## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
The consistent first-place ranking across all dimensions highlights its competitiveness and dominance in achieving optimal solutions. When compared to its GSK-based variants (Tables 17 and 18 and Figure 13) and other well-known metaheuristics (Tables 19 and 20 and Figure 14), ATMALS-GSK consistently performs better. The Wilcoxon test demonstrates that it significantly outperforms most of the competing methods across all problem dimensions (\(D=10\) to \(100\)), with p-values often much lower than \(0.05\), especially when compared to the original GSK and its variants. Although ATMALS-GSK performs similarly to improved variants like AGSK and APGSK at lower dimensions (e.g., for \(D=10\)), it becomes noticeably superior as the problem size increases. It also clearly outperforms popular metaheuristics like IADE, GJO-JOS, QSCA, and cSM in nearly all cases. These results show that ATMALS-GSK is not only effective but also consistently reliable across a wide range of benchmark problems and dimensions. ### Parametric Study of ATMALS-GSK

This section provides a parametric study of the proposed ATMALS-GSK algorithm to evaluate the individual effect of each contribution to the overall performance. To achieve this purpose, we compare ATMALS-GSK with its five components (ATMALS-GSK-1 to ATMALS-GSK-5). For each algorithm, one change among the five adaptive parameters (\(K\), \(K_{r}\), \(K_{f}\), \(P\), and \(P_{LS}\)) is enabled, while keeping the other four parameters fixed as the basic GSK algorithm. The aim is to show that our algorithm ATMALS-GSK (with five adaptive parameters) is superior to ATMALS-GSK-1, ATMALS-GSK-2, ATMALS-GSK-3, ATMALS-GSK-4, and ATMALS-GSK-5 (each considering just one parameter as adaptive and fixing the other parameters). In the following, we test the individual components of the ATMALS-GSK algorithm to assess the impact and efficiency of the modifications made to the GSK algorithm and to verify that ATMALS-GSK performs as expected. The ATMALS-GSK algorithm includes five main modifications compared to the original GSK algorithm: GSK with adaptive \(K\) (ATMALS-GSK-1), GSK with adaptive \(K_{r}\) (ATMALS-GSK-2), GSK with adaptive \(K_{f}\) (ATMALS-GSK-3), GSK with adaptive \(P\) (ATMALS-GSK-4), and GSK with adaptive local search mechanism \(P_{LS}\) (ATMALS-GSK-5). These modifications influence the convergence and balance the exploration and exploitation aspects of the ATMALS-GSK algorithm. To assess the effectiveness of the five ATMALS-GSK components, these versions are compared with ATMALS-GSK and GSK on CEC 2017 with \(D=30\). The average results of ATMALS-GSK, ATMALS-GSK-1, ATMALS-GSK-2, ATMALS-GSK-3, ATMALS-GSK-4,

Figure 14: Rank analysis of the proposed ATMALS-GSK algorithm against the existing metaheuristic algorithms, based on the Friedman test. ATMALS-GSK-5, and GSK over 51 runs on different problems are reported in Table 21. The results show that all components impact the efficiency of the combined modifications and confirm that the ATMALS-GSK algorithm functions as intended. It is evident that the ATMALS-GSK algorithm significantly outperforms others in F1, F3, F4, F7, F8, F10, F11, F12, F13, F14, F16, F17, F18, F20, F21, F22, F23, F24, F25, F28, F29, and F30. \begin{table}
\begin{tabular}{c c c c c c c c} \hline



The statistical analysis results are provided in Table 22 and Figure 15. The Wilcoxon test results in Table 22 indicate that the proposed ATMALS-GSK algorithm surpasses its five versions in 22, 24, 23, 22, and 19 functions, which results in success ratios of ATMALS-GSK against ATMALS-GSK-1 to ATMALS-GSK-5 by 75.9%, 82.7%, 79.3%, 82.7%, and 65.5%, respectively. The rank analysis using the Friedman test in Figure 15 demonstrates the effectiveness of applying all modifications of the ATMALS-GSK algorithm together. According to the mean and overall ranks in Figure 15, ATMALS-GSK secures the top rank when compared with GSK and the ATMALS-GSK components.