## 5 Computational Experiment
### Computational Results of Benchmark Instances  
In this section, we compare the performance of LPT, NEH, GA, TS, DDQN, and PPO without disjunctive graph representation and our approach to the Taillard benchmark dataset. Table 8 shows the _makespan_ achieved by each algorithm for each problem scale. To highlight the differences in results, we have bolded the best results in the table. On the whole, reinforcement learning is better than metaheuristic and heuristic algorithms. But also, as a reinforcement learning algorithm, our method consistently achieves the smallest _makespan_ across all problem instances, surpassing the performance of DDQN and PPO methods. Particularly, when the number of workstations is fixed at five and the number of blocks is 20, the disparities between different algorithms are minimal. However, as the number of blocks increases, our algorithm proves more efficient in time-saving. For instance, compared to the PPO algorithm, our algorithm can save 0 h, 13 h, and 33.5 h, respectively, when the number of blocks is 20, 50, and 100. With the increasing number of blocks, the superiority of our algorithm becomes increasingly prominent.  
Table 9 presents the computational time of our approach and all the baseline methods. From Table 9, it can be observed that LPT remains instantaneous in solving instances of any scale. Although our method is slower than LPT in terms of computation speed, it outperforms other baseline methods, which is acceptable in practical industrial production. Furthermore, when the number of workstations is the same, our model demonstrates the ability to save more computational time compared to the other five methods, excluding the LPT approach, as the number of blocks increases. For instance, when compared to the PPO  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Blocks**} & \multicolumn{2}{c}{**Heuristic**} & \multicolumn{2}{c}{**Metaheuristic**} & \multicolumn{2}{c}{**Reinforcement Learning**} & \multirow{2}{*}{**Ours**} \\ \cline{2-2} \cline{4-7}  & **LPT** & **NEH** & **GA** & **TS** & **DDQN** & **PPO** \\ \hline
25 & \(-1.21\) & 0.91 & 3.15 & 6.30 & 0.24 & 0.32 & 0 \\
50 & \(-1.54\) & 1.33 & 4.78 & 8.58 & 0.24 & 0.31 & 0 \\
75 & \(-2.32\) & 2.03 & 5.62 & 10.78 & 0.45 & 0.70 & 0 \\
100 & \(-3.83\) & 3.60 & 6.39 & 11.79 & 0.51 & 0.84 & 0 \\
125 & \(-4.99\) & 5.53 & 7.54 & 13.34 & 0.69 & 1.05 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparative analysis of the differences in computation time among the algorithms (s).  
Figure 7: Line graph depicting the differences in computation time among the algorithms.  
algorithm, which does not use disjunctive graph representation for states, our algorithm achieves respective reductions in computational time of 0.07 s, 0.23 s, and 0.54 s when the number of blocks is 20, 50, and 100. Moreover, the rate of time savings surpasses the rate of block increase significantly. This observation further underscores our model's capacity to economize computational time as the instance size expands.  
Based on the results presented in Tables 8 and 9, our method consistently exhibits the shortest computational time and achieves the smallest makespan across all test cases. Consequently, when considering the comprehensive performance of the model, it can be confidently concluded that our approach continues to surpass the baseline methods in the Tarillard benchmark test.