# Airport Field Path Optimization Method Based on Conflict Hotspot Avoidance Mechanism
## 2 Modelling
### Path Optimization Model  
To establish a path optimization model for the conflict hotspot area of the airport field, the model parameters are defined as follows:  
\(F\): denotes the set of all aircraft in the specified time, \(F=\{f_{1},f_{2},\cdots,f_{k}\}\);  
\(N\): denotes the set of nodes of the airport field topology network;  
\(L_{ab}\): denotes the length of the glide path from node \(a\) to node \(b\), where \(a,b\in N\);  
\(T_{ia}\): denotes the actual time at which the aircraft \(f_{i}\) needs to avoid the hotspot area to reach the node \(a\) or \(T_{ia}=0\) if aircraft \(f_{i}\) does not reach the node \(a\);  
\(t_{ia}\): denotes the time when the aircraft \(f_{i}\) reaches the node \(a\) without avoiding the hotspot area, or \(t_{ia}=0\) if the aircraft \(f_{i}\) does not reach the node \(a\);  
\(t_{ij}^{a}\): denotes the minimum safe interval, \(t_{ij}^{a}=40\) s;  
\(v_{i}\): denotes the glide speed of the aircraft, taking \(v_{i}=50\) km/h;  
\(C_{ab}\): indicates whether there is a feasible glide path from node \(a\) to node \(b\). \(C_{ab}=1\) indicates that there is a feasible glide path from node \(a\) to node \(b\), otherwise \(C_{ab}=0\);  
\(R_{iab}\): indicates the glide direction of the aircraft, \(R_{iab}=1\) indicates that the aircraft \(f_{i}\) glides from node \(a\) to node \(b\), otherwise \(R_{iab}=0\);  
\(Z_{ija}\): denotes the order in which two aircraft arrive at a node, \(Z_{ija}=1\) denotes that aircraft \(f_{i}\) arrives at the node \(a\) before the aircraft, otherwise \(Z_{ija}=0\);  
\(N^{i}\):denotes the glide path of an aircraft \(f_{i}\), consisting of a series of field network nodes, \(N^{i}=\left\{o^{i},n_{1}^{i},n_{2}^{i},\cdots,n_{k_{i}^{i}}^{i},d^{i}\right\}\);  
\(M^{i}\): denotes the set of field network nodes that aircraft \(f_{i}\) has passed through: \(M^{i}=\left\{o^{i},n_{1}^{i},n_{2}^{i},\cdots,n_{k_{i}^{i}}^{i},d^{i}\right\}\), \(o^{i}\) is the starting point, and \(d^{i}\) is the endpoint. Figure 2: Conflict resolution process. In this paper, the shortest taxiing time for all aircraft to taxi to the endpoint within the specified time is taken as the objective, and the objective function is constructed as follows:  
\[\text{min}T=\sum_{f_{i}\in F}\sum_{a,b\in N}\frac{R_{iab}L_{ab}}{v_{i}}+(T_{ia}- t_{ia}) \tag{12}\]  
Aircraft taxiing needs to satisfy taxi path constraints, safety interval constraints, head-to-head constraints, and overrun constraints. (1) Glide path constraints  
\[R_{iab}\leq C_{ab},\forall f_{i}\in F,\forall a,b\in N \tag{13}\]  
\[\sum_{a\in N}R_{iab}-\sum_{s\in N}R_{ibs}=\left\{ \begin{array}{ll}1,b=d^{i} \\ -1,b=o^{i}\\ 0,otherwise\end{array} \right.,\forall f_{i}\in F,\forall a,b,s\in N \tag{14}\]  
Equation (13) indicates that the aircraft glide path must be feasible and Equation (14) indicates that the aircraft glide path must be continuous. (2) Safety interval constraints  
\[T_{ja}-T_{ia}\geq t_{ij}^{e}\forall f_{i},f_{j}\in F,\forall a\in N \tag{15}\]  
Equation (15) indicates that the distance between any two aircraft passing through the unity node must satisfy the minimum safe separation. (3) Head-to-head constraints  
\[\left\{ \begin{array}{ll}Z_{ija}-Z_{jib}\leq 2-\left(R_{iab}+R_{jba}\right)\\ Z_{ija}-Z_{jib}\geq-2+\left(R_{iab}+R_{jba}\right)\end{array} \right.,\forall f _{i},f_{j}\in F\wedge i\
eq j,\forall a,b\in N \tag{16}\]  
Equation (16) indicates that head-to-head conflict is not permitted between any two aircraft. (4) Beyond constraints  
\[\left\{ \begin{array}{ll}Z_{ija}-Z_{jib}\leq 2-\left(R_{iab}+R_{jab}\right)\\ Z_{ija}-Z_{jib}\leq-2+\left(R_{iab}+R_{jab}\right)\end{array} \right.,\forall f _{i},f_{j}\in F\wedge i\
eq j,\forall a,b\in N \tag{17}\]  
Equation (17) indicates that a tailgating conflict between any two aircraft is not permitted.