Multi-criteria Decision Making Methods with Bipolar Fuzzy Sets

ISBN-13: 9789819905683 / Angielski

Muhammad Akram;Shumaiza; Jos? Carlos Rodr?guez Alcantud

Opis

This monograph discusses the theoretical and practical development of multicriteria decision making (MCDM). The main purpose of MCDM is the construction of systematized strategies for the "optimisation" of feasible options, as well as the justification of why some alternatives can be declared "optimal". However, at time, we must make decisions in an uncertain environment and such inconvenience gives rise to a much more elaborate scenario. This book highlights models where this lack of certainty can be flexibly fitted in and goes on to explore valuable strategies for making decisions under a multiplicity of criteria. Methods discussed include bipolar fuzzy TOPSIS method, bipolar fuzzy ELECTRE-I method, bipolar fuzzy ELECTRE-II method, bipolar fuzzy VIKOR method, bipolar fuzzy PROMETHEE method, and two-tuple linguistic bipolar fuzzy Heronian mean operators. This book is a valuable resource for researchers, computer scientists, and social scientists alike.

Spis treści

1 TOPSIS and ELECTRE I Methodologies: Bipolar Fuzzy Formulations . . . . . . . . . . . . 17

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17

1.2 Bipolar Fuzzy Sets . . . . . . . . . . . . .18

1.3 Multi-criteria Decision Making Methods . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . 19

1.4 Bipolar Fuzzy TOPSIS Method . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 22

1.5 Bipolar Fuzzy ELECTRE I Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.6 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.7 Bipolar Fuzzy Extended TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

References . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2 TOPSIS Method with Trapezoidal Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . 45

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2 Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3 Bipolar Fuzzy Linguistic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Ranking of Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 (α, β)-Cut of Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 TOPSIS Method Based on Trapezoidal Bipolar Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . 58

2.7 Trapezoidal Bipolar Fuzzy Information System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3 VIKOR Method with Trapezoidal Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Trapezoidal Bipolar Fuzzy VIKOR Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 Comparative Analysis with Trapezoidal Bipolar Fuzzy TOPSIS . . . . . . . . . . . . . . . . . . . . . . 85

3.4 Comparison of Trapezoidal Bipolar Fuzzy VIKOR with Fuzzy VIKOR . . . . . . . . . . . . . . . . 88

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Extended VIKOR Method Based on Complex Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . 91

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Complex Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Structure of Complex Bipolar Fuzzy VIKOR Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.6 Merits of the Presented Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5 Beyond ELECTRE I: A Bipolar Fuzzy ELECTRE II Method . . . . . . . . . . . . . . . . . . . . . 121

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Bipolar Fuzzy ELECTRE II Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3 Comparative Study and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3.1 Bipolar Fuzzy TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3.2 Bipolar Fuzzy ELECTRE I Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3.3 Comparison of bipolar fuzzy ELECTRE II Method with fuzzy ELECTRE II Method139

5.4 Insights and Limitations of the Method Proposed in this Chapter . . . . . . . . . . . . . . . . . . . . . 140

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6 Extended PROMETHEE Method with Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . 143

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 Bipolar Fuzzy PROMETHEE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2.1 Preference Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2.2 The Bipolar Fuzzy PROMETHEE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3 Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.4 Insights of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7 Enhanced Decision-Making Method with Two-Tuple Linguistic Bipolar Fuzzy Sets 165

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.2 The 2-Tuple Linguistic Bipolar Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.3 The 2-Tuple Linguistic Bipolar Fuzzy Heronian Mean Aggregation Operators . . . . . . . . . . 171

7.4 An approach to MAGDM Problem with 2-Tuple Linguistic Bipolar Fuzzy Information . . 178

7.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.6 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.7 Advantages of the proposed strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

O autorze

MUHAMMAD AKRAM is Professor at the Department of Mathematics, University of the Punjab, Lahore, Pakistan. He earned his Ph.D. in fuzzy mathematics from the Government College University, Lahore, Pakistan. His research interests include numerical algorithms, fuzzy graphs, fuzzy algebras, and fuzzy decision support systems. He has published 10 monographs and  485 research articles in international peer-reviewed journals.  According to reports from Stanford University for years 2020, 2021 and 2022, Dr. Akram is ranked in the top 2% of scientists for years 2020, 2021 and 2022 in the world in artificial intelligence and image processing. He has been on the editorial of 15 international academic journals and reviewer/referee for 155 international journals, including Mathematical Reviews (USA) and Zentralblatt MATH (Germany). Under his supervision, 17 Ph.D. students have completed their research and he is currently guiding five more towards the same.

SHUMAIZA is a research scholar at the University of Punjab, Lahore, Pakistan, from where she also has earned her M.Phil. degree in Mathematics. She has won the HEC indigenous scholarship for her Ph.D. program and has published 15 research articles in renowned international peer-reviewed journals. Her area of research is fuzzy systems.

JOSÉ CARLOS R. ALCANTUD is Full Professor at the Department of Economics and Economic History, University of Salamanca, Spain, since 2010, and has been heading the BORDA Research Unit since 2015. He received his Ph.D. in Mathematics in 1996 from the University of Santiago de Compostela, Spain and M.Sc. in Mathematics in 1991 from the University of Valencia, Spain. His research interests include social choice theory, mathematical economics, and fuzzy theory and decision support systems. He has published over 200 papers on these topics and has supervised 3 Ph.D. students on their research.