Fekete and Schepers’ Graph-based Algorithm for the Two-Dimensional Orthogonal Packing Problem Revisited

Ferreira, Eduarda Pinto; Oliveira, Jonice

doi:10.1007/978-3-8349-9777-7_2

Cited by 5 publications

(7 citation statements)

References 5 publications

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“…The authors refer that this characterization can be easily extended to higher dimensional problems. Ferreira and Oliveira (2005) presented some degenerated cases that can occur when applying Fekete and Schepers procedure. Ferreira and Oliveira (2008) presented an additional property to the graph‐based algorithm to avoid degenerative packing situations, verifying that the packing is actually constrained inside the object boundaries.…”

Section: Solution Approachesmentioning

confidence: 99%

An introduction to the two‐dimensional rectangular cutting and packing problem

Oliveira

Gamboa

Silva

2022

Int Trans Operational Res

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Cutting and packing problems have been widely studied in the last decades, mainly due to the variety of industrial applications where the problems emerge. This paper presents an overview of the solution approaches that have been proposed for solving two‐dimensional rectangular cutting and packing problems. The main emphasis of this work is on two distinct problems that belong to the cutting and packing problem family. The first problem aims to place onto an object the maximum‐profit subset of items, that is, output maximization, while the second one aims to place all the items using as few identical objects as possible, that is, input minimization. The objective of this paper is not to be exhaustive but to provide a solid grasp on two‐dimensional rectangular cutting and packing problems by describing their most important solution approaches.

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Section: Solution Approachesmentioning

confidence: 99%

An introduction to the two‐dimensional rectangular cutting and packing problem

Oliveira

Gamboa

Silva

2022

Int Trans Operational Res

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“…Discussion. The proof of Theorems 1 and 2 shows an interesting property of the graph model: given a feasible system of graphs G k = (I, E k ), k = 1, d, and restoring a corresponding packing layout, the fact e = (ij) ∈ E k does not necessarily mean that items i and j strictly overlap in projection on axis k. An illustrating example was given in [FO08], where such cases are called degenerate, Figure 3. Note that in the problem setting of [FSvdV07] it is impossible to demand that W = (9, 9), w 1 = (4, 5, 4, 2, 3), w 2 = (4, 5, 5, 4, 4).…”

Section: The Interval-graph Model and A Simplificationmentioning

confidence: 99%

“…Having integer item sizes, we may restrict our search to graph systems G k = (I, E k ), k = 1, d with the property P4 (Visibility, [FO08]):…”

Section: 2mentioning

confidence: 99%

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

Belov

Rohling

2013

Operations Research

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Abstract. We consider the feasibility problem OPP in higher-dimensional orthogonal packing: given a set of d-dimensional (d ≥ 2) rectangular items, decide whether all of them can be orthogonally packed in the given rectangular container without rotation. The 1D 'bar' LP relaxation of OPP reduces the latter to a 1D cutting-stock problem where the packing of each stock bar represents a possible 1D stitch through an OPP layout. The dual multipliers of the LP provide us with another kind of powerful bounding information (conservative scales). We investigate how the set of possible 1D packings can be tightened using the overlapping information of item projections on the axes, with the goal to tighten the relaxation. We integrate the bar relaxation into an interval-graph algorithm for OPP, which operates on such overlapping relations. Numerical results on 2D and 3D instances demonstrate the efficiency of tightening leading to a speedup and stabilization of the algorithm.

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“…Hence there are still symmetry issues in this model. Furthermore Ferreira and Oliveira [20] noticed that some so-called degenerated packing classes are unnecessarily enumerated by Fekete and Schepers' algorithm. A binary matrix M ∈ M n,m (IB) has the consecutive ones property if for every row i and k ≤ k ′ , M ik = 1 and M ik ′ = 1 implies M il = 1 for all k ≤ l ≤ k ′ (for every row, the set of 1s occur consecutively).…”

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confidence: 99%

Consecutive Ones Matrices for Multi-dimensional Orthogonal Packing Problems

Joncour

Pêcher

2011

J Math Model Algor

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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Distributed under a Creative Commons Attribution-ShareAlike| 4.0 International License

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Fekete and Schepers’ Graph-based Algorithm for the Two-Dimensional Orthogonal Packing Problem Revisited

Cited by 5 publications

References 5 publications

An introduction to the two‐dimensional rectangular cutting and packing problem

An introduction to the two‐dimensional rectangular cutting and packing problem

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

Consecutive Ones Matrices for Multi-dimensional Orthogonal Packing Problems

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