# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 7 Computational Results
### Comparative Analysis of Different Models  
In independent operation mode, no cross-line trains are operated, while in full cross-line operation mode, no small-crossing trains are included. To evaluate the effectiveness of the proposed hybrid cross-line operation mode that integrates both express and local trains, a comparative analysis with the two aforementioned modes is conducted. Table 2 shows the train timetable objective function values and the ratio of each metric for this article's cross-line, full cross-line, and independent modes of operation. As can be seen from the table regarding the implementation of this article's cross-line mode of operation compared to the independent operation of the case, the objective function value decreases by 5%, the train operating costs increases by 8%, the total cost of passenger trips decreases by 6%, and the magnitude of passenger trip cost is greater than the train operating cost, indicating that this article's cross-line mode of operation, although increasing train operating costs, does so in exchange for a significant reduction in the total cost of passenger trips. The ratio of total passenger travel cost to train operating cost is reduced by 13%, indicating that the article cross-line operation mode can effectively reduce the target value and improve the efficiency of passenger transportation per unit of train operating cost. Combining the results of the comparison between full cross-line operation and independent operation, where a 2% increase in the objective function value resulted in only a 3% \begin{table}
\begin{tabular}{c c c c} \hline
**Coefficient of Fluctuation \(\theta\)** & \multicolumn{2}{c}{**The Value of the Objective Function**} & \multicolumn{1}{c}{**Gap**} \\  & **Gurobi** & **E-ALNS** & **Gap** \\ \hline
0.8 & 2,558,611 & 2,640,264 & 3.19\% \\
1.0 & 3,192,764 & 3,289,185 & 3.02\% \\
1.5 & 4,758,355 & 4,940,115 & 3.82\% \\ \hline \end{tabular}
\end{table} 
Table 1: The solution of example. Figure 4: Convergence curve of the algorithms. reduction in the total passenger travel cost versus train operating cost, it is found that the article cross-line operation model is more effective. As illustrated in Figure 5a,b, the full cross-line operation mode has the highest objective function value, total passenger travel cost, and passenger penalty cost among the three modes. Although it reduces the passenger transfer cost, in order to control the total train operating cost when the unit cost of train operation becomes higher, it reduces the number of trains, resulting in an increase in passenger penalty cost. It shows that the cost structure of this model is unbalanced, making it difficult to balance the contradiction between supply and demand, and it should be used when the demand for cross-line direct passenger flow is oversized and the short-distance passenger flow on this line is extremely low. At the same time, it shows that on the basis of cross-line operation, an appropriate increase in the number of small-crossing trains can further reduce passenger travel costs and access more passengers. As shown in Figure 6, this article's cross-line operation has the lowest number of unsuccessful trips and the shortest waiting time per capita, indicating that the effective integration of line resources through train cross-line operation significantly optimizes the efficiency of passenger trips, can provide direct service for cross-line passengers, and reduces the transfer walk time, transfer waiting time, and on-board travel time for some passengers. It also shows that this article's cross-line operations are more advantageous than stand-alone operations in terms of increasing the number of successful passenger trips and reducing passenger transfers. It shows that the addition of crossover trains to independent operations can effectively reduce passenger travel costs and achieve efficient travel for more passengers.