## 5 Numerical Experiments
### Small-Scale Case Study  
The small-scale experiment is implemented on an outbound railway corridor within the time horizon \([0,\ 90]\) min. The railway corridor consists of six stations and five segments, as shown in Figure 3, wherein stations are indexed from the starting station of railway corridor station 1 to ending station 6 along the outbound direction. The length of each segment and travel time of trains across each segment are also displayed in Figure 3. There are five existing trains in the off-peak period. The stop plans and operational zones of these existing trains are illustrated in Figure 4. The passenger distribution plans of these existing trains are listed in Figure 4. Each train has different colored rectangles because of the different passenger distribution of each train. The length of the whole \begin{table}
\begin{tabular}{c c} \hline \hline
**Original Nonlinear Constraints** & **Reformulated Linear Constraints** \\ \hline Constraints (4)–(5) & Constraints (43) and (44) \\ Constraints (7)–(11) & Constraints (45)–(49) \\ Constraints (12) & Constraints (50) and (51) \\ Constraints (13) and (14) & Constraints (52) and (53) \\ Constraints (15) & Constraints (54)–constraints (59) \\ Constraints (16)–(21) & Constraints (60) \(\sim\)(65) \\ Constraints (24) and (25) & Constraints (66) and (67) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Reformulation. rectangle represents the maximum capacity of each train, and the width represents the total running distance of train. In Figure 4, the different colored rectangles represent passengers for different OD. The length of each colored rectangle represents the number of passengers for this OD pair. The width of each color rectangle represents the distance of the OD pair. To clarify the number of passengers for each OD in each train, the number in each rectangle denotes the number of passengers that the train takes for this OD pair. The number of empty seats in each train is also noted in these figures of passenger distribution. The passenger demand for each OD of the railway corridor in peak periods is given in Table 6. Obviously, transport capacity provided by the existing trains cannot meet the passenger demand. To meet passenger demand in peak periods, we intend to add five additional trains according to Equation (1). The additional trains must maintain the departure headway to avoid collisions. The minimum departure headway \(l_{min}^{d}\) is set to 2 min. The maximum loading capacity of each train is 120. To leave enough time for passengers to board, we set the minimum dwelling time \(d_{min}\) to 1 min. The capacity-occupancy ratio is set to 0.8 to make full use of the transport capacity. Since our operational plan is meant to be passenger-responsive, the third objective of passenger demand has the highest priority in our experiments. The second objective of dwelling time has the second priority and the importance of the first objective is the lowest. Thus, the weight parameters of the three objectives \(\delta_{1}\), \(\delta_{2}\), and \(\delta_{3}\) are set to 0.1, 0.2, and 0.7. To normalize the three objectives, we first implement two experiments maximizing and minimizing the first objective and get the values of \(F_{1 min}\) and \(F_{1 max}\), i.e., 0 and 714. Similarly, the value of \(F_{2 min}\) and \(F_{2 max}\) are 0 and 166, and \(F_{3 min}\) and \(F_{3 max}\) are 408 and 186,930. Based on the above parameters, the optimal solution of the model is obtained. The value of the first objective function of trains' running distance is 750 km. The value of the second objective function of dwelling time is 12 min. The value of the third function is 408 unit.km. Although we set a high weight for the third objective, 17 passengers cannot travel from origin (station 5) to destination (station 6) on train. To better illustrate the experimental results, the operational zones, stop plans, timetables, and passenger distribution for additional trains are shown in Figures 5-7.