## 5 Results
### Performance of Three PSO-Based Algorithms and Seven DE-Based Algorithms
We also used the free software KEEL [62] to apply the Friedman test to obtain average rankings for the ten algorithms proposed in this paper based on the results of the experiments for POP = 30 and POP = 50. To perform the Friedman test for the results when POP = 30, the data in Tables 12 and 13 were first transformed into rank data. Then, the rank data were used as the inputs to perform the Friedman test. The average rankings of the ten algorithms proposed in this paper are shown in Table 14. The results in Table 14 indicate that the best three algorithms were the NSDE, DE-3, and DE-1 algorithms. This was consistent with our analysis. The Friedman statistics showed significant differences between the algorithms, with a value of 48.103636 and a \(p\)-value = 0. In addition to the comparison of performance based on the average fitness function values, we also compared the efficiency of the ten algorithms. Tables 15 and 16 show the average number of generations required to find the best solutions for all runs of each algorithm when the population size was 30. Figure 5 shows the average number \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Ranking** \\ \hline
**PSO** & 6.85 \\
**CLPSO** & 8.55 \\
**CenPSO** & 9.1 \\
**NSDE** & 3 \\
**DE-1** & 3.15 \\
**DE-2** & 6.25 \\
**DE-3** & 3 \\
**DE-4** & 4.7 \\
**DE-5** & 5.75 \\
**DE-6** & 4.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Average rankings of the ten algorithms obtained by Friedman test for \(NP=30\). Figure 4: Average fitness function values for \(r_{D}=r_{P}=0.1\) when \(NP=30\). By comparing Table 15 with Table 16, it can be seen that the average number of generations required to find the best solutions by the six DE-based algorithms was much lower than that required by the PSO, CLPSO, and CenPSO algorithms. The six DE-based algorithms were more efficient than the three variants of the PSO algorithms when the population size was 30. Figure A1 through Figure A10 in Appendix B show the convergence speeds of simulation runs for each algorithm and each test case. Based on the comparison above, the seven DE-based algorithms were more efficient than the PSO, CLPSO, and CenPSO algorithms when the population size was 30. For common parameter setting 2, the population size was 50. The average fitness function values obtained by applying the PSO, CLPSO, CenPSO, and DE-7(NSDE) algorithms are listed in Table 17. Regarding the three PSO-based algorithms, the average fitness function value obtained by the PSO algorithm was the same as that of the CLPSO algorithm and the CenPSO algorithm for Case 1. The average fitness function value obtained by the PSO algorithm was also the same as that of the CLPSO algorithm and CenPSO algorithm for Case 2. The average fitness function values obtained by the PSO algorithm were higher than those of the CenPSO algorithm for Case 2. The average fitness function values obtained by the PSO algorithm were higher than those of the CLPSO and CenPSO algorithms for all other cases (Case 3 through Case 10). By comparing the average fitness function values obtained by the PSO algorithm and the DE-7(NSDE) algorithm, we found that the average fitness function values obtained by the DE-7(NSDE) algorithm were equal to those of the PSO algorithm for Case 1, Case 2, Case 4, and Case 5. Moreover, the average fitness function values obtained by the DE-7(NSDE) algorithm were greater than those of the PSO algorithm for all other cases (Case 3, Case 6, Case 7, Case 8, Case 9, and Case 10). This indicated that the performance of the DE-7(NSDE) algorithm was either as good as or better than that of the PSO algorithm for all the test cases when the population size was 50. When the population size was 50, the average fitness function values obtained by applying the six DE algorithms are listed in Table 18. The average fitness function values obtained by the DE-1 algorithm and the DE-3 algorithm were the same as those of the NSDE algorithm for all test cases.