## 5 Computational Experiments
### Results and Analysis  
The optimal value of Model III obtained by the Gurobi solver is 3030 s, the values of parameters \(T^{\text{min}}\_r\) and \(T^{\text{min}}\_\text{dev}\) are 2024 s and 1006 s, respectively, the solving time is 0.177 s, and the corresponding Gap is 0%. The rescheduled timetable is shown in Figure 9, which can be used as the result of the third solution approach or the result of the second solution approach when the two tolerance coefficients are both 0. At the same time, following the process of the first solution approach, the values of the two parameters are substituted into Model II. The optimal value of Model II is 836.898, the solving time is 1.216 s, the Gap is 0%, and the rescheduled timetable is shown in Figure 10.  
In Figures 9 and 10, the cross-line backup train is located between the second and third planned trains (i.e., Ta2 and Ta3, and Tb2 and Tb3), but there are differences in arrival and departure times at different stations. In the two rescheduled timetable, both the previous train and the rear train of the backup train are affected, but neither has a significant deviation. It is worth noting that in the rescheduled timetable shown in Figure 9, the number of transfer passengers served by the cross-line backup train is 708, while for the rescheduled timetable shown in Figure 10, the corresponding value is 836. Figure 9: Train rescheduled timetable (\(\epsilon^{di}=0\), \(\epsilon^{ef}=0\)). Regarding the impact of the deviation tolerance and running efficiency tolerance on the objective function value, the results shown in Figure 11 are obtained based on the second solution approach. It can be intuitively observed that when the parameters are small, the objective function value is also small, and as the two increase synchronously, there is a trend of the objective function value increasing. For example, when both are taken as 0, the objective function value is 708.734; When both are taken as 30%, the objective function value is 836.898. According to Figure 11, for this experiment, the recommended adjustment range for the two coefficients in the second solution approach is [0, 30%]. Figure 11: Impact of parameter changes on the objective function value. Figure 10: Train rescheduled timetable (\(\epsilon^{di}=20\%\), \(\epsilon^{ef}=20\%\)). After tripling the demand (i.e., passenger flow), a calculation is conducted by dynamically adjusting parameters according to the second solution approach, and it is found that the optimal values of the objective function are all 846.389. Meanwhile, the calculation result of the third solution approach is also 846.389. These results indicate that the larger demand reduces the impact of two parameters on the objective function value. When the demand is high, the number of passengers served by the train is also limited due to the limited capacity of the train, and the results obtained are consistent with the actual situation. Therefore, when in great demand, the rescheduled timetable for train operation can only be obtained based on Model III or the third solution approach. To further analyze the advantages and disadvantages of the strategy of adding a cross-line backup service to the large transfer passenger flow, we compared it with the corresponding results of the original timetable. The relevant results of the two are shown in Table 2.  
According to Table 2, it can be seen that in terms of the number of passengers transferring out at the interchange station (A5) for different planned trains, the maximum value of the rescheduled timetable is 605, the minimum value is 428, and the average value is 489. The maximum value of the original timetable is 669, the minimum value is 428, and the average value is 559. Therefore, the impact of the transfer passenger flow of each planned train in the rescheduled timetable on adjacent lines is relatively smaller. At the same time, the maximum number of stranded passengers obtained from the rescheduled timetable is also more ideal than the original timetable (e.g., Station A4, 170 \(<\) 1001).