## 4 Experimental Results
### The Planetary Gear Train Optimization
To enhance the evaluation of algorithmic performance in addressing the planetary gearbox MOO problem within a three-dimensional context, this study utilizes the Spacing metric. This metric is employed to assess the spread or diversity of solutions across the Pareto front, particularly when the problem's complexity is less apparent compared to a two-dimensional scenario. This metric is defined as  
\[S=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(\overline{d}-d_{i}\right)^{2}}, \tag{44}\]  
where \(n\) is the number of non-dominated solutions, \(d_{i}\) is the Euclidean distance between the \(i\)th non-dominated vector and its nearest vectors in the Pareto set, and \(\overline{d}\) is the mean of all \(d_{i}\). A higher spacing metric value indicates a more evenly distributed set of solutions. Therefore, the appropriate three-dimensional Pareto-optimal curves, for the case when three objectives have been considered, are depicted in Figure 8, where, for the objectives, the contact ratio \(\epsilon_{a}\), gearbox efficiency \(\eta_{gb}^{H}\), and contact stress \(\sigma_{H}\) have been taken into the account. Figure 7: Pareto-optimal curves obtained using HMOBPSO and NSGA-II algorithms, for two combinations of objective functions (**a**) \(\epsilon_{a}\) and \(\sigma_{H}\) and (**b**) \(\epsilon_{a}\) and \(\sigma_{F}\). From the results in Figure 8, we observe that all considered objectives are conflicting, where the increase in one objective leads to a decrease in the other two. In this regard, for comparison purposes, an ideal solution (which does not belong to the Pareto front but minimizes all objectives) is extracted as a tuple \(\left(\epsilon_{\alpha},\sigma_{H},\eta_{gb}^{H}\right)=\left(1.25,614.8,0.995\right)\), and the compromise solution obtained using the HMOBPSO algorithm is \(\left(1.29,633,0.9925\right)\), and the solution obtained using NSGA-II is \(\left(1.33,630,0.9903\right)\). The Spacing parameter of the NSGA-II algorithm is \(3.25\), while the proposed HMOBPSO achieved a higher spacing value of \(4.99\), showing that it is more efficient for the considered problem. Next, we consider the following objectives: the contact ratio \(\epsilon_{\alpha}\), gearbox efficiency \(\eta_{gb}^{H}\) and root stress \(\sigma_{F}\) in the MOO problem, with the obtained Pareto curves displayed in Figure 9.  
From Figure 9, we can see that for the conflicting objectives, the ideal solution is a tuple \(\left(\epsilon_{\alpha},\sigma_{F},\eta_{gb}^{H}\right)=\left(1.25,817.7,0.9949\right)\), while the compromise solution obtained using the HMOBPSO algorithm is a tuple \(\left(1.3,860,0.9912\right)\) and the solution given using the NSGA-II algorithm is \(\left(1.31,863,0.9894\right)\). The spread indicator for the NSGA-II algorithm is  
Figure 8: Three-dimensional Pareto-optimal curves obtained using the considered algorithms for the case with three objective functions, including \(\epsilon_{\alpha}\), \(\sigma_{H}\) and \(\eta_{gb}^{H}\). Figure 9: Three-dimensional Pareto-optimal curves obtained using the considered algorithms for the case with three objective functions, including \(\epsilon_{\alpha}\), \(\sigma_{F}\) and \(\eta_{gb}^{H}\). 9.82, while for HMOBPSO, it is 17.77. This shows that the HMOBPSO is better suited for the considered optimization problem. Upon examination of the data supplied in Table 5 and considering the findings from the Pareto-optimal fronts displayed in Figures 6-9, it is observed that in the majority of cases, the optimal solution is achieved using the HMOBPSO algorithm described in this paper. The best efficiency achieved is 0.9949, which is achieved with the proposed algorithm for the case of objectives \(\varepsilon_{\alpha}\) and \(\eta_{\beta\beta}^{H}\). In all other objective combinations, the best efficiency is achieved using the HMOBPSO algorithm. When compared to the industrial gearbox reference, the proposed approach demonstrates a notable enhancement in various aspects.