## 2 Problem Description
In Plan 1, the additional train stops at station 2 and station 3, while the train in plan 2 just stops at station 2 and skips station 3. We assume that when passengers have two choices of ways to reach their destination, we can distribute the passengers equally to each train. If we adopt plan 1, passengers going from station 1 to station 3 have two choices, which means that \(N/2\) passengers take existing trains to station 3 and \(N/2\) passengers take the additional trains in plan 1 to station 3. Meanwhile, only \(N/2\) passengers going from  
Figure 1: Adding additional train on a railway corridor in the peak period. station 1 to station 2 can board the train at station 1. Then, passengers from station 1 to station 2 get off at station 2, and the rest of the capacity of the additional train at station 2 is \(N/2\), which results in only \(N/2\) passengers boarding the train to go from station 2 to station 4. In consequence, \(N/2\) passengers going from station 2 to station 4 cannot get to their destination, and \(N/2\) seats are empty in the existing train from station 1 to station 3. Passengers from station 1 to station 3 cannot take the additional train in plan 2 because the train does not stop at station 3. \(N\) passengers take the additional train in plan 2 from station 1 to station 2, then \(N\) passengers take this train from station 3 to station 5 to get their destination. It is known from Table 2 that, although the operating cost of plan 1 is higher than that of plan 2, the number of unsatisfied passengers is greater. Another comparison between plan 3 and plan 4 is implemented. We know from Figure 2 that the operational zone in plan 3 is longer than that in plan 4. However, there are \(N/2\) passengers going from station 1 to station 2 who can board the additional train in plan 3 because passengers going from station 1 to station 3 can reach their destination. Meanwhile, there are \(N\) passengers who can take the additional train from station 1 to station 2 in plan 2. As listed in Table 2, although the operational zone in plan 3 covers more segments and stations, both the unstratified demand and the operating cost are higher than in plan 4. The two comparisons show that more stops and a longer operational zone do not always load more passengers and may cause more empty seats and higher operating costs. This example illustrates the necessity of the optimization of the operational zone and stop plan for additional trains, which can improve passenger satisfaction, cost-saving, and utilization of transport capacity. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Origin/Terminal Station** & **1** & **2** & **3** & **4** & **5** \\ \hline
1 & & N & N & \\
2 & & & & N & \\
3 & & & & N & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Passenger demand for each origin-destination (OD) pair. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Operation Zone** & **Operational Cost** & **Loading Passengers** & **Unsatisfied Passenger Demand** \\ \hline Off-peak & \(E\) & \(2N\) & \(2N\) \\ \hline Plan 1 & \(E+3c+2s\) & \(3N\) & \(N\) \\ \hline Plan 2 & \(E+3c+s\) & \(4N\) & \(0\) \\ \hline Plan 3 & \(E+2c+s\) & \(2.5N\) & \(0.5N\) \\ \hline Plan 4 & \(E+c\) & \(3N\) & \(N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparison of plan 1, plan 2, plan 3, and plan 4. Figure 2: An illustration of operation zones and stop plans for additional trains. However, we cannot ignore the importance of optimizing timetables for additional trains. The timetable has a great impact on the operational zone, stop plan, and passenger distribution. The departure and arrival times of additional trains must maintain the time interval between additional and existing trains; we call that time interval headway because the added train must avoid conflicting with existing trains and additional trains. In a finite time horizon, the number of trains that can be added is finite because of headway.