## 3 Results
### Multi Variable Crashworthiness and Weight Optimization of Vehicle Body
#### 3.2.3 Hybrid Algorithm for Roll Crashworthiness and Weight Optimization  
Currently, the popular global optimization methods used for solving nonlinear vehicle collision problems are particle swarm optimization (PSO) [34] and the non-dominated sorting genetic algorithm (NSGA-II) [35]. However, despite the fast convergence rate of the PSO algorithm, it is prone to local optimal solutions [36]. On the other hand, NSGA-II has good convergence [37], but its population diversity and the non-uniformity of the Pareto distribution may impact decision-makers' choices for the final solution [38]. Therefore, considering the strongly nonlinear objective function and the four approximate models proposed in this paper, we propose a hybrid approach that combines the artificial immune system (ARTIS) and PSO, known as the artificial immune particle swarm optimization (AI-PSO) algorithm, to optimize the rollover crashworthiness and weight. ARTIS is a distributed model of an artificial immune system proposed by [PERSON], which consists of several nodes that simulate lymph nodes. Each node contains multiple detectors and can independently perform immune functions. Although the clonal selection algorithm based on this model is a common immune algorithm, it may not be efficient for solving the four approximate models due to the significant computation required for sorting and screening. To address this issue, PSO can be used to improve the clonal selection algorithm. By combining the strengths of the two algorithms, the resulting hybrid algorithm can achieve more effective optimization. This concept of a hybrid algorithm has been used in various research studies. The basic PSO algorithm can be enhanced by incorporating cross-mutation operations from the artificial immune system and controlling particle flight speed. During the algorithm iteration, if the similarity of the particle swarm exceeded a certain threshold, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Responses** & **MRE** & **R2** & **Radj2** \\ \hline SEA & 0.0427 & 0.9603 & 0.9548 \\ \(\omega\) & 0.0616 & 0.9412 & 0.9330 \\ In & 0.0135 & 0.9404 & 0.9320 \\ M & 0.0151 & 0.9584 & 0.9526 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Precision error analysis results. Figure 12: Three-dimensional post-processing display results of the KRG model: **(a)** SEA; **(b)**\(\omega\); **(c)** In; **(d)** M.  
cross-mutation operations were performed on the swarm with a certain probability. Additionally, if the particle velocity was less than a certain value, the particle was controlled in order for it to regain its flying speed and move away from the gathering point. The specific steps of the algorithm were as follows:  
1. The learning factors \(c_{1}\) and \(c_{2}\), the number of particle populations M, and the maximum number of iterations were determined. 2. The initial population was randomly generated, the velocity and position of particles were initialized, and the iteration counter t was set to 1. 3. The optimal positions \(P^{t}_{i}\) of the particles and the optimal positions \(P^{t}_{g}\) of the entire particle swarm were calculated. 4. The particle position speed was updated using a formula, and the fitness \(\mathrm{aff}^{t}_{k}\) of each particle was calculated. \[x_{i}^{(t+1)}=x_{i}^{t}+v_{i}^{(t+1)}\] (13) \[v_{i}^{(t+1)}=X(wv_{i}^{(t)}+c_{1}r_{1}(p_{i}^{(t)}-x_{i}^{(t)}))+c_{2}r_{2}(p _{g}^{(t)}-x_{i}^{(t)})\] (14) \[\mathrm{aff}^{t}_{k}=(1+\frac{\sum_{i=1}^{m}\left(w_{i}|x_{0}^{*}(i)-\tau_{i }(i)|\right)^{2}}{\sum_{i=1}^{m}\left(w_{i}|x_{0}^{-}(i)-\tau_{i}(i)|\right)^{2 }})^{-1}\] (15) where \(X=2/\left|2-\phi-\sqrt{\phi^{2}-4\phi}\right|\), \(\phi=c_{1}+c_{2}\), \(r_{1}\), \(r_{2}\) are random numbers between 0 and 1. 5. The similarity of particles in the population \(\xi\) was calculated. If \(\xi\geq\xi_{\mathrm{om}}\), cross-mutation operations were performed with a certain probability. 6. The velocity \(v_{i}^{t}\) of each particle in the population was calculated.