## 2 Objective Function
#### 3.4.2 Express Cargo Transportation Section
\begin{array}{c}C_{delay}=\hskip 14.226378 pteday\times\sum_{p\in P}f_{p} \times Delay^{p}\\ \hskip 14.226378 pt\times[\sum_{(s,t,i;s^{\prime},t^{\prime},i^{\prime})\in B_{ p}/\big{(}B^{vir}_{p}+B^{train}_{p})}y_{p}(s,t,i;s^{\prime},t^{\prime},i^{ \prime})\times T(s,t,i;s^{\prime},t^{\prime},i^{\prime})\\ \hskip 14.226378 pt-\Big{(}T^{M_{p}}-T^{start}_{p}\Big{)}\Big{]}\times\Big{(}e^{ fixin}_{M_{p}}\times f_{p}+e^{flein}_{M_{p}}\times f_{p}\times q\Big{)}\end{array}  \tag{27}\]  
5 Express Cargo Retention Cost  
\[C_{vir}=e^{pir}\times\sum\limits_{p\in P}\sum\limits_{(s,t,i;s^{\prime},t^{ \prime},i^{\prime})\in B^{vir}_{p}}y_{p}\big{(}s,t,i;s^{\prime},t^{\prime},i^{ \prime}\big{)}\times\Big{(}e^{fixin}_{M_{p}}\times f_{p}+e^{flein}_{M_{p}} \times f_{p}\times q\Big{)} \tag{28}\]  
3. Constraints  
(1) Express Cargo Flow Balance Constraint  
Equation (29) represents the fundamental flow balance constraint in network optimization. Given the relatively small demand for high-speed rail express services and the current limited infrastructure for express cargo at high-speed rail stations, the model ensures that the express cargo flow is non-splittable. This approach aims to reduce the risk of lost or missing items during transportation and enhance the reliability of cargo flow, as reflected in Equation (29). \[\sum_{(s,t,i):(s,t,i;s^{\prime},t^{\prime},i^{\prime})\in B}y_{p}\big{(}s,t, i;s^{\prime},t^{\prime},i^{\prime}\big{)}-\sum_{(s,t,i):(s^{\prime},t^{\prime},i^{ \prime}\big{)}\in B}y_{p}\big{(}s^{\prime},t^{\prime},i^{\prime};s,t,i\big{)}= \begin{cases}-1&(s^{\prime},t^{\prime},i^{\prime})=\delta_{p}\\ 1&(s^{\prime},t^{\prime},i^{\prime})=\varepsilon_{p}\,\forall p\in P\\ 0&\text{otherwise}\end{cases}  \tag{29}\](2) Express Loading and Unloading Time Constraint  
Considering that express shipments must undergo loading and unloading operations at the departure, destination, and transfer stations, the time required for these operations is proportional to the volume of express cargo. Additionally, the loading and unloading time is constrained by the dwell time of the transporting and connecting trains. Equation (30) indicates that the required loading and unloading time for express cargo \(p\) is related to the number of express items \(f_{p}\). The feasibility of performing loading and unloading operations at a station is constrained by the dwell time \((s,t,i;s^{\prime},t^{\prime},i^{\prime})\) of the carrying train i at the stopping arc \(T(s,t,i;s^{\prime},t^{\prime},i^{\prime})\). \[T\big{(}s,t,i;s^{\prime},t^{\prime},i^{\prime}\big{)}-\sum_{p:(s,t,i;s^{\prime },t^{\prime},i)\in B_{p}^{p}}y^{p}\big{(}s,t,i;s^{\prime},t^{\prime},i^{\prime} \big{)}\times f_{p}\times q\times h\geq 0\ \forall i\in I,\forall\big{(}s,t,i;s^{\prime},t^{\prime},i^{\prime}\big{)} \in A_{i}^{dwell} \tag{30}\]  
(3) The Constraint on the Number of Trans-Shipments for Express Cargo  
High-speed rail express transport mainly targets time-sensitive and high-value goods. The number of trans-shipments and the associated loading and unloading operations directly impact the transport efficiency and quality of the express cargo. Therefore, Equation (31) indicates that this model limits the number of trans-shipment operations during the high-speed rail express transport to one, ensuring the quality of service for express shipments. \[\sum_{(s,t,i;s^{\prime},t^{\prime},i^{\prime})\in B_{p}^{trans}}y_{p}\big{(}s,t,i;s^{\prime},t^{\prime},i^{\prime}\big{)}\leq 1\ \ \ \forall p\in P,\forall s\in S/\big{\{}or_{p},de_{p}\big{\}} \tag{31}\]  
(4) Delay Determination  
The timeliness of high-speed rail express delivery is a primary concern. Thus, the delay rate of express shipments is a crucial factor for companies when evaluating transport route options. Equation (32) determines whether express shipment \(p\) is delayed. If the total transport time for shipment \(p\) is less than its corresponding delivery time limit, the delay indicator variable \(Delay^{p}\) is 0; otherwise, it is 1. \[Delay^{p}=