# An Integrated Optimization Method of High-Speed Railway Rescheduling Problem at the Network Level
## 3 Multi-Objective Integrated Optimization Model
### Objectives
#### 3.3.3 Constraints of Passenger Demand  
We stipulate that the affected passenger demands can be served by affected trains and the standby train sets, while they cannot be served by normal trains.  
The number of passengers in each section of the train set or standby train set cannot be more than the seating capacity after serving affected passengers, as Constraint (28) shows.  
\[\left\{\begin{array}{l}\sum_{c\in C^{aff}}\sum_{t\in T^{aff}}\sum_{x\in X^{ diff}}tr^{c,t,x}_{c^{\prime},t^{\prime}}\times PN^{c,t,x}\times PBS^{c,t,x}_{j, 1+1}\leq SC_{i}\times\alpha\\ \sum_{c\in C^{aff}}\sum_{t\in T^{aff}}\sum_{x\in X^{aff}}tr^{c,t,x}_{b}\times PN ^{c,t,x}\times PBS^{c,t,x}_{j,1+1}\leq SC_{b}\times\alpha\\ \forall c^{\prime}\in C^{aff},t^{\prime}\in T^{aff},\forall i\in I^{aff},\forall b \in B,\forall j,j+1\in J\end{array}\right. \tag{28}\]  
In constraint (28), \(PBS^{c,t,x}_{j,j+1}\) indicates whether the passenger demand \(x\) of train \(t\) in circulation \(c\) passes through section \(j,j+1\), \(PBS^{c,t,x}_{j,j+1}=1\) represents that passenger demand \(x\) passes through section \(j,j+1\); otherwise, \(PBS^{c,t,x}_{j,j+1}=0\). \(SC_{b}\) is the seating capacity of standby train set \(b\). \(SC_{i}\) is the seating capacity of train set \(i\).  
Constraint (29) ensures that the departure time of the train set or standby train set at the station cannot be earlier than the scheduled departure time of the passenger demand it serves.  
\[\left\{\begin{array}{l}PDDT^{c,t,x}_{DPS^{c,t,x}}-PDT^{c,t}_{DPS^{c,t,x}} \leq(1-tr^{c,t,x}_{c^{\prime},t^{\prime}})\times M\\ PDDT^{c,t,x}_{DPS^{c,t,x}}-PDT^{c,t}_{DPS^{c,t,x}}\leq(1-tr^{c,t,x}_{b})\times M \\ \forall c,c^{\prime}\in C,\forall t,t^{\prime}\in T,\forall b\in B,\forall x \in X,\forall j\in J\end{array}\right. \tag{29}\]  
Constraints (30) and (31) express that if one train serves the passenger demand at one station, the train must stop at that station and must stop at the destination station of the passenger demand.  
\[\left\{\begin{array}{l}aws^{c,t}_{DPS^{c,t,x}}\geq tr^{c,t,x}_{c^{\prime},t }\\aws^{c,t,x}_{DPS^{c,t,x}}\geq tr^{c,t,x}_{b}\end{array}\right.,\forall c,c^{ \prime}\in C^{aff},\forall t,t^{\prime}\in T^{aff},\forall b\in B,\forall x\in X ^{aff},\forall j\in J \tag{30}\]  
\[\left\{\begin{array}{l}aws^{c,t}_{APS^{c,t,x}}\geq tr^{c,t,x}_{c^{\prime},t }\\aws^{c,t,x}_{APS^{c,t,x}}\geq tr^{c,t,x}_{b}\end{array}\right.,\forall c,c^{ \prime}\in C^{aff},\forall t,t^{\prime}\in T^{aff},\forall b\in B,\forall x\in X ^{aff},\forall j\in J \tag{31}\]  
In Constraints (29)-(31), \(PDDT^{c,t,x}_{j}\) represents the departure time for passenger demand \(x\) of train \(t\) in circulation \(c\) at station \(j\). \(DPS^{c,t,x}_{j,j+1}\) is the departure station for passenger demand \(x\) of train \(t\) in circulation \(c\). \(APS^{c,t,x}\) is the arrival station for passenger demand \(x\) of train \(t\) in circulation \(c\). \(\alpha\) is the acceptable maximum coefficient of overcrowding.  
The parametric of the model are valued as follows Table 4: