## 5 Numerical Experiments
### Comparison of Flexible and Fixed Composition Mode  
In this section, we set \(k\) as a constant, \(k=8\), to obtain a multi-period line planning model under the fixed composition and compared it with our model to illustrate that flexible composition is more suitable for fluctuating demand. Table 5 summarizes the influence of the two composition modes on the relevant indicators of the train line planning under the conditions of relatively tight (\(N=60\)) and relatively sufficient (\(N=140\)) reserved carriage resources. It can be seen that under the same conditions of train carriage resources: (i) the frequency under the flexible composition mode is about twice as much that under the fixed composition mode, and the service capacity can be increased by more than 2%, and the economic benefit can be increased by more than 1.9%; (ii) the number of trains in the fixed composition mode is relatively small, and the choice of passengers is more concentrated, so the average train passenger load ratio is slightly higher than that of the flexible composition mode. This result confirms that the flexible composition mode can help to improve the social and economic benefits simultaneously. Figure 8: The number of operational carriages and average load factor at each time period. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Number of**} & \multicolumn{2}{c}{**\(N=60\)**} & \multicolumn{2}{c}{**\(N=140\)**} \\ \hline \multirow{2}{*}{**Composition Mode**} & **Fixed Composition Mode** & **Flexible Composition Mode** & **Fixed Composition Mode** & **Flexible Composition Mode** \\ \hline \multirow{2}{*}{Profit (RMB)} & 1,966,768.93 & 2,043,111.35 & 2,136,572.70 & 2,177,145.44 \\  & \(51\) & 100 & 59 & 113 \\ \multicolumn{1}{c}{The proportion of} & 89.87 & 91.88 & 97.13 & 99.35 \\ \multicolumn{1}{c}{\multirow{2}{*}{passengers served (\%)}} & \multirow{2}{*}{75.46} & \multirow{2}{*}{72.58} & \multirow{2}{*}{72.88} & \multirow{2}{*}{72.64} \\ \cline{1-1} \cline{5-5}  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Indicators of the line planning. Figure 9 shows the comparison of the frequency and service capacity at each time period under different composition modes when \(N=60\). We can see that:  
1. When the flexible composition mode is adopted, the frequency of each time period is significantly higher than that of the fixed composition mode, and the passenger service capacity, especially in the peak hours of passenger flow, is also significantly higher than that of the fixed composition mode. The data show that the utilization efficiency of carriages of the flexible composition mode is higher than that of the fixed composition mode. The average transportation of each carriage under the flexible composition mode is 1609.35 person-times/day, which is higher than 1574.15 person-times/day under the fixed composition mode. The reason is that the number of remaining carriages in the same period is more than that of the fixed composition mode, and the number of carriages of the trains that are dispatched under the flexible composition mode is not unique, while that under the fixed marshalling mode must be 8. Thus, the utilization efficiency under the flexible composition mode is higher. For example, during the time period from 18:00 to 19:00 in the upstream direction, when the flexible composition mode is adopted, the number of remaining carriages is 21, so three trains with seven carriages can be operated, while when the fixed composition mode is adopted, the number of remaining carriages is 16, so only two trains with eight carriages can be operated. 2. Compared with the fixed composition mode, the train frequency under the flexible marshalling mode is more susceptible to the fluctuation of passenger flow. This is because the line capacity under flexible marshalling is more flexible.