## 5 Numerical Experiments
### Analysis of the Impact of the Number of Reserved Carriages on Train Operation  
The transportation supply that is provided at each time period of a day should coincide with the distribution of passenger, to meet the travel demand of passenger flow and improve the transportation revenue. Since the fluctuating characteristics of passenger demand will have a great impact on the number of operational trains at each time period and the turnover of trains, the number of reserved carriages will affect the train line planning. In order to analyze the influence of the number of reserved carriages on the train that are running and rail line profit, we set the number of reserved carriages \(N=\{60,70,\ldots 140\}\) and the solution results are shown in Figure 5.  
Figure 5: The trend of the objective function and the proportion of passengers that are served. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number of**} & \multicolumn{2}{c}{**Gurobi**} & \multicolumn{2}{c}{**The Improved Round Heuristic**} \\ \cline{2-6}  & \multirow{2}{*}{**Objective Function**} & \multicolumn{2}{c}{**Computational Time (s)**} & \multirow{2}{*}{**Mip-Gap (\%)**} & \multirow{2}{*}{**Objective Function**} & **Computational Time (s)** \\ \hline
50 & 327,196.28 & 60.75 & * & 296,276.22 & 13.74 \\
100 & 727,691.93 & 87.75 & * & 660,637.31 & 15.41 \\
150 & 1,206,145.08 & 376.88 & * & 1,096,329.19 & 18.60 \\
200 & 1,469,070.17 & 1800.00 & 0.02 & 1,336,637.90 & 20.77 \\
250 & - & 1800.00 & - & 1,701,417.40 & 22.71 \\ \hline \hline \end{tabular}  
*: less than Gurobi default toleration (0.0001); -: no feasible solution was found in the given time.  
\end{table} 
Table 3: Performance comparison of the improved round heuristic and Gurobi. As can be seen from Figure 5, with the increase in the number of reserved carriages, the rail line profit and the number of passengers that are served gradually increases, and the waiting time cost of unserved passengers who choose to travel in the next period gradually decreases. This is because when the carriage resources are increased, the impact of resource constraints on the line planning scheme is weakened, and the railway department can dispatch more trains, serve more passengers, and obtain more benefits. It should be noted that the objective function has a significant change when the carriage resources increase from 60 to 110. After that, the increase of carriage resources has little impact on the objective function. This indicates that 60 carriages are not enough to meet the given demand, which limits the railway company to provide a higher level of service. While 110 carriages are put into service, 99.22% of the demand can be served. Therefore, the car resources have little impact on the line planning scheme. Therefore, it is suggested that the railway company should consider investing 110 carriages to obtain satisfactory line revenue. Next, we take \(N=110\) as an example and analyze the relevant indicators of the optimized line planning. The corresponding indicators of the all-day train line planning scheme are shown in Table 4. It can be seen from Table 4 that the number of trains in the downstream direction is more than that in the upstream direction, which is caused by the asymmetry of demand. The average load factor of the train in the upstream direction is 70.03%, and that in the downstream direction is 72.20%, indicating that the passenger congestion is low and the trip is comfortable. The detailed line planning scheme of peak hours is shown in Figure 6, in the upstream direction from 18:00 to 21:00 and the downstream direction from 7:00 to 10:00. It can be seen from Figure 6 that the service frequencies of the station S4, S8, and S11 are more than those of other stations in the upstream direction, and that of station S4, S6, and S8 are more than those of other stations in the downstream direction, which is also related to the asymmetry of passenger demand.