# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 7 Computational Results
### Sensitivity Analysis  
(1) Transfer walking time  
As the transfer walking time increases, the model cost structure changes significantly. From a ratio of 0.97 for a transfer walking time of 2 min to 1.08 for a transfer walking time of 10 min, the train operating costs show an upward trend, indicating that the model needs to mitigate the loss of accessibility due to the decrease in interchange efficiency by increasing the number of train runs. Too long or too short transfer walking time brings about an increase in the objective function value. The most optimal effect is achieved when the transfer walking time is 5 min. As shown in Figure 7b, at 2 min transfer walking time, the number of transfers rises to 1.55 times that of the 5 min transfer walking time scenario, indicating that riders tend to choose the faster interline transfer path. As shown in Figure 7a, When the transfer time rises to 10 min, the number of transfers drops to 0, and all passengers abandon the transfer option altogether, resulting in a complete disappearance of both the transfer cost and the transfer path. This phenomenon reflects a clear threshold of transfer acceptance, indicating that there is a psychological tolerance threshold for transfer walking time, beyond which passengers will completely switch to non-transfer paths. Figure 6: Comparison of headcount and time data under the three modes of operation. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{\(\tau_{s}^{tra}\)} & **Ratio of Objective** & **Ratio of Train** & **Ratio of Total** \\  & **Function Value to** & **Operating Cost to** & **Passenger Travel** \\  & **Baseline** & **Baseline** & **Cost to Baseline** \\ \hline
2 & 0.99 & 0.97 & 0.99 \\
3 & 0.99 & 1.04 & 1 \\
5 & 1 & 1 & 1 \\
7 & 0.98 & 1.06 & 0.98 \\
10 & 0.98 & 1.08 & 0.97 \\ \hline \end{tabular}
\end{table} 
Table 3: Results for different transfer walking time. (2) The unit penalty cost  
Penalty costs are generated for passengers due to not being served, and the results of the metrics are shown in Table 4 when the penalty costs are changed. The unit penalty cost is set to 300, 500, 700, 900, and 1100, respectively, and the penalty cost = 700 is set as the baseline value. From the table, it can be seen that the unit penalty cost will lead to the objective function value and the total travel cost of passengers. As can be seen in Figure 8, when the unit penalty cost is elevated, the direction of optimization at this time is to reduce the number of unserved passengers, but at the same time, due to the increase in the unit penalty cost, even if the number of passengers is reduced, the total penalty cost still shows a continuous upward trend. (3) The unit passenger waiting cost  
Passengers waiting for a train at a station incur waiting costs, and when the unit waiting cost increases, the passenger waiting cost and waiting time are shown in Figure 9. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{_cpenalty_} & **Ratio of Objective** & **Ratio of Train** & **Ratio of Total** \\  & **Function Value to** & **Operating Cost to** & **Passenger Travel** \\  & **Baseline** & **Baseline** & **Cost to Baseline** \\ \hline
300 & 0.93 & 0.92 & 0.93 \\
500 & 0.96 & 1.02 & 0.95 \\
700 & 1 & 1 & 1 \\
900 & 1.04 & 0.98 & 1.05 \\
1100 & 1.07 & 1.01 & 1.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for different unit penalty cost for unserved passengers. Figure 8: Comparison of metrics for different unit penalty cost for unserved passengers. Figure 7: **(a)** Passenger transfer cost for different transfer walking time; (b) Comparison of headcount and time data for different transfer walking time. By adjusting the train schedule, the waiting time is reduced, but the passenger waiting cost shows an increasing trend, indicating that the unit waiting cost has a significant impact on the passenger waiting cost.