## 4 Experiments
### Experiment Results  
The proposed method consists of two parts, where RL is used to generate high-quality initial solutions and \(IG_{h}\) is used to optimize the solutions obtained by RL. The role of RL methods is similar to that of heuristics, so RL is compared with some classical heuristics such as NEH [4], earliest-due-date first (EDD) [19], and smallest late work insertion (SLW) [18]. A comparison of the effects of these methods is shown in Table 5. The table shows a description of the problem instances, the average relative difference percent (ARDP) values and average computation time (ACT) of each method. The \(RDP\) is calculated as Equation (22):  
\[RDP=\frac{Obj_{x}-Obj_{min}}{m\times n\times 10}\times 100\% \tag{22}\]  
where \(Obj_{x}\) is the objective function value of the solution obtained by method \(x\), \(Obj_{min}\) represents the minimum objective function value obtained by all methods. The ARDP can indicate the difference between the objective function values obtained by several methods solving the same set of benchmarks, and a value of 0 means that the current method obtains the smallest late work among several methods. In short, the smaller the ARPD value, the better the performance of the algorithm and the higher the quality of the solutions obtained. As can be seen from Table 5, the RL algorithm using the sampling decoding strategy performs better than the other heuristics on problems of all sizes. Since EDD uses the simplest dispatch rule, the computation time of the EDD heuristic can be neglected, but the largest ARDP value 100.54 is obtained. The NEH algorithm is a widely used heuristic. The computation time of RL-Greedy is slightly increased compared to NEH, but the ARDP value is reduced by half (16.79 to 9.9). Box plot 10 visually shows the distribution of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & \(d\) & \(T_{p}\) & **AIP** \\ \hline
1 & 1 & 1 & 30.23 \\
2 & 1 & 2 & 30.42 \\
3 & 1 & 3 & 31.34 \\
4 & 1 & 4 & 27.65 \\
5 & 2 & 1 & 33.87 \\
6 & 2 & 2 & 37.54 \\
7 & 2 & 3 & 40.18 \\
8 & 2 & 4 & 39.11 \\
9 & 3 & 1 & 35.54 \\
10 & 3 & 2 & 37.29 \\
11 & 3 & 3 & 38.93 \\
12 & 3 & 4 & 33.74 \\
13 & 4 & 1 & 29.76 \\
14 & 4 & 2 & 33.66 \\
15 & 4 & 3 & 35.85 \\
16 & 4 & 4 & 29.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of the orthogonal test. Figure 9: Effect of parameter settings on algorithm performance. (**(a)** is the orthogonal test result for \(d\), (b)** is the orthogonal test result for \(t_{p}\)). achieved RDPs and skewness by displaying the data quartiles (or percentiles) and averages. The box plots generally include the five-number summary of a set of data: the minimum score, first quartile, median, third quartile, and maximum score [44]. As can be seen in Figure 10, the RL-based method is significantly better than other heuristics. Furthermore, the combination of RL-Sample and \(IG_{h}\) is compared with other metaheuristic algorithms such as hybrid cuckoo search algorithm (HCS) [45], discrete artificial bee colony algorithm (DABC) [46], and NEH + IG algorithmA [42]. The hyperparameters of other algorithms use the default settings in the corresponding article. The running times of the IG-based algorithm for solving problems of 20, 50, 100, and 200 scales are 2 s, 4 s, 8 s, and 16 s, respectively; the number of iterations for the other algorithms is fixed at 500. The experimental results of several metaheuristics are shown in Table 6, where BR indicates the best RDP value, AR indicates the average RDP value. The BR value can represent the optimization ability of the algorithm, and the AR represents the stability of the algorithm. Table 6 shows that the RL + IG\({}_{\text{h}}\) algorithm achieves the smallest BR and AR values and runs in much less time than the HCS and DABC algorithms (especially for problems above the 50-job scale), RL + IG\({}_{\text{h}}\) showed a 42% improvement in AR compared to NEH + IG, and the BR obtained by RL + IG\({}_{\text{h}}\) outperforms the NEH + IG method on all size benchmarks.