## 4 Experimental Study
### Experimental Results  
Table 3 and Figure 6 illustrate the comparison results of the profits obtained from three different algorithms. The results are the average number of 10 runs. It is evident that when the task scale is within 300, algorithm profits do not differ much. However, as the size of the instances increases, the differences between the algorithms become more pronounced. These differences are attributed to improvements in the KG-PHLS algorithm, which include enhanced methods for constructing new solutions during local search, the incorporation of tabu strategies, and the use of FP-growth-based data mining methods. These enhancements boost the algorithm's optimization and global exploration capabilities, enabling exceptional performance in solving large-scale instances.  
Table 4 and Figure 7 present the comparison results of the computation time among three different algorithms. The results are the average times on ten independent runs. Concerning computation time, the ALNS-I algorithm exhibits lower efficiency than the other two algorithms when the instance size exceeds 75. The difference in  
As the instance size increases beyond 175, the KG-PHLS algorithm demonstrates significantly higher efficiency than the ALNS-TPF and ALNS-I algorithms. This outcome is due to the knowledge-guided new solution construction method and parallel computing. By leveraging domain-specific expertise or prior information, the algorithm can focus more effectively on regions of the solution space that may contain high-quality solutions, thereby reducing the search for invalid solutions. Through parallel computing, more tasks can be processed within the same time frame, accelerating the algorithm's execution speed.  
Higher deletion ratios enable the algorithm to escape local optima and explore the solution space more extensively, potentially leading to superior solutions. Conversely, lower deletion ratios restrict the algorithm's exploration range, increasing the likelihood of becoming trapped in local optima and resulting in lower yields. We tested these parameters at data scales of 300 and 400. Similar characteristics apply to the size of the deferred table: a large deferred table can constrain the algorithm's exploration capabilities. At the same time, a deferred table that is too small may prematurely trap the algorithm in local optima. Moreover, deferred table size influences algorithm stability somewhat, although this effect was not consistently significant during our experiments.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Number of Tasks**} & \multicolumn{3}{c}{**Computation Times (s)**} \\  & **ALNS-I** & **ALNS-TPF** & **KG-PHLS** \\ \hline
50 & 0.06 & **0.05** & 1.40 \\
75 & **0.35** & 0.47 & 2.42 \\
100 & 16.52 & **3.28** & 4.22 \\
125 & 51.01 & 8.80 & **6.25** \\
150 & 77.38 & 22.30 & **8.32** \\
175 & 96.54 & 27.06 & **26.80** \\
200 & 117.23 & 42.49 & **31.53** \\
225 & 143.85 & 65.95 & **41.46** \\
250 & 155.44 & 72.96 & **45.83** \\
275 & 214.68 & 62.97 & **54.00** \\
300 & 227.15 & 87.64 & **63.51** \\
325 & 235.21 & 131.92 & **73.17** \\
350 & 266.89 & 133.55 & **79.87** \\
375 & 288.70 & 161.92 & **84.54** \\
400 & 333.43 & 175.20 & **106.26** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Computation times with different algorithms.  
Figure 7: Results comparison on computation time.  
From Table 5, we observed that increasing the deletion ratio and extending the deferred table did not significantly improve outcomes despite increasing computational time. Therefore, we opted for the parameter combination \(k_{n}=0.1S\) and \(U_{L}=0.05S\), where \(S\) denotes the data scale as our preferred setup.