## 5 Numerical Examples
### Comparative Experiments
#### 5.2.1 Experiment 1
It is noted that each experiment has been executed 10 times and the best value per set of 10 executions is recorded. The table includes the number of AMRs (\(m\)), travel distance (\(\sum d_{ij}\)), and objective value, where \(f_{VNS}\) and \(f_{TS}\) are the best objective values obtained by VNS and TS, respectively. Let \(G_{1}=(f_{VNS}-f_{TS})/f_{VNS}*100\). If \(G_{1}<0\), the VNS algorithm performs better. The running time of both algorithms is set to be equal. The experimental results presented in Table 8 reveal a comparison between the TS algorithm and the VNS algorithm. The findings indicate that the VNS algorithm generates a higher quality solution than the TS algorithm (\(G_{1}=-1.26\%\)). It is worth noting that if the distribution of request nodes is relatively random, such as R101 and R201, the VNS algorithm obtains the optimal solution that is inferior to the TS algorithm. However, for the remaining four instances, the VNS algorithm generates higher quality solutions than TS algorithm. The results of this experiment verify that the VNS algorithm is effective. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{6}{c}{**CPLEX**} & \multicolumn{6}{c}{**VNS**} \\ \cline{2-13}  & \multirow{2}{*}{\(m\)} & \multirow{2}{*}{\(\sum d_{ij}\)} & \multirow{2}{*}{**Time (s)**} & & \multicolumn{2}{c}{\(m\)} & \multicolumn{2}{c}{\(\sum d_{ij}\)} & \multicolumn{2}{c}{**Time (s)**} & \multicolumn{2}{c}{\(f=\xi_{1}m+\xi_{2}\sum d_{ij}\)} & \multirow{2}{*}{**Gap** (\%)} \\ \cline{3-3} \cline{5-13}  & & & & & & & & & & & & & \\ \cline{3-3} \cline{5-13}  & & & & & & & & & & & & & \\ \hline R101-15 & 12 & 248.91 & 1800 & 362.48 12 & 12 & 603 & 611.0 & 31.06 & 32.03 & 366.02 & 366.11 & 0.02 & \(-0.66\) \\ R201-15 & 3 & 327.35 & 12.53 & 93.27 3 & 3 & 3.6 & 397.4 & 378.1 & 24.59 & 25.05 & 93.97 & 111.78 & 15.9 & \(-0.75\) \\ C101-15 & 4 & 104.80 & 1800 & 121.04 4 & 4.6 & 220 & 270.1 & 25.37 & 26.17 & 122 & 140.7 & 13.2 & \(-0.95\) \\ C201-15 & 2 & 189.26 & 8.47 & 61.89 2 & 2 & 2.6 & 211.2 & 245.3 & 23.16 & 24.84 & 62.11 & 80.45 & 22.7 & \(-0.35\) \\ RC102-15 & 12 & 186.94 & 1800 & 361.86 12 & 12.2 & 868.2 & 881.1 & 30.69 & 31.44 & 368.6 & 374.8 & \(1.65\) & \(-1.88\) \\ RC201-15 & 3 & 198.30 & 12.51 & 91.98 3 & 3.2 & 310 & 350 & 22.42 & 23.97 & 93.10 & 99.50 & 6.43 & \(-1.22\) \\ Average & & & & & & & & & & & & 10.0 & \(-0.96\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison between the CPLEX and the VNS algorithms for the small instances. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{3}{c}{**TS**} & \multicolumn{3}{c}{**VNS**} & \multirow{2}{*}{\(G_{1}\)} \\ \cline{2-2} \cline{4-7}  & \(m\) & \(\sum d_{ij}\) & \(f_{TS}\) & \(m\) & \(\sum d_{ij}\) & \(f_{VNS}\) \\ \hline R101 & 87 & 121.92 & 2731.93 & 87 & 122.2 & 2732.2 & 0.01 \\ R201 & 47 & 82.791 & 1492.79 & 47 & 83.59 & 1493.6 & 0.05 \\ C101 & 62 & 148.6 & 2008.6 & 62 & 145.3 & 2005.3 & \(-0.16\) \\ C201 & 37 & 109.8 & 1219.8 & 35 & 118.6 & 1168.6 & \(-4.38\) \\ RC102 & 87 & 148.31 & 2758.31 & 86 & 147.1 & 2727.1 & \(-1.14\) \\ RC201 & 51 & 108.9 & 1638.9 & 50 & 107.8 & 1607.8 & \(-1.93\) \\ Average & 61.83 & 120.05 & 1975.1 & 61.2 & 120.7 & 1955.8 & \(-1.26\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison between the TS and the VNS algorithms for the large instances.