## 4 Experimental Design and Results
### Case 1: The Performance of HEMOGWO on Benchmark Functions
To further evaluate the significance of performance differences between HEMOGWO and other algorithms in terms of the GD and IGD metrics, the Wilcoxon Signed-Rank Test (WSRT) [29] was employed for statistical comparison. The WSRT results are summarized in the last row of Tables 5 and 6, respectively. In these tables, \"+\" indicates that HEMOGWO performs significantly better than the compared algorithm, \"\(\approx\)\" denotes no statistically significant difference, and \"\(-\)\" indicates inferior performance. In terms of the GD metric, the comparison between HEMOGWO and SMOGWO yielded a result of 11/4/2, indicating that although SMOGWO demonstrates some competitiveness on a few functions, HEMOGWO still outperforms it on the majority of test cases. When compared with MOGWO, HEMOGWO shows superior performance in most instances, with a notable margin of improvement. Against MOEAD_DE, HEMOGWO achieved a result of 16/0/1, reflecting its higher convergence accuracy on nearly all test functions compared to this classic differential evolution-based multi-objective algorithm. Furthermore, HEMOGWO consistently outperformed NSGA-III and MOPSO across all 17 test functions, establishing an overwhelming and absolute advantage. Regarding the IGD metric, while HEMOGWO and SMOGWO exhibited comparable performance on some functions, HEMOGWO demonstrated overall superiority in terms of solution distribution and stability. Compared with MOGWO, HEMOGWO's advantage became more pronounced, further validating the effectiveness of the proposed improvements in search depth and diversity control over the original MOGWO. Notably, HEMOGWO achieved complete dominance when compared with NSGA-III, MOPSO, and MOEAD_DE, which strongly supports its significant advantages in achieving balanced and representative solution sets. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Function** & & **HEMOGWO** & **SMOGWO** & **MOGWO** & **MOEAD\_DE** & **MOPSO** & **NSGA-III** \\ \hline \multirow{2}{*}{UF1} & Mean & \(6.42\times 10^{-4}\) & \(\mathbf{3.73\times 10^{-4}}\) & \(5.48\times 10^{-4}\) & \(2.11\times 10^{-2}\) & \(6.11\times 10^{-2}\) & \(5.90\times 10^{-3}\) \\  & Std & \(\mathbf{3.38\times 10^{-4}}\) & \(1.16\times 10^{-3}\) & \(4.81\times 10^{-4}\) & \(7.86\times 10^{-3}\) & \(2.38\times 10^{-2}\) & \(1.03\times 10^{-2}\) \\  & Mean & \(\mathbf{5.33\times 10^{-4}}\) & \(8.98\times 10^{-4}\) & \(1.15\times 10^{-3}\) & \(3.97\times 10^{-3}\) & \(6.68\times 10^{-3}\) & \(4.74\times 10^{-3}\) \\  & Std & \(\mathbf{1.96\times 10^{-4}}\) & \(2.89\times 10^{-4}\) & \(4.24\times 10^{-4}\) & \(2.11\times 10^{-3}\) & \(1.57\times 10^{-3}\) & \(1.19\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of GD metric on benchmark functions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Function** & & **HEMOGWO** & **SMOGWO** & **MOGWO** & **MOEAD\_DE** & **MOPSO** & **NSGA-III** \\ \hline UF1 & Mean & \(\mathbf{1.80\times 10^{-3}}\) & \(2.26\times 10^{-3}\) & \(3.33\times 10^{-3}\) & \(1.10\times 10^{-1}\)  
To provide a more intuitive comparison of the solution quality obtained by different algorithms, Figure 2 illustrates the optimal solution distributions of six algorithms on three representative benchmark functions: UF2, ZDT3, and CF1. In the plots, black solid squares represent the PF\({}_{\text{F}}\)ure, while red solid circles denote the sets of optimal solutions obtained by each algorithm. The visual analysis reveals that, on the UF2 function, HEMOGWO demonstrates superior convergence and solution set coverage compared to SMOGWO and MOGWO. This is particularly evident in the latter portion of the Pareto front, where the solution distribution of HEMOGWO aligns more closely with the true Pareto-optimal front. In contrast, SMOGWO and MOGWO exhibit noticeable deviations in certain regions, and their overall solution distributions are less uniform, indicating limitations in maintaining population diversity.