## 3 Primary Train-Set Circulation Plan
### Profit Function  
The operating profit of the train-set is related to the full load ratio. The full load ratio of section \(j\) can be calculated as:  
\[F_{j}=\frac{Q_{j}}{C_{j}} \tag{3}\]  
where \(Q_{j}\) means the quantity of passengers of section \(j\), and \(C_{j}\) means the sum of the capacity of all train-sets of section \(j\). The full load ratio of a section can be calculated on the basis of historical passenger transport data. The passenger travel demands of different sections and times are different. Therefore, the full load ratio of the train-sets operating in these sections will also be different. By comparing the full load ratio of train-sets, we can observe the passenger demands in this section. At the same time, the full load ratio of the train-set directly reflects the profits of the train-set. The high load ratio of the train-set shows that it has a high income. More train-sets are needed for the sections with high travel demand for passengers. For the section with less travel demand, train-sets can be operated less appropriately. Therefore, the profits of train-set operation come from the quantity of passengers. In this paper, we evaluate the profits based on the full load ratio of train-sets in the section. It represents the profits generated by a train-set passing through a section. This benefit is used as a reference when calculating the stop position of the train-set under the condition of satisfying the state constraint at the end of the transition time. The profit is related to the train-set operating direction. The train-set operating direction of stations at both ends of the HSR line is fixed. Moreover, there are two directions for train-sets at intermediate stations. Figure 5 shows the train-set direction of the stations at both ends and the train-set direction of the intermediate station. \begin{table}
\begin{tabular}{c c} \hline \hline
**Subscripts** & **Description** \\ \hline \(i\) & Train-set \(i\in I\),where \(I\) is the set of train-sets, \(|I|=n\). \\ \(j\) & Section \(j\in J\), where \(J\) is the set of sections, \(|J|=m\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General subscripts. \begin{table}
\begin{tabular}{c c} \hline \hline
**Subscripts** & **Description** \\ \hline \(p_{j}^{l}\) & The profit of train-set \(i\) when train-set \(i\) passing section \(j\). \\ \(C^{i}\) & The capacity of train-set \(i\). \\ \(C_{j}\) & The total capacity of all train-sets in section \(j\). \\ \(D_{i}\) & The depart time of train-set \(i\). \\ \(E_{j}\) & The ticket price of section \(j\). \\ \(Q_{j}\) & The quantity of passengers in section \(j\). \\ \(F_{j}\) & The average full load ratio of section \(j\). \\ \(S_{j}^{l}\) & Whether train-set \(i\) passing section \(j\). \\ \(T_{s}\) & The starting time of the operation hours. \\ \(T_{e}\) & The end time of the operation hours. \\ \(T_{j}^{l}\) & The running time of train-set \(i\) in section \(j\). \\ \(T_{d}\) & The departure interval \\ \(u_{i}\) & Whether train-set \(i\) goes in the direction \(u\). \\ \(d_{i}\) & Whether train-set \(i\) goes in the direction \(d\). As shown in Figure 5, train-set 1 can only perform the operation line in the A-G direction at station A, like the blue line. The same train-set 3 can only perform the G-A direction at G station, like the red line. There are two options for train-set 2 departing from station C. One is to perform the C-A direction, like the green line. Moreover, the other is to perform the direction of C-G, like the yellow line. Thus, we sum the profit of each section to evaluate the operating profit. The profit function \(P(i)\) for train-set \(i\) is as follows:  
\[P(i)=\sum_{j}\Big{(}P_{j}^{i}\times S_{j}^{i}\Big{)} \tag{4}\]  
where \(S_{j}^{i}\) means that whether train-set \(i\) passing section \(j\), \(P_{j}^{i}\) means the profit of train-set \(i\) pass through section \(j\).