## 3 Mathematical Model
### Mathematical Model
#### 3.3.1 Objective Functions  
For the objective function, this model takes into account the revenue as well as the operating costs of high-speed rail express business, including five aspects: freight revenue, transportation cost, loading and unloading cost, transit cost, and delay penalty. \begin{table}
\begin{tabular}{c c c} \hline
**Type** & **Symbol** & **Definition** \\ \hline \(\varepsilon\) & Time granularity, unit: min \\ \(s^{p}\) & Unit weight of unit transport distance tariff of \(p\) type products, \(p\in P\), \\ \(d^{e}\) & Station distance for OD pair \(e\), unit: km \\ \(c_{tr}^{e,r}\) & The transportation cost per unit distance of the selected path \(r\) for unit weight cargo \\  & between OD pair \(e\), \(e\in E\), \(r\in R_{e}\), unit: yuan (RMB) \\  & Loading and unloading costs per unit weight of cargo, including the process of \\ \(c_{l}\) & loading and unloading cargo at the origin, destination, and transit stations, \\  & unit: yuan (RMB) \\ \(l^{e,r}\) & Required times of loading and unloading for cargoes choosing path \(r\) between OD \\  & pair \(e\), \(e\in E\), \(r\in R_{e}\) \\  & The transit costs incurred by the transit of goods per unit weight in the course of \\  & \(c_{sto}\) & transportation, including the costs of storage and handling incurred by the transit, \\  & unit: yuan/kg \\  & \(sto^{e,r}\) & Required transfer times for cargoes choosing path \(r\) between OD pair \(e\), \(e\in E\), \(r\in R_{e}\) \\ Parameters & \(n^{a}\) & Transport capacity of the transportation arc segment \(a\) that is related to the transport \\  & \(ef\) & Average efficiency of loading and unloading operations at the platform of \\  & high-speed railway stations, unit: kg/min \\ \(f_{e,t}^{p}\) & Freight traffic volume of express product \(p\) with shipping time \(t\) between OD pair \(e\), \\  & \(e\in E\), \(p\in P_{e}\)\(t\in T_{e}^{p}\), unit:kg \\ \(\alpha_{e,t}^{p}\) & The promised arrival time of freight flow \(f_{e,t}^{p}\), that is, the interval from the shipment \\  & time \(t\) to the promised latest arrival time \(t^{\prime}\), \(e\in E\), \(p\in P_{e}\)\(t\in T_{e}^{p}\), unit: min \\  & The actual arrival time of the freight flow \(f_{e,t}^{p}\) choosing path \(r\), that is, the interval \\ \(\beta_{e,t}^{p,r}\) & from the shipment time \(t\) to the actual arrival time \(t^{\prime}\) for the \(f_{e,t}^{p}\) choosing path \(r\), \\  & \(r\in R_{e}\), unit: min \\ \(\gamma_{e,t}^{p,r}\) & Whether the selection path \(r\) of freight flow \(f_{e,t}^{p}\) is delayed, 1 for yes, 0 otherwise \\ \(\tau_{e,t}^{p,r}\) & Delay penalty factor of the selection path \(r\) of freight flow \(f_{e,t^{\prime}}^{p}\), \(p\in P\) \\ \(\theta^{p}\) & Delay penalty cost threshold factor of express product \(p\), that is, ratio of the \\  & maximum penalty charge to the freight rate for that category of goods, \(p\in P\) \\ \(t_{p}^{\pi}\) & The critical delay time acceptable for express products, unit: min \\  & Time length of loading arc and unloading arc \(\alpha\), which equals the dwell time at the \\  & corresponding station, which marks the upper time limit of high-speed rail express \\  & loading and unloading operations, \(a\in A_{ul}\cup A_{l0}\), unit: min \\ \hline Decision & \(x_{e,t}^{p,r}\) & 0–1 variable, if the path \(r\) is selected by freight flow \(f_{e,t}^{p}\), then \(x_{e,t}^{p,r}=1\), \\ Variables & \(x_{e,t}^{p,r}\) & otherwise \(x_{e,t}^{p,r}=0\) \\  & \(y_{e,t}^{p,r,a}\) & Integer variable, the traffic volume on the arc \(a\) of freight flow \(f_{e,t}^{p}\) choosing path \(r\) \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._1 Freight revenue  
The revenue source of the high-speed rail express business for railway operators is mainly the freight revenue from conducting express business.