## 2 Materials and Methods
### Definition of Optimization Problem  
Without loss of generality, a multi-objective optimization problem with n decision variables and m objective variables can be expressed as:  
\[\left\{ \begin{matrix}\text{miny}=\text{F}(\text{x})=[\text{f}_{1}(\text{x}),\text {f}_{2}(\text{x}),\dots,\text{f}_{\text{n}}(\text{x})]^{\text{T}}\\ \text{s.t.g}_{\text{i}}(\text{x})\leq 0,\text{i}=1,2,\dots,\text{q}\\ \text{h}_{\text{j}}(\text{x})=0,\text{j}=1,2,\dots,\text{p}\end{matrix} \right. \tag{5}\]  
where x = (x\({}_{1}\), x\({}_{2}\),..., x\({}_{n}\)) \(\in\) X \(\subset\) R\({}^{n}\) is the n-dimensional decision vector, X is the n-dimensional decision space, y = (y\({}_{1}\), y\({}_{2}\),..., y\({}_{n}\)) \(\in\) Y \(\subset\) R\({}^{m}\) is the m-dimensional target vector, and Y is the m-dimensional target space. Objective function F(x) defines m mapping functions from the decision space to the objective space; g\({}_{\text{i}}(\text{x})\leq 0\), (i = 1, 2,..., q) defines q inequality constraints; and h\({}_{\text{j}}(\text{x})\) = 0, (j = 1, 2,..., p) defines p equality constraints. In general, the sub-objectives of a multi-objective optimization problem are contradictory. It is impossible to achieve the optimal values of multiple sub-objectives simultaneously, but only to coordinate and compromise among them. Therefore, the solution of the multi-objective optimization problem is not unique, but exists in a set of optimal solutions consisting of many Pareto optimal solutions. In this study, the artificial immune particle swarm optimization (AI-PSO) algorithm was used to solve the multi-objective optimization problem. In the generated Pareto solution set, the sum of the distances between each non-inferior solution and all other non-inferior solutions in all objective function spaces was calculated, and the non-inferior solution with the shortest distance was selected as the optimization solution, that is, the method achieving the shortest distance. The crashworthiness and weight optimization process, based on unreplicated saturated factorial design, is shown in Figure 7.  
Analysis Factors and Design Objectives  
Based on the results of the testing and simulation analysis, it is evident that the left and right sides of the A pillar of the vehicle, as well as the top cover, are the main areas where deformation and intrusion occur. The cross-beam structure located at the top of the vehicle serves as the primary supporting structure, and it is susceptible to bending failure and deformation during a rollover event, which can lead to encroachment on the passenger's living space [29; 30]. Therefore, the front stop cross-beam of the roof is considered to be the key structure for optimizing the crashworthiness and weight of the vehicle during a rollover event. To investigate the impact of changes in thickness on the crashworthiness of vehicles, the thicknesses of 19 body parts, including the A pillar and front bumper beam, were taken as the analysis variables in this paper. The initial thickness values are provided in Table 1, and the 19 analysis factors are depicted in Figure 8, considering symmetry. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c} \hline \hline \begin{tabular}{c} **Analysis** \\ **Factors** \\ \end{tabular}  & **A\({}_{1}\)** & **A\({}_{2}\)** & **A\({}_{3}\)** & **A\({}_{4}\)** &  
Figure 8: Nineteen analysis factors of van rollover. Figure 7: The flow chart for the crashworthiness design with unreplicated saturated factorial design. In evaluating the crash performance of the vehicle, the relevant indicators for front and side crashes typically involve the crashworthiness and energy absorption characteristics of the structures. Hence, this paper primarily focuses on the evaluation indicators of rollover performance by considering the survival space invasion and energy absorption of structures. The displacement of the top cover serves as a measure of the invasion of the living space, which is recorded as \"In\".