## 5 Results of Comparing Algorithms
### Effectiveness Comparison  
This study compares the effectiveness of the four algorithms to see how effective each algorithm optimizes the two defined objective functions. Table 11 shows the average and worst fitness values for both capacity and distance functions (fcapacity, distance), as well as the execution times obtained for 30 runs of each algorithm. In all 30 cases, AMOSA was the best one optimizing both objective functions, while MSPSO derived extreme values compare to MOABC and NSGA-II. As shown in Table 2, the Kruskal-Wallis test provided very strong evidence of a difference between the mean ranks of the four methods, optimizing the fcapacity and fdistance. The \(p\)-values of both functions are smaller than alpha = 0.05. This means that all algorithms perform differently in terms of optimizing the two objective functions. For effectiveness evaluation in Table 2, the results from the Kruskal-Wallis test shows very strong evidence of a difference (\(p\) = 0.000 and \(p\) = 0.000) between the mean ranks of the four algorithms in both optimizing fcapacity and fdistance. As shown in Table 3, the pairwise comparison using the Conover-Iman test was carried out to compare the four algorithms and we notice strong evidence of the difference between MSPSO and the other three algorithms, regarding the minimum fitness of capacity as well as distance functions. The asterisk symbol in Table 3 shows where the \(p\)-value is less than alpha (\(\alpha\) = 0.05), indicating a significant difference between a pair of algorithms in terms of the quality of solutions obtained (see Table 11). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline AMOSA & \\ Number of population & 100 \\ Number of iterations & 500 \\ Tmax & 100 \\ Tmin & \(10\times 10^{-3}\) \\ Alpha (\(\alpha\)) & 0.9 \\ MOABC & \\ Colony size & 100 \\ Number of iterations & 500 \\ Inertia Weight (w1) & 0.7 \\ Inertia Weight (w2) & 0.8 \\ MSPSO & \\ Number of particles & 100 \\ Number of iterations & 500 \\ Acceleration constant (c1 = c2) & 1.49 \\ Inertia Weight (w) & 0.72 \\ NSGA-II & \\ Number of chromosomes & 100 \\ Number of iterations & 500 \\ cross-mutate rate & 0.9 \\ mutation rate & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameter values of the four algorithms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Effectiveness**} & \multicolumn{2}{c}{**Efficiency (Fitness Variation)**} & \multicolumn{2}{c}{**Efficiency (Execution Time)**} \\ \cline{2-5}
**KW Test** & \begin{tabular}{c} **Cost of** \\ **Fcapacity** \\ \end{tabular} & \begin{tabular}{c} **Cost of** \\ **Fdistance** \\ \end{tabular} & \begin{tabular}{c} **Cost of** \\ **Fcapacity** \\ \end{tabular} & \begin{tabular}{c} **Cost of** \\ **Fdistance** \\ \end{tabular} & \
\begin{tabular}{c} **Cost of** \\ **Fdistance** \\ \end{tabular} &
\begin{tabular}{c} **Cost of** \\ **Fdistance** \\ \end{tabular} \\ \hline Chi-Square & 88.809 & 98.016 & 674.13 & 162.42 & 105.028 \\ \(p\)-value & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(p\)-values of the Kruskal–Wallis test for evaluating the effectiveness and efficiency of four algorithms in both optimizing capacity and distance function. The box plot presents the average cost of two objectives for the four algorithms (Figure 5). Figure 5a,b show that AMOSA and MOABC are the algorithms with the minimum average cost for both objectives. NSGA-II is the third in optimizing both objective functions, while MSPSO has a significantly higher average cost. Figure 6 shows that AMOSA returns the high number of solutions while MOABC returns the small number of solutions in the final Pareto front. AMOSA and NSGA-II effectively converge faster to the minimum fitness compare to MOABC and MSPSO. The large size of AMOSA's solutions is due to its capacity for archiving and clustering solutions that control diversity among the non-dominated solutions.