## 4 Results
### Comparative Analysis of Local Search Operators
#### 4.2.3 Performance Analysis on Large-Sized Instances (\(n=500\))  
The results for larger instances (\(n=500\)) are presented in this section, focusing on the performance of four local operator configurations within the Whale Optimization Algorithm. As shown in Table 6, each configuration's maximum and average values are reported alongside computation times and their standard deviations. A Kruskal-Wallis test (\(p<0.05\)) confirmed that these configurations produce statistically different outcomes in both the average and maximum values. Subsequently, Table 7 provides post hoc [PERSON]'s test comparisons, illustrating which specific operator pairs exhibit significant differences. Collectively, these analyses offer a comprehensive view of how each configuration behaves on instances of size \(n=500\). Within Table 6, the \"All Operator\" configuration again demonstrates notably strong results, often surpassing the other three configurations in terms of both maximum and average performance. For certain instances, the \"Add\" or \"Swap\" configurations yield competitive outcomes, but their results typically carry greater variability. In contrast, the \"Drop\" configuration tends to produce considerably lower average values, which is further highlighted by the large gap in some instances. This performance gap and increased variability observed in the \"Drop\" configuration may be due to its limited ability to recover from suboptimal moves, as it solely removes items without providing opportunities for corrective actions such as adding or swapping. Consequently, this approach might miss promising regions in the solution space, leading to less consistent outcomes. Additionally, the superior performance of the \"All Operator\" configuration suggests that combining diverse local moves fosters a more balanced search strategy, which is particularly beneficial as problem size increases, thereby enhancing the algorithm's scalability and robustness. These findings reinforce the idea that simultaneously leveraging multiple local operators (\"All Operator\") can better explore the solution space, leading to improved overall outcomes at this instance scale. The post hoc [PERSON]'s test results shown in Table 7 substantiate the overall Kruskal-Wallis findings by pinpointing exactly where the significant differences occur. Notably, \"Drop\" diverges substantially from the other configurations in most of the pairwise comparisons (e.g., \"Add vs. Drop\", \"All Operator vs. Drop\", and \"Swap vs. Drop\"). Meanwhile, both \"Add vs. All Operator\" and \"Add vs. Swap\" comparisons show strong statistical significance, supporting the notion that \"All Operator\" remains a top performer across multiple measures. In contrast, certain pairs (like \"Drop vs. Swap\") occasionally demonstrate no significant difference in specific tests, suggesting that those methods can converge to similar results under particular conditions, albeit at generally lower performance levels than \"All Operator\". A visual summary of these outcomes for \(n=500\) can be provided in a figure analogous to Figure 3, now adapted for larger instances (e.g., Figure 4). The violin plots would illustrate the distribution and spread of %GAP in both the averages and maximums for each operator, while the scatter plots would relate %GAP to the best-known values. These visualizations highlight the pronounced difference in performance among configurations and underscore the relatively consistent advantage of \"All Operator\". By combining the statistical and graphical insights, it becomes clear that the richer local search provided by the \"All Operator\" configuration often yields superior solutions and reduced variability for \(n=500\).