊗LST (D) will be used for determining the ⊗LFT (F). Contrary, it will be considered that the activity E is superior to activity D, and its ⊗LST (E) will be used for determining the ⊗LFT (F). In this case, one should consider even the independence/non-independence relationship among the grey numbers. In the case of independent numbers, the upper value of the ⊗LST will be considered when determining the ⊗LFT of its predecessor activity. After the forward and the backward steps are completed, we will have the values of the ⊗EST , ⊗EFT , ⊗LST and ⊗LFT for all the activities and also, we will know the completion time as being the ⊗EFT or the ⊗LFT of the last activity. As in the case of any other network diagram represented using AoN, dummy starting/ending nodes will be added in the case in which the network has more than one staring/ending node. The duration of the fictive (dummy) nodes will be equal to [0, 0]. In the same manner as in the case of the classical project management problems, in this case, the critical activities are defined as the activities for which the ⊗EST = ⊗LST or ⊗EFT = ⊗LFT . The succession of the critical activities from the start of the project to the end, gives the critical path/paths. The length of the critical path is given by the sum of the durations of the activities included in the critical path, being equal to the project completion time. For the case in which the AoA representation is used, the theoretical background remains the same and one should pay a particular attention to the relationship among the involved interval grey numbers, with respect to their independence or nonindependence. Therefore, we invite the reader to explore the AoA representation both from theoretical and practical point of view.

Chen et al.’s Project Scheduling Approach Chen et al. [7] proposed a similar approach to project schedule as Xie [6]. The authors discuss the independence of the grey numbers and use the grey critical path method for determining the ⊗EST , ⊗EFT , ⊗LST and ⊗LFT for all the activities within the project. As in other project scheduling approaches, the authors assume that an activity cannot be interrupted once it begins, each activity can start only after its predecessor activity/activities end and that after the duration of an activity is estimated (using in this case the grey numbers), it cannot be further changed [7]. Additionally, the authors state that the durations of all the activities are grey numbers, characterized by an upper and a lower bound [7]. A graph representation is used for describing the precedence of the activities.

270

7 Complex Projects Management with Interval Grey Numbers

The graph is traversed two times, first from left to right and the values for the ⊗EST and ⊗EFT are determined, and second, from left to right and the values for ⊗LFT and ⊗LST are calculated for all the activities. As in Xie [6], in Chen et al. [7] the forward step implies the calculus of the ⊗EST and ⊗EFT , being known that the moment for starting the project is considered the zero moment. For the determination of the ⊗EFT for an arbitrary selected activity A the classical formula is used, keeping in mind that ⊗EST , ⊗EFT and ⊗d are grey numbers: ⊗EFT (A) = ⊗EST (A) + ⊗d (A) As for the determination of the ⊗EST of an activity, the values of the ⊗EFT of its direct predecessors are considered, choosing the maximum among them [7]. The approach is similar to Xie [6]. One should note that, as we are comparing grey numbers, the maximum among the concurrent ⊗EFT is determined through a grey comparison, considering the threshold of 0.5. A difference from the approach proposed by Xie [6] appears in Chen et al. [7] on the backward step. Here, when determining ⊗LST , instead of considering the upper ⊗LFT in the situation in which the duration of the activity is independent, Chen et al. [7] propose the use of the following formula in all the cases: ⊗LST (A) = ⊗ LFT (A) − ⊗d (A) The difference between the non-independence/independence of the grey numbers when determining the ⊗LST should only be considered when performing the subtraction (in a grey/white manner). As for the ⊗LFT , it is determined by choosing the minimum among the direct successor activities [7]. Again, as we are dealing with grey numbers, a grey comparison should be considered, with a threshold of 0.5.

Peng et al.’s Project Scheduling Approach Another approach to project scheduling is provided by Peng et al. [8] using the process of whitenization of the grey numbers. The authors started from the idea that an interval grey number can be described by four points, noted a, b, c, d , corresponding to the left increase function, noted L(t), the peak area, and the right decrease function, noted R(t)—as presented in Fig. 7.9 [8]. Thus, the classical type of a whitenization weight function describing an interval grey number ⊗ = [c, a; b, d], noted f⊗ (t), has the following form [8]: ⎧ ⎨ L(t), t ∈ [c, a) f⊗ (t) = 1, t ∈ [a, b] ⎩ R(t), t ∈ (b, d ]

Theoretical and Practical Approach to Project Scheduling with Grey … Fig. 7.9 Whitenization grey function

271

f ⊗ (t)

1 L(t)

c

R(t)

a

b

d

t

Peng et al. [8] noted that, in practical applications, the functions L(t) and R(t) are often simplified and expressed through straight lines. As a result, the above form of the whitenization function becomes [8]: ⎧ t−c ⎪ ⎨ a−c , t ∈ [c, a) f⊗ (t) = 1, t ∈ [a, b] ⎪ ⎩ d −t , t ∈ (b, d ] d −b Based on this function, one can determined the expected grey duration of a grey activity, noted with A, by applying [8]: EG(A) =

∫dc f⊗ (t)tdt ∫dc f⊗ (t)dt

Furthermore, in the case in which the expression of the f⊗ (t) is unknown, the following formula can be used for EG(A) [8]: EG(A) =

a + 2b + 2c + d 6

Also, in the case in which an interval grey number is only described by the a and b values, ⊗ = [a; b], namely in the case in which c = a and d = b, the EG(A) becomes: EG(A) =

a + 2b + 2a + b 3a + 3b a+b = = 6 6 2

Having these elements related to the expected grey duration of an activity, after the transfer of the interval grey numbers into expected grey duration, Peng et al. [8] propose the following steps to solve the project scheduling problem when grey interval numbers are involved:

272

7 Complex Projects Management with Interval Grey Numbers

• Determine the revised grey duration of each activity During this step the value obtained for the EG is revised by dividing it by 2. This division is proposed by the authors as it is supposed that the managers will add buffer to all the tasks, which may lead to the waste of the buffers [8]. Therefore, the revised grey duration, noted RD, of each activity is determined. The formula for an arbitrary chosen activity, A, is: RD(A) =

EG(A) 2

• Identify the critical path based on the revised grey duration of each activity Having the RD determined for all the activities, the network diagram is constructed, and the times associated with each activity are determined. As a result of the forward and backward steps, the duration of the project is computed, and the critical activities and the critical path are identified. • Determine the α values for all the activities of the project In order to achieve this step, four sub-steps are needs. The first one refers to the calculation of the position weight, noted PW , for each activity. According to Peng et al. [8], the position weight of an activity, A, represents the distance between the project starting time and the middle point of the duration of A divided by the project completion time: PW (A) =

EST (A) + RD(A) − PST 2 CT

where CT is the project completion time and PST is the project starting time (usually equal to zero). The second sub-step refers to the calculus of the degree of grey for each activity, noted DG. The formula provided by Peng et al. [8] reads as follows: DG(A) =

∫a L(t)dt + (b − a) + ∫db R(t)dt ∫dc fA (t) = c d −c d −c

Third sub-step is given by the selection of a proper decision parameter, noted with β. This selection depends on the appetite of the decision-maker to uncertainty. The bigger the β, the higher the uncertainty associated with the activity duration [8]. Fourth and last step represents the computation of the α value for each activity. The α is strictly connected with the idea of an α-cut which divides the area presented in Fig. 7.9 into two parts (an upper and a lower part). Figure 7.10 presents the α-cut line in the context of linear whitenization of a grey function [8]. The determination of the α for an activity A is given by [8]: α(A) = 1 − (PW (A)∗β + DG(A)∗(1 − β))

Theoretical and Practical Approach to Project Scheduling with Grey … Fig. 7.10 The α-cut line

273

f ⊗ (t)

1

α -cut line

α L(t)

c

R(t)

a

b

d

t

• Calculate the buffers for all the activities Three values are determined in this step: FI (A) which denotes the area bounded by the fA (t), FI α (A) which denotes the remainder of the fA (t) area after applying the α-cut and the buffer for activity A, noted BF(A). The formulas for the three variables are provided by Peng et al. [8]: FI (A) = FI α (A) =

b+d −c−a 2

(d + n − m − c)∗αA 2

with: m = a − (a − c)(1 − α) and n = b + (d − b)(1 − α), having the positions as presented in Fig. 7.11 [8]. and: BF(A) =

Fig. 7.11 The position of m and n points after applying the α-cut

FI α (A) ∗ EG(A) FI (A)

f ⊗ (t)

1

α -cut line

α L(t)

c

R(t)

ma

bn

d

t

274

7 Complex Projects Management with Interval Grey Numbers

Table 7.6 Project activities, immediate predecessors, and their associated duration

Activity code

Duration (t.u.) ⊗d

Immediate predecessors

A

–

[2, 3]

B

A

[1, 2]

C

A

[2, 5]

D

B

[4, 6]

E

B, C

[2, 4]

F

C

[5, 6]

G

E

[4, 7]

H

D

[3, 4]

I

F, G

[3, 4]

J

H, I

[2, 5]

• Schedule the project considering the critical chain thinking process According to this step, one should find the critical path and should schedule the project in accordance to the critical path thinking process [8].

Practical Approach to Project Scheduling with Grey Numbers Let’s see in the following how the project scheduling problem can be approached when the duration of the activities is represented by interval grey numbers. For this, let us consider a project described by the activities in Table 7.6. In order to determine the project completion time, the times associated with the activities, the critical activities and the critical path/paths, the AoN representation will be used. The resulting network diagram which includes the activities and their associated grey durations (⊗d ) are presented in Fig. 7.12. B

D

H

[1,2]

[4,6]

[3,4]

A

E

G

J

[2,3]

[2,4]

[4,7]

[2,5]

C

F

I

[2,5]

[5,6]

[3,4]

Fig. 7.12 The AoN network representation for the project in Table 7.6

Theoretical and Practical Approach to Project Scheduling with Grey …

275

The values for the ⊗EST , ⊗EFT , ⊗LST and ⊗LFT are calculated in conformity with the approaches proposed by Xie [6] and Chen et al. [7] as discussed in the following:

Solution Using Xie’s Approach For the forward step, the ⊗EST and ⊗EFT of all the activities are determined by traversing the network diagram from left to right, starting with activity A and ending with activity J . We will highlight in the following how the two values are determined for the activities of the project: Activity A starts at the moment zero, so the ⊗EST (A) = [0, 0]. The ⊗EFT is determined by adding the duration of activity A to its ⊗EST : ⊗EFT (A) = ⊗EFT (A) + ⊗d (A) = [0, 0] = [2, 3] = [2, 3]. Activities B and C are the direct successors of activity A, so their ⊗EST is equal to the ⊗EFT of activity A—they cannot start as long as activity A is still in progress. Therefore, we have: ⊗EST (B) = ⊗EFT (A) = [2, 3] and ⊗EST (C) = ⊗EFT (A) = [2, 3]. The ⊗EFT for B and C is determined adding the duration of the activities to their ⊗EST . Thus, we have: ⊗EFT (B) = ⊗EST (B) + ⊗d (B) = [2, 3] + [1, 2] = [3, 5] and ⊗EFT (C) = ⊗EST (C) + ⊗d (C) = [2, 3] + [2, 5] = [4, 8]. The start of activities D and F depends on the ⊗EFT of the activities B and C. As a result, we have that: ⊗EST (D) = ⊗EFT (B) = [3, 5] and ⊗EST (F) = ⊗EFT (C) = [4, 8]. The ⊗EFT of the activities D and F are determined as: ⊗EFT (D) = ⊗EST (D) + ⊗d (D) = [3, 5] + [4, 6] = [7, 11] and ⊗EFT (F) = ⊗EST (F) + ⊗d (F) = [4, 8] + [5, 6] = [9, 14]. For activity E, as its ⊗EST depends on the ⊗EFT of both the activities B and C, we will have to compare the ⊗EFT (B) = [3, 5] and ⊗EFT (C) = [4, 8] in order to determine which one is superior. After performing the grey comparison, it results that: p(⊗EFT (B) > ⊗EFT (C)) = 0.0625 < 0.5, resulting that ⊗EFT (C) is superior to ⊗EFT (B). Therefore, ⊗EST (E) = ⊗EFT (C) = [4, 8], while ⊗EFT (E) = ⊗EST (E) + ⊗d (E) = [4, 8] + [2, 4] = [6, 12]. In the case of activities G and H , as they depend only on activity E, respectively activity D, their ⊗EST will depend simply on the ⊗EFT of the mentioned activities. So, we have: ⊗EST (G) = ⊗EFT (E) = [6, 12] and ⊗EST (H ) = ⊗EFT (D) = [7, 11]. The ⊗EFT for activities G and H are: ⊗EFT (G) = ⊗EST (G) + ⊗d (G) = [6, 12] + [4, 7] = [10, 19] and ⊗EFT (H ) = ⊗EST (H ) + ⊗d (H ) = [7, 11] + [3, 4] = [10, 15]. Activity I depends on both activities G and F. Thus, for determining ⊗EST (I ), we need to compare ⊗EFT (G) = [10, 19] and ⊗EFT (F) = [9, 14]: p(⊗EFT (G) > ⊗EFT (F)) = 0.8222 > 0.5. Therefore: ⊗EST (I ) = ⊗EFT (G) = [10, 19] and ⊗EFT (I ) = ⊗EST (I ) + ⊗d (I ) = [10, 19] + [3, 4] = [13, 23]. Last, for activity J we have to determine: p(⊗EFT (H ) > ⊗EFT (I )) = 0.040 < 0.5. As a result, ⊗EST (M ) = ⊗EFT (I ) = [13, 23] and ⊗EFT (J ) = ⊗EST (J ) + ⊗d (J ) = [13, 23] + [2, 5] = [15, 28].

276

7 Complex Projects Management with Interval Grey Numbers [2,3]

B

[3,5]

[3,5]

[1,2]

[0,0]

A

D

[7,11]

[4,6]

[2,3]

[4,8]

[2,3]

E

[6,12]

C

[4,8]

H

[10,15]

[3,4]

[6,12]

[2,4]

[2,3]

[7,11]

G

[10,19]

[13,23]

[4,7]

[4,8]

[2,5]

F [5,6]

[9,14]

J

[15,28]

[2,5]

[10,19]

I

[13,23]

[3,4]

Fig. 7.13 The forward step

After filling-in the ⊗EST and ⊗EFT of all the activities, we know that the project completion time is equal to [15, 28]—Fig. 7.13. Next, the backward step should be completed in order to get the ⊗LST and ⊗LFT of all the activities and to determine the critical path/paths and the critical activities. The ⊗LFT for the last activity, J , equals its ⊗EFT , ⊗LFT (J ) = ⊗EFT (J ) = [15, 28] while the value of the ⊗LST is determined by subtracting the ⊗d (J ) from the ⊗LFT (J ), ⊗LST (J ) = ⊗LFT (J ) − ⊗d (J ) = [15, 28] − [2, 5] = [13, 23]. The ⊗LFT for activity I equals the ⊗LST of its direct successor activity J , ⊗LFT (I ) = ⊗EFT (J ) = [13, 23], while ⊗EFT (I ) is determined by the ⊗d (I ) from the ⊗LFT (I ), ⊗LST (I ) = ⊗LFT (I )−⊗d (I ) = [13, 23]−[3, 4] = [10, 19]. On the other hand, the ⊗LST for activity H is calculated taking into account the fact that the duration of H is independent with ⊗LFT and therefore, the ⊗LFT (H ) = [23, 23], while ⊗LST (H ) = Upper(⊗LFT (H )) − ⊗d (H ) = [23, 23] − [3, 4] = [19, 20]. We will proceed in the same manner for the remainder of the activities. Additional to checking the independence/non-independence condition, in the case of the activities with multiple successors, the minimum value of their successors ⊗LST should be considered—this situation occurs in the case of A, B and C activities. The resulting network diagram with all times calculated for all the activities is presented in Fig. 7.14. The critical activities are highlighted in light red.

Solution Using Chen et al.’s Approach The steps presented above for the forward step held the same even in the case of Chen et al. [7] approach (Fig. 7.13). Differences appears only in the determination of the ⊗LST and ⊗LFT of some of the activities during the backward step. For the activities that have a duration that it is non-independent with ⊗LFT , the calculus of the ⊗LST is the same as in Xie [6] approach. As a result, the values for the ⊗LST and ⊗LFT for the activities A, C, E, G, I and J remains the same as in the

Theoretical and Practical Approach to Project Scheduling with Grey … [2,3]

B

[3,5]

[6,7]

[1,2]

[8,8]

[3,5]

D

[7,11]

[7,11]

[14,16] [4,6] [20,20]

A

[2,3]

[4,8]

E

[6,12]

[6,12]

[0,0]

[2,3]

[2,3]

[4,8]

[2,4]

[6,12]

[6,12] [4,7] [10,19]

C

[4,8]

[2,3]

[2,5]

[4,8]

[4,8]

F

H

[10,15]

[19,20] [3,4] [23,23]

[0,0]

[2,3]

277

G

[10,19]

[9,14]

[13,23]

[15,28]

[13,23] [2,5] [15,28]

[10,19]

[13,14] [5,6] [19,19]

J

I

[13,23]

[10,19] [3,4] [13,23]

Fig. 7.14 The backward step using Xie [6] approach

solution provided through the use of Xie [6] approach. As for the remainder of the nodes, namely B, D, F and H , the ⊗LFT of these activities is equal to the ⊗LST of their direct successor activities, while the ⊗LST is determined by subtracting their duration using a grey approach. Let us discuss the ⊗LST and ⊗LFT values obtained for nodes H and B: For node H , the ⊗LFT (H ) = ⊗LST (J ) = [13, 23], while ⊗LST (H ) = ⊗LFT (H ) − ⊗d (H ) = [13, 23] − [3, 4] = [9, 20]. Node B has two direct successor nodes, D and E, thus, first we have to compare ⊗LST (D) with ⊗LST (E) and to determine p(⊗LST (D) > ⊗LST (E)) = 0.7692 > 0.5 thus, as we need to choose the minimum value, the ⊗LST (E) will be selected, which conducts to ⊗LFT (B) = ⊗LST (E) = [4, 8]. Furthermore, as the duration of activity B is independent with the ⊗LFT (B), the ⊗LST (B) = ⊗LFT (B) − ⊗d (B) as grey numbers, conducting to ⊗LST (B) = [2, 7]. The resulting network diagram is presented in Fig. 7.15. The critical activities, which for the critical path, are highlighted in light red.

[2,3]

B

[3,5]

[3,5]

D

[2,7]

[1,2]

[4,8]

[3,16] [4,6]

[7,11]

[9,20]

[9,20] [3,4] [13,23]

[0,0]

A

[2,3]

[4,8]

E

[6,12]

[6,12]

[0,0]

[2,3]

[2,3]

[4,8]

[2,4]

[6,12]

[6,12] [4,7] [10,19]

F

H

[7,11]

[2,3]

C

[4,8]

[4,8]

[2,3]

[2,5]

[4,8]

[4,14] [5,6] [10,19]

G

[9,14]

Fig. 7.15 The backward step using Chen et al. [7] approach

[10,15]

[10,19]

[13,23]

J

[15,28]

[13,23] [2,5] [15,28]

[10,19]

I

[13,23]

[10,19] [3,4] [13,23]

278

7 Complex Projects Management with Interval Grey Numbers

A first observation when comparing the two results obtained through the use of Xie [6] approach and Chen et al. [7] approach is that the duration of the project, its associated critical activities and the resulting critical path are the same. Second, the ⊗EST , ⊗EFT , ⊗LST and ⊗LFT of all the critical activities are the same, while for the non-critical activities only the values for ⊗EST and ⊗EFT are the same. Differences appear in the values of the ⊗LST and ⊗LFT for the non-critical activities. However, if we refer to the ⊗EST and ⊗LST of the non-critical activities, one can observe that Lower(⊗LST ) and Upper(⊗LFT ) are the same in both approaches (e.g. Lower(⊗LST (F)) = 2 and Upper(⊗LFT (F)) = 19). This information is important as, given the fact that the difference between the two approaches appear only in the case of the non-critical activities, which can be delayed with a certain amount of time without delaying the project completion time, it remains in the “hands” of the manager to proper schedule these activities in order not to pass the maximum moment of time at which these activities can start. The same observation can be made for the Lower(⊗EFT ) and Upper(⊗LFT ), which are the same in both approaches. In fact, Chen et al. [7] mention in their paper the limitations of the grey critical path approach that derives from the work under grey information conditions, especially in the case in which the number of activities is large. Even more, the authors state that their proposed approach properly determines the critical path, which has been observed even from the numerical example presented above.

Further Analysis We invite the reader to solve the numerical application by considering the approach proposed by Peng et al. [8], keeping in mind that, in this case, we will have for the duration of each activity: c = a and d = b. If interested, the reader can consider the other approaches proposed in the scientific literature for applying on the proposed project management problem.

Concluding Remarks As shown during this chapter, grey systems theory is used by the researchers in the field to address the complexity of the project scheduling which derives from the uncertainty related to the duration of the involved activities. Thus, in the papers published in this field, the researchers have assumed grey durations for the activities and have tried to determine the times associated with the start/end of each activity, highlighting the critical activities, critical path/paths and determining the minimum completion time. Studying the scientific literature, it can be observed that the papers can be divided into three main categories: using interval grey numbers and differentiate between the independence/non-independence of the activities duration on the latest finishing

References

279

time, proceeding to the whitenization of the grey numbers or using linear models with grey numbers. The common point of all the approaches was the determination of the times associated with the activities, the critical activities, the critical/paths and the project completion time. Some of the researches have focused even on considering multiple objectives (other than minimizing the project completion time) or considering the limited resources for finishing the activities. Based on the proposed approaches, it can be observed that they offer reasonable solutions to the project scheduling problem. Considering the critical activities and the project completion time, the solution offered by the approaches proposed in the field is, in general, a good one, given the uncertainty associated with the activities’ duration. As for the non-critical activities, the situation needs further research as a common ground has not been achieved yet. With all these, the advancements made in the field are considerable and they open the ground for new research which can bring more light to the complex project management problem under uncertainty.

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13. Zhao, Q.-Q., Chen, H.-Z., Yu, B.: Study on a new grey CPM under incomplete information. In: 2009 IEEE International Conference on Grey Systems and Intelligent Services (GSIS 2009). pp. 373–377. IEEE, Nanjing, China (2009). https://doi.org/10.1109/GSIS.2009.5408289 14. Zhongmin, S., Xizu, Y.: Critical path for a grey interval project network. In: 2009 IEEE International Conference on Grey Systems and Intelligent Services (GSIS 2009), pp. 697–701. IEEE, Nanjing, China (2009). https://doi.org/10.1109/GSIS.2009.5408223 15. Xie, N., Liu, S.: Novel methods on comparing grey numbers. Appl. Math. Model. 34, 415–423 (2010). https://doi.org/10.1016/j.apm.2009.05.001

Chapter 8

Hybrid Approaches Featuring Grey Systems Theory

Introduction This chapter is focusing on the papers from the economics and social sciences field that have used a mix approach in their researches, combining the advantages brought by the grey systems theory with the ones provided by fuzzy theory, genetic algorithms, neural networks or rough sets. A bibliometric analysis is conducted, followed by a brief review of the top-5 most cited articles from each category. In order to select the papers that have used hybrid approaches, we have started from the 869 articles database extracted through the use of the following keywords in title, abstract or keywords section: • “grey system”, “grey numbers”, “grey cluster”, “grey control”, “grey decision”, “grey incidence”, “grey model”, “grey theory”, “grey sequence”, “grey prediction” keywords and of • “gray system”, “gray numbers”, “gray cluster”, “gray control”, “gray decision”, “gray incidence”, “gray model”, “gray theory”, “gray sequence”, “gray prediction” keywords and manually selected for ensuring that the papers are from the economics and social sciences area. More information regarding the database extraction and analysis has been provided in Chap. 1. The time period for which the data have been extracted from Web of Science database is 1982—the year the first paper has been written of grey systems theory— and 2021—the year marking 40-years of grey systems theory. As expected, the hybridization papers would have a publishing year after 1987—the year in which the first paper on an economic theme has been published. In this chapter, as we are interested in the hybrid approaches, a further search has been made on the title, abstract or keywords section using the “fuzzy”, “neural”, “genetic” and “rough” keywords. The resulting lists have been further read in order

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Delcea and L.-A. Cotfas, Advancements of Grey Systems Theory in Economics and Social Sciences, Series on Grey System, https://doi.org/10.1007/978-981-19-9932-1_8

281

282

8 Hybrid Approaches Featuring Grey Systems Theory

to ensure the fact that the four mentioned theories used for hybridization have been used in the selected papers. The data has been analyzed through bibliometrics, content, visualization and citation analysis. Biblioshiny software developed by Aria and Cuccurullo [1] has been used for data visualization and analysis. The software can be used by installing the bibliometrix R package and by adding the following text: library(bibliometrix) and biblioshiny(). In the following, the four resulted databases after applying the additional keywords are analyzed and discussed. In order to ensure comparability—for the interested parties—with the analysis provided in Chap. 1, the titles and the contents of the common tables and figures have been kept in the same format.

Grey-Fuzzy Approaches in Economics and Social Sciences As a result of the additional condition related to the existence of word “fuzzy” in the keywords the number of documents retained into analysis has been equal to 53. These papers are analyzed in the following.

Dataset Overview First, it can be observed that the timespan for the 53 selected documents is 2021– 2021—marking a 20-years period in which the grey systems theory has been used along with fuzzy theory in the area of economics and social sciences (Table 8.1). In terms of sources, 42 journals have been identified, indicating that there are journals in which the papers featuring hybrid approaches have been published more than once. The value 7.37 recorded for the average years from publication shows that more papers have been written in the recent years than at the start of 2001. Table 8.1 Main information about data related to grey-fuzzy approaches

Indicator

Value

Timespan

2001:2021

Sources (journals, books, etc.)

42

Documents

53

Average years from publication

7.37

Average citations per documents

24.11

Average citations per year per doc.

2.906

References

2110

Grey-Fuzzy Approaches in Economics and Social Sciences

283

The average citations per documents is 24.11, which can be considered a high value when compared to the average citations per document one encounters for the papers written in the economics and social sciences area. This indicator is comparable to the value obtained for the average citations per document in the case of economics and social sciences papers which use grey systems (23.87). As expected, even the average citations per year per document is taking a high value of 2.906—Table 8.1, while the number of references is equal to 2110. A total number of 216 author’s keywords have been extracted, conducting to an approximative value of 4 keywords/document—Table 8.2. The number of keywords plus—the index terms automatically generated from the titles of cited articles—is equal to 152 (approximatively 2.87 keywords plus/document). In terms of authors and authors collaboration, the data in Table 8.3 have been extracted. In the 53 documents, there are 141 author appearances. There are 12 authors of single-authored documents and 14 single-authored documents, showing that some of the single-authors have authored more than one paper. The number of documents per author is 0.449, while the authors per document are 2.23. The value of the co-authors per document is 2.66, smaller than in the case of the papers written in the economics and social sciences area that use grey systems theory (with an average of 3.14 co-authors on document). This value can be due to the fact that the single-author papers are more (in percentage) than the one written in the economics and social sciences with grey systems theory (26.4% vs. 11.2%) (Table 8.4). Table 8.2 Document contents

Table 8.3 Authors

Table 8.4 Authors collaboration

Indicator

Value

Keywords plus (ID)

152

Author’s keywords (DE)

216

Indicator

Value

Author appearances

141

Authors of single-authored documents

12

Authors of multi-authored documents

106

Indicator

Value

Single-authored documents

14

Documents per author

0.449

Authors per document

2.23

Co-authors per documents

2.66

Collaboration index

2.72

284

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.1 Annual scientific production evolution

The collaboration index is 2.72, which can be considered a normal value for the papers written in the economics and social sciences area. The evolution of the annual scientific production is depicted in Fig. 8.1. As expected from the data presented in Table 8.1 related to the average years from publication, it was expected that a higher number of papers to be in the 2011–2021 period than in the 2001–2011 period. Even more, it can be observed from Fig. 8.1 that there has been recorded a peak for the year 2020 in which 9 papers have been written. Furthermore, the annual growth rate has been determined to be equal to 9.89%, recording a good value, even though it is a smaller than the 21.7% growth rate recorded for the economics and social sciences papers with grey systems theory. Figure 8.2 provides the evolution of the annual average article citations per year evolution. It can be noticed that the values for this indicator are smaller than 10 in most of the analyzed years, except for 2009, when a value of 20.2 has been recorded.

Sources Considering the journals in which the grey-fuzzy hybrid papers have been published in, the top-6 most relevant journals based on the number of publications have been extracted. The remainder of 36 sources have had 1 paper published each and have not been listed in Fig. 8.3.

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Fig. 8.2 Annual average article citations per year evolution

Fig. 8.3 Top-6 most relevant journals

The journal with the higher number of published papers in the area of economics and social sciences with a grey-fuzzy approach is Journal of Intelligent and Fuzzy Systems with 5 publications, followed by Expert Systems with Applications with 4 papers. Other 4 journals have published 2 papers each: Energies, International Journal of Advanced Manufacturing Technology, Journal of Grey System and Sustainability—Fig. 8.3.

286

8 Hybrid Approaches Featuring Grey Systems Theory

As expected, the two journals dedicated to the grey systems research have published papers on grey-fuzzy approach in the economics and social sciences area: Journal of Grey System—2 papers, and Grey Systems—Theory and Application journal—1 paper.

Authors The top-18 authors based on the number of papers published using a grey-fuzzy approach for their papers in the area of economics and social sciences are listed in Fig. 8.4. The remainder of the authors (until 141) have published only one paper and are not listed in Fig. 8.4. The author with the most papers is Mahmoudi A (4 papers), followed by Hu Y.C., Liu S.F. and Tavakkoli-Moghaddam R. (with 3 papers each). The authors with 2 papers are: Ali S.M., Bagherpour M., Hajiagha S.H.R., Hashemi S.S., Javed S.A., Jiang P., Karuppiah K., Li P., Liu J., Mousavi S.M., Sankaranarayanan B., Tsaur R.C., Tseng M.L., and Wang Y.F. Some of the mentioned authors (e.g. Professor Liu SF and Professor Javed SA) are top contributors even on the dataset extracted for the grey systems theory presented in Chap. 1. The most relevant affiliations in terms of number of articles are presented in Fig. 8.5. As expected, the Nanjing University of Aeronautics and Astronautics— which led the top universities in the case of grey systems applications presented in Chap. 1—shares the first place with Islamic Azad University, both of them

Fig. 8.4 Top-18 authors based on number of documents

Grey-Fuzzy Approaches in Economics and Social Sciences

287

Fig. 8.5 Top-18 most relevant affiliations

having 5 papers published in the area of economics and social sciences with grey-fuzzy approach. Southeast University and University of Tehran have 4 papers each—Fig. 8.5. As expected, the country with the most contribution in this area is China—Fig. 8.6, with a MCP (Multiple Country Publications—an inter-country collaboration index) of 2 papers and SCP (Single Country Publications—an intra-country collaboration index) of 25 papers. Iran has a MCP of 2 papers and a SCP of 6 papers, while India has a MCP of 2 papers and a SCP of 1 paper. The remainder of the countries have either only a MCP indicator (e.g. Australia, Malaysia and Romania—1 paper each) or only a SCP indicator (e.g. Turkey with 7 papers; Canada, Thailand and USA—1 paper each). It should be noted that the countries of all the corresponding authors are listed in Fig. 8.6.

Papers’ Analysis The top-5 most cited papers have been selected in order to perform a more in depth analysis in terms of number of authors, the country of the authors, total citations per year and the content of the paper. Table 8.5 presents the information extracted for the top-5 most cited papers.

288

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.6 Top-10 most relevant corresponding author’s country

Top-5 Papers Overview The total citations per year (TCY) is determined by dividing the number of citations to the number of years since the paper has been published, while the Normalized TC (NTC) is an indicator that provide equal credit of citations to all the authors of the paper that cites a document. The most cited paper belongs to Tseng [2]—262 citations, with a TCY of 18.71 and a NTC of 1.00, followed by the papers of Wang [3]—186 citations, a TCY of 8.85 and a NTC of 1.00, and Wang [4]—with 130 citations, a TCY of 6.84 and a NTC of 1.00. It can be observed that all the three papers have a NTC of 1.00 as they are single-authored papers. Furthermore, the paper written by Tseng [2] has also been listed in the top-10 papers in the area of economics and social sciences with grey systems discussed in Chap. 1, being place on the 7th place based on the number of citations. Nevertheless, the papers listed on the 4th and 5th place in the top-5 obtained a high interest from the research community expressed through the number of citations: Golmohammadi and Mellat-Parast [5] with 91 citations and Zarbakhshnia et al. [6] with 54 citations.

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Table 8.5 Top-5 most global cited documents Table Paper (first no. author, year, journal, reference)

Number Country/countries Digital Total of object citations authors identifier (TC) (DOI)

Total Normalized citations TC (NTC) per year (TCY)

1

Tseng M.L., 2009, Expert Systems with Applications, [2]

1

Taiwan

https:// 262 doi.org/ 10.1016/ j.eswa. 2008. 09.011

18.71

1.00

2

Wang Y.F., 2002, Expert Systems with Applications, [3]

1

Taiwan

https:// 186 doi.org/ 10.1016/ S0957417 4(01)000 47-1

8.85

1.00

3

Wang C.H., 2002, Tourism Management, [4]

1

Taiwan

https:// 130 doi.org/ 10.1016/ S0261517 7(03)001 32-8

6.84

1.00

4

Golmohammadi 2 D., 2012, International Journal of Production Economics, [5]

USA

https:// 91 doi.org/ 10.1016/ j.ijpe. 2012. 01.025

8.27

1.86

5

Zarbakhshnia N., 2020, Journal of Cleaner Production, [6]

Iran, Australia, Denmark

https:// 54 doi.org/ 10.1016/ j.jclepro. 2019. 118461

18.00

4.81

4

Top-5 Papers Review and Brief Summary A brief review of the top-5 papers is provided in the following, while a summary of the title, grey systems theory main elements considered in the work, type of data used, purpose and contribution to theory or practice is provided in Table 8.6. As the paper of Tseng [2] has been discussed in Chap. 1, we will no longer provide a review in this chapter. The same situation occurs for the paper written by Golmohammadi and Mellat-Parast [5] on supplier selection which has been discussed in Chap. 3. Wang [3] proposed a system written in Visual BASIC for predicting stock price. The system featured the existence of a graphical display tool and of a prediction agent.

Wang, Y.F., 2002, Expert Systems with Applications, [3]

Wang, C.H., 2002, Predicting tourism Tourism Management, demand using fuzzy [4] time series and hybrid grey theory

Golmohammadi, D., 2012, International Journal of Production Economics, [5]

Zarbakhshnia, N., 2020, Journal of Cleaner Production, [6]

2

3

4

5

Taiwan tourist arrivals from Hong Kong, USA and Germany

Taiwan stock market

Questionnaire

Data

Third-party reverse logistic providers for car parts manufacturing company

Grey relational analysis, Suppliers of products in Grey numbers auto industry

Grey model

Grey model

Grey numbers

Grey systems theory main elements

A novel hybrid multiple Grey numbers attribute decision-making approach for outsourcing sustainable reverse logistics

Developing a grey-based decision-making model for supplier selection

Predicting stock price using fuzzy grey prediction system

A causal and effect decision making model of service quality expectation using grey-fuzzy DEMATEL approach

Tseng, M.L., 2009, Expert Systems with Applications, [2]

1

Title

Paper (first author, year, journal, reference)

No.

Table 8.6 Brief summary of the content of top-5 most global cited documents

Both

Both

Both

Theoretical/practical approach

To propose a new multiple attribute decision-making approach

Both

To develop a model for Both solving the supplier selection problem

To predict the tourism demand

To analyze the stock market and to predict the stock market price

To rank the criteria in customer expectations

Purpose

290 8 Hybrid Approaches Featuring Grey Systems Theory

Grey-Fuzzy Approaches in Economics and Social Sciences

291

In order to conduct the predictions, the GM (1, 1) model is used for predicting the stock price, using a dataset that has been previously pre-classified and fuzzified using fuzzy theory. Experimental results are conducted for data extracted from Taiwan stock market and the effectiveness of the proposed system is demonstrated [3]. As a drawback, the author mentions the fact that most of the model’s parameters are defaulted and cannot be changed by the user [3]. Wang [4] applies fuzzy time series theory and five linguistic values for fuzzifying the historical data extract for the Taiwan tourist arrivals from Hong Kong, USA and Germany. Two models are considered for predictions: the GM (1, 1) model and the Markov-improved model. Based on the simulations results, the author state that the fuzzy time series are suitable to be used along with both GM (1, 1) and Markov-improved model [4]. Furthermore, the author showed that the GM (1, 1) provides more accurate results when used on Hong Kong and USA data, while Markov-improved model offers better results when applied to the Germany dataset. Zarbakhshnia et al. [6] propose a new multiple attribute decision-making approach based on fuzzy analytic hierarchy process (fuzzy AHP) and grey multi-objective optimization. Both fuzzy and grey numbers have been used in the approach and their associate mathematics has been considered. The proposed approach has been applied to a company that assembles engine, gearbox, and axle, and produces main parts of passenger car powertrain [6]. Nine third-party reverse logistic providers for car parts manufacturing company have been considered and ranked based on the proposed approach. The authors conclude that their approach provide better results than the conventional MOORA (multi-objective optimization on the basis of ratio analysis) approach. Considering the information in Table 8.6, it can be observed that the top-5 papers have brough contribution to both theory and practice in the fields they were addressing. Grey numbers and grey model have been the main elements from the grey systems theory that have been used along with fuzzy theory in the selected applications.

Words Analysis The top-10 most frequently used words in the author’s keywords listed in Table 8.7 are underlying on one side the theories used in the selected scientific papers (“grey theory”, “grey systems theory”, “fuzzy”, “grey numbers”, “grey relational analysis”, “dematel”) and on the other side the applicability areas (“project management”, “decision making”, “supplier selection”, “earned value management”). Furthermore, we have analyzed the word clouds for the top-50 words included in keywords plus, keywords, title and abstract—Fig. 8.7. Regarding the top-50 words included in keywords plus, the most encountered word has been “model”—14 times, followed by “optimization”—8 times, “management”—6 times, and “risk”—5 times. The words “algorithm” and “topsis” occurred 4 times each. The remainder of the words listed in Fig. 8.7—section (A)—have had a frequency of up to 3 times.

292

8 Hybrid Approaches Featuring Grey Systems Theory

Table 8.7 Top-10 most frequent words in authors’ keywords

Words

Occurrences

Grey theory

6

Grey system theory

5

Fuzzy

4

Grey numbers

4

Project management

4

Decision making

3

Grey relational analysis

3

Supplier selection

3

Dematel

2

Earned value management

2

(a) Top-50 words based on keywords plus

(b)

Top-50 words based on authors’ keywords

(c)

(d)

Top-50 words based on abstract

Top-50 words based on title

Fig. 8.7 Top-50 words based on keywords plus (a), authors’ keywords (b), title (c), abstract (d)

In the case of the top-50 words based on author’s keywords (Fig. 8.7—section (B)), it can be observed that the most frequently used words are “grey theory”—6 times, “grey systems theory”—5 times, “fuzzy”—4 times, “grey numbers”—4 times and “project management—4 times. All the other words in this section have appeared up to 3 times. For the titles, the frequency of the most-used words are: “grey”—24 times, “fuzzy”—23 times, “model”—16 times, “approach”—9 times, “forecasting”—9 times. The remainder of the words in Fig. 8.7—section (C) have had a frequency of up to 8 times.

Grey-Fuzzy Approaches in Economics and Social Sciences

293

A high frequency for the selected 50-words is encountered in the case of the words extracted from the abstracts: “fuzzy”—119 times, “grey”—98 times, “model—90 times, “method”—84 times. The other words listed in Fig. 8.7—section (D) have had a frequency of up to 61 times. Based on the word clouds it can be underlined, once more, the purpose of the selected papers and the main theories used for the methodological approach. A bigrams and trigrams analysis is provided in Tables 8.8 and 8.9 based on the top-10 most frequent words associations. From Table 8.8 it can be observed that some of the bigrams present in the abstract are also present in the title (e.g. “grey prediction”, “grey system” and “fuzzy grey”) and they are related to the methodological aspects provided in the papers. Other bigrams refer to the applications in which the grey-fuzzy approach has been Table 8.8 Top-10 most frequent bigrams in abstracts and titles Bigrams in abstracts

Occurrences

Bigrams in titles

Occurrences

Supply chain

11

Fuzzy grey

6

Grey prediction

10

Grey model

4

Grey theory

10

Demand forecasting

3

Intuitionistic fuzzy

10

Grey prediction

3

Proposed model

10

Grey system

3

Circular economy

9

Tourism demand

3

Grey system

9

Decision_making approach

2

Prediction model

9

Demand prediction

2

Economy practices

8

Evaluation method

2

Fuzzy grey

8

Goal programming

2

Table 8.9 Top-10 most frequent trigrams in abstracts and titles Trigrams in abstracts

Occurrences

Trigrams in titles

Occurrences

Circular economy practices

8

Fuzzy grey model

3

Fuzzy set theory

7

Grey system theory

2

Fuzzy risk analysis

5

Tourism demand forecasting

2

Grey relational analysis

5

Aggregate production planning

1

Grey system theory

5

Ahp grey theory

1

Internet public opinion

5

Annual biofuel production

1

Project cash flow

5

Attribute decision_making approach

1

Additional query terms

4

Chain collaborative quality

1

Analytic hierarchy process

4

Chinas annual biofuel

1

Supply chain management

4

Circular economy practices

1

294

8 Hybrid Approaches Featuring Grey Systems Theory

used, such as “supply chain”, “circular economy”, “economy practices”, “demand forecasting”, “tourism demand”, and “demand prediction”. Table 8.9 provides the top-10 trigrams for the abstract and titles. The selected trigrams provide more insight on the practical applications in which the grey-fuzzy approach has been utilized: “circular economy practices”, “internet public opinion”, “project cash flow”, “supply chain management”, “tourism demand forecasting”, “aggregate production planning”, “annual biofuel production”, and “chain collaborative quality”.

Mixed Analysis A mixed analysis in terms of countries, authors and journals is provided in Fig. 8.8. As underlined in the previous analyses in this sub-chapter, the most-contributing countries to the number of papers written using grey-fuzzy approach are China and Iran.

Fig. 8.8 Three-fields plot: countries (left), authors (middle), journals (right)

Grey-Neural Networks Approaches in Economics and Social Sciences

295

Grey-Neural Networks Approaches in Economics and Social Sciences Based on the search of word “neural” on the dataset comprising the papers extracted from ISI Web of Science database in the field of economics and social sciences through the use of grey systems, 44 papers have been extracted and retained in the analysis. Another step in which the presence of the neural networks in the selected papers has been performed manually by reading the papers’ titles, keywords and abstracts. The results are discussed below.

Dataset Overview The timespan for the 44 selected papers has been 1998–2021 (Table 8.10), the moment in which the first hybrid grey-neural networks paper has been written, 1998, being three-years earlier than the moment the first hybrid grey-fuzzy paper has been written, 2001. The number of journals in which the papers have been published equals 35. The papers count for 1219 references. The average years from publication is equal to 6, which makes us believe that most of the papers included in the dataset have a recent publication date—Table 8.10. The average citations per document is 14.55, lower than in the case of the hybrid grey-fuzzy papers, 24.11, and lower than in the case of the papers featuring only a grey approach, 23.87. The average citations per year per document is 2.106, lower than in the fuzzy-grey papers, where 2.906 has been recorded, and lower than in the grey papers, with a value of 3.53. In terms of keywords plus, 90 items have been identified, while for the authors’ keywords, 149 items have been extracted—Table 8.11. The authors of the selected papers have been 126, with an author appearance value of 146—Table 8.12. Table 8.10 Main information about data related to grey-neural networks approaches

Indicator

Value

Timespan

1998:2021

Sources (journals, books, etc.)

35

Documents

44

Average years from publication

6

Average citations per documents

14.55

Average citations per year per doc

2.106

References

1219

296 Table 8.11 Document contents

8 Hybrid Approaches Featuring Grey Systems Theory Indicator Keywords plus (ID) Author’s keywords (DE)

Table 8.12 Authors

Value 90 149

Indicator

Value

Authors

126

Author appearances

145

Authors of single-authored documents

3

Authors of multi-authored documents

123

Among them, 3 have been authors of single-authored documents and 123 have been authors of multi-authored documents. Among the hybrid grey-neural networks papers, 8 have been single-authored documents—Table 8.13. As a result of the above values, the following indicators have been determined: documents per author—with a value of 2.93, co-authors per document—with 3.35, and collaboration index—of 3.51. The value of the collaboration index is higher than in the two previously discussed datasets: hybrid grey-fuzzy papers—with 2.72, and grey papers—with 2.23. The annual scientific production is depicted in Fig. 8.9. As expected, based on the data in Table 8.10, a great amount of the selected papers has been written during the recent years. As it can be observed from Fig. 8.9 the year with the most published papers is 2021—10 papers. The annual growth rate has been of 21.15%, higher than in the case of the hybrid grey-fuzzy papers, 9.89%, and comparable with the value recorded for the general grey dataset, 21.7%. The annual average article citations per year evolution is depicted in Fig. 8.10, where a series of peaks can be observed, the most prominent belonging to the year 2005 with a value of 8.2. Table 8.13 Authors collaboration

Indicator

Value

Single-authored documents

8

Documents per author

0.341

Authors per document

2.93

Co-authors per documents

3.35

Collaboration index

3.51

Grey-Neural Networks Approaches in Economics and Social Sciences

Fig. 8.9 Annual scientific production evolution

Fig. 8.10 Annual average article citations evolution

297

298

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.11 Top-5 most relevant journals

Sources The journals with the highest number of publications are depicted in Fig. 8.11: Expert Systems with Applications—with 5 papers, Sustainability—with 3 papers, Energies, Energy, International Journal of Advanced Manufacturing Technology— with 2 papers each. The rest of the journals (until 30) have published a paper each and are not listed in Fig. 8.11. None of the two journals dedicated to the research in the area of grey systems have had a paper on the economics and social sciences area in which a grey-neural network approach has been used.

Authors The authors with the most publications are: Hu YC—11 papers, Jiang P—4 papers, and Choi T.M., Hui C.L., Li G.D., Masuda S., Nagai M. and Yu Y.—with 2 papers each (Fig. 8.12). The remainder of the authors have only one publication and are not listed in Fig. 8.12. The most relevant affiliations based on the number of published papers are provided in Fig. 8.13. Among them, the Fujian Agriculture and Forestry University takes the first place with 13 papers, being shortly followed by Chung Yuan Christian University with 11 papers. The countries from which the corresponding author comes are four as presented in Fig. 8.14 (China, Japan, Australia and Belgium). As expected, China has the highest value for both the MCP (5 documents) and SCP indicators (35 documents). Japan

Grey-Neural Networks Approaches in Economics and Social Sciences

299

Fig. 8.12 Top-8 authors based on number of documents

Fig. 8.13 Top-14 most relevant affiliations

has 1 document for MCP and 1 document for SCP, while Australia and Belgium have a MCP equal to 1.

300

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.14 Top-4 most relevant corresponding author’s country

Papers’ Analysis The top-5 most cited documents have been extracted based on the number of citations. As for the 5th place, there have been 3 papers with an equal number of citations, in the following, we will discuss them all.

Top-5 Papers Overview The information related to the number of authors, DOI, the countries of the authors, the total number of citations, the total number of citations per year, and the normalized total number of citations are presented in Table 8.14. The paper placed on the 1st place has a total number of citations of 139, while the papers on the 2nd—5th place have a number of citations ranging between 31 and 39.

Top-5 Papers Review and Brief Summary A review of the 7 papers placed in top-5 based on the number of citations is provided in this section. Lai et al. [7] aim at creating a model which could help the product designers to determine the best combination of form elements for achieving a desirable product image. To this extent, the authors start by using grey relational analysis in order to determine which are the most influential elements of a product form from a given

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301

Table 8.14 Top-5 most global cited documents Table Paper (first no. author, year, journal, reference)

Number Country/countries Digital Total of object citations authors identifier (TC) (DOI)

Total Normalized citations TC (NTC) per year (TCY)

1

Lai H.H., 3 2005, Computers and Operations Research, [7]

Taiwan, Australia https:// 139 doi.org/ 10.1016/ j.cor. 2004. 03.021

7.72

1.00

2

Liu C., 2016, 6 Expert Systems with Applications, [8]

China

https:// 39 doi.org/ 10.1016/ j.eswa. 2015. 09.052

5.57

1.00

3

Li J., 2018, Energy, [9]

4

China, USA

https:// 38 doi.org/ 10.1016/ j.energy. 2017. 12.042

7.60

2.53

4

Meng M., 2011, Energies, [10]

3

China

https:// 33 doi.org/ 10.3390/ en4 101495

2.75

1.05

5

Chen P.W.D., 4 2013, Applied Mathematics and Information, [11]

Taiwan

https:// doi.org/ 10. 12785/ amis/ 072L12

31

3.10

1.67

6

Choi T.M., 4 IEEE Transactions on Systems, Man and Cybernetics, Part C (Applications and Reviews), [12]

China

https:// 31 doi.org/ 10.1109/ TSMCC. 2011. 2176725

2.82

1.94

(continued)

302

8 Hybrid Approaches Featuring Grey Systems Theory

Table 8.14 (continued) Table Paper (first no. author, year, journal, reference) 7

Number Country/countries Digital Total of object citations authors identifier (TC) (DOI)

Wu W.W., 1 2011, Expert Systems with applications, [13]

Taiwan

https:// 31 doi.org/ 10.1016/ j.eswa. 2011. 04.096

Total Normalized citations TC (NTC) per year (TCY) 2.58

0.99

product image [7]. Then, using a hybrid approach, the grey model, GM (1, 9), and neural networks are put together for predicting and suggesting the best form design combination. The approach is utilized on a sample made by 33 mobile phones, containing 27 form types, and concluded that the grey relational analysis can be included in the neural network model in the particular case in which designers are interested mainly on the most influential elements in form design of mobile phones [7]. Liu et al. [8] proposed a grey neural networks model starting from the classical grey model, GM (1, 1), to which an input layer of back-propagation neural networks is applied. The model is compared to the classical GM (1, 1) and to an improved GM (1, 1) obtained based on the idea that in building a prediction model, it is more likely that the results of the simulations to be affected by short-term rather than long-term data. The data considered for testing have been the sales of the Hunan HX Office and School Furniture Factory—affected by the snowstorm in South China in 2008. The results of the comparison show that the proposed model outperforms the other considered models. As a limitation of the study, the authors mentioned the consideration of a single supply chain enterprise [8]. Li et al. [9] proposed 26 combination models for increasing the prediction accuracy and for eliminating the over-fitting problem in the case of oil consumption in China. GM (1, 1) and grey neural networks have been employed in the proposed models. The authors show that the considered combination models perform better than the traditional approaches. A particular model, the TCM-NNCT (traditional combination method with the proposed no negative constraint theory) proved to be the most feasible and effective when used on oil consumption data [9]. Meng et al. [10] considered historical data related to the monthly electric energy consumption in China. These data have been decomposed and reconstructed considering their frequency. The GM (1, 1) is used for analyzing the trend, while the periodical waves are calculated based on radial basis function to the neural networks field (RBF NNs). The forecasting results are obtained by adding the values resulted from GM and RBF NNs [10]. The authors argue that the proposed approach provide good performances in terms of expected risk and forecasting precision. Chen et al. [11] uses survey questionnaires regarding the service quality of an e-business seller and apply four approaches in order to determine the most suitable

Grey-Neural Networks Approaches in Economics and Social Sciences

303

one. The four approaches are: the principal component analysis (PCA), the fruit fly optimized grey model neural network (FOAGMNN), the grey neural network model (GNNM) and multiple regression (ML). Based on the considered data, the authors determined that FOAGMNN provides the best classification forecast capability, while offering the fastest error convergence [11]. Choi et al. [12] examine various forecasting models: artificial neural networks, GM(1,1), Markov regime switching, and a hybrid GM(1,1)-ANN model for determining the most appropriate one in the case of color trend forecasting of the fashionable products. Based on the data extracted from a fashion company, it has been determined that GM (1, 1)-ANN model provides the best results when confronted with limited data [12]. Wu [13] aims to enhance rankings trustworthiness and proposes a solution that incorporated data envelopment analysis (DEA), grey systems theory and artificial neural network for producing sensitive rankings for travel and tourism competitiveness. The rankings are further merged through the use of Borda count methodology. The author states that the approach is comprehensive and can be used for proper identification of the best performer. Furthermore, the ranked list can be useful for the interested parties (e.g. policy makers) [13]. Table 8.15 provides a brief summary of the content of the above-discussed papers in terms of title, the elements used from grey systems theory, purpose and either the contributions are mainly theoretical or practical or both. Based on the information in Table 8.15 it can be observed that almost all the selected papers have used grey model in the hybridization process with the neural networks. As for the contribution of the papers, it has been observed that the papers have proposed a theoretical approach to the problem under investigation, but have also apply it on real data, bringing both a theoretical and a practical approach to the field.

Words Analysis From the word analysis when considering the authors’ keywords (Table 8.16), it can be observed that the most common words have been “neural network”—16 times, “grey prediction”—10 times, and “grey model”—7 times. These results are in line with the expectations if we consider the above review on the most cited documents. Further a word cloud analysis is conducted for top-50 words in keywords plus, authors’ keywords, title and abstract. The results are depicted in Fig. 8.15. In terms of keywords plus, “algorithm” appears 8 times, “china” and “energy consumption” 7 times each, and “electricity consumption”, “model” and “optimization” 6 times each. All the other words in Fig. 8.15—section (A) appears of up to 5 times. Regarding the authors’ keywords, the most frequent words are “neural network”— 16 times, “grey prediction”—10 times, “grey model”—7 times (Fig. 8.15—section (B)).

Paper (first author, year, journal, reference)

Lai H.H., 2005, Computers and Operations Research, [7]

Liu C., 2016, Expert Systems with Applications, [8]

Li J., 2018, Energy, [9]

Meng M., 2011, Energies, [10]

No.

1

2

3

4

Forecasting monthly electric energy consumption using feature extraction

Analysis and forecasting of the oil consumption in China based on combination models optimized by artificial intelligence algorithms

An improved grey neural network model for predicting transportations disruptions

Form design of product image using grey relational analysis and neural network models

Title

Grey model

Grey model

Grey model

Grey relational analysis, Grey model

Grey systems theory main elements

Table 8.15 Brief summary of the content of top-5 most global cited documents

Monthly electric energy consumption data

To provide a model for helping the enterprises better predict market demand after transportation disruptions

To provide a model for forecasting the electric energy consumption

Both

Both

Both

(continued)

Theoretical/practical approach

To create a model which Both could help the product designers to determine the best combination of form elements for achieving a desirable product image

Purpose

Oil consumption data To propose combination models that provide a desirable forecasting result

Sales of Hunan HX Office and School Furniture Factory

Sample mobile phones

Data

304 8 Hybrid Approaches Featuring Grey Systems Theory

Choi T.M., IEEE Transactions on Systems, Man and Cybernetics, Part C (Applications and Reviews), [12]

Wu W.W., 2011, Expert Systems with applications, [13]

6

7

Grey model

Grey model

Grey systems theory main elements

Beyond travel and Grey model tourism competitiveness ranking using DEA, GST, ANN and Borda count

Color trend forecasting of fashionable products with very few historical data

Chen P.W.D., 2013, Using fruit fly Applied Mathematics optimization algorithm and Information, [11] optimized grey model neural network to perform satisfaction analysis for e-business service

5

Title

Paper (first author, year, journal, reference)

No.

Table 8.15 (continued)

Travel and tourism competitiveness ranking

Sales data from a fashion company

Survey questionnaires

Data

Both

Theoretical/practical approach

To provide a solution for Both produce sensible travel and tourism competitiveness ranking

To examine and compare Both different models in order to determine the most appropriate one for color trend forecast

To determine the best service satisfaction detection model

Purpose

Grey-Neural Networks Approaches in Economics and Social Sciences 305

306

8 Hybrid Approaches Featuring Grey Systems Theory

Table 8.16 Top-10 most frequent words in authors’ keywords

Words

Occurrences

Neural network

16

Grey prediction

10

Grey model

7

bp neural network

6

Neural networks

5

Artificial intelligence

4

Energy demand

4

Genetic algorithm

4

Arima

3

Artificial neural network

3

(a) Top-50 words based on keywords plus

(b) Top-50 words based on authors’ keywords

(c)

(d)

Top-50 words based on title

Top-50 words based on abstract

Fig. 8.15 Top-50 words based on keywords plus (a), authors’ keywords (b), title (c), abstract (d)

In the title, the most used word is “grey”—22 times, followed by “neural”— 21 times, “forecasting” and “prediction”—20 times each, and “network”—17 times (Fig. 8.15—section (C)). Considering the abstract, the most frequent word is “model”—156 times. “Prediction” appears 103 times, while “forecasting” and “grey” are used 83 times each. The words “neural” and “network” appear 57, respectively 44 times (Fig. 8.15—section (D)). The bigram analysis in abstracts revealed the top-10 most frequent bigrams, as listed in Table 8.17. As expected, most of them refer to “neural network”, “grey

Grey-Neural Networks Approaches in Economics and Social Sciences Table 8.17 Top-10 most frequent bigrams in abstracts and titles

307

Bigrams in abstracts

Occurrences

Bigrams in titles

Occurrences

Neural network

38

Neural network

17

Grey prediction

28

Grey prediction

8

Prediction model

28

Demand forecasting

6

Grey model

21

Prediction models

5

bp neural

17

Energy demand

4

Neural networks

15

Grey neural

4

Prediction accuracy

15

Prediction model

4

gm model

13

Artificial intelligence

3

Prediction models

13

Functional_link net

3

Time series

12

Grey model

3

prediction” and “prediction model”. Two of them, namely to “neural network” and “grey prediction” are listed as the top bigrams even in the titles—Table 8.17. The analysis of the most frequent trigrams revealed similar terms as the top term in abstract and titles: “grey prediction model” (in abstract, with an occurrence of 15) and “grey prediction models” (in titles, with an occurrence of 5). Both in abstracts and in the titles, combination of elements belonging to grey systems theory and neural networks can be encountered (Table 8.18).

Mixed Analysis The three fields plot in Fig. 8.16 reveals the connection between the top-authors’ county, their name and the journals they have published in. As it has been noticed from the previous analysis conducted on the selected papers, China has been the country with the highest scientific production, followed by Japan—Fig. 8.16.

308 Table 8.18 Top-10 most frequent trigrams in abstracts and titles

8 Hybrid Approaches Featuring Grey Systems Theory Trigrams in abstracts

Occurrences

Trigrams in titles Occurrences

Grey prediction model

15

Grey prediction models

5

bp neural network

14

Grey neural network

4

Grey prediction models

8

Energy demand forecasting

3

Artificial neural network

7

Forecasting tourism demand

2

Neural network model

6

Gray neural network

2

Proposed grey prediction

6

Grey relational analysis

2

Computer information service

5

Interval grey prediction

2

Grey model gm

5

Neural network model

2

Grey relational analysis

5

Algorithm optimized grey

1

Grey system theory

5

Artificial neural network

1

Grey-Genetic Algorithms Approaches in Economics and Social Sciences The next dataset to be discussed is related to the use of grey systems theory along with genetic algorithms approaches in the scientific papers from the economics and social sciences area.

Dataset Overview The result of the search using the “genetic” keyword performed on the initial dataset from Chap. 1 generated a dataset containing 22 scientific papers—Table 8.19.

Grey-Genetic Algorithms Approaches in Economics and Social Sciences

309

Fig. 8.16 Three-fields plot: countries (left), authors (middle), journals (right)

Table 8.19 Main information about data related to grey-genetic algorithm approaches

Indicator

Value

Timespan

2005:2021

Sources (journals, books, etc.)

19

Documents

22

Average years from publication

6.41

Average citations per documents

36.05

Average citations per year per doc

4.20

References

626

The timespan for the selected papers has been 2005—2021, which have been published in 19 different journals. The number of references in the 22 papers is equal to 626. The average years from publication is 6.41, which gives us a hint related to the fact that most of the papers have been written in the time period which approaches year 2021. The average citations per document is 36.05, higher than in the grey papers from the initial dataset, 23.87, showing a high interest from the scientific community to the works published using a grey-genetic algorithms approach—Table 8.19. A number of 64 keywords plus have been extracted, accompanied by 72 author’s keywords—Table 8.20.

310 Table 8.20 Document contents

Table 8.21 Authors

Table 8.22 Authors collaboration

8 Hybrid Approaches Featuring Grey Systems Theory Indicator

Value

Keywords plus (ID)

64

Author’s keywords (DE)

72

Indicator

Value

Authors

55

Author appearances

62

Authors of single-authored documents

3

Authors of multi-authored documents

52

Indicator

Value

Single-authored documents

3

Documents per author

0.4

Authors per document

2.5

Co-authors per documents

2.82

Collaboration index

2.74

The number of authors is equal to 55, while the authors appearances number is 62—Tables 8.21 and 8.22. There are 3 authors of single-authored documents and 3 single-authored documents—Tables 8.22 and 8.23. The rest of 52 authors have authored papers in collaboration with other authors. The value for the documents for author, 0.4, is comparable with the one recorded for the general dataset in Chap. 1, namely 0.485. The number of co-authors per document is 2.82, an expected value for the papers written in the economics and social sciences field, while the collaboration index is 2.74—Table 8.22. The annual scientific production is presented in Fig. 8.17. The years 2017 and 2018 mark the years with the highest number of papers, namely 5 papers. There are also three periods of time in which no paper has been written using the grey-genetic algorithms in the area of economics and social sciences as it can be observed from Fig. 8.17. The annual growth rate id 8.01%, lower than on the general grey systems dataset where a 21.7% annual growth has been recorded. In terms of citations (Fig. 8.18), it can be observed an increase in the citations number in 2020, reaching a 20 citations value.

Grey-Genetic Algorithms Approaches in Economics and Social Sciences

311

Table 8.23 Top-5 most global cited documents Table Paper (first no. author, year, journal, Reference)

Number Country/countries Digital Total of object citations authors identifier (TC) (DOI)

Total Normalized citations TC (NTC) per year (TCY)

1

Lee Y.S., 2 2011, Energy Conversion and Management, [14]

Taiwan

https:// 203 doi.org/ 10.1016/ j.enc onman. 2010. 06.053

16.92

1.59

2

Huang S.J., 2008, European Journal of Operational Research, [15]

3

Taiwan

https:// 136 doi.org/ 10.1016/ j.ejor. 2007. 07.002

9.07

1.01

3

Wang C.H., 2 2008, Applied Mathematics and Computation, [16]

Taiwan

https:// 132 doi.org/ 10.1016/ j.amc. 2007. 04.080

8.80

0.99

4

Lee Y.S., 2012, Applied Energy, [17]

2

Taiwan

https:// 68 doi.org/ 10.1016/ j.ape nergy. 2012. 01.063

6.18

1.00

5

Hsu L.C., 1 2011, Expert Systems with Applications, [18]

Taiwan

https:// 52 doi.org/ 10.1016/ j.eswa. 2011. 04.192

4.33

0.41

Sources The top-3 most relevant journals based on the number of published scientific papers are: Applied Mathematics and Computation, Applied Soft Computing, and Journal of Intelligent and Fuzzy Systems—all of them with 2 papers published each (Fig. 8.19). The remainder of 16 sources, have published one paper each. Among these sources, we can name the two journals dedicated to the grey systems: Grey Systems—Theory and Application, and Journal of Grey System.

312

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.17 Annual scientific production evolution

Fig. 8.18 Annual average article citations evolution

Grey-Genetic Algorithms Approaches in Economics and Social Sciences

313

Fig. 8.19 Top-3 most relevant journals

Authors The top-8 authors based on the number of publications are presented in Fig. 8.20. The author with the most scientific papers is Hu Y.C.—3 papers, followed by Hsu L.C., Jiang P., Alee Y.S., Tong L.I., and Tsai J.F.—with 2 papers each. The remainder of the authors, not listed in Fig. 8.20 have published only 1 paper in the area of hybrid grey-genetic algorithms. The universities with the highest number of papers are listed in Fig. 8.21. Fujian Agriculture and Forestry University occupies the first place with 4 articles, followed by Chung Yuan Christian University, and Ling Tung University—with 3 papers each.

Fig. 8.20 Top-8 authors based on number of documents

314

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.21 Top-9 most relevant affiliations

Based on the corresponding’s author country, 5 main countries have been identified: China, India, Lithuania, Thailand and Turkey—Fig. 8.22.

Fig. 8.22 Top-20 most relevant corresponding author’s country

Grey-Genetic Algorithms Approaches in Economics and Social Sciences

315

As we are already accustomed, China occupies the first place, with a SCP equal to 18 documents. India, Thailand and Turkey have each a SCP equal to 1 document, while Lithuania has a MCP equal to 1 document.

Papers’ Analysis A top-5 papers analysis is provided in the following in terms of general information and brief review.

Top-5 Papers Overview The data regarding the top-5 cited papers is summarized in Table 8.23. The most-cited document belong to Lee and Tong [14], counting for 203 citations. High values in terms of citations, above 100 citations, are recorded for the papers listed on the second and third place—136 citations and respectively 132 citations. Good values for the citations, given the economic specificity of the authored papers, are obtained for the papers on the fourth and fifth place—68, respectively 52 citations. As presented in Table 8.23, two of the top-5 papers—first and fourth place—have been co-authored by the same authors (namely Lee Y.S. and Tong L.I.). From Table 8.23 it can be observed that all the 5 papers included belong to the authors from Taiwan, which was expected given the information in Fig. 8.22.

Top-5 Papers Review and Brief Summary The above-mentioned top-5 papers are briefly discussed in this section. Lee and Tong [14] propose an improved grey forecasting model in which elements taken from genetic algorithms have been included in order to minimize the forecasting errors. The authors mentioned the need for the proposed approach as the traditional grey model, GM (1, 1), applied on energy consumption data, offered large forecasting errors. The approach has been applied to China energy consumption data. Four models are tested in the same conditions: GM (1, 1), the proposed approach, grey forecasting model proposed by some other authors and a simple linear regression. The results show that on energy consumption data, especially in the case in which the data is small, the proposed model based on grey and genetic algorithm outperforms the other three considered models, providing a low forecasting error [14]. Huang et al. [15] focus on the project management practices related to efforts needed in software development. The authors state the need for an accurate estimate of the efforts in software development and combines the advantages of grey relational analysis and genetic algorithms for deciding the appropriate weights when addressing this issue. The proposed approach is compared with case-based reasoning (CBR), classification and regression trees (CART) and artificial neural networks (ANN),

316

8 Hybrid Approaches Featuring Grey Systems Theory

proving its advantages. The data simulations are made based on COCOMO database and Albrecht dataset [15]. Wang and Hsu [16] considers the grey model to forecast the output of the high tech industry and the genetic algorithms for estimating the parameters of the forecasting model. The resulting approach is used for forecasting the technology industrial output in Taiwan. The authors show that the forecasting results are better when the hybrid approach is used instead of the classical grey forecasting model [16]. Lee and Tong [17] propose a new model based on grey systems theory and genetic algorithms which has a dynamic option. The authors apply the model on both Chinese and USA data related to energy consumption and compares the results with six other models. Among the compared models, the authors have included their model proposed in Lee and Tong [14]—discussed above. The authors have proven that the dynamic hybrid model proposed in the current paper [17] is more accurate and reliable than other forecasting models. Hsu [18] started from the idea that, even though, a series of forecasting models have been developed over time, there are still drawbacks as each one of them needs specific conditions of application. In this context, the author proposed an improvement of the GM (1, 1) model by means of genetic algorithms. The model is compared with three other grey forecasting models, namely the classical GM (1, 1), rolling GM (1, 1) and transformed GM (1, 1). Based on the forecasting results, the author state that the proposed model outperforms the other three grey models bot on in-sample and out-of-sample forecasting [18]. Additionally, the author mentions that the proposed approach can be useful in improving the accuracy of the short-term forecasts [18]. The summary of the discussed papers is provided in Table 8.24. As resulted from the papers’ review and from the data in Table 8.24, it can be observed that most of the papers—4 of 5—have used the genetic algorithms hybridization in conjunction with grey model, and only one paper has referred to grey relational analysis. In terms of theoretical versus practical approach, all the papers have proposed a model obtained through hybridization and have applied it to real data, contributing to both the theory and the practice from the field the papers were referring to.

Words Analysis Top-10 most frequent words in authors’ keywords show that “genetic algorithm” appears 13 times, “genetic algorithms” and “genetic programming” appear 2 times each—Table 8.25. Other popular words are related to grey systems theory and energy consumption. Word clouds have been extracted based on top-50 words in keywords plus, keywords, title and abstract—Fig. 8.23. The most frequent word in keywords plus is “optimization”—6 times, in authors’ keywords is “genetic algorithm”—13 times, in title is “grey”—17 times, in abstract is “model”—73 times.

Grey model

Lee Y.S., 2011, Energy Conversion and Management, [14]

Huang S.J., 2008, European Journal of Operational Research, [15]

Wang C.H., 2008, Applied Mathematics and Computation, [16]

Lee Y.S., 2012, Applied Forecasting nonlinear Energy, [17] time series of energy consumption using a hybrid dynamic model

Hsu L.C., 2011, Expert Systems with Applications, [18]

1

2

3

4

5

Using improved grey forecasting models to forecast the output of opto-electronics industry

Using genetic algorithms grey theory to forecast high technology industrial output

Grey model

Grey model

Grey model

Integration of the grey Grey relational relational analysis with analysis genetic algorithm for software effort estimation

Forecasting energy consumption using a grey model improved by incorporating genetic programming

Grey systems theory main elements

Paper (first author, year, Title journal, reference)

No.

Table 8.24 Brief summary of the content of top-5 most global cited documents

Taiwan opto-electronics industry data

Chinese and USA energy consumption data

Taiwan’s integrate circuit industry

COCOMO database and Albrecht dataset

Chinese energy consumption data

Data

Theoretical/practical approach

Both

Both

To propose an Both improved grey model for the opto-electronics industry

To propose a modified Both grey forecasting model for energy consumption

To provide a high-precision forecasting model for the technology industrial output

To estimate the effort in software development from a project management perspective

To propose a modified Both grey forecasting model for energy consumption

Purpose

Grey-Genetic Algorithms Approaches in Economics and Social Sciences 317

318

8 Hybrid Approaches Featuring Grey Systems Theory

Table 8.25 Top-10 most frequent words in authors’ keywords

Words

Occurrences

Genetic algorithm

13

Grey model

5

Forecasting

4

Energy consumption

3

Neural network

3

Genetic algorithms

2

Genetic programming

2

Grey forecasting model

2

Grey prediction

2

Grey system theory

2

(a) Top-50 words based on keywords plus

(b) Top-50 words based on authors’ keywords

(c)

(d)

Top-50 words based on title

Top-50 words based on abstract

Fig. 8.23 Top-50 words based on keywords plus (a), authors’ keywords (b), title (c), abstract (d)

Considering the bigrams in abstracts and titles, 5 bigrams are common in the two categories in top-10, namely “energy consumption”, “genetic algorithm”, “grey prediction”, “grey model”, “prediction model”—Table 8.26. Based on the trigrams analysis, it can be observed that the main trigrams are referring to the grey model, forecasting and energy—Table 8.27.

Grey-Genetic Algorithms Approaches in Economics and Social Sciences

319

Table 8.26 Top-10 most frequent bigrams in abstracts and titles Bigrams in abstracts

Occurrences

Bigrams in titles

Occurrences

Energy consumption

17

Grey model

7

Genetic algorithm

15

Demand forecasting

4

Grey prediction

11

Energy consumption

4

Forecasting model

10

Genetic algorithm

4

Grey model

10

Grey prediction

4

Natural gas

10

Energy demand

2

Prediction model

10

Forecasting nonlinear

2

gm model

8

Functional_link net

2

Prediction accuracy

8

Model improved

2

Experimental results

7

Prediction model

2

Table 8.27 Top-10 most frequent trigrams in abstracts and titles Trigrams in abstracts

Occurrences

Trigrams in titles

Occurrences

Grey prediction model

5

Energy demand forecasting

2

Natural gas consumption

5

Forecasting nonlinear time

2

Genetic algorithm ga

4

Grey model improved

2

Grey forecasting model

4

Grey prediction models

2

Railway freight volume

4

Nonlinear time series

2

Software effort estimation

4

Adaptive nonlinear grey

1

Food supply chain

3

Algorithm_generalized neural network

1

Forecast energy consumption

3

Algorithms grey theory

1

Forecasting model based

3

awng_bp prediction technique

1

Grey model gm

3

Combining grey model

1

Mixed Analysis The three fields plot featuring the authors’ countries, their names and the journals they have published to limited to a number of 20 items is provided in Fig. 8.24. As expected from the previous discussed data, China leads the authors’ country list, having the greatest contribution when considering the grey genetic algorithm papers in the area of economics and social sciences.

320

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.24 Three-fields plot: countries (left), authors (middle), journals (right)

Grey-Rough Sets Approaches in Economics and Social Sciences The last search performed on the grey systems dataset provided in Chap. 1 has been made by adding the condition that the papers should contain “rough” either in title, abstracts or keywords. The resulting dataset is discussed in this section.

Dataset Overview The search results by using the additional condition related to “rough” word, conducted to a small dataset, made by only 5 papers—Table 8.28. The 5 papers have been published in 4 journals in the time period 2008—2011 and have referenced 172 papers. The average years from publication indicator is equal to 12.2, which corroborated with the timespan, 2008—2011, makes us think that the papers have been published mostly at the end of the timespan (year 2011). This situation is different from the other three searches performed above, where the hybridization papers have been mostly published in the recent years of each period’s timespan and the timespan was ending in 2021. Also, the timespan period is small, being made of only 4 years, and it is placed far away from the current year.

Grey-Rough Sets Approaches in Economics and Social Sciences Table 8.28 Main information about data related to grey-rough sets approaches

321

Indicator

Value

Timespan

2008:2011

Sources (journals, books, etc.)

4

Documents

5

Average years from publication

12.2

Average citations per documents

143.8

Average citations per year per doc

10.91

References

172

Another interesting observation is related to the value recorded for the average citations per document, which is equal to 143.8, outperforming all the other values recorded in the case of the hybrid grey-fuzzy sets, grey-neural networks and greygenetic algorithms selected papers (Table 8.28). As a result, even the values of the average citations per year per document indicator are higher than the ones recorded for the other datasets discussed in this Chapter, reaching a value of 10.91—Table 8.28. The keywords plus are equal to 12, while the author’s keywords are equal to 20—Table 8.29. For the 8 papers, a number of 9 authors have been extracted, with an author appearances value of 10—Table 8.30. There is only one author of single-authored documents and one single-authored documents—Tables 8.30 and 8.31. The other indicators listed in Table 8.31 (documents per author, authors per document, co-authors per document, collaboration index) are falling in a normal range for the economics and social sciences papers. As expected, the highest value for the annual scientific production is recorded for 2011—a year located at the end on the timespan 2008—2011. In 2011, there have been written 2 papers using a hybrid grey- rough sets approach. The other 3 papers have been written in 2008, 2009 and 2010—Fig. 8.25. Table 8.29 Document contents

Table 8.30 Authors

Indicator

Value

Keywords plus (ID)

12

Author’s keywords (DE)

20

Indicator Authors Author appearances

Value 9 10

Authors of single-authored documents

1

Authors of multi-authored documents

8

322 Table 8.31 Authors collaboration

8 Hybrid Approaches Featuring Grey Systems Theory Indicator

Value

Single-authored documents

1

Documents per author

0.556

Authors per document

1.8

Co-Authors per documents

2

Collaboration index

2

Fig. 8.25 Annual scientific production evolution

Given the evolution of the number of papers, the annual growth rate is 25.99%. The average article citations per year is presented in Fig. 8.26. Considering the data in Fig. 8.26, it can be observed that only one peak is recorded in 2010, when the value of the average citations per year indicator has been 42.2.

Sources There are 4 journals who have published the selected papers—Fig. 8.27.

Grey-Rough Sets Approaches in Economics and Social Sciences

323

Fig. 8.26 Annual average article citations evolution

Fig. 8.27 The journals

The journal with the highest number of papers, namely 2 papers, is Expert Systems with Applications. The remaining 3 journals have published 1 paper each: International Journal of Advanced Manufacturing Technology, International Journal of Production Economics, and Journal of Grey System.

324

8 Hybrid Approaches Featuring Grey Systems Theory

Fig. 8.28 The authors and the number of documents

Authors The 9 authors are listed in Fig. 8.28. It can be observed that there is only one author who has published two papers on the hybrid grey-rough sets, namely Sarkis J. The affiliations of the authors are presented in Fig. 8.29. There are 5 universities to which the 9 authors belong to. The university with the highest number of papers in the economics and social sciences area when hybrid grey-rough sets are used is Ling Tung University—with 3 papers. Two university have a record of 2 papers each, namely Clark University, and Kanagawa University—Fig. 8.29. The countries of the corresponding authors are listed in Fig. 8.30. The countries with the highest number of papers are China and USA, each of them having 2 papers. Japan has only 1 paper written with co-authors from the same country—Fig. 8.30.

Papers’ Analysis In this section, the top-5 papers based on the number of citations are discussed.

Top-5 Papers Overview The top-paper based on the number of citations belong to Bai and Sarkis [19], with a number of citations equal to 506—Table 8.32. This paper has been listed even

Grey-Rough Sets Approaches in Economics and Social Sciences

325

Fig. 8.29 The authors’ affiliations

Fig. 8.30 The corresponding author’s country

in Chap. 1 among the top-10 papers that use grey systems theory in the area of economics and social sciences on the second position. The other 4 papers listed in top-5 received a number of citations between 2 and 85 citations. Various countries for the contributing authors are identified, as results from Table 8.32.

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Table 8.32 Top-5 most global cited documents Table Paper (first no. author, year, journal, reference)

Number Country/countries Digital Total of object citations authors identifier (TC) (DOI)

Total Normalized citations TC (NTC) per year (TCY)

1

Bai C., 2010, International Journal of Production Economics, [19]

2

China, USA

https:// 506 doi.org/ 10.1016/ j.ijpe. 2009. 11.023

38.92

1.00

2

Huang K.Y. 2009, Expert Systems with Applications [20]

2

Taiwan

https:// 85 doi.org/ 10.1016/ j.eswa. 2008. 06.103

6.07

1.00

3

Li G.D., 2007, 3 International Journal of Advanced Manufacturing Technology [21]

Japan

https:// 67 doi.org/ 10.1007/ s001700060910-y

4.47

1.00

4

Bai C., 2011, Expert Systems with Applications, [22]

2

China, USA

https:// 59 doi.org/ 10.1016/ j.eswa. 2011. 02.137

4.92

1.95

5

Wang K.C., 2011, The Journal of grey system, [23]

1

Taiwan

NA

0.167

0.066

2

Top-5 Papers Review and Brief Summary The above-mentioned top-5 papers are discussed and summarized in Table 8.33. Furthermore, as a review is provided in Chap. 1 of this book for the paper authored by Bai and Sarkis [19], and the paper is also mentioned and discussed in Chap. 3 as the topic of the paper is related to supplier selection, we will not discuss this paper here. The same observation applies to the paper written by Li et al. [21] on the supplier selection problem, listed on the third position in the top-5, which will no longer be presented here as it has been described in Chap. 3. Huang and Jane [20] combine the moving average autoregressive exogenous prediction model with elements taken from grey systems theory and tough sets. The grey relational analysis and the GM (1, N) model are incorporated in the proposed

Evaluating supplier development programs with grey based rough set methodology

The evaluation of e-commerce homepage design through grey-based Kansei engineering scheme

Huang K.Y. 2009, Expert Systems with Applications [20]

Li G.D., 2007, International Journal of Advanced Manufacturing Technology [21]

Bai C., 2011, Expert Systems with Applications, [22]

Wang K.C., 2011, The Journal of grey system, [23]

2

3

4

5

A grey-based rough decision-making approach to supplier selection

A hybrid model for stock market forecasting and portfolio selection based on ARX, grey system and RS theories

Synthetic data

Data

Grey model

Grey numbers

E-commerce homepages

Synthetic data

Grey relational analysis, Synthetic data Grey numbers

Grey relational analysis, New Taiwan Economy Grey model electronic stock data

Grey numbers

Bai C., 2010, International Journal of Production Economics, [19]

1

Integrating sustainability into supplier selection with grey system and rough set methodologies

Grey systems theory main elements

Paper (first author, year, Title journal, reference)

No.

Table 8.33 Brief summary of the content of top-5 most global cited documents Theoretical/practical approach

Both

Both

To evaluate the e-commerce homepages in terms of users’ preference

Both

To identify the Both important practices and programs related to suppliers’ performance

To provide a solution to the supplier selection problem

To create an automatic stock market forecasting and portfolio selection mechanism

To expand the supplier Both selection methodology by adding sustainability attributes

Purpose

Grey-Rough Sets Approaches in Economics and Social Sciences 327

328

8 Hybrid Approaches Featuring Grey Systems Theory

approach. The grey relational analysis is combined with rough sets for processing the data, which is further analyzed using the grey multivariate model. The proposed approach is compared with the GM (1, 1) model on electronic stock data. The model proposed by Huang and Jane [20] outperforms the GM (1, 1) model, yielding a greater rate of return for the selected stocks. Bai and Sarkis [22] aims to identify the important practices and programs related to suppliers’ performance. To this extent, the authors provide a comprehensive review of the main category of supplier development practices and activities from the scientific literature. In terms of methodology, the grey numbers are used along with the rough sets for determining the most effective determinants on the performance outcomes for the supplier development programs. Wang [23] starts from the Kensei engineering scheme and designs an approach in which combines the grey systems theory and rough sets on the purpose of connecting the content of the e-commerce web pages and the users’ preferences. The author state that the proposed approach offers a good prediction ability [23]. In the case of the hybrid grey-rough sets papers, it can be observed that several elements from grey systems theory (grey numbers, grey model, grey relational analysis) have been used in hybridization.

Words Analysis The top words extracted from the authors’ keywords reveal the use of various elements from the grey systems theory and rough sets—Table 8.34. The word clouds presented in Fig. 8.31 highlight the most used words in keywords plus, authors’ keywords, titles, and abstracts. The bigrams and trigrams analysis provided in Tables 8.35 and 8.36 highlight once more the use of the grey systems theory along with the rough sets in the selected papers. Table 8.34 Top-10 most frequent words in authors’ keywords

Words

Occurrences

Grey relational analysis

2

Grey system

2

Rough set

2

Supply chain

2

arx model

1

Benchmarking

1

Environment

1

gm (1 n)

1

Grey-based rough set

1

Grey system theory

1

Grey-Rough Sets Approaches in Economics and Social Sciences

329

(a) Top-50 words based on keywords plus

(b) Top-50 words based on authors’ keywords

(c)

(d)

Top-50 words based on title

Top-50 words based on abstract

Fig. 8.31 Top-50 words based on keywords plus (a), authors’ keywords (b), title (c), abstract (d)

Table 8.35 Top-10 most frequent bigrams in abstracts and titles Bigrams in abstracts

Occurrences

Bigrams in titles

Occurrences

Rough set

7

Grey system

2

Grey system

4

Rough set

2

Set theory

4

Supplier selection

2

Supplier selection

4

arx grey

1

Supply chain

4

Based rough

1

Design elements

3

Decision_making approach

1

System theory

3

Development programs

1

Applications conclude

2

E_commerce homepage

1

Grey relational

2

Engineering scheme

1

Relational analysis

2

Evaluating supplier

1

Mixed Analysis The three-fields plot in Fig. 8.32 show the connection between the three countries, China, Japan and USA, the authors of the selected papers and the journals the papers have been published.

330

8 Hybrid Approaches Featuring Grey Systems Theory

Table 8.36 Top-10 most frequent trigrams in abstracts and titles Trigrams in abstracts

Occurrences

Trigrams in titles

Occurrences

Rough set theory

4

arx grey system

1 1

Grey system theory

3

Based rough set

arx prediction model

2

E_commerce homepage 1 design

Contribution includes introduction

2

Evaluating supplier development

1

Future research directions

2

Grey based rough

1

Grey relational analysis

2

grey_based kansei engineering

1

Approach financial data

1

grey_based rough decision_making

1

Approach takes advantage

1

Kansei engineering scheme

1

Approach utilizes grey

1

Portfolio selection based

1

Automatic stock market

1

Rough decision_making 1 approach

Fig. 8.32 Three-fields plot: countries (left), authors (middle), journals (right)

Concluding Remarks

331

Concluding Remarks The present chapter has been dedicated to the hybrid approaches in the scientific literature in which grey systems theory has been used along with other intelligent artificial techniques for modelling different situations related to the economics and social sciences field. In terms of grey hybridization, four main theories have been chosen, namely fuzzy sets theory, neural networks, genetic algorithm and rough sets. The first hybrid paper in which grey systems theory has been used along with one of the four-mentioned theories has been written in 1998. The paper has used a grey-neural networks approach. The dataset with the highest number of articles has been the grey-fuzzy dataset (53 papers). The second largest dataset was the grey-neural networks dataset (44 papers), followed by the grey-genetic algorithms dataset (22 papers) and grey-rough sets dataset (5 papers). From the analysis of the four datasets corresponding to the four hybridization theories, it has been observed that different timespans and article growing rates have been encountered. Considering the growing rates, the highest value is encountered in the grey-rough sets dataset (25.99%), followed by grey-neural networks (21.15), grey-fuzzy (9.89%) and grey-genetic algorithms (8.01%). On the other hand, in terms of timespan, it can be observed that while grey-fuzzy articles have a 2001– 2021 timespan, grey-neural network have a 1998–2021 timespan, and grey-genetic algorithms have a 2005–2021 timespan, the grey-rough sets have a visibly reduced timespan, namely 2008–2011. The papers in the selected datasets have addressed various aspects related to economics and social sciences area and have brought their contribution to both the theory and to the practice of the discussed subjects. The analysis provided in this chapter has limitations, mostly related to the data collection process. The papers have been limited to the English written papers published in the ISI Web of Science and extracted using some specific keywords. Extending the list of the keywords, the language of the papers, the type of the papers (e.g. including besides the scientific papers published in journals, the papers presented at conferences and published in proceedings or the books and book chapters), the database (e.g. using Scopus along with the ISI Web of Science), the number of the papers would have been different. We hope that the results made in the area of economics and social sciences through the use of grey systems and presented in this book will bring some light for the interest parties and will form a starting base for the documentation and research made by the reader. Thank you!

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