## 3 Modeling of Apron Support Vehicle Operation Scheduling Problem  
### Problem Description  
The actual apron support vehicle operation is very complicated, involving a lot of support equipment and support personnel. For better understanding of the problem, the apron support vehicle operation is simplified into 7 stages, as shown in Figure 1.  
The scheduling problem of apron support vehicles is that \(m\) types of support vehicles are used to serve \(n\) aircraft. Each aircraft has to go through P support processes, and each support process requires a certain operating time. At the same time, each support vehicle can only support one aircraft, and one operation of each aircraft can only be completed by one support vehicle. The sequence of support operations for each aircraft and the support process of the support vehicle is random. The purpose of the support operation scheduling is to determine the process combination of all aircraft and support vehicles under the condition that a certain number of support vehicles are satisfied, so as to minimize the total support operation time. According to the actual situation of the apron support vehicle operation, the following assumptions are made for the scheduling problem of the apron support vehicle operation:  
1. The travel time of the support vehicle is ignored, and only the support operation time is calculated;
2. The problems of aircraft towing, bomb hanging, calibration of inertial navigation, and inspection and acceptance are free from resource constraints;
3. Each support operation of the aircraft requires only one type of support vehicle for one service;
4. For a certain type of aircraft, the operating hours of the guaranteed vehicles are the same;
5. The contents of support operations for all aircraft are independent of each other, and there is no mutual restraint between aircraft;
6. A guaranteed vehicle can only serve one aircraft at the same time;
7. Each aircraft must complete all support operations;
8. The support vehicle is allowed to be idle and the aircraft to wait for the support vehicle;
9. Once the vehicle starts, the support operation to a certain aircraft starts, and there is no interruption until the service support operation is completed;
10. The resources carried by each support vehicle are sufficient or can be replenished in time, without affecting the aircraft support mission. \begin{table}
\begin{tabular}{c c} \hline \hline
**Specialty** & **Main Work** \\ \hline Mechanical specialty & Responsible for aircraft structure, engine, operating system, and so on. Complete the filling of oil, hydraulic oil and lubricating oil, nitrogen, etc. with the assistance of the refueling truck, air conditioner, and other supporting vehicles. \\ \hline Avionics specialty & Responsible for the aircraft control system, fire control system, communication and navigation equipment, etc. Complete the inspection and collation with the assistance of power vehicles. \\ \hline Ordnance specialty & Responsible for aircraft’s seats, pylons, aircraft guns, etc. Complete the loading and unloading of weapons with the assistance of the loader of air-launched weapons. \\ \hline Ad hoc specialty & Responsible for cockpit electronics, instruments, special equipment, and so on with the assistance of power vans and oxygen filling vehicles. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main content of each major of the crew. In this paper, the guarantee processes such as refueling, charging, and oxygenation are selected as the research objects. Before the start of the support task, all support vehicles should prepare for the approach and resources in advance. In comparison with the support time, the scheduling time of support vehicles among support aircraft can be neglected. Simultaneously, the same support operation time of the same model has a small difference, so it can be regarded as equal.