ABSTRACT Two integer linear programming models are developed for the unrestricted vehicle routing problem with two-dimensional loading constraints. The first one is a complete model, and the other uses valid inequalities to guarantee that routes are connected and respect the two-dimensional loading constraints. The models are solved with a branch-and-cut algorithm. Computational experiments on benchmark instances showed the complete model has allowed optimal solutions for 5% of the instances, while the second model optimally solved 64% of the instances. Given the superior performance of the second model, we adapted it to handle the sequential variant of the problem, which is harder, and then optimal solutions were obtained for 46% of the instances within the given time limit. The second model compared with a branch-and-cut algorithm from the literature found identical or better solutions for all the instances.

ABSTRACT This paper presents an approach to deal with horizontal cargo stability in container loading problems. Cargo stability has been explored mainly with support factors that constrain the minimum area of each box’s faces to be supported by other boxes. On the other hand, we propose an approach based on the static equilibrium of rigid bodies to check the static stability of a given packing. The approach is used as a cutting plane routine in a branch-and-cut framework to the single container loading problem. This framework considers the resolution of an integer linear programming model to obtain feasible packings next checked with the proposed approach to avoid unstable packings. The computational experiments consider 180 benchmark instances on which stable solutions of the proposed approach have better container fill rates than the support factor approach. In terms of lateral support, the proposed approach provides the minimum value inferior to 70% on average, which is satisfactorily smaller and less restrictive than the full support. Results also indicate that more unstable solutions emerge from refined grids and fewer types of boxes available.