## 4 Results
### Comparative Analysis of Local Search Operators
#### 4.2.2 Performance Analysis on Medium-Sized Instances (\(n=250\))  
The results for medium-sized instances (\(n=250\)) are presented in this section, highlighting the performance of four local operator configurations within the Whale Optimization Algorithm framework. Table 4 details the maximum and average values obtained across configurations, along with the associated computation times and standard deviations. Statistical significance among configurations was assessed using the Kruskal-Wallis test, with \(p\)-values confirming significant differences (\(p<0.05\)) in both the average and maximum results. Pairwise comparisons, shown in Table 5, were performed using the post-hoc Dunn's test, which identified the statistically significant differences between specific operator pairs for both the average and maximum values. Finally, Figure 3 visually summarizes the performance across configurations. Scatter plots contrast the %GAP of average and maximum values relative to the best-known solution, while violin plots emphasize the distribution and variability of %GAP values. These visualizations and analyses underline the effectiveness and variability of each configuration, particularly the superior performance of the \"All Operator\" configuration in many cases. The results presented in Table 4 provide a detailed comparison of the four local operator configurations applied to medium-sized instances (\(n=250\)) within the Whale Optimization Algorithm framework. The table highlights the maximum and average values achieved by each configuration, as well as the associated computation times and standard deviations. Notably, the \"All Operator\" configuration consistently outperforms the other configurations in terms of both the maximum and average values, with statistically significant differences confirmed by the Kruskal-Wallis test (\(p<0.05\)). These differences can be attributed to the synergistic effect of combining multiple local operators, which enables a more balanced search strategy that both explores diverse regions of the solution space and exploits promising areas more effectively. Although this integrated approach may slightly increase computation time, the resultant improvement in solution quality is a favorable trade-off. In practical applications, particularly for medium-sized instances of the Multi-Demand Multidimensional Knapsack Problem, this robust performance translates into more reliable and consistent outcomes, thereby enhancing the overall effectiveness of the optimization process. These findings suggest that the \"All Operator\" configuration offers a more robust exploration of the solution space. This result underscores the effectiveness of the \"All Operator\" configuration in optimizing performance for this instance size and algorithmic setup. The results in Table 5 summarize the pairwise comparisons of values between the four local operator configurations using the post hoc Dunn's test. This analysis complements the overall significance identified by the Kruskal-Wallis test, providing detailed insights into which specific operator pairs exhibit statistically significant differences. The \"Add\" configuration shows significant differences with the \"All Operator\" configuration in both the average and maximum values (\(p<0.05\)), underscoring the superior performance of the latter. Similarly, significant differences are observed between the \"All Operator\" and \"Drop\" configurations, as well as the \"All Operator\" and \"Swap\" configurations. These findings reaffirm the effectiveness of the \"All Operator\" configuration, particularly when compared to the other approaches. In contrast, the \"Add\" and \"Swap\" configurations show no significant differences in performance, suggesting similar behaviors in the tested instances.