Proof of Concept 2: MCDA in Aircraft Evaluation
6.3 Evaluation Results using ELECTRE I
When ELECTRE I is utilized to solve the business aircraft evaluation problem, it requires a decision matrix as input data and weighting factors as the presentation of DM’s preference infor-mation. For this example, the decision matrix is shown in matrixD, where each row corresponds to one business jet alternative, and each column corresponds to one decision criterion. In the first round of evaluation, equal weighting factors are considered, as shown in vectorW.
D=
0.2396 870 1466 84.2333 4.0500 7.63 55 0.2720 952 1567 82.4333 2.3556 8.22 39 0.2264 870 1854 86.7333 3.1000 7.75 82 0.2624 870 1545 86.1000 3.4375 7.66 78
W = [0.1429 0.1429 0.1429 0.1429 0.1429 0.14290.1429 ]T
The stepwise calculations of ELECTRE I are presented in detail in the following subsection, based on the methodology description in Subsection 2.3.4 .
6.3 Evaluation Results using ELECTRE I
6.3.1 Stepwise Calculations of ELECTRE I
There are two kinds of criteria: benefit criteria and cost criteria. Bigger values of benefit criteria and smaller values of cost criteria are preferred. In the business aircraft evaluation problem, benefit criteria are high-speed cruise speed (C2), cabin volume per passenger (C5), product support level (C6), and manufacturer’s reputation (C7), while fuel consumption per seat kilometer (C1), take-off field length (C3), and noise (C4) are cost criteria. Before conducting the normalization, cost criteria are transformed into benefit criteria by taking the reciprocal values.
1. Normalize the decision matrixD.
Dn=
0.5178 0.4881 0.5423 0.5035 0.6149 0.4879 0.4175 0.4561 0.5341 0.5073 0.5145 0.3577 0.5257 0.2960 0.5480 0.4881 0.4288 0.4890 0.4707 0.4956 0.6225 0.4728 0.4881 0.5145 0.4926 0.5219 0.4899 0.5921
2. Calculate the weighted normalized decision matrix Dnw.
Dnw=
0.0740 0.0697 0.0775 0.0720 0.0879 0.0697 0.0597 0.0652 0.0763 0.0725 0.0735 0.0511 0.0751 0.0423 0.0783 0.0697 0.0613 0.0699 0.0673 0.0708 0.0890 0.0676 0.0697 0.0735 0.0704 0.0746 0.0700 0.0846
3. Determine the concordance and discordance sets.
For instance, for the pair of alternativesA1 and A2, the set of decision criteria is divided into two disjoint subsets. The concordance set C12 is composed of all criteria which support thatA1 is preferred to A2. The discordance set D12 is the complementary set of the concordance setC12, with respect to the decision criteria set{1,2,3,4,5,6,7}.
C12={1,3,5,7} D12={2,4,6}
C41={2,6,7} D41={1,3,4,5}
C42={1,3,5,7} D42={2,4,6}
C43={2,3,4,5} D43={1,6,7}
4. Calculate the concordance matrix Mconcordance.
Each element of the concordance matrix is calculated by the sum of criteria weights which are contained in the concordance set. For example, the elementMconcordance12 betweenA1 and A2 is calculated by Equation 6.2.
Mconcordance= 5. Calculate the discordance matrixMdiscordance.
Each element of the discordance matrix reflects the degree to which one alternative is worse than the other. For instance, the elementMdiscordance12 between A1 and A2 is calculated by Equation 6.3.
max{0.0088,0.0066,0.0050,0.0016,0.0368,0.0054,0.0174}
= 0.0066
0.0368 = 0.1793 (6.3)
6. Determine the concordance dominance matrix Mconcordance dominance.
A concordance threshold c needs to be chosen to perform the concordance test. In this study, the average value of the elements in the concordance matrix Mconcordance is used, c= 0.5359. For instance, A1 possibly dominates alternativeA2, if Mconcordance12 ≥c. In
6.3 Evaluation Results using ELECTRE I
this example, Mconcordance12 ≥ c (0.5716 ≥ 0.5359), thus, the concordance test is passed and the element of the concordance dominance matrix is 1. Otherwise, the element is 0.
Mconcordance dominance=
7. Determine the discordance dominance matrixMdiscordance dominance.
A discordance threshold d needs to be chosen to perform the discordance test. In this study, the average value of the elements in the discordance matrix Mdiscordance is used, d= 0.7240. For instance,A1 possibly dominatesA2, ifMdiscordance12 ≤d. In this example, Mdiscordance12 ≤d(0.1793≤0.7240), thus, the discordance test is passed and the element of the discordance dominance matrix is 1. Otherwise, the element is 0.
Mdiscordance dominance=
8. Aggregate the dominance matrix Maggregated dominance.
The aggregated dominance matrix is calculated by an element-to-element product of the concordance dominance matrix and the discordance dominance matrix.
9. Eliminate the dominated alternatives.
In the aggregated dominance matrix, the element 1 in the column indicates that this alternative is dominated by other alternatives. In this example, it can be identified that A1 is dominated byA3,A2 is dominated byA1 and A4. Thus, A1 andA2 are dominated alternatives and can be excluded by ELECTRE I.
It can be obtained that when weighting factors are evenly distributed among the seven cri-teria,A1 andA2are dominated byA3andA4. In other words,A1 (Bombardier Challenger 300) and A2 (Cessna Citation X) should be excluded from the candidates of business jets. But the outranking relationship betweenA3 (Gulfstream G200) and A4 (Hawker H4000) cannot be identified in the current set of weighting factors.
6.3.2 Typical Weighting Scenarios for ELECTRE I
Weighting factors play an important role in the decision analysis process. In this study, in order to better simulate DM’s preference information, typical weighting scenarios for the seven crite-ria are generated from eleven levels of experimental design. The weighting factors for the seven criteria are the combination of seven numbers from the set [0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1], with the constraint that the sum is one. Since the seven decision criteria need to be considered simultaneously in the decision analysis process, all the seven numbers are required to be big-ger than zero. Thus, 84 sets of weighting factors are generated and attached in Table C.4 in Appendix C.3.
The weighting factors reflect the relative importance of the decision criteria. For instance, the first row in Table C.4 is [0.4 0.1 0.1 0.1 0.1 0.1 0.1]. This set of weighting factors indicates thatC1 (fuel consumption per seat kilometer) is the most important decision criterion, and the other six decision criteria have the same level of importance. The other 83 sets of weighting factors have similar explanations.
The evaluation results using ELECTRE I for the 84 sets of weighting factors are summarized in Table 6.7. It is observed that when the DM takes into account all the seven criteria, A4 has the highest frequency to be a non-dominated alternative, and A2 has the highest frequency to be a dominated alternative. Therefore, it can be concluded that for the scenario considered in this study, A2 (Cessna Citation X) should be excluded from the candidates of business jets and A4 (Hawker H4000) should be recommended for the business aviation customer to purchase.
Table 6.7: Evaluation Results for 84 Sets of Weighting Factors using ELECTRE I A1 A2 A3 A4
Non-dominated times 50 34 51 59
Dominated times 34 50 33 25
Non-dominated frequency 59.52% 40.48% 60.71% 70.24%
Dominated frequency 40.48% 59.52% 39.29% 29.76%