## 3 Utilized Design Optimization Algorithms
### Grey Wolf Optimizer (GWO)
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Design Variables** & **Opt. Alg.** & **Mount 1** & **Mount 2** & **Mount 3** & **Mount 4** \\ \hline \multirow{4}{*}{Optimized Orientation (deg)} & GWO & (0.1, 18., 0) & (\(-0.1\), \(-18.0\)) & (\(28.7\), \(5.2\), 0) & (\(-28.7\), \(-5.2\), 0) \\ \cline{2-6}  & GA & (0, 50, 0) & (0, \(-50, 0\)) & (0.1, 50, 0) & (\(-0.1\), \(-50, 0\)) \\ \cline{2-6}  & SQP & (0, 30.2, 0) & (0, \(-30.2\), 0) & (1.3, 50, 0) & (\(-1.3\), \(-50, 0\)) \\ \hline \multirow{4}{*}{Optimized Position (in)} & GWO & (8, \(-9\), \(-7\)) & (8, \(-9\), 7) & (\(-11\), \(-6\), \(-3\)) & (\(-11\), \(-6\), 3) \\ \cline{2-6}  & GA & (10.5, \(-5\), \(-3\)) & (10.5, \(-5\), 3) & (\(-11\), \(-6\), \(-6\).4) & (\(-11\), \(-6\), 6.4) \\ \cline{1-1} \cline{2-6}  & SQP & (8, \(-9\), \(-3\)) & (8, \(-9\), 3) & (\(-13.1\), \(-6\), \(-4.7\)) & (\(-13.1\), \(-6\), 4.7) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Optimization results for mounts’ positions and orientations (12-DOF model—combined loading). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Opt. Algorithm** & \multicolumn{5}{c}{**Damping Coefficients**} \\ \hline  & 0.0114 & 0.0120 & 0.0218 & 0.0263 & 0.0399 & 0.0472 \\ \cline{2-6}  & 0.0798 & 0.932 & 0.1071 & 0.1100 & 0.1474 & 0.3326 \\ \hline \multirow{4}{*}{GA} & 00.78 & 0.195 & 0.0234 & 0.0256 & 0.0400 & 0.0410 \\ \cline{2-6}  & 0.0763 & 0.0896 & 0.1144 & 0.1400 & 0.1474 & 0.29997 \\ \hline \multirow{4}{*}{SQP} & 0.0087 & 0.0151 & 0.0218 & 0.0256 & 0.0360 & 0.0400 \\ \cline{2-6}  & 0.0914 & 0.1020 & 0.1201 & 0.1344 & 0.1474 & 0.3103 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Optimization results of damping coefficients (12-DOF model—combined loading). (combined loading), where GWO obtained the smallest (optimal) transmitted force of 92.9407 lb. compared to 288.0116 lb. and 434.3781 lb. using GA and SQP, respectively, as shown in Figure 18.

Figure 17: Minimized transmitted force for the 12-DOF model (shaking loads only). Figure 18: Minimized transmitted force for the 12-DOF model (combined loading). In contrast to GA and SQP, GWO smoothly and precisely converged to a global optimal solution, as illustrated in Figure 15. For the proposed PMS model, this reflects the efficiency and resilience of the swarm intelligence optimization algorithm. The convergence characteristics of the second PMS model when combined loads (shaking and road loads) are considered are shown in Figure 16. Even though this is a complex model, GWO outperformed GA and SQP. As shown in Figure 16, even the beginning of the search in GWO was better than the optimum to which GA and SQP converged. GWO's capacity to handle complex PMS design optimization models is demonstrated in this way. When swarm intelligence optimization techniques are applied to complex PMS models, the computing cost is reduced, and the outcomes are improved. It is worth mentioning that GA did not work well on MPS models since the objective function value did not change much over many iterations, as seen in Figures 15 and 16. In other words, new generations and children do not appear to have made much progress. The SQP convergence trend shows that the algorithm has trouble finding a global best solution. In both Figures 15 and 16, the objective function evaluation did not smoothly converge from high to low values, but rather traveled up and down until it reached what the algorithm believed is a global optimum. This is because SQP is a local search method, and the best solution (which could be local or global) is determined by the search beginning point. In [41], the authors gave a performance comparison of GA, SQP, and other evolutionary and swarm optimization techniques. Figures 17 and 18 show that GWO performed better than the GA and SQP algorithms in terms of the amount of transmitted force for both 12-DOF models, the first model that considers only shaking loads and the second model that considers the combined loads. GWO was able to minimize the transmitted loads to 1.5981 lb.