## 6 Conclusions  
The main goal of this paper is to formulate and solve the electric vehicle routing problem considering time windows and recharging stations under energy consumption uncertainty (EVRPTW-ECU), which is very common in real life. We establish a robust optimization model for this problem and develop a novel hybrid adaptive large neighborhood search metaheuristic (NHALNS) to solve it. Numerical experiments demonstrate that the proposed NHALNS can solve the small-scale instances optimally and outperform some recently developed algorithms, such as LNS + SP adapted from [PERSON] et al. [9] and Hybrid VNS/TS from [PERSON] et al. [23], in both effectiveness and efficiency. Moreover, compared with the deterministic model and the traditional Min-Max robust optimization model, the proposed robust optimization model of this paper can produce the solution which effectively achieves the tradeoff between the cost performance and risk avoidance.  
Logistic distribution involving electronic vehicles is an increasingly popular issue in current logistic management. Although plenty of investigations have been conducted in this area, there are still many problems unsolved, especially those from real applications. The work of this article mainly considers the uncertain consumption of electricity and makes some extensions based on the existing investigations. Of course, there are some other factors relative to the practical demands worthwhile of being considered. For example, various charging techniques for electric vehicles have been developed recently which enable the charging process to be more efficient and flexible. Hence, how to plan the transportation  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Inst** & **TC (D)** & **TC (R)** & **TC (W)** & **Gap (TC (R)/TC (W))** & **D@W** \\ \hline c103C15 & 2694.25 & 2725.22 & 2731.45 & \(-\)0.23\% & 2849.75 \\ c106C15 & 2620.46 & 2661.25 & 2664.35 & \(-\)0.12\% & \(-\) \\ c208C15 & 2648.72 & 2679.09 & 2683.18 & \(-\)0.15\% & 2779.13 \\ rc108C15 & 3900.37 & 3945.51 & 3946.37 & \(-\)0.02\% & 4066.70 \\ rc202C15 & 2713.29 & 2769.65 & 2775.67 & \(-\)0.22\% & \(-\) \\ c101\_21 & 15,267.77 & 15,429.22 & 15,612.47 & \(-\)1.17\% & 15,732.14 \\ r101\_21 & 24,192.97 & 25,660.31 & 25,931.43 & \(-\)1.05\% & \(-\) \\ r103\_21 & 19,046.96 & 19,298.93 & 19,511.90 & \(-\)1.09\% & \(-\) \\ r201\_21 & 8629.40 & 9584.84 & 11,283.29 & \(-\)15.05\% & \(-\) \\ r202\_21 & 8193.33 & 9407.09 & 10,823.47 & \(-\)13.09\% & \(-\) \\ r203\_21 & 7980.15 & 8131.28 & 10,706.00 & \(-\)24.05\% & \(-\) \\ rc101\_21 & 22,027.11 & 22,716.77 & 24,803.43 & \(-\)10.59\% & 24,807.63 \\ rc102\_21 & 21,943.43 & 21,993.52 & 24,962.41 & \(-\)11.89\% & 25,835.14 \\ rc103\_21 & 19,303.11 & 19,385.54 & 20,796.89 & \(-\)6.79\% & 21,164.98 \\ rc201\_21 & 10,932.03 & 12,036.92 & 13,688.79 & \(-\)12.07\% & \(-\) \\ \hline \end{tabular}
\end{table}
Table 7: Results for the deterministic and robust model.  
routes for the electric vehicles with multiple charging modes (or different energy supply modes) under uncertainty is deserving of further study. We need to further explore efficient operators to improve the search efficiency of the proposed algorithm.  
Conceptualization, D.W.; investigation, D.W. and H.Z.; methodology, D.W. and W.Z.; validation, D.W.; writing--original draft preparation, all authors. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of China [grant number: 71471007]; Youth Foundation of Beijing Wuzi University [grant number: 2023 XJQN18].  
Not applicable.  
The data that support the findings of this study are available from the corresponding author upon reasonable request.  
The authors declare no conflicts of interest.