## 4 Model Formulation
### Lower Model Construction  
#### 4.3.1 Objective Functions  
The lower model aims to optimize the equilibrium of load factor between full-length short-turn train for the trains running on short-turn routes. The lower model takes the number of marshalled trains as the decision variable.  
Since the objective function of this model considers the load factor equilibrium of trains running on short-turn routes, it mainly considers passengers who will pass through short routes, including second, third, fourth, and fifth kinds of passenger. This paper regards passengers who can only take the full-length train as the first category of passenger, including the second and third kinds of passenger. Passengers who can take the full-length train or short-turn train are regarded as the second category of passenger, including the fourth kind of passenger. The fifth kind of passenger, who can either 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, is regarded as the third category of passenger.  
Passenger flow of the first category of passenger can be calculated using Equations (26)-(28):  
\[M_{1}=M_{1}^{+}+M_{1}^{-}\,, \tag{26}\]  
\[M_{1}^{+}=\sum_{i=1}^{a-1}\sum_{j=a}^{N}q_{ij}, \tag{27}\]  
\[M_{1}^{-}=\sum_{i=b+1}^{N}\sum_{j=1}^{b}q_{ij}. \tag{28}\]  
\(M_{1}^{+}\) and \(M_{1}^{-}\) are the passenger flow of the first category of passenger in the up direction and down direction.  
Passenger flow of the second category of passenger can be calculated using Equations (29)-(31):  
\[M_{2}=M_{2}^{+}+M_{2}^{-}\,, \tag{29}\]  
\[M_{2}^{+}=\sum_{i=a}^{b-1}\sum_{j=i+1}^{b}q_{ij}, \tag{30}\]  
\[M_{2}^{-}=\sum_{i=a+1}^{b}\sum_{j=a}^{i-1}q_{ij}. \tag{31}\]  
\(M_{2}^{+}\) and \(M_{2}^{-}\) are the passenger flow of the second category of passenger in the up direction and down direction.  
Passenger flow of the third second category of passenger can be calculated using Equations (32)-(34):  
\[M_{3}=M_{3}^{+}+M_{3}^{-}\,, \tag{32}\]  
\[M_{3}^{+}=\sum_{i=a}^{b-1}\sum_{j=b+1}^{N}q_{ij}, \tag{33}\]  
\[M_{3}^{-}=\sum_{i=a+1}^{b}\sum_{j=1}^{a-1}q_{ij}. \tag{34}\]  
\(M_{3}^{+}\) and \(M_{3}^{-}\) are the passenger flow of the third category of passenger in the up direction and down direction.  
For the second category of passenger, the probability of taking a full-length train or short-turn train is equal to the proportion of the train departure frequency of full-length trains and short-turn trains.  
The probability of the second category of passenger taking the full-length train can be obtained by the following formula:  
\[\beta_{1}=f_{1}/(f_{1}+f_{2}). \tag{35}\]  
Probability of the second category of passenger taking the short-turn train can be obtained by the following formula:  
\[\beta_{2}=1-\beta_{1}. \tag{36}\]  
Passengers in the third category who choose to take a short-turn train first have to transfer halfway to a full-length train. As a result, some passengers prefer not to board the train and instead continue to wait for the full-length train when short-turn trains arrive. In this paper, it is assumed that the probability of passengers choosing not to get on the short-turn train is \(p\), which is taken as 0.2.  
Probability of the third category of passenger taking the full-length train can be calculated as follows:  
\[\beta_{1}^{\prime}=\beta_{1}+\beta_{2}\cdot p. \tag{37}\]  
Probability of the third category of passenger taking the short-turn train can be calculated as follows:  
\[\beta_{2}^{\prime}=\beta_{2}\cdot(1-p). \tag{38}\]  
The equilibrium of load factor between the full-length train and short-turn train can be calculated as follows:  
\[\varphi=\bigg{(}\frac{M_{1}+\beta_{1}\cdot M_{2}+\beta_{1}^{\prime}\cdot M_{3} }{2\cdot f_{1}\cdot(n_{1}+n_{2})\cdot C}-\frac{\beta_{2}\cdot M_{2}+\beta_{2}^ {\prime}\cdot M_{3}}{2\cdot f_{2}\cdot n_{2}\cdot C}\bigg{)}^{2}. \tag{39}\]  
The objective function of the lower function is designed as follows:  
\[\min W_{3}=\varphi. \tag{40}\]