## 4 Results
### Comparative Analysis of Local Search Operators
#### 4.2.3 Performance Analysis on Large-Sized Instances (\(n=500\))
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} & \multicolumn{1}{c}{**Model Description**} \\ \hline \hline
**Model Description** &  
In conclusion, for these larger problem sizes, the \"All Operator\" configuration demonstrates a robust capability to produce higher-quality solutions compared to the other operators. The statistical tests confirm that many of these advantages are significant, with the post hoc [PERSON]'s test further detailing the operator pairs responsible for the performance gaps. Despite some competitive instances for \"Add\" or \"Swap\", the overall evidence supports the conclusion that combining multiple local operators typically leads to the better exploitation and exploration of the solution space, making it a compelling strategy for \(n=500\) instances in the Whale Optimization Algorithm. To validate the results using more than one algorithm, the Grey Wolf Optimizer (GWO) [27] was employed to solve the challenging MDMKP problem with a population size of \(n=30\). The metaheuristic optimization algorithm, inspired by the social behavior and hunting strategies of grey wolves, follows a leadership hierarchy consisting of the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Comparison** & **Avg \(p\)-Value** & **Max \(p\)-Value** & **Avg Sig.** & **Max Sig.** \\ \hline Add vs. All Operator & \(1.43\times 10^{-4}\) & \(1.14\times 10^{-4}\) & Yes & Yes \\ Add vs. Drop & \(1.39\times 10^{-10}\) & \(6.56\times 10^{-8}\) & Yes & Yes \\ Add vs. Swap & \(3.05\times 10^{-9}\) & \(1.57\times 10^{-7}\) & Yes & Yes \\ All Operator vs. Drop & \(7.70\times 10^{-12}\) & \(5.99\times 10^{-10}\) & Yes & Yes \\ All Operator vs. Swap & \(2.45\times 10^{-9}\) & \(3.21\times 10^{-8}\) & Yes & Yes \\ Drop vs. Swap & 1.00 & 1.00 & No & No \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Post hoc Dunn’s test results for pairwise comparisons between operators (\(n=500\)). Avg \(p\)-value and Max \(p\)-value indicate significance levels, with corresponding “Yes” or “No” for significant differences. Figure 4: Comparison of %GAP across different operators for \(n=500\) when evaluating average and maximum values relative to the best-known solution. The left panel presents scatter plots of %GAP for average results, and the right panel shows %GAP for maximum outcomes. Below these, the violin and scatter plots illustrate the distribution of %GAP, emphasizing the performance and variability among the operators. following four types of wolves: alpha, beta, delta, and omega. The first three guide the search process, which is structured in three main phases as follows: searching for prey, encircling the prey, and attacking the prey. To assess its effectiveness in addressing large-scale instances of the MDMKP problem, the algorithm was tested on large-scale instances under different local operator configurations. The number of search agents (_SearchAgents_no_) was set to five, defining the population size, while the maximum number of iterations (_Max_iter_) was established at 800, determining the number of updates in the agents' positions. The control parameter \(a\) was utilized to balance exploration and exploitation and was defined as \(a=2-\frac{2\times l}{Max\_iter}\), where \(l\) represents the current iteration number, starting at \(a=2\). Two coefficients, \(A\) and \(C\), were used to adjust the wolves' movements during the optimization.