## 3 Model Formulation
### Constraints  
(1) Constraint of train-station matching  
Due to the categorization of trains and stations, trains with different classes need to stop at stations with specific levels to provide diverse services for passengers. As for train \(l\in L\), it can only stop at stations in its candidate stop set \(CS_{l}\). As for station \(v\in V\), if it  
Figure 1: Illustration of capacity utilization impact of stop patterns. belongs to \(CS_{l}\), then the coefficient between \(l\) and \(v\) is \(\epsilon(l,v)=1\); otherwise, \(\epsilon(l,v)=0\). Thus, the constraint of train-station matching can be shown as  
\[x_{i}^{l}\cdot\left[\epsilon\left(l,v_{i}^{l}\right)-1\right]=0,i=1,2,\cdots,h (l),l\in L \tag{1}\]  
(2) Constraints of service frequency  
Stop planning should satisfy the direct travel desires of passengers, so the direct services for a certain O-D pair should reach the required frequency. As for train \(l\in L\), it can serve O-D pair \((r,s)\in RS\) only if two conditions are satisfied: (a) train \(l\) simultaneously stops at stations \(r\) and \(s\), and (b) train \(l\) passes station \(r\) first and then passes station \(s\). Let \(\sigma_{v}^{l}\) denote the coefficient between train \(l\) and station \(v\), which means if train \(l\) stops at station \(v\), then \(\sigma_{v}^{l}=1\), otherwise, \(\sigma_{v}^{l}=0\); denote \(\sigma_{rs}^{l}\) as the coefficient between the passing station order of train \(l\) and O-D pair \((r,s)\), which means if train \(l\) passes station \(r\) first and then passes station \(s\), then \(\sigma_{rs}^{l}=1\), otherwise, \(\sigma_{rs}^{l}=0\). The constraint of O-D service frequency is  
\[\sum_{l\in L}\sigma_{r}^{l}\cdot\sigma_{s}^{l}\cdot\sigma_{rs}^{l}\geq n_{rs }^{0}\cdot(r,s)\in RS \tag{2}\]  
where \(n_{rs}^{0}\) is the required direct service frequency between O and D \((r,s)\). Meanwhile, the service frequency for each station should also satisfy a certain service level, i.e.,  
\[\sum_{l\in L}\sigma_{v}^{l}\geq n_{v}^{0},v\in V \tag{3}\]  
where \(n_{v}^{0}\) is the required service frequency for station \(v\). (3) Constraint of stop ratio  
Trains with different classes have different travel speeds to satisfy diverse service levels. As for a certain train class, the stop ratio should not exceed a specific range to ensure required service quality [49; 52]. As for train \(l\in L\), the highest stop ratio of the corresponding train class is denoted by \(r(R_{l})\), so then the constraint is  
\[\sum_{i=1}^{h(l)}x_{i}^{l}/|V_{l}|\leq r(R_{l}),l\in L \tag{4}\]  
(4) Constraints of required stops  
In practical operation, trains need to stop at some specific stations to ensure necessary circulation, maintenance, and service [1]. Each train needs to stop at both the starting station and the ending station, i.e.,  
\[x_{1}^{l}=1,l\in L \tag{5}\]  
\[x_{h(l)}^{l}=1,l\in L \tag{6}\]  
Meanwhile, to ensure the required service level, trains generally stop at high-level stations. For example, trains are almost certain to stop at provincial stations in China due to the huge passenger volume and the political and economic importance of provincial capitals [49]. Assume that stations are classified into \(H\) levels; then, the station set of level \(h=1,2,\cdots,H\) is denoted by \(V(h)\). In this paper, trains need to stop at level-one stations to ensure enough services for important stations.