## 5 Results and Discussion  
The optimization results are summarized in Tables 1 and 2. The optimal design variables obtained by the hybrid metaheuristic algorithm (HMA) [37] and their comparison with the initial values are provided in Table 2. Significant changes in \(D_{d}\) and \(D_{z}\) values highlight the high sensitivity of these variables to system performance.  
A comparison of the predicted values from the metamodels and the actual simulation-based results is provided in Table 2, showing excellent agreement with relative errors below 2.24%. This level of accuracy is within the acceptable range for surrogate-based optimization models and is sufficient to capture global design trends, confirming the accuracy and reliability of the metamodel [38].  
For a more intuitive visualization of the optimization effectiveness, Figure 15 provides a comparison between the initial and the optimized objective values for the two objective functions, namely \(\mu_{\text{RMS}}+3\sigma_{\text{RMS}}\) and \(max\ F_{cs}\). The logarithmic left y-axis displays the magnitude of objective values, while the right y-axis shows the percentage improvement after optimization.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design** & **Initial** & **Optimal Value** & **Change** \\
**Variable** & **Value** & **(by HMA)** & **Change** \\ \hline \(L_{1}\) (m) & 0.4 & 0.5282 & 32.05\% \\ \(L_{3}\) (m) & 0.23 & 0.3498 & 52.09\% \\ \(Dz\) (m) & 0.003 & 0.005794 & 93.13\% \\ \(Dd\) (m) & 0.005 & 0.01196 & 139.20\% \\ \(nd\) & 3 & 2 & \(-\)33.33\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimal design variables recommended by HMA compared with the initial values.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Objective** &
\begin{tabular}{c} **Metamodeler** \\ **Predicted** \\ \end{tabular} & **Verification** & **Relative Error (\%)** \\ \hline \(\mu_{\text{RMS}}+3\sigma_{\text{RMS}}\) (m/s\({}^{2}\)) & 25.46 & 25.50 & 0.17\% \\ \(max\ F_{cs}\) (N) & 6936.84 & 7096.00 & 2.24\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between predicted and verified optimal objective values.  
Figure 14: Metamodel prediction performance for the two objective functions: (**a**) acceleration robustness (\(\mu_{\text{RMS}}+3\sigma_{\text{RMS}}\), m/s\({}^{2}\)); (**b**) peak lateral force (\(max\ F_{cs}\), \(N\)).  
The figure shows that all three objectives experienced substantial reductions after optimization. Specifically, \(\mu_{\text{RMS}}+3\sigma_{\text{RMS}}\) decreased by 53.3%, respectively, reflecting significant improvements in ride comfort and acceleration robustness across different operating conditions. Most notably, the maximum value of \(F_{cs}\) reduced by 81.4%, indicating a remarkable reduction in the side force acting on the suspension system. This not only enhanced the structural safety of the system but also minimized potential damage to the internal components due to lateral loads.  
Overall, the results confirmed the effectiveness of the HMA-based multi-objective optimization framework, which significantly improved the performance robustness and structural reliability of the ISU suspension system under uncertain operating conditions.