## 4 Model Formulation
### Upper Model Construction  
#### 4.2.1 Objective Functions  
To minimize passenger travel time and enterprise operation cost, the upper model adopts the departure frequency of the train and the location of the turn-back station of the short-turn route as the decision variables. Both \(T_{w}\) and \(L_{s}\) are minimized. 1. Calculation of travel cost of passengers \(T_{w}\)  
As passengers are insensitive to time when they are on the train, this paper takes the waiting time as the travel cost of passengers. Due to the difference in the section of the departure station and the terminal station of the passenger travel, the types of trains that passengers can take are also different. Therefore, passengers can be classified according to the type of train they take, and then the waiting time for each type of passenger is calculated. This paper mainly classifies passengers in the up direction, as shown in Figure 2, which can be divided into 6 kinds. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \(q_{ij}\) & The passenger flow from station \(i\) to station \(j\) \\ \(T_{k}\) & Calculation period (1 h) \\ \(L_{1N}\) & Total length of the line \\ \(Lab\) & Length of short-turn route \\ \(m,n\) & A random positive integer \\ \(f_{0}\) & Minimum departure frequency \\ \(f_{m}\) & Maximum departure frequency \\ \(Z\) & Number of vehicles used \\ \(Z_{0}\) & Number of vehicles used in a single train operation plan \\ \(R_{i}\) & Operation time of train in section \(i\) \\ \(S_{j}\) & Stop time of the train at station \(j\) \\ \(t_{z}\) & Turn-back time of trains \\ \(\lceil\ \rceil\) & Round up \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definition of upper-layer model parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \(q_{ij}\) & The passenger flow from station \(i\) to station \(j\) \\ \(\beta_{1}\) & Proportion with the second category of passengers taking the full-length train \\ \(\beta_{2}\) & Proportion with the second category of passengers taking the short-turn train \\ \(\beta_{1}^{\prime}\) & Proportion with the third category of passengers taking the full-length train \\ \(\beta_{2}^{\prime}\) & Proportion with the third category of passengers taking the short-turn train \\ \(p\) & Probability of passengers choosing not to get on the short-turn train \\ \(n_{0}\) & Maximum number of marshalled vehicles \\ \(\gamma_{\max}^{\prime}\) & Maximum load factor of the trains \\ \(\gamma_{\min}\) & Lower limit with load factor of the trains \\ \(\gamma_{\max}\) & Upper limit with load factor of the trains \\ \(\mathbb{C}\) & Capacity of each train \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definition of the lower model parameters. 1. The first kind of passenger The starting station and terminal station of the first kind of passenger are in section I. This kind of passenger can only take the full-length train. 2. The second kind of passenger The starting station of the second kind of passenger is in section I and their terminal station is in section II. This kind of passenger can only take the full-length train. 3. The third kind of passenger The starting station of the third kind of passenger is in section I and their terminal station is in section III. They can only take the full-length train. 4. The fourth kind of passenger The starting station and terminal station of the fourth kind of passenger are in section II. This kind of passenger can take the full-length train or short-turn train. 5. The fifth kind of passenger The starting station of the fifth kind of passenger is in section II and terminal station of this kind of passenger is in section III. They can take the full-length train to the terminal station directly, or take the short-turn train first and then change to take the full-length train. 6. The sixth kind of passenger The starting station and terminal station of the sixth kind of passenger are in section III. They can only take the full-length train.