Applied Optimization,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. 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 1st 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 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.

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