## 4 Experimental Results
### The Benchmark Results
The results of numerical simulation computed over 30 independent runs over ten multi-objective test functions of the CEC2009 benchmark and DTLZ-I test problems are presented in Table 3 in the form of mean and standard deviation values of IGD metrics. Here, the best results obtained are marked in bold and second best are underlined. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Test Instances** & **DEMO** & **NSGA-II** & \
\begin{tabular}{c} **MO\_Ring\_PSO\_SCD** \\ **mean** \(\pm\)**std** \\ \end{tabular} &
\begin{tabular}{c} **MHPSO** \\ **MHF500E** \\ \end{tabular} \\ \hline \hline MMF1 & \(4.86\times 10^{-2}\pm 6.48\times 10^{-2}\) & \(8.75\times 10^{-2}\pm 1.42\times 10^{-1}\) & \(6.43\times 10^{-2}\pm 6.58\times 10^{-2}\) & \(\mathbf{3.35\times 10^{-2}\pm 3.86\times 10^{-2}}\) \\ MMF2 & \(1.89\times 10^{-1}\pm 9.18\times 10^{-2}\) & \(1.85\times 10^{-1}\pm 7.75\times 10^{-2}\) & \(\mathbf{1.59\times 10^{-2}\pm 5.32\times 10^{-3}}\) & \(1.94\times 10^{-1}\pm 7.17\times 10^{-2}\) \\ MMF3 & \(9.01\times 10^{-1}\pm 2.98\times 10^{-1}\) & \(2.46\times 10^{-1}\pm 8.67\times 10^{-2}\) & \(1.69\times 10^{-1}\pm 1.40\times 10^{-1}\) & \(\mathbf{1.42\times 10^{-1}\pm 1.68\times 10^{-1}}\) \\ MMF4 & \(2.48\times 10^{-1}+1.11\times 10^{-1}\) & \(\mathbf{7.37\times 10^{-2}\pm 3.07\times 10^{-2}}\) & \(1.22\times 10^{-1}\pm 6.79\times 10^{-2}\) & \(9.25\times 10^{-2}\pm 3.28\times 10^{-2}\) \\ MMF5 & \(\mathbf{9.64\times 10^{-2}\pm 1.56\times 10^{-2}}\) & \(2.30\times 10^{-2}\pm 3.01\times 10^{-2}\) & \(6.13\times 10^{-2}\pm 1.33\times 10^{-1}\) & \(1.81\times 10^{-2}\pm 5.46\times 10^{-2}\) \\ MMF6 & \(4.62\times 10^{-2}\pm 8.10\times 10^{-2}\) & \(4.92\times 10^{-2}\pm 6.94\times 10^{-2}\) & \(4.43\times 10^{-2}\pm 8.14\times 10^{-2}\) & \(\mathbf{4.36\times 10^{-2}\pm 3.83\times 10^{-2}}\) \\ MMF7 & \(1.23\times 10^{-2}\pm 4.06\times 10^{-3}\) & \(3.96\times 10^{-2}\pm 5.86\times 10^{-2}\) & \(6.51\times 10^{-2}\pm 7.11\times 10^{-2}\) & \(\mathbf{1.12\times 10^{-2}\pm 3.60\times 10^{-3}}\) \\ MMF8 & \(2.40\times 10^{-1}\pm 1.66\times 10^{-1}\) & \(1.45\times 10^{-1}\pm 1.00\times 10^{-1}\) & \(\mathbf{8.58\times 10^{-2}\pm 9.26\times 10^{-2}}\) & \(9.22\times 10^{-2}\pm 8.35\times 10^{-2}\) \\ MMF9 & \(2.06\times 10^{-1}\pm 2.11\times 10^{-1}\) & \(1.25\times 10^{-1}\pm 9.39\times 10^{-2}\) & \(7.44\times 10^{-2}\pm 4.93\times 10^{-2}\) & \(\mathbf{5.99\times 10^{-2}\pm 4.03\times 10^{-2}}\) \\ MMF10 & \(4.57\times 10^{-1}\pm 3.37\times 10^{-1}\) & \(2.50\times 10^{-1}\pm 1.21\times 10^{-1}\) & \(\mathbf{1.21\times 10^{-1}\pm 1.34\times 10^{-1}}\) & \(\mathbf{8.41\times 10^{-2}\pm 9.24\times 10^{-2}}\) \\ DTLZ-I & \(1.69\times 10^{0}\pm 1.97\times 10^{0}\) & \(2.56\times 10^{1}\pm 1.86\times 10^{1}\) & \(3.31\times 10^{1}\pm 7.71\times 10^{1}\) & \(\mathbf{1.46\times 10^{0}\pm 1.80\times 10^{0}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The mean and standard deviation values of IDG metrics for different considered optimization algorithms. Based on the results in Table 2, it can be concluded that the proposed MHPSODE algorithm performed the best on functions\(f_{1}\), \(f_{3}\), \(f_{6}\), \(f_{7}\), \(f_{9}\) and \(f_{10}\), while it performed the second best on \(f_{4}\), \(f_{5}\) and \(f_{8}\). It can be observed that only on \(f_{2}\) the proposed algorithm did not perform satisfactorily, where MO_Ring_PSO_SCD is the best performing algorithm while NSGA-II is the second best. Furthermore, it can be observed that MHPSODE achieved the best performance on DTLZ-I test problem. To perform the statistical pair-wise comparison of the optimization performance of the proposed MHPSODE and other considered algorithms, the obtained results have been analyzed using [PERSON] signed-rank test. In this regard, in Table 4 the statistical results of applying [PERSON]'s signed-rank test on CEC2009 benchmark problems have been presented. From the results in Table 4, it can be concluded that the proposed MHPSODE algorithm has significantly better performance than DEMO and NSGA-II algorithms.