## 3 Formulation
### Formulations of Constraints  
#### Constraints of Operational Zone  
Not all added additional trains need to travel across the whole rail corridor. Designing the operational zone requires selecting the origin station and terminal station for additional trains. Formulating some constraints to design the appropriate operational zones for the added trains makes sense. First, it is known that only one origin station and terminal station exists during an additional train's one trip. Constraint (3) is formulated to restrict that, as follows, i.e.,  
\[\sum_{i\in S}z_{  
In this model, we do not consider the dwelling time at the origin station and terminal station, and we assume that the departure time is equal to the arrival time at trains' origin station and terminal station, as in Constraints (10) and (11).  
\[(D_{ki}^{a}-A_{ki}^{a})z_{ki}^{I}=0\ k\in K_{2},\ i\in S. \tag{10}\]  
\[(D_{ki}^{a}-A_{ki}^{a})z_{ki}^{T}=0\ k\in K_{2},\ i\in S. \tag{11}\]  
It is assumed that the traveling time of each segment is fixed. Thus, the arrival time of train \(k\) at station \(i+1\) is the sum of the departure time of train \(k\) and the traveling time of the segment \((i,i+1)\) (Constraint (12)). Note that station \(i+1\) must be within the operational zone. Thus, we introduce two terms \(\sum_{i\in S,P^{\prime}\leq i}z_{ki^{\prime}}^{I}\) and \(\sum_{i\in S,P^{\prime}>i}z_{ki^{\prime}}^{T}\) to make sure that Constraint (12) makes sense, with only station \(i+1\) within the operational zone. \(\sum_{i\in S,P^{\prime}\leq i}z_{ki^{\prime}}^{I}\) and \(\sum_{i\in S,P^{\prime}>i}z_{ki^{\prime}}^{T}\) both being equal to 1 means that the origin station of train \(k\) is station \(i\) or before station \(i\), and the terminal station of train \(k\) is after station \(i\). \[\left(A_{k,i+1}^{a}-D_{k,i}^{a}\right)\cdot\sum_{i^{\prime}\in S,P^{\prime}\leq i }z_{ki^{\prime}}^{I}\cdot\sum_{i^{\prime}\in S,P^{\prime}>i}z_{ki^{\prime}}^{T }=t_{i,i+1}\ k\in K_{2},i\in S/\{|S|\}. \tag{12}\]  
If additional train \(k\) stops at station \(i\) within its operational zone, the dwelling time must be greater than the minimum dwelling time \(d_{min}\) to leave enough time for passengers to board and alight. Therefore, station \(i\) must be within the operational zone, not including the origin station or terminal station, because we do not consider the dwelling time at the origin station and terminal station. On the other hand, the departure time and arrival time of train \(k\) are the same at station \(i\), at which train \(k\) does not stop. We formulate the above situation via Constraints (13) and (14). \[D_{k,i}^{a}-A_{k,i}^{a}\geq d_{min}\cdot y_{ki}\cdot\sum_{i^{\prime}\in S,P^{ \prime}<i}z_{ki^{\prime}}^{I}\cdot\sum_{i^{\prime}\in S,P^{\prime}>i}z_{ki^{ \prime}}^{T}\ k\in K_{2},i\in S/\{1,|S|\}. \tag{13}\]  
\[\left(D_{k,i}^{a}-A_{k,i}^{a}\right)\ast(1-y_{ki})=0\ k\in K_{2},i\in S. \tag{14}\]  
Headway is the time interval of departure time or arrival time of trains at the same station. Trains must keep a certain headway from each other to ensure safety. As the traveling time of each segment is fixed, we just need to formulate the headway constraints of departure time to limit both the departure time and arrival time. As we allow for overtaking at intermediate stations in the model, the departure order of trains is different at different stations. Thus, the departure headway between two trains must be related to their departure order. The departure order variable \(\mu_{kf}^{i}\) is introduced to the model, and \(\mu_{kf}^{i}\) equal to 1 means that train \(k\) is before train \(f\). In the study, we cannot only consider the headway between additional trains; the headway between additional trains and existing trains cannot be ignored. The departure headway constraints are classified into two sets. The first set of constraints is of headway between an existing train and the additional train, and the other set is of headway between two additional trains. Constraints (15) and (16) formulate the constraints of headway between additional trains.