## 4 Experimental Results
### The Planetary Gear Train Optimization
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Algorithm** & \(\mathbf{z_{1}}\) & \(\mathbf{z_{2}}\) & \(\mathbf{z_{3}}\) & \(\mathbf{m}\) **mm** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{x_{3}}\) & \(\mathbf{a}\) **mm** & \(\mathbf{\eta_{gb}^{H}}\) & \(\mathbf{V}\) **mm\({}^{3}\)** & \(\mathbf{\varepsilon_{\alpha}}\) \\ \hline \multicolumn{10}{l}{Objectives \(\varepsilon_{\alpha}\) and \(\sigma_{F}\)} \\ \hline \multicolumn{10}{l}{HMOBPSO} & 19 & 22 & 60 & 10 & 0.70 & 0.50 & 0.50 & 215.31 & 0.9873 & \(3.99\times 10^{9}\) & 1.27 \\ NSGA-II & 19 & 22 & 97 & 10 & 0.70 & 0.50 & 0.50 & 215.31 & 0.9842 & \(1.62\times 10^{10}\) & 1.27 \\ \hline \multicolumn{10}{l}{Objectives \(\varepsilon_{\alpha}\) and \(\sigma_{H}\)} \\ \hline \multicolumn{10}{l}{HMOBPSO} & 20 & 23 & 104 & 10 & 0.70 & 0.50 & 0.50 & 225.35 & 0.9914 & \(2.28\times 10^{10}\) & 1.29 \\ NSGA-II & 22 & 43 & 105 & 10 & 0.70 & 0.50 & 0.50 & 335.78 & 0.9850 & \(7.43\times 10^{10}\) & 1.40 \\ \hline \multicolumn{10}{l}{Objectives \(\varepsilon_{\alpha}\) and \(\eta_{gb}^{H}\)} \\ \hline \multicolumn{10}{l}{HMOBPSO} & 20 & 35 & 45 & 10 & 0.70 & 0.50 & 0.50 & 285.62 & 0.9949 & \(3.76\times 10^{9}\) & 1.35 \\ NSGA-II & 22 & 43 & 45 & 10 & 0.70 & 0.49 & 0.50 & 335.69 & 0.9933 & \(5.99\times 10^{9}\) & 1.40 \\ \hline \multicolumn{10}{l}{Objectives \(\varepsilon_{\alpha}\), \(\sigma_{H}\) and \(\eta_{gb}^{H}\)} \\ \hline \multicolumn{10}{l}{HMOBPSO} & 20 & 23 & 47 & 10 & 0.69 & 0.50 & 0.50 & 225.28 & 0.9925 & \(2.23\times 10^{9}\) & 1.29 \\ NSGA-II & 21 & 28 & 45 & 10 & 0.70 & 0.50 & 0.50 & 255.51 & 0.9903 & \(2.79\times 10^{9}\) & 1.33 \\ \hline \multicolumn{10}{l}{Objectives \(\varepsilon_{\alpha}\), \(\sigma_{F}\) and \(\eta_{gb}^{H}\)} \\ \hline \multicolumn{10}{l}{HMOBPSO} & 19 & 22 & 46 & 10 & 0.70 & 376.00 & 0.38 & 214.35 & 0.9912 & \(1.84\times 10^{9}\) & 1.30 \\ NSGA-II & 22 & 22 & 45 & 10 & 0.70 & 0.49 & 0.50 & 230.32 & 0.9894 & \(2.12\times 10^{9}\) & 1.31 \\ \hline \multicolumn{10}{l}{} \\ \hline \end{tabular}
\end{table} 
Table 5: The obtained values of design variables for the considered MOO problem of the planetary gearbox and parameters of the reference industrial gearbox. Figure 6: The appropriate Pareto-optimal curves obtained using the HMOBPSO and NSGA-II algorithms for the case where the contact ratio \(\varepsilon_{\alpha}\) in combination with gearbox efficiency are taken as objectives. As in the previous case, we observe from Figure 7 that the considered paired objectives are conflicting, as the reduction in gearbox dimensions inevitably leads to the increase in appropriate stress. In the case of combination \(\epsilon_{a}\) and \(\sigma_{F}\), the ideal solution is \(1.25\) for the contact ratio and \(924.6\) N/mm\({}^{2}\) for the bending stress. In this regard, the compromise solution for this case is represented as a tuple \((\epsilon_{a},\sigma_{F})\) and is \((1.27,817.7)\), which is obtained using the HMOBPSO algorithm, and \((1.27,924.7)\) for the NSGA-II algorithm. For the combination of objectives \(\epsilon_{a}\) and \(\sigma_{H}\), the ideal solution is represented as a tuple \((\epsilon_{a},\sigma_{H})\) and is \((1.25,614.8)\). The compromise solution obtained using the proposed algorithm is \((1.29,634)\), while for the NSGA-II, it is \((1.4,632)\). By analyzing the three-dimensional Pareto-optimal curves for objectives such as contact ratio \(\epsilon_{a}\), gearbox efficiency \(\eta_{gb}^{H}\), and gear strength (\(\sigma_{F}\) and \(\sigma_{H}\)), engineers can gain valuable insights into the trade-offs between these objectives. These insights can inform decision making and help engineers make more informed choices based on their priorities. For example, if the contact ratio is of utmost importance, the design engineer can identify the region on the Pareto curve that maximizes this objective while still considering the trade-offs with gearbox efficiency and strength.