## 4 Results and Discussion
### Results of Optimization  
First, the best-performing neural network models were identified, and their outputs were used as inputs in the next optimization phase. The regression plots shown in Figure 20  
Figure 19: Sensitivity matrix. Figure 18: Force–displacement curves of cases 19–27 and the optimized model. represent a statistical method used to model and predict the relationships between a dependent variable and one or more independent variables. The fitting curve uses regression analysis methods to find the mathematical model that best fits the data points. This analysis is a widely used tool for understanding the distribution and trend of a dataset, predicting future values or studying the relationships between variables. Then, a multi-objective optimization method was applied using the best neural network models obtained. This optimization method aims to find the best combination of design variables in terms of crashworthiness and lightweight. This step was performed through MATLAB R2022b software, and the optimized design variables were defined. The optimization process resulted in 35 Pareto-optimal solutions, as shown in Figure 21, which is an essential concept for identifying, analyzing and selecting optimal solutions in multi-objective optimization problems. After that, the TOPSIS method is implemented, and the function is equal to each objective, to reach the most balanced solution among these Pareto-optimal results. As a result of the optimization, the optimal solution was observed at iteration No. 32, which is marked with a red point. The optimal parameters are given in Table 6.  
The training process was carried out to minimize the mean square error (MSE). To improve the performance of the model and ensure the reliability of the results, 50 ANNs were trained with different initial weights. This approach reduced the risk of becoming \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline
**Iteration No.** & **p1 (mm)** & **p2 (mm)** & **p3 (degree)** & **Peak Force (N)** & **Mass (g)** & **Absorbed Energy (J)** \\ \hline
32 & 67.88 & 7.23 & 92.35 & 43,074.34 & 5812.27 & 633.65 \\ \hline \end{tabular}
\end{table} 
Table 6: Optimum parameters. Figure 20: Regression plots of (**a**) mass (kg), (**b**) force (kN), and (**c**) absorbed energy (kJ). stuck in local minima and resulted in a more accurate model. As shown in Figure 22, R\({}^{2}\) and MSE values were continuously monitored to measure the test performance. Figure 21: Three objective Pareto front curves with normalized data. Figure 22: Multiple regression plots obtained from ANN: (**a**) training, (**b**) test, and (**c**) all. The integrated approach applied in this study resulted in optimal solutions that balanced conflicting objectives. ANN models successfully model the complex relationships between design parameters and performance outputs. These models were evaluated via R\({}^{2}\) values and mean square error (MSE) metrics, and high levels of accuracy were achieved. NSGA-II optimization effectively explores a large design space and produces several Pareto-optimal solutions. These solutions provide alternatives reflecting different design preferences. For example, some solutions minimize the peak load and mass while sacrificing absorbed energy, whereas others maximize absorbed energy while increasing other parameters. The TOPSIS application made it possible to select the most balanced solution among the Pareto-optimal solutions. The optimal solution selected provided a reasonable balance between the three performance parameters. The results obtained in this study are summarized in Table 7. The results of this study show that the proposed methodology is a powerful tool for solving complex engineering problems. This approach can be applied not only to optimize the battery side profile, but also to solve similar multi-objective optimization problems. According to the results of the analysis, the peak force was found to be 45,909 N, and the absorbed energy was found to be 658 J.