# Time-Dependent Vehicle Routing Problem with Drones Under Vehicle Restricted Zones and No-Fly Zones
## 7 Conclusions and Future Researches  
This paper investigates the time-dependent vehicle routing problem with drones in vehicle-restricted zones and no-fly zones (TDVRPD-VRZ-NFZ). The following conclusions are drawn: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**TN**} & \multicolumn{3}{c}{**Case 2**} & \multicolumn{3}{c}{**Case 3**} & \multicolumn{3}{c}{**Gap/\%**} \\ \cline{2-9}  & **KI/K2** & **DI/D2** & **CI/C2/C** & **KI/K2** & **DI/D2** & **CI/C2/C** & **Case2** & **Case3** \\ \hline C02 & 2/7 & 232.5/78.5 & 789.8/15.7/805.5 & 2/7 & 153.5/74.6 & 766.2/150.7/81.2 & 1.7 & \(-14\) \\ C04 & 2/10 & 212.9/115.9 & 784.3/233.8/807.6 & 2/12 & 162.4/127.8 & 768.9/25.7/794.6 & \(-3.7\) & \(-5.2\) \\ C06 & 3/20 & 608.4/209.3 & 1263.2/42.1/1305.3 & 3/26 & 384.7/300.5 & 1195.4/60.1/1255.0 & 1.6 & \(-2.3\) \\ C08 & 5/32 & 462.8/296.9 & 1939.5/597.1/1999.0 & 5/33 & 384.1/3313.3 & 1915.6/630.1/1978.6 & \(-0.7\) & \(-1.7\) \\ \hline \hline \end{tabular} Note: gap = (C(case2/case3) - C(case1))/C(case1) \(\times\) 100%.  
\end{table} 
Table 4: Results under different vehicle speed scenarios. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**TN**} & \multicolumn{3}{c}{**Case 2**} & \multicolumn{3}{c}{**Case 3**} & \multicolumn{3}{c}{**Gap/\%**} \\ \cline{2-9}  & **KI/K2** & **DI/D2** & **CI/C2/C** & **KI/K2** & **DI/D2** & **CI/C2/C** & **Case2** & **Case3** \\ \hline C02 & 2/7 & 149.8/64.2 & 764.5/12.7/777.2 & 2/8 & 298.7/131.3 & 809.3/264.8/385.7 & \(-1.9\) & 5.5 \\ C04 & 2/12 & 260.8/91.0 & 798.5/18.3/816.8 & 2/13 & 474.8/174.6 & 862.9/35.1/898.0 & \(-2.6\) & 7.1 \\ C06 & 3/17 & 339.9/194.1 & 1182.3/390.1/212.3 & 3/32 & 747.2/287.1 & 1304.9/57.7/1362.6 & \(-4.9\) & 6.1 \\ C08 & 5/29 & 354.5/254.7 & 1906.7/51.2/1957.9 & 6/35 & 373.8/339.3 & 2272.5/58.2/2330.7 & \(-2.7\) & 15.8 \\ avg & & & & & & \(-3.5\) & 10.1 \\ \hline \hline \end{tabular} Note: gap = (C(case2/case3) - C(case1))/C(case1) \(\times\) 100%.  
\end{table} 
Table 5: Results under different restricted zones and no-fly zones scenarios. Figure 9: Different vehicle speeds. (1) A mixed-integer programming model is established with the objective of minimizing delivery costs, considering constraints such as time-varying road networks, vehicle-restricted zones, no-fly zones, and vehicle-drone load energy consumption in the TDVRPD-VRZ-NFZ. Although this increases the complexity of solving the problem, it more closely reflects the delivery impacts caused by policies and environmental factors in real-world scenarios, thus enriching the research on the VRPD. (2) Gurobi is used to validate the correctness of the proposed model. Since TDVRPD-VRZ-NFZ is an NP-hard problem, Gurobi is only suitable for solving small-scale instances. The design of HGAVNS-LM provides a reliable algorithmic framework for solving TDVRPD-VRZ-NFZ. A comparison analysis with GAVNS and HGA [11] using instances of different scales shows average improvement rates of 4.7% and 12.3%, respectively, indicating that the proposed algorithm can effectively reduce delivery costs, optimize delivery plans, and improve economic efficiency. Additionally, ablation experiments combined with a learning mechanism further demonstrate the effectiveness of the learning mechanism in solving TDVRPD-VRZ-NFZ. (3) In the sensitivity analysis of different vehicle speeds, this study fits the vehicle speed using a nine-term polynomial with an R\({}^{2}\) value of 0.9908. By comparing the results with those in reference [38] and a fixed-speed scheme, deviations are observed in delivery distance, delivery cost, and the number of vehicle-drone routes. This highlights the practical significance of considering time-varying road networks in route planning for logistics companies. (4) In the sensitivity analysis of vehicle-restricted zones and no-fly zones, the presence of these zones leads to an increase in delivery costs by 1.9% to 4.9%. Therefore, it is necessary for companies to consider this factor in their route planning.