## 2 Construction of the Spatiotemporal Network for Subway Vehicle Maintenance and Scheduling  
### Fundamental Elements  
The spatiotemporal network for subway vehicle maintenance and scheduling proposed in this study consists of two hierarchical layers: the maintenance layer and the scheduling layer. Within this network, the maintenance layer encompasses the vehicle maintenance process, including both preventive and routine maintenance activities, while the scheduling layer pertains to the operational scheduling of subway vehicles, such as trip timing and route adjustments. Each layer comprises fundamental elements, including operation start nodes, operation end nodes, and various operational tasks performed by subway vehicles. These elements form the foundational framework of the spatiotemporal network, where a sequence of operations from the start node to the end node defines the paths within the network. The final spatiotemporal network structure is established based on the aforementioned elements, paths, and hierarchical layers. This network provides a clearer understanding of subway vehicle operations and maintenance processes, serving as a crucial reference for optimizing subway operations. By leveraging this framework, subway vehicle operation plans can be optimized, the overall operational efficiency can be enhanced, and maintenance quality can be improved, thereby offering essential decision-making support for subway system management. The proposed spatiotemporal network for subway vehicle maintenance and scheduling is represented as a set \(W=\{D,L\}\), where \(D\) denotes the set of nodes within the network, and \(L\) represents the set of arcs. The fundamental elements of the network are explained as follows: \begin{table}
\begin{tabular}{c c c c} \hline
**Research** & **Main Research** & **Objective** & **Limitations** \\ \hline \multirow{6}{*}{Maintenance and scheduling optimization} & \multirow{2}{*}{[PERSON] et al. [21]} & Established a linear integer programming model based on spatiotemporal networks to optimize costs & Focused mainly on operational costs; did not detail maintenance time arrangements \\ \cline{2-4}  & & Optimized to reduce maintenance equipment occupancy time & Only optimized maintenance equipment; did not combine with train scheduling \\ \cline{2-4}  & & Studied the impact of maintenance on subway vehicle scheduling & Focused on case studies; did not propose a general optimization framework \\ \cline{2-4}  & & Established a daily train scheduling model to optimize maintenance time arrangements & Focused on static scheduling; did not consider dynamic adjustments \\ \cline{2-4}  & & Optimized train scheduling by incorporating maintenance constraints & High solution complexity; did not fully consider spatiotemporal network characteristics \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._1. Spatiotemporal network nodes: In this network, the set of operational nodes is defined as \(D=\left\{D_{y},D_{j},D_{s},D_{e}\right\}\), where \(D_{y}\) denotes vehicle scheduling operations, \(D_{j}\) represents vehicle maintenance operations, \(D_{s}\) corresponds to the operation start node, and \(D_{e}\) indicates the operation end node, marking the conclusion of the planned operational cycle. 2. Spatiotemporal network arcs: Within the spatiotemporal network, subway vehicle maintenance, scheduling, and task transitions are represented by maintenance operation arcs, scheduling operation arcs, and state transition arcs, respectively. The horizontal axis denotes time, capturing the start and end times of each operational process, while the vertical axis represents space, indicating the corresponding operational locations. The set of arcs is given as \(D=\left\{L_{y},L_{j},L_{ex},L_{end}\right\}\), where \(L_{y}\) denotes scheduling operation arcs, \(L_{j}\) represents maintenance operation arcs, \(L_{s}\) corresponds to task transition arcs, \(L_{ex}\) signifies state transition arcs, and \(L_{end}\) marks operation termination arcs.