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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
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Kluwer Academic Publishers
Applied Optimization Series, Vol. 44
ISBN 0-7923-6607-7
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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Visit Dr. Triantaphyllou's Homepage
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Special Issue of the Journal "Comps & IE"
on:
Multi-Criteria Decision Making in IE
A Special Issue of Comps. and IE!
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Dr. E. Triantaphyllou (trianta@lsu.edu).
