Pairwise Comparisons Method (eBook)

Preface 7
Introduction 8
References 11
Contents 13
Part I Pairwise Comparisons Method—Theory 17
1 Preliminaries 18
1.1 Fuzzy Sets 18
1.2 Extension Principle 20
1.3 Binary Relations, Valued Relations, and Fuzzy Relations 21
1.4 Fuzzy Quantities, Fuzzy Numbers, and Fuzzy Intervals 22
1.5 Matrices with Fuzzy Elements 24
1.6 Abelian Linearly Ordered Groups 26
References 29
2 Pairwise Comparison Matrices in Decision-Making 31
2.1 Historical Remarks 31
2.2 State of the Art 32
2.3 Problem Definition 33
2.4 Multiplicative Pairwise Comparisons Matrices 34
2.5 Methods for Deriving Priorities from Multiplicative Pairwise Comparison Matrices 37
2.5.1 Eigenvector Method (EVM) 38
2.5.2 Arithmetic Mean Method (AMM) 40
2.5.3 Least Squares Method (LSM) 42
2.5.4 Logarithmic Least Squares Method (LLSM)/Geometric Mean Method (GMM) 43
2.5.5 Fuzzy Programming Method 47
2.6 Desirable Properties of the Priority Vector 48
2.7 Alternative Approach to Derivation of the Priority Vector 54
2.7.1 (Problem 0) 54
2.7.2 Transformation to (Problem ?) 55
2.7.3 Solving (Problem ?) 56
2.7.4 Illustrative Example 59
2.8 Additive Pairwise Comparison Matrices 61
2.8.1 Deriving Priority Vector from Additive PCM 63
2.9 Fuzzy Pairwise Comparison Matrices 68
2.9.1 Some Relations Between Fuzzy Pairwise Comparison Matrices 70
2.9.2 Methods for Deriving Priorities from PCF Matrices 71
2.10 Conclusion 74
References 75
3 Pairwise Comparisons Matrices on Alo-Groups in Decision-Making 80
3.1 Unified Framework for Pairwise Comparisons Matrices over ALO-Groups 80
3.1.1 Introduction 80
3.1.2 Continuous Alo-Groups over a Real Interval 81
3.1.3 Pairwise Comparison Matrices over a Divisible Alo-Group 86
3.2 Desirable Properties of the Priority Vector 88
3.3 Deriving Priority Vector by Solving an Optimization Problem 95
3.3.1 Transformation to (-Problem ?) 96
3.3.2 Solving (-Problem ?) 97
3.4 Generalized Geometric Mean Method (GGMM) 98
3.5 Measuring Consistency of PCM in Alo-Groups 102
3.5.1 Multiplicative Alo-Group 102
3.5.2 Measuring the Inconsistency of PCMs on Alo-Groups 104
3.6 Strong Transitive and Weak Consistent PCM 106
3.6.1 Special Notation 106
3.6.2 -Transitive PCM 107
3.6.3 Weak–Consistent PCM 109
3.6.4 Strong–Transitive PCM 111
3.6.5 Examples 112
3.7 Pairwise Comparison Matrix with Missing Elements 113
3.7.1 Formulation of the Problem 113
3.7.2 Missing Elements of Matrix 115
3.7.3 Problem of Missing Elements in PC Matrices Based on Optimization 116
3.7.4 Particular Cases of PC Matrices with Missing Elements 119
3.7.5 Case L={(1,2),(2,3),@?????,(n-1,n)} 120
3.7.6 Case L={(1,2),(1,3),@?????,(1,n)} 121
3.7.7 Incompleteness Index 123
3.8 Incompleteness—Conclusions 124
3.9 What Is the Best Evaluation Method for Pairwise Comparisons: A Case Study 125
3.9.1 Introduction to Case Study 125
3.9.2 Three Evaluation Systems 125
3.9.3 The Experiment 129
3.9.4 Results of the Experiment 132
3.9.5 Discussion and Conclusions 133
References 134
4 Pairwise Comparisons Matrices with Fuzzy and Intuitionistic Fuzzy Elements in Decision-Making 137
4.1 Introduction 137
4.2 Preliminaries 139
4.3 FPC Matrices, Reciprocity, and Consistency 141
4.4 Desirable Properties of the Priority Vector 151
4.5 Priority Vectors 156
4.6 Measuring Inconsistency of FPC Matrices 159
4.7 Pairwise Comparisons Matrices with Intuitionistic Fuzzy Elements 162
4.7.1 Introduction 162
4.7.2 Preliminaries 164
4.7.3 Pairwise Comparison Matrices with Elements Being Intuitionistic Fuzzy Intervals 166
4.7.4 IFPC Matrices, Reciprocity, and Consistency 168
4.7.5 Priority Vectors of IFPC Matrices 173
4.7.6 Measuring Inconsistency of IFPC Matrices 176
4.8 Conclusion 179
References 180
5 Stochastic Approaches to Pairwise Comparisons Matrices in Decision-Making 183
5.1 Introduction 183
5.2 Basic Models 184
5.3 Linear Models 185
5.3.1 Thurstone–Mosteller Model 187
5.3.2 Bradley–Terry Model 188
5.3.3 Logarithmic Least Squares and the Normal Distribution 189
5.4 Direct Approaches 191
5.4.1 The Kullback–Leibler Distance 193
5.5 Conclusion 195
References 196
Part II Pairwise Comparisons Method—Applications in Decision Making 198
6 Applications in Decision-Making: Analytic Hierarchy Process—AHP Revisited 199
6.1 Introduction 199
6.2 Applications of AHP 200
6.3 Establishing Priorities 201
6.3.1 Normalization of Criteria 204
6.3.2 Basic Scale 205
6.3.3 Calculation of Weights from the Matrix of Pairwise Comparisons 205
6.3.4 Consistency of a PCM 207
6.4 Synthesis 208
6.5 Case Study: Optimal Choice of a Passenger Car 210
6.6 AHP Procedure: Seven Steps in Decision-Making 211
6.7 Case Study: Optimal Choice of a Passenger Car—Continuation from Sect.6.5 217
References 220
7 Applications in Practical Decision-Making Methods: PROMETHEE and TOPSIS 222
7.1 Introduction to PROMETHEE 222
7.2 Formulation of the Problem 223
7.3 Preference Functions 224
7.4 Case Study: Optimal Choice of Personal Computer 228
7.5 Introduction to TOPSIS Method 229
7.6 Description of the TOPSIS Method 230
7.7 The Algorithm 233
7.8 Application of TOPSIS: An Example 234
7.9 Conclusion of Applications of PCMs in Practical Decision-Making Problems 236
References 237
Appendix Index 238
Index 238