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 on previous algorithms from the literature and solving for the first time a number of open instances. Summary of Contribution: We address the two-dimensional bin packing problem (2D-BPP), which calls for packing a set of rectangular items into a minimal set of larger rectangular bins. The 2D-BPP is a very difficult generalization of the standard one-dimensional bin packing problem, and it has been widely studied in the past because it models many real-world applications, including production, warehouse management, and transportation. We solve the 2D-BPP with a combinatorial Benders decomposition that is based on a model in which the two-dimensional items and bins are 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 it is enriched with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The algorithm we developed has been extensively tested on the most well-known benchmark set from the literature, which contains 500 instances. It behaved very well, improving on average upon previous algorithms, and solving for the first time a number of open instances. We analyzed in detail several configurations before obtaining the best one and discussed several insights from this analysis in the manuscript.
Highlights
In the Two-dimensional Bin Packing Problem (2D-BPP), we are given a set of rectangular items and a large number of rectangular bins
We believe this paper has a number of interesting contributions: (i) we develop a new exact algorithm for the 2D-BPP that adopts a different approach from the ones available in the literature; (ii) we gather together state-of-the-art techniques to effectively tackle Cutting & Packing (C&P) problems, including combinatorial cuts (Codato and Fischetti 2006), preprocessing techniques (Boschetti and Montaletti 2010), dual feasible functions (Alves et al 2016), conservative scales (Belov et al 2013), valid inequalities and lifting techniques (Kaparis and Letchford 2010); (iii) we solve for the first time a number of open instances from the literature; (iv) we show how the developed ideas can be adjusted to solve other difficult well-known C&P problems
To what done by Pisinger and Sigurd (2007), we let our exact algorithm run for one hour
Summary
In the Two-dimensional Bin Packing Problem (2D-BPP), we are given a set of rectangular items and a large number of rectangular bins. Cote et al (2014a) used a primal decomposition method to solve the Two-Dimensional Strip Packing Problem (2D-SPP), the problem of orthogonally packing a given set of items without overlapping in a strip of given width and infinite height, by minimizing the height used for the packing Their approach is different from the one presented in this paper as it involves solving in the master problem a relaxation in which the items can be cut into vertical slices that are packed contiguously in the strip, and checking in the sub-problem if all slices of an item can be packed at the same vertical height, for each item, building a feasible packing, if any. Relevant hints for future research directions, both on the 2D-BPP and on other problems, are provided in the concluding Section 9
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
