An optimization model for the vehicle routing problem with practical three-dimensional loading constraints

In this paper, we present an integer linear programming model for the vehicle routing problem that considers real-world three-dimensional (3D) loading constraints. In this problem, a set of customers make requests of goods that are wrapped up in boxes, and the objective is to find minimum cost delivery routes for a set of identical vehicles that, departing from a depot, visit all customers only once and return to the depot. Apart from the usual 3D container loading constraints that ensure the boxes are packed completely inside the vehicles and the boxes do not overlap each other in each vehicle, the problem also takes into account constraints related to the vertical stability of the cargo, multidrop situations, and load-bearing strength of the boxes (including fragility). Computational tests with the proposed model were performed using an optimization solver embedded into a modeling language. The results validate the model and show that it is only able to handle problems of a moderate size. However, this model will be useful to motivate other researchers to explore approximate solution approaches to solve this problem, such as decomposition methods, relaxation methods, heuristics, among others, as well as to treat other variants of the problem, such as when time windows or a heterogeneous fleet are present, among others.