## 4 Experimental Results
### The Planetary Gear Train Optimization
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Algorithm** & \(z_{a}\) & \(z_{g}\) & \(z_{b}\) & \(m_{m}\)**[mm]** & \(x_{a}\) & \(x_{g}\) & \(x_{b}\) & \(\eta_{gb}^{H}\) & \(V\)**[mm\({}^{3}\)]** & \(a\)**[mm]** \\ \hline \multicolumn{10}{l}{Multi-objective optimization with gearbox center distance and efficiency as objectives} \\ \hline MHPSODE & 22 & 44 & 45 & 6 & 0.25 & 0.251 & 0.25 & 0.9908 & \(2\times 10^{9}\) & 150.5 \\ NSGA-II & 21 & 52 & 56 & 6 & 0.25 & 0.25 & 0.2545 & 0.9887 & \(2.2\times 10^{9}\) & 221 \\ \hline \multicolumn{10}{l}{Multi-objective optimization with gearbox center distance and bending stress} \\ \hline MHPSODE & 22 & 32 & 102 & 6 & 0.25 & 0.25 & 0.4147 & 0.9735 & \(2.99\times 10^{9}\) & 140 \\ NSGA-II & 22 & 37 & 105 & 6 & 0.25 & 0.25 & 0.274 & 0.9717 & \(3.98\times 10^{9}\) & 180 \\ \hline \multicolumn{10}{l}{Multi-objective optimization with gearbox center distance and contact stress} \\ \hline MHPSODE & 18 & 22 & 45 & 10 & 0.5571 & 0.25 & 0.25 & 0.969 & \(3\times 10^{9}\) & 138 \\ NSGA-II & 18 & 22 & 50 & 12 & 0.2848 & 0.25 & 0.494 & 0.965 & \(3.86\times 10^{9}\) & 245 \\ \hline \multicolumn{10}{l}{Multi-objective optimization with gearbox volume and efficiency as objectives} \\ \hline MHPSODE & 17 & 22 & 45 & 10 & 0.25 & 0.25 & 0.41 & 0.9912 & \(2.45\times 10^{9}\) & 199 \\ NSGA-II & 18 & 23 & 45 & 12 & 0.25 & 0.25 & 0.9689 & \(3.1\times 10^{9}\) & 251 \\ Industrial gearbox & & & & & & & & & \\ - & 20 & 37 & 94 & 9 & - & - & 0.97 & \(3.21\times 10^{9}\) & 285 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The obtained values of design variables for the considered MOO problem of planetary gearbox and parameters of the reference industrial gearbox. The obtained Pareto curves for the case when the considered objectives are the center distance and contact stress \(\sigma_{H}\) are depicted in Figure 5.  
Since, the considered objectives are conflicting, the ideal solution is determined where center distance is 40 mm and \(\sigma_{H}\) is 615 N/mm\({}^{2}\). Therefore, we are adopting center distance of 138 mm and contact stress of 1070 N/mm\({}^{2}\) as the compromise solution obtained by MHPSODE algorithm. Figure 4: The obtained Pareto frontier for the center distance and bending stress \(\sigma_{F}\) as objectives. Figure 3: Illustration of the Pareto frontier for the objective functions: center distance and planetary gearbox efficiency. Furthermore, the minimization of volume of the planetary gearbox, and therefore its mass, is of great importance in applications where the lightweight constructions are required, such as in aerospace industry. In this regard, the stated MOO problem has been carried out, where the considered objectives are the volume of the gears in the gearbox and the power loss of the planetary gearbox. Figure 6 shows the Pareto curves for this case obtained from proposed MHPSODE and NSGA-II algorithms. Figure 5: The obtained Pareto frontier for the center distance and contact stress \(\sigma_{H}\) as objectives. Figure 6: The obtained Pareto frontier for the volume of the gears and planetary gearbox efficiency as objectives. From the results depicted in Figure 6, it can be observed that the considered objective functions are conflicting. Analyzing the obtained Pareto set, starting from the left to right and moving along the Pareto set, it can be observed that with the increase in the value of gearbox volume, leads to simultaneous decrease in the value of the planetary gearbox efficiency, and vice versa. Therefore, the ideal solution is volume of gears \(1.28\times 10^{9}\) mm\({}^{3}\) and gearbox efficiency of 0.995%. The point on the Pareto front closest to the ideal solution, a compromise solution, is volume of gears \(2.45\times 10^{9}\) mm\({}^{3}\) and efficiency of 0.9912% obtained form MHPSODE algorithm. Analyzing the results shown in Table 2, and taking into the consideration the results presented in Figures 3-6, it can be observed that the best solution is obtained using the proposed MHPSODE algorithm.