# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 3 Mathematical Model
### Objective Function
#### 3.1.1 Passenger Travel Costs  
To accurately calculate passenger travel costs, we divide passengers in Table 3. Class 1 represents passengers who can reach their destination directly without transfer, while Class 2 represents passengers who need to transfer to reach their destination. Additionally, passengers are further subdivided based on their origin-destination (OD) pairs, accounting for varying train frequencies in different zones. For instance, in Class 1, passengers traveling online 1 with their OD not entirely within zone II are categorized as Case 1a. Consequently, we further classify Class 1 into five cases, denoted as \(P_{1}=\{1a,1b,1c,1d,1e\}\), as illustrated in Figure 3. Similarly, Class 2 is subdivided into three cases, denoted as \(P_{2}=\{2a,2b,2c\}\), as illustrated in Figure 4.  
Generally, passenger travel costs mainly include the resistance time for type-\(p\) passengers \(RT_{p}\), in-vehicle time for type-\(p\) passengers \(TT_{p}\), and additional VC time for type-\(p\) passengers \(VT_{p}\). Among them, \(RT_{p}\) includes the waiting time and walking transfer time; the waiting time is set as half of the train departure interval as the passenger arrival is uniformly distributed, i.e., \(30/f^{m,n}\). Notably, when trains from different lines enter collinear zones, VC is performed, wherein two trains operate in the form of one train. Consequently, for passengers who can travel on both local and cross-line trains, the passenger waiting time is calculated using the greater of the two trains' departure frequencies. \(TT_{p}\) is mainly influenced by the origin and destination stations of passengers. In addition, in CO, the uncertainty of the train's arrival time may lead to additional VC time for the trains of different lines during the VC process. Thus, \(VT_{p}\) represents additional VC time for type-\(p\) passengers. The critical point will be the first station where the VC occurs (including cross-line turn-back station 2, \(a,1,b\); transfer station \(1p\), which is represented by \(a,\)\(b,\)\(tra\)). Additional VC time applies to passengers who either originate from or pass critical points. Additional VC time is proportional to the number of VC and is determined by lower train departure frequency between the local line and the cross-line services. The passenger travel cost is calculated using Equation (1). \[W=\gamma\cdot\sum_{p\in P}\left(RT_{p}+TT_{p}+VT_{p}\right)\text{, }\forall p\in P \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Passenger Class**} & **OD Distribution** & **Frequencies** \\ \hline \multirow{4}{*}{Class 1} & Case 1a & OD is within _line_ 1, without OD in zone II & \(f_{1}\) \\  & Case 1a & OD in zone II & \(f_{1}\),\(f_{3}\) \\  & Case 1a & OD is zone II and IV & \(f_{3}\) \\  & Case 1a & OD is within _line_ 2, without OD in zone IV & \(f_{2}\) \\  & Case 1a & OD in zone IV & \(f_{2}\),\(f_{3}\) \\ \hline \multirow{4}{*}{Class 2} & Case 2a & OD in zone I, III, and IV & \(f_{1}\),\(f_{2}\),\(f_{3}\) \\  & Case 2b & OD in zone I, III, and V & \(f_{1}\),\(f_{2}\) \\ \cline{1-1}  & Case 2c & OD in zone II and V & \(f_{1}\),\(f_{2}\),\(f_{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The table of passenger classification. The detailed calculation process for each type of passenger is as follows. 1. The travel costs of the direct passengers  
Case 1a: The passengers in line 1, excluding those whose ODs are both within zone II (see Figure 3a). The resistance time for type-1\(a\) passengers \(RT_{1a}\), in-vehicle time for type-1\(a\) passengers \(TT_{1a}\), and additional VC time for type-1\(a\) passengers \(VT_{1a}\) can be calculated as follows. In addition, \(q^{m,n}\) represents the number of passengers from \(m\) to \(n\), and \(t^{m,n}\) represents the running time of the train from \(m\) to \(n\).