# Drone Arc Routing Problems and Metaheuristic Solution Approach
## 5 Computational Results
### Computational Results for DARP-DD  
Seven different test problems were generated for the DARP-DD and the information about these test problems was detailed in the previous section. The performance of the proposed metaheuristic algorithm was tested in solving these problems. The parameter values of the simulated annealing algorithm were determined as initial temperature value \((T_{0})\), final temperature value \((T_{f})\), and cooling coefficient \((\infty)\) value \((T_{0},\ T_{f},\ \infty)=(100,0.1,0.99)\). Since the initial solution of the algorithm is randomly generated, the initial temperature value is not varied according to the size of the problem. The number of iterations \((I)\) to be operated at each temperature was determined based on the problem size and is shown in Table 5.  
In order to sufficiently explore the search space for each temperature value of the algorithm, the number of iterations is increased as the size of the problem increases. The URL heuristic algorithm was used to evaluate the quality of the results obtained with the proposed simulated annealing algorithm for the DARP-DD. The URL algorithm was proposed by [PERSON] et al. [44] in the literature. In this algorithm, a giant tour is first created with path scanning with random link. Therefore, the five rules of the path scanning algorithm are not used in this algorithm, and it differs from the path scanning algorithm proposed by [PERSON] et al. [45]. During the giant tour creation phase, path scanning with random link adds the feasible unserved edge that is nearest to the last node added. Then, the created giant tour is divided into vehicle routes using [PERSON]'s splitting algorithm [46]. Moreover, the URL algorithm shows that 4 to 11 executions are enough to perform better than the path scanning algorithm [47]. The suggested simulated annealing algorithm and URL algorithm for the DARP-DD was run 10 times for each test problem and the best obtained results are presented in Table 6. \begin{table}
\begin{tabular}{c c c} \hline
**Test Problems** & **Number of Required Edges** & **Number of Iterations** \\ \hline DARP-1 & 17 & 200 \\ DARP-2 & 28 & 300 \\ DARP-3 & 44 & 750 \\ DARP-4 & 64 & 1250 \\ DARP-5 & 84 & 2000 \\ DARP-6 & 100 & 2500 \\ DARP-7 & 200 & 5000 \\ \hline \end{tabular}
\end{table} 
Table 5: The number of iterations in the SA algorithm. Table 6 presents the results obtained for seven different DARP-DD test problems. The first four columns of Table 6 contain information about the test problem. The number of drones and the energy capacity of the drones in the test problems are given in the fifth column. Their energy capacities are presented in Joules and the number of drones used in each test problem is reported. In the sixth column, the obtained results for the DARP-DD are given, which are the total traveled distances by drones. The seventh column shows the computation times given in seconds. The last two columns in Table 6 present the results obtained with the URL algorithm. Table 6 demonstrates that the proposed metaheuristic algorithm was capable of solving differently sized problems within reasonable computation times. For example, the DARP-2 problem with 6 components, 28 required edges, and 406 non-required edges was solved in about 66 s using four drones with an energy capacity of 1000 Joules. The total traveled distance by the drones was 2429.58 units for this problem. The DARP-7 problem includes 34 components, 200 required edges, and 18,915 non-required edges. For the DARP-7 problem, the solution was obtained in about 1 h by using 21 drones with an energy capacity of 1500 Joules. The total distance traveled by the drones was obtained as 20,137.39 units. Examining the results presented in Table 6, it can be seen that the proposed simulated annealing algorithm is more successful and achieves better solutions than the URL algorithm. For example, for the DARP-1 problem, the total distance traveled with the proposed algorithm is 1204.46 units, while the total distance traveled with the URL algorithm is 1324.12 units.