Preface
Introduction
References
Contents
Part I Pairwise Comparisons Method—Theory
1 Preliminaries
1.1 Fuzzy Sets
1.2 Extension Principle
1.3 Binary Relations, Valued Relations, and Fuzzy Relations
1.4 Fuzzy Quantities, Fuzzy Numbers, and Fuzzy Intervals
1.5 Matrices with Fuzzy Elements
1.6 Abelian Linearly Ordered Groups
References
2 Pairwise Comparison Matrices in Decision-Making
2.1 Historical Remarks
2.2 State of the Art
2.3 Problem Definition
2.4 Multiplicative Pairwise Comparisons Matrices
2.5 Methods for Deriving Priorities from Multiplicative Pairwise Comparison Matrices
2.5.1 Eigenvector Method (EVM)
2.5.2 Arithmetic Mean Method (AMM)
2.5.3 Least Squares Method (LSM)
2.5.4 Logarithmic Least Squares Method (LLSM)/Geometric Mean Method (GMM)
2.5.5 Fuzzy Programming Method
2.6 Desirable Properties of the Priority Vector
2.7 Alternative Approach to Derivation of the Priority Vector
2.7.1 (Problem 0)
2.7.2 Transformation to (Problem ε)
2.7.3 Solving (Problem ε)
2.7.4 Illustrative Example
2.8 Additive Pairwise Comparison Matrices
2.8.1 Deriving Priority Vector from Additive PCM
2.9 Fuzzy Pairwise Comparison Matrices
2.9.1 Some Relations Between Fuzzy Pairwise Comparison Matrices
2.9.2 Methods for Deriving Priorities from PCF Matrices
2.10 Conclusion
References
3 Pairwise Comparisons Matrices on Alo-Groups in Decision-Making
3.1 Unified Framework for Pairwise Comparisons Matrices over ALO-Groups
3.1.1 Introduction
3.1.2 Continuous Alo-Groups over a Real Interval
3.1.3 Pairwise Comparison Matrices over a Divisible Alo-Group
3.2 Desirable Properties of the Priority Vector
3.3 Deriving Priority Vector by Solving an Optimization Problem
3.3.1 Transformation to (-Problem ε)
3.3.2 Solving (-Problem ε)
3.4 Generalized Geometric Mean Method (GGMM)
3.5 Measuring Consistency of PCM in Alo-Groups
3.5.1 Multiplicative Alo-Group
3.5.2 Measuring the Inconsistency of PCMs on Alo-Groups
3.6 Strong Transitive and Weak Consistent PCM
3.6.1 Special Notation
3.6.2 -Transitive PCM
3.6.3 Weak–Consistent PCM
3.6.4 Strong–Transitive PCM
3.6.5 Examples
3.7 Pairwise Comparison Matrix with Missing Elements
3.7.1 Formulation of the Problem
3.7.2 Missing Elements of Matrix
3.7.3 Problem of Missing Elements in PC Matrices Based on Optimization
3.7.4 Particular Cases of PC Matrices with Missing Elements
3.7.5 Case L={(1,2),(2,3),@汥瑀瑯步渠,(n-1,n)}
3.7.6 Case L={(1,2),(1,3),@汥瑀瑯步渠,(1,n)}
3.7.7 Incompleteness Index
3.8 Incompleteness—Conclusions
3.9 What Is the Best Evaluation Method for Pairwise Comparisons: A Case Study
3.9.1 Introduction to Case Study
3.9.2 Three Evaluation Systems
3.9.3 The Experiment
3.9.4 Results of the Experiment
3.9.5 Discussion and Conclusions
References
4 Pairwise Comparisons Matrices with Fuzzy and Intuitionistic Fuzzy Elements in Decision-Making
4.1 Introduction
4.2 Preliminaries
4.3 FPC Matrices, Reciprocity, and Consistency
4.4 Desirable Properties of the Priority Vector
4.5 Priority Vectors
4.6 Measuring Inconsistency of FPC Matrices
4.7 Pairwise Comparisons Matrices with Intuitionistic Fuzzy Elements
4.7.1 Introduction
4.7.2 Preliminaries
4.7.3 Pairwise Comparison Matrices with Elements Being Intuitionistic Fuzzy Intervals
4.7.4 IFPC Matrices, Reciprocity, and Consistency
4.7.5 Priority Vectors of IFPC Matrices
4.7.6 Measuring Inconsistency of IFPC Matrices
4.8 Conclusion
References
5 Stochastic Approaches to Pairwise Comparisons Matrices in Decision-Making
5.1 Introduction
5.2 Basic Models
5.3 Linear Models
5.3.1 Thurstone–Mosteller Model
5.3.2 Bradley–Terry Model
5.3.3 Logarithmic Least Squares and the Normal Distribution
5.4 Direct Approaches
5.4.1 The Kullback–Leibler Distance
5.5 Conclusion
References
Part II Pairwise Comparisons Method—Applications in Decision Making
6 Applications in Decision-Making: Analytic Hierarchy Process—AHP Revisited
6.1 Introduction
6.2 Applications of AHP
6.3 Establishing Priorities
6.3.1 Normalization of Criteria
6.3.2 Basic Scale
6.3.3 Calculation of Weights from the Matrix of Pairwise Comparisons
6.3.4 Consistency of a PCM
6.4 Synthesis
6.5 Case Study: Optimal Choice of a Passenger Car
6.6 AHP Procedure: Seven Steps in Decision-Making
6.7 Case Study: Optimal Choice of a Passenger Car—Continuation from Sect.6.5
References
7 Applications in Practical Decision-Making Methods: PROMETHEE and TOPSIS
7.1 Introduction to PROMETHEE
7.2 Formulation of the Problem
7.3 Preference Functions
7.4 Case Study: Optimal Choice of Personal Computer
7.5 Introduction to TOPSIS Method
7.6 Description of the TOPSIS Method
7.7 The Algorithm
7.8 Application of TOPSIS: An Example
7.9 Conclusion of Applications of PCMs in Practical Decision-Making Problems
References
Appendix Index
Index
