## 6 Case Validation and Discussion
### Effectiveness of GNN-Based Planning Model  
The model proposed in this paper for solving the workshop planning problem is a combination of Graph Attention Networks and Deep Reinforcement Learning, i.e., GAT-DRL. In order to validate the computational effectiveness of the planning model in the actual scheduling environment, a synthetic dataset consisting of standard arithmetic examples and workshop planning samples are employed as the GNN training, validation and testing dataset. The training dataset has a total of four scales corresponding to the number of jobs \(n=\{15,\ 20,\ 30\}\) and machines \(m=\{15,20\}\), where each operation \(o_{ij}\) contains available machine \(|\mathcal{M}|\) and workhour values \(p_{ij}^{k}\) sampled integrally from the set of machines \(U(1,\ m)\) and the set of workhours \(U(1,\ 99)\), respectively. The PPO hyperparameters of the model, in accordance with the optimal scheme of comparable research [10; 11; 12; 13; 14; 40], are configured with the total number of iterations \(I=1000\), the instance batch size (training environment) \(B=4\), the step size of each iteration of policy update \(K=10\), the learning rate \(lr=3\times 10^{-4}\), the discount rate \(\gamma=0.98\), and the clipping parameter \(\epsilon=0.2\). The policy loss coefficients, the entropy coefficients as well as the critics' loss coefficients correlate to 2, 0.01, and 1. Regarding the GAT itself, the number of attention layers \(L=2\), and the operation attention network and the machine attention network are both set up with four attention heads per layer each, while ELU is used as the activation function \(\sigma\). The sizes of the MLPs involved are uniformly 64. The model with its  
Figure 12: Multi-station collaborative distribution schematic. parameter settings has been applied in practical engineering deployed in the server of the OE manufacturing enterprise with the DT system, which has been packaged as a module and constitutes one of the crucial functions in the application service layer. The entire test is done in a test computer with an Intel i9-12900H CPU and NVIDIA RTX 3070 Ti GPU to verify that the model has efficient computational performance and generalizability under the initial conditions, ignoring the influence of the parameters that have been trained in the actual project. The benchmark analyzes the gap in the optimal solution, stability, and speed against GAT-DRL, with simultaneous PDRs [11] and Genetic Algorithms (GA) [45] solving the makespan of scheduling instances as well as the computation time. In particular, the PDRs are evaluated against four rules, namely, First-in-First-out (FIFO), Shortest Processing Time First (SPT), Longest Processing Time First (LPT), and Most Work Remaining (MWKR), which have the best overall performance. The mathematical model of the comparison algorithm is subject to Section 4.1. The training curves of GAT-DRL on \(15\times 15\), \(20\times 15\), \(20\times 20\) and \(30\times 20\) instances are illustrated in Figure 13, which reveals that the model accommodates different sizes of datasets and converges rapidly, verifying it acclimates to varying sizes of scheduling problems and obtains high-quality solutions. In addition, with the support of IAM, the converged mean validated makespan is stable with minor fluctuations. For the trained GAT-DRL model, multiscale performance tests have been conducted in this paper based on the TA dataset and a set of synthetic datasets with equivalent actual computational sizes in the workshop. The tests are executed for a total of eight sizes, corresponding to solving small-scale (\(15\times 15\) & \(20\times 15\)), medium-scale (\(30\times 15\), \(20\times 20\) & \(30\times 20\)), and large-scale (\(50\times 20\), \(100\times 20\) & \(100\times 12\)) scheduling problems. Each algorithm in each scale carries out 100 group tests, records the makespan minimum in each group, and statistics the average computation time along with the error fluctuation gap \(\Delta=\left(c_{max}/c^{*}_{max}-1\right)\times 100\%\).