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

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This book is gratefully dedicated to all my students;
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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 Applying the Dual Approach . . . . . . . . . . . 122

7.4 Some Numerical Results for Problems of
Different Sizes . . . . . . . . . . . . . . . . . . . . . . . .. 124
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .. 128

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 <,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
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 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

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 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 . . . . . . . . . . . . . . . . . 257
13.8.1 Description of the Computational
Results . . . . . . . . . . . . . . . . . . . . . . . .. 258
13.8.2 Analysis of the Computational
Results . . . . . . . . . . . . . . . . . . . . . . . .. 261
13.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

14 Conclusions and Discussion for Future Research .....•• 263

14.1 The Study of MCDM Methods:
Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . 263
14.2 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 Introduction to Multi-Criteria Decision Making • • .. 1

Figure 1-1: A Typical Decision Matrix . . . . . . . . . . . . . . . . . . 3
Figure 1-2: A Taxonomy of MCDM methods (according to
Chen and Hwang [1991]) . . . . . . . . . . . . . . . . . . . 4

2 Multi-Criteria Decision Making Methods ••••••.• 5

3 Quantification of Qualitative Data for

MCDM Problems ••.•••••...••...•.•••.. 23
Figure 3-1: Actual Comparison Values . . . . . . . . . . . . . . . . . . 37
Figure 3-2: Maximum, Average, and Minimum CI Values of
Random CDP Matrices When the Original
Saaty Scale is used . . . . . . . . . . . . . . . . . . . . . . . 42
Figure 3-3: Inversion Rates for Different Scales and Size
of Set (Class 1 Scales) . . . . . . . . . . . . . . . . . . . . . 46
Figure 3-4: Indiscrimination Rates for Different Scales
and Size of Set (Class 1 Scales) . . . . . . . . . . . . . . . 47
Figure 3-5: Inversion Rates for Different Scales and Size
of Set (Class 2 Scales) . . . . . . . . . . . . . . . . . . . . . 48
Figure 3-6: Indiscrimination Rates for Different Scales
and Size of Set (Class 2 Scales) . . . . . . . . . . . . . . . 49
Figure 3-7: The Best Scales . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 3-8: The Worst Scales . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Deriving Relative Weights from Ratio Comparisons . 57

Figure 4-1: Average Residual and CI versus Order of Set
When the Human Rationality Assumption is Used
(the Results Correspond to 100 Random Observations) . 70
Figure 4-2: Average Residual and CI versus Order of Set
When the Eigenvalue Method is Used
(the Results Correspond to 100 Random Observations) . 71

5 Deriving Relative Weights from Difference

Comparisons . . . . • • • . • . . • • • • • • . • . . . • . • . . 73
xiv MCDM Methods: A Comparative Study, by E. Triantaphyllou

6 A Decomposition Approach for Evaluating Relative

Weights Derived from Comparisons ••••••••••• 87
Figure 6-1: Partitioning of the n(n-1)/2 Pairwise
Comparisons ...... . . . . . . . . . . . ... . . . . . . . . 90
Figure 6-2: Error Rates Under the LP Approach for Sets
of Different Size as a Function of the
Available Comparisons. . . . . . . . . . . . . . . . . . .. 106
Figure 6-3: Error Rates Under the Non-LP Approach for Sets
of Different Size as a Function of the
Available Comparisons. . . . . . . . . . . . . . . . . . .. 107
Figure 6-4: Error Rates Under the LP Approach for Sets
of Different Size as a Function of the
Common Comparisons . . . . . . . . . . . . . . . . . . ., 108
Figure 6-5: Error Rates Under the Non-LP Approach for Sets
of Different Size as a Function of the
Common Comparisons . . . . . . . . . . . . . . . . . . .. 109
Figure 6-6: Error Rates for the two Approaches as a
Function of the Available Comparisons. . . . . . . . .. 110
Figure 6-7: Error Rates for the two Approaches as a
Function of the Common Comparisons . . . . . . . . .. 111

7 Reduction of Pairwise Comparisons Via a

Duality Approach •....•................ 115
Figure 7-1: Total Number of Comparisons and Reduction
Achieved When the Dual Approach is Used.
The Number of Criteria n = 5 . . . . . . . . . . . . . . 125
Figure 7-2: Total Number of Comparisons and Reduction
Achieved When the Dual Approach is Used.
The Number of Criteria n = 10 . . . . . . . . . . . . . . 125
Figure 7-3: Total Number of Comparisons and Reduction
Achieved When the Dual Approach is Used.
The Number of Criteria n = 15 . . . . . . . . . . . . . . 126
Figure 7-4: Total number of Comparisons and Reduction
Achieved When the Dual Approach is Used.
The Number of Criteria n = 20 . . . . . . . . . . . . . . 126
Figure 7-5: Net Reduction on the Number of
Comparisons When the Dual Approach is used.
Results for Problems of Various Sizes . . . . . . . . . . 127
Figure 7-6: 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
with the Lowest Weight for Different Size
Problems Under the WPM Method . . . . . . . . . . . . 154

9 Evaluation of Methods for Processing a

Decision Matrix and Some Cases
of Ranking Abnormalities . . . . . . . . . . . . . . . . . 177
Figure 9-1: Contradiction Rate (%) Between the
XVI MCDM Methods: A Comparative Study, by E. Triantaphyllou

WSM and the AHP . . . . . . . . . . . . . . . . . . . . .. 184

Figure 9-2: Contradiction Rate (%) Between the
WSM and the Revised AHP . . . . . . . . . . . . . . . .. 185
Figure 9-3: Contradiction Rate (%) Between the
WSM and the WPM . . . . . . . . . . . . . . . . . . . . . 185
Figure 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. . . . . . . . . . . . . . . . . . . . . . . .. 191
Figure 9-5: 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
Figure 9-6: Contradiction Rate (%) Between the WSM
and TOPSIS Method . . . . . . . . . . . . . . . . . . . . . 196
Figure 9-7: 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 xvii

Any Alternative When Alternatives are

Considered Together and in Pairs.
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 Introduction to Multi-Criteria Decision Making . . .. 1

2 Multi-Criteria Decision Making Methods .•.••••. 5

3 Quantification of Qualitative Data for

MCDM Problems ••••.•.•.•.•••••.••.... 23
Table 3-1: Scale of Relative Importances
(according to Saaty[1980]) . . . . . . . . . . . . . . . . . . . 27
Table 3-2: Scale of Relative Importances
(According to Lootsma[1988]) . . . . . . . . . . . . . . . . 28
Table 3-3: Two Exponential Scales . . . . . . . . . . . . . . . . . . . . 29

4 Deriving Relative Weights from Ratio

Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Table 4-1: RCI Values of Sets of Different Order n . . . . . . . . . . 59
Table 4-2: Data for the Second Extensive Numerical Example ... 66
Table 4-3: Comparison of the Weight Values for
the Data in Table 4-2 . . . . . . . . . . . . . . . . . . . . . . 67
Table 4-4: 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

5 Deriving Relative Weights from Difference

Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Table 5-1: Proposed Similarity Scale . . . . . . . . . . . . . . . . . . . 77

6 A Decomposition Approach for Evaluating Relative

Weights Derived from Comparisons ..••....... 87
Table 6-1a: Computational Results, Part A . . . . . . . . . . . . . .. 101
Table 6-1b: Computational Results, Part B . . . . . . . . . . . . . .. 102
Table 6-1c: Computational Results, Part C . . . . . . . . . . . . . .. 103
Table 6-1d: Computational Results, Part D . . . . . . . . . . . . . .. 104
xx MCDM Methods: A Comparative Study, by Eo Triantaphyllou

7 Reduction of Pairwise Comparisons Via a

Duality Approach . . . . . . . . . . . . . . . • • . . . . . 115

8 A Sensitivity Analysis Approach

for MCDM Methods . . . . . . . . . . . . . . . . . . . . 131
Table 8-1: 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
Table 8-2: Current Final Preferences 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 139
Table 8-3: 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
Table 8-4: 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
Table 8-5: 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
Table 8-6: 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
Table 8-7: All Possible K Values for the WPM Example 0 0 0 0 0 0 145
Table 8-8: 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
Table 8-9: 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
Table 8-10: 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
Table 8-11: Sensitivity Coefficients sens(aij) for each aij
Performance Measure 0 0 0 • • • • • • • • • • • • • • • •• 160
Table 8-12: Decision Matrix for Numerical Example . . . . . . . .. 162
Table 8-13: Initial Ranking . 0 0 • • • • • • • 0 0 • 0 0 • • • 0 • • • • •• 162
Table 8-14: Threshold Values 7/ioj.k (%) in Relative
Terms for the WPM Example o. o. 0 0 0 •• 0 •• 0 •• 163
Table 8-15: Criticality Degrees b.. lij (in %) for each aij
Measure of Performance . . . . 0 • • • 0 • • • 0 0 • • • •• 164
Table 8-16: Sensitivity Coefficients sens(aij) for each aij
Measure of Performance . . . . . . . . . . . . . . o. 0 • • 164

9 Evaluation of Methods for Processing a

Decision Matrix and Some Cases
of Ranking Abnormalities . . . . . . . . . . . . . . . . . 177
Table 9-1: Contradiction Rate (%) Between the
WSM and the AHP . 0 • • • • • • • • 181 0 • • • • 0 •• 0 0 • 0

Table 9-2: Contradiction Rate (%) Between the

WSM and the Revised AHP . . . . . . . . . . . . . 182 0 • • 0
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, multi-
objective 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
xxiv MCDM Methods: A Comparative Study, by E. Triantaphyllou

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
xxvi MCDM Methods: A Comparative Study, by E. Triantaphyllou

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
Preface xxvii

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
xxviii MCDM Methods: A Comparative Study. by E. Triantaphyllou

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: trianta@fsu.edu
Personal Web page: http://www.imse. fsu.edu/vangeiis/