# A Two-Stage Optimization Model for Airport Stand Allocation and Ground Support Vehicle Scheduling
## 4 Modeling
### The Stand Allocation Problem Considering Turnaround Resource Constraints
#### 4.2.1 Objectives of the First-Stage Model  
1. \(\quad\) Minimize the Taxi Fuel Consumption for Arriving and Departing Flights. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Notation** & **Meaning** \\ \hline \multirow{3}{*}{sets} & \(R\) & the set of runways: 0 represents Runway 15, and 1 represents Runway 16. \\ \cline{2-3}  & \(F\) & the set of flights \\ \cline{2-3}  & \(G\) & the set of parking positions \\ \hline \multirow{3}{*}{parameters} & \(t^{d}_{r_{R}g}\) & the taxing time from runway \(r_{m}\) to parking position \(g\) of departure flight \\ \cline{2-3}  & \(t^{d}_{g\tau_{r}}\) & the taxing time from runway \(r_{n}\) to parking position \(g\) of arrival flight \\ \cline{2-3}  & \(S^{i}_{i}\) & the aircraft size of flight \(i\), \(S^{f}_{i}\in\{1,2,3\}\), corresponding to aircraft types C, D, and E. \\ \cline{2-3}  & \(v_{g}\) & the time taken for the ferry vehicles to travel back and forth between the parking position \(g\) and the terminal. \\ \cline{2-3}  & \(T^{i}\) & The turnaround time of flight \(i\) \\ \hline \multirow{3}{*}{variables} & \(Sa_{i}\) & The arrival time of flight \(i\) \\ \cline{2-3}  & \(\delta^{g}\) & the minimum safety interval for the stand \(g\) \\ \cline{2-3}  & \(\theta^{t}\) & the service interval of the towing tractor \\ \cline{2-3}  & \(Sd_{i}\) & The departure time of flight \(i\) \\ \hline \multirow{3}{*}{variables} & \(x_{ig}\) & binary decision variable, equal to 1 if flight \(i\) is assigned to parking position \(g\), otherwise equal to 0 \\ \cline{2-3}  & \(y_{igg}\) & binary decision variable, equalto1 if flight \(j\) is assigned to parking position \(g\) immediately after flight \(i\), \\ \cline{2-3}  & \(y_{igg}\) & otherwise equal to 0 \\ \cline{2-3}  & \(z_{jit}\) & binary decision variable, equal to 1 if both flight \(i\) and flight \(j\) are assigned to apron \(t\), otherwise equal to 0 \\ \cline{2-3}  & \(l^{2}_{ij}\) & the time interval between the departure time of flight \(i\) from the parking position and the arrival time \\ \cline{2-3}  & \(x_{ig}\) & binary decision variable, equalto1 if flight \(i\) is assigned to parking position \(g\), otherwise equal to 0 \\ \cline{2-3}  & \(y_{igg}\) & binary decision variable, equalto1 if flight \(j\) is assigned to parking position \(g\) immediately after flight \(i\), \\ \cline{2-3}  & \(z_{jit}\) & binary decision variable, equal to 1 if both flight \(i\) and flight \(j\) are assigned to apron \(t\), otherwise equal to 0 \\ \cline{2-3}  & \(z_{jit}\) & the time interval between the departure time of flight \(i\) from the parking position and the arrival time \\ \cline{2-3}  & \(l^{2}_{ij}\) & of flight \(j\) at the parking position \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sets, parameters, and variables of the stand allocation model. Figure 3: Two-Stage optimization model for stand allocation and ground support vehicle scheduling. The objective function is shown in Equation (1). When flights are assigned to different stands, the taxi-in and taxi-out time required vary. Excessive taxi time leads to increased fuel consumption, decreased passenger satisfaction, significantly higher operating costs for airlines, and intensified environmental pollution caused by fuel emissions. Therefore, optimizing taxi fuel consumption is crucial to achieving a balance between corporate profitability, operational efficiency, and social responsibility. Aircraft taxi fuel consumption is primarily influenced by the aircraft type and taxi time. [PERSON] [35] analyzed the average taxi fuel consumption of different aircraft types in his study. This paper focuses on C, D, and E class aircraft, and the relevant data are summarized in Table 2.  
\[min\ Z_{1}=\sum_{i\in F}\sum_{g\in G}x_{ig}\left(t_{m\,g}^{a}+t_{g\,n}^{d}\right) \tag{1}\]  
Minimizing the Ferry Vehicles Driving Time for Remote Stands. The objective function is shown in Equation (2). During peak hours, due to the limited availability of contact stands, there are more flights than these stands can accommodate.