## 3 Mathematical Model
### Problem Description and Assumption  
The time nodes and space nodes of the space-time-state network correspond to the space-time network; the difference is that the space-time network is mainly used to describe the operation process of the train and thus used to generate the alternative set of express transportation solutions, as shown in Figure 1; the space-time-state network mainly portrays the cargo in the station, inter-station state, and the corresponding temporal and spatial nodes. In Figure 1, we use a small example to show the space-time network where the time range is 5-24 h, the space range is station A-D, and there are five alternative paths included. Note that Figure 1 is in fact the visualization result of Table 1.  
The process of high-speed rail express in the whole shipping process includes the following cases: (1) goods are stored at the station; (2) goods are loaded at the station; (3) goods are unloaded at the station; (4) goods are transported between stations; and (5) goods follow the train to stop at the station. Therefore, it is necessary to generate cargo service arcs for the above-mentioned cases in the space-time-state network. \begin{table}
\begin{tabular}{c c c c} \hline
**OD** & **Path Number** & **Transportation Path Scheme** & **Collection of Arc Segments** & **Transportation Time/h** \\ \hline \multirow{4}{*}{A–D} & 1 & A \(\underline{k}\)\(\underline{l}\) D & [a1, a2, a3] & 7 \\  & 2 & A \(\underline{k}\)\(\underline{l}\) C \(\underline{k}\)\(\underline{3}\) D & [a1, a2, a16, a12, a13] & 14 \\  & 3 & A \(\underline{k}\)\(\underline{2}\) C \(\underline{k}\)\(\underline{3}\) D & \{a4, a5, a6, a8, a17, a12, a13] & 11 \\  & 4 & A \(\underline{k}\)\(\underline{2}\) B \(\underline{k}\)\(\underline{3}\) D & [a4, a5, a7, a10, a11, a12, a13] & 11 \\  & 5 & A \(\underline{k}\)\(\underline{3}\) D & [a9, a10, a11, a12, a13] & 8 \\ \hline \end{tabular}
\end{table} 
Table 1: Alternative paths within the A–D section. Figure 1: Space-time network diagram representing the alternative set of express transportation solutions. The following is the schematic diagram of the space-time-state network designed in this paper for describing the whole shipping process of the cargo. The cargo state layer is added to the two-dimensional space-time network to form a three-dimensional space-time-state network, where the time range is 5-24 h, the space range is station A-D, and the cargo state includes the departure layer, the storage layer, and the arrival layer. Figures 2 and 3 correspond to the two scenarios of the no transit storage process and the transit storage process, respectively. Figure 3: Time-space-state network diagram of goods under the A–D interval alternative path (with transit). Figure 2: Space-time-state network diagram of goods based on alternative paths between sections A–D (no transit). Among them, the logical relationship between the state layer and the cargo service arc is the following: (1) If the arc segment stays at the storage layer, it is a storage arc; (2) if the arc segment migrates from the storage layer to the departure layer, it is a loading arc; (3) if the arc segment migrates from the departure layer to the arrival layer, it is a transportation arc; (4) if the arc segment migrates from the arrival layer to the storage layer, it is an unloading arc; and (5) if the arc segment migrates from the arrival layer to the departure layer, it is a stopping arc. As shown in Figure 2, {b1, b2, b3, b4, b5, b6} is a transit-free path in the space-time-state network, which corresponds to path one of the alternative transportation schemes, as shown in Table 1, indicating that the cargo chooses this alternative for transportation.