Stability constraints in a 3D knapsack problem with non parallelepipedic items

Abstract

This paper concerns stability issues that we encountered while solving an industrial pallet loading problem. The practical problem can be modeled as a three-dimensional knapsack with original constraints related to the stability of the pallet: when the placement of the boxes on the pallet is computed, the weight of the boxes and the placement sequence are not known.Despite this lack of information, the placement must remain stable throughout the construction of the pallet. Even more important, the major difference with classical placement problems is that the boxes are parallelepipedic rectangle with edge reduction (the upper surface may be smaller than the bottom face).In this paper, we review stability constraints used in the literature and the assumptions under which these constraints can be used and we detail their adequacy to the problem we consider. We then propose a new stability constraint which takes into account the practical features, which can be integrated into a placement algorithm and which is not too restrictive.Numerical experiments on benchmarks from the literature show that this constraint, added to a classical placement algorithm, obtains results which are similar to algorithms from the literature dedicated to a specific stability constraint for parallelepipedic boxes. By adjusting the benchmark to take into account the practical problem specificities, we show that this new stability constraint is perfectly suited to the industrial problem and obtains very good results compared to classical constraints. Therefore, it has been integrated in the placement software developed by the company.