## 2 Methods
### Modeling
#### 2.1.2 Model Construction  
Under the above assumptions, this paper constructed a multi-UAV collaborative task assignment and routing problem with simultaneous delivery and pick-up and time windows (MVTARPSDPTW), considering the factors of UAV load, energy consumption, cargo quality, and volume. The descriptions of relevant symbols in the model are shown in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(B\) & Set of all demand points. & \(L^{k}_{i-1}\) & Loading weight (including the starting point) when UAV \(k\) leaves node \(i-1\). \\ \hline \(K\) & Set of all UAVs. & \(R_{i}\) & Required weight at node \(i\). \\ \hline \(x^{k}_{ij}\) & Decision variable: determine whether & \(L^{k}_{i}\) & Retrieve the weight of goods at node \(i\). \\ \hline \(\varpi\) & Unit transportation cost of drones. & \(L_{0k}\) & Loading weight when UAV \(k\) leaves the starting point. \\ \hline \(Load^{k}_{ij}\) & The real loading weight of UAV \(k\) on the path from node \(i\) to node \(j\). & \(Tw^{k}_{i}T^{k}_{i}T^{k}_{i}T^{k}_{i}\) & Represent the sum of the time window, the sum of the loading weight, and the sum of the loading volume violated by drone \(k\) at node \(i\), respectively. \\ \hline \(dis(i,j)\) & Euclidean distance between node \(i\) and node \(j\). & \(\lambda,\beta,\gamma\) & Represent the penalty coefficients of the violation of the time window, the loading weight, and the loading volume, respectively. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptions of relevant symbols. 1. Objective  
First, consider the minimum cost of all UAV delivery tasks, which consists of three parts: UAV transportation, start-up, and maintenance. The sub-objectives are as follows:  
\[sumTD=\min\left(\omega\cdot(\sum_{k\in K}\sum_{(i,j)\in B}x_{ij}^{k}\cdot dis(i, j)\cdot Load_{ij}^{k})+NV\cdot(\psi+\wp)\right) \tag{1}\]  
\[x_{ij}^{k}=\left\{ \begin{array}{ll}1,d\text{rone k starts from node i to node j}\\ 0,others\end{array} \right. \tag{2}\]  
\[x^{k}=\left\{ \begin{array}{ll}1,for\ \forall(i,j)\in B,only\ \exists x_{ij}^{k}=1\\ 0,others\end{array} \right. ;\forall k\in K \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(x^{k}\) & Be used to judge whether drone \(k\) is used. & \(x_{i}^{k}\) & Be used to judge whether node \(i\) is served by drone \(k\). \\ \hline
**NV** & The number of drones used. & \(Time_{i}^{k}\) & \begin{tabular}{c} Represents the time when the drone \(k\) arrives at node \(i\). \\ \end{tabular}  \\ \hline \(\psi\) & Unit start-up cost. & \([a,b]\) & The left and right time windows of node \(i\). \\ \hline \(\wp\) & Unit maintenance cost. & \(Lc_{i}^{k}\) & The loading weight of drone \(k\) at node \(i\). \\ \hline \(En_{leave_{a}}\) & The energy remaining of UAV \(k\) after completing the delivery task. & \(C^{k}\) & The maximum loading weight of drone \(k\). \\ \hline \(En_{leave_{a}}\) & Average of UAV energy remaining. & \(V^{k}\) & The maximum loading volume of drone \(k\). \\ \hline \(m\) & Number of drones. & \(Lv_{i}^{k}\) & The loading volume of drone \(k\) at node \(i\). \\ \hline \(En_{k}\) & The initial energy of UAV \(k\). & \(w_{i-1}^{k}\) & The service start-time of drone \(k\) to node \(i-1\). \\ \hline \(En_{c_{i}}\) & Energy consumption of UAV \(k\). & \(s_{i-1}\) & The service time of node \(i-1\). \\ \hline \(En_{c_{i\mu}}\) & Energy consumption of UAV \(k\) along the path from node \(i\) to node \(j\). & \(\Delta^{-}(i)\) & Set of arcs pointing to node \(i\). \\ \hline \(m_{ULAV_{i}}\) & The weight of UAV \(k\). & \(\Delta^{+}(i)\) & Set of arcs pointed by \(i\). \\ \hline \(g\) & Acceleration of heavy force. & \(x_{i-1,i}^{k}\) & \begin{tabular}{c} Determine whether drone \(k\) is from node \(i-1\) to node \(i\). \\ \end{tabular}  \\ \hline \(\rho\) & Air density. & \(d_{i-1,i}\) & Distance from node \(i-1\) to node \(i\). \\ \hline \(\xi\) & The area of the rotating blade disc. & \(x_{0j}^{k}\) & \