# Collaborative Allocation Method of En-Route Network Resources Based on Stackelberg Game Model
## 3 Initial Resource Allocation Multi-Objective Optimization Model  
The first stage of the collaborative resource allocation game process for the en-route network is to establish a multi-objective allocation model with the objective of global optimality and calculate the initial resource allocation scheme. The problem of allocating route resources to affected flights is a large-scale, multi-objective, and multi-constrained resource allocation problem. Suppose the set of affected flights of airlines is \(f\), the sets of route options and time slots are \(C\) and \(T\), respectively, and there are \(K\) FCAs in each route. Let \(f\), \(c\), \(k\), and \(t\) denote the flight \(f\), the FCA \(k\) in the route \(c\), and the time slot \(tf(\in F;c\in C;k\in K;t\in T)\), then the decision variables for the resource allocation problem are \(x^{f,k}_{c,t}\)[16]:  
\[x^{f,k}_{c,t}=\left\{ \begin{array}{rl}1,&\text{flight $f$ is assigned to the route option $c$ and}\\ &\text{departs from the FCA $k$ on this route before time $t$}\\ &0,&\text{else}\end{array} \right. \tag{1}\]  
Introduce the following auxiliary variables:  
\[\lambda^{f}_{c}=\left\{ \begin{array}{rl}1,&\text{flight $f$ submitted route option $c$}\\ 0,&\text{flight $f$ has not submitted route option $c$}\end{array} \right. \tag{2}\]  
The definitions of other parameters involved in the model are as follows:  
\(\delta_{k,k+1}\): the time required for the affected flight to move from the FCA \(k\) to the FCA \(k+1\) adjacent to it on the same route. \(\widetilde{x}^{f,k}_{c}\): the time required for the affected flight \(f\) to pass through the FCA \(k\) on route \(c\). \(t^{f,k}_{c}\): the time assigned for the affected flight \(f\) to enter the FCA \(k\) on route \(c\). \(\widetilde{t}^{f,k}_{c}\): the time submitted by the airline for the affected flight \(f\) to enter the FCA \(k\) on route \(c\), i.e., the ETA (in actual implementation, airlines only need to submit the time for the affected flight \(f\) to enter the first FCA on route \(c\), and the time for entering subsequent FCAs can be calculated based on the distance and flight speed). \(\omega^{f}_{c}\): a binary variable (0-1) indicating whether the affected flight \(f\) is assigned to route \(c\). \(T^{k}_{c}\): The set of available time windows for the FCA \(k\) in the route \(c\). Among them, the earliest available time window is \(\overline{T^{k}_{c}}\), while the latest available time window is \(\underline{T^{k}_{c}}\), \(t\in T^{k}_{c}\). \(M^{k}_{c}\): the capacity of the FCA \(k\) on the route \(c\) within a specified time frame. \(\Omega^{k}_{c}\): the set composed of the first FCA on route \(c\). \(F\): the set of affected flights, \(f\in F\). \(F_{a}\): the set of affected flights for the airline \(a\). \(C\): the set of routes, \(c\in C\). \(r_{f}\): the equivalent flight volume of flight \(f\). \(\beta\): the ratio of the unit cost of air delay to the unit cost of ground delay. The optimization objectives of this study need to simultaneously consider efficiency and fairness. It is not only aimed at minimizing the total delay costs for all affected flights to maximize the utilization of restricted airspace but also involves ensuring that the allocated delay times for individual airlines are as close as possible. Objective functions and constraint conditions are shown below. \[\min E=\sum_{f\in F}\sum_{e\in C}\sum_{k\in\Omega_{c}^{0}}(t_{c}^{f,k}-\bar{t}_{c}^ {f,k})\cdot\omega_{c}^{f}+\beta\sum_{f\in F}\sum_{e\in C}\sum_{k\
otin\Omega_{c}^ {0}}(t_{c}^{f,k}-\bar{t}_{c}^{f,k})\cdot\omega_{c}^{f} \tag{3}\]  
\[\min T=\sum_{f\in F_{x}}\frac{d_{f}}{q_{a}}\times\lg(n\times\frac{d_{f}}{q_{a}}) \tag{4}\]  
\[\sum_{t\in T_{c}^{k}}\sum_{k}x_{c,t}^{f,k}=\lambda_{c}^{f},\forall f\in F,c\in C \tag{5}\]  
\[t_{c}^{f,k}=t-\bar{\delta}_{c}^{f,k},x_{c,t}^{f,k}-x_{c,t-1}^{f,k}=1,\forall f \in F,c\in C \tag{6}\]  
\[t_{c}^{f,k}\geq\bar{t}_{c}^{f,k}\cdot\lambda_{c}^{f},\forall f\in F,c\in C,t \in T_{c}^{k} \tag{7}\]  
\[\omega_{c}^{f}=\left\{