## 4 Experimental Results
### The Planetary Gear Train Optimization  
Here, we present the results of numerical simulations conducted on the optimization model of the planetary gearbox defined in Section 2. For reference purposes and for comparison, the obtained gearbox parameters are compared with an example from the reference literature outlined in AGMA ([PERSON] et al., 2017), whose parameters are shown in Table 4. Firstly, the analysis of conflicting objectives is performed, where we have concluded that the objectives gearbox volume \(V\) (Equation (2)), center distance \(a\), and contact ratio \(\epsilon_{a}\) (Equation (3)) are not conflicting objectives and that a linear relationship can be established between them, with a correlation coefficient of 0.6, as shown in the Figure 5.  
Therefore, for the next analysis, only one of these objectives will be taken into consideration in combination with other conflicting objectives. Thus, we take into consideration several combinations of objectives, including:  
* Contact ratio (defined in Equation (3)) in combination with the gearbox efficiency \(\eta_{gb}^{H}\) defined in Equation (18),
* Contact ratio \(\epsilon_{a}\) with the bending stress \(\sigma_{F}\) defined in Equation (4)
* Contact ratio with the contact stress \(\sigma_{H}\) defined in (8),  
Figure 5: The non-conflicting relationship between gearbox volume \(V\), center distance \(a\), and contact ratio \(\epsilon_{a}\). On the figures the red line represents the fitted linear relationship. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Population Size \(N_{P}\)** &
\begin{tabular}{c} **Max. Number of** \\ **Iterations** \\ \end{tabular} & \multicolumn{2}{c}{**Other Relevant Parameters**} \\ \hline HMOBPSO & & & \(c_{1}=2\); \(c_{2}=2\); \(c=0.01\); \(a=0.25\) \\ NSGA-II & 100 & 10,000 & \(p_{c}=0.9\)—crossover probability; \(p_{m}=0.33\)—mutation probability; \(\eta_{c}=2\) crossover index \\ CMOPSO & & & \(\gamma=10\) size of elite particle set \\ MOPSO & & & Repository size = 100; Mutation rate = 0.5 \\ MOBOA & & & Switch probability \(-0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Specification of parameters of the optimization algorithm used in numerical experiments. as well as three objective problems, which include the following combinations:  
* Contact ratio \(\varepsilon_{\alpha}\) with the gearbox efficiency \(\eta_{gb}^{H}\) and bending stress \(\sigma_{F}\), and
* Contact ratio \(\varepsilon_{\alpha}\) with the gearbox efficiency \(\eta_{gb}^{H}\) and contact stress \(\sigma_{H}\). As stated in the introduction, since the majority of papers employ the NSGA-II algorithm for engineering multi-objective optimization, we have performed numerical simulations with NSGA-II and proposed the HMOBPSO algorithm for comparison. The parameters used in the simulations are \(N_{P}=100\) particles and a maximum number of generations of 10,000. The remaining parameters of these optimization algorithms are displayed in Table 3. The simulations are performed using MATLAB software on a computer with 3.2 GHz CPU and 16 GB of RAM. Since the considered objectives are conflicting, there is no single solution that can be extracted; rather, a Pareto set of solutions is obtained and presented. Therefore, for comparison, an ideal solution \(\mathbf{x}_{ideal}=(x_{ideal,1},x_{ideal,2},\dots,x_{ideal,M})\) is firstly generated as follows:  
\[x_{ideal,i}=min(f_{i}),\quad\forall i=1,2,\dots,M \tag{43}\]  
where \(M\) is the number of objectives. Since solution \(\mathbf{x}_{ideal}\) does not belong to the obtained Pareto set, for the comparison, we obtained a compromise solution, as the solution closest to the Pareto front to the ideal solution, with the Euclidean distance serving as the measure of proximity. The compromise solutions, derived through numerical simulation for various combinations of objective functions described above, are presented in Table 5. These solutions were obtained using both the proposed HMOBPSO algorithm and the widely used NSGA-II algorithm.