## 5 Numerical Experiments and Analysis
### Computational Results and Analysis
#### 5.2.2 Comparison Between NHALNS and Other Heuristics
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Inst**} & \multicolumn{4}{c}{**TC (NHALNS)**} & \multicolumn{4}{c}{**TC (LNS + SP)**} & \multicolumn{4}{c}{**Gap (\%)**} \\ \cline{2-11}  & **Best** & **Avg** & **SD** & **Time (s)** & **Best** & **Avg** & **SD** & **Time (s)** & **TC** & **Time** \\ \hline c101\_21 & 15,386.45 & 15,390.25 & 10.14 & 4113.65 & 15,386.45 & 15,389.45 & 6.02 & 9031.33 & 0.00 & \(-\)54.45 \\ c102\_21 & 15,361.41 & 15,372.01 & 14.17 & 4001.65 & 14,210.99 & 14,212.19 & 3.60 & 12,512.91 & 0.08 & \(-\)68.02 \\ c103\_21 & 12,926.37 & 12,926.37 & 0.00 & 5396.92 & 12,906.23 & 12,910.63 & 9.08 & 10,726.93 & 0.00 & \(-\)49.69 \\ c104\_21 & 12,812.92 & 12,816.12 & 9.60 & 5016.19 & 11,786.63 & 11,787.03 & 1.20 & 11,964.81 & 0.09 & \(-\)58.08 \\ c105\_21 & 12,994.03 & 12,995.23 & 3.60 & 5121.66 & 14,177.29 & 14,177.29 & 0.00 & 8067.58 & \(-\)0.08 & \(-\)36.52 \\ r101\_21 & 25,622.42 & 25,622.42 & 0.00 & 4414.41 & 25,582.63 & 25,584.43 & 2.89 & 8560.59 & 0.00 & \(-\)48.43 \\ r102\_21 & 21,858.73 & 21,858.93 & 0.60 & 6115.91 & 22,996.23 & 22,996.43 & 0.60 & 10,448.70 & \(-\)0.05 & \(-\)41.47 \\ r103\_21 & 19,294.48 & 19,294.78 & 0.90 & 8025.97 & 20,418.58 & 20,418.78 & 0.60 & 13,093.19 & \(-\)0.06 & \(-\)38.70 \\ r104\_21 & 15,474.68 & 15,475.28 & 1.80 & 5031.07 & 16,690.11 & 16,690.91 & 2.40 & 13,279.45 & \(-\)0.07 & \(-\)62.11 \\ r105\_21 & 19,406.86 & 19,406.86 & 0.00 & 5693.99 & 20,513.66 & 20,514.36 & 2.10 & 9734.61 & \(-\)0.05 & \(-\)41.51 \\ rc101\_21 & 22,130.62 & 22,131.02 & 1.20 & 5495.84 & 23,313.34 & 23,314.04 & 2.10 & 9105.41 & \(-\)0.05 & \(-\)39.64 \\ rc102\_21 & 21,989.54 & 21,989.64 & 0.30 & 4600.63 & 21,981.39 & 21,981.69 & 0.90 & 13,888.33 & 0.00 & \(-\)66.87 \\ rc103\_21 & 24,521.43 & 24,521.73 & 0.90 & 4869.08 & 19,356.55 & 19,357.55 & 3.00 & 10,200.26 & 0.27 & \(-\)52.27 \\ rc104\_21 & 16,845.49 & 16,846.49 & 3.00 & 6040.07 & 18,059.76 & 18,060.06 & 0.90 & 16,055.42 & \(-\)0.07 & \(-\)62.38 \\ rc105\_21 & 20,689.04 & 20,689.04 & 0.00 & 4960.18 & 20,713.66 & 20,714.66 & 3.00 & 11,366.81 & 0.00 & \(-\)56.36 \\ \hline \hline \end{tabular} 
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**St.**} & \multicolumn{4}{c}{**Gap (TC)** = ((Avg (NHALNS) - Avg (LNS + SP))/Avg(LNS + SP)) \(\times\) 100%. Gap (Time) = ((Time (NHALNS) - Time (LNS + SP))/Time (LNS + SP)) \(\times\) 100%. Next, we make a more comprehensive comparison for the proposed algorithm with other sophisticated heuristic methods. The VNS algorithm combined with Tabu Search (TS) (Hybrid VNS/TS) is widely adopted to solve the vehicle routing problems, such as the EVRPTW [23], VRP with Simultaneous Pickup and Delivery with Time Limit [48], the MDGVRP [49], and so on. Hence, Hybrid VNS/TS is also selected for the comparison with NHALNS. The termination criterion is to run both algorithms for 2000 s. The results are shown in Table 6, where N.V is the number of EVs used. Clearly, NHALNS can achieve a solution with remarkably higher performance within the same computation time for all the instances, compared with Hybrid VNS/TS. Another notable finding is that the number of required EVs optimized by our algorithm is significantly less than that optimized by Hybrid VNS/TS. This implies that the utilization of the vehicles can be effectively improved by NHALNS, which is also an important concern in practice. Figure 2: Convergence of NHALNS and LNS + SP. (**a**) The convergence of the two algorithms for instance c104_21. (**b**) The convergence of the two algorithms for instance c105_21. The comparison for the convergence pattern of the two algorithms is illustrated in Figure 3, as evidenced by instance r112_21. It is further confirmed that NHALNS is much more efficient in solving EVRPTW-ECU than Hybrid VNS/TS, especially for large-sized instances.