Abstract

Abstract

Abstract
Two-dimensional rectangular strip packing problems belong to the broader class of Cutting and Packing (C&P) problems, in which small items are required to be cut from or packed on a larger object, so that the waste (unused regions of the large object) is minimized. C&P problems differ from other combinatorial optimization problems by the intrinsic geometric constraints: items may not overlap and have to be fully contained in the large object. This survey approaches the specific C&P problem in which all items are rectangles, therefore fully characterized by a width and a height, and the large object is a strip, i.e. a rectangle with a fixed width but an infinite height, being the problem’s goal to place all rectangles on the strip so that the height is minimized. These problems have been intensively and extensively tackled in the literature and this paper will focus on heuristic resolution methods. Both the seminal and the most recent approaches (from the last decade) will be reviewed, in a rather tutorial flavor, and classified according to their type: constructive heuristics, improvement heuristics with search over sequences and improvement heuristics with search over layouts. Building on this review, research gaps are identified and the most interesting research directions pointed out. © 2016 Brazilian Operations Research Society.

Abstract

Abstract
In this paper we consider the problem of minimizing the waste of textile material in a Portuguese home textile manufacturing company. The company has a vertical structure covering the different production stages of the home textile, from weaving until the finished products. Production planning comprises different decisions: the definition of the widths and lengths of the fabric rolls to be produced, the number of fabric rolls to be used from stock or purchased and the definition of the cutting patterns to be applied to each width of the fabric roll, so that the waste is minimized. We propose a MIP model, solved by a column generation method, to tackle the problem.

Abstract
Two-dimensional irregular strip packing problems are cutting and packing problems where small pieces have to be cut from a larger object, involving a non-trivial handling of geometry. Increasingly sophisticated and complex heuristic approaches have been developed to address these problems but, despite the apparently good quality of the solutions, there is no guarantee of optimality. Therefore, mixed-integer linear programming (MIP) models started to be developed. However, these models are heavily limited by the complexity of the geometry handling algorithms needed for the piece non-overlapping constraints. This led to pieces simplifications to specialize the developed mathematical models. In this paper, to overcome these limitations, two robust MIP models are proposed. In the first model (DTM) the non-overlapping constraints are stated based on direct trigonometry, while in the second model (NFP - CM) pieces are first decomposed into convex parts and then the non-overlapping constraints are written based on nofit polygons of the convex parts. Both approaches are robust in terms of the type of geometries they can address, considering any kind of non-convex polygon with or without holes. They are also simpler to implement than previous models. This simplicity allowed to consider, for the first time, a variant of the models that deals with piece rotations. Computational experiments with benchmark instances show that NFP CM outperforms both DTM and the best exact model published in the literature. New real-world based instances with more complex geometries are proposed and used to verify the robustness of the new models.