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methods dominate the few solutions obtained by IMOEA/DTL. However, some solu- tions of IMOEA/DTL dominate all solutions obtained by MOEA/D-MR, NSGA-II, and SPEA2. As shown in Table 8, the solutions obtained by IMOEA/DTL have better perfor- mance than those obtained by MOEA/D-MR, NSGA-II, and SPEA2 in C -metric. Table 9 shows the p -value in the Wilcoxon signed-rank test of the C -metric between two algo- rithms. P1 represents the p -value of the Wilcoxon signed-rank test of the C -metric between IMOEA/DTL and MOEA/D-MR, P2 represents the p -value of the Wilcoxon signed-rank test of the C -metric between IMOEA/DTL and NSGA-II, and P3 represents the p -value of the Wilcoxon signed-rank test of the C -metric between IMOEA/DTL and SPEA2. P1, P2, and P3 are less than 0.05 in 16 scheduling problems, indicating significant differences between IMOEA/DTL and MOEA/D-MR, IMOEA/DTL and NSGA-II, IMOEA/DTL and SPEA2. Tables 8 and 9 indicate that IMOEA/DTL is superior to MOEA/D-MR, NSGA-II, and SPEA2 in terms of C -metric.
Table8. C-metric of the solutions obtained by MOEA/D-MR, NSGA-II, SPEA2, and IMOEA/DTL.
Problems
C ( W , Z )
C ( Z , W )
C ( X , Z )
C ( Z , X )
C ( Y , Z )
C ( Z , Y )
Job1 Job2 Job3 Job4 Job5 Job6 Job7 Job8 Job9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.98
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Job10 Job11 Job12 Job13 Job14 Job15 Job16
0.98
Table9. The p -value of the Wilcoxon signed-rank test of the C -metric between two algorithms.
Problems
P1
P2
P3
Job1 Job2 Job3 Job4 Job5 Job6 Job7 Job8 Job9
0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020
0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020
0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020
Job10 Job11 Job12 Job13 Job14 Job15 Job16
Table 10 lists the average value of HV -metric of the solutions obtained by IMOEA/DTL, MOEA/D-MR, SPEA2, and NSGA-II. The solutions obtained by IMOEA/DTL have a larger average HV -metric than those obtained by MOEA/D-MR, SPEA2, and NSGA-II in all scheduling problems, indicating that the proximity and diversity of the solutions of
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