## 2 Models
### Parcel-Receiving Station and Public Air-Route Planning
#### 2.4.1 Upper-Layer Model  
1. Objective function  
The upper-layer model of the logistics UAV parcel-receiving station location problem can be expressed as a multi-objective optimization problem as:  
\[\text{min}C_{1}=N_{s} \tag{10}\]  
\[\text{min}C_{2}=\frac{\sum\limits_{i\in D}v_{i}d_{i}}{\sum\limits_{i\in D}v_{ i}} \tag{11}\]  
\[\text{min}C_{3}=\frac{1}{\sum\limits_{i\in D}v_{i}}\sqrt{\sum\limits_{i\in D}v_ {i}{(d_{i}-\overline{d})}^{2}} \tag{12}\]  
Function (10) refers to the cost of selected parcel-receiving stations count, which reflects the economy of the location result. The smaller the result of Function (10), the fewer the number of the location result and the stronger the economy. Function (11) refers to the cost of the average picking distance, which reflects the practicality of the location result. The smaller the result of Function (11), the shorter the picking distance of unit demand and the stronger the practicability. Function (12) refers to the cost of picking fairness, which reflects the fairness of the location result. The smaller the result of Function (12), the smaller the picking distance gap per unit demand and the stronger the fairness. \begin{table}
\begin{tabular}{c c} \hline \multicolumn{2}{c}{**Notations**} \\ \hline \(v_{i}\) & the demand volume of demand point \(i\) \\ \(N_{s}^{\text{max}}\) & the maximum allowable number of parcel-receiving stations in the planning area \\ \(d_{\text{max}}\) & the maximum acceptable picking distance for a customer \\ \(C_{\text{min}}\) & the minimum capacity of parcel-receiving stations \\ \(C_{\text{max}}\) & the maximum capacity of parcel-receiving stations \\ \(\gamma\) & the maximum acceptable negative air-range optimization rate \\ \hline \multicolumn{2}{c}{**Sets**} \\ \hline \(S\) & the set of selected parcel-receiving stations \\ \(D\) & the set of demand points \\ \(E\) & the set of connecting edges between warehouse and stations \\ \(R_{s}\) & the set of edges through which the air route between warehouse and station s passes \\ \(G^{i}\) & the set of grids through which edge \(e_{i}\) passes \\ \hline \multicolumn{2}{c}{**Variables**} \\ \hline \(x_{ij}\) & matching decision variable; equals 1 when the demand point \(i\) matches the parcel-receiving station \(j\), \\  & otherwise 0 \\ \(r_{kw}\) & edge-passing decision variable; equals 1 when the air route of station \(w\) passes edge \(k\), otherwise 0 \\ \(N_{s}\) & the number of selected parcel-receiving stations from alternative locations \\ \(n_{e}^{k}\) & the number of grids through which edge \(k\) passes \\ \(n_{s}\) & the number of grids through which the single air route of station \(s\) passes \\ \(d_{i}\) & the Manhattan distance from demand point \(i\) to its matching parcel-receiving station \\ \(d_{ij}\) & the Manhattan distance from demand point \(i\) to parcel-receiving station \(j\) \\ \(\overline{d}\) & the average Manhattan distance from unit demand to its matching parcel-receiving station \\ \(C_{j}\) & the service demand volume of parcel-receiving station \(j\) \\ \(|R_{s}|\) & the number of edges in the set \(R_{s}\) \\ \hline \end{tabular}
\end{table} 
Table 4: New parameters for parcel-receiving station and public air-route planning formulation defined. Constraints  
\[d_{i}=\underset{j\in S}{\text{min}}d_{ij}\leq d_{\text{max}} \tag{13}\]  
\[N_{s}\leq N_{s}^{\text{max}} \tag{14}\]  
\[\sum_{i\in D}\sum_{j\in S}x_{ij}=1 \tag{15}\]  
\[C_{\text{min}}\leq C_{j}=\sum_{i\in D}v_{i}x_{ij}\leq C_{\text{max}} \tag{16}\]  
Equation (13) indicates the maximum picking-distance constraint. Each demand point \(i\) of \(D\) should be matched to the nearest station \(j\) of \(S\), and the picking distance between \(i\) and \(j\) cannot exceed the maximum acceptable picking distance \(d\text{max}\). Since the ground roads are mostly square-format roads, Manhattan distance is used when calculating the picking distance.