Packing cylinders and rectangular parallelepipeds with distances between them into a given region

Stoyan, Yu. G.; Chugay, A. M.

doi:10.1016/j.ejor.2008.07.003

Cited by 27 publications

(16 citation statements)

References 19 publications

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“…We do not follow this approach, as it seems to us to be too complicated to implement the eigenvalue computation and the two-different-root condition in an NLP model. Another techniques to deal with non-overlap conditions in cutting or packing problems is the phi-function approach; cf Chernov et al [2] for 2D cutting and packing problems, Stoyan and Chugay [17] for solving 3D packing problems, or Romanova et al [16] for covering problems. However, in this approach, only local optimality can be proven and it is not clear how this approach could be translated into a declarative NLP model.…”

Section: Non-overlap Conditions For Ellipsoidsmentioning

confidence: 99%

Packing ellipsoids into volume-minimizing rectangular boxes

Kallrath

2015

J Glob Optim

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A set of tri-axial ellipsoids, with given semi-axes, is to be packed into a rectangular box; its widths, lengths and height are subject to lower and upper bounds. We want to minimize the volume of this box and seek an overlap-free placement of the ellipsoids which can take any orientation. We present closed non-convex NLP formulations for this ellipsoid packing problem based on purely algebraic approaches to represent rotated and shifted ellipsoids. We consider the elements of the rotation matrix as variables. Separating hyperplanes are constructed to ensure that the ellipsoids do not overlap with each other. For up to 100 ellipsoids we compute feasible points with the global solvers available in GAMS. Only for special cases of two ellipsoids we are able to reach gaps smaller than 10 −4 .

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Section: Non-overlap Conditions For Ellipsoidsmentioning

confidence: 99%

Packing ellipsoids into volume-minimizing rectangular boxes

Kallrath

2015

J Glob Optim

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“…where W = U ∈ R d : Ψ (U ) 0 (25) and Ψ (U ) 0 denotes the system of inequalities specifying all the relevant constraints.…”

Section: Constraintsmentioning

confidence: 99%

“…[11]), we also apply strips as simpler enclosing shapes, see [22,25], and thus achieve a high speed in choosing an initial layout.…”

Section: Findmentioning

confidence: 99%

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Mathematical model and efficient algorithms for object packing problem

Chernov

Stoyan

Romanova

2010

Computational Geometry

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130

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The article is devoted to mathematical models and practical algorithms for solving the cutting and packing (C&P) problem. We review and further enhance the main tool of our studies -phi-functions. Those are constructed here for 2D and 3D objects (unlike other standard tools, such as No-Fit Polygons, which are restricted to the 2D geometry). We also demonstrate that in many realistic cases the phi-functions can be described by quite simple formulas without radicals and other complications. Lastly, a general solution strategy using the phi-functions is outlined and illustrated by several 2D and 3D examples.

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“…To solve the problem (2.6) a modification of the Zoutendjik method of feasible directions [16,20] and the concept of ε-active inequalities [15] are used. The modification is realized by the classical iterative formula…”

Section: Local Optimizationmentioning

confidence: 99%

Packing congruent hyperspheres into a hypersphere

Stoyan

Yaskov

2011

J Glob Optim

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The paper considers a problem of packing the maximal number of congruent nD hyperspheres of given radius into a larger nD hypersphere of given radius where n = 2, 3, . . . , 24. Solving the problem is reduced to solving a sequence of packing subproblems provided that radii of hyperspheres are variable. Mathematical models of the subproblems are constructed. Characteristics of the mathematical models are investigated. On the ground of the characteristics we offer a solution approach. For n ≤ 3 starting points are generated either in accordance with the lattice packing of circles and spheres or in a random way. For n > 3 starting points are generated in a random way. A procedure of perturbation of lattice packings is applied to improve convergence. We use the Zoutendijk feasible direction method to search for local maxima of the subproblems. To compute an approximation to a global maximum of the problem we realize a non-exhaustive search of local maxima. Our results are compared with the benchmark results for n = 2. A number of numerical results for 2 ≤ n ≤ 24 are given.

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Packing cylinders and rectangular parallelepipeds with distances between them into a given region

Cited by 27 publications

References 19 publications

Packing ellipsoids into volume-minimizing rectangular boxes

Packing ellipsoids into volume-minimizing rectangular boxes

Mathematical model and efficient algorithms for object packing problem

Packing congruent hyperspheres into a hypersphere

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