## 5 Numerical Experiments
### Additional Experiments  
The influences of the three parameters of our proposed model, i.e., the capacity-occupancy ratio of each train \(\delta\), the minimum headway \(h_{min}\), and the minimum dwelling time \(d_{min}\), on the optimal solution are analyzed. (i) _Capacity-occupancy ratio_  
In the model, we introduce a coefficient of the capacity-occupancy ratio, \(\delta\), to control the capacity-occupancy ratio. This coefficient is used to restrict the number of passengers along each segment belonging to the train's operational zone. Although a higher capacity-occupancy ratio for a single train  
Figure 8: Passenger distribution for existing trains. is better, the impact of different capacity-occupancy ratios on the final operational plan is not known. Thus, we implement several experiments using capacity-occupancy ratio from 0.5 to 1.0 and obtain the corresponding operational plan. The experimental environment and other parameters are the same as in the small-scale case study. For computational accuracy, the relative MIP gap tolerance is set to 0.01 for the set of experiments. The detailed computational results are listed in Table 7.  
In Table 7, some indexes of additional trains, i.e., the number of trains, the total running distance of trains, the total dwelling time, the number of stops, and unsatisfied passengers, are displayed to evaluate the operational plan. Clearly, when \(\delta\) is from 0.5 to 0.8, although the stop plan is changed slightly depending on the capacity-occupancy ratio, the number of additional trains, number of unsatisfied passengers, and running distance are the same. The reason is that the objective function of passenger demand plays a role in ensuring the capacity-occupancy ratio to some extent. The third objective function can keep the capacity-occupancy ratio high. When \(\delta\) increases to 0.9 and 1.0, the number of additional trains decreases, and the number of unsatisfied passengers increases significantly. In the operational plan of 0.9 and 1.0, only four trains are added and the corresponding total running distance is decreased, which means lower costs for the railway company. However, the expectation of decision maker is not only for a lower cost, but also meeting more passenger demand. The number of unsatisfied passengers increases significantly, which is contrary to our aim. Generally, a high capacity-occupancy ratio means fewer trains loading more passengers. However, in terms of designing the operational plan for multiple trains, a higher ratio is not necessarily better. Despite the number of passengers on some trains being large in the operational plan, the remaining passengers are not enough to guarantee at-capacity occupancy for other trains and so they are canceled. Thus, choosing a suitable \(\delta\) value that is good for cost and passenger demand is important. (ii) _The minimum dwelling timed\({}_{min}\)_  
The parameter of the minimum dwelling time \(d_{min}\) is proposed to offer time for passengers to get on/off. Passengers need more dwelling time to board and alight, while the passengers staying on the train hope for less dwelling time. Next, we will test the optimal solutions under the value of \(d_{min}\) from 1 to 6 min. Some meaningful indexes of optimal solutions are displayed in Table 8. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\delta\)} & **The Number** & **Running** & \multirow{2}{*}{**Dwelling Time**} & **The Number** & **Unsatisfied** \\  & **of Trains** & **Distance** & & **of Stops** & **Passengers** \\ \hline
0.5 & 5 & 750 & 10 & 4 & 17 \\ \hline
0.6 & 5 & 750 & 11 & 5 & 17 \\ \hline
0.7 & 5 & 750 & 11 & 5 & 17 \\ \hline
0.8 & 5 & 750 & 12 & 6 & 17 \\ \hline
0.9 & 4 & 600 & 5 & 5 & 208 \\ \hline
1.0 & 4 & 576 & 5 & 5 & 207 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimized solutions obtained under different capacity-occupancy ratio \(\delta\).