## 5 Experimental Results and Analysis
### Selection of Search Methods  
In Section 3.8, we outline the design of four distinct search methods to select elements for element updates. Subsequently, we conduct experiments to compare the performance  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline \(sp_{cm1}\) & \(3.11\times 10^{11}\) & \(1.35\times 10^{11}\) & \(1.93\times 10^{11}\) & \(2.18\times 10^{11}\) & \(8.71\times 10^{10}\) \\ \(sp_{cm2}\) & \(3.32\times 10^{11}\) & \(1.83\times 10^{11}\) & \(2.07\times 10^{11}\) & \(2.34\times 10^{11}\) & \(6.00\times 10^{10}\) \\ \(sp_{cm3}\) & \(2.78\times 10^{11}\) & \(1.31\times 10^{11}\) & \(1.88\times 10^{11}\) & \(1.94\times 10^{11}\) & \(5.66\times 10^{10}\) \\ \(sp_{cm4}\) & \(2.89\times 10^{11}\) & \(1.87\times 10^{11}\) & \(2.22\times 10^{11}\) & \(2.33\times 10^{11}\) & \(4.79\times 10^{10}\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Statistical results of HV indicator of the algorithm using special \(p_{c}\), \(p_{m}\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline \(L=5,q=20\) & \(3.25\times 10^{11}\) & \(2.79\times 10^{11}\) & \(2.93\times 10^{11}\) & \(2.98\times 10^{11}\) & \(1.25\times 10^{10}\) \\ \(L=10,q=10\) & \(3.07\times 10^{11}\) & \(9.97\times 10^{10}\) & \(1.80\times 10^{11}\) & \(2.06\times 10^{11}\) & \(8.21\times 10^{10}\) \\ \(L=20,q=5\) & \(2.85\times 10^{11}\) & \(1.63\times 10^{11}\) & \(2.48\times 10^{11}\) & \(2.34\times 10^{11}\) & \(5.04\times 10^{10}\) \\ \(L=50,q=2\) & \(2.85\times 10^{11}\) & \(1.63\times 10^{11}\) & \(2.48\times 10^{11}\) & \(2.34\times 10^{11}\) & \(5.04\times 10^{10}\) \\ \hline \hline \end{tabular}
\end{table}
Table 11: Statistical results of HV indicator of the algorithms with different \(L\) and \(q\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline \(p_{cm1}\) & \(3.25\times 10^{11}\) & \(2.79\times 10^{11}\) & \(2.93\times 10^{11}\) & \(2.98\times 10^{11}\) & \(1.25\times 10^{10}\) \\ \(p_{cm2}\) & \(3.04\times 10^{11}\) & \(1.01\times 10^{11}\) & \(2.33\times 10^{11}\) & \(2.36\times 10^of these four methods and identify the most effective one for integration into the final MOFECO-MS.  
For experimentation purposes, we fix the parameters as follows: \(L=5\), \(q=20\), \(l=L\times 60\%\), \(p_{c}=0.9\), \(p_{m}=0.9\), \(N=100\) and \(c_{max}=N\times 1000\). We select the instance \(a280\_n279\) for testing to identify the optimal method for updating elements. The experimental findings are summarized in Table 13.  
From Tables 4 and 13, we can see that SM4 obtains the best results, which indicates that when expanding the exploration, the \(\mathbf{x}_{bj^{\prime}}(k)\) adopts \(\mathbf{x}_{bj^{\prime}}(k)\) for the best search effect. Because in expanded exploration using the \(\mathbf{x}_{bj^{\prime}}(k)\) can explore more solutions and improve the diversity of the algorithm, in narrowed exploitation the \(\mathbf{x}_{gs}(k)\) is also best to use \(\mathbf{x}_{bj^{\prime}}(k)\), and in narrowed exploitation, \(\mathbf{x}_{bj^{\prime}}(k)\) is used because it may be better than \(\mathbf{x}_{bj}(k)\) and has more possibilities than \(\mathbf{x}_{bj}(k)\), so using \(\mathbf{x}_{bj^{\prime}}(k)\) can make the comprehensive performance of the algorithm better. Thus, in the MOFECO-MS algorithm, we choose SM4 to update the elements.  
Figure 7 shows the pareto front corresponding to the median HV values obtained by each search method after 15 runs of the MOFECO-MS algorithm with these four search methods.  
The results in Figure 7 are also consistent with those in Table 13. Since the objective function of this paper is to minimize, the closer the frontier curve is to the lower left, the better the algorithm effect will be. From Figure 7, the accuracy and distribution of each search method can also be seen intuitively, and SM4 is obviously better than the other three methods.