## 4 Experiment
### Experimental Results and Comparison  
To validate the effectiveness of the proposed method, the experiments primarily evaluate performance in terms of scheduling quality and response time. The scheduling quality is measured by the makespan \(C_{max}\), while the response time refers to the duration required to generate a complete scheduling plan. As presented in Table 6, the bolded values indicate the optimal makespan results and the proposed method achieved the minimum makespan across all five instances, demonstrating an optimization improvement of 5.6% to 7.3% compared to the DQN method and up to a 35.6% enhancement over traditional composite dispatching rules. This superior performance is attributed to the ability of the HEDG to effectively model the dynamic interactions between operations and machines, as well as the PPO algorithm's proficiency in learning strategies for global optimization objectives. In contrast, traditional dispatching rules rely on local heuristic strategies, which makes it challenging to capture and manage complex dependencies inherent in the scheduling process. As shown in Table 7, the proposed method also exhibits favorable response times. Although slightly higher than those of traditional rules, the increase remains modest even as the instance size grows, ensuring that response times scale smoothly with larger problem sizes. Moreover, in practical production environments, these response times are still within acceptable limits. Figure 6 further illustrates that the proposed method achieves an effective balance between scheduling quality and response time, highlighting its practicality and efficiency in real-world manufacturing scenarios. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **S1** & **S2** & **S3** & **S4** & **S5** \\ \hline Proposed Method & **1306** & **1672** & **2125** & **2894** & **3548** \\ DQN & 1384 & 1786 & 2293 & 3117 & 3821 \\ SPT + NINQ & 1553 & 1967 & 2538 & 3476 & 4259 \\ SPT + WINQ & 1602 & 2041 & 2615 & 3568 & 4387 \\ MWKR + NINQ & 1815 & 2284 & 2947 & 4132 & 5083 \\ MWKR + WINQ & 1879 & 2371 & 3062 & 4289 & 5295 \\ EDD + NINQ & 1658 & 2102 & 2719 & 3664 & 4512 \\ EDD + WINQ & 1723 & 2176 & 2825 & 3791 & 4678 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of scheduling results across instances. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Dimension of node embedding & 16 \\ Dimension of the MLP hidden layer & 128 \\ Dimension of policy network and value network & 64 \\ Number of layers in policy network and & 3 \\ value network & 4 \\ Number of iterations & 2000 \\ Batch size & 20 \\ Discount factor & 0.99 \\ Optimizer & Adam \\ Learning rate & \(3\times 10^{-4}\) \\ Clipping ratio & 0.2 \\ Policy loss coefficient & 1 \\ Value loss coefficient & 0.5 \\ Exploration strategy & linear decay from 0.5 to 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model parameter settings. To further validate the effectiveness of the proposed scheduling method, a detailed performance comparison was conducted using the well-established Brandimarte benchmark dataset [45]. This dataset is widely recognized for evaluating scheduling algorithms and includes 10 problem instances of varying sizes within the FJSP. The dataset serves as a robust test case, reflecting both smaller and larger scale scheduling environments. For this analysis, our method, based on the HEGNN-PPO, was compared with several state-of-the-art methods. The comparison methods include two categories:  
DRL-Based Methods:  
* AC-SD Method [46]: An end-to-end DRL framework based on an improved pointer network and attention mechanisms, considering both static and dynamic features. * DRL-AC Method [47]: A reinforcement learning approach based on an actor-critic architecture, utilizing parameterized priority rules and Beta distribution action sampling. Rule-Based Heuristic Methods:
* Combinations of Shortest Processing Time (SPT + NINQ), Most Work Remaining (MWKR + NINQ), and Earliest Due Date (EDD + NINQ) rules.