# Optimization of Truck-Drone Parcel Delivery Using Metaheuristics
## 4 Results
### Results of the Proposed GRASP Algorithm for TSP-D
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Size** & **Cost** & **Runtime** & \(Swap\) & \(TwoOpt\) & \
\begin{tabular}{c} \(Swap\) \\ Success \\ \end{tabular} &
\begin{tabular}{c} \(TwoOpt\) \\ Success \\ \end{tabular} \\ \hline \multicolumn{8}{c}{**Uniform Distribution**} \\ \hline \multirow{8}{*}{\(\alpha=1\)} & 10 & 281.46 \(\pm\) 1.38 & 4.05 & 5107.89 & 4709.60 & 338.84 & 0.02 \\  & 20 & 363.83 \(\pm\) 1.64 & 11.13 & 4141.19 & 4058.06 & 89.94 & 0.13 \\  & 50 & 554.10 \(\pm\) 0.57 & 57.01 & 3848.66 & 3847.08 & 23.01 & 0.07 \\  & 75 & 647.18 \(\pm\) 1.12 & 121.85 & 3804.58 & 3784.04 & 9.60 & 0.13 \\  & 100 & 731.32 \(\pm\) 0.78 & NA & 12,867.77 & 12,851.60 & 13.75 & 0.17 \\  & Avg. & 516.99 \(\pm\) 1.10 & 48.51 & 5954.02 & 5850.07 & 95.03 & 0.10 \\ \hline \multirow{8}{*}{\(\alpha=2\)} & 10 & 224.74 \(\pm\) 3.67 & 2.32 & 4246.95 & 4010.12 & 105.2 & 0.00 \\  & 20 & 306.39 \(\pm\) 4.55 & 4.98 & 3963.70 & 3888.41 & 43.81 & 0.04 \\  & 50 & 472.88 \(\pm\) 2.90 & 18.89 & 3807.67 & 3806.22 & 11.17 & 1.43 \\  & 75 & 551.92 \(\pm\) 3.12 & 32.69 & 3795.46 & 3791.46 & 7.98 & 1.42 \\  & 100 & 632.30 \(\pm\) 2.41 & NA & 15,166.70 & 15,204.00 & 19.08 & 3.36 \\  & Avg. & 437.65 \(\pm\) 3.33 & 14.72 & 6196.10 & 6140.04 & 37.45 & 1.25 \\ \hline \multicolumn{8}{c}{**Single Center**} \\ \hline \multirow{8}{*}{\(\alpha=1\)} & 10 & 385.49 \(\pm\) 8.14 & 4.235 & 5500.51 & 5027.52 & 484.10 & 1.00 \\  & 20 & 546.13 \(\pm\) 3.91 & 11.19 & 4172.03 & 4085.86 & 101.39 & 0.44 \\  & 50 & 776.97 \(\pm\) 1.28 & 56.69 & 3838.70 & 3835.01 & 21.20 & 0.11 \\  & 75 & 1039.74 \(\pm\) 1.47 & 121.95 & 3789.25 & 3812.06 & 11.97 & 0.10 \\  & 100 & 1238.28 \(\pm\) 1.60 & NA & 15,177.63 & 15,201.60 & 23.89 & 0.10 \\  & Avg. & 797.32 \(\pm\) 3.28 & 48.52 & 6495.62 & 6392.45 & 128.51 & 0.35 \\ \hline \multirow{8}{*}{\(\alpha=2\)} & 10 & 277.00 \(\pm\) 6.17 & 2.39 & 4406.14 & 4143.45 & 158.16 & 5.07 \\  & 20 & 402.92 \(\pm\) 7.09 & 4.53 & 3920.74 & 3856.99 & 33.13 & 2.35 \\ \cline{1-1}  & 50 & 604.79 \(\pm\) 6.22 & 13.22 & 3787.35 & 3792.57 & 7.09 & 1.19 \\ \cline{1-1}  & 75 & 851.01 \(\pm\) 7.03 & 23.03 & 3774.44 & 3775.38 & 3.83 & 0.83 \\ \cline{1-1}  & 100 & 995.72 \(\pm\) 9.10 & NA & 15,145.04 & 15,128.24 & 10.71 & 5.29 \\ \cline{1-1}  & Avg. & 626.29 \(\pm\) 7.12 & 10.80 & 6206.74 & 6139.33 & 42.58 & 2.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results obtained with the proposed GRASP approach, HCLS variant. Table 3 summarizes the results obtained using the proposed GRASP approach with the SA variant. The values reported in the table represent the average values of the best known solutions obtained over the ten runs for each of the ten different instances in each of the (type, size, \(\alpha\)) combinations. The column definitions are the same as those used in Table 2.  
Table 3 shows that, as the instance size increased, the success ratios of both moves also increased. Nonetheless, the \(Swap\) move was still more successful than the \(TwoOpt\) move in all cases, except for the case (type = single-center, size = 75, \(\alpha=2\)). The standard deviation of the solution cost in uniform instances tended to be slightly lower than that for single-centered instances. We also observed that the standard deviation of the solution cost in instances in which \(\alpha=1\) tended to be lower, compared to that for instances having \(\alpha=2\). As can be seen in Figure 10, overall, the GRASP SA variant performed worse than or approximately equal to the GRASP HCLS variant. The overall average solution cost for the models obtained using the GRASP HCLS variant was 517.79 _secs_, compared to 556.90 _secs_ for the models generated using the GRASP SA variant. The standard deviation of the solution cost obtained with the GRASP SA models was higher than that of the GRASP HCLS model. A possible explanation for the large gap between runs of instances of the same size in the former is that the LS with SA accepts sub-optimal moves. The Wilcoxon signed-rank test was used to compare the models obtained with the two GRASP variants.