## 4 Experimental Results
### The Benchmark Results
However, in comparison with the MO_Ring_PSO_SCD algorithm it is observed that proposed algorithm achieves similar performance, while in this case MHPSODE algorithm still provides higher \(R^{+}\) values than \(R^{-}\) in all considered cases. Furthermore, the numerical experiments have been carried out to study the effectiveness of hybridization proposed in this paper by comparing the overall performance of all considered algorithms. In this regard, the average ranks according to [PERSON] for the considered algorithms for different objective functions of CEC2009 benchmark problems have been presented in Table 5.  
In this table, the rank of the best performing algorithm is denoted in bold, and the second best is underlined. From the results in Table 5, it can be observed that the \(p\) values computed using the [PERSON] test for different objective functions are less than significance \(\alpha=0.05\). This means that there is significant difference between the performance of different considered algorithms, which agrees with the conclusions drawn from statistical comparison using [PERSON]'s test. Therefore, the obtained results of the statistical comparison demonstrate the effectiveness of the hybrid algorithm proposed in this paper. Finally, the average computation times of the considered algorithms in searching for the global optimum of the MOO problem have been determined on the computer with 1.6 GHz CPU and 8 GB of RAM. Therefore, the comparison between the average computation times of the MHPSODE, MO_Ring_PSO_SCD, DEMO and NSGA-II algorithms in solving considered CEC2009 benchmark problems are shown in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & \(R^{+}\) & \(R^{-}\) & \(p\)**-Value** & **Dec.** \\ \hline MHPSODE vs. DEMO & 30,818 & 14,333 & 0.00 & \(+\) \\ MHPSODE vs. & 24,799 & 20,352 & 0.1345 & \(\approx\) \\ MO\_Ring\_PSO\_SCD & 27,930 & 17,221 & 0.000 & \(+\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The statistical results of the Wilcoxon’s test between HPSODE and other considered algorithms on CEC2009 benchmark. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Algorithm** & **MMF1** & **MMF2** & **MMF3** & **MMF4** & **MMF5** & **MMF6** & **MMF7** & **MMF8** & **MMF9** & **MMF10** & **Mean ranking** & **Rank** \\ \hline MHPSODE & 2.93 & **1.43** & **1.66** & 2.13 & 2.5 & **1.8** & 3.4 & **1.96** & **2.00** & **1.73** & 2.16 & 1 \\ MO\_Ring\_PSO\_SCD & 2.76 & 1.56 & 2.00 & 2.47 & 3.13 & 2.63 & 3.26 & 2.00 & 2.3 & 1.87 & 2.4 & 2 \\ DEMO & 2.5 & 3.5 & 2.6 & **1.53** & 2.73 & 2.6 & 2.13 & 2.63 & 3.03 & 2.97 & 2.62 & 3 \\ NSGA-II & **1.8** & 3.5 & 3.73 & 3.86 & **1.63** & 2.97 & **1.2** & 3.4 & 2.66 & 3.43 & 2.82 & 4 \\ Friedman \(p\) value & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average ranks computed using the Friedman statistical test for all algorithms across all problems using CEC2009 benchmark, with the significance level \(\alpha=0.05\). From the results presented in Table 6, it can be observed that the DEMO algorithm has the fastest implementation among the considered algorithms, while the MO_Ring_PSO_SCD algorithm has the worst implementation. The NSGA-II algorithm has consistent average execution times among all considered MOO functions, while the execution time of proposed MHPSODE algorithm depends on the complexity of the problem which is solved. Since the considered planetary gearbox optimization problem is not a real-time task, and based on the conclusions drawn from the analysis of the performance of the algorithms on MOO problems, the higher execution time of the proposed MHPSODE provides a compromise between the required design parameters of planetary gearbox and average computation time.