## 4 Experimental Design and Results
### Case 2: The Performance of HEMOGWO on SCOS Problems
The statistical results indicate that the HEMOGWO algorithm achieves the best performance in terms of the GD metric across the majority of tested problem scales. However, in certain small- to medium-scale scenarios--such as 10-20, 10-50, and 15-50--its GD mean values are higher than those of competing \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 & \(\mathbf{1.30\times 10^{-3}}\) & \(1.53\times 10^{-3}\) & \(1.77\times 10^{-3}\) & \(4.41\times 10^{-3}\) & \(7.04\times 10^{-3}\) & \(2.53\times 10^{-3}\) \\  & Std & \(8.77\times 10^{-4}\) & \(\mathbf{3.60\times 10^{-4}}\) & \(4.41\times 10^{-4}\) & \(5.06\times 10^{-4}\) & \(9.37\times 10^{-4}\) & \(2.68\times 10^{-3}\) \\  & Mean & \(\mathbf{2.48\times 10^{-3}}\) & \(2.67\times 10^{-3}\) & \(4.62\times 10^{-3}\) & \(1.20\times 10^{-2}\) & \(1.32\times 10^{-2}\) & \(4.34\times 10^{-3}\) \\  & Std & \(\mathbf{1.12\times 10^{-3}}\) & \(3.53\times 10^{-3}\) & \(1.73\times 10^{-3}\) & \(1.45\times 10^{-3}\) & \(1.61\times 10^{-3}\) & \(4.50\times 10^{-4}\) \\  & Mean & \(\mathbf{3.03\times 10^{-3}}\) & \(8.10\times 10^{-3}\) & \(1.21\times 10^{-2}\) & \(2.24\times 10^{-2}\) & \(2.24\times 10^{-2}\) & \(9.04\times 10^{-3}\) \\
5-100 & Std & \(2.59\times 10^{-3}\) & \(2.87\times 10^{-3}\) & \(\mathbf{1.92\times 10^{-3}}\) & \(6.79\times 10^{-3}\) & \(6.77\times 10^{-3}\) & \(4.97\times 10^{-3}\) \\  & Mean & \(1.77\times 10^{-2}\) & \(\mathbf{1.17\times 10^{-2}}\) & \(1.79\times 10^{-2}\) & \(4.15\times 10^{-2}\) & \(5.84\times 10^{-2}\) & \(2.09\times 10^{-2}\) \\  & Std & \(\mathbf{5.72\times 10^{-2}}\) & \(6.55\times 10^{-2}\) & \(5.96\times 10^{-2}\) & \(6.80\times 10^{-2}\) & \(9.14\times 10^{-2}\) & \(9.17\times 10^{-2}\) \\
10-50 & Mean & \(4.35\times 10^{-2}\) & \(5.71\times 10^{-2}\) & \(4.00\times 10^{-2}\) & \(7.05\times 10^{-2}\) & \(1.02\times 10^{-2}\) & \(\mathbf{2.11\times 10^{-2}}\) \\  & Std & \(\mathbf{2.33\times 10^{-3}}\) & \(1.22\times 10^{-2}\) & \(1.74\times 10^{-2}\) & \(2.26\times 10^{-2}\) & \(2.91\times 10^{-2}\) & \(2.35\times 10^{-3}\) \\  & Mean & \(\mathbf{7.89\times 10^{-2}}\) & \(1.31\times 10^{-1}\) & \(1.66\times 10^{-1}\) & \(1.44\times 10^{-1}\) & \(2.19\times 10^{-1}\) & \(1.38\times 10^{-1}\) \\
10-100 & Std & \(\mathbf{2.02\times 10^{-2}}\) & \(3.31\times 10^{-2}\) & \(3.29\times 10^{-2}\) & \(4.07\times 10^{-2}\) & \(5.06\times 10^{-2}\) & \(3.62\times 10^{-2}\) \\  & Mean & \(\mathbf{3.10\times 10^{-2}}\) & \(4.05\times 10^{-2}\) & \(5.03\times 10^{-2}\) & \(9.05\times 10^{-2}\) & \(1.33\times 10^{-1}\) & \(4.75\times 10^{-2}\) \\
15-20 & Std & \(1.28\times 10^{-2}\) & \(1.70\times 10^{-2}\) & \(1.14\times 10^{-2}\) & \(2.65\times 10^{-2}\) & \(3.49\times 10^{-2}\) & \(\mathbf{1.11\times 10^{-2}}\) \\  & Mean & \(7.86\times 10^{-2}\) & \(7.90\times 10^{-2}\) & \(\mathbf{6.41\times 10^{-2}}\) & \(1.26\times 10^{-1}\) & \(1.73\times 10^{-1}\) & \(9.27\times 10^{-2}\) \\  & Std & \(4.61\times 10^{-2}\) & \(\mathbf{2.62\times 10^{-2}}\) & \(3.08\times 10^{-2}\) & \(2.75\times 10^{-2}\) & \(3.80\times 10^{-2}\) & \(4.64\times 10^{-2}\) \\
15-100 & Mean & \(\mathbf{7.42\times 10^{-2}}\) & \(1.03\times 10^{-1}\) & \(1.34\times 10^{-1}\) & \(2.24\times 10^{-1}\) & \(2.88\times 10^{-1}\) & \(1.18\times 10^{-1}\) \\ \(+/{\approx}/{-}\) & & & \(6/2/1\) & \(6/1/2\) & \(9/0/0\) & \(9/0/0\) & \(8/0/1\) \\ \hline \hline \end{tabular} Bold indicates the best result under each indicator.  
\end{table} 
Table 8: Results of GD metric on SCOS problems.