## 5 Results of Comparing Algorithms
### Efficiency Comparison  
To evaluate the efficiency of the four algorithms in terms of convergence speed and execution time the Kruskal-Wallis test was used. The convergence speed of an algorithm is evaluated based on the fitness variation. This criterion shows how the algorithm converges toward the optimum solution through a number of iterations, while the execution time reveals how fast the algorithm is in terms of running time. The results of the efficiency criteria are presented in Table 2. The \(p\)-values in Table 2 show that there is a very significant difference in convergence speed (fitness variation rate) between the algorithms for both objective functions. This is identified by \(p\)-value = 0.000, which is less than alpha = 0.05. The post hoc tests using the Conover test were carried out for pairwise convergence comparison, and the results are presented in Table 4. From Table 4, we found that there are statistically significant differences between all algorithms when optimizing the capacity function (\(p<0.05\)). Regarding fitness variation of distance function, only two paired comparisons of MOABC-MSPSO and AMOSA-NSGA-II did not show significant differences (\(p>0.05\)) among six paired comparisons. Figure 7 shows a boxplot of fitness variation of both objectives and average execution time for the four algorithms. By assessing box plot a and b in Figure 7, we notice that AMOSA outperforms the other three algorithms, with a minimum average cost for both capacity and distance functions. However, Figure 7c demonstrates that NSGA-II is the fastest algorithm compared to the three others. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Index**} & \multicolumn{2}{c}{**Fcapacity**} & \multicolumn{2}{c}{**Fdistance**} \\ \cline{2-5}  & **t-Statistic** & \(p\)**-Value** & **t-Statistic** & \(p\)**-Value** \\ \hline AMOSA-MOABC & \(-\)11.272 & 0.000 * & \(-\)7.710 & 0.000 * \\ AMOSA-MSPSO & \(-\)23.040 & 0.000 * & \(-\)8.554 & 0.000 * \\ MOABC-MSPSO & \(-\)18.221 & 0.000 * & \(-\)1.309 & 0.467 \\ AMOSA-NSGA-II & \(-\)3.055 & 0.006 * & \(-\)1.575 & 0.340 \\ MOABC-NSGA-II & 12.716 & 0.000 * & 9.495 & 0.000 * \\ MSPSO-NSGA-II & 30.921 & 0.000 * & 10.799 & 0.000 * \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of fitness values of capacity and distance function (fcapacity & distance) for convergence speed. Figure 6: The final Pareto fronts of four algorithms with normalized fitness values. (**a**) Pareto front of AMOSA, (**b**) Pareto front of MOABC, (**c**) Pareto front of MSPSO, (**d**) Pareto front of NSGA-II. This shows that the algorithm with high convergence speed is not always the one with the shortest execution time. The execution time is mostly influenced by the size of the population and the number of iterations. For AMOSA, 500 iterations have increased the computation time compared with when the algorithm runs of 100 iterations. Figure 8 presents the convergence trends of the four algorithms, for both capacity and distance objective functions. The best fitness values of the two functions were normalized in order to facilitate their comparisons. With the progress of the algorithms, the convergence speed is reduced until the optimal solutions are attained. The mean fitness variation of MSPSO is higher compared to that of AMOSA, NSGA-II, and MOABC as shown in Figure 7a,b. Note that for AMOSA, the number of iterations displayed in Figure 8a did not attain 500 as for other algorithms. This is due to its nested loops that also iterate the cooling rate (from high temperature to lower temperature). To avoid the repetitions of solutions, we only retrieved the minimum fitness value obtained after 500 iterations of every degree of the cooling temperature (set to 100 \({}^{\circ}\)C). In general, the convergence speed of AMOSA and NSGA-II are higher (better) followed by MOABC and MSPSO. The reason for smoother convergence of NSGA-II is the crossover and mutation operators that influence to obtain the best survivors (offspring) for the next generation.