Two-echelon vehicle routing problem (2E-VRP) is an NP-hard combinatorial optimization problem and a basic mathematical model of modern city logistics. While it is difficult to obtain the optimal solution of 2E-VRP, this study finds a breakthrough that the structure of the optimal route planning for 2E-VRP is usually an embedded Hamiltonian graph. In the graph, routes can be drawn in a planar graph as Hamiltonian circuits without intersections. Based on this finding, an embedded Hamiltonian graph-guided heuristic algorithm is proposed to solve 2E-VRP. As a crucial part of the algorithm, an initialization scheme is designed to search for the farthest vertices from each route and insert the rest of the vertices. In the satellite-adjustment process, a dynamic adjustment for satellites scheme is proposed to adjust the state of satellites. The two schemes aim to construct Hamiltonian circuits with few intersections. Experiments have been conducted on 207 instances to demonstrate the effect of the proposed algorithm on solving 2E-VRP. Experimental results show that the proposed algorithm can obtain more solutions of 2E-VRP with significantly smaller objective-function values. Furthermore, the number of intersections in routes generated by the proposed algorithm is much less than those obtained by the compared algorithms. With the use of the two schemes, the embedded Hamiltonian graph-guided heuristic algorithm significantly outperforms the compared algorithms for 2E-VRP.

中文翻译：

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一种嵌入式哈密顿图引导启发式​​算法解决两级车辆路径问题


两梯队车辆路径问题（2E-VRP）是一个NP难组合优化问题，是现代城市物流的基本数学模型。虽然2E-VRP的最优解很难获得，但本研究的突破在于2E-VRP的最优路径规划的结构通常是嵌入的哈密顿图。在图中，路线可以在平面图中绘制为没有交叉点的哈密顿回路。基于这一发现，提出了一种嵌入式哈密顿图引导启发式​​算法来求解 2E-VRP。作为算法的关键部分，初始化方案被设计为搜索距每条路线最远的顶点并插入其余的顶点。在卫星调整过程中，提出了一种卫星动态调整方案来调整卫星的状态。这两种方案旨在构建具有很少交点的哈密顿电路。在207个实例上进行了实验，证明了所提出的算法在求解2E-VRP上的效果。实验结果表明，该算法能够以更小的目标函数值获得更多的2E-VRP解。此外，所提出的算法生成的路线中的交叉口数量比对比算法获得的交叉口数量少得多。通过使用这两种方案，嵌入式哈密顿图引导启发式​​算法显着优于 2E-VRP 的对比算法。