# Time-Dependent Vehicle Routing Problem with Drones Under Vehicle Restricted Zones and No-Fly Zones
## 4 Mathematical Model
### Drone-Related Constraints  
\[d_{u}y_{inj}^{u_{k}}\leq Q^{2},\forall i\in V,w\in V/0,j\in V,k\in K \tag{18}\]  
\[B^{2}-\big{(}e_{u_{k}iw}+e_{u_{k}wj}+e_{u_{k}ij}\big{)}y_{inj}^{u_{k}}\geq B^{2} \epsilon,i\in V,w\in V/0,j\in V,k\in K \tag{19}\]  
\[\sum_{w\in V/0}\sum_{j\in V}y_{inj}^{u_{k}}\leq 1,i\in V,k\in K \tag{20}\]  
\[\sum_{i\in V}\sum_{w\in V/0}y_{inj}^{u_{k}}\leq 1,\ j\in V,k\in K \tag{21}\]  
\[T_{u_{l}i}^{d}+4t^{4}+\frac{l_{u_{k}iw}}{v_{u}}+t^{2}+\frac{l_{u_{k}w}}{v_{u}} -M(1-\sum_{j\in V}y_{inj}^{u_{k}})\leq T_{u_{k}j}^{4},\forall i\in V,\forall w \in V/0,\forall j\in V,\forall k\in K \tag{22}\]  
\[T_{u_{l}i}^{d}+4t^{4}+\frac{l_{u_{k}iw}}{v_{u}}+t^{2}+\frac{l_{u_{k}w}}{v_{u}} +M(1-\sum_{j\in V}y_{inj}^{u_{k}})\geq T_{u_{k}j}^{4},\forall i\in V,\forall w \in V/0,\forall j\in V,\forall k\in K \tag{23}\]  
\[\min\Big{\{}T_{h_{2}}^{f}-T_{u_{l}i^{\prime}}^{d},T_{u_{k}w}^{d}-T_{h_{2}}^{s} \Big{\}}\leq M\Big{(}1-y_{inj}^{u_{k}}\Big{)},i\in V_{2},w\in V_{1}\cup V_{3},j\in V_{1}\cup V_{3},k\in K,h_{2}\in H_{2} \tag{24}\]  
\[\min\Big{\{}T_{h_{2}}^{f}-T_{u_{l}i^{\prime}}^{d},T_{u_{l}j}^{a}-T_{h_{2}}^{s} \Big{\}}\leq M\Big{(}1-y_{inj}^{u_{k}}\Big{)},i\in V,w\in V_{2},j\in V,k\in K,h_{2}\in H_{2} \tag{25}\]  
\[\min\Big{\{}T_{h_{2}}^{f}-T_{u_{l}i^{\prime}}^{d},T_{u_{l}j}^{a}-T_{h_{1}}^{s} \Big{\}}\leq M\Big{(}1-y_{inj}^{u_{k}}\Big{)},i\in V/V_{2},w\in V_{1}\cup V_{3 },j\in V_{2},k\in K,h_{2}\in H_{2} \tag{26}\]  
\[\min\Big{\{}T_{h_{2}}^{f}-T_{u_{l}i^{\prime}}^{d},T_{u_{l}j}^{a}-T_{h_{1}}^{s} \Big{\}}\leq M\Big{(}1-y_{inj}^{u_{k}}\Big{)},i\in V_{2},w\in V_{1}\cup V_{3},j\in V_{2},k\in K,h_{2}\in H_{2} \tag{27}\]  
Constraint (18) represents that the demand at the customer \(w\) served by the drone must not exceed its maximum payload capacity. Constraint (19) specifies that the remaining battery charge of the drone after completing a delivery must not be lower than its minimum state of charge. Constraints (20) and (21) state that the drone can launch and recover at the same customer point no more than once. Constraints (22) and (23) represent the relationship between the departure and arrival times of the drone between the takeoff point and the recovery point. Constraint (24) specifies that if the drone's takeoff point is within a no-fly zone and the service or landing point is outside the no-fly zone, the drone's takeoff time must be later than the end of the no-fly period, or the arrival time at the service point must be earlier than the start of the no-fly period. Constraint (25) indicates that if the drone's service point is within a no-fly zone, the drone's takeoff time must be later than the end of the no-fly period, or the arrival time at the landing point must be earlier than the start of the no-fly period. Constraint (26) specifies that if the drone's landing point is within a no-fly zone and the takeoff or service point is outside the no-fly zone, the drone must leave the service point no later than the end of the no-fly period, or its arrival time at the landing point must be earlier than the start of the no-fly period. Constraint (27) indicates that if both the drone's takeoff and landing points are within a no-fly zone and the service point is outside the no-fly zone, the drone must leave the takeoff point no later than the end of the no-fly period, or its arrival time at the landing point must be earlier than the start of the no-fly period.