Lexicographic optimization for the multi-container loading problem with open dimensions for a shoe manufacturer

Abstract

Motivated by a real-world application, we present a multi-container loading problem with 3-open dimensions. We formulate it as a biobjective mixed-integer nonlinear program with lexicographic objectives in order to reflect the decision maker's optimization priorities. The first objective is to minimize the number of containers, while the second objective is to minimize the volume of those containers. Besides showing the NP-hardness of this sequential optimization problem, we provide bounds for it which are used in the three proposed algorithms, as well as, on their evaluation when a certificate of optimality is not available. The first is an exact parametric-based approach to tackle the lexicographic optimization through the second objective of the problem. Nevertheless, given that the parametric programs correspond to large nonlinear mixed-integer optimizations, we present a heuristic that is entirely mathematical-programming based. The third algorithm enhances the solution quality of the heuristic. These algorithms are specifically tailored for the real-world application. The effectiveness and efficiency of the devised heuristics is demonstrated with numerical experiments.

Last Mile Delivery Box Packing

“…Vieira et al (2021) approach the same problem and objective, that is, to choose box sizes, but in their case, each box has all three dimensions open. In a following work, Vieira and Carvalho (2022), extend the problem to consider multi-container loading with open dimensions. Furthermore, the authors use a bi-objective mixed-integer non-linear function to optimize container volume and their number.…”