## 4 The Passenger Travel Demand Decomposition and Stop Plan
### Passenger Travel Demand Decomposed  
If a train departs, arrives, or passes through a station, we call the train serves the station. The service plan defines the minimum quantity of train service at each station.  
For stations at both ends of the railway, we use the following method to calculate \(S_{k}\), the minimum quantity of train-set service:  
\[S_{k}=\frac{Q_{k}}{C} \tag{14}\]  
where \(Q_{k}\) means the number of passengers that station \(k\) served in one day. \(C\) means the average capacity of a train. In this paper we default that all the train-set has the same capacity, hence \(C\) is equal to the capacity of a single train-set.  
Considering that the train-set has carried passengers before it arrives at the intermediate station, we need to consider the average full load ratio between adjacent stations of a section when calculating the quantity of train service for the intermediate station. It should be noted that the section here is different from Section 3.3. Section 3.3 is for boundary stations, and the section here is for adjacent stations. The corresponding relationship of the full load ratio and the station is shown in Figure 8.  
As shown in Figure 8, \(F_{A-B}\) is the average of section A-B, hence the number of passengers train-set can transport in station B can be written as:  
\[C_{B}=C\times(1-F_{A-B}) \tag{15}\]  
Therefore, we use the following method to calculate the minimum quantity of train service for an intermediate station \(k\) in section \(j\):  
\[S_{k}=\frac{Q_{k}}{C_{k}}=\frac{Q_{k}}{C\times(1-F_{k})} \tag{16}\]  
\(F_{k}\) means the average full load ratio of a train at section \(k\), which is connected with station \(k\). The full load ratio of trains is different in different directions at the same station.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Subscripts** & **Description** \\ \hline \(S_{k}\) & The minimum quantity of train service at station \(k\). \\ \(Q_{k}\) & The number of passengers that station \(k\) served in one day. \\ \(Q^{L}\) & The number of passengers on the railway line that we are concerned about. \\ \(Q^{T}\) & The total amount of passengers served on all the railway lines passing through station \(k\) in one day. \\ \(C\) & The average capacity of a train. \\ \(O^{J,z}\) & Whether the \(z-th\) scheme in type \(l\) is operated. \\ \(V_{l}^{L,z}\) & Whether the \(z-th\) scheme in type \(l\) stops at station \(k\). \\ \(V_{l}\) & The maximum number of stops of scheme in type \(l\). \\ \(V_{k}\) & The minimum number of train service at station \(k\). \\ \(U^{l}\) & The maximum number of schemes of type \(l\). \\ \hline \hline \end{tabular}
\end{table}
Table 4: Description of the parameters.  
Figure 8: The corresponding relationship of full load ratio and adjacent stations.  
There are some stations that serve several railway lines, which are called hub stations. For these stations, the calculation method for the minimum quantity of train-sets has two types similarly. For hub stations at both ends of the railway, we write the formula as:  
\[S_{k}=\frac{Q_{k}\times\frac{Q^{L}}{Q^{T}}}{C} \tag{17}\]  
And for other hub stations, we write the formula as:  
\[S_{k}=\frac{Q_{k}\times\frac{Q^{L}}{Q^{T}}}{C\times(1-F_{k})} \tag{18}\]  
\(Q^{T}\) means the total amount of passengers served on all the railway lines passing through station \(k\) in one day. \(Q^{L}\) means the number of passengers on the railway line we are concerned about.