## 2 Methods
### Modeling
#### 2.1.2 Model Construction
\begin{array}{l}0,a_{i}<Time_{i}^{k}<b_{i}\\ \lambda_{1},Time_{i}^{k}<a_{i}\\ \lambda_{2},b_{i}<Time_{i}^{k}\end{array} \right. \tag{13}\]  
\[Tw_{i}^{k}=\left\{ \begin{array}{l}0,a_{i}<Time_{i}^{k}<b_{i}\\ (a_{i}-Time_{i}^{k})\cdot x_{i}^{k},Time_{i}^{k}<a_{i}\\ (Time_{i}^{k}-b_{i})\cdot x_{i}^{k},b_{i}<Time_{i}^{k}\end{array} \right. ; \forall i\in B,\forall k\in K \tag{14}\]  
\[Tl_{i}^{k}=\max\left(Lc_{i}^{k}-C^{k},0\right),\forall i\in B,\forall k\in K \tag{15}\]  
\[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]  
\[Time_{i}^{k}=w_{i-1}^{k}+s_{i-1}+\frac{x_{i-1,i}^{k}\cdot d_{i-1,i}}{v^{k}},i- 1\in\Delta^{-}(i) \tag{17}\]  
where, \(sumLW\) is the total penalty. Equation (11) calculates the sum of the time window penalty, loading weight penalty, and loading volume penalty. Equation (12) determines whether node \(i\) is on the path of drone \(k\). Equation (13) represents the value of the time window penalty coefficient \(\lambda\). Equation (14) calculates the duration of delay/early completion (the task was not completed within the time window). Equation (15) calculates the drone's weight overloaded at node \(i\). Equation (16) calculates the volume of drone k overloaded at node \(i\). Equation (17) calculates the time for drone k to reach node \(i\). 2. Constraints  
1. The number of drones carrying out tasks shall not exceed the total number of drones owned. \[NV\leq m\] (18)
2. UAV's service rules: Equation (19) indicates that each demand point can only be assigned to one path. Equation (20) indicates that a demand point can only be served by one drone. Equation (20) indicates that a drone can serve multiple demand points and at least one demand point. \[\sum_{k\in K}\sum_{j\in\Delta^{+}(i)}x_{ij}^{k}=1;\forall i,j\in B\] (19) \[\sum_{k\in K}x_{i}^{k}=1,i\in B\] (20) \[1\leq\sum_{i\in B}x_{i}^{k},k=1,2,\ldots,r\] (21)
3. Traffic constraints of a UAV on the path: Equation (22) indicates that the drone can only reach one node from the starting point, meaning only one drone can be on that path. Equation (23) indicates that drones returning from a particular node to the starting point can only have one drone on that path. Equation (24) constrains the flow balance on this path. \[\sum_{j\in\Delta^{+}(0)}x_{0j}^{k}=1,\forall j\in B,\forall k\in K \tag{22}\] \[\sum_{i\in\Delta^{-}(0)}x_{i0}^{k}=1,\forall i\in B,\forall k\in K\] (23) \[\sum_{i\in\Delta_{-}(j)}x_{ij}^{k}-\sum_{i\in\Delta^{+}(j)}x_{ji}^{k}=0, \forall(i,j)\in B,\forall k\in K \tag{24}\]
4. Continuous constraints on UAV travel time: During the process of the drone completing the task from the previous to the current demand point, there is no other time consumption except for service and travel time. Equation (25) can ensure no time loss for the drone during the process, ensuring the continuity of time. \[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]0 (25)
5. Load relation constraint: Equation (26) represents the actual loading weight of the drone \(k\) after serving the first demand point on the route. Equation (27) represents the actual loading weight of the drone \(k\) after completing service at any demand point on the route (excluding the first demand point). \[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]1 (26) \[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]2 (27)
6. The drone's flight speed cannot exceed the maximum speed limit. \[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]3 (28)
7. The drone's flight time cannot exceed the maximum time limit. \[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]4 (29)
8. The flying distance of drones shall not exceed the maximum distance limit. \[Tv_{i}^{k}=\max\left(Lv_{i}^{k}-V^{k},0\right),\forall i\in B,\forall k\in K \tag{16}\]5 (30)
9. The total volume of cargo loaded by drones shall not exceed the maximum capacity of the drone cargo hold.