## 4 Case and Discussion
### Discussion
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(\text{min}\,J_{1}\) & \(\text{min}\,J_{3}\) & Method 1 & Method 2 & Method 3 & SSO Method \\ \hline \(J_{1}/\left(\min(J_{1})\right)\) & 1 & 65.9354 & 1.4879 & 1.5170 & 1.0161 & 1.5031 \\ \(J_{3}/\left|\min(J_{3})\right|\) & 78.7690 & 1 & 1.5731 & 1.5448 & 71.4381 & 1.5581 \\ Computing time & 18.44 s & 18.61 s & 17.34 s & 21.42 s & 7715.89 s & 263.52 s \\ \hline \end{tabular}
\end{table} 
Table 5: \(J_{1}\) and \(J_{3}\) values under different methods. Figure 7: Search time and optimization results under different parameter \(\Delta c\) using the SSO method. Figure 8 has represented the distribution characteristics of the different methods on the pareto front and the stepper search process of the SSO method under the coordinate system consisting of \((J_{1}/\left|\min(J_{1})\right|,J_{3}/\left|\min(J_{3})\right|)\) and \((J_{1}/\left|\min(J_{1})\right|,J_{4}/\left|\min(J_{4})\right|)\), respectively. With the first optimization objective value decreasing, the second optimization objective value rises. Further, the pareto front exhibits two components with significant differences in the slope. However, the slope changes exhibited by the pareto front are different, which means that if the above multiple objectives are comprehensively considered to form a multi-objective optimization problem, the slope changes in different objective function directions will be different. Table 6 and Figure 9 exhibit the comparison results under the dual objectives of \((J_{1},J_{4})\) with single-objective optimization, methods 1, 2 and 3 and the SSO method, respectively. In Table 6, the optimization results from methods 1, 2 and 3 and the SSO method have achieved effective balance in two objective function directions. It is different from \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\min J_{1}\) & \(\min J_{4}\) & Method 1 & Method 2 & Method 3 & SSO Method \\ \hline \(J_{1}/\left|\min(J_{1})\right|\) & 1 & 49.5009 & 1.0906 & 1.1398 & 1.5947 & 1.1016 \\ \(J_{4}/\left|\min(J_{4})\right|\) & \(-\)0.5872 & \(-\)1 & \(-\)0.7795 & \(-\)0.8228 & \(-\)1.0000 & \(-\)0.7900 \\ Computing time & 18.44 s & 14.03 s & 11.96 s & 13.15 s & 2251.62 s & 109.45 s \\ \hline \hline \end{tabular}
\end{table} 
Table 6: \(J_{1}\) and \(J_{4}\) values under different methods. Figure 8: The distribution of optimization results \((J_{1}/\left|\min(J_{1})\right|,J_{3}/\left|\min(J_{3})\right|)\) at the pareto front. Table 6. However, the results of the solving time are the same. Methods 1, 2 and 3 decrease by 0.0906, 0.13982 and 0.5947 in the \(J_{1}/\mid\min(J_{1})\mid\) direction and improve by 0.2205, 0.1772 and 0 in the \(J_{4}/\min(J_{4})\) direction. Then, the SSO method reduces 0.1016 in the \(J_{1}/\mid\min(J_{1})\mid\) direction and enhances 0.21 in the \(J_{4}/\mid\min(J_{4})\mid\) direction. The above results show that the SSO method can find equitable recommended solutions more consistently than other methods. Compared to Figures 5 and 8, the pareto front curve in Figure 9 has more stages of slope variation. The optimization results of method 3 differ significantly from other methods and is almost equivalent to the results of the single optimization objective \(J_{4}\). It has caused an imbalance between the two optimization objectives' functions. Then, the result of method 1 is at the intersection of two slopes. It means that the changes in the two optimization objective functions will vary. Combining the values of Table 6, it can be seen that the optimization effect of the SSO method is better. Additionally, based on the results in Figures 5, 8 and 9, it can be seen that the SSO method can be applied to different forms of pareto front format. The adaptability of the method is good.