# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 3 Mathematical Model
### Objective Function
#### 3.1.1 Passenger Travel Costs
\forall m\
eq n \tag{19}\]  
\[TT_{2a}=(\sum\limits_{m\in S_{\text{III}},\text{III}}\sum\limits_{n\in S_{ \text{IV}}}q^{m,n}+\sum\limits_{m\in S_{\text{IV}}}\sum\limits_{n\in S_{\text{III }}}q^{m,n})\cdot t^{m,n}\quad\forall m\
eq n. \tag{20}\]  
\[VT_{2a}=VT_{2a1}+VT_{2a2} \tag{21}\]  
\[VT_{2a1}= \begin{cases}(\sum\limits_{m\in S_{\text{III}}}\sum\limits_{n\in S_{ \text{IV}}}q^{m,n}+\sum\limits_{m\in S_{\text{IV}}}\sum\limits_{n\in S_{\text{III }}}q^{m,n})\cdot t^{vc}\cdot\dfrac{f_{3}}{f_{1}+f_{3}}&f_{1}\geq f_{3}\\ (\sum\limits_{m\in S_{\text{III}}}\sum\limits_{n\in S_{\text{IV}}}q^{m,n}+\sum \limits_{m\in S_{\text{IV}}}\sum\limits_{n\in S_{\text{III}}}q^{m,n})\cdot t ^{vc}\cdot\dfrac{f_{1}}{f_{1}+f_{3}}\cdot t^{vc}&f_{1}<f_{3}\end{cases} \forall m\
eq n \tag{22}\]  
\[VT_{2a2}= \begin{cases}(\sum\limits_{n\in S_{\text{III}},\text{III}}q^{a,n}+ \sum\limits_{m\in S_{\text{IV}}}\sum\limits_{n\in S_{\text{III}}}q^{m,n})\cdot t ^{vc}\cdot\dfrac{f_{2}}{f_{2}+f_{3}}&f_{2}\geq f_{3}\\ (\sum\limits_{n\in S_{\text{III}},\text{III}}q^{a,n}+\sum\limits_{m\in S_{ \text{IV}}}\sum\limits_{n\in S_{\text{III}}}q^{m,n})\cdot t^{vc}\cdot\dfrac{f_ {3}}{f_{2}+f_{3}}&f_{3}<f_{2}\end{cases} \forall m\
eq n \tag{23}\]  
Case 2b: Passengers are in zones I, III, and V (see Figure 4b). The resistance time for type-\(2b\) passengers \(RT_{2b}\), in-vehicle time for type-\(2b\) passengers \(TT_{2b}\), and additional VC time for type-\(2b\) passengers \(VT_{2b}\) can be calculated as follows. For type-\(2b\) passengers, additional VC time is required at the transfer station for train 1 and train 3. Furthermore, some passengers require additional VC time at station \(a\) for train 2 and train 3. Consequently, the calculation is divided into two components, as represented in Equations (27) and (28). \[RT_{2b}=(\sum\limits_{m\in S_{\text{III}},\text{III}}\sum\limits_{n\in S_{ \text{V}}}q^{m,n}+\sum\limits_{m\in S_{\text{IV}}}\sum\limits_{n\in S_{\text{III }}}q^{m,n})\cdot(\dfrac{30}{f_{1}}+\dfrac{30}{f_{2}})+t_{walk}\quad\forall m\
eq n \tag{24}\]  
\[TT_{2b}=(\sum\limits_{m\in S_{\text{III}},\text{III}}\sum\limits_{n\in S_{ \text{V}}}q^{m,n}+\sum\limits_{m\in S_{\text{V}}}\sum\limits_{n\in S_{\text{III }}}q^{m,n})\cdot t^{m,n}\forall m\
eq n \tag{25}\]  
\[VT_{2b}=VT_{2b1}+VT_{2b2} \tag{26}\]  
\[VT_{2b1}=\begin{cases}(\sum\limits_{m\in S_{\text{III}}}\sum\limits_{n\in S_{ \text{V}}}q^{m,n}+\sum\limits_{m\in S_{\text{V}}}\sum\limits_{n\in S_{\text{III }}}q^{m,n})\cdot t^{vc}\cdot\dfrac{f_{3}}{f_{1}+f_{3}}&f_{1}\geq f_{3}\\ (\sum\limits_{m\in S_{\text{III}}}\sum\limits_{n\in S_{\text{V}}}q^{m,n}+\sum \limits_{m\in S_{\text{V}}}\sum\limits_{n\in S_{\text{III}}}q^{m,n}  
\[VT_{2h2}= \begin{cases}(\sum\limits_{m\in\mathbb{S}_{\text{II}}}\sum\limits_{n\in \mathbb{S}_{\text{V}}}q^{m,n}+\sum\limits_{m\in\mathbb{S}_{\text{V}}}\sum\limits _{n\in\mathbb{S}_{\text{II}}}q^{m,n})\cdot t^{\text{rec}}\cdot\frac{f_{2}}{f_{2} +f_{3}}&f_{2}\geq f_{3}\\ (\sum\limits_{m\in\mathbb{S}_{\text{II}}}\sum\limits_{n\in\mathbb{S}_{\text{V}}} q^{m,n}+\sum\limits_{m\in\mathbb{S}_{\text{V}}}\sum\limits_{n\in\mathbb{S}_{\text{II}}}q^{m,n}) \cdot t^{\text{rec}}\cdot\frac{f_{3}}{f_{2}+f_{3}}&f_{3}<f_{2}\end{cases} \forall m\
eq n \tag{28}\]  
Case 2c: Passengers requiring transfers and the OD within zones II and V. The resistance time for the type-\(2c\) of passengers \(RT_{2c}\), in-vehicle time for type-\(2c\) passengers \(TT_{2c}\), and additional VC time for type-\(2c\) passengers \(VT_{2c}\) can be calculated as follows. For type-\(2c\) passengers, additional VC time is required at the transfer station for train 1 and train 3. Furthermore, some passengers require additional VC time at station \(a\) for train 2 and train 3. Consequently, the calculation is divided into two components, as represented in Equations (32) and (33). \[