Introducing an approach based on an advanced hybrid model for the two-dimensional bin packing problem

A Computational Study of Lower Bounds for the Two Dimensional Bin Packing Problem

In this paper, we propose a global optimization method based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) involving cutting plane techniques for solving two-dimensional Bin packing and Strip packing problems. Given a set of rectangular items, we consider problems of allocating each item to larger rectangular standardized units. In two-dimensional bin packing problem, these units are finite rectangles, and the objective is to pack all the items into the minimum number of units. In two-dimensional strip packing problem, there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. These problems are characterized as BLP (Binary Linear Programming) problems. Thanks to exact penalty technique in DC Programming, the BLP can be reformulated as polyhedral DC program which can be efficiently solved via the proposed DC programming approach. Computational experiments on large-scale dataset involving up to 200 items show the good performance of our algorithm.KeywordsTwo-dimensional packingBin packingStrip packingBLPDC ProgrammingDCACutting plane techniques

Heuristics for the two-dimensional irregular bin packing problem with limited rotations

This paper proposes an adaptation, to the two-dimensional irregular bin packing problem of the Djang and Finch heuristic (DJD), originally designed for the one-dimensional bin packing problem. In the two-dimensional case, not only is it the case that the piece’s size is important but its shape also has a significant influence. Therefore, DJD as a selection heuristic has to be paired with a placement heuristic to completely construct a solution to the underlying packing problem. A successful adaptation of the DJD requires a routine to reduce computational costs, which is also proposed and successfully tested in this paper. Results, on a wide variety of instance types with convex polygons, are found to be significantly better than those produced by more conventional selection heuristics.

Two-dimensional bin packing problem (2BPP) is a classical problem in operations research and has been extensively investigated in the past years. The 2BPP can be classified according to choices of the orientation (fixed orientation or 90 degrees rotated) and the cut type (guillotine or non-guillotine cut). Many mathematical models and solution methods have been proposed for the problem. However, due to the difficulty in representing constraints, no exact mathematical model is presented for the two-dimensional non-guillotine bin packing problem with free rotation (2BPP-NR). In this paper, we present two Mixed-integer programming (MIP) models for the 2BPP-NR. Computational experiments are conducted on random problem instances and comparative analyses of the proposed models are presents. Results show the Model II outperforms the Model I in both number of constraints and computational time.

A lookahead matheuristic for the unweighed variable-sized two-dimensional bin packing problem

In this paper, we present a two-dimensional irregular bin packing problem (2DIBPP) that takes into account the slit distance and allows the pieces to rotate freely. The target is to arrange a specified collection of pieces with irregular shapes into a minimal number of bins. Firstly, we develop a mathematical model for the 2DIBPP that considers slit distance and free rotation of the pieces, and an equidistant edge expansion approach is then proposed to handle the slit distance. Secondly, a two-stage method is implemented to get a finite collection of promising rotation angles, effectively decreasing the search neighbourhood. Thirdly, we decompose the 2DIBPP into two sub-problems: piece assignment and packing. The Partial Bin Packing (PBP) strategy is employed in the allocation stage, and we adopt an overlap minimization method to pack the pieces into an individual bin. Finally, we use a local search (LS) algorithm to advance the quality of the solutions by adjusting the piece assignment across bins. Experimental evidence exhibits that our approach is competitive in most instances of the literature, with four better results in five benchmark instances.

A goal-driven approach to the 2D bin packing and variable-sized bin packing problems

A hybrid feasibility constraints-guided search to the two-dimensional bin packing problem with due dates

Two-dimensional online bin packing with rotation

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.

An agent-based approach to the two-dimensional guillotine bin packing problem

An iteratively doubling binary search for the two-dimensional irregular multiple-size bin packing problem raised in the steel industry

An iteratively doubling local search for the two-dimensional irregular bin packing problem with limited rotations

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.