# Slot Optimization Based on Coupled Airspace Capacity of Multi-Airport System
## 3 Multi-Airport System for Cooperative Optimization of Flight Schedules
### Multi-Airport System for Cooperative Optimization of Flight Schedules
#### 3.3.2 Objective
Each airport's unit flight deviation, as shown in Equation (3), is determined by dividing each airport's deviation by the number of flights and determining fairness by the variance of the airports within the multi-airport system, as shown in Equation (4). \[a_{u}=\frac{\sum\limits_{m\in M}\sum\limits_{t\in T}\left(t_{m_{0}}-t_{m} \right)\cdot X_{m}^{t}}{C_{u}} \tag{2}\]  
\[\mathrm{N}=\frac{\sum\limits_{u}\mathbf{a}_{u}}{|\mathrm{U}|} \tag{3}\]  
\[Z_{2}=min\frac{\sum\limits_{u\in(0,n)}\left(N-a_{u}\right)^{2}}{|U|} \tag{4}\]  
(3) Fairness of airlines' consideration of congestion contributions  
It is obviously unfair for airlines with more off-peak time resources to bear a larger slot deviation in multi-airport system airport slot optimization. Therefore, in the multi-airport system optimization process, the fairness of airlines is reflected by the congestion contribution of each airline, i.e., adding the dimension of \"congestion\" to \"contribution\" to define the fairness of considering congestion. Under this rule, airlines with more flights during congested hours will generate more time deviations than stipulated by the original fairness indicator. Therefore, the objective function of the fairness of airlines considering congestion contribution is designed to assess the equilibrium of airport slot allocation by calculating the difference between the fairness index of the airlines and the overall fairness index of the industry. The goal is to minimize the difference between the two, in order to achieve the highest standard of fairness. The fairness objective function expression is shown in Equation (5). \[Z_{3}=min\left\{max\left|\mu_{i}-\frac{\sum\limits_{i\in I}\mu_{i}}{|I|}\right|\right\} \tag{5}\]  
\[\mu_{i}=\frac{\sum\limits_{m\in M_{i}}\sum\limits_{m\in T}|t_{m_{0}-t_{m}}|X_{m }^{t}}{\sum\limits_{m\in M_{i}}\sum\limits_{m\in T}\sum\limits_{m\in T}|t_{m_{0} -t_{m}}|X_{m}^{t}} \tag{6}\]  
where \(\mu_{i}\) indicates the fairness of airline \(i\), with the numerator being the proportion of all slot deviations of airline i to the total slot deviations, as shown in Equation (6). \(v_{m}^{t}\) in the denominator is a known value of 0-1; \(v_{m}^{t}=1\) indicates that time period \(t\) during which flight \(m\) occurs is congested (demand is greater than capacity); the flight \"contributed\" to the congestion at time t; otherwise, it is 0. A congested slot is defined as a period when the demand by flights exceeds the handling capacity of the airport. The denominator is the two components, congested and non-congested slots, and the share of airlines in each of these components is calculated and subsequently weighted and summed as a way of adjusting the weights of the slot deviations. For the value of \(\mu_{i}\), the discussion is illustrated in three cases:\(\mu_{i}=1\): Treatment received by airline \(i\) is considered fair;  
\(\mu_{i}<1\): Airline \(i\) enjoys preferential treatment;  
\(\mu_{i}>1\): Airline \(i\) suffers from unfair treatment.