# Optimization of Truck-Drone Parcel Delivery Using Metaheuristics
## 4 Results
### Results of the Proposed GRASP Algorithm for TSP-D  
The GRASP HCLS variant experiment used 200 test instances, \((10,20,50,75,100)\) in size, of uniform and single-center type, and with a relative truck/drone speed of \(\alpha=1\) or \(\alpha=2\). All parameters were initialized to the settings stated in Table 1.  
Table 2 summarizes the results obtained using the proposed GRASP approach with HCLS. The values reported in the table represent average values of the best-known solutions obtained over the ten runs for each of the ten different instances for each of the (type, size, \(\alpha\)) combinations. The table is divided according to the distributions; that is, uniform followed by single-center. In each distribution, the results are sectioned by the relative drone/truck speed (\(\alpha\)). In the table, the column Size represents the instance size, the column Cost represents the average trip duration of the obtained TSP-D solution (in _secs_), the column Runtime represents the average running time (in _secs_), the column \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Value** & **Description** \\ \hline \(maxTrials\) & \(n\times 10\) & Max. number of iterations in GRASP \\ \(p_{m}\) & 0.5 & Move selection probability in LS \\ \(\beta\) & 0.02 & Learning rate for updating \(p_{m}\) \\ \(windowSize\) & \(\frac{n}{2}\) & Window size for updating \(p_{m}\) \\ \(maxIter\) & \(n\) & Max. number of iterations in HCLS \\ _convergence_ & \(n\times 0.3\) & Max. number of iterations without improvement in LS \\ \(T_{0}\) & 500 & Initial temperature in SA \\ \(T_{f}\) & 0.01 & Final temperature in SA \\ \(\gamma\) & 0.95 & SA cooling schedule coefficient \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Algorithm parameter initialization. \(Swap\) represents the average number of times in which a \(Swap\) move was applied, the column \(TwoOpt\) represents the average number of times that a \(TwoOpt\) move was applied, the column \(Swap\) Success represents the average number of times that a \(Swap\) move was successful (i.e., it improved the TSP-D solution and decreased its cost), and finally, the column \(TwoOpt\) Success represents the average number of times that a \(TwoOpt\) move was successful. Moreover, instances of size 100 were run on different machines; hence, the run time is not reported (NA). Table 2 shows that the \(Swap\) move was used more successfully in smaller-sized instances than in larger-sized instances, while the \(TwoOpt\) move was more successful in larger-sized instances. Nonetheless, the \(Swap\) move was more successful than the \(TwoOpt\) move overall. The standard deviation of the solution cost in instances with a single-uniform distribution tended to be lower than that observed in instances with a single-center distribution. In addition, the standard deviation of the solution cost in instances in which \(\alpha=1\) tended to be lower than instances in which \(\alpha=2\). Concerning the computational time, when \(\alpha=1\), the algorithm ran for longer. The run time did not differ between uniform and single-center instances. Notably, the moves were often not accepted in the GRASP HCLS variant, which suggests that it became stuck in a local optimum. To address this problem, the GRASP SA variant for LS, in which sub-optimal moves can be accepted, was examined. In this experiment, 160 test instances of sizes \((10,20,50,75)\), with both uniform and single-center types and vehicle speed configurations of \(\alpha=1\) and \(\alpha=2\) were utilized. All parameters were initialized to the settings stated in Table 1.