## 4 Solving Sample Problems in Larger Sizes with MOPSO and NSGA II Algorithms
### Examining the \(t\)-Test Test on the Averages of the Spacing Index  
Table 16 presents the statistical comparisons using the \(t\)-Test on the means of the diversity distance indicator. Additionally, Figure 17 illustrates the comparisons of the mean diversity distance indicators across all sample problems, categorized by the NSGA II and MOPSO algorithms. The results of Table 16 and the value of the obtained P test statistic (0.205 value) show that there is no significant difference between the average spacing index obtained by NSGA II and MOPSO algorithms. The observation of the graph in Figure 17 shows that the NSGA II algorithm has performed better than the MOPSO algorithm in obtaining the dist \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **Sample Size** & **Mean** & **Standard** & **95\% Confidence** & **\(t\)-Test** & **\(p\)-Value** \\  & & & **Deviation** & **Interval** & **Statistic** & **\(p\)-Value** \\ \hline NSGA II & 15 & 0.623 & 0.152 & & & & \\ MOPSO & 15 & 0.557 & 0.148 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Output results of the \(t\)-Test test on average spacing index. Figure 16: Box plot to confirm or reject the null hypothesis for the means of the maximum spread indicator [23]. results mean that the dispersion of the first and second objective function results in the NSGA II algorithm is more and more regular than the MOPSO algorithm. The graph of Figure 18 also completes the results of Table 16 and shows the rejection of hypothesis one and the absence of significant differences in the comparison of the averages of the spacing index. Examining the \(t\)-Test on average computing time. Table 17 shows the output results of the \(t\)-Test test on average computing time. Also, Figures 19 and 20 show the comparative graph of average computing time in each sample problem and also the box graph for rejecting or accepting the null hypothesis in the \(t\)-Test. According to the value of the P-test statistic obtained from the comparisons of the \(t\)-Test test on the computing time averages, it can be acknowledged that there is no significant difference between the computing time averages obtained from solving sample problems with NSGA II and MOPSO algorithms. Figure 17: Comparison of average spacing index in example problems with meta-heuristic algorithms [23]. Figure 18: Box plot to confirm or reject the null hypothesis for the averages of distance index [23]. In Figure 19, we can see that the computational time increases exponentially with the increase in the size of the sample problems, which is proof of the NP-Hardness of the problem body. However, the MOPSO algorithm is better than the NSGA II algorithm in terms of computing time for medium-sized problems, but with the increase in the size of the problem, the computing time obtained by this algorithm has greatly increased. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **Sample Size** & **Mean** & \begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & \
\begin{tabular}{c} **95\% Confidence** \\ **Interval** \\ \end{tabular} &
\begin{tabular}{c} **\(t\)-Test** \\ **Statistic** \\ \end{tabular} & \(p\)**-Value** \\ \hline NSGA II & 15 & 844 & 730 & (\(-483\times 88\)) & 1.48 & 0.160 \\ MOPSO & 15 & 1041 & 1220 & (\(-483\times 88\)) & 1.48 & 0.160 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Output results of the \(t\)-Test test on average computing time. Figure 20: Box plot for confirming or rejecting the null hypothesis for computing time averages [23]. Figure 19: Comparison of computing time averages in example problems with meta-heuristic algorithms [23]. Finally, the diagram in Figure 20 shows that the null hypothesis is placed in the 95% confidence interval, which is a reason for the one hypothesis. Therefore, it can be concluded that there is no significant difference between the average computing time obtained by NSGA II and MOPSO algorithms. Table 18 also shows an overview of the significant difference between the average comparison indicators.