Processes 2021 , 9 , 274
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IMOEA/DTL are best in all solutions. As shown in Table 10, the solutions obtained by MOEA/D-MR, NSGA-II, and SPEA2 are inferior to that obtained by IMOEA/DTL in terms of HV -metric. The p -value of the Wilcoxon signed-rank test of the HV -metric between two algorithms is shown in Table 11. P1 represents the p -value of the Wilcoxon signed-rank test of the HV -metric between IMOEA/DTL and MOEA/D-MR, P2 represents the p -value of the Wilcoxon signed-rank test of the HV -metric between IMOEA/DTL and NSGA-II, and P3 represents the p -value of the Wilcoxon signed-rank test of the HV -metric between IMOEA/DTL and SPEA2. P1, P2, and P3 are less than 0.05, demonstrating the significant differences between IMOEA/DTL and MOEA/D-MR, IMOEA/DTL and NSGA-II, and IMOEA/DTL and SPEA2. Tables 10 and 11 indicate that IMOEA/DTL is superior to MOEA/D-MR, NSGA-II, and SPEA2 in terms of HV -metric.
Table 10. Average values of hypervolume indicator ( HV -metric) of the solutions obtained by MOEA/D-MR, NSGA-II, SPEA2, and IMOEA/DTL.
Problems
MOEA/D-MR NSGA-II
SPEA2
IMOEA/DTL
3.46 × 10 7 1.75 × 10 7 7.02 × 10 7 5.33 × 10 7 3.33 × 10 7 3.40 × 10 7 4.22 × 10 7 2.40 × 10 7 5.99 × 10 7 2.66 × 10 7 4.41 × 10 7 4.43 × 10 7 5.43 × 10 7 3.32 × 10 7 4.83 × 10 7 8.19 × 10 7
3.32 × 10 7 1.65 × 10 7 6.38 × 10 7 5.04 × 10 7 3.06 × 10 7 3.53 × 10 7 3.74 × 10 7 2.01 × 10 7 5.44 × 10 7 2.62 × 10 7 4.49 × 10 7 4.07 × 10 7 5.43 × 10 7 3.27 × 10 7 4.54 × 10 7 7.78 × 10 7
3.42 × 10 7 1.58 × 10 7 6.60 × 10 7 5.15 × 10 7 3.09 × 10 7 3.37 × 10 7 4.03 × 10 7 2.07 × 10 7 5.75 × 10 7 2.60 × 10 7 4.45 × 10 7 4.18 × 10 7 5.46 × 10 7 3.35 × 10 7 4.62 × 10 7 8.15 × 10 7
5.43 × 10 7 2.79 × 10 7 9.82 × 10 7 8.27 × 10 7 4.92 × 10 7 5.87 × 10 7 6.29 × 10 7 3.65 × 10 7 8.79 × 10 7 4.32 × 10 7 7.30 × 10 7 6.58 × 10 7 8.77 × 10 7 5.16 × 10 7 7.67 × 10 7 1.19 × 10 7
Job1 Job2 Job3 Job4 Job5 Job6 Job7 Job8 Job9
Job10 Job11 Job12 Job13 Job14 Job15 Job16
Table11. The p -value of the Wilcoxon signed-rank test of the HV -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
In order to intuitively show the scheduling schemes obtained by different algorithms, some Gantt charts of scheduling schemes obtained by different algorithms are provided, as shown in Figures 4–6. Each scheduling scheme corresponds to a solution in the Pareto front. We only provide Gantt charts of Job1, Job8, and Job16, and the Gantt charts for
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