## 3 The Algorithm
### SaNSDE Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & **I** & **K** & **DE1** & **DE2** & **DE3** & **DE4** & **DE5** & **DE6** \\ \hline
1 & 4 & 3 & 320.4/23.7 & 320.4/96.3 & 320.4/113.8 & 320.4/31.5 & 321/24.1 & 321/493.1 \\
2 & 5 & 5 & 42.8/2.3 & 42/25.5 & 40.5/20.5 & 40.4/764.4 & 41.5/1093.9 & 40.3/810.9 \\
3 & 5 & 5 & 51.7/98.3 & 42.8/2604.1 & 51/221.6 & 41.6/934.3 & 49.4/755.2 & 53/1428.6 \\
4 & 10 & 10 & 534.06/31.6 & 540.6/32.8 & 542.2/894.3 & 539/77.3 & 536.66/90.2 & 539/119.2 \\
5 & 10 & 10 & 717.93/2846.6 & 621.36/3302.6 & 701.53/5920.3 & 593.53/5588.3 & 538.62/4369.4 & 694.59/3550.2 \\
6 & 10 & 20 & 16743035.1 & 1602.44/466.8 & 1737/4950.5 & 1615.6/3021.6 & 1574.6/2961.2 & 1596.8/2059 \\
7 & 20 & 20 & 1544.9/5964.4 & 1110.7/5285 & 1381.8/4779.9 & 1194.1/5950.2 & 1078/3609.4 & 1440.3/5512 \\
8 & 30 & 10 & 661.7/1439.1 & 354.6/2993.8 & 678.2/2655.8 & 525.9/2668.5 & 532.9/2401.8 & 601.2/3066.8 \\
9 & 30 & 20 & 725.1/1760.3 & 657.8/389.8 & 732.6/1093.8 & 721.9/1305 & 693.6/2318.3 & 706.6/1458.9 \\
10 & 40 & 10 & 730.8/1771.2 & 566.1/4057.5 & 700.1/1883.1 & 689.8/2653.2 & 623.1/1862 & 629.1/2833.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average fitness function values for DE1, DE2, DE3, DE4, DE5 and DE6 with \(NP=10\). Figure 5: Average fitness function values for discrete SaNSDE (\(LP=1000\)), NSDE and PSO with \(NP=10\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **I** & **K** & **SaNSDE** & **NSDE** & **PSO** \\ \hline
1 & 4 & 3 & 321/17.8 & 319.8/13 & 321/25.6 \\
2 & 5 & 5 & 42/133 & 42/28 & 42/14.5 \\
3 & 5 & 5 & 55/1731.8 & 51/99.6 & 54/4039.1 \\
4 & 10 & 10 & 539/70.9 & 536.66/28.3 & 542.2/133.5 \\
5 & 10 & 10 & 715.08/5327.1 & 616.06/145.7 & 485.12/4946.1 \\
6 & 10 & 20 & 1717/5140.6 & 1580.7/149.7 & 1310.4/4603.3 \\
7 & 20 & 20 & 1557.9/4860.4 & 1070.7/178.3 & 256.1/4832.2 \\
8 & 30 & 10 & 726.9/3750.3 & 639.4/134.8 & 434.1/4327.8 \\
9 & 30 & 20 & 768.8/2453.2 & 731.4/140.9 & 461.3/4841.6 \\
10 & 40 & 10 & 780.1/3175.5 & 677.7/140.9 & 303.9/4999.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fitness function values for discrete SaNSDE, NSDE and PSO with population size \(NP=10\). In terms of performance, the results of Table 7 indicate that the SaNSDE algorithm either outperforms or performs as well as the NSDE algorithm and PSO algorithm in all instances. The results of Table 8 indicate that the SaNSDE algorithm either outperforms or performs as well as the six standard DE algorithms for most instances, with the exceptions of Case 5 and Case 7. The number of best solutions found by the SaNSDE algorithm, the NSDE algorithm, DE1 algorithm, DE2 algorithm, DE3 algorithm, DE4 algorithm, DE5 algorithm and DE6 algorithm are 8, 4, 3, 7, 3, 2, 1, 1 and 2, respectively. Therefore, in terms of the number of best solutions found, the SaNSDE algorithm outperforms the other algorithms. Tables 9 and 10 show the results for the population size \(NP\) = 50, obtained by applying the SaNSDE algorithm, a variant of the neighborhood search-based DE (NSDE) algorithm, the PSO algorithm and six standard DE algorithms. The average fitness function values in Tables 9 and 10 are shown in the bar charts of Figures 9 and 10, respectively. Figure 8: Average fitness function values for SaNSDE (\(LP\) = 1000), DE1, DE2, DE3, DE4, DE5 and DE6 with \(NP\) = 30. Figure 7: Average fitness function values for discrete SaNSDE (\(LP\) = 1000), NSDE and PSO with \(NP\) = 30. In terms of performance, the results of Table 9 indicate that the SaNSDE algorithm either outperforms or performs as well as the NSDE algorithm and PSO algorithm for all instances. The results of Table 10 indicate that the SaNSDE algorithm either outperforms or performs as well as the six standard DE algorithms in most instances, with the exceptions of Case 5, Case 6 and Case 7.