26
Multi-Criteria Decision Making Methods: A Comparative Study
Multi-Criteria Decision Making Methods: A Comparative Study
Applied Optimization
Volume 44
Series Editors:
Panos M. Pardalos University of Florida, US.A.
Donald Hearn University of Florida, US.A.
The titles published in this series are listed at the end of this volume.
Multi-Criteria Decision Making Methods: A Comparative Study
by
Evangelos Triantaphyllou Department of Industrial and Manufacturing Systems Engineering, College of Engineering, Louisiana State University, Baton Rouge, Louisiana, US.A.
SPRINGER-SCIENCE+BUSINESS MEDIA B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4838-0 ISBN 978-1-4757-3157-6 (eBook) DOI 10.1007/978-1-4757-3157-6
Printed an acid-free paper
AU Rights Reserved © 2000 Springer Science+Business Media Dordrecht OriginaIly published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1 st edition 2000 No part ot the matenal protecteo by thlS copynght not1ce may be reproouceO or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any infonnation storage and retrieval system, without written pennission from the copyright owner
This book is gratefully dedicated to all my students; of the past, the present, and the future.
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
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
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
viii MCDM Methods: A Comparative Study, by E. Triantaphyllou
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
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
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
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
Table of Contents ix
7.3.2 7.4
7.5
8 8.1 8.2
8.3.
8.3.1 8.3.2
8.3.2.1
8.3.2.2
8.3.2.3 8.3.2.4
8.3.3 8.4
8.4.1 8.4.2
8.4.2.1
8.4.2.2
8.4.2.3
8.4.2.4
8.5
Applying the Dual Approach ........... 122 Some Numerical Results for Problems of Different Sizes . . . . . . . . . . . . . . . . . . . . . . . .. 124 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .. 128
A Sensitivity Analysis Approach for MCDM Methods •••• Background Information . . . . . . . . . . . . . . . . . . . Description of the Two Major Sensitivity Analysis Problems . . . . . . . . . . . . . . . . . . . . . . . Problem 1: Determining the Most Critical
Criterion ....................... . Definitions and Terminology .......... . Some Theoretical Results in Determining the Most Critical Criterion ........... .
Case (i): Using the WSM or the AHP Method ............... . An Extensive Numerical Example
131 131
133
135 135
137
137
for the WSM Case ............ 138 Case (ii): Using the WPM Method . . 142 An Extensive Numerical Example for the WPM Case ............ 143
Some Computational Experiments ........ 145 Problem 2: Determining the Most Critical aij
Measure of Performance ................... 155 Definitions and Terminology ........... 155 Determining the Threshold Values <,j,k . . . . . . . . . . . . . . . . . . . . .. 157
Case (i): When Using the WSM or the AHP Method ............ 157 An Extensive Numerical Example When the WSM or the AHP Method is Used . . . . . . . . . .. 158 Case (ii): When Using the WPM Method ................... 161 An Extensive Numerical Example When the WPM Method is Used . . .. 161
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .. 165
x MCDM Methods: A Comparative Study, by E. Triantaphyllou
Appendix to Chapter 8 . . . . . . . • . . . . . . . . . . . . . . .. 167 8.6 Calculation of the 01 12 Quantity When
the AHP or the WSM Method is Used ..... 167 8.7 Calculation of the 01•1•2 Quantity When
the WPM Method is Used ............. 169 8.8 Calculation of the 7 3.4.5 Quantity When
the WSM Method is Used . . . . . . . . . . . .. 170 8.9 Calculation of the 7 3•4,5 Quantity When
the AHP Method is Used ............. 171 8.10 Calculation of the 7 3,4,5 Quantity When
the WPM Method is Used . . . . . . . . . . . .. 174
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
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
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
Table of Contents xi
11.4 11.5 11.6 11.6.1
11.6.2
11.7
Some Computational Results ................ 223 A Multiplicative Version of the AHP . . . . . . . . . . . 228 Results from Two Real Life Case Studies ........ 230
Comparative Ranking Analysis of the "Bridge Evaluation" Problem ........ 230 Comparative Ranking Analysis of the "Site Selection" Problem ........... 232
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
12 Fuzzy Sets and Their Operations ..•.......••••... 235 12.1 Background Information . . . . . . . . . . . . . . . . . . . 235 12.2 Fuzzy Operations ....................... 236 12.3 Ranking of Fuzzy Numbers . . . . . . . . . . . . . . . .. 238
13 13.1 13.2 13.3 13.4 13.5 13.6 13.7
13.7.1
13.7.2
13.8 13.8.1
13.8.2
13.9
14 14.1
14.2
Fuzzy Multi-Criteria Decision Making • • . . . . . . . . • . . . 241 Background Information . . . . . . . . . . . . . . . . . . . 241 The Fuzzy WSM Method .................. 242 The Fuzzy WPM Method .................. 244 The Fuzzy AHP Method . . . . . . . . . . . . . . . . . . . 245 The Fuzzy Revised AHP Method ............. 247 The Fuzzy TOPSIS Method . . . . . . . . . . . . . . . . . 248 Two Fuzzy Evaluative Criteria for Fuzzy MCDM Methods ................... 250
Testing the Methods by Using the First Fuzzy Evaluative Criterion ............ 251 Testing the Methods by Using the Second Fuzzy Evaluative Criterion ............ 255
Computational Experiments . . . . . . . . . . . . . . . . . 257 Description of the Computational Results . . . . . . . . . . . . . . . . . . . . . . . .. 258 Analysis of the Computational Results . . . . . . . . . . . . . . . . . . . . . . . .. 261
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Conclusions and Discussion for Future Research .....•• 263 The Study of MCDM Methods: Future Trends ......................... 263 Lessons Learned . . . . . . . . . . . . . . . . . . . . . . .. 263
xii MCDM Methods: A Comparative Study, by E. Triantaphyllou
References ..................................... 267
Subject Index ........................... . . . . . . .. 275
Author Index ................................... 283
About the Author ................................ 289
LIST OF FIGURES
1 Figure 1-1: Figure 1-2:
2
3
Figure 3-1: Figure 3-2:
Figure 3-3:
Figure 3-4:
Figure 3-5:
Figure 3-6:
Figure 3-7: Figure 3-8:
4 Figure 4-1:
Figure 4-2:
5
Introduction to Multi-Criteria Decision Making • • .. 1 A Typical Decision Matrix .................. 3 A Taxonomy of MCDM methods (according to Chen and Hwang [1991]) ................... 4
Multi-Criteria Decision Making Methods ••••••.• 5
Quantification of Qualitative Data for MCDM Problems ••.•••••...••...•.•••.. 23 Actual Comparison Values .................. 37 Maximum, Average, and Minimum CI Values of Random CDP Matrices When the Original Saaty Scale is used ....................... 42 Inversion Rates for Different Scales and Size of Set (Class 1 Scales) . . . . . . . . . . . . . . . . . . . . . 46 Indiscrimination Rates for Different Scales and Size of Set (Class 1 Scales) ............... 47 Inversion Rates for Different Scales and Size of Set (Class 2 Scales) ..................... 48 Indiscrimination Rates for Different Scales and Size of Set (Class 2 Scales) ............... 49 The Best Scales ......................... 51 The Worst Scales ........................ 52
Deriving Relative Weights from Ratio Comparisons . 57 Average Residual and CI versus Order of Set When the Human Rationality Assumption is Used (the Results Correspond to 100 Random Observations) . 70 Average Residual and CI versus Order of Set When the Eigenvalue Method is Used (the Results Correspond to 100 Random Observations) . 71
Deriving Relative Weights from Difference Comparisons . . . . • • • . • . . • • • • • • . • . . . • . • . . 73
xiv
6
Figure 6-1:
Figure 6-2:
Figure 6-3:
Figure 6-4:
Figure 6-5:
Figure 6-6:
Figure 6-7:
7
Figure 7-1:
Figure 7-2:
Figure 7-3:
Figure 7-4:
Figure 7-5:
Figure 7-6:
MCDM Methods: A Comparative Study, by E. Triantaphyllou
A Decomposition Approach for Evaluating Relative Weights Derived from Comparisons ••••••••••• 87 Partitioning of the n(n-1)/2 Pairwise Comparisons ...... . . . . . . . . . . . ... . . . . . . . . 90 Error Rates Under the LP Approach for Sets of Different Size as a Function of the Available Comparisons. . . . . . . . . . . . . . . . . . .. 106 Error Rates Under the Non-LP Approach for Sets of Different Size as a Function of the Available Comparisons. . . . . . . . . . . . . . . . . . .. 107 Error Rates Under the LP Approach for Sets of Different Size as a Function of the Common Comparisons . . . . . . . . . . . . . . . . . . ., 108 Error Rates Under the Non-LP Approach for Sets of Different Size as a Function of the Common Comparisons . . . . . . . . . . . . . . . . . . .. 109 Error Rates for the two Approaches as a Function of the Available Comparisons. . . . . . . . .. 110 Error Rates for the two Approaches as a Function of the Common Comparisons . . . . . . . . .. 111
Reduction of Pairwise Comparisons Via a Duality Approach •....•................ 115 Total Number of Comparisons and Reduction Achieved When the Dual Approach is Used. The Number of Criteria n = 5 .............. 125 Total Number of Comparisons and Reduction Achieved When the Dual Approach is Used. The Number of Criteria n = 10 .............. 125 Total Number of Comparisons and Reduction Achieved When the Dual Approach is Used. The Number of Criteria n = 15 .............. 126 Total number of Comparisons and Reduction Achieved When the Dual Approach is Used. The Number of Criteria n = 20 .............. 126 Net Reduction on the Number of Comparisons When the Dual Approach is used. Results for Problems of Various Sizes .......... 127 Percent (%) Reduction on the Number of Comparisons When the Dual Approach is used. Results for Problems of Various Sizes .......... 127
List of Figures xv
8 A Sensitivity Analysis Approach for MCDM Methods .................... 131
Figure 8-1: Frequency of the time that the PT Critical Criterion is the Criterion with the Highest Weight . . . . . . . . . . . . . . . . . . . . .. 149
Figure 8-2: Frequency of the time that the PT Critical Criterion is the Criterion with the Lowest Weight ...................... 149
Figure 8-3: Frequency of the time that the PA Critical Criterion is the Criterion with the Highest Weight . . . . . . . . . . . . . . . . . . . . .. 150
Figure 8-4: Frequency of the time that the PA Critical Criterion is the Criterion with the Lowest Weight ...................... 150
Figure 8-5: Frequency of the time that the AT Critical Criterion is the Criterion with the Highest Weight . . . . . . . . . . . . . . . . . . . . .. 151
Figure 8-6: Frequency of the time that the AT Critical Criterion is the Criterion with the Lowest Weight ...................... 151
Figure 8-7: Frequency of the time that the AA Critical Criterion is the Criterion with the Highest Weight . . . . . . . . . . . . . . . . . . . . .. 152
Figure 8-8: Frequency of the time that the AA Critical Criterion is the Criterion with the Lowest Weight ................. 152
Figure 8-9: Frequency of the time that the AT and PT Definitions point to the Same Criterion. . . . . . . . .. 153
Figure 8-10: Frequency of the time that the AA and PA Definitions point to the Same Criterion. . . . . . . . .. 153
Figure 8-11: Frequency of the time that the AT, PT, AA, and PA Definitions point to the Same Criterion Under the WSM Method .................. 154
Figure 8-12: Rate that the AT Criterion is the one
9
Figure 9-1:
with the Lowest Weight for Different Size Problems Under the WPM Method ............ 154
Evaluation of Methods for Processing a Decision Matrix and Some Cases of Ranking Abnormalities ................. 177 Contradiction Rate (%) Between the
XVI
Figure 9-2:
Figure 9-3:
Figure 9-4:
Figure 9-5:
Figure 9-6:
Figure 9-7:
MCDM Methods: A Comparative Study, by E. Triantaphyllou
WSM and the AHP . . . . . . . . . . . . . . . . . . . . .. 184 Contradiction Rate (%) Between the WSM and the Revised AHP . . . . . . . . . . . . . . . .. 185 Contradiction Rate (%) Between the WSM and the WPM ..................... 185 Rate of Change (%) of the Indication of the Optimum Alternative When a Non-Optimum Alternative is Replaced by a Worse one. The AHP Case. . . . . . . . . . . . . . . . . . . . . . . .. 191 Rate of Change (%) of the indication of the Optimum Alternative When a Non-Optimum Alternative is Replaced by a Worse one. The Revised AHP Case ................... 191 Contradiction Rate (%) Between the WSM and TOPSIS Method ..................... 196 Rate of Change (%) of the Indication of the Optimum Alternative When aNon-Optimum Alternative is Replaced by a Worse one. The TOPSIS Case . . . . . . . . . . . . . . . . . . . . . .. 196
Figure 9-8: Indication of the Best MCDM Method According to Different MCDM Methods. . . . . . . . . . . . . . .. 198
10 A Computational Evaluation of the Original and the Revised AHP . . . . . . . . . . . . . . . . . . .. 201
Figure 10-1: The Failure Rates are Based on 1,000 Randomly Generated Problems. The AHP Case . . . . . . . . . . . 210
Figure 10-2: The Failure Rates are Based on 1,000 Randomly Generated Problems. The Revised AHP Case ..... 211
11 More Cases of Ranking Abnormalities When Some MCDM Methods Are Used •............... 213
Figure 11-1: Contradiction Rates on the Indication of the Best Alternative When Alternatives are Considered Together and in Pairs. The Original AHP Case ................... 225
Figure 11-2: Contradiction Rates on the Indication of the Best Alternative When Alternatives are Considered Together and in Pairs. The Ideal Mode (Revised) AHP Case . . . . . . . . . . . 225
Figure 11-3: Contradiction Rates on the Indication of
List of Figures
Any Alternative When Alternatives are Considered Together and in Pairs.
xvii
The Original AHP Case ................... 226 Figure 11-4: Contradiction Rates on the Indication of
Any Alternative When Alternatives are Considered Together and in Pairs. The Ideal Mode (Revised) AHP Case . . . . . . . . . . . 226
Figure 11-5: Contradiction Rates on the indication of Any Alternative When Alternatives are Considered in Pairs. The Original AHP Case ................... 227
Figure 11-6: Contradiction Rates on the indication of Any Alternative When Alternatives are Considered in Pairs. The Ideal Mode AHP Case ................. 227
12 Fuzzy Sets and Their Operations ............ 235 Figure 12-1: Membership Functions for the Two Fuzzy
Alternatives AJ and A2 .................... 239
13 Fuzzy Multi-Criteria Decision Making . . . . . . . . . 241 Figure 13-1: Membership Functions of the Fuzzy Alternatives
AI' A2, and A3 of Example 13-1 According to the Fuzzy WSM Method . . . . . . . . . . . . . . . . . 243
Figure 13-2: Membership Functions of the Fuzzy Alternatives AJ, A27 and A3 of Example 13-2 According to the Fuzzy WPM Method . . . . . . . . . . . . . . . . . 244
Figure 13-3: Contradiction Rate R11 When the Number of Fuzzy Alternatives is Equal to 3 ........ . . . . . . 259
Figure 13-4: Contradiction Rate R11 When the Number of Fuzzy Alternatives is Equal to 21 ............. 259
Figure 13-5: Contradiction Rate R21 When the Number of Fuzzy Alternatives is Equal to 3 . . . . . . . . . . . . . . 260
Figure 13-6: Contradiction Rate R21 When the Number of Fuzzy Alternatives is Equal to 21 ............. 260
Figure 13-7: Contradiction Rate R12 When the Number of Fuzzy Alternatives is Equal to 3 . . . . . . . . . . . . . . 261
14 Conclusions and Discussion for Future Research .. 263
LIST OF TABLES
1
2
3
Table 3-1:
Table 3-2:
Table 3-3:
4
Table 4-1: Table 4-2: Table 4-3:
Table 4-4:
5
Table 5-1:
6
Table 6-1a: Table 6-1b: Table 6-1c: Table 6-1d:
Introduction to Multi-Criteria Decision Making . . .. 1
Multi-Criteria Decision Making Methods .•.••••. 5
Quantification of Qualitative Data for MCDM Problems ••••.•.•.•.•••••.••.... 23 Scale of Relative Importances (according to Saaty[1980]) ................... 27 Scale of Relative Importances (According to Lootsma[1988]) ................ 28 Two Exponential Scales .................... 29
Deriving Relative Weights from Ratio Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 RCI Values of Sets of Different Order n . . . . . . . . . . 59 Data for the Second Extensive Numerical Example ... 66 Comparison of the Weight Values for the Data in Table 4-2 . . . . . . . . . . . . . . . . . . . . . . 67 Average Residual and CI Versus Order of Set and CR When the Human Rationality Assumption (HR) and the Eigenvalue Method (EM) is used. Results Correspond to 100 Random Observations .... 69
Deriving Relative Weights from Difference Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Proposed Similarity Scale ................... 77
A Decomposition Approach for Evaluating Relative Weights Derived from Comparisons ..••....... 87 Computational Results, Part A . . . . . . . . . . . . . .. 101 Computational Results, Part B . . . . . . . . . . . . . .. 102 Computational Results, Part C . . . . . . . . . . . . . .. 103 Computational Results, Part D . . . . . . . . . . . . . .. 104
xx
7
8
Table 8-1:
Table 8-2: Table 8-3:
Table 8-4:
Table 8-5:
Table 8-6: Table 8-7: Table 8-8:
Table 8-9:
Table 8-10:
Table 8-11:
Table 8-12: Table 8-13: Table 8-14:
Table 8-15:
Table 8-16:
9
Table 9-1:
Table 9-2:
MCDM Methods: A Comparative Study, by Eo Triantaphyllou
Reduction of Pairwise Comparisons Via a Duality Approach ...............••..... 115
A Sensitivity Analysis Approach for MCDM Methods .................... 131 Decision Matrix for the Numerical Example on the WSM 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 139 Current Final Preferences 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 139 All Possible 0k.i,j Values (Absolute Change in Criteria Weights) 0 0 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 140 All Possible Olk.i,j Values (Percent Change in Criteria Weights) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 141 Decision Matrix for the Numerical Example on the WPM 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 143 Current Ranking 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 144 All Possible K Values for the WPM Example 0 0 0 0 0 0 145 Decision Matrix and Initial Preferences for the Example 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 158 Threshold Values 7 /i•j•k (%) in Relative Terms for the WSM/AHP Example 0 0 0 0 0 0 0 0 0 0 0 0 159 Criticality Degrees j),,iij (%) for each aij Performance Measure 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 160 Sensitivity Coefficients sens(aij) for each aij Performance Measure 0 0 0 • • • • • • • • • • • • • • • •• 160 Decision Matrix for Numerical Example . . . . . . . .. 162 Initial Ranking . 0 0 • • • • • • • 0 0 • 0 0 • • • 0 • • • • •• 162 Threshold Values 7/ioj.k (%) in Relative Terms for the WPM Example o. 0 0 0 •• 0 •• 0 •• o. 163 Criticality Degrees b.. lij (in %) for each aij Measure of Performance . . . . 0 • • • 0 • • • 0 0 • • • •• 164 Sensitivity Coefficients sens(aij) for each aij Measure of Performance . . . . . . . . . . . . . . 0 • • o. 164
Evaluation of Methods for Processing a Decision Matrix and Some Cases of Ranking Abnormalities ................. 177 Contradiction Rate (%) Between the WSM and the AHP . 0 •••••••• 0 •••• 0 •• 0 0 • 0 181 Contradiction Rate (%) Between the WSM and the Revised AHP . . . . . . . . . . . . . 0 • • 0 182
List of Tables xxi
Table 9-3: Contradiction Rate (%) Between the WSM and the WPM ..................... 183
Table 9-4: Rate of Change (%) of the Indication of the Optimum Alternative When a Non-Optimum Alternative is Replaced by a Worse One. The AHP Case . . . . . . . . . . . . . . . . . . . . . . . . . 188
Table 9-5: Rate of Change (%) of the Indication of the Optimum Alternative When a Non-Optimum Alternative is Replaced by a Worse One. The Case of the Revised AHP ............... 188
Table 9-6: Summary of the Computational Results . . . . . . . . . . 190 Table 9-7: Contradiction Rate (%) Between the WSM and
the TOPSIS Method . . . . . . . . . . . . . . . . . . . . . . 194 Table 9-8: Rate of Change (%) of the Indication of the
Optimum Alternative When a Non-Optimum Alternative is Replaced by a Worse One. The TOPSIS Case . . . . . . . . . . . . . . . . . . . . . . . 195
10 A Computational Evaluation of the Original and the Revised AHP .................... 201
Table 10-1: The Failure Rates are Based on 1,000 Randomly Generated Problems. The AHP Case ........... 208
Table 10-2: The Failure Rates are Based on 1,000 Randomly Generated Problems. The Revised AHP Case ..... 209
11 More Ranking Abnormalities When Some MCDM Methods Are Used ................ 213
Table 11-1: Priorities and Rankings of the Alternatives in the "Bridge Evaluation" Case Study [Saaty, 1994] ..... 231
12 Fuzzy Sets and Their Operations ...•........ 235
13 Fuzzy Multi-Criteria Decision Making ......... 241
14 Conclusions and Discussion for Future Research .. 263
FOREWORD
Multi-Criteria Decision Making (MCDM) has been one of the fastest growing problem areas during at least the last two decades. In business, decision making has changed over the last decades. From a single person (the Boss!) and a single criterion (profit), decision environments have developed increasingly to become multi-person and multi-criteria situations. The awareness of this development is growing in practice. In theory many methods have been proposed and developed since the sixties to solve this problem in numerous ways.
Two main theoretical streams can be distinguished. First, multiobjective decision making models which assume continuous solution spaces (and therefore are based on continuous mathematics), try to determine optimal compromise solutions and generally assume, that the problem to be solved can be modeled as a mathematical programming model. This is primarily the realm of theoreticians since continuous mathematics is very elegant and powerful and readily allows for many modifications of a basic model or method. Unfortunately mathematical programming does not solve the majority of MCDM-problems in practice, and so these nice and powerful methods are only of limited value for the practitioner. The second stream focuses on problems with discrete decision spaces, i.e. with countable few decision alternatives and basically uses approaches from discrete mathematics, which are mathematically not as elegant as the former. This stream is often called "Multi-Attribute Decision Making". In this book the more general term MCDM is used. These models do not try to compute an optimal solution, but they try to determine via various ranking procedures either a ranking of the relevant actions (decision alternatives) that is "optimal" with respect to several criteria, or they try to find the "optimal" actions amongst the existing solutions (decision alternatives). Even though this type of problem is much more relevant and frequent in practice, there are many fewer methods available and their quality is much harder to determine than in the continuous case. Therefore, the question "Which is the best method for a given problem?" has become one of the most important but also most difficult to answer.
This is exactly where the book of Dr. Triantaphyllou has its focus and why it is that important. Rather than suggesting another MCDM method without any convincing justification, he concentrates on the best known and most frequently used methods. He extensively compares them and makes the reader aware of quite a number of "abnormalities" of some of the methods of which users are often not conscious. He also considers very critically the touchiest points in solving real MCDM problems, namely, quantification of
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qualitative data, deriving weights from ratio and difference comparisons, and especially sensitivity analysis of MCDM methods. This to me seems as valuable or even more so than suggesting a new method which may solve another variant of the MCDM problem. At the end of the book Fuzzy MCDM methods are described and evaluated.
What makes this book so valuable and different from other MCDM books is, that even though the analyses are very rigorous, the results are described very clearly and are understandable even to the non-specialist. Also, very extensive numerical studies and comparisons are presented, which are hard to find in any other text that I know. This book, in fact, provides a unique perspective into the core of MCDM methods and practice. The presented theoretical and empirical analyses are complementary to each other, thus allowing the reader to gain a deep theoretical and practical insight into the topics covered in this book. In addition to this, the author offers at the end of each chapter and at the end of the book suggestions for further research and I can only hope, that his suggestions will be accepted by many scientists.
Dr. Triantaphyllou has been involved in MCDM for almost two decades. He has become internationally known as one of the leading experts in the field and he is, therefore, qualified as hardly anybody else to write this book. I can only congratulate him on his achievement and hope that many practitioners will benefit from this excellent book and that scientists will accept his suggestions for further research as fascinating challenges.
Aachen, Germany, April 2000
Hans-Jtirgen Zimmermann
PREFACE
Probably the most perpetual intellectual challenge in science and engineering is how to make the optimal decision in a given situation. This is a problem as old as mankind. In some ancient civilizations people attempted to solve complex and risky decision problems by seeking advice from priests or the few knowledgeable individuals. In ancient Egypt it was believed that only the kings and the upper clergy could find what is the best solution to a given problem. In classical Greece oracles served a similar purpose.
Many centuries passed since then. Today mankind has replaced the old methods with modern science and technology. The development of scientific disciplines such as operations research, management science, computer science, and statistics, in combination with the use of modern computers, are nothing but aids in assisting people in making the best decision for a given situation. Theories such as linear programming, dynamic programming, hypothesis testing, inventory control, optimization of queuing systems, and multi-criteria decision making have as a common element the search for an optimal decision (solution).
Among the previous methods, there is one class of methods which probably has captured the attention of most of the people for most of the time. This is multi-criteria decision making (MCDM). That is, given a set of alternatives and a set of decision criteria, then what is the best alternative? This problem may come in many different forms. For instance, the alternatives or the criteria may not be well defined, or even more commonly, the -related data may not be well defined. In many real life cases it may even be impossible to accurately and objectively quantify the pertinent data. Often a decision problem can be structured as a multi-level hierarchy. Also, it is not unusual to have a case in which all or part of the data are stochastic or even fuzzy.
The central decision problem examined in this book is how to evaluate and rank the performance of a finite set of alternatives in terms of a number of decision criteria. It is assumed that the decision maker is capable of expressing his/her opinion of the performance of each individual alternative in terms of each one of the decision criteria. The problem then is how to rank the alternatives when all the decision criteria are considered simultaneously.
In the main treatment the data are assumed to be deterministic. In the latter part of this book we also consider the case in which the data are fuzzy. That is, this book does not consider stochastic or probabilistic data. Although this may sound restricted, nevertheless it captures many real life situations, for stochastic data are difficult to be obtained or individual decision makers
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feel uncomfortable dealing with them. The author of this book became actively involved with research in this
area of decision making when he was a graduate student at Penn State University, more than seventeen years ago. What has captured his attention since the early days was the plethora of alternative methods for solving the same MCDM problem. In most cases the authors and supporters of these methods have identified some weaknesses of the previous methods and then they propose a new method claiming to be the best method. As a result, today a decision maker has an array of methods which all claim that they can correctly solve a given MCDM problem. The subjectivity and the tremendous conceptual complexity involved in many MCDM problems make the problem of comparing MCDM methods a challenging and urgent one.
This book presents the research experiences of the author gathered during a long search in finding which is the best MCDM method. Although the final goal of determining the best method seems to be unattainable and utopian, some useful lessons have been learned in the process and are presented here in a comprehensive and systematic manner.
A methodology has been developed for evaluating MCDM methods. This methodology examines methods for estimating the pertinent data and methods for processing these data. A number of evaluative criteria and testing procedures have been developed for this purpose. What became clear very soon is that there is no single method which outperforms all the other methods in all aspects. Therefore, the need which rises is how one can conclude which one is the best method. However, for one to answer the problem of which is the best MCDM method, he/she will first need to use the best MCDM method! Thus, a decision paradox is reached.
This is the main reason why a comparative approach is needed in dealing with MCDM methods. By simply stating various MCDM theories and methods one fails to capture the very real and practical essence of MCDM. The present book attempts to bridge exactly this gap. Although not every MCDM method has been considered in this book, the procedures followed here can be easily expanded to deal with any MCDM method which examines the problem of evaluating a discrete set of alternatives in terms of a set of decision criteria.
This book provides a unique perspective into the core of MCDM methods and practice. It provides many theoretical foundations for the behavior and capabilities of various MCDM methods. This is done by describing a number of lemmas, theorems, corollaries, and by using a rigorous and consistent notation and terminology. It also presents a rich collection of examples, some of which are extensive. A truly unique characteristic of this book is that almost all theoretical developments are accompanied by an extensive empirical analysis which often involved the
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solution of hundreds of thousands or millions of simulated test MCDM problems. The results of these empirical analyses are tabulated, graphically depicted, and analyzed in depth. In this way, the theoretical and empirical analyses presented in this book are complementary to each other, so the reader can gain both a deep theoretical and practical insight of the covered subjects. Another unique characteristic of this monograph is that at the end of almost each chapter there is description of some possible research problems for future research. It also presents an extensive and updated bibliography and references of all the subjects covered. These are very valuable characteristics for people who wish to get involved with new research in MCDM theory and applications. Some of the findings of these comparative analyses are so startling and counter intuitive, that are presented as decision making paradoxes.
Therefore, this book can provide a useful insight for people who are interested in obtaining a deep understanding of some of the most frequently used MCDM methods. It can be used as a textbook for senior undergraduate or graduate courses in decision making in engineering and business schools. It can also provide a panoramic and systematic exposure to the related methods and problems to researchers in the MCDM area. Finally, it can become a valuable guidance for practitioners who wish to take a more effective and critical approach to problem solving of real life multi-criteria decision making problems.
The arrangement of the chapters follows a natural exposition of the main subjects in MCDM theory and practice. Thus, the first two chapters provide an outline and background information of the most popular MCDM methods used today. These are the weighted sum model (WSM) , the weighted product model (WPM), the analytic hierarchy process (AHP) with some of its variants, and the ELECTRE and TOPSIS methods.
The third chapter provides an exposition of some ways for quantifying qualitative data in MCDM problems. This includes discussions on the elicitation of pairwise comparisons and the use of different scales for quantifying them. Chapters four to seven describe some different approaches for extracting relative priorities from pairwise comparisons and also of ways for reducing the number of the required judgments.
Chapter eight is the longest one and it deals with a unified sensitivity analysis approach for MCDM methods. Since no real life decision problem can be considered completely analyzed without a sensitivity analysis, this is a critical subject. As with most of the chapters, this chapter provides an in depth theoretical and empirical analysis of some key sensitivity analysis problems.
Chapters nine to eleven deal with the comparison of different MCDM methods and procedures. Chapter nine presents a comparison of different
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ways for processing a decision matrix. Chapter ten presents a computational study of the AHP and the Revised AHP. Chapter eleven presents some new cases of ranking irregularities when the AHP and some of its additive variants are used. One can claim that these new cases of ranking irregularities are strongly counter intuitive. They have been analyzed both theoretically and empirically.
Chapters twelve and thirteen present some fundamental concepts of fuzzy decision making. As always, the treatments here are accompanied with extensive comparative empirical analyses. Finally, some conclusions and possible directions for future research are discussed in the last chapter.
ACKNOWLEDGMENTS
The research and the writing of this book would never had been accomplished without the decisive help and inspiration from a number of people to which the author is deeply indebted. His most special thanks go to his first M.S. Advisor and Mentor, Professor Stuart H. Mann, currently the Dean of the W.F. Harrah College of Hotel Administration at the University of Nevada. The author would also like to thank his other M.S. Advisor Professor Panos M. Pardalos currently at the University of Florida and his Ph.D. Advisor Professor Allen L. Soyster, currently the Dean of Engineering at the Northeastern University. He would also like to thank Professors F.A. Lootsma, S. Konz, J. Elzinga, T.L. Saaty, L.G. Vargas, E.H. Forman, Dr. P. Murphy; the CEO of InfoHarvest, Inc., and the late Drs. I.H. LaValle, Dr. H.K. Eldin, and C.L. Hwang. The author is also highly appreciative for the excellent comments made by Professor H.-J. Zimmermann.
Many thanks go to his colleagues at LSU; especially to Dr. E. McLaughlin; Dean Emeritus of the College of Engineering at LSU and to Dr. L.W. Jelinski; Vice-Chancellor for Graduate Studies and Research at LSU for her support and inspiration. The author is also very indebted to his distinguished colleague Dr. Tryfon T. Charalampopoulos at LSU. Many special thanks are also given to the Editor Mr. John Martindale at Kluwer Academic Publishers for his encouragement and incredible patience and to his graduate student Vetle I. Torvik for his always thoughtful comments. The author would also like to recognize here the significant contributions to his published papers made by the numerous and anonymous referees and editors of the journals in which his papers have been published.
The author would also like to acknowledge his most sincere gratitude to his graduate and undergraduate students, which have always provided him with unlimited inspiration, motivation and joy.
Evangelos (Vangelis) Triantaphyllou
April 2000
Department of Industrial and Manufacturing Systems Engineering 3128 CEBA Building
Louisiana State University Baton Rouge, LA 70803-6409, U.S.A.
E-mail: [email protected] Personal Web page: http://www.imse. fsu.edu/vangeiis/
