## 3 Methodology
### Construction of the Spatio-Temporal Network
#### 3.3.2 Construction of Spatio-Temporal Network for Express Cargo
\[B_{p}^{vir}=\big{\{}\big{(}s,t,0;s^{\prime},t^{\prime},0\big{)}\big{|}(s,t,0)= \delta_{p},\big{(}s^{\prime},t^{\prime},0\big{)}=\epsilon_{p}\big{\}} \tag{10}\]  
(7) Express Dedicated Train Arcs \(B_{p}^{train}\)  
Express dedicated train arcs connect the virtual origin \(\delta_{p}\), the virtual destination \(\epsilon_{p}\), and the dedicated train transport nodes \(\theta^{s}\). For any express cargo \(p\in P\), when the express cargo is transported by a dedicated high-speed train, there exists an express dedicated train arc connecting the virtual origin \((s,t,0)=\delta_{p}\), the virtual destination \((s^{\prime},t^{\prime},0)=\epsilon_{p}\), and the dedicated train transport nodes \(\theta^{s}\), as well as connecting the dedicated train transport nodes \(\theta^{s}\) and \(\theta^{s^{\prime}}\), as shown in Equation (11). \[ \begin{array}{ll}B_{p}^{train}=&\big{\{}(s,t,0;s,t^{\prime},ttrain^{\prime}) \big{|}(s,t,0)=\delta_{p};(s,t^{\prime},ttrain^{\prime})=\phi^{\sigma_{p}} \big{\}}\cup\\ &\big{\{}(s,t,ttrain^{\prime};s^{\prime},t^{\prime},0)\big{|}(s,t,train^{\prime })=\phi^{\sigma_{p}};(s^{\prime},t^{\prime},0)=\epsilon_{p}\big{\}}\cup\\ &\big{\{}(s,t^{\prime},train^{\prime};s^{\prime},t^{\prime},train^{\prime}) \big{|}(s,t^{\prime},train^{\prime})=\theta^{s},s\in S_{p}\big{\}}\end{array}  \tag{11}\]  
For \(p\in P\), taking express cargo \(p\) as an example, express cargo \(p\) can be transported by optional trains \(i\in I_{p}\), by dedicated express trains, or by remaining at the origin location. The research objective of this model includes determining the number of departures of high-speed rail express trains without specifying the exact departure times. Therefore, when express cargo \(p\) is transported by a dedicated express train, the nodes passed by are abstract nodes rather than specific departure and arrival nodes at the stations. When express cargo \(p\) is limited by train capacity, loading/unloading capacity, or loading/unloading time, for the convenience of model construction, express cargo \(p\) uses a virtual arc \((s,t,i;s^{\prime},t^{\prime},t^{\prime})\in B_{p}^{vir}\) to represent its transport path. However, in actual operations, express cargo \(p\) does not depart and remains in the warehouse at the origin location. When express cargo \(p\) can be transported by train, assuming that the optional sub-spatio-temporal network for express cargo \(p\) is as shown in Figure 6, the starting point of the transport path for express cargo \(p\) is the virtual origin \(\delta_{p}\), and the endpoint of the transport path is the virtual destination \(\epsilon_{p}\). The intermediate optional nodes are \((s,t,i)\in N_{p}\), and the optional arcs for the express transport path are \((s,t,i;s^{\prime},t^{\prime},i^{\prime})\in B_{p}\). Due to potential constraints on transport capacity or factors such as delivery deadlines and transport costs, the transport path of express cargo \(p\) from the departure station to the destination station may need to consider whether to transfer at certain stations. When express cargo \(p\) is transported directly without any transfer stations, the transport path is as shown in Figure 6. The transport path for express cargo \(p\) starts from the virtual origin \(\delta_{p}\), first undergoes loading operations through the loading and unloading  
Figure 6: Spatio-temporal network diagram (no transfer example). arc, follows the operational path of the selected interval of train ii, undergoes unloading operations at the destination station through the loading and unloading arc, and finally reaches the virtual destination \(\varepsilon_{p}\). Express cargo \(p\) uses only train ii for transportation.