## 2 Problem Modeling
### Constraints  
\[\sum_{i\in C}z_{i}^{k}+\sum_{i\in C}\sum_{I\in L}z_{il}^{u}=1 \tag{8}\]  
\[\sum_{i\in C\smallsetminus\{n+1\}}\sum_{j\in C}x_{ij}=\sum_{j\in C}z_{j}^{k} \tag{9}\]  
\[\sum_{i\in C\smallsetminus\{n+1\}}\sum_{j\in C}y_{ij}\geq\sum_{j\in C}z_{jl}^{u },\forall I\in L \tag{10}\]  
\[\sum_{I\in L}y_{ij}+\sum_{I\in L}y_{jil}\leq 1,\forall(i,j)\in E \tag{11}\]\[1-\mathrm{M}\times\big{(}3-\mathrm{x_{ij}}-\mathrm{y_{ihl}}-\mathrm{z_{hl}^{u}}\big{)} \leq\mathrm{m_{il}^{f}}\leq\sum\mathrm{x_{ij}},\forall i\in C,j\in\{C:j\
eq i \},h\in\{C:h\
eq i\
eq j\} \tag{12}\]  
\[1-\mathrm{M}\times\big{(}3-\mathrm{x_{ji}}-\mathrm{y_{hill}}-\mathrm{z_{hl}^{u}} \big{)} \leq\mathrm{m_{il}^{r}}\leq\sum\mathrm{x_{ji}},\forall j\in C,i\in\{C:i \
eq j\},h\in\{C:h\
eq j\
eq i\} \tag{13}\]  
\[0\leq\mathrm{q_{il}^{u}}\leq Q,\forall i\in C,l\in L \tag{14}\]  
\[\mathrm{m_{ii}^{f}}\mathrm{e_{ii}^{u}}=E \tag{15}\]  
\[\mathrm{m_{il}^{r}}\mathrm{e_{ii}^{u}}=0.1\times E \tag{16}\]  
\[\mathrm{t_{i}^{k}}\geq\mathrm{\dot{f}_{i}^{k}}+\frac{\mathrm{d_{ij}^{k}}}{ \mathrm{v_{k}}}-\mathrm{M}\times\big{(}1-\mathrm{x_{ij}}\big{)},\forall(i,j) \in E \tag{17}\]  
\[\mathrm{\dot{f}_{i}^{k}}\geq\mathrm{t_{i}^{k}}+\mathrm{\dot{r}_{i}^{k}}- \mathrm{M}\times\Big{(}1-\mathrm{z_{i}^{k}}\Big{)},\forall i\in C \tag{18}\]  
\[\mathrm{t_{jl}^{u}}\geq\mathrm{f_{il}^{u}}+\frac{\mathrm{d_{ij}^{u}}}{ \mathrm{v_{u}}}-\mathrm{M}\times\Big{(}1-\mathrm{y_{ij}}\Big{)},\forall(i,j) \in E \tag{19}\]  
\[\mathrm{f_{il}^{u}}\geq\mathrm{t_{il}^{u}}-\mathrm{M}\times(1-\mathrm{z_{il}^ {u}}),\forall i\in C \tag{20}\]  
\[\mathrm{f_{il}^{u}}\leq\mathrm{f_{i}^{k}}-\mathrm{M}\Big{(}1-\mathrm{m_{il}^{ f}}\Big{)} \tag{21}\]  
\[\mathrm{t_{il}^{u}}\geq\mathrm{t_{i}^{k}}+\mathrm{M}(1-\mathrm{m_{il}^{r}}) \tag{22}\]  
The constraint Equation (8) stipulates that each customer demand point may be served exclusively by either a vehicle or a drone on a single occasion. Equation (9) asserts that all customer points traversed by vehicles must be serviced by those vehicles. Equation (10) specifies that all customer points attended to by drones must be included in the drone routes. Constraint (11) ensures that each path arc utilized by the drone is traversed only once. Equations (12) and (13) impose restrictions on the dynamic collaborative take-off and landing points, mandating that each drone launch and recovery point must be accessed by a vehicle, and that each drone delivery route must serve a minimum of one customer point. Equation (14) delineates the load-carrying constraint, which states that upon completion of the drone route and departure from any customer point, the load must not exceed the maximum allowable limit. Equation (15) indicates that the power at the launch point of any drone route must satisfy the maximum endurance time requirement. Equation (16) specifies that when the drone returns to the meeting point along any route, the power must meet at least 10% of the maximum endurance time. Equations (17) and (18) establish the time constraints for the vehicle's arrival at and departure from a node, respectively. Similarly, Equations (19) and (20) outline the time constraints for the drone's arrival at and departure from a node, respectively. Equation (21) constrains the departure time of the vehicle at the drone take-off point to occur after that of the drone. Finally, Equation (22) stipulates that, at the drone landing point, when the vehicle encounters the drone, the vehicle's arrival time must not exceed that of the drone.