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Introduction to Cutting and Packing Optimization pp 123–156Cite as

Orthogonal Packing Feasibility, Two-Dimensional Knapsack Problems

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 263))

Abstract

The Orthogonal Packing Feasibility Problem and the Orthogonal Knapsack Problem are basic problems in two- and higher-dimensional cutting and packing. For given dimensionality d ≥ 2, we consider a set of d-dimensional rectangular items (pieces) that need to be packed into the given container. The input data describe the container sizes, the item sizes, and, in case of a knapsack problem, the profit (value) coefficient of any item.

The d-dimensional Orthogonal Packing Problem (dOPP) is the feasibility problem: decide whether all the m pieces can orthogonally be packed into the container without rotations. The d-dimensional Orthogonal Knapsack Problem (dOKP) asks for a subset of items of maximal total profit which can orthogonally be packed into the container without rotations.

In the following, we describe a basic nonlinear and some integer linear programming models (for simplicity, frequently for d = 2). Afterwards we discuss some necessary conditions for the feasibility of an OPP instance. We continue with a short description of the graph-theoretical approach of Fekete and Sche pers (2004). Moreover, we also present some results concerning statements on the packability of a set of rectangular items into a container. Finally, we propose a (general) branch-and-bound algorithm based on the contour concept which allows to regard various additional restrictions and to construct fast heuristics.

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Authors and Affiliations

  1. Institute for Numerical Mathematics, TU Dresden, Dresden, Germany

    Guntram Scheithauer

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  1. Guntram Scheithauer
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Scheithauer, G. (2018). Orthogonal Packing Feasibility, Two-Dimensional Knapsack Problems. In: Introduction to Cutting and Packing Optimization. International Series in Operations Research & Management Science, vol 263. Springer, Cham. https://doi.org/10.1007/978-3-319-64403-5_5

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