## 4 Proposed Algorithm
### Main Framework of the Proposed Algorithm
### Comparison with Other Algorithms

Tables 1 and 2 record the mean values and standard deviations of \(PSP\) and \(HV\), respectively. The best results are highlighted in bold, and the results are statistically analyzed by [PERSON]'s rank sum test. From Table 1, it is observed that the ZS-MMBSO significantly outperforms the SMPSO_MM, the MMGPE, the MMODE_CSCD, and the ZS-MO_Ring_PSO_SCD in terms of \(PSP\) on all MMOPs. The main reason may be that the ZS can greatly reduce the problem complexity, and the BSO algorithm has good global search capability. Namely, the proposed algorithm can integrate the advantages of \"soft isolation\" and \"hard isolation\" to maintain the population diversity. Therefore, compared with other competitors, the ZS-MMBSO can preserve more equivalent Pareto optimal solutions. Compared to the MMODE_ICD, the ZS-MMBSO wins 20 test functions. Clearly, the overall performance of the proposed algorithm is significantly better than that of the MMODE_ICD. For the other two remaining MMOPs (i.e., the SYM_PART simple and the Omni_test), the MMODE_ICD performs better. For the SYM_PART simple, most PSs are distributed in the boundary regions of subspaces, thus it is difficult for the ZS-MMBSO to search equivalent Pareto optimal solutions. For the Omni_test, the distribution of PSs in different subspaces is imbalanced, thus the proposed algorithm may waste computing resources. To visualize the performance comparison of all algorithms, the Friedman test rankings are illustrated in Figure 1 and the ZS-MMBSO owns the best ranking among all compared algorithms. The above statistical results indicate the superiority of the ZS-MMBSO in decision space. Table 2 lists the \(HV\) results for these comparison algorithms. It is observed that the ZS-MMBSO outperforms the MMGPE on all test functions in terms of \(HV\). Compared with the ZS-MO_Ring_PSO_SCD and SMPSO_MM, the proposed algorithm achieves better results on 21 test functions and similar result on one test function. The ZS-MMBSO is superior to the MMODE_CSCD on 20 test functions. For the MMF10, the proposed algorithm is surpassed by the MMODE_CSCD. The reason is that the global PS of MMF10 is simple, and it is easier for MMODE_CSCD to find the PF approximation. The MMODE_ICD performs better than the ZS-MMBSO on the MMF1_e, but their difference is small. In general, the proposed algorithm is competitive in the objective space. One of the reasons is that the ZS-MMBSO integrates the \"soft isolation\" and the \"hard isolation\" to find more PSs, which the population diversity in both decision space and objective space will be improved. The novel individual generation strategy can achieve a trade-off between global exploration and local exploitation, which can balance diversity and convergence. However, the performance improvement of the proposed algorithm in the objective space is not more significant than that in the decision space. In the MMOPs, there are multiple equivalent solutions in the decision space that are corresponding to the same objective value. It is easier for MMOEAs to find a good PF approximation than to locate equivalent solutions in the decision space. As a result, the performance difference of the comparison algorithms in the objective space is relatively small. Figure 2 shows [PERSON]'s test results of the overall performance ranking. The ZS-MMBSO gets the first ranking, followed by MMODE_ICD, MMODE_CSCD, ZS-MO_Ring_PSO_SCD, SMPSO_MM, and MMMGPE. The above observations fully indicate the ZS-MMBSO performs better in the objective space. Figure 1: Performance rankings of all compared algorithms in terms of \(PSP\) values. Figure 2: Performance rankings of all compared algorithms in terms of \(HV\) values. Based on the above analyses, the ZS-MMBSO performs the best among all competitors in both decision space and objective space. The proposed algorithm is an effective tool to solve different types of MMOPs. ### Experimental Analysis

5.3.1.