## 6 Conclusions  
This study addresses the challenges in joint optimization modeling and solving caused by the interdependence between subway vehicle scheduling and maintenance planning. A joint optimization method for preventive maintenance and train scheduling based on spatiotemporal network graphs is proposed. First, a multi-level spatiotemporal state network graph is constructed for subway vehicle maintenance and scheduling using spatiotemporal network theory, where paths represent the temporal and spatial transitions of various operations throughout the planning period. Second, a joint optimization model is established, comprehensively considering the constraints of preventive maintenance and scheduling requirements for subway vehicles to achieve coordinated optimization of the maintenance and scheduling plans. Finally, an improved genetic algorithm is employed to solve the model and obtain the optimal scheduling and maintenance plan.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Statistical** & **Best Value** & **Worst Value** & **Average Value** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline Maintenance & 21.5 & 25 & 22.95 & 1.21 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparison of the results of the number of maintenance days in ten operations.  
Figure 13: Comparison diagram of maintenance times before and after optimization.  
The experimental results show that the proposed method can effectively address the difficulties in modeling and solving the joint optimization of subway vehicle maintenance and scheduling, enabling efficient coordinated optimization of both. Specifically, compared to traditional methods, this approach improves vehicle utilization by 7.97% and reduces the number of maintenance shutdowns by 14, simultaneously lowering maintenance costs and improving the scheduling efficiency, fully meeting the actual operational needs.  
However, there are still some limitations in this study. First, the subway operation environment is complex and contains uncertain factors, such as unexpected failures and temporary scheduling adjustments, which are not sufficiently considered in the current model. Second, the optimization algorithm's solving efficiency could be improved, especially in large-scale spatiotemporal networks with numerous constraints. Future research will focus on improvements in the following areas: (1) developing efficient optimization algorithms capable of handling large-scale spatiotemporal networks and complex constraints to enhance computational efficiency; (2) combining deep reinforcement learning techniques to achieve intelligent subway vehicle status assessment and scheduling decisions, improving the model's adaptability and autonomous optimization ability; and (3) exploring methods that combine graph neural networks with reinforcement learning to improve the solving speed in large-scale networks and enhance the model's adaptability to complex scheduling environments.  
Conceptualization, C.L.; methodology, Z.X.; software, M.M.; validation, W.L.; writing--original draft preparation, C.L.; writing--review and editing, Z.X. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
Author [PERSON] was employed by the company Shanghai Hengjun Technology Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.