## 5 Discussion
### Impact of Parameter Uncertainty  
To evaluate the robustness of the optimization outcomes under parameter uncertainty, we introduced uncertainty analysis for the inertia constants and damping coefficients through Monte Carlo (MC) simulations. For each Pareto front (considering only those with \(\zeta\) constraints), the ranges \(H_{\text{min}}\), \(H_{\text{max}}\), and \(D_{\text{min}}\)\(D_{\text{max}}\) were determined from all solutions on the respective Pareto front. Two cases of MC simulations were performed: we generated 100,000 samples, randomly varying each \(H_{i}\) and \(D_{i}\) within the intervals \(H_{\text{min}}\) to \(H_{\text{max}}\), and \(D_{\text{min}}\) to \(D_{\text{max}}\) in the first case, and 100,000 samples within the extended intervals \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Metric** & **Run** & \(\boldsymbol{J_{\text{1 Knee}}}\) & \(\boldsymbol{J_{\text{1 Knee}}}\) & **Hypervolume** & **Elapsed Time (s)** \\ \hline \multirow{6}{*}{EV\({}_{\text{max}}\)} & 1 & 0.8849, 0.7430 & 0.1487 & 427.05 \\ \cline{2-5}  & 2 & 0.8854, 0.7055 & 0.1587 & 469.30 \\ \cline{2-5}  & 3 & 0.8801, 0.7300 & 0.1542 & 577.45 \\ \cline{2-5}  & 4 & 0.9479, 0.7918 & 0.1277 & 523.52 \\ \cline{2-5}  & 5 & 0.8860, 0.7697 & 0.1417 & 518.21 \\ \hline Mean & & 0.8969, 0.7480 & 0.1462 & 503.11 \\ \hline \multirow{6}{*}{TR} & \multirow{6}{*}{2} & 0.0286, 0.0337 & 0.0121 & 57.23 \\ \cline{2-5}  & 1 & 0.6673, 0.9991 & 0.1450 & 397.07 \\ \cline{2-5}  & 2 & 0.7695, 0.9540 & 0.1455 & 429.07 \\ \cline{2-5}  & 3 & 0.8139, 0.9468 & 0.1413 & 381.40 \\ \cline{2-5}  & 4 & 0.7561, 0.9426 & 0.1560 & 410.07 \\ \cline{2-5}  & 5 & 0.6962, 0.9986 & 0.1400 & 555.94 \\ \hline \multirow{2}{*}{Mean} & \multirow{6}{*}{5} & 0.7406, 0.9682 & 0.1456 & 434.71 \\ \cline{2-5}  & & 0.0587, 0.0283 & 0.0063 & 69.99 \\ \hline \multirow{6}{*}{DET} & 1 & 0.8083, 0.9896 & 0.1107 & 650.16 \\ \cline{2-5}  & 2 & 0.8304, 0.9652 & 0.1125 & 1051.71 \\ \cline{1-1} \cline{2-5}  & 3 & 0.8382, 0.9615 & 0.1188 & 706.75 \\ \cline{1-1} \cline{2-5}  & 4 & 0.9452, 0.9722 & 0.0835 & 1146.55 \\ \cline{1-1} \cline{2-5}  & 5 & 0.8258, 0.9835 & 0.1031 & 674.02 \\ \hline Mean & & 0.8496, 0.9744 & 0.1057 & 845.84 \\ \hline \multirow{2}{*}{Std} & \multirow{2}{*}{0.0546, 0.0119} & \multirow{2}{*}{0.0136} & \multirow{2}{*}{234.50} \\ \cline{1-1} \cline{2-5}  & & 0.0546, 0.0119 & 0.0136 & 234.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computational performance (optimization with damping ratio constraints). in the second case, i.e., \(H_{\rm min}-20\%\) to \(H_{\rm max}+20\%\) and \(D_{\rm min}-20\%\) to \(D_{\rm max}+20\%\). In each iteration, samples with unstable responses were discarded. For each valid sample, the damping ratios \(\zeta_{i}\) and the objective function values \(J_{1}\) and \(J_{2}\) were computed. The indices of all MC-based samples were stored based on whether the damping ratios \(\zeta_{i}\) satisfied or violated the constraint \(\zeta_{\rm min}=\zeta_{0}\), which is the constraint value used during the optimization process. The obtained results for each Gramian-based controllability metric in the first case are illustrated in Figure 10, along with Pareto fronts with and without damping ratio constraints. As the uncertainty is limited to the original parameter bounds (13), many solutions satisfy the damping ratio constraint. However, the level of robustness depends on the controllability metric used. EV\({}_{\rm min}\) and DET show moderate robustness, with tight clustering around the Pareto front and minimal constraint violations. EV\({}_{\rm 2 min}\) shows lower robustness, as the samples are more dispersed and deviate further from the Pareto front with more frequent damping violations. In the case of EV\({}_{\rm max}\), samples are close to the Pareto front; however, constraint violations are also observed. In the case of the TR metric, samples are clustered the closest to the constrained Pareto front, indicating more robustness.