## 2 Materials and Methods
### Multi-Objective Optimization Using Pareto Simulated Annealing Algorithm  
The research suggests that in practical engineering multi-objective optimization problems, it is nearly impossible to find a single solution that optimizes all objectives. Traditional multi-objective optimization methods, such as the weighted sum method and goal programming, often require differentiation and are prone to becoming trapped in local optima. Simulated annealing (SA) is a randomized algorithm used for solving optimization problems.  
The SA algorithm has a global optimal search capability, allowing it to escape local optima and adapt to complex optimization problems such as irregular and multi-peak issues. It requires relatively few parameters to be tuned, mainly the initial temperature and the cooling rate, making it easy to set up and use. The randomness of the algorithm allows for a broad exploration of the solution space in the initial phase, and as the temperature decreases, it refines the search, balancing the needs of exploration and exploitation. The algorithm is simple to implement and can be combined with other optimization algorithms to enhance performance. Based on the SA algorithm's compatibility with other optimization methods, a simulated annealing algorithm combined with Pareto optimization can generate a Pareto optimal solution set, providing a selection of solutions that meet design requirements.  
The core idea of solving the Pareto optimal solution set is to address the following multi-objective optimization problem [41]:  
\[\min f(x)=f_{1}(x),f_{2}(x),\cdots,f_{n}(x)\]  
\[s.t.g(x)=0,h(x)\leq 0,x\in X\subseteq R_{n} \tag{9}\]  
If the variables \(\mathrm{x_{1}}\in R_{n}\) and \(\mathrm{x_{2}}\in R_{n}\) (where Rn represents the variable constraint set), and for all objective functions \(\mathrm{f_{i}(x_{1})}\leq\mathrm{f_{i}(x_{2})}\) ( i = 1, 2,\(\ldots\), n), and there exists at least one objective function such that \(\mathrm{f_{i}(x_{1})}<\mathrm{f_{i}(x_{2})}\), then \(\mathrm{x_{1}}\) dominates \(\mathrm{x_{2}}\), denoted as \(\mathrm{x_{1}}>\mathrm{x_{2}}\). If there is no variable x and y in Rn such that y > x, then x is a non-dominated variable in Rn, and such a variable is the Pareto optimal solution in a multi-objective optimization problem. Pareto optimal solutions are often not only 1 optimal solution, but in the form of a set, so the key to multi-objective optimization is to find out as many optimal solutions as possible. The core process of multi-objective optimization of the Pareto SA algorithm is shown in Figure 4 [42].  
Based on the Pareto optimal solution set, the SA algorithm is introduced to form the Pareto SA algorithm, which, according to the Metropolis criterion, makes the solution space accept the optimal solution while accepting the deterioration value with a certain probability in order to expand the global search capability, and the traditional single-objective optimization methods are often unable to deal with trade-offs among multiple objectives efficiently in the face of the complex multi-objective optimization problems. For this reason, this study combines the SA algorithm with the Pareto optimization approach, whose key lies in accepting the probability as the distance between the current solution and the new solution in the space. Its probability expression is as follows:  
\[p\big{(}x\geq x^{\prime}\big{)}=\begin{cases}1&f(x^{\prime})<f(x)\\ \exp\left[\frac{\sqrt{f^{2}(x^{\prime})-f^{2}(x)}}{T}\right]&f(x^{\prime})\geq f (x)\end{cases} \tag{10}\]