# A Two-Stage Optimization Model for Airport Stand Allocation and Ground Support Vehicle Scheduling
## 4 Modeling
### Multi-Type Ground Support Vehicle Scheduling Model
#### 4.3.3 Constraints of the Second-Stage Model  
The constraints of the second-stage model include both the common vehicle constraints and the coordination constraints between different vehicle types.  
\[\sum_{n\in N_{v}}\psi_{in}^{v}=1,\forall i,j\in F^{v},\forall v\in V \tag{15}\]  
\[\sum_{i\in F^{v}}\varphi_{ijn}^{v}-\sum_{k\in F^{v}}\varphi_{ikn}^{v}=1,\forall j \in F^{v},\forall v\in V,\forall n\in N_{v} \tag{16}\]  
\[\sum_{n\in N_{v}}\varphi_{n}^{v}\leq N_{v},\forall v\in V \tag{17}\]  
\[ST_{in}^{v}+T^{v}+\frac{s_{ij}^{v}}{c}-M\Big{(}1-\varphi_{ijn}^{v}\Big{)} \leq ST_{jn}^{v},\forall i,j\in F^{v},\forall n\in N_{v},\forall v\in V \tag{18}\]  
\[RT_{i}^{v}\psi_{in}^{v}\leq ST_{in}^{v}\psi_{in}^{v}\leq DT_{i}^{v}\psi_{in}^ {v},\forall i\in F^{v},\forall n\in N_{v},\forall v\in V \tag{19}\]  
\[AT_{in}^{1}\leq AT_{in}^{2},\forall i\in F^{v},\forall n\in N_{v} \tag{20}\]  
\[ST_{in}^{3}\geq ST_{in}^{5}+T^{5},\;\forall i\in F^{a},\forall n\in N_{v} \tag{21}\]  
\[ST_{in}^{5}\geq ST_{in}^{3}+T^{3},\;\forall i\in F^{a},\forall n\in N_{v} \tag{22}\]  
\[0\leq C_{in}\leq C^{4} \tag{23}\]  
\[C_{jn}\geq C_{in}-M\Big{(}1-\varphi_{ijn}^{v}+\varphi_{n}^{v}\Big{)}-d_{j}, \forall i,j\in F^{v},\forall n\in N_{v} \tag{24}\]  
\[AT_{in}^{6}\geq AT_{in}^{5},\forall i\in F^{v},\forall n\in N_{v} \tag{25}\]  
\[ST_{in}^{5}=ST_{in}^{6}=\left\{\begin{matrix}AT_{in}^{6}&AT_{in}^{6}\geq RT_{ i}^{v}\\ RT_{i}^{v}&AT_{in}^{6}<RT_{i}^{v}\end{matrix}\right. \tag{26}\]  
Equation (15) guarantees that each flight must be served by one vehicle for each type of service. Equation (16) guarantees that each flight has a predecessor and a successor in the sequence served by one vehicle. Equation (17) limits the available number of vehicles. Equation (18) is the serving sequence constraint on two adjacent flights served by one vehicle. Equation (19) means that the time a vehicle begins servicing a flight must be within the flight's service time window. Equation (20) is the collaborative constraint between ferry vehicles and passenger stairs, indicating that the passenger stairs must arrive before the ferry vehicles. Equations (21) and (22) are the collaborative constraints between the ferry vehicle and the fuel truck. For arriving flights, the fuel truck must begin its service after the ferry vehicle has completed its task. For departing flights, the fuel truck must finish its service before the ferry vehicle starts its operation. Equation (23) constrains the water is restricted to be no more than \(C^{4}\). Equation (24) constrains that if the water truck does not return to the resource supply point to refill after serving the flights \(i\), the water remaining in that vehicle needs to meet the restriction. Equation (25) constrains that if the water truck returns to the resource supply point to refill after serving the flights \(i\), the water fluid remaining in that vehicle needs to meet the restriction. Equations (26) and (27) are the collaborative constraints between the baggage belt loader and the baggage tractor,indicating that the baggage belt loader must arrive before the baggage tractor, and service starts after both the baggage belt loader and the baggage tractor have arrived.