Subota I. Optimal ellipse packing problem: mathematical models and solution methods

The subject of the research is the optimal ellipse packing process taking into account ellipse continuous rotations and allowable distances. The objective of the research is improving of the efficiency of solutions of optimization ellipse packing problems by developing of constructive tools of mathematical modeling and efficient optimization algorithms. The research methods employ analytical geometry, geometric design methods, methods of nonlinear programming and nonsmooth optimization. The developed tools, models and methods for optimal ellipse packing problems allow us to search for local-optimal solutions which are better in comparison with the known published results. The theoretical and practical results of the thesis makes it possible to place a large number of ellipses (more then 100) in the minimal size (an area, a radius, a coefficient of homothety) container (a rectangle, a circle or an ellipse) taking into account allowable distances. Scientific novelty: the tools of mathematical modeling and computer simulation are developed in the form of new classes of phi-functions and quasi-phi-functions, the new mathematical models are derived in the form of nonlinear programming problems and the efficient algorithms are developed for solving the ellipse packing problems, employing methods of nonlinear programming and nonsmooth optimization. The tools of mathematical modeling are used in the educational process, in the future we plan to use the results of the research in the industry. The research results can be used in logistics, powder metallurgy industry, mineral industry, molecular dynamics and cutting of industrial materials problem.