## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Fun.** & **LADE [50]** & **GJO-JOS [51]** & **QQSCA [52]** & **cSM [53]** & **FOWPC [54]** & **ATMALS-GSK** \\ \hline
1 & \(9.46\times 10^{4}\pm 1.85\times 10^{3}\) & \(3.90\times 10^{4}\pm 1.10\times 10^{4}\) & \(3.27\times 10^{4}\pm 3.372\times 10^{3}\) & \(8.14\times 10^{3}\pm 1.28\times 10^{4}\) & \(1.88\times 10^{4}\pm 2.20\times 10^{3}\) & \(2.55\pm 4.79\) \\ \hline
3 & \(3.68\times 10^{4}\pm 8.20\times 10^{3}\) & \(5.20\times 10^{4}\pm 1.10\times 10^{4}\) & \(3.00\times 10^{4}\pm 3.85\times 10^{-1}\) & \(4.83\times 10^{4}\pm 2.26\times 10^{4}\) & \(4.23\times 10^{4}\pm 9.62\times 10^{3}\) & \(4.91\times 10^{4}\pm 1.26\times 10^{4}\) \\ \hline
4 & \(7.98\times 10^{4}\pm 1.23\times 10^{-10}\) & \(7.70\times 10^{2}\pm 5.20\times 10^{4}\) & \(4.04\times 10^{2}\pm 2.27\) & \(2.31\times 10^{2}\pm 2.27\times 10^{1}\) & \(1.52\times 10^{2}\pm 4.63\times 10^{4}\) & \(4.90\pm 1.45\times 10^{1}\) \\ \hline
5 & \(1.41\times 10^{3}\pm 3.34\times 10^{1}\) & \(1.00\times 10^{3}\pm 1.70\times 10^{2}\) & \(5.11\times 10^{2}\pm 2.32\) & \(5.00\times 10^{2}\pm 6.65\times 10^{3}\) & \(3.11\times 10^{2}\pm 5.11\times 10^{4}\) & \(9.70\times 10^{4}\pm 2.17\times 10^{1}\) \\ \hline
6 & \(6.62\times 10^{2}\pm 1.18\) & \(6.40\times 10^{2}\pm 9.70\) & \(6.00\times 10^{2}\pm 3.58\times 10^{-3}\) & \(6.53\times 10^{-5}\pm 1.06\times 10^{-4}\) & \(2.96\times 10^{-1}\pm 8.85\times 10^{-2}\) & \(1.24\times 10^{-2}\pm 2.73\times 10^{-2}\) \\ \hline
7 & \(2.59\times 10^{3}\pm 2.92\times 10^{2}\) & \(1.50\times 10^{3}\pm 2.30\times 10^{2}\) & \(7.23\times 10^{2}\pm 3.62\) & \(7.07\times 10^{2}\pm 7.70\times 10^{1}\) & \(3.43\times 10^{2}\pm 3.11\times 10^{1}\) & \(1.68\times 10^{2}\pm 1.91\times 10^{1}\) \\ \hline
8 & \(1.66\times 10^{3}\pm 1.28\times 10^{2}\) & \(1.30\times 10^{3}\pm 1.80\times 10^{2}\) & \(8.09\times 10^{2}\pm 2.70\) & \(5.19\times 10^{2}\pm 7.77\times 10^{1}\) & \(3.28\times 10^{2}\pm 4.38\times 10^{1}\) & \(9.65\times 10^{4}\pm 1.12\times 10^{1}\) \\ \hline
9 & \(2.99\times 10^{4}\pm 7.15\times 10^{2}\) & \(4.50\times 10^{4}\pm 1.60\times 10^{4}\) & \(9.00\times 10^{2}\pm 1.88\times 10^{-5}\) & \(1.84\times 10^{4}\pm 5.08\times 10^{3}\) & \(7.71\times 10^{4}\pm 2.00\times 10^{3}\) & \(1.05\times 10^{4}\pm 3.14\) \\ \hline
10 & \(1.57\times 10^{4}\pm 1.19\times 10^{3}\) & \(1.60\times 10^{4}\pm 5.70\times 10^{3}\) & \(1.28\times 10^{3}\pm 1.05\times 10^{2}\) & \(1.18\times 10^{4}\pm 1.00\times 10^{3}\) & \(1.05\times 10^{4}\pm 7.56\times 10^{2}\) & \(2.44\times 10^{4}\pm 1.01\times 10^{3}\) \\ \hline
11 & \(3.40\times 10^{3}\pm 1.16\times 10^{3}\) & \(2.80\times 10^{3}\pm 2.90\times 10^{2}\) & \(1.10\times 10^{3}\pm 8.65\times 10^{-1}\) & \(1.29\times 10^{3}\pm 2.17\times 10^{2}\) & \(2.70\times 10^{2}\pm 5.01\times 10^{1}\) & \(1.76\times 10^{2}\pm 6.28\times 10^{1}\) \\ \hline \end{tabular}
\end{table} 
Table 16: Comparison of the average and standard deviation results (Mean \(\pm\) STD) of ATMALS-GSK with existing metaheuristics over 51 runs for \(D=100\) on CEC 2017. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Fun. ** & **LADE [50]** & **GJO-JOS [51]** & **QQSCA [52]** & **cSM [53]** & **FOWPC [54]** & **ATMALS-GSK** \\ \hline
23 & \(2.97\times 10^{3}\pm 3.90\times 10^{1}\) & \(2.90\times 10^{3}\pm 5.40\times 10^{3}\) & \(2.61\times 10^{3}\pm 2.88\) & \(1.44\times 10^{4}\pm 3.65\times 10^{3}\) & \(5.40\times 10^{2}\pm 8.26\times 10^{3}\) & \(4.39\times 10^{2}\pm 1.27\times 10^{1}\) \\ \hline
24 & \(3.13\times 10^{3}\pm 3.48\times 10^{1}\) & \(3.10\times 10^{3}\pm 6.90\times 10^{3}\





### Statistical Analysis

The statistical analysis of the CEC 2017 benchmark functions using the Wilcoxon and Friedman tests (shown in Tables 17-20 and Figures 13 and 14) supports the strong performance of ATMALS-GSK. The Wilcoxon test results in Table 17 demonstrate that the proposed ATMALS-GSK algorithm surpasses GSK, AGSK, APGSK, FDBAGSK, and eGSK algorithms in 87, 86, 91, 77, and 76 cases out of all 29 \(\times\) 4 = 116 cases (29 functions over four dimensions).