A Primal Decomposition Algorithm for the Two-dimensional Bin Packing Problem

TL;DR

This paper introduces a novel combinatorial Benders decomposition approach for the 2D Bin Packing Problem, effectively solving previously unsolved instances and outperforming existing algorithms through advanced preprocessing and cutting techniques.

Contribution

It presents a new decomposition algorithm based on area representation and no-good cuts, significantly improving solution quality for complex bin packing instances.

Findings

Successfully solved previously unsolved instances

Outperformed existing algorithms on benchmark sets

Enhanced solution process with preprocessing and cutting techniques

Abstract

The Two-dimensional Bin Packing Problem calls for packing a set of rectangular items into a minimal set of larger rectangular bins. Items must be packed with their edges parallel to the borders of the bins, cannot be rotated and cannot overlap among them. The problem is of interest because it models many real-world applications, including production, warehouse management and transportation. It is, unfortunately, very difficult, and instances with just 40 items are unsolved to proven optimality, despite many attempts, since the 1990s. In this paper, we solve the problem with a combinatorial Benders decomposition that is based on a simple model in which the two-dimensional items and bins are just represented by their areas, and infeasible packings are imposed by means of exponentially-many no-good cuts. The basic decomposition scheme is quite naive, but we enrich it with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The resulting algorithm behaved very well on the benchmark sets of instances, improving on average upon previous algorithms from the literature and solving for the first time a number of open instances.