## 5 Numerical Experiments
### Small-Scale Case Study
Figure 5 illustrates the operational zone and stop plan for the additional trains and existing trains. The red one represents an additional train, and the black one is an existing train. S1 is station 1, E1 is the first existing train to arrive at its terminal station, and A1 is the first additional train to arrive at its terminal station. In Figure 5, a solid dot denotes a train that needs to stop at this station, and a hollow dot denotes a train that goes through this station without stopping. Figure 6 shows the timetables of additional trains and existing trains. Similarly, the red line represents the timetable of the additional train, and the black line \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Origin\textbackslash\{Terminal Station}** & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **Total** \\ \hline S1 & - & 380 & 160 & 89 & 82 & 200 & 911 \\ S2 & - & - & 210 & 79 & 102 & 214 & 605 \\ S3 & - & - & - & 120 & 150 & 60 & 330 \\ S4 & - & - & - & - & 157 & 83 & 240 \\ S5 & - & - & - & - & - & 521 & 411 \\ Total & - & 380 & 370 & 288 & 491 & 1078 & 2607 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Passenger demand for each OD in peak period. Figure 3: A small-scale outbound railway corridor. represents the timetable of the existing train. In Figure 6, the existing trains are indexed from E1 to E5 depending on the arrival time, and additional trains are indexed from A1 to A5. The passenger distribution of the five additional trains is illustrated in Figure 7. Since the passenger distribution for existing trains is optimized jointly with additional trains in the model, the passenger distribution for existing trains has changed compared with the data in Figure 4. The new passenger distribution for existing trains is illustrated in Figure 7.  
Figure 4: Passenger distribution for existing trains. It is known from Figure 5 that the five additional trains have different operational zones and stop plans. There four kinds of operational zones for these additional trains. Train A3 has a long-distance operational zone from S1 to S6. To cut down on unnecessary running distance, the other trains do not need to select the starting station or ending station of the corridor as their origin station or terminal station. As a result, the operational zone of trains A1 and A2 is from S2 to S6. Although trains A1 and A2 have the same operational zone, they have different stop plans. Train A2 just serves the passengers from S2 to S6, with no stops at intermediate stations. Train A1 stops at S5 to serve passengers departing or arriving there. It is seen in Figure 6 that the additional trains can maintain the departure headway in relation to existing trains to avoid collisions. It is noted that additional train A3 is overtaken by existing trains E2 at S3 and E3 at S5. Train A3 arrives at S3 after 32 min and waits at S3 for 4 min to make way for train E2. Train E2 goes through S3 after 34 min without stopping. Train A3 departs from S3 36 min after E2 departs from S3 to maintain the departure-arrival headway of 2 min with relation to train E2. Next, train A3 arrives at S5 after 69 min and waits there for 4 min to make way for train A3. Figure 5: Stop plans for existing trains and additional train. Figure 6: Timetables for existing trains and additional train. E3. In the end, train A3 arrives at S6 after 88 min. Therefore, our proposed model has the ability to choose an overtaking strategy to insert more additional trains between the existing trains and maintain the necessary headway. We know from the timetables of trains in Figure 6 that a decision maker must select suitable times and stations to insert the additional trains to maintain headway and meet passenger demand as much as possible. Timetabling decisions affect the operational zones and stop plans of additional trains and further affect passenger distribution. Thus, considering the timetables of additional trains in the operational plan is essential. Figure 7 shows the passenger distribution of these additional trains.