## 3 Methodology
### Resilience of Air Flights
#### 3.1.2 Weights of Airport Routes  
The weights of each route for an airport hub are determined based on the average daily flight volume, route length, and the reduction in airport capacity due to the absence of a particular route.  
(1) Average Daily Flight Volume  
The average daily flight volume for each route is calculated based on an airport's daily operating time of 16 h, as shown in Equation (5):  
\[ADFV_{k}=Hourly\ Flight\ Capacity_{k}\times 16 \tag{5}\]  
(2) Flight Distance  
The flight distance from the starting node to the destination node can be obtained from actual flight data.  
(3) Flow Stability  
The flow stability of each route \(RAI^{k}\) considers the impact of a reduced route capacity on the maximum airport traffic flow, as expressed in Equation (6).  
\[RAI^{k}=\frac{\int_{u_{0}}^{u_{T}}(F_{\text{max}}^{k}(u_{0})-F_{\text{max}}^{ k}(u))du}{\int_{u_{0}}^{u_{T}}F_{\text{max}}^{k}(u_{0})du} \tag{6}\]  
Based on the definition of robustness, RAI evaluates and quantifies the impact of localized (node-level) capacity reductions on the maximum network traffic flow [35]. InEquation (6), \(u\) is the degradation parameter which represents the potential reduction in route capacity. \(F_{\max}^{k}(u)\) is the maximum network traffic flow of an airport when the capacity of route \(k\) reduces \(u\). Finding the maximum network traffic flow for a given degradation is equivalent to solving a linear optimization problem. Let the capacity of route \(k\) be \(c_{k}\), and the objective function for the airport's maximum traffic flow is given by the following:  
\[\max\sum\
olimits,flight_{n} \tag{7}\]  
Constraints:  
Traffic volume of each route  
The volume of each route must be less than or equal to the route capacity:  
\[\sum\limits_{n\in N}\delta_{nk}flight_{n}\leq c_{k} \tag{8}\]  
Non-negativity  
The traffic volume of flights on each route must be greater than or equal to 0:  
\[\sum\limits_{n\in N}\delta_{nk}flight_{n}\geq 0 \tag{9}\]  
where \(flight_{n}\) represents the \(n\)th flight of the airport on one operating day.  
\(\delta_{nk}\) is a 0-1 binary variable used to determine whether a flight belongs to route \(k\), which is expressed in Equation (10):  
\[\delta_{nk}=\begin{cases}0,n\
otin k,\\ 1,n\in k.\end{cases} \tag{10}\]  
Since the Grey Relational Analysis (GRA) method is used to analyze the degree of correlation between factors in a system, it is particularly suitable for analyzing small samples and uncertain systems. Therefore, in this study, the GRA method is used to determine the total weight of each route. After normalizing the data, the reference sequence (ideal value) is set to 1, and the absolute difference between the reference sequence and the comparison sequence is calculated, as shown in Equation (11):  
\[\Delta_{kj}=\left|1-X_{kj}\right|,\,j=1,2,3 \tag{11}\]  
The correlation coefficient is calculated according to the formula for the Grey Relational Degree, as shown in Equation (12):  
\[\varepsilon_{kj}=\frac{min_{k}min_{j}\Delta_{kj}+\rho max_{k}max_{j}\Delta_{ kj}}{\Delta_{kj}+\rho max_{k}max_{j}\Delta_{kj}},\,\,j=1,2,3 \tag{12}\]  
where \(\rho\) is the distinguishing coefficient, generally taken as 0.5. The correlation coefficients are then weighted and summed to obtain the Grey Relational Degree of each comparison sequence, as shown in Equation (13):  
\[\gamma_{k}=\frac{1}{3}\sum\limits_{j=1}^{3}\varepsilon_{kj} \tag{13}\]