## 4 Case Study
### Numerical Application of Route Resilience  
In this numerical example, the raw data for each fight are planned departure time, planned arrival time, actual departure time, and actual arrival time. The time window of flight resilience \(\Delta t\) is 15 min. Step 1. Calculation of KPIs  
In Step 1, the route data between Xi'an Xianyang Airport and Changsha ranging from 15:00 to 16:00 3 March 2023 are chosen to quantify the flight resilience, which contains four time windows \(\Delta t_{i},i=1,\cdots,4\). (1) Operation performance  
* The number of delayed flights from 15:00 to 16:00: \(n_{11}(\Delta t_{1})=1\), \(n_{11}(\Delta t_{2})=0\), \(n_{11}(\Delta t_{3})=3\), \(n_{11}(\Delta t_{4})=1\);
* The number of canceled flights from 15:00 to 16:00: \(n_{12}(\Delta t_{1})=1\), \(n_{12}(\Delta t_{2})=0\), \(n_{12}(\Delta t_{3})=1\), \(n_{12}(\Delta t_{4})=0\). (2) Economic performance  
* Profit loss caused by delays from 15:00 to 16:00: The unit cost of delay per flight \(C_{31/d}\) is assumed to be 7000 RMB/min. Therefore,  
\[CD_{21}(\Delta t_{1})=n_{11}(\Delta t_{1})\cdot\overline{d_{31}(\Delta t_{1}) }\cdot C_{31/d}=1\cdot 21\cdot 7000=147,000;\]  
\[CD_{21}(\Delta t_{2})=0;\]  
\[CD_{21}(\Delta t_{3})=3\cdot 15\cdot 7000=315,000;\]  
\[CD_{21}(\Delta t_{1})=1\cdot 32\cdot 7000=224,000.\]  
where \(\overline{d_{31}(\Delta t_{1})}\) is the average delay time of each flight in \(\Delta t_{i}\). * Profit loss caused by flight cancellations from 15:00 to 16:00: The average profit loss \(\overline{p_{21/ap}}\) caused by a flight cancellation is assumed to be 50,000 RMB. Therefore,  
\[PL_{21}(\Delta t_{1})=n_{12}(\Delta t_{1})\cdot\overline{pl_{21/ap}}=1\cdot 50,000=50,000;\]  
Figure 6: Pareto frontier of airport capacity every 15 min. \[PL_{21}(\Delta t_{2})=0;\] \[PL_{21}(\Delta t_{3})=50,\!000;\] \[PL_{21}(\Delta t_{4})=0. \tag{3}\]  
Route capacity in Xi'an Xianyang Airport  
Based on the concept of the Pareto frontier, the 15-min transportation capacity of this route equals the planned number of arrivals and departures. Therefore,  
\[c(\Delta t_{1})=4,\;c(\Delta t_{2})=6,\;c(\Delta t_{3})=5,\;c(\Delta t_{4})=4.\]  
Step 2. Weights of Airport Routes  
In Step 2, to calculate the route weight between Xi'an Xianyang Airport and Changsha, all data of this route (from 23 February 2023 to 15 April 2023) are applied. (1) Average daily flight volume  
First, the planned arrivals and departures in one hour are obtained. Then, the Pareto frontier of this route is drawn to evaluate its hourly flight capacity, which can be regarded as 19 flights/hour, as shown in \[CD_{21}(\Delta t_{2})=0;\]0  
Therefore, the average daily flight volume of this route is:  
\[CD_{21}(\Delta t_{2})=0;\]1  
(2) Route distance  
The flight distance between Xi'an Xianyang Airport and Changsha is 810 km. (3) Flow stability  
For the flow stability of the route between Xi'an Xianyang Airport and Changsha, in Equation (6), the degradation in route capacity \(u\) starts from 0. \(F_{\text{max}}^{k}(u)\) is the maximum network traffic flow of an airport when the route capacity reduces \(u\). According to Equations (7)-(10), the calculation method of \(F_{\text{max}}^{k}(u)\) is as follows:  
* Capacity reduction process: Start by reducing the route capacity for each period, beginning with 19 flights/hour and decreasing stepwise to 0. * Filter flight data: Filter flights exceeding each hour's new capacity after reducing capacity. For example, if the capacity is reduced to 15 flights/hour, remove flights from the actual data surpassing this hourly capacity (i.e., only keep the first 15 flights each hour if the data contains more). \[CD_{21}(\Delta t_{2})=0;\]2 Hourly flight capacity estimation of the route between Xi’an Xianyang Airport and Changsha. * Daily Maximum Network Traffic Flow Calculation: After filtering, sum up the remaining flights once the hourly data has been filtered for all daily hours. This gives the maximum network traffic flow for the entire day at the reduced capacity level. Repeat this for each capacity degradation.