## 5 Numerical Experiments
### Large-Scale Case Study  
In this section, we will test the performance by applying the proposed model to designing an operational plan for high-speed trains on the Beijing-Shanghai high-speed railway corridor in China. The Beijing-Shanghai railway corridor is the main railway corridor connecting Beijing and Shanghai and one of the country's most important corridors. The experimental data in this large-scale study are real-world data. Before implementing the large-scale experiment, we first introduce the input experimental data. We implement this experiment within \([0,720]\) min on the Beijing-Shanghai corridor. As shown in Figure 9, the Beijing-Shanghai railway corridor has a total length of 1318 km and consists of 24 stations and 23 segments [29]. The details on the distance of each segment and the traveling time of high-speed trains in each segment are listed in Table 11[30]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{y}_{min}\)} & \multicolumn{2}{c}{**The Number of Trains**} & \multicolumn{2}{c}{**Running Distance**} & \multirow{2}{*}{**Dwelling Time**} & \multicolumn{2}{c}{**The Number of Stops**} & \multicolumn{1}{c}{**Unsatisfied Passengers**} \\ \hline
10 & 5 & 750 & 12 & 6 & 17 \\ \hline
20 & 5 & 750 & 12 & 6 & 17 \\ \hline
30 & 5 & 750 & 12 & 6 & 17 \\ \hline
40 & 5 & 750 & 11 & 5 & 32 \\ \hline
50 & 5 & 750 & 11 & 5 & 32 \\ \hline
60 & 5 & 690 & 5 & 4 & 142 \\ \hline
70 & 5 & 690 & 5 & 4 & 142 \\ \hline
80 & 5 & 690 & 5 & 4 & 142 \\ \hline
90 & 4 & 600 & 4 & 3 & 210 \\ \hline
100 & 4 & 576 & 3 & 2 & 252 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Optimized solutions obtained under the minimum dwelling time \(y_{min}\). Figure 9: Map of Beijing-Shanghai high-speed railway corridor. There are 24 stations along the Beijing-Shanghai railway corridor. Since Tianjin South Station and Tianjin North Station both belong to the city of Tianjin, we do not consider the difference in passenger demand between the two stations; their passenger demand is combined and regarded as the passenger demand at Tianjin Station. Thus, there are totally \(23\times 22\)/\(2=253\) OD pairs. The OD passenger demand associated with stations on the Beijing-Shanghai railway corridor in peak periods is listed in Table 12a,b [30]. In this experiment, we designate 42 existing trains for the off-peak period. The timetables of these 42 existing trains are illustrated in Figure 10, where the black lines represent the timetables of existing trains. The stop plans for existing trains are also illustrated in Figure 11. The rolling stock used on the Beijing-Shanghai high-speed railway corridor is usually CR400 AF. There are three marshalling types of CR400 AF: eight marshalling types, 16 marshalling types, and 17 marshalling types, and their capacity are 576, 1193, and 1283. In the peak period, rolling stock with large capacity is employed. Thus, the maximum loading capacity of trains is set to 1200 unit in this paper. The total passenger demand along the Beijing-Shanghai corridor is 132,358. The total passenger demand distributed for existing trains is 112,601. There are still 19,757 passengers who cannot travel by train in the peak period. Based on the passenger demand in the peak period and the number of existing trains, we intend to add four trains. The minimum departure headway \(l_{min}^{d}\) is set to 2 min. The minimum dwelling time \(d_{min}\) is set to 2 min. To avoid wasting the transport capacity, the capacity-occupancy ratio for both existing trains and additional trains is set to 0.7. In our paper, because the purpose of adding additional trains is to meet passenger demand in peak periods, the objective of passenger demand has the highest priority in this experiment. The weight of the third objective \(\delta_{3}\) is set to 0.7. However, the cost to the railway company and the service for in-train passengers cannot be ignored.