Orthogonal packing of identical rectangles within isotropic convex regions

Birgin, Ernesto G.; Lobato, Rafael D.

doi:10.1016/j.cie.2010.07.004

Cited by 32 publications

(20 citation statements)

References 34 publications

(66 reference statements)

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“…It is important to mention that most of the computational effort is spent when solving the problem with m = m * and trying to solve the problem with m = m * +1. See, for example, [10,14,17] where exhaustive numerical experiments support this claim.…”

Section: Packing the Maximum Number Of Ellipses Within A Rectanglementioning

confidence: 71%

Packing ellipsoids by nonlinear optimization

2015

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In this paper, continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space are introduced. Two different models for the non-overlapping and models for the inclusion of ellipsoids within half-spaces and ellipsoids are presented. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, illustrative numerical experiments are presented.

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Section: Packing the Maximum Number Of Ellipses Within A Rectanglementioning

confidence: 71%

Packing ellipsoids by nonlinear optimization

2015

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“…The process stops when m is such that a feasible solution of (5-8) cannot be found, and the solution found for (5-8) with m * = m − 1 is considered as the solution of packing as many identical balls as possible. Similar strategies have been employed in [5,6,8,9] where empirical justifications for the use of this kind of sequential process of increasing m one by one, instead of other strategies such as bisection, are given.…”

Section: Maximizing the Number Of Identical Circles Within A Given Elmentioning

confidence: 99%

“…. , m. We prefer (5)(6)(7)(8) in our experiments because all the constraints are formulated as polynomials. The unknowns of the nonlinear feasibility problem (5-8) are (u i , v i ), i = 1, .…”

Section: Given Circles Within a Given Ellipsementioning

confidence: 99%

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Packing circles within ellipses

Birgin

Bustamante

Callisaya

et al. 2013

Int Trans Operational Res

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. AbstractThe problem of packing circles within ellipses is considered in the present paper. A new ellipse-based system of coordinates is introduced by means of which a closed formula to compute the distance of an arbitrary point to the boundary of an ellipse exists. Nonlinear programming models for some variants of 2D and 3D packing problems involving circular items and elliptical objects are given. The resulting models are medium-sized highly nonlinear challenging nonlinear programming problems for which a global solution is sought. For this purpose, multistart strategies are carefully and thoroughly explored. Numerical experiments are exhibited.

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“…Some recent examples are in [2,27,35,17,13,22,48]. Nonlinear-programming-based method for packing rectangles within arbitrary convex regions, considering different types of positioning constraints, were presented in [10,11,8,38].…”

Section: Introductionmentioning

confidence: 99%

Generating unconstrained two-dimensional non-guillotine cutting patterns by a Recursive Partitioning Algorithm

Birgin

Lobato

Morábito

2012

Journal of the Operational Research Society

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In this study, a dynamic programming approach to deal with the unconstrained twodimensional non-guillotine cutting problem is presented. The method extends the recently introduced recursive partitioning approach for the manufacturer's pallet loading problem. The approach involves two phases and uses bounds based on unconstrained two-staged and non-staged guillotine cutting. The method is able to find the optimal cutting pattern of a large number of problem instances of moderate sizes known in the literature and a counterexample for which the approach fails to find known optimal solutions was not found. For the instances that the required computer runtime is excessive, the approach is combined with simple heuristics to reduce its running time. Detailed numerical experiments show the reliability of the method.

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Orthogonal packing of identical rectangles within isotropic convex regions

Cited by 32 publications

References 34 publications

Packing ellipsoids by nonlinear optimization

Packing ellipsoids by nonlinear optimization

Packing circles within ellipses

Generating unconstrained two-dimensional non-guillotine cutting patterns by a Recursive Partitioning Algorithm

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