Abstract
The problems of optimum ellipse packing belong to the class of NP-hard problems. The issues of development of efficient algorithms based on application of local and global optimization methods, construction of adequate mathematical models based on the analytical description of the ellipse interrelations taking into account their continuous translations and rotations are of vital importance. In this article, the problem of packing of sets of ellipses in a given region taking into account conditions of nonintersection and technological restraints which are concretized in the conditions of the applied problem is formulated. The model of packing of a set of ellipses in a rectangle of minimum dimensions is constructed. Continuous ellipse rotations and translations are allowed, the possibility of availability of minimum admissible distances between them is assumed. New quasi-phi-functions are constructed for modeling of the relations of ellipse nonintersection and to define belonging of an ellipse to the container. The algorithm of search for locally optimal solutions is modified. It consists of two stages: generation of the regions of feasibility which contain the starting point and local optimization in the constructed region of feasibility. Only the algorithm step concerning construction of quasi-phi-functions is subjected to modification. It is necessary to notice that the algorithm have shown its efficiency when the quantity of ellipses does not exceed the value of 400. The model of the individual-and-flow movement of individuals approximated by ellipses with specification of technological restraints is constructed. The method of local optimization is given. Examples of computer modeling of the problems assigned in the work are given.
Highlights
The cutting and packing problems, known as the problems of optimum allocation [1, 2], are the study subject in computational geometry and methods of their solution represent the new trend in the operations research theory
This paper provides for a rather complete overview of literature on the problems of ellipse packing
The approach to modeling of ellipse placement in a rectangle of minimum dimensions consisting in the model and the solution method modification was further improved due to the use of the proposed functions
Summary
The cutting and packing problems, known as the problems of optimum allocation [1, 2], are the study subject in computational geometry and methods of their solution represent the new trend in the operations research theory. This class of problems has a wide range of scientific and practical applications. The class of problems connected with packing of given ellipse sets into a rectangular region of minimum dimensions is considered. Such problems arise in powder metallurgy when movement of loose substances is modeled and in logistics when modeling optimum packs of cylindrical objects with elliptic bases
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