## 4 Solving Sample Problems in Larger Sizes with MOPSO and NSGA II Algorithms
### Investigation of the \(t\)-Test on the Means of the First Objective Function  
Table 12 displays the output results of the \(t\)-Test conducted on the averages of the first objective function. Additionally, Figures 7 and 8 show comparative plots of the averages of the first objective function for each sample problem, along with box plots indicating the acceptance or rejection of the null hypothesis in the \(t\)-Test.  
According to Table 12 and considering the \(p\)-value, it is observed that there is a significant difference between the mean values of the first objective function obtained from solving with NSGA II and MOPSO algorithms. With these interpretations, based on the minimization of the first objective function, it can be concluded that in this criterion, the MOPSO algorithm has achieved better results compared to the NSGA II algorithm.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample Problem**} & \multirow{2}{*}{\begin{tabular}{c} **First** \\ **Objective** \\ **Functions** \\ \end{tabular} } & \multicolumn{2}{c}{**Sum of the Second**} & \multirow{2}{*}{\begin{tabular}{c} **Number of** \\ **and Third** \\ **Objective** \\ **Functions** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Maximum** \\ **Feasible** \\ **Solutions** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Diversity** \\ **Distance** \\ **Indicator** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Computational** \\ **Time** \\ \end{tabular} } \\  & & & & & & \\ \hline
1 & 633,806.72 & 62,601.70 & 9 & 270,273.91 & 0.37 & 36.46 \\
2 & 778,692.87 & 78,381.54 & 19 & 585,593.25 & 0.77 & 108.00 \\  
According to the observations in Figure 9, it can be inferred that the MOPSO algorithm has achieved better results compared to the NSGA II algorithm in sample problems (12) to (15). This indicates that the performance of the MOPSO algorithm in obtaining results for the first objective function will be even better in much larger dimensions.  
According to the box plot in Figure 10, since the null hypothesis is not within the obtained interval, there is a significant difference between the means of the first objective function obtained from solving with NSGA II and MOPSO algorithms.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **Sample Size** & **Mean** & **Standard** & **95\% Confidence** & **\(t\)-Test** & **\(p\)-Value** \\ \hline NSGA II & 15 & 5,626,072 & 3,852,039 & (4041 \(\times\) 53,686) & 2.49 & 0.026 \\ MOPSO & 15 & 4,497,208 & 3,821,250 & (4041 \(\times\) 53,686) & 2.49 & 0.026 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Results of the \(t\)-Test output on the means of the first objective function.  
Figure 10: Box plot for confirming or rejecting the null hypothesis for the means of the first objective function [23].  
Figure 9: Comparison of mean values of the first objective function in sample problems with metaheuristic algorithms [23].  
With a \(p\)-value of 0.584 obtained from Table 13, it can be concluded that there is no significant difference between the means of the second objective function. Therefore, to compare the most efficient algorithm, multi-criteria decision-making methods such as TOPSIS should be used.  
Figure 11 displays the comparison of mean values of the second objective function in different sample problems, indicating no significant difference among the obtained results. Therefore, it is not straightforward to assess the performance of the algorithms in achieving the second objective function results.  
Figure 12 also complements the results presented in Table 13. Given that the null hypothesis falls within the 95% confidence interval, it can be concluded that there is no significant difference between the means of the second objective function obtained by the NSGA II and MOPSO algorithms.