## 5 Case Study
### Parameter Settings
The solution belongs to the solution set of the current parameter combination. 2. The solution is not dominated by any solution in the benchmark reference set. Divide this count by the total number of solutions in the benchmark reference set to obtain the solution contribution rate of the parameter combination in the current round. Step 4: Aggregate data across rounds. Compute the arithmetic mean of the contribution rates obtained by each parameter combination over 50 experiments to generate the final comprehensive performance value of that parameter combination. Step 5: Performance ranking and evaluation. Compare the final performance values of all parameter combinations. A higher value indicates that a larger proportion of effective non-dominated solutions were generated during optimization, meaning better algorithm performance. Following the above steps, the orthogonal table L25 (\(3^{5}\)) corresponding to the parameter levels in Table 3, along with the response values, is shown in Table 4[37]. Specifically, Table 4 displays the results of a designed orthogonal experiment (L25(\(3^{5}\))) evaluating different combinations of three key algorithm parameters: population size, crossover rate, and mutation rate. Additionally, by further processing the data in Table 4, we calculate the average performance value for each parameter at different levels. Sorting these average performance values allows us to determine which parameter has the greatest impact on the performance of the NSGA-II algorithm. Table 5 summarizes the average performance and range analysis for each parameter level, which supports identifying the most influential parameters. This indicates that among the three parameters, population size has the most significant impact on the performance of the NSGA-II algorithm, and an excessively small population size can negatively affect algorithm performance. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{3}{c}{**Parameters**} & \multicolumn{3}{c}{**Performance**} & \multicolumn{3}{c}{**Parameters**} & \multirow{2}{*}{**Performance**} \\ \cline{2-2} \cline{4-9}  & \(N_{upper}\) & \(P_{c}\) & \(P_{m}\) & & **Value** & & \(N_{upper}\) & \(P_{c}\) & \(P_{m}\) & Value \\ \hline
1 & 50 & 0.6 & 0.05 & 0.22 & 14 & 200 & 0.85 & 0.05 & 12.98 \\
2 & 50 & 0.7 & 0.06 & 0.15 & 15 & 200 & 0.9 & 0.06 & 10.67 \\
3 & 50 & 0.8 & 0.07 & 0.67 & 16 & 300 & 0.6 & 0.08 & 5.87 \\
4 & 50 & 0.85 & 0.08 & 0.76 & 17 & 300 & 0.7 & 0.09 & 4.04 \\
5 & 50 & 0.9 & 0.09 & 1.06 & 18 & 300 & 0.8 & 0.05 & 9.54 \\
6 & 100 & 0.6 & 0.06 & 2.66 & 19 & 300 & 0.85 & 0.06 & 8.47 \\
7 & 100 & 0.7 & 0.07 & 1.55 & 20 & 300 & 0.9 & 0.07 & 7.45 \\
8 & 100 & 0.8 & 0.08 & 3.94 & 21 & 500 & 0.6 & 0.09 & 8.24 \\
9 & 100 & 0.85 & 0.09 & 2.78 & 22 & 500 & 0.7 & 0.05 & 6.34 \\
10 & 100 & 0.9 & 0.05 & 3.45 & 23 & 500 & 0.8 & 0.06 & 7.88 \\
11 & 200 & 0.6 & 0.07 & 7.82 & 24 & 500 & 0.85 & 0.07 & 8.76 \\
12 & 200 & 0.7 & 0.08 & 9.46 & 25 & 500 & 0.9 & 0.08 & 6.56 \\
13 & 200 & 0.8 & 0.09 & 8.09 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Orthogonal table of parameters and performance values. Furthermore, Figure 8 more clearly illustrates the impact of variations in the three parameters on algorithm performance, based on the calculations in Table 5. Specifically, Figure 8 visualizes the relationship between each parameter level and the corresponding average performance, providing an intuitive understanding of how the parameters affect the model outcomes. As shown in Figure 8, the algorithm achieves the best performance when the population size is set to 200, the crossover probability is set to 0.85, and the mutation probability is set to 0.05. Therefore, this combination is identified as the optimal algorithm parameter setting.