# Drone Arc Routing Problems and Metaheuristic Solution Approach
## 5 Computational Results
### Performance of the Simulated Annealing Algorithm Components  
The performance of the components of the suggested algorithm was analyzed using the DARP-6 test problem. This problem has 17 components, 100 required edges, and 4753 non-required edges. The parameter values of the simulated annealing algorithm were determined as the initial temperature value \((T_{0})\), the final temperature value \((T_{f})\), the cooling rate \((\infty)\), and the number of iteration \((I)\) values \((T_{0},\,T_{f},\,\infty,I)=(100,0.1,0.99,2500)\)  
Figure 7: The obtained drone routes for MS-DARP-DD-1. When the DARP-6 problem was solved with the suggested algorithm, eight drones with an energy capacity of 1500 Joules were used in the obtained solution and the total distance traveled by the drones was 6916.44 units. The computation time of the algorithm for this problem is approximately 1319 s. The convergence graph of the suggested algorithm for the DARP-6 problem is presented in Figure 8.  
Figure 8 shows the convergence graph of the suggested algorithm for DARP-6 based on the objective function values obtained between the initial and the final temperature values. Since the initial solution of the algorithm is randomly generated, the search process is initiated with solutions having very large objective function values. The objective function value of the first feasible solution obtained at the initial temperature values is 9017. However, the algorithm's well-designed components allowed for it to quickly converge to high-quality solution regions, and the objective function values were improved. For example, the obtained solution's objective function value was 7354 when the algorithm's temperature value was 47, and 7145 when the temperature value decreased to 21. At the end of the algorithm, a solution with a value of 6916.44 was reached owing to the intensification mechanism, which allowed for the algorithm to search more thoroughly for quality solution regions at lower temperature levels. Furthermore, the cooling rate in the geometric cooling schedule was set as close to 1 as possible. Thus, the algorithm was able to thoroughly search the solution space at each temperature value. This also enabled the algorithm to access the quality solution regions in the solution space. A study was conducted on the impact of the dynamic penalty function, which is another component of the algorithm and used as a constraint-handling strategy, on the solution quality. When the DARP-6 problem was solved using the dynamic penalty function within the proposed algorithm's structure, the objective function value was obtained as 6916.44. On the other hand, the penalty coefficient value (\(k\)) was taken as 1 in the dynamic penalty function, thus obtaining a generic (fixed) penalty function. Then, the DARP-6 problem was solved using the generic penalty function within the proposed algorithm's structure. In this case, the objective function value was obtained as 7084.11. Therefore, it can be said that the dynamic penalty function improves solution quality by approximately 2.4% for DARP-6. Moreover, the same study was conducted for other test problems. On examining the results, it was found that the dynamic penalty function improves the solution quality by approximately 0.5% for DARP-1, DARP-2, and DARP-3; by approximately 1.2% for DARP-4 and DARP-5; and by approximately 2% for DARP-6 and DARP-7. The initial temperature value and the number of iterations performed at each temperature were analyzed. The proposed algorithm was run for the solution of the DARP-6 problem with different initial temperatures and the number of iterations to be performed at each temperature. The obtained results of this experiment are presented in Table 9.  
Figure 8: Convergence graph of the proposed algorithm for the DARP-6 problem.