Abstract: Both vehicle routing problem (VRP) and bin packing problem (BPP) are important, typical combinatorial optimization problems, and are frequently and independently studied in the past few decades. In recent years, some attention has been devoted to their combined optimization, called 2L-CVRP (two-dimensional capacitated vehicle routing problem), which is a combination of two important NP-hard optimization problems in distribution logistics. In the 2L-CVRP, clients demand are defined as sets of rectangular weighted items, while vehicles have a weight capacity and a rectangular two-dimensional loading surface. The objective of the 2L-CVPR is to serve all the customers and load the corresponding items into the vehicles, through a road network, with minimum total cost.
Problem description:In this paper, a multi-depots capacitated vehicle routing problem with time window where client demand is composed of two-dimensional weighted items is addressed (2L-CVRP-MDTM).The model and an improved hybrid algorithm to solve the problem are proposed. The hybrid algorithm GLS-QPSO is composed by the Quantum-behaved Particle Swarm Optimization algorithm which is applied to solve the vehicle routing problem and Guided Local Search algorithm which is used to identify the feasible packing solutions.
Method: New loading heuristic rules are proposed in Guided Local Search algorithm based on the principle of minimize the waste to get better results of the matching relationship between the items and the corresponding packing positions in shorter time. Through the algorithm we can get the tours, and identify the clients covered by the vehicles, as well as vehicles in the routes. All the items demanded by the clients need to be loaded in the vehicles with the related constrains. In fact, there are two aspects that packing algorithm needs to consider. One is to determine the next loading item, and another is to determine the feasible loading position.
Computational experiments: Twenty of these 2L-CVRP standard instances are selected randomly, and loading surface (L,W) is fixed as (40,20) for all instances. The 20 selected instances are changed to two depots instances to testify the proposed algorithm in this paper by adding a new depot. We calculate the center of the clients, and then select a point between the first depot and the center randomly as the second depot.
Results: We compare the improved hybrid algorithm with the known best solutions, and the other is based on the new instances which include multi-depots with time window constrain. We calculate the center of the clients, and then select a point between the first depot and the center randomly as the second depot. The results of the numerical experiments show that the proposed hybrid algorithm has better performances in terms of both the calculation results and computing time comparing with the selected recent literature. The computational results show that the proposed algorithm is effective in finding high quality solutions and the practical recharging option may significantly improve the 2L-CVRP-MDTM which a new variant of the 2L-CVRP considering vehicles start from several different depots with time window constrains.

Key words: vehicle routing problem, two-dimensional packing problem, Quantum-behaved Particle Swarm Optimization algorithm, multi-depots, time window