## 2 Optimisation Modelling of Gate Assignments
### Determine the Objective Function  
Airports, passengers, and airlines all have a stake in the allocation of airline slots. The preassignment of airline slots should take into account flight delays, weather effects, and other disruptions from the perspective of airport operations management. To improve the robustness of the preallocation, the gate assignment model must be solved with the ability to handle dynamic changes, minimise the number of idle slots, and minimise the number of conflicts between real-time operating flights. As a result, minimising real-time flight conflicts is chosen as an optimisation goal. Because the number of airport boarding bridges is limited, as the number of flights increases, some flights will be assigned to more distant aprons. To access the boarding area, passengers must take a bus. This causes inconvenience to passengers while also increasing the workload for dispatchers. As a consequence, optimising flight occupancy will reduce wasted boarding bridge slots and improve passenger satisfaction. Furthermore, due to the promotion of \"green transportation\" and the ongoing rise in international oil prices, airlines have faced pressures due to fuel consumption costs in recent years. The cost of aircraft fuel accounts for approximately 15.5 percent of all airline operating costs [28]. As a result, reducing fuel consumption and lowering transportation costs have been critical issues for airlines to address. The choice of gate directly affects the taxiing costs of the aircraft from the landing runway to the gate in the gate assignment problem so minimising fuel consumption is used as an optimisation objective. At the same time, although the aircraft typically wants to arrive at the assigned gate using the least amount of fuel, the gate reached at this point may be a more distant gate. As a result, the overall goal is to maximise the boarding bridge rate while minimising aircraft taxiing fuel consumption. In summary, the detailed description of the identified optimisation objectives is as follows. 1. Minimise real-time flight conflicts  
In this paper, the multiobjective function optimisation uses the linear weighting method. Minimising the number of real-time running flight conflicts during gate preallocation can improve the flight punctuality rate. The mathematical model of the conflicting objective function is shown below. \[F_{1}=\min(\sum\limits_{k\in N}\sum\limits_{u\in M}x_{ku}+\sum\limits_{k\in N }\sum\limits_{u\in M}\psi_{ku}) \tag{1}\]  
where \(N\) is the set of flights, \(M\) is the set of aircraft slots, and \(x_{ku}\) is a variable taken as 0 or 1. \(x_{ku}=1\) if flight \(k\) is assigned to slot \(u\) and \(x_{ku}=0\) otherwise. \(\psi_{ku}\) is the penalty factor when a flight is assigned to a distant slot and is 0 if assigned to a near slot and 1 when assigned to a distant slot. The robustness of the preassignment scheme can be improved by minimising the number of conflicts in real-time flights, which can equalise the efficiency of the utilisation of aircraft resources, and by improving the normalisation rate of flight releases in terms of minimising idle time. The minimisation of the slot idle time objective function is shown as follows. \[\overline{t}_{k}=\frac{\sum\limits_{k=1}^{N}t_{k}}{m} \tag{2}\]  
\[\min F_{2}=\frac{\sum\limits_{k=1}^{N}\left(t_{k}-\overline{t}_{k}\right)^{2} }{m} \tag{3}\]  
where \(t_{k}\) denotes the slot idle time between flight \(k\) and flight \(k-1\). The dataset in this paper contains \(m\) flights and assuming that the slot is closed after the arrival of the last flight, there are a total of \(m\) slot idle times. 2. Maximise the boarding bridge rate  
The objective function for maximising the number of flight stops at the boarding bridge is shown below. \[F_{3}=\max\sum\limits_{k\in N}\sum\limits_{u\in M}x_{ku}p_{u} \tag{4}\]  
where \(p_{u}\) is a variable that takes the value of 0 or 1. It takes the value of 1 when the flight is assigned to a near gate and 0 when it is assigned to a far gate. 3.