## 3 Methodology
### Construction of the Spatio-Temporal Network
#### 3.3.2 Construction of Spatio-Temporal Network for Express Cargo
For any express cargo \(p\in P\), at its transit station \(s\in S_{p}\), there exists an express loading and unloading arc from the storage node at the logistics center or warehouse at station \((s,t,i)\in\epsilon_{i}^{s}\) to the departure node \((s,t^{\prime},i)\in\rho_{i}^{s}\), as well as from the arrival node \((s,t,i)\in\theta_{i}^{s}\) to the storage node \((s,t^{\prime},i)\in\epsilon_{i}^{s}\); express loading and unloading arcs can be expressed as \((s,t,i)\rightarrow(s,t^{\prime},i)\), as shown in Equation (6). \[B_{p}^{lo}= \left\{ \begin{aligned} &(s,t,\text{store}\prime;s,t^{ \prime},i)\Big{|}(s,t,i;s,t^{\prime},i)\in A_{i}^{dwell},i\in I_{p},s\in S_{p} \Big{\}}\cup\\ &\big{\{}(s,t,i;s,t^{\prime},\text{store}\prime)\Big{|}(s,t,i;s,t ^{\prime},i)\in A_{i}^{dwell},i\in I_{p},s\in S_{p}\Big{\}}\end{aligned} \right. \tag{6}\]  
(3) Express Travel Arcs \(B_{p}^{travel}\)  
Express travel arcs connect the express transport nodes. For any express cargo \(p\in P\), there exist corresponding train travel arcs \((s,t,i;s^{\prime},t^{\prime},i)\in A_{i}^{travel}\) and train dwell arcs \((s,t,i;s^{\prime},t^{\prime},i)\in A_{i}^{dwell}\) that can be occupied by the selected train \(i\in I_{p}\), as shown in Equation (7). \[B_{p}^{travel}=\Big{\{}\big{(}s,t,i;s^{\prime},t^{\prime},i)\Big{|}\big{(}s,t, i;s^{\prime},t^{\prime},i)\in A_{i}^{travel}\cup A_{i}^{dwell},i\in I_{p}\Big{\}} \tag{7}\]  
(4) Express Transfer Arcs \(B_{p}^{trans}\)  
Express transfer arcs connect the storage nodes. For any express cargo \(p\in P\), at its passing station \(s\in S_{p}/\big{\{}de_{p},or_{p}\big{\}}\), for any train \(i,i^{\prime}\in I_{p}^{st}\) that can be occupied at station \(s\), there exists an express transfer arc from the departure node at station \(s\), \((s,t,\text{store}\prime)\in\epsilon_{i}^{s}\), corresponding to the storage node \((s,t,\text{/store}\prime)\in\epsilon_{i}^{s}\) and the arrival node at station \(s\); express transfer arcs can be expressed as \((s,t,\text{store}\prime)\rightarrow(s^{\prime},t^{\prime},\text{store}\prime)\), corresponding to the storage node \((s,t^{\prime},\text{store}\prime)\in\epsilon_{i^{\prime}}^{s}\) as shown in Equation (8). \[B_{p}^{trans}=\left\{\big{(}s,t,\text{store}\prime;s,t^{\prime},\text{store} \prime\big{)}\Big{|}\big{(}s,t,i)\in\rho_{i}^{s},(s,t^{\prime},i^{\prime})\in \theta_{i^{\prime}}^{s},\right. \tag{8}\]  
(5) Express End Arc \(B_{p}^{end}\)  
Express end arcs connect the departure nodes at the destination station and the virtual destination. For any express cargo \(p\in P\), at its destination station \(s=de_{p}\in S_{p}\), there exists an express end arc from the departure node at the destination station \(de_{p}\), \((s,t,i)\in\rho_{i}^{d_{i}}\) corresponding to the storage node \((s,t,\text{/store}\prime)\in\epsilon_{i^{\prime}}^{d_{i}}\) and the virtual destination \((s^{\prime},t^{\prime},0)=\varepsilon_{p}\); express end arcs can be expressed as \((s,t,i^{\prime}\text{store}\prime)\rightarrow(s^{\prime},t^{\prime},0)\), as shown in Equation (9). \[B_{p}^{trans}=\left\{\big{(}s,t,\text{/store}\prime;s,t^{\prime},\text{store} \prime\big{)}\Big{|}\big{(}s,t,i)\in\rho_{i}^{s},(s,t^{\prime},i^{\prime})\in \theta_{i^{\prime}}^{s},\right. \tag{9}\](6) Express Virtual Arcs \(B_{p}^{vir}\)  
Express virtual arcs connect the virtual origin and the virtual destination. For any express cargo \(p\in P\), when the express cargo is not transported due to network capacity constraints, there exists an express virtual arc from the virtual origin \((s,t,0)=\delta_{p}\) to the virtual destination \((s^{\prime},t^{\prime},0)=\epsilon_{p}\); express virtual arcs can be expressed as \((s,t,0)\rightarrow(s^{\prime},t^{\prime},0)\), as shown in Equation (10).