# Drone Arc Routing Problems and Metaheuristic Solution Approach
## 5 Computational Results
### Computational Results for MS-DARP-DD
For example, in DARP-7, the largest-sized test problem, the objective function value obtained with the proposed algorithm is 20724.5, but the objective function value provided by the URL algorithm is 24718.4. Therefore, an approximately 20% better result was achieved for the DARP-7 instance with the proposed algorithm. Moreover, an average of 14% better results were obtained for the other test problems. Additionally, the number of drones used with the URL algorithm is higher for all test problems. In this table, no information is provided on the computing time of the URL algorithm, because it is a heuristic method, and the computing time is quite short. For each test problem, the computing time is less than three seconds. The obtained drone routes for the DARP-1 problem when all required edges are serviced just once are shown in Figure 6. In the same problem, the drone routes obtained when (\(4-6\)) is serviced twice, edge (\(10-8\)) twice, edge (\(1-2\)) three times, and the other required edges once are presented in Figure 7. Furthermore, the drone routes for  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Test Problems**} & \multirow{3}{*}{ \begin{tabular}{c} **Number of** \\ **Required** \\ **Edges** \\ \end{tabular}  } & \multicolumn{3}{c}{**Proposed Simulated Annealing Algorithm**} & \multicolumn{3}{c}{**URL Algorithm**} \\ \cline{3-7}  & & \begin{tabular}{c} **Number of** \\ **Drones (Drone** \\ **Capacity)** \\ \end{tabular}  & \begin{tabular}{c} **Results** \\ **(Objective** \\ **Function** \\ **Values)** \\ \end{tabular}  & \begin{tabular}{c} **Computation** \\ **Times** \\ **(Seconds)** \\ \end{tabular}  & \ \begin{tabular}{c} **Number of** \\ **Drones (Drone** \\ **Capacity)** \\ \end{tabular}  & \begin{tabular}{c} **Results** \\ **(Objective** \\ **Function** \\ **Values)** \\ \end{tabular}  \\ \hline DARP-1 & 17 & 5 (500 J) & 1417.3 & 32.5 & 6 (500 J) & 1551.81 \\ DARP-2 & 28 & 5 (1000 J) & 2707.22 & 70.2 & 6 (1000 J) & 3048.65 \\ DARP-3 & 44 & 7 (1000 J) &  
the DARP-6 problem, where all required edges are serviced only once, are presented in Appendix A. In the same problem, the drone routes obtained when edge (\(60-61\)) is serviced twice, edge (\(70-71\)) five times, edge (\(95-96\)) three times, and the other required edges once are presented in Appendix B. In Figure 7, 21 required edges were serviced by using five drones with an energy capacity of 500 Joules. The edges in each drone's route are shown in a different color. Dashed edges indicate deadheading on the relevant edge. Drone routes were obtained as follows:  
_First drone route (blue-colored route)_: \(\mathrm{d}(\mathrm{depot}-4)-(4-6)-(6-5)-(5-4)-\mathrm{d}(4-\mathrm{depot})\). _Second drone route (red-colored route)_: \(\mathrm{d}(\mathrm{depot}-2)-(2-3)-\mathrm{d}(3-11)-(11-12)-\mathrm{d}(12-6)-(6 -4)-\mathrm{d}(4-\mathrm{depot})\). _Third drone route (green-colored route)_: \(\mathrm{d}(\mathrm{depot}-2)-(2-1)-(1-2)-(2-1)-(1-3)-\mathrm{d}(3-\mathrm{ depot})\). _Forth drone route (purple-colored route)_: \(\mathrm{d}(\mathrm{depot}-16)-(16-15)-(15-14)-(14-13)-(13-18)-(18-17)-(17-16)- \mathrm{d}(16-8)-(8-10)-\mathrm{d}(10-\mathrm{depot})\). _Fifth drone route (orange-colored route)_: \(\mathrm{d}(\mathrm{depot}-10)-(10-8)-(8-7)-(7-9)-(9-10)-\mathrm{d}(10-\mathrm{ depot})\). When the obtained drone routes are evaluated, it is revealed that Drone-3 served the edge (\(1-2\)) three times in succession. The edge (\(4-6\)) was served by Drone-1 and Drone-2, whereas the edge (\(10-8\)) was served by Drone-4 and Drone-5. In other words, the multiple service demands of these edges were met by different drones. Examining the drone routes reveals that multiple service to the required edges need not be provided by the same drone. Multiple service can be carried out by the same drone if it is advantageous with regard to the traveled distance. On the other hand, the number of required edges on the graph increases in the case of multiple services and the number of drones increases to five.