# Time-Dependent Vehicle Routing Problem with Drones Under Vehicle Restricted Zones and No-Fly Zones
## 4 Mathematical Model
### Vehicle-Drone Collaborative Delivery Constraints  
\[\sum_{i\in V}\sum_{k\in K}x_{kiw}+\sum_{i\in V}\sum_{j\in V}\sum_{k\in K}y_{ inj}^{u_{k}}=1,\forall w\in V/0 \tag{28}\]\[\sum_{i\in V}\sum_{j\in V}d_{j}x_{kij}+\sum_{i\in V/0}\sum_{j\in V}\sum_{w\in V/0 }y_{inyj}^{u_{k}}\leq Q^{1},\forall k\in K \tag{29}\]  
\[\sum_{m\in V}x_{kim}+\sum_{n\in V}x_{knij}\geq 2y_{inji}^{u_{k}}\forall i\in V, \forall j\in V,\forall k\in K,\forall w\in V/0 \tag{30}\]  
\[\max\Bigl{\{}T_{ki}^{a},T_{u_{ki}}^{a}\Bigr{\}}+t^{3}-M\Biggl{(}1-\sum_{w\in V /0}\sum_{j\in V}y_{inyj}^{u_{k}}\Biggr{)}\leq T_{u_{ki}}^{d},\forall i\in V, \forall k\in K \tag{31}\]  
\[\max\Bigl{\{}T_{ki}^{a},T_{u_{ki}}^{a}\Bigr{\}}+t^{3}+M\Biggl{(}1-\sum_{w\in V /0}\sum_{j\in V}y_{inj}^{u_{k}}\Biggr{)}\geq T_{u_{ki}}^{d},\forall i\in V, \forall k\in K \tag{32}\]  
\[n_{lk}\geq n_{jk}-M\Biggl{(}3-\sum_{w\in V/0}y_{inyj}^{u_{k}}-\sum_{m\in V/0} \sum_{n\in V}y_{lmn}^{u_{k}}-x_{kil}\Biggr{)},\forall i\in V,\forall l\in V,\forall k\in K \tag{33}\]  
\[Q_{ki}-d_{i}-\sum_{w\in V/0}d_{w}y_{inj}^{u_{k}}-M\Bigl{(}1-x_{kij}\Bigr{)} \leq Q_{kj},\forall i\in V,\forall j\in V,\forall k\in K \tag{34}\]  
\[Q_{ki}-d_{i}-\sum_{w\in V/0}d_{w}y_{inj}^{u_{k}}+M\Bigl{(}1-x_{kij}\Bigr{)} \geq Q_{kj},\forall i\in V,\forall j\in V,\forall k\in K \tag{35}\]  
\[\max\Biggl{\{}T_{ki}^{a}+t^{1},T_{u_{ki}}^{a}\cdot\sum_{a\in V}\sum_{b\in V/0}y _{abi}^{u_{k}}\Biggr{\}}+t^{3}\Biggl{(}\sum_{m\in V/0}\sum_{n\in V}y_{imm}^{u_ {k}}\Biggr{)}+M\Biggl{(}1-\sum_{j\in V}x_{kij}\Biggr{)}\geq T_{ki}^{d},\, \forall i\in V,\,\forall k\in K \tag{36}\]  
\[\max\Biggl{\{}T_{ki}^{a}+t^{1},T_{u_{ki}}^{a}\cdot\sum_{b\in V/0}\sum_{a\in V} y_{abi}^{u_{k}}\Biggr{\}}+t^{3}\Biggl{(}\sum_{m\in V/0}\sum_{n\in V}y_{ imm}^{u_{k}}\Biggr{)}-M\Biggl{(}1-\sum_{j\in V}x_{kij}\Biggr{)}\leq T_{ki}^{d},\forall i\in V /0,\forall k\in K \tag{37}\]  
Constraint (28) indicates that each customer point must be serviced either by a vehicle or a drone. Constraint (29) specifies that the total demand of the customer points served by the vehicle and its carried drones must not exceed the vehicle's maximum payload capacity. Constraint (30) states that if a drone is launched from a node, services a customer, and is then recovered at the same node, the vehicle must also pass through these two nodes. Constraints (31) and (32) express that the departure time of the vehicle at the drone's takeoff point is equal to the time of vehicle-drone rendezvous plus the duration for loading goods and changing batteries. Constraint (33) ensures that the drone will not be launched from the vehicle again during the journey. Constraints (34) and (35) describe the relationship between the changes in the vehicle's payload as it departs from nodes \(i\) and \(j\). Constraints (36) and (37) state that the vehicle's departure time at a customer point is equal to the maximum of the vehicle's arrival time and the drone's arrival time, plus the vehicle's service time at the customer point. If the customer point is the drone's takeoff point, the time must also include the drone's loading and battery swapping duration.