An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure

Abstract

We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.

A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting

“…Approximate algorithms for this problem have also been proposed by Morabito and Arenales (1996), Cung et al (2000), Belov and Scheithauer (2006), Hifi and M'Hallah (2006), Gongalves (2007) and Cui (2008). Finally, we note that there are also a variety of exact and heuristic methods for solving non-guillotine problems, including Beasley (1985b), Baldacci and Boschetti (2007) and Alvarez-Valdes et al (2007).…”

Bidirectional best-fit heuristic considering compound placement for two dimensional orthogonal rectangular strip packing

“…Gilmore & Gomory (1961) developed a linear programming approach to solve very small strip packing problems to optimality. Christofides & Whitlock (1977) and Beasley (1985) used methods based on tree-search to solve the guillotine and non-guillotine variants of the strip packing problems respectively. The approach of Christofides & Whitlock (1977) was improved by Hifi & Zissimopoulos (1997) and further by Cung et al (2000) however solving large instances was still impractical in a reasonable amount of time.…”

Mathematical models and decomposition algorithms for cutting and packing problems

“…In the following we adopt, for the sake of clarity, the modeling approach originally developed by Beasley [26], where the variables represent the coordinates at which the items are packed in the bin/strip. As it will be clear later, such variables correspond to the decisions that the user has to take when using our visual tool.…”