## 3 Methodology
### Construction of the Spatio-Temporal Network
#### 3.3.2 Construction of Spatio-Temporal Network for Express Cargo  
Let \(G=(N,B,I)\) represent the spatio-temporal network for express cargo, where \(N\) denotes the set of nodes in the spatio-temporal network, \(B\) represents the set of arcs in the spatio-temporal network, and \(I\) indicates the set of states in the spatio-temporal network, including the initial layer, storage layer, dedicated train layer, and train layer. Each express shipment task can be viewed as a sub-spatio-temporal network for express cargo, \(G_{p}=\big{(}N_{p},B_{p},I_{p}\big{)}\), where \(N_{p}\) is the set of nodes in the spatio-temporal network that can be occupied by express cargo \(p\), \(B_{p}\) is the set of arcs in the spatio-temporal network that can be occupied by express cargo \(p\), and \(I_{p}\) is the set of trains in the spatio-temporal network that can be utilized by express cargo \(p\). A visual representation of the express cargo spatio-temporal network is shown in Figure 5.  
1. Construction of Network Nodes  
For each express cargo \(p\in P\), its spatial attribute of the nodes depends on the transit stations \(s\in S_{i}\), the temporal attribute constructed by the time window \(t\in T\), and the state attribute built by the train \(i\in I_{p}\). To ensure the completeness and solvability of the express cargo transport path, virtual origin and destination nodes are introduced. Additionally, considering that express cargo transport relies on train travel, the nodes for express cargo transport include the arrival and departure nodes of the selected trains \(i\in I\). Therefore, the set of nodes in the spatio-temporal network for express cargo \(N_{p}\) includes the virtual origin \(\delta_{p}\), the set of express transport nodes \(\big{\{}\theta_{i}^{s}\cup\rho_{i}^{s},i\in I_{p},s\in S_{i}\big{\}}\), the set of storage nodes \(\big{\{}\epsilon_{i}^{s}\in\theta_{i}^{s}\cup\rho_{i}^{s},i\in I_{p},s\in S_{ i}\big{\}}\), the dedicated train nodes \(\big{\{}\delta^{s},s\in S_{p}\big{\}}\), and the virtual destination \(\varepsilon_{p}\). The set of nodes in the sub-spatio-temporal network for express cargo \(\varepsilon_{p}\) can be expressed as \(N_{p}=\big{\{}\delta_{p},\varepsilon_{p}\big{\}}\cup\big{\{}\epsilon_{i}^{s} \cup\theta_{i}^{s}\cup\theta_{i}^{s},i\in I_{s},s\in S_{i}\big{\}}\). Thus, for any node \((s,t,i)\) and \((s^{\prime},t^{\prime},i^{\prime})\) in the spatio-temporal network, \((s,t,i),(s^{\prime},t^{\prime},i^{\prime})\in N\). 2. Construction of Network Arcs  
For any express cargo \(p\in P\), the sub-spatio-temporal network \(B_{p}\) includes seven types of arcs: express start arcs \(B_{p}^{start}\), express travel arcs \(B_{p}^{travel}\), express loading and unloading  
Figure 5: Network diagram of spatio-temporal state of express delivery goods. arcs \(B_{p}^{lo}\), express transfer arcs \(B_{p}^{trans}\), express end arcs \(B_{p}^{end}\), express dedicated train arcs \(B_{p}^{train}\), and express virtual arcs \(B_{p}^{vir}\). Hence, \(B_{p}=B_{p}^{start}+B_{p}^{travel}+B_{p}^{lo}+B_{p}^{trans}+B_{p}^{end}+B_{p}^{ train}+B_{p}^{vir}\). (1) Express Start Arcs \(B_{p}^{start}\)  
Express start arcs connect the virtual origin and the storage node at the origin station. For any express cargo \(p\in P\), at its origin station \(s^{\prime}=or_{p}\in S\), there exists an express start arc \((s,t,0)=\delta_{p}\) to the storage node at the origin station \(or_{p}\), \((s^{\prime},t^{\prime},\text{store}\prime)\in\epsilon_{i}^{or_{p}}\), as shown in Equation (5). \[B_{p}^{start}=\Big{\{}\big{(}s,t,0;s^{\prime},t^{\prime},\text{store}\prime \big{)}\Big{|}(s,t,0)=\delta_{p},\big{(}s^{\prime},t^{\prime},\text{store} \prime\big{)}\in\epsilon_{i}^{or_{p}},i\in I_{p}\Big{\}} \tag{5}\]  
(2) Express Loading and Unloading Arcs \(B_{p}^{lo}\)  
Express loading and unloading arcs connect the arrival nodes, departure nodes, and storage nodes at the origin, destination, and intermediate transfer stations.