We study pseudo-polynomial formulations for the classical bin packing and cutting stock problems. We first propose an overview of dominance and equivalence relations among the main pattern-based and pseudo-polynomial formulations from the literature. We then introduce reflect, a new formulation that uses just half of the bin capacity to model an instance and needs significantly less constraints and variables than the classical models. We propose upper and lower bounding techniques that make use of column generation and dual information to compensate reflect weaknesses when bin capacity is too high. We also present non-trivial adaptations of our techniques that solve two interesting problem variants, namely, the variable sized bin packing problem and the bin packing problem with item fragmentation. Extensive computational tests on benchmark instances show that our algorithms achieve state of the art results on all problems, improving upon previous algorithms and finding several new proven optimal solutions.
M. Delorme and M Iori. Enhanced pseudo-polynomial formulations for bin packing and cutting stock problems. Technical report OR-17-6, DEI “Guglielmo Marconi”, Alma Mater Studiorum Università di Bologna, Italy, 2017.