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