## 5 Numerical Experiments
### Additional Experiments
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(d_{min}\)} & **The Number** & **Running** & \multirow{2}{*}{**Dwelling Time**} & **The Number** & **Unsatisfied** \\  & **of Trains** & **Distance** & & **of Stops** & **Passengers** \\ \hline
1 & 5 & 750 & 12 & 6 & 17 \\ \hline
2 & 5 & 750 & 17 & 4 & 31 \\ \hline
3 & 5 & 690 & 22 & 3 & 141 \\ \hline
4 & 5 & 690 & 22 & 2 & 154 \\ \hline
5 & 5 & 690 & 25 & 2 & 154 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Optimized solutions obtained under the minimum dwelling time \(d_{min}\). When the minimum dwelling time changes from 1 to 5, the number of trains is still five, but the running distance is decreased. With \(d_{min}\) increasing, the dwelling time also increases, and the number of stops decreases. To insert more additional trains into the original operational plan and avoid conflicts with existing trains, the number of stops need to be reduced. Also, the operational zones of trains are shortened. As a result, the number of unsatisfied passengers increases, with the value of \(d_{min}\) also increasing. The worst result is when the number of trains is three and \(d_{min}\) is 6: the number of unsatisfied passengers rises to 400. Thus, a suitable dwelling time at a station where there are passengers getting on/off is indispensable. However, more dwelling time is not always better. _The headway \(h_{min}\)_  
In the real world, railway company operators always set a certain headway, which is not only to avoid conflicts between trains, but also to leave enough time for operators and workers to prepare. The influence of headway on the operational plan needs to be investigated. The experimental results with different values of \(h_{min}\) are listed in Table 9.  
When the headway is from 1 min to 2 min, there seems to be no difference between the two operational plans. However, when we look a little more carefully at the results, the total dwelling time changes from 9 min to 12 min. The reason is that one of the additional trains dwells at one station for 4 min to give way to the existing train. That is to say that when the headway become longer, some operations, e.g., overtaking, will be performed to insert more additional trains and load more passengers. The situation is more obvious as the headway increases. When the headway increases to 6 min, all five trains cannot be inserted within the limited time horizon, and one additional train is cancelled. Meanwhile, the number of unsatisfied passengers rises to 261. It is known from this set of experiments that suitable headway can leave enough time for operators to prepare and avoid collisions. Therefore, the value of the headway should be selected carefully in real-world operations. _The minimum number of barding and alighting persons for stopping, \(y_{min}\)_  
\(y_{min}\) is the minimum number of bounding and alighting persons required for a train to stop. Stopping consumes energy to accelerate and decelerate, which increases the railway company's costs. Therefore, operators are very serious about designing a stop plan. The required minimum number of persons boarding and alighting for stopping is proposed in the model, which means that the train cannot stop at a station until the sum of boarding and alighting persons is up to a certain threshold. We select 10 different values of \(y_{min}\), from 10 to 100, and the corresponding solutions are listed in Table 10. From Table 10, the number of trains, running distance, dwelling time, and number of stops tends to decrease, while the number of unsatisfied passengers tends to increase. On the face of it, increasing the threshold for stopping can decrease the dwelling time, the number of stops, and the number of trains, which can reduce costs. However, more passengers may not be able to board the trains. Hence, in the real world, selecting the threshold for stopping is important for operators. In addition, more passenger demand is met, which can bring about benefits for the railway company.