A Comparative Study, by Evangelos Triantaphyllou (original) (raw)
Multi-Criteria Decision Making Methods: A Comparative Study
by Evangelos Triantaphyllou, Ph.D.
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 Observations)...............71
5 Deriving Relative Weights from Difference Comparisons......73
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
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
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 a Non-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 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 A1 and A2....................................239
13 Fuzzy Multi-Criteria Decision Making......................241 Figure 13-1: Membership Functions of the Fuzzy Alternatives A1, A2, and A3 of Example 13-1 According to the Fuzzy WSM Method...................................243 Figure 13-2: Membership Functions of the Fuzzy Alternatives A1, A2, 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
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