A Unique Book on Multi-Criteria Decision Making (MCDM) Methods Multi-Criteria Decision Making Methods:
A Comparative Study



by Evangelos Triantaphyllou, Ph.D.


A monograph published on November 2000 (and a reprint in June of 2002) by

Kluwer Publishers Logo Kluwer Academic Publishers

Applied Optimization Series, Vol. 44
ISBN 0-7923-6607-7





TABLE OF CONTENTS
List of Figures................................................xiii 
List of Tables..................................................xix 
Foreword......................................................xxiii 
Preface.........................................................xxv 
Acknowledgments.................................................xxix 

 
1	Introduction to Multi-Criteria Decision Making............1
1.1		Multi-Criteria Decision Making:
		A General Overview................................1
1.2		Classification of MCDM Methods....................3 

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2	Multi-Criteria Decision Making Methods....................5
2.1		Background Information............................5
2.2		Description of Some MCDM Methods..................5 
2.2.1			The WSM Method............................6
2.2.2			The WPM Method............................8
2.2.3			The AHP Method............................9 
2.2.4			The Revised AHP Method...................11
2.2.5			The ELECTRE Method.......................13
2.2.6			The TOPSIS Method........................18

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3 	Quantification of Qualitative Data for 
	MCDM Problems............................................23
3.1		Background Information...........................23
3.2		Scales for Quantifying Pairwise Comparisons......25
3.2.1			Scales Defined on the Interval [9, 1/9]..26 
3.2.2			Exponential Scales.......................28
3.2.3			Some Examples of the Use of 
                        Exponential Scales.......................29
3.3		Evaluating Different Scales......................32
3.3.1			The Concepts of the RCP and CDP 
                        Matrices.................................32
3.3.2			On The Consistency of CDP Matrices.......35
3.3.3			Two Evaluative Criteria..................43
3.4		A Simulation Evaluation of Different Scales......44
3.5		Analysis of the Computational Results............50
3.6		Conclusions......................................53

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4    Deriving Relative Weights from Ratio Comparisons............57
4.1         Background Information...............................57
4.2         The Eigenvalue Approach..............................58
4.3         Some Optimization Approaches.........................60
4.4         Considering The Human Rationality Factor.............61 
4.5         First Extensive Numerical Example....................65  
4.6         Second Extensive Numerical Example...................66 
4.7         Average Error per Comparison for Sets 
            of Different Size....................................67 
4.8         Conclusions..........................................72

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5    Deriving Relative Weights from Difference Comparisons.......73 
5.1         Background Information...............................73
5.2         Pairwise Comparisons of Relative Similarity..........76
5.2.1               Quantifying Pairwise Comparisons  
                    of Relative Similarity.......................76  
5.2.2               Processing Pairwise Comparisons 
                    of Relative Similarity.......................77 
5.2.3               An Extensive Numerical Example...............79  
5.3         Conclusions..........................................85

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6    A Decomposition Approach for Evaluating Relative 
     Weights Derived from Comparisons............................87
6.1         Background Information...............................87
6.2         Problem Description..................................88
6.3         Two Solution Approaches..............................91
6.3.1               A Simple Approach............................91
6.3.2               A Linear Programming Approach................92
6.4         An Extensive Numerical Example.......................95
6.5         Some Computational Experiments.......................97
6.6         Analysis of the Computational Results...............100
6.7         Conclusions.........................................112

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7    Reduction of Pairwise Comparisons Via a 
     Duality Approach...........................................115
7.1         Background Information..............................115
7.2         A Duality Approach for Eliciting Comparisons........116
7.3         An Extensive Numerical Example......................120
7.3.1               Applying the Primal Approach................121
7.3.2               Applying the Dual Approach..................122
7.4         Some Numerical Results for Problems of 
            Different Sizes.....................................124
7.5         Conclusions.........................................128

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8    A Sensitivity Analysis Approach for MCDM Methods...........131
8.1         Background Information..............................131
8.2         Description of the Two Major Sensitivity 
            Analysis Problems...................................133
8.3.        Problem 1:  Determining the Most 
            Critical Criterion..................................135
8.3.1               Definitions and Terminology.................135
8.3.2               Some Theoretical Results in Determining 
                    the Most Critical Criterion.................137
8.3.2.1                      Case (i):  Using the WSM
                             or the AHP Method..................137
8.3.2.2                      An Extensive Numerical Example 
                             for the WSM Case...................138
8.3.2.3                      Case (ii):  Using the WPM 
                             Method.............................142
8.3.2.4                      An Extensive Numerical Example 
                             for the WPM Case...................143
8.3.3               Some Computational Experiments..............145
8.4         Problem 2:  Determining the Most Critical aij 
            Measure of Performance..............................155
8.4.1               Definitions and Terminology.................155
8.4.2               Determining the Threshold Values d/i,j,k.....157
8.4.2.1                      Case (i):  When Using the WSM 
                             or the AHP Method..................157
8.4.2.2                      An Extensive Numerical Example 
                             When the WSM or the 
                             AHP Method is Used.................158
8.4.2.3                      Case (ii):  When Using the WPM
                             Method.............................161
8.4.2.4                      An Extensive Numerical Example  
                             When the WPM Method is Used........161
8.5         Conclusions.........................................165

     Appendix to Chapter 8......................................167
8.6         Calculation of the t1,1,2 Quantity When
            the AHP or the WSM Method is Used...................167
8.7         Calculation of the t1,1,2 Quantity When 
            the WPM Method is Used..............................169
8.8         Calculation of the d3,4,5 Quantity When 
            the WSM Method is Used..............................170
8.9         Calculation of the d3,4,5 Quantity When 
            the AHP Method is Used..............................171
8.10        Calculation of the d3,4,5 Quantity When 
            the WPM Method is Used..............................174

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9    Evaluation of Methods for Processing a Decision Matrix 
     and Some Cases of Ranking Abnormalities....................177
9.1         Background Information..............................177
9.2         Two Evaluative Criteria.............................177
9.3         Testing the Methods by Using the First
            Evaluative Criterion................................179
9.4         Testing the Methods by Using the Second
            Evaluative Criterion................................186
9.5         Analysis of the Computational Results...............192
9.6         Evaluating the TOPSIS Method........................194
9.7         Conclusions.........................................197

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10   A Computational Evaluation of the Original and the
     Revised AHP................................................201
10.1       Background Information...............................201
10.2       An Extensive Numerical Example.......................202
10.3       Some Computational Experiments.......................206
10.4       Conclusions..........................................212

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11   More Cases of Ranking Abnormalities When Some 
     MCDM Methods Are Used......................................213
11.1       Background Information...............................213
11.2       Ranking Irregularities When Alternatives Are
           Compared Two at a Time...............................215
11.3       Ranking Irregularities When Alternatives Are 
           Compared Two at a Time and Also as a Group...........220
11.4       Some Computational Results...........................223
11.5       A Multiplicative Version of the AHP..................228
11.6       Results from Two Real Life Case Studies..............230
11.6.1              Comparative Ranking Analysis of
                    the "Bridge Evaluation" Problem.............230
11.6.2              Comparative Ranking Analysis of
                    the "Site Selection" Problem................232
11.7       Conclusions..........................................233

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12   Fuzzy Sets and Their Operations............................235 
12.1       Background Information...............................235 
12.2       Fuzzy Operations.....................................236 
12.3       Ranking of Fuzzy Numbers.............................238 

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13   Fuzzy Multi-Criteria Decision Making.......................241
13.1       Background Information...............................241
13.2       The Fuzzy WSM Method.................................242
13.3       The Fuzzy WPM Method.................................244
13.4       The Fuzzy AHP Method.................................245
13.5       The Fuzzy Revised AHP Method.........................247
13.6       The Fuzzy TOPSIS Method..............................248
13.7       Two Fuzzy Evaluative Criteria for 
           Fuzzy MCDM Methods...................................250
13.7.1              Testing the Methods by Using the First
                    Fuzzy Evaluative Criterion..................251
13.7.2              Testing the Methods by Using the Second 
                    Fuzzy Evaluative Criterion..................255
13.8       Computational Experiments............................258
13.8.1              Description of the Computational 
                    Results.....................................258 
13.8.2              Analysis of the Computational 
                    Results.....................................261 
13.9       Conclusions..........................................269 

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14   Conclusions and Discussion for Future Research.............263
14.1       The Study of MCDM Methods: Future Trends.............263   
14.2       Lessons Learned......................................264

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References......................................................267


Subject Index...................................................275


Author Index....................................................283


About the Two Editors................................................289

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