# Time-Dependent Vehicle Routing Problem with Drones Under Vehicle Restricted Zones and No-Fly Zones
## 3 Problem Description
### Detour Method  
As shown in Figure 2a, consider the example of a vehicle-restricted zone between two customers to specifically analyze the detour distance. The detour distance depends \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Sets**} \\ \hline \multirow{4}{*}{\(V\)} & Set of nodes, \(V=\{0\}\cup V_{1}\cup V_{2}\cup V_{3}\), which 0 is the depot, \(V_{1}\) is the set of customers within the no-fly zone, \(V_{3}\) is the set of customers outside both the vehicle-restricted zone and the no-fly zone. \\ \multirow{4}{*}{\(H_{1}\)} & Set of vehicle-restricted zones \\ \multirow{2}{*}{\(H_{2}\)} & Set of no-fly zones \\ \multirow{2}{*}{\(K\)} & Set of vehicles \\ \multirow{2}{*}{\(U_{k}\)} & Set of drones \\ \hline \multicolumn{2}{c}{**Parameters**} \\ \hline \(\begin{bmatrix}T_{s},T_{f}\\ T_{h_{1}},T_{h_{2}}^{\prime}\end{bmatrix}\) & The depot time window \\ \(\begin{bmatrix}T_{h_{1}}^{*},T_{h_{1}}^{\prime}\end{bmatrix}\) & Restricted hours in vehicle restriction zone \(h_{1}\) \\ \(\begin{bmatrix}T_{h_{2}}^{*},T_{h_{2}}^{\prime}\end{bmatrix}\) & Restricted hours in no-fly zone \(h_{2}\) \\ \multirow{2}{*}{\(l_{ij}\)} & The Euclidean distance of \(arc(i,j)\), km \\ \(d_{i}\) & The demand of customer \(i\), kg \\ \(v^{\mu}\) & Drone speed, km/h \\ \(v(T)\) & Vehicle speed at time \(T\), km/h \\ \(t^{1}\) & Vehicle service time, s \\ \(t^{2}\) & Drone service time, s \\ \(t^{3}\) & Drone launch time, s \\ \(t^{4}\) & Time required for the drone to ascend from the ground to horizontal flight altitude or descend from horizontal flight altitude to the ground, s \\ \(c_{1}\) & The dispatch cost per vehicle with the carried drone, yuan \\ \(c_{2}\) & The electricity price, yuan/kWh \\ \(Q^{1}\) & Vehicle payload capacity, kg \\ \(Q^{2}\) & Drone payload capacity, kg \\ \(B^{1}\) & Vehicle battery capacity, kWh \\ \(B^{2}\) & Drone battery capacity, kWh \\ \(M\) & A sufficiently large positive number \\ \(\omega\) & The road detour coefficient \\ \(\varepsilon\) & The minimum allowable state of charge for electric vehicle and drone batteries \\ \hline \multicolumn{2}{c}{**Intermediate variable**} \\ \hline \(Q_{ki}\) & \(k\) remaining load after serving node \(i\), kg \\ \(e_{ij}\) & \(k\) energy consumption of \(arc(i,j)\), kWh \\ \(e_{u,ij}\) & The \(u_{k}\) energy consumption of \(arc(i,j)\), kWh \\ \(e_{u,i}\) & The \(u_{k}\) hover energy consumption at node \(i\), kWh \\ \(t_{hij}\) & \(k\) travel time of \(arc(i,j)\), s \\ \(l_{hij}\) & \(k\) travel distance of \(arc(i,j)\), km \\ \(l_{hij}\) & The \(u_{k}\) travel distance of \(arc(i,j)\), km \\ \(T_{hj}^{d}\) & The \(k\) arrival time at node \(i\) \\ \(T_{hi}^{d}\) & The \(k\) departure time from node \(i\) \\ \(T_{hi,i}^{u}\) & The \(u_{k}\) arrival time belonged to \(k\) at node \(i\) \\ \(T_{u_{i},i}^{u_{k}}\) & The \(u_{k}\) departure time belonged to \(k\) from node \(i\) \\ \hline \multicolumn{2}{c}{**Decision variables**} \\ \hline \(x_{kij}\) & 1 if \(k\) travels \(arc(i,j)\), and 0 otherwise \\ \(y_{i,inj}^{u_{k}}\) & 1 if \(u_{k}\) belonged to \(k\) travels \(arc(i,w,j)\), and 0 otherwise \\ \(z_{kijh,t}\) & 1 if \(k\) travels \(arc(i,j)\) through the vehicle-restricted zone \(h_{1}\) during a restricted \\ \multicolumn{2}{c}{time period, and 0 otherwise} \\ \(z_{ijh,t}^{u_{k}}\) & 1 if \(u_{k}\) belonged to \(k\) travels \(arc(i,j)\) through the drone no-fly zone \(h_{2}\) during a restricted time period, and 0 otherwise \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and variables of the TDVRPD-VRZ-NFZ model. on the chosen detour method, as illustrated by the three detour methods in Figure 2a. The vertices of the rectangular vehicle-restricted zone are labeled as \(a\), \(b\), \(c\), and \(d\) in a clockwise direction. Select the detour method based on the number of line segments out of the four connecting the vertex of the vehicle-restricted zones to the customer points that intersect the restricted zone.