## 3 Research Methods
### Differential Evolution Algorithm (DE)  
This research used the differential evolution (DE) algorithm as the method of local search in order to escape the current local solution and make the solution more diversified in the update process of the SSO algorithm. The characteristics of the DE algorithm are fast convergence, simple parameters, and a strong local search ability. Additionally, its update mechanism can make the SSO algorithm escape, that is, it can only be replaced in its own solution, the local best solution, or the global best solution. Although the SSO algorithm also has the opportunity to completely randomize the value and escape the current local, it can easily become an infeasible solution due to its characteristics. In addition, the DE algorithm randomly selects the solutions in each set of solutions to update, which means a certain percentage of the solutions will not be updated so that it can retain some of its characteristics and strengthen the variability between the solution and the solution when a well-performing solution enters the DE algorithm. The local search strategy of this study is as follows and following Equations (36)-(38). \[V_{i}^{G+1}=pX_{i_{1}}^{G}+F\times(pX_{r_{2}}^{G}-pX_{r_{3}}^{G})\quad r1\
eq r2 \
eq r3,andi\in N_{sol} \tag{36}\]  
\[u_{i,j,j}^{G+1}= \begin{cases}v_{i,t,j}^{G+1}&\text{if }rand(t,j)\leq CR\text{ or }j=mb(i,t)\\ px_{i,t,j}^{G}&\text{otherwise}\end{cases}  \tag{37}\]  
\[X_{i}^{G+1}= \begin{cases}U_{i}^{G+1}&\text{if }F(U_{i}^{G+1})\leq F(U_{i}^{G}) \\ X_{i}^{G}&\text{otherwise}\end{cases}  \tag{38}\]  
where  
\[N_{sol}:\] The total number of solutions in the group, \[i=1,2,\ldots,N_{sol}.\]  
\[N_{sur}:\] The number of variables, \[j=1,2,\ldots,N_{sur}.\]  
\[N_{gen}:\] Total number of iterations, \[G=1,2,\ldots,N_{gen}.\]  
\[N_{time}:\] Total number of time periods, \[t=1,2,\ldots,N_{time}.\]  
\[F(X):\] Fitness function value. \[V_{i}^{G+1}:\] \[pX_{i}^{G}= \begin{cases}pX_{i,t,1}^{G},pX_{i,t,2}^{G},\ldots,pX_{i,t,N_{ sur}}^{G}\end{cases} \] represents the local best solution in the \[i\] th group  
\[pX_{i}^{G}:\] \[pX_{i}^{G}:\] \[\text{of the }G\text{th iteration}.\]  
\[F:\] Mutation factor, which users can adjust by themselves. \[CR:\] Evolution parameter, which is between 0 and 1. The solutions of the ith group for the \[j\]th variable in the time period \[t\] obtained after mating of the \[G+1\]th iteration. \[u_{i,t,j}^{G+1}:\] \[pX_{i}^{G}:\]0 \[pX_{i}^{G}:\]1 \[pX_{i}^{G}:\]2 \[pX_{i}^{G}:\]3  
This research applied some modifications to the original DE algorithm, that is, the mutation solution did not use the solution \(X_{i}^{G}\) left by this generation of SSO for the mutation but instead used the local best solution \(pX_{i}^{G}\) to perform mutation. The reason for selecting the local best solution is that it is difficult for DE to find solutions that are better solutions than those left by the original SSO in the process of local search because the of the fast convergence nature of SSO when the original DE uses the solution \(X_{i}^{G}\) obtained by SSO after iteration to make mutations. It makes the local search inefficient and fails to show the characteristics of DE that can enhance the diversity of solutions. Therefore, this researchreplaced \(X_{i}^{G}\) with the local best solution \(pX_{i}^{G}\) for mutation when updating with the DE algorithm to facilitate local search and update, retained the updated solution of DE with a higher probability, and enhanced the diversity of the solution space of the entire algorithm. The following examples for \(N_{gen}=10\), \(N_{var}=2\), \(N_{sol}=8\), and \(N_{time}=2\) are used to explain the update process when the DE algorithm is used for local search:  
1. Before entering the stage of local search, it should first select the best solutions of the top \(A\) percentage to enter the local search, and set the action of local search after every \(B\) iterations. For this example, \(A=50\%\), \(B=1\), and \(g=1\) as shown in \[pX_{i}^{G}:\]4
2.