## 4 Experimental Design and Results
### Case 2: The Performance of HEMOGWO on SCOS Problems
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Function** & & **HEMOGWO** & **SMOGWO** & **MOGWO** & **MOEAD\_DE** & **MOPSO** & **NSGA-III** \\ \hline
5-20 & Mean & \(\mathbf{1.30\times 10^{-3}}\) & \(8.01\times 10^{-2}\) & \(1.77\times 10^{-3}\) & \(4.41\times 10^{-3}\) & \(7.04\times 10^{-3}\) & \(2.53\times 10^{-3}\) \\  & Std & \(8.77\times 10^{-4}\) & \(3.34\times 10^{-2}\) & \(\mathbf{4.41\times 10^{-4}}\) & \(5.06\times 10^{-4}\) & \(9.37\times 10^{-4}\) & \(2.68\times 10^{-3}\) \\
5 algorithms, and its standard deviations are also relatively larger. This suggests that, in these specific cases, HEMOGWO does not exhibit superior convergence accuracy or stability. As the problem scale increases, particularly in medium-to-large configurations like 10-100 and 15-100, HEMOGWO once again demonstrates a clear advantage, producing solution sets that more closely approximate the true Pareto front and exhibit greater consistency. In contrast, SMOGWO, MOGWO, and NSGA-III display competitive performance only under limited scales, with less consistent results overall. Regarding the IGD metric, HEMOGWO maintains a dominant position. Except for three problem scales (5-100, 10-100, and 15-50), it achieves the lowest IGD mean values in all other test scenarios. Notably, in small-scale problems such as 5-20 and 10-20, HEMOGWO significantly outperforms all baseline algorithms, highlighting its superior ability in maintaining balanced solution set distributions. Moreover, the standard deviation analysis shows that HEMOGWO generally exhibits lower variability across most problem scales, further validating its robustness and algorithmic stability. In comparison, MOPSO performs worst in terms of IGD. It frequently suffers from poor convergence and solution sets that deviate from the Pareto front, likely due to its tendency to become trapped in local optima and its limited capacity to maintain population diversity. The WSRT results based on GD and IGD metrics indicate that HEMOGWO achieved a complete win against MOEAD_DE and MOPSO in terms of GD, demonstrating a significant advantage in the accuracy of approximating the true Pareto front. When compared with NSGA-III, HEMOGWO attained a favorable result of 8/0/1, showing only a marginal performance gap in one test scenario. Although a few ties and isolated losses were observed in comparisons with SMOGWO and MOGWO, HEMOGWO still secured six wins in both cases, indicating superior overall performance. Regarding the IGD metric, HEMOGWO also exhibited strong distribution capability. Except for one loss and two ties against SMOGWO, it achieves a decisive victory in all comparisons with MOGWO, MOEAD_DE, MOPSO, and NSGA-III, particularly securing a perfect record in the comparisons with MOEAD_DE and MOPSO. To provide an intuitive comparison of the optimization performance of each algorithm on SCOS problems, Figure 3 shows the distribution of Pareto-optimal solutions under different problem scales after normalization. From the distribution patterns, it is evident that the optimal solutions obtained by the HEMOGWO algorithm are mainly concentrated along the outer boundary of the true Pareto front. This suggests its significant advantage in convergence, demonstrating its ability to effectively approximate the global optimal front. The results confirm HEMOGWO's strong capability to balance solution diversity with proximity to the true Pareto front. SMOGWO and NSGA-III exhibit a nondominated solution set with a distribution pattern similar to that of HEMOGWO. The uniformity and coverage of their solutions further highlight the algorithm's adaptability and effectiveness in solving complex SCOS scenarios. In contrast, the solution sets generated by MOPSO and MOEAD_DE show clear deviations from the true Pareto front, with this divergence becoming more pronounced as the problem scale increases. Table 10 presents the Friedman test rankings of the compared algorithms based on the GD and IGD performance indicators.