LP Bounds in an Interval-Graph Algorithm for Orthogonal-Packing Feasibility

2014

Operations Research

We study the strip packing problem, in which a set of two-dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height, with the aim of minimizing the height used. The problem is important because it models a large number of real-world applications, including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste, scheduling problems in which tasks require a contiguous subset of identical resources, and container loading problems arising in the transportation of items that cannot be stacked one over the other.The strip packing problem has been attacked in the literature with several heuristic and exact algorithms, nevertheless, benchmark instances of small size remain unsolved to proven optimality. In this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort. Our method is based on a Benders' decomposition, in which in the master we cut items into unit-width slices and pack them contiguously in the strip, and in the slave we attempt to reconstruct the rectangular items by fixing the vertical positions of their unit-width slices. If the slave proves that the reconstruction of the items is not possible, then a cut is added to the master, and the algorithm is reiterated.We show that both the master and the slave are strongly -hard problems and solve them with tailored preprocessing, lower and upper bounding techniques, and exact algorithms. We also propose several new techniques to improve the standard Benders' cuts, using the so-called combinatorial Benders' cuts, and an additional lifting procedure. Extensive computational tests show that the proposed algorithm provides a substantial breakthrough with respect to previously published algorithms.

Section: Preprocessing and Boundsmentioning

confidence: 99%

Combinatorial Benders' Cuts for the Strip Packing Problem

Côté

Dell’Amico

2014

Operations Research

“…In the BPPC, a certain number of copies might exist for an item, and all copies should be packed in consecutive bins. Starting from Martello et al [36], the BPPC has been used as a relaxation for two-dimensional cutting and packing problems, either within branch-and-bound algorithms (see, e.g., Alvarez-Valdes et al [1] and Belov and Rohling [7]) or in combinatorial Benders decompositions (see, e.g., Côté et al [17] and Delorme et al [22]).…”

Section: Literature and Related Problemsmentioning

confidence: 99%

“…The objective function (7) minimizes the number of packing patterns (bins) used, and constraints (8) ensure that each item is packed in one bin.…”

Section: An Exponential-size Modelmentioning

confidence: 99%

A branch-and-price algorithm for the temporal bin packing problem

Dell’Amico

Furini

Computers & Operations Research

2020

We study an extension of the classical Bin Packing Problem, where each item consumes the bin capacity during a given time window that depends on the item itself. The problem asks for finding the minimum number of bins to pack all the items while respecting the bin capacity at any time instant. A polynomial-size formulation, an exponential-size formulation, and a number of lower and upper bounds are studied. A branch-and-price algorithm for solving the exponential-size formulation is introduced. An overall algorithm combining the different methods is then proposed and tested trough extensive computational experiments.

“…The 2OPP arises as a subproblem in many two-dimensional C&P problems, such as knapsack, bin packing, and strip packing. It has been tackled with several algorithms, including, e.g., constraint programming techniques by Clautiaux et al (2008), and mixed methods by Mesyagutov et al (2012) and Belov and Rohling (2013). Here we solve it first by means of branch-and-bound algorithms and then by primal decomposition techniques.…”

Section: Application Ii: Non-exact Two-stage Cutting Stock Problemmentioning

confidence: 99%

The Meet-in-the-Middle Principle for Cutting and Packing Problems

Côté

INFORMS Journal on Computing

2018

Cutting and packing (C&P) is a fundamental research area that models a large number of managerial and industrial optimization issues. A solution to a C&P problem basically consists in a set of one-or multi-dimensional items packed in/cut from one or more bins, by satisfying problem constraints and minimizing a given objective function. Normal patterns are a well-known C&P technique used to build solutions where each item is aligned to the bottom of the bin along each dimension. The rationale in their use is that they can reduce the search space while preserving optimality, but the drawback is that their number grows consistently when the number of items and the size of the bin increase. In this paper we propose a new set of patterns, called meet-in-the-middle, that lead to several interesting results. Their computation is achieved with the same time complexity as that of the normal patterns, but their number is never higher, and in practical applications it frequently shows reductions of about 50%. The new patterns are applied to improve some state-of-the-art C&P techniques, including arc-flow formulations, combinatorial branch-andbound algorithms, and mixed integer linear programs. The efficacy of the improved techniques is assessed by extensive computational tests on a number of relevant applications. Keywords. Cutting and packing, normal patterns, meet-in-the-middle, cutting stock, orthogonal packing. Acknowledgement. The second author acknowledges financial support by CAPES/Brazil under grant PVE n o A007/2013. Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et opinions contenus dans cette publication ne reflètent pas nécessairement la position du CIRRELT et n'engagent pas sa responsabilité.