## 4 Experimental Results
### The Planetary Gear Train Optimization  
The results of the multi-objective planetary gearbox optimization problem, formulated in Section 2, have been presented in this subsection. In this regard, the minimization of the volume, center distance, safety factor for bending stress and contact stress, and minimization of power losses have been treated as the objectives. Results obtained from the formulated MOO problem have been compared with the reference gearbox outlined in the ISO/AGMA standards [44]. To perform comparison, the ISO VG 220 oil has been used in the numerical simulations, as referenced in the standard for the industrial gearbox. Numerical results, such as the corresponding number of teeth on planet, central and ring gear, module as well as the profile shift coefficients obtained from the numerical simulation using the proposed MHPSODE and NSGA-II algorithms have been shown in Table 2, where the values for the reference industrial gearbox are also presented for comparison. Due to the conflicting nature of multiple objectives, the result of the MOO problem is not a single unique solution, rather a set of solutions, e.g., a Pareto set. To extract a single solution, which can be compared to the reference values, the ideal solution must be determined. The ideal solution \(\mathbf{x}_{ideal}\) represents the minimum best value for each individual objective, regardless of compliance with the rest of objective functions. Seeing that the ideal solution does not belong to the Pareto set, it is obtained as the approximation of the true ideal value from the obtained objective values on Pareto frontier. Therefore, the solutions, named compromise solution, of the considered MOO problem, presented in Table 2 are obtained as the results on the Pareto curve coset to the ideal solution, where the measure for closeness is the Euclidean distance. The Pareto optimal curve for the MOO problem obtained with the proposed MHPSODE and well-known NSGA-II algorithms, where the objectives are the power loss of the planetary gear set and center distance between sun and planet gears, is depicted in Figure 3.  
From the Figure 3 it is observed that efficiency of the planetary gearbox and the center distance between sun and planet gears are conflicting objectives. The ideal solution for the center distance and power efficiency are 91 mm and 0.995%, respectively. Therefore, the compromise solution obtained by MHPSODE algorithm (marked with red diamond), as the solution which belongs to the Pareto set and is closest to the ideal solution is 150.5 mm for the center distance and 0.9908% for the gearbox efficiency. Next, the center distance between sun and planet gears and the bending stress are considered to be the objectives of the MOO problem. Therefore, the corresponding Pareto curves obtained with the proposed and NSGA-II algorithms are depicted in in Figure 4.  
As in the previous case, from the Figure 4 it is observed that considered objectives are conflicting. The ideal solution is 89.92 mm for center distance and 367 \(\mathrm{N/mm^{2}}\) for bending stress. Therefore, the compromise solution for this case is center distance 140 mm and bending stress 996.8 \(\mathrm{N/mm^{2}}\), obtained from the MHPSODE algorithm.