## 4 Results
### Experimental Results and Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cases** & **MODMSA** & **SPEA2** & **NSGA-II** & **MOEA/D** & **I-MOMA** & **D-MOMA** \\ \hline Case400\_100 & **91.16** & 89.31 & 88.66 & 83.72 & 90.22 & 90.34 \\ Case400\_200 & **90.84** & 87.38 & 85.86 & 79.05 & 88.86 & 88.67 \\ Case400\_300 & **89.94** & 84.81 & 82.33 & 75.63 & 87.37 & 86.50 \\ Case400\_400 & **89.83** & 81.03 & 80.60 & 72.97 & 85.53 & 84.05 \\ \hline Case600\_150 & **89.40** & 80.63 & 79.21 & 74.67 & 83.42 & 83.75 \\ Case600\_300 & **88.29** & 76.33 & 72.57 & 69.16 & 82.10 & 79.29 \\ Case600\_450 & **87.04** & 71.45 & 67.67 & 65.76 & 76.68 & 76.59 \\ Case600\_600 & **85.38** & 69.32 & 65.49 & 61.78 & 73.97 & 73.12 \\ \hline Case800\_200 & **87.53** & 72.59 & 71.16 & 67.11 & 77.89 & 76.45 \\ Case800\_400 & **84.85** & 65.19 & 64.44 & 61.23 & 71.12 & 71.33 \\ Case800\_600 & **82.12** & 61.04 & 60.28 & 57.41 & 67.28 & 65.69 \\ Case800\_800 & **79.65** & 56.70 & 55.56 & 52.90 & 61.31 & 61.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The maximum profit rates of the non-dominated solutions obtained by all multi-objective scheduling algorithms in different test cases. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cases** & **MODMSA** & **SPEA2** & **NSGA-II** & **MOEA/D** & **I-MOMA** & **D-MOMA** \\ \hline Case400\_100 & **0** & 3863 & 3820 & 4443 & 3871 & 3836 \\ Case400\_200 & **0** & 3930 & 3947 & 4587 & 3983 & 3984 \\ Case400\_300 & **0** & 3901 & 3968 & 4511 & 3868 & 3953 \\ Case400\_400 & **0** & 3984 & 4001 & 4615 & 3  
The hypervolumes (HVs) of all multi-objective scheduling algorithms in different test cases are shown in Table 4. The HV is an indicator used to measure the performance of the multi-objective optimization algorithm, and a larger HV reflects better comprehensive performance. The objective function values of the non-dominated solution are normalized before calculating the HV. Comparing the HVs of non-dominated solutions obtained by all algorithms, it is easy to find that the MODMSA performs best in terms of comprehensive performance. Specifically, compared to other multi-objective algorithms, the MODMSA improved by 527% on average in terms of the hypervolume. Furthermore, experiments were conducted to evaluate the roles of the proposed improvement procedures, i.e., the minimum perturbation random initialization, the adaptive perturbation control strategy, and the profit-oriented local search algorithm. All algorithms were compared in the test cases with medium load. The comparison algorithms are briefly described as follows:  
* MODMSA-INI, which applies the random initialization proposed in [44] and reserves the other procedures of the MODMSA. * MODMSA-PPCS, which is a variant of the MODMSA, but without the adaptive perturbation control strategy. * MODMSA-LS, which is a variant of the MODMSA, but without the profit-oriented local search algorithm. As shown in Figure 8, the Pareto fronts MODMSA-INI and the MODMSA-PPCS are predominantly distributed in the regions with high profit and high perturbation, which demonstrates that the proposed initialization strategy and the adaptive perturbation control strategy are crucial in preserving the solution diversity. The non-dominated solutions solved by the MODMSA-LS are distributed in the region with lower profit and fewer perturbation, which shows that the local search algorithm can effectively improve the profit of solutions. Regarding the maximum profit obtained, the MODMSA-INI and MODMSA-PPCS can obtain higher profit than the MODMSA as the search space of the MODMSA is limited at the beginning stage. In addition, the MODMSA-LS can find better non-dominated solutions than the MODMSA in the region of low profit and fewer perturbations, which is because the local search algorithm in the MODMSA introduces additional perturbation. However, only the MODMSA can obtain non-dominated solutions with superior performance in quality and diversity. Therefore, every improvement procedure is necessary and indispensable in the proposed MODMSA.