## 3 The Problem and Solution of the BiOP
### A Bi-Objective Optimization Method Based on SSO  
According to the mathematical model described in Expression (14), this paper has proposed a SSO method for solving the BiOP. Figure 3 shows the stepper search process of the SSO method. Suppose the current search point is \(a\), whose coordinates are obtained by using the single-objective optimization model described in the interior-point method optimization Expression (14) under the condition \(r_{1}=c\), that is, point \(a=\min(r_{2}\,|\,r_{1}=c)\). Figure 3: Schematic diagram of the SSO method. Modifying the value of \(r_{1}\) in the direction of the objective function \(f_{1}(\mathbf{X})\) with a step size \(\Delta c\), \(r_{1}{=}\)\(r_{1}+\Delta c\), it can be seen that the point a will move to the point c, \(\min(r_{2}\,|\,r_{1}=c+\Delta c)\). At this point, if the absolute value of the change in the direction of the objective function \(f_{2}(\mathbf{X})\) is greater than or equal to \(\min(f_{2}(\mathbf{X}))\times\Delta c\), which means that the change in the optimization objective function \(f_{2}(\mathbf{X})\) is greater under the change in the objective function \(f_{1}(\mathbf{X})\), then the point \(c\) will be the starting point of the next search. If the absolute value of the change in the direction of the objective function \(f_{2}(\mathbf{X})\) is less than \(\min(f_{2}(\mathbf{X}))\times\Delta c\), then this means that the change in the optimization objective function \(f_{2}(\mathbf{X})\) is less than the change in the objective function \(f_{1}(\mathbf{X})\), and the search step should be narrowed down (the dichotomy can be utilized to change \(\Delta c\)), that is, \(r_{1}=r_{1}+0.5\times\Delta c\). Additionally, the SiOP described in Expression (14) is optimized once again by using the interior point method, and point a will move to \(\min(r_{2}\,|\,r_{1}=c+0.5\times\Delta c)\) at point b. Again, compare the amount of change in the optimization objective functions \(f_{1}(\mathbf{X})\) and \(f_{2}(\mathbf{X})\). By continuously completing the above process, when the deviation value of the variation of the optimization objective functions \(f_{1}(\mathbf{X})\) and \(f_{2}(\mathbf{X})\) is small or reaches the set maximum number of iterative searches, the search process will be stopped, and the result of the last search will be the final optimization result. Based on the stepper search approach depicted in Figure 3, the pseudo-code of the SSO algorithm has been listed in Algorithm 1. ```
Input: In the DES described in this article, the BiOP has been shown in mathematical relation (1)-(13), and the mathematical model for transforming the BiOP of the DES is shown in Expression (16). The initial step parameter \(\Delta c\) and the deadline of stepper search are defined. The deadline of the stepper search is the maximum number of iterations \(N_{\max}\) and stop threshold \(\lambda\). Assuming that the iteration \(N\) is 0,
1. Optimize two single optimization objective functions independently. The minimum value of each optimization objective is \(\min(J_{1}(\mathbf{X}))\) and \(\min(J_{2}(\mathbf{X}))\), respectively;
2. \(r_{1}^{N+1}{\leftarrow}\Delta c\) optimize Expression (16), and \(\min(r_{2}^{N}\,|\,r_{1}^{N})\) is obtained. 3. Calculate \(J_{1}^{N}(\mathbf{X})\) and \(J_{2}^{N}(\mathbf{X})\) under the above \(r_{1}^{N}\) and \(r_{2}^{N}\) condition,
4. While \(N<N_{\max}\)
5. \(r_{1}^{N+1}{\leftarrow}r_{1}^{N}\)\(\Delta c\) optimize Expression (16), and \(\min(r_{2}^{N+1}\,|\,r_{1}^{N+1})\) is obtained. 6.