## 3 Mathematical Model
### Model Formulation
#### 3.2.1 Objective Function
\begin{table}
\begin{tabular}{c l} \hline \hline
**Sets** & **Definition** \\ \hline \(S\) & The set of stations in the corridor, \(s,m,n\in S=\{1,2,\ldots,|S|\}\). \\ \(T\) & The set of time periods, \(t\in T=\{1,2,\ldots,|T|\}\). \\ \(K\) & The set of trains, \(i,j\in K=\{1,2,\ldots,|K|\}\). \\ \hline
**Parameters** & **Definition** \\ \hline \(r_{l,m}\) & The pure running time of trains from station \(m\) to station \(m+1,\ m\in\{1,2,\ldots,|S|-1\}\). \\ \(Q_{t}^{m,n}\) & The number of passengers expecting to travel from station \(m\) to station \(n\) at time \(t\). \\ \(t_{ac}\) & Acceleration additional time. \\ \(t_{de}\) & Deceleration additional time. \\ \(t_{sb}^{ub}\) & Maximum dwell time of the train at the station. \\ \(t_{sb}^{b}\) & Minimum dwell time of the train at the station. \\ \(num\) & Maximum number of stops for the train. \\ \(num\) & Minimum number of stops for the train. \\ \(h_{dp}\) & Minimum headway between two adjacent trains, of which the former one departs from the station and the latter \\  & one passes through the same station. \\ \(h_{pd}\) & Minimum headway between two adjacent trains, of which the former one passes through the station and the latter \\  & one departs from the same station. \\ \(h_{pp}\) & Minimum headway between two adjacent trains, which both pass through the same station. \\ \(h_{dd}\) & Minimum headway between two adjacent trains, which both depart from the same station. \\ \(h_{ap}\) & Minimum headway between two adjacent trains, of which the former one arrives at the station and the latter one passes through the same station. \\ \(h_{pa}\) & Minimum headway between two adjacent trains, of which the former one passes through the station and the latter \\  & one arrives at the same station. \\ \(h_{aa}\) & Minimum headway between two adjacent trains, which both arrive at the same station. \\ \(U\) & A large positive number. \\ \(C\) & Train seat capacity. \\ \(\mu\) & Maximal seat occupancy rate. \\ \(\theta_{f}\) & Weights of each part of objective, \(f\in\{1,2\}\). \\ \(\Delta T\) & Maximum allowable departure time deviation of passengers. \\ \hline
**Variables** & **Definition** \\ \hline \(a_{t}^{m}\) & Arrival time of train \(i\) at station \(m,\ m\in\{2,3,\ldots,|S|\}\). \\ \(d_{t}^{m}\) & Departure time of train \(i\) from station \(m,\ m\in\{1,2,\ldots,|S|-1\}\). \\ \(p_{t}^{m}\) & Binary variable : if train \(i\) stops at station \(m\), equals 1; otherwise, 0. \\ \(qrc_{t,n}^{m}\) & Number of remaining passengers expecting to travel from station \(m\) to station \(n\) at time \(t\). \\ \(q_{t,t}^{a,n}\) & Number of passengers assigned on train \(i\) who expect to travel from station \(m\) to station \(n\) at time \(t\). \\ \(x_{i,t}^{m,n}\) & Binary variable : if there are passengers assigned on train \(i\) who expect to travel from station \(m\) to station \(n\) at time \(t\) \\  & equals 1; otherwise, 0. \\ \(y_{ij}^{m}\) & Binary variable : if train \(i\) departs from station \(m\) earlier than train \(j\), equals 1; otherwise, 0. \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Sets, parameters and variables used in the model.