## 3 Formulation
### Decision Variables  
There are two types of decision variables in the model--binary variable and integer variable. The first binary variable \(x_{k}\) is used to determine whether additional train \(k\) is added to the operational plan in the peak period. We use the binary variable \(\mu_{kf}^{i}\) to determine the departure order of trains, which indicates the overtaking of trains at stations. Binary variable \(y_{ki}\) indicates the stop plan for the train. Another two binary variables, \(z_{ki}^{I}\) and \(z_{ki^{\prime}}^{T}\) are used to determine the operational zone for train \(k\). \(D_{ki}^{a}\) and \(D_{ki}^{a}\) are two integer variables to determine the timetables for additional trains. \(q_{ij}^{k}\) is the integer variable to indicate the distribution of passenger demand of each OD pair along the railway corridor.  
It is worth mentioning that the value of decision variable \(x_{k}\) determines the number of additional trains. \(N_{K_{2}}^{e}\) denotes the number of expected additional trains, and we set the value of \(N_{K_{2}}^{e}\) according to Equation (1). \(\sum\limits_{i^{\prime}\leq i,j>i}Q_{i^{\prime}j}\ /\ C_{k}\) represents the required number of trains to meet the passenger demand along segment \((i,i+1)\). The maximum required number among all segments is the expected additional trains. However, not all of the additional trains can be added into the operational plan because it is subject to limited transport resources in peak periods. The value of decision variable \(x_{k}\) is equal to 1, which means train \(k\) is added to the operational plan. Otherwise, train \(k\) is canceled. Thus, the number of additional trains \(N_{K_{2}}\) in the operational plan is equal to the value of \(\sum\limits_{k\in K_{2}}x_{k}\), as in Equation (2).  
\[N_{K_{2}}^{e}=max_{i\in S}\{\sum\limits_{i^{\prime}\leq i,j>i}Q_{i^{\prime}j}\ /\ C_{k}-N_{K_{1}}\} \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Definitions** \\ \hline \(x_{k}\) & selection indicator for adding additional train \(k\in K_{2}\):= 1 if train \(k\) is added in peak period, \(=0\) otherwise; \\  & selection indicator for the departure order of train \(k\in K_{2}\) and \\ \(\mu_{kf}^{i}\) & train \(f\in K_{1}\ \cup K_{2}\) at station \(i\):=1 if train \(k\) depart from station \\  & earlier than train \(f\) before, =0 otherwise. \\ \(y_{ii}\) & selection indicator for stop plan for additional train \(k\in K_{2}\) for \\  & station \(i\in S\):=1 if train \(k\) stop at station \(i\), = 0 otherwise. \\  & selection indicator for origin station for additional train \(k\in K_{2}\) \\ \(z_{ki}^{I}\) & for station \(i\in S\) /\(||S||\):= 1 if train \(k\) select train \(i\in S\) /\(|S|\) as its \\  & origin station, = 0 otherwise. \\  & selection indicator for terminal station additional train \(k\in K_{2}\) \\ \(z_{ki}^{T}\) & for station \(i\in i\in S\) /\(|1\):= 1 if train \(k\) select train \(i\) as its \\  & terminal station, = 0 otherwise; \\ \(D_{ki}^{a}\) & departure time of train \(k\) at station \(i\), where \(k\in K_{2}\) and \\ \(A_{ki}^{a}\) & arrival time of train \(k\) at station \(i\), where \(k\in K_{2}\) and \\  & \(i\in S_{l}\cup\ S\) /\(1\) \\ \(q_{ij}^{k}\) & the number of passengers take additional train \(k\) from station \(i\) \\  & to station \(j\), \(k\in K_{2}\), \(i,j\in S\). \\ \(\rho_{ij}^{\prime\ast}\) & the number of passengers take existing train \(k\) from station \(i\) to \\  & station \(j\) in peak period, \(k\in K_{1}\), \(i,j\in S\). \\ \hline \hline \end{tabular}
\end{table}
Table 4: Decision variables.  
\[N_{K_{2}}=\sum_{k\in K_{2}}x_{k} \tag{2}\]