## Control Systems and Computers, N5, 2016, Article 1

##### DOI: https://doi.org/10.15407/usim.2016.05.003

Upr. sist. maš., 2016, Issue 5 (265), pp. 3-9.

UDC 519.6

A.N. Danilinadjunct, National University of Civil Protection of Ukraine (Kharkiv)

V.V. Komyak, PhD (Eng.), National University of Civil Protection of Ukraine (Kharkiv), E-mail: vlad1m1r@list.ru,

V.M. Komyak, Doctor (Eng.), National University of Civil Protection of Ukraine (Kharkiv), E-mail: vkomyak@ukr.net,

A.V. Pankratov,  Doctor (Eng.), Institute of Mechanical Engineering of A.M. Podgorny NAS Ukraine (Kharkiv), E-mail: impankratov@mail.ru

### The Ellipses Packing in a Rectangle of the Minimal Size

Introduction. In this paper, we deal with the optimal ellipse packing problem, which is a part of operational research and computational geometry. Ellipse packing problems have a variety of real world applications. However, the problem has received relatively little attention in the literature so far. Finding high quality ellipse packings is a difficult computational problem.

Methods and results. The problem is formulated as follows: the set of ellipses is to be placed without overlapping, free of any orientation restrictions, on a rectangular plate such that the area of the rectangle is minimized.

The ellipses are described by the coordinates of their center and an orientation angle to allow their rotation. Two types of constraints are necessary to be modeled: the non-overlapping ellipses and the bounds placing the ellipses inside the rectangle.

A new quasi-phi-function is developed for an analytical description of non-overlapping ellipses. An additional variable is introduced for each quasi-phi-function. Already known phi-function is applied to hold the containment constraint.

A new mathematical programming formulations for this problem in the form of a nonlinear programming problem are presented. The constrained optimization problem is NP-hard nonlinear programming problem. The solution space Q has a complicated structure. A matrix of the inequality system which specifies Q is strongly sparse and has a block structure. Our LOFRT procedure allows us to reduce the problem with O(n2) inequalities and O(n2)-dimensional solution space to a sequence of sub-problems, each with O(n)inequalities and a O(n)-dimensional solution subspace. This reduction is of a paramount importance, since it deals with non-linear optimization problems.

Conclusion. The ellipse packing problem in the form of a nonlinear programming problem is formulated and an efficient solution algorithm is developed. The presented algorithm allows to reduce the computational cost of the researched problem.

Keywords: packaging, ellipses, continuous turns, quasi-phi-function, mathematical model, nonlinear optimization.

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