## 5 Conclusions
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Algorithm** & **MMF1** & **MMF2** & **MMF3** & **MMF4** & **MMF5** & **MMF6** & **MMF7** & **MMF8** & **MMF9** & **MMF10** & **Total avg. Time** \\ \hline
**MHPSODE** & \(1.79\times 10^{2}\) & \(2.08\times 10^{2}\) & \(7.72\times 10^{1}\) & \(1.50\times 10^{1}\) & \(1.04\times 10^{1}\) & \(1.27\times 10^{1}\) & \(1.65\times 10^{2}\) & \(2.99\times 10^{1}\) & \(8.83\times 10^{1}\) & \(1.10\times 10^{1}\) & \(7.96\times 10^{1}\) \\
**NSGA-II** & \(3.47\times 10^{2}\) & \(2.16\times 10^{1}\) & \(3.01\times 10^{1}\) & \(1.87\times 10^{1}\) & \(5.99\times 10^{1}\) & \(4.33\times 10^{1}\) & \(3.86\times 10^{1}\) & \(1.90\times 10^{1}\) & \(1.86\times 10^{1}\) & \(1.90\times 10^{1}\) & \(2.99\times 10^{1}\) \\
**DEMO** & \(2.94\times 10^{2}\) & \(2.89\times 10^{2}\) & \(3.21\times 10^{2}\) & \(2.79\times 10^{2}\) & \(4.13\times 10^{2}\) & \(3.34\times 10^{2}\) & \(3.26\times 10^{2}\) & \(5.79\times 10^{2}\) & \(3.21\times 10^{2}\) & \(3.75\times 10^{2}\) & \(3.53\times 10^{2}\) \\
**MO\_Ring\_PSO\_SCD** & \(2.25\times 10^{2}\) & \(1.86\times 10^{2}\) & \(1.61\times 10^{2}\) & \(1.39\times 10^{2}\) & \(2.50\times 10^{2}\) & \(2.60\times 10^{2}\) & \(2.38\times 10^{2}\) & \(1.88\times 10^{2}\) & \(1.93\times 10^{2}\) & \(1.76\times 10^{2}\) & \(2.02\times 10^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average computation times of the considered algorithms on CEC2009 benchmark functions. **Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**  
\(K_{A}\)Application factor [-]  
\(a_{gg}\)Center distance between sun and planet gears [mm]  
\(Y_{e}\)Contact ratio factor [-]  
\(Z_{e}\)Contact ratio factor [-]  
\(K_{\varphi}\)Dynamic factor[-]  
\(\eta_{0}\)Dynamic viscosity of oil \([\mathrm{N}\cdot\mathrm{s}/\mathrm{m}^{2}]\)  
\(Z_{E}\)Elasticity factor \([\sqrt{\mathrm{N}/\mathrm{mm}^{2}}]\)  
\(K_{Fa}\)Face load factor [-]  
\(\mathbb{F}\)Feausible domain of the optimization problem  
\(b\)Gear width [mm]  
\(\beta\)Helix angle \([^{\circ}]\)  
\(Z_{N}\)Life factor for contact stress [-]  
\(Z_{L}\)Lubricant factor [-]  
\(m_{n}\)Module [mm]  
\(E\)Modulus of elasticity of gears \([\mathrm{kN}/\mathrm{mm}^{2}]\)  
\(\alpha_{n}\)Normal pressure angle \([^{\circ}]\)  
\(n_{w}\)Number of planet gears [-]  
\(z_{g}\)Number of teeth of planet gear [-]  
\(z_{b}\)Number of teeth of ring gear [-]  
\(z_{a}\)Number of teeth of sun gear [-]  
\(d_{a(g)}\)Outside diameter of planet gear [mm]  
\(d_{a(b)}\)Outside diameter of ring gear [mm]  
\(d_{a(a)}\)Outside diameter of sun gear [mm]  
\(d_{s}\)Outside bearing diameter [mm]  
\(D\)Outside diameter of ring gear [mm]  
\(\eta_{gb}^{H}\)Planetary gearbox efficiency [-]  
\(x_{g}\)Profile shift coefficient of planet gear [-]  
\(x_{b}\)Profile shift coefficient of ring gear [-]  
\(x_{a}\)Profile shift coefficient of sun gear [-]  
\(d_{(}a)\)Pitch circle of sun gear [mm]  
\(d_{(}g)\)Pitch circle of planet gear [mm]  
\(Y_{R\ell T}\)Relative surface factor [-]  
\(Z_{R}\)Roughness factor affecting surface durability [-]  
\(Y_{ST}\)Stress concentration factor [-]  
\(K_{H\alpha}\)Transverse load factor [-]  
\(\alpha_{t}\)Transverse pressure angle \([^{\circ}]\)  
\(\alpha_{c}\)Viscosity-pressure coefficient [mm\({}^{2}/\mathrm{N}]\)  
\(V\)Volume of gears [mm\({}^{3}]\)  
\(Z_{H}\)Zone factor [-]