## 4 Experimental Design and Results
### Case 1: The Performance of HEMOGWO on Benchmark Functions  
This section performs a performance evaluation of the proposed HEMOGWO algorithm using 17 widely used benchmark functions in multi-objective optimization research. The benchmark functions include UF1-UF7, ZDT1-ZDT3, and CF1-CF7. The relevant experimental parameters are listed in Table 4. All parameter values are based on the original literature of each algorithm or their commonly used configurations in the field of multi-objective optimization, unless otherwise specified, with no special tuning applied. To comprehensively assess the algorithm's performance, two commonly used metrics, GD and IGD, are selected. A comparison is made between the HEMOGWO, SMOGWO, MOGWO, MOEAD_DE, MOPSO, and NSGA-III algorithms. The computational results of these algorithms on the GD and IGD metrics are presented in Tables 5 and 6, where each result is obtained from 30 independent runs. The best result for each metric is highlighted \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters** \\ \hline HEMOGWO & MaxIt:300, Population size:100, Archive size:100, nGrid:10, Inflation rate:0.1, Leader selection \\  & pressure:4, Deletion selection pressure:2, Stagnation threshold:10, \(\beta\):1.5. \\ SMOGWO & MaxIt:300, Population size:100, Archive size:100, nGrid:10, Inflation rate:0.1, Leader selection \\  & pressure:4, Deletion selection pressure:2, \\  & MaxIt:300, Population size:100, Archive size:100, nGrid:10, Inflation rate:0.1, Leader selection \\  & pressure:4, Deletion selection pressure:2. \\ MOEAD_DE & MaxIt:300, size of population:100, Archive size:100, Neighborhood Size:20, F:0.5, CR:0.5. \\  & MaxIt:300, Population size:100, Archive size:100, nGrid:10, Inflation rate:0.1, Leader selection \\  & pressure:4, Deletion selection pressure:2, Inertia weight:0.5, Inertial weight damping rate:0.99, \\  & Personal learning coefficient:1, Global learning coefficient:2. \\ NSGA-III & MaxIt:300, Population size:100, Archive size:100, Crossover percentage:0.5, Mutation \\  & percentage:0.4, Mutation rate:0.02. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The parameters setting for cases 1 and 2.  
in bold, and \"Mean/Std\" represents the mean and standard deviation of the results, which are used to evaluate the stability and convergence performance of the algorithms. Based on the statistical results presented in the table, HEMOGWO achieved the best GD values on all benchmark functions except UF1, UF5, CF1, and CF6, demonstrating its significant advantage in solving most test problems, particularly those in the ZDT series. SMOGWO also performed well on several functions, especially obtaining the best results on UF5 and CF5. MOGWO showed the most outstanding performance on CF6, while MOEAD_DE, though only achieving optimal performance on CF1, exhibited excellent results on that function. In contrast, MOPSO and NSGA-III did not achieve the best GD values on any of the tested functions. Regarding the IGD metric, HEMOGWO again exhibited strong performance, achieving the best IGD values on 13 out of the 17 benchmark functions, except for UF3, UF5, CF2, and CF5. SMOGWO outperformed others on these four functions and showed comparable performance to HEMOGWO on the remaining benchmarks, indicating strong competitiveness. However, MOGWO, MOEAD_DE, MOPSO, and NSGA-III performed poorly overall in terms of IGD, failing to achieve the best result on any function, and showed limitations in both convergence and solution set distribution. Although HEMOGWO did not consistently attain the smallest standard deviation across all functions, it outperformed other algorithms in terms of average performance and result stability on most test functions. These findings verify the effectiveness of the proposed strategies in enhancing global search capability, improving solution quality, and increasing robustness.