## 5 Conclusions  
In this study, the impact of four local operator configurations--Add, Drop, Swap, and All Operator--on the performance of metaheuristic algorithms in solving the MDMKP was investigated. The experiments were conducted using the Whale Optimization Algorithm (WoA) with a specialized binarization process and a repair mechanism. Each local operator configuration was evaluated on small (\(n=100\)), medium (\(n=250\)), and large (\(n=500\)) benchmark instances. The results consistently indicated that the use of multiple local oper  
Figure 5: Sensitivity analysis results for the parameter \(L\). (**a**) %GAP of maximum value. (**b**) %GAP of average maximum value.  
ators in the \"All Operator\" configuration led to significantly better performance compared to individual operators. Higher average and maximum solution values were achieved, while maintaining greater stability. In contrast, the \"Drop Operator\" was found to be less effective in large-scale settings. Although \"Add\" and \"Swap\" performed competitively on certain instances, their results exhibited greater variability. Statistical analyses, including the [PERSON]'s post hoc tests, confirmed that these differences were statistically significant in most pairwise comparisons.  
To further validate these findings, an additional study was conducted using the Grey Wolf Optimizer (GWO) under the same four local operator configurations. The results again indicated that the \"All Operator\" configuration yielded the best performance for large-scale instances, reinforcing the conclusion that incorporating multiple local moves improves both solution quality and consistency. Additionally, a sensitivity analysis was performed on the local search iteration parameter (\(L\)), varying its value from 1000 to 2000 in increments. The results showed a consistent reduction in the percentage gap (%GAP) for both the maximum and average maximum values as \(L\) increased. The most reliable improvements were observed at \(L=2000\), where the lowest %GAP values were consistently recorded across all instances.  
Several directions for future research can be considered. The development of adaptive or learning-based approaches that dynamically adjust operator usage based on the search state could enhance the balance between exploration and exploitation, particularly in large-scale or real-time optimization contexts. Additionally, extending the \"All Operator\" strategy to other combinatorial optimization problems (e.g., knapsack variants, scheduling, or routing) could provide further insights into its broader applicability and robustness. Investigating parallelization strategies may also be a promising avenue, especially for large-scale problems, given the computational complexity of advanced local operators combined with sophisticated metaheuristics.  
Conceptualization, J.G. and P.M.; Methodology, I.C. and H.P.; Software, J.G. and I.C.; Formal analysis, J.G., I.C., P.M. and H.P.; Investigation, J.G. and I.C.; Resources, P.M.; Writing--original draft, J.G. and I.C.; Writing--review & editing, P.M.; Supervision, H.P.; Funding acquisition, H.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors. The authors declare no conflicts of interest.