# Optimization of Multi-Day Flexible EMU Routing Plan for High-Speed Rail Networks
## 2 Model Description
### Model Construction  
#### 2.2.1 Construction of the EMU Circulation Connection Network  
To support the formulation of the multi-day EMU routing plan model, we constructed a directed connection network that systematically represents the movement and operational status of EMUs. This network abstracts key operational elements--such as depots, stabling yards, and scheduled train services--into vertices and establishes a set of directed arcs to represent feasible transitions under operational and technical constraints. Maintenance requirements are embedded through dedicated maintenance arcs. Figure 4 illustrates a simplified version of the connection network based on the routing scenario shown in Figure 3. For clarity, only a subset of feasible arcs is displayed.  
Figure 3: The flexible operation scheme.  
Figure 2: The fixed operation scheme.  
**Network vertices:**  
1. _Source/sink vertices:_ These are virtual vertices representing the start and end points of every feasible circulation path.
2. _Depot vertices:_ Each EMU depot is modeled as a vertex with associated parameters, including the maximum number of daily inspections, the storage capacity, and the set of associated stations.
3. _Stabling yard vertices:_ These represent stations equipped with overnight stabling capabilities. Each vertex is defined by its maximum capacity and the corresponding set of stations.
4. _Train service vertices:_ Each scheduled train operation is modeled as a vertex. These are divided into short-formation and long-formation categories. Each vertex includes attributes such as departure/arrival station and time, formation type, distance, and duration.  
**Network Arcs:**  
1. _Virtual entry and exit arcs:_ Connect the source vertex with depot or stabling vertices, indicating the initiation and termination of EMU movement paths.
2. _Depot service arcs:_ Represent direct connections between depots and train service vertices, corresponding to EMUs starting or ending service at a depot.
3. _Deadhead arcs:_ Denote non-revenue-generating movements (empty runs) between train service terminals and depot-connected stations.
4. _Overnight stabling arcs:_ Represent connections from terminal train services to yard vertices when overnight storage is possible and servicing facilities are available.
5. _Connection arcs:_ Link two train service vertices, allowed only when the terminal station of the first matches the departure station of the second and the time gap meets the minimum turnaround requirements. Long-formation services may be split into equivalent short-formation services to allow for flexible coupling or decoupling.
6. _Maintenance arcs:_ Represent train-to-depot connections for scheduled inspection. These arcs are feasible when the train's terminal station is associated with the depot and sufficient time is allowed for inspection.  
Each complete circulation of an EMU corresponds to a path starting and ending at the source vertex, traversing a series of arcs that represent service execution, stabling, and maintenance. This network structure allows the routing problem to be formulated as an arc-based multi-commodity flow model.