An Embedded Hamiltonian Graph-Guided Heuristic Algorithm for Two-Echelon Vehicle Routing Problem

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.