A branch-and-cut approach for the vehicle routing problem with loading constraints

We consider a transport system problem which involves route planning for vehicles with containers attached, and cargo accommodation. The specific problem considered is the vehicle routing problem with loading constraints. Its input consists of a weighted undirected graph where vertices corresponds to clients and edges costs represents the cost to move the vehicle along that edge. Each vertex has a demand for a set of items (which are two or three dimensional rectangles). There is also a special vertex that represents a depot where all items are stored at first. The problem is to determine a set of routes, each one leaving the depot, with minimum total cost, such that all clients are visited exactly once. The corresponding cargo for the clients of each route must fit into the container. As it would be too expensive to remove and rearrange the cargo at each point in the route, it is interesting that the goods to be unloaded in a client must be removed without moving the remaining cargo. These are called unloading constraints. In this paper we describe a branch-and-cut algorithm for the vehicle routing problem with loading constraints, using several techniques to prune the search tree. The problem of finding feasible loadings into the bins are solved using techniques adapted from the literature. These techniques involves branch-and-bound, constraint programming and metaheuristics. The efficiency of the algorithms are evaluated by an experimental analysis with instances from the literature, and compared with other algorithms.