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by Jaroslav Ramík (Author)
About the Author
Prof. Jaroslav Ramik, Ph.D. is a professor of Mathematics, Statistics and Operations Research at the Silesian University Opava, School of Business Karvina, Czech Republic. He is a head of Department of Mathematical methods in Economics. His professional interests include optimization methods in economics and decision making. He is an author of 6 books in English and more than 50 papers listed in WoS. He is active in the Czech Society for Operations Research where he served as a former president of the Society in two time periods.
About this Book
This book examines relationships between pairwise comparisons matrices. It first provides an overview of the latest theories of pairwise comparisons in decision making, discussing the pairwise comparison matrix, a fundamental tool for further investigation, as a deterministic matrix with given elements. Subsequent chapters then investigate these matrices under uncertainty, as a matrix with vague elements (fuzzy and/or intuitionistic fuzzy ones), and also as random elements. The second part of the book describes the application of the theoretical results in the three most popular multicriteria decision-making methods: the Analytic Hierarchy Process (AHP), PROMETHEE and TOPSIS. This book appeals to scholars in areas such as decision theory, operations research, optimization theory, algebra, interval analysis and fuzzy sets.
Brief Contents
Part I Pairwise Comparisons Method—Theory
1 Preliminaries 3
1.1 Fuzzy Sets 3
1.2 Extension Principle 5
1.3 Binary Relations, Valued Relations, and Fuzzy Relations 6
1.4 Fuzzy Quantities, Fuzzy Numbers, and Fuzzy Intervals 7
1.5 Matrices with Fuzzy Elements 9
1.6 Abelian Linearly Ordered Groups 11
References 14
2 Pairwise Comparison Matrices in Decision-Making 17
2.1 Historical Remarks 17
2.2 State of the Art. 18
2.3 Problem Definition 19
2.4 Multiplicative Pairwise Comparisons Matrices 20
2.5 Methods for Deriving Priorities from Multiplicative Pairwise Comparison Matrices 23
2.6 Desirable Properties of the Priority Vector 34
2.7 Alternative Approach to Derivation of the Priority Vector 40
2.8 Additive Pairwise Comparison Matrices 47
2.9 Fuzzy Pairwise Comparison Matrices 54
2.10 Conclusion 60
References 61
3 Pairwise Comparisons Matrices on Alo-Groups in Decision-Making 67
3.1 Unified Framework for Pairwise Comparisons Matrices over ALO-Groups 67
3.2 Desirable Properties of the Priority Vector 75
3.3 Deriving Priority Vector by Solving an Optimization Problem 82
3.4 Generalized Geometric Mean Method (GGMM) 85
3.5 Measuring Consistency of PCM in Alo-Groups 89
3.6 Strong Transitive and Weak Consistent PCM 93
3.7 Pairwise Comparison Matrix with Missing Elements 100
3.8 Incompleteness—Conclusions 111
3.9 What Is the Best Evaluation Method for Pairwise Comparisons: A Case Study 112
References 121
4 Pairwise Comparisons Matrices with Fuzzy and Intuitionistic Fuzzy Elements in Decision-Making 125
4.1 Introduction 125
4.2 Preliminaries 127
4.3 FPC Matrices, Reciprocity, and Consistency. 129
4.4 Desirable Properties of the Priority Vector 139
4.5 Priority Vectors 144
4.6 Measuring Inconsistency of FPC Matrices 147
4.7 Pairwise Comparisons Matrices with Intuitionistic Fuzzy Elements 150
4.8 Conclusion 167
References 168
5 Stochastic Approaches to Pairwise Comparisons Matrices in Decision-Making 171
5.1 Introduction 171
5.2 Basic Models 172
5.3 Linear Models 173
5.4 Direct Approaches 179
5.5 Conclusion 183
References 184
Part II Pairwise Comparisons Method—Applications in Decision Making
6 Applications in Decision-Making: Analytic Hierarchy Process—AHP Revisited 189
6.1 Introduction 189
6.2 Applications of AHP. 190
6.3 Establishing Priorities 191
6.4 Synthesis 198
6.5 Case Study: Optimal Choice of a Passenger Car 200
6.6 AHP Procedure: Seven Steps in Decision-Making 201
6.7 Case Study: Optimal Choice of a Passenger Car—Continuation from Sect. 6.5 207
References 210
7 Applications in Practical Decision-Making Methods: PROMETHEE and TOPSIS 213
7.1 Introduction to PROMETHEE 213
7.2 Formulation of the Problem. 214
7.3 Preference Functions 215
7.4 Case Study: Optimal Choice of Personal Computer 219
7.5 Introduction to TOPSIS Method 220
7.6 Description of the TOPSIS Method 221
7.7 The Algorithm 224
7.8 Application of TOPSIS: An Example 225
7.9 Conclusion of Applications of PCMs in Practical Decision-Making Problems 227
References 228
Index 229
Pages: 231 pages
Publisher: Springer; 1st ed. 2020 edition (February 24, 2020)
Language: English
ASIN: B08549KZQ2
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