## 4 Experimental Results
### The Benchmark Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Instances** & **HMOBPSO** & **NSGA-II** & **MOBOA** & **CMOPSO** & **MOPSO** \\  & & & **Mean \(\pm\) Std** & & \\ \hline DTLZ1 & \(2.5878\times 10^{-1}\) & \(6.8853\times 10^{0}\) & \(2.1367\times 10^{0}\) & \(1.3832\times 10^{1}\) & \(4.3650\times 10^{0}\) \\  & (\(2.78\times 10^{-1}\)) & (\(1.77\times 10^{0}\)) & (\(6.27\times 10^{0}\)) & (\(4.54\times 10^{0}\)) & (\(2.50\times 10^{0}\)) \\ DTLZ2 & \(6.9384\times 10^{-2}\) & \(5.6932\times 10^{-2}\) & \(3.5930\times 10^{-1}\) & \(5.8342\times 10^{-2}\) & \(1.2624\times 10^{-1}\) \\  & (\(2.53\times 10^{-3}\)) & (\(7.66\times 10^{-4}\)) & (\(2.92\times 10^{-2}\)) & (\(1.06\times 10^{-3}\)) & (\(2.13\times 10^{-2}\)) \\ DTLZ3 & \(7.6168\times 10^{0}\) & \(1.1711\times 10^{2}\) & \(3.2117\times 10^{2}\) & \(1.3933\times 10^{2}\) & \(1.3230\times 10^{2}\) \\  & (\(3.99\times 10^{0}\)) & (\(1.57\times 10^{1}\)) & (\(5.48\times 10^{1}\)) & (\(3.99\times 10^{1}\)) & (\(5.20\times 10^{1}\)) \\ DTLZ4 & \(1.4369\times 10^{-1}\) & \(6.2684\times 10^{-2}\) & \(8.8102\times 10^{-1}\) & \(1.4057\times 10^{-1}\) & \(3.0302\times 10^{-1}\) \\  & (\(1.99\times 10^{-1}\)) & (\(6.03\times 10^{-3}\)) & (\(1.13\times 10^{-1}\)) & (\(2.53\times 10^{-1}\)) & (\(2.65\times 10^{-1}\)) \\ DTLZ5 & \(6.0880\times 10^{-3}\) & \(5.3969\times 10^{-2}\) & \(2.6824\times 10^{-1}\) & \(7.3418\times 10^{-3}\) & \(9.1614\times 10^{-3}\) \\  & (\(3.45\times 10^{-4}\)) & (\(5.44\times 10^{-3}\)) & (\(3.37\times 10^{-2}\)) & (\(5.74\times 10^{-4}\)) & (\(1.39\times 10^{-3}\)) \\ DTLZ6 & \(1.0085\times 10^{-2}\) & \(2.6649\times 10^{-1}\) & \(7.8499\times 10^{0}\) & \(1.5201\times 10^{-1}\) & \(9.1875\times 10^{0}\) \\  & (\(5.19\times 10^{-2}\)) & (\(2.64\times 10^{-1}\)) & (\(4.91\times 10^{-1}\)) & (\(4.00\times 10^{-1}\)) & (\(1.11\times 10^{-1}\)) \\ DTLZ7 & \(1.0292\times 10^{-1}\) & \(2.8526\times 10^{-1}\) & \(6.7820\times 10^{0}\) & \(1.4424\times 10^{-1}\) & \(6.9118\times 10^{0}\) \\  & (\(5.36\times 10^{-2}\)) & (\(1.01\times 10^{-1}\)) & (\(5.19\times 10^{-1}\)) & (\(1.85\times 10^{-1}\)) & (\(7.16\times 10^{-1}\)) \\ ZDT1 & \(1.2544\times 10^{-2}\) & \(2.4370\times 10^{-2}\) & \(1.6425\times 10^{0}\) & \(5.0456\times 10^{-3}\) & \(1.6634\times 10^{0}\) \\  & (\(2.28\times 10^{-3}\)) & (\(1.62\times 10^{-2}\)) & (\(1.30\times 10^{-1}\)) & (\(3.85\times 10^{-4}\)) & (\(9.22\times 10^{-2}\)) \\ ZDT2 & \(2.7702\times 10^{-2}\) & \(2.3539\times 10^{-2}\) & \(2.5048\times 10^{0}\) & \(4.6474\times 10^{-3}\) & \(3.1771\times 10^{0}\) \\  & (\(4.03\times 10^{-2}\)) & (\(3.32\times 10^{-2}\)) & (\(1.38\times 10^{-1}\)) & (\(3.57\times 10^{-4}\)) & (\(2.18\times 10^{-1}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The obtained mean and standard deviation values of the IGD metric for all used algorithms over all considered ZDT and DTLZ benchmark problems. Figure 10: Visual representation of the mean of IGD metric from Table 6.  
From the results in Table 7, it can be concluded that the proposed HMOBPSO algorithm achieved similar performance to the NSGA-II and CMOPSO algorithms, while still providing higher \(R^{+}\) values than \(R^{-}\) in these cases. Compared to the MOBOA and MOPSO, it clearly outperformed these algorithms. Table 8 displays the mean rankings of the analyzed algorithms for various objective functions of the ZDT benchmark problems, as per [PERSON]'s methodology, which is performed to evaluate the efficiency of the proposed HMOBPSO algorithm with regard to other state-of-the-art multi-objective optimization algorithms taken for analysis. Here, we display in bold the rank of the best-performing algorithm for the considered problem, while the second-best algorithm is underlined. When analyzing the results in Table 8, we observe that the obtained \(p\) values are lower than the significance \(\alpha=0.05\), meaning that there exists a significant difference between the performance of algorithms taken for consideration.