## 2 Construction of the Spatiotemporal Network for Subway Vehicle Maintenance and Scheduling
### Establishment of the Subway Vehicle Maintenance and Scheduling Spatiotemporal Network  
To visually represent subway vehicle scheduling and maintenance operations, a multi-level spatiotemporal network diagram is constructed, incorporating three stations: Stations A-C. Station B is directly connected to the subway depot, indicating that trains undergo scheduling and maintenance tasks at this location, while Stations A and C serve as turnaround points. With the fundamental elements and operational paths of the spatiotemporal network established, a multi-level subway vehicle maintenance and scheduling spatiotemporal state network diagram, as illustrated in Figure 5, is developed to describe train scheduling and maintenance processes. This diagram consists of two layers: the scheduling layer and the maintenance layer. By leveraging the spatiotemporal state network diagram, subway vehicle scheduling and maintenance can be optimized. Specifically, station locations and quantities can be adjusted, maintenance sites can be determined, and appropriate scheduling cycles can be formulated to generate a spatiotemporal network diagram that aligns with real-world operational needs.  
In the constructed spatiotemporal state network for subway vehicle maintenance and scheduling, \"minutes\" are used as the smallest time unit for scheduling. The scheduling cycle begins at 00:00 on the first day and spans 14 days, with the final time point set at 14 \(\times\) 1440 min. In this diagram, train state transition moments are represented as nodes, while various operational states are depicted as arcs. These include scheduling operation arcs, which indicate scheduling states, and maintenance operation arcs, which represent maintenance states. By integrating these elements, a comprehensive and structured spatiotemporal network for preventive maintenance and train scheduling is formed. This network encapsulates essential operational stages and time intervals, serving as a theoretical foundation for analyzing and optimizing subway vehicle maintenance and scheduling. Moreover, it provides a basis for developing an integrated optimization model for preventive maintenance and scheduling, facilitating more effective management and planning of subway vehicle operations and maintenance.  
Through an in-depth analysis of the spatiotemporal network, the integrated optimization of preventive maintenance and train scheduling can be divided into two key steps. First, based on the transition process between train scheduling and preventive maintenance states, a spatiotemporal network diagram for subway vehicle maintenance and scheduling is constructed. This diagram represents the complete execution process of all operational stages from the start to the end of the planning period using paths within the spatiotemporal network. Second, an integrated optimization model for preventive maintenance and train scheduling is formulated. This model is built upon the set of feasible spatiotemporal network paths generated in the first step and is developed based on optimization objectives and constraints. The optimal scheduling plan is then obtained using path search algorithms.