# Time-Dependent Vehicle Routing Problem with Drones Under Vehicle Restricted Zones and No-Fly Zones
## 6 Numerical Experiments
### Model Validation  
Based on nine small-scale test cases, Gurobi is used to solve the mathematical model, and the results are compared with those obtained from HGAVNS-LM. The maximum solving time for Gurobi is set to 1200 s, and the results are presented in Table 2. In the table, TN represents the test case's name. N is the number of customers. N1 is the number of customers in restricted zones. N2 is the number of customer points in no-fly zones. K1 is the number of vehicle routes. K2 is the number of drone routes. D1 is the total travel distance for vehicles, in kilometers. D2 is the total travel distance for drones, in kilometers. C is the total delivery cost, in CNY. C1 is the delivery cost for vehicles, in CNY. C2 is the total delivery cost for drones, in CNY. RT is the runtime, in seconds. A comparison shows that the deviation between the solutions from Gurobi and HGAVNS-LM ranges from \(-0.2\%\) to \(0.3\%\), indicating that the solutions are very close. However, the average solving time for Gurobi is \(666.1\) s, while HGAVNS-LM requires only \(2.1\) s, resulting in a \(99.68\%\) reduction in solving time.  
For a customer scale of 6, the solution quality from Gurobi and HGAVNS-LM is nearly identical. However, when the customer scale increases to 8 or more, the exact algorithm used by Gurobi leads to a significant increase in computational complexity, causing the solving time to rise substantially. In some instances, such as A07, A08, and A09, Gurobi is unable to obtain the optimal solution within the set maximum time limit.  
In conclusion, the correctness of the model is validated through comparison with Gurobi, but HGAVNS-LM demonstrates clear advantages in both solving time and computational efficiency.  
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**TN**} & \multirow{2}{*}{**N/N1/N2**} & \multicolumn{4}{c}{**Gurobi**} & \multicolumn{4}{c}{**HGAVNS-LM**} & \multirow{2}{*}{**Gap/\%**} \\ \cline{2-3} \cline{5-10}  & & **K1/K2** & & & & & & & & & \\ \hline A01 & 6/1/1 & 1/2 & 41.8/15.3 & 372.6/31.375.7 & 3.6 & 1/2 & 41.8/15.3 & 372.6/3.1/375.7 & 1.5 & 0.0 \\ A02 & 6/1/1 & 1/2 & 37.2/16.7 & 371.2/34.3/374.6 & 7.8 & 1/2 & 37.2/16.7 & 371.2/34.3/374.6 & 1.3 & 0.0 \\ A03 & 6/2/1 & 1/3 & 56.6/28.4 & 377.1/5.7/382.8 & 12.4 & 1/3 & 56.6/28.4 & 377.1/5.7/382.8 & 1.6 & 0.0 \\ A04 & 8/2/1 & 1/3 & 37.4/30.3 & 371.3/6.1/377.4 & 680.9 & 1/3 & 37.4/30.3 & 371.3/6.1/377.4 & 2.1 & 0.0 \\ A05 & 8/2/1 & 1/3 & 59.3/29.9 & 377.9/6.0/383.9 & 710.4 & 1/3 & 59.3/29.9 & 377.9/6.0/383.9 & 1.5 & 0.0 \\ A06 & 10/3/2 & 1/3 & 59.8/32.8 & 378.1/6.6/384.7 & 980.0 & 1/3 & 64.8/30.8 & 379.5/6.2/385.7 & 2.3 & 0.3 \\ A07 & 10/3/1 & 1/3 & 53.5/29.2 & 376.2/5.9/382.1 & 1200 & 1/3 & 52.0/27.5 & 376.1/5.5/381.6 & 2.6 & \(-0.1\) \\ A08 & 12/3/2 & 1/4 & 63.0/46.3 & 379.1/9.3/388.4 & 1200 & 1/4 & 62.4/43.8 & 378.9/8.8/387.7 & 2.8 & \(-0.2\) \\ A09 & 12/3/2 & 1/4 & 49.0/42.3 & 374.8/8.5/383.3 & 1200 & 1/4 & 46.2/43.2 & 374.0/8.7/382.7 & 3.3 & \(-0.2\) \\ Avg & & & & & 666.1 & & & & & & \\ \hline \hline \end{tabular} Note: gap = (CHGAVNS-LM) \(-\) C(Gurobi)/C(Gurobi) \(\times\) 100%.  
\end{table}
Table 2: Model validation results.  
Figure 8: Signal-to-noise ratio response diagram.