## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
This results in success ratios of ATMALS-GSK against GSK, AGSK, APGSK, FDBAGSK, and eGSK of 75%, 74.1%, 78.4%, 66.4%, and 65.5%, respectively. Furthermore, the results in Table 19 indicate that ATMALS-GSK exceeds IADE, GJO-JOS, QCSCA, cSM, and FOWFO, recording success ratios of 96.6%, 96.6%, 87.1%, 91.4%, and 5 \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{6}{c}{**Ranks**} & \multirow{2}{*}{**Mean Rank**} & \multirow{2}{*}{**Overall Rank**} \\ \cline{2-3} \cline{5-8}  & \(D\) = 10 & \(D\) = 30 & \(D\) = 50 & \(D\) = 100 & \multicolumn{1}{c}{**Mean Rank**} & \multicolumn{1}{c}{**Overall Rank**} \\ \hline ATMALS-GSK & 3.37 & 1.84 & 1.93 & 1.82 & 2.24 & 1 \\ \hline eGSK & 3.53 & 2.24 & 2.76 & 2.86 & 2.84 & 2 \\ \hline FDBAGSK & 2.81 & 3.66 & 3.09 & 3.82 & 3.34 & 3 \\ \hline AGSK & 2.87 & 4.36 & 4.41 & 3.93 & 3.89 & 4 \\ \hline GSK & 4.86 & 4.16 & 4.28 & 3.58 & 4.22 & 5 \\ \hline APGSK & 3.53 & 4.74 & 4.53 & 4.96 & 4.44 & 6 \\ \hline \(p\)-value & 0.0003 & 0.0001 & 0.0000 & 0.0000 & 0.0000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Friedman test of the ATMALS-GSK algorithm with the GSK variants. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**D**}} & \multirow{2}{*}{**Methods**} & \multirow{2}{*}{**R+**} & \multirow{2}{*}{**R –**} & \multirow{2}{*}{**\(p\) Value**} & \multirow{2}{*}{**+**} & \multirow{2}{*}{**=**} & \multirow{2}{*}{**Dec.**} \\ \hline \multirow{8}{*}{10} & \multirow{8}{*}{ATMALS-GSK vs.} & IADE & 435 & 0 & 0.0000 & 29 & 0 & 0 & + \\ \cline{3-8}  & & GJO-JOS & 435 & 0 & 0.0000 & 29 & 0 & 0 & + \\ \cline{3-8}  & & QSCCA & 435 & 0 & 0.0000 & 29 & 0 & 0 & + \\ \cline{3-8}  & & cSM & 435 & 0 & 0.0000 & 29 & 0 & 0 & + \\ \cline{3-8}  & & FOWFO & 143 & 292 & 0.1104 & 14 & 0 & 15 & \(\approx\) \\ \hline \multirow{8}{*}{30} & \multirow{8}{*}{ATMALS-GSK vs.} & IADE & 435 & 0 & 0.0000 & 29 & 0 & 0 & + \\ \cline{3-8}  & & GJO-JOS & 435 & 0 & 0.0000 & 29 & 0 & 0 & + \\ \cline{3-8}  & & QSCCA & 425 & 10 & 0.0000 & 28 & 0 & 1 & + \\ \cline{3-8}  & & cSM & 406 & 0 & 0.0000 & 28 & 1 & 0 & + \\ \cline{3-8}  & & FOWFO & 205.5 & 229.5 & 0.7983 & 16 & 0 & 13 & \(\approx\) \\ \hline \multirow{8}{*}{50} & \multirow{8}{*}{ATMALS-GSK vs.} & IADE & 433 & 2 & 0.0000 & 28 & 0 & 1 & + \\ \cline{3-8}  & & GJO-JOS & 415 & 20 & 0.0000 & 27 & 0 & 2 & + \\ \cline{3-8}  & & QSCCA & 322 & 113 & 0.0229 & 24 & 0 & 5 & + \\ \cline{3-8}  & & cSM & 392 & 43 & 0.0000 & 26 & 0 & 3 & + \\ \cline{3-8}  & & FOWFO & 217 & 218 & 1.00 & 15 & 0 & 14 & \(\approx\) \\ \hline \multirow{8}{*}{100} & \multirow{8}{*}{ATMALS-GSK vs.} & IADE & 383 & 52 & 0.0001 & 26 & 0 & 3 & + \\ \cline{3-8}  & & GJO-JOS & 396 & 39 & 0.0000 & 27 & 0 & 2 & + \\ \cline{3-8}  & & QSCCA & 276 & 159 & 0.2132 & 20 & 0 & 9 & \(\approx\) \\ \cline{3-8}  & & cSM & 360 & 75 & 0.0013 & 23 & 0 & 6 & + \\ \cline{3-8}  & & FOWFO & 236 & 199 & 0.7013 & 18 & 0 & 11 & \(\approx\) \\ \hline \hline \end{tabular}
\end{table} 
Table 19: [PERSON] signed-rank test of the ATMALS-GSK algorithm with existing metaheuristics. The Friedman test results in Tables 18 and 20 further showcase the effectiveness of the proposed ATMALS-GSK algorithm. In these tables, \"Ranks\" provide the average Friedman test rankings of the different algorithms over 51 runs for each dimension (\(D\) = 10, 30, 50, and 100). Furthermore, \"Mean Rank\" calculates the average rankings for all dimensions. Finally, \"Overall Rank\" provides the sorted rank of the different algorithms based on their Mean Rank. Furthermore, Figures 13 and 14 visually reinforce the outcomes derived from the Friedman test shown in Tables 18 and 20, in terms of the \"Mean Rank\" and \"Overall Rank\" of the different algorithms. In Figure 13, based on the \"Overall Rank\" results, ATMALS-GSK secures the top rank when compared with its GSK-based variants across all dimensions. Similarly, Figure 14 presents a comparative ranking against other metaheuristics, where ATMALS-GSK again outperforms existing metaheuristics by a significant margin.