## 2 Methods
### Modeling
#### 2.1.2 Model Construction
\begin{tabular}{c} Determine whether drone \(k\) is from starting point and reaches node \(j\). \\ \end{tabular}  \\ \hline \(n\) & The number of UAV rotors. & \(x_{i0}^{k}\) & \begin{tabular}{c} Determine whether drone \(k\) is from other nodes and returns to the starting point. \\ \end{tabular}  \\ \hline \(d_{ij}\) & Distance from node \(i\) to node \(j\). & M & An infinite number. \\ \hline \(\wp_{k}\) & Speed of drone \(k\). & \(V_{i}\) & The volume of cargo \(i\). \\ \hline \(\wp_{k\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\_\\_\_\\_\\\_\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\[NV=\sum\limits_{k\in K}x^{k} \tag{4}\]  
where, \(sumTD\) is the total cost. Equation (1) calculates the total cost of UAVs performing delivery tasks. Equations (2) and (3) represent the range of values for decision variables. Equation (4) calculates the number of drones used to perform delivery tasks. When using UAVs, it is essential to find the right balance. To measure this balance, we can look at the remaining power of each UAV after completing delivery tasks. If the remaining power is evenly distributed among all UAVs, their utilization rates are similar, and there is no overuse or underuse of any UAV. The sub-goals for achieving this balance are as follows:  
\[En_{differ}=\min\frac{\sum\limits_{k=1}^{m}\left(|En_{leave_{k}}-En_{leave_{ k}}|\right)}{m} \tag{5}\]  
\[En_{leave_{k}}=En_{k}-En_{leave_{k}},\forall k\in K \tag{6}\]  
\[En_{c_{k}}=\sum\limits_{(i,j)\in B}En_{c_{ijk}}\cdot x_{ij}^{k} \tag{7}\]  
\[En_{c_{ijk}}=\left((m_{UAV_{k}}+Load_{ij}^{k})^{\frac{3}{2}}\cdot\frac{g^{3}}{ \sqrt{2\rho\xi n}}\cdot\frac{d_{ij}}{\
u_{k}},\forall(i,j)\in B,\forall k\in K\right. \tag{8}\]  
\[Load_{ij}^{k}=L_{t}^{k}=L_{t-1}^{k}-R_{i}+L_{t}^{k},\forall i-1,i\in B,\forall k\in K \tag{9}\]  
\[L_{0k}=\sum\limits_{i\in B}R_{i}\sum\limits_{j\in\Delta^{+}(i)}x_{ij}^{k}, \forall k\in K \tag{10}\]  
where, \(En_{differ}\) is an indicator used to measure the balance of drone usage. Equation (5) first calculates the difference between the remaining energy of each drone after executing the task and the average remaining energy; then, these differences are summed up, and the average value is taken. The data obtained from this can indicate the degree of use of each drone in a specific distribution plan. Equation (6) calculates the remaining energy value of drone k after completing the task. Equation (7) calculates the energy consumption of drone \(k\) in executing tasks. Equation (8) calculates the energy consumption of each small segment of the drone delivery route. Equation (9) indicates that the actual cargo loading weight of the drone on section \(i\to j\) is equal to the cargo loading weight when the drone leaves point \(i\), which is equal to the loading weight when leaving node \(i-1\) minus the required weight at node \(i\) plus the recovered weight at node \(i\). Equation (10) represents the weight of the cargo carried by the drone when departing from the starting point. To more genuinely reflect the power consumption of the actual delivery of the UAV, consider the dynamic change of the power consumption of the UAV during the distribution process and calculate the power consumption of each section of the distribution route using Formula (8) [26], as shown in Figure 1.  
Figure 1: Schematic diagram of energy consumption calculation. The minimum punishment in the UAV distribution process is ensuring timely and safe material delivery to the demand point. These penalties include punishment for exceeding the time window, load weight, and load volume limits. The sub-objectives are as follows:  
\[sumLW=\min\sum_{k\in K}\sum_{i\in B}\left(Tw_{i}^{k}\cdot\lambda+Tl_{i}^{k} \cdot\beta+Tv_{i}^{k}\cdot\gamma\right) \tag{11}\]  
\[x_{i}^{k}=\left\{ \begin{array}{l}1,for\ \forall(i,j)\in B,only\ \exists x_{ij}^{k}=1\\ 0,others\end{array} \right. \tag{12}\]  
\[\lambda=\left\{