## 3 The Problem and Solution of the BiOP  
### The Problem of the BiOP  
The first section shows that the optimal control problem of the DES can be described as a bi-objective quadratic programming problem with a large number of complex constraints. This mathematical optimization problem can be described as Expressions (14) and (15),where \(\mathbf{X}=[x_{1},x_{2},\ldots,x_{m}]\) is a vector composed of optimization variables \(x_{i}\), \(i=1\), \(2\), \(\ldots\), \(m\), while \(f_{1}(\mathbf{X})\) and \(f_{2}(\mathbf{X})\) respectively represent two optimization objective functions, \(g_{k}(\mathbf{X})\) represents the \(k\)-th inequality constraint, \(h_{l}(\mathbf{X})\) represents the \(l\)-th equality constraint and \(x_{i}^{\text{lb}}\) and \(x_{i}^{\text{ub}}\) respectively represent the lower and upper bounds of the optimization variable \(x_{i}\). \[\underset{\mathbf{X}}{\text{minimize}}\quad F(\mathbf{X})=[f_{1}(\mathbf{X}),f_{2}(\mathbf{X})] \tag{14}\]  
\[\text{Subject}\quad\text{to}\quad \begin{cases}\quad g_{k}(\mathbf{X})\leq 0,k=1,2, \ldots,K\\ \quad h_{l}(\mathbf{X})\leq 0,l=1,2,\ldots,L\\ x_{i}^{\text{lb}}\leq x_{i}\leq x_{i}^{\text{ub}},i=1,2,\ldots,m\end{cases}  \tag{15}\]  
In a MOP, there is generally no solution to ensure that both optimization objectives are the most global optimal. The way to solve such optimization problems is to use the pareto optimization method to obtain a series of non-dominated solutions, and there is no definite dominance relationship between different optimization solutions rather than a global optimal solution. Figure 2 shows the results of analyzing pareto optimality using two optimization objectives as examples. From Figure 2, it can be observed that if each objective is independently optimized, a global optimum can be obtained. The points marked by these global optimum values are defined as the optimal points in the multi-objective function space, such as point A in the figure. However, this optimal point is not achievable in multi-objective optimization problems. Pareto optimization is used to obtain a series of non-dominated solutions that form a convex surface in the space of multi-objective functions, which is the pareto front. Obviously, it is difficult to choose the most reasonable solution among many non-dominated solutions on the pareto front. However, it is worth thinking about in this paper that non-dominated solutions on the pareto front, such as points B and C, show great differences in their slopes \(\text{d}f_{1}(\mathbf{X})/\text{d}f_{1}(\mathbf{X})\,|\,x_{=\,X_{\text{B}}}\) or \(x_{\text{C}}\). In the neighborhood of point B, when \(f_{1}(\mathbf{X})\) changes slightly, \(f_{2}(\mathbf{X})\) will change dramatically. In the neighborhood of point C, on the contrary, when \(f_{1}(\mathbf{X})\) changes greatly, \(f_{2}(\mathbf{X})\) hardly changes. It is from this phenomenon that this paper carries out a study from the perspective of the degree of the change of the two optimization objective functions and proposes a bi-objective optimization method based on stepper search to complete the problem of selecting the non-dominated solutions on the pareto front. Figure 2: Block diagram for describing a bi-objective optimization problem. Based on the above analysis, it can be seen that the method proposed in this paper needs to address the following two issues. (1) How to define the change in the slope of the pareto front. (2) How to find the point with the largest slope change. It is very unfortunate that pareto front parsing expressions cannot be built in the currently available methods, and the above problems can only be solved by means of numerical analysis. Therefore, in the SSO method, the most crucial part is to transform the description form of the BiOP, and the transformed optimization problem is described as shown in Expression (16). \[