No abstract

The packing of different circles in a circular container under balancing and distance conditions is considered. Two problems are studied: the first minimizes the container’s radius, while the second maximizes the minimal distance between circles, as well as between circles and the boundary of the container. Mathematical models and solution strategies are provided and illustrated with computational results.

The article considers mathematical models and algorithms for optimal balancedsparse packing of spheres and cubes into spherical and cubic containers. A balanced sparse (allowable distances between objects are specified) packing of objects into an outer container is such a packing that the center of gravity of thefamily of objects coincides with the center of the outer container, and the distances between the objects as well as the distances from them to the outer container are not less than the predetermined values. Mathematical models, sequential and parallel algorithms for solving problems of finding a balanced sparsepacking of balls of different radii into spherical and cubic containers are given.A mathematical model of the problem of finding a balanced sparse packing ofcubes into a cube of minimum volume, provided that the sides of all cubes areparallel to the coordinate axes, and a description of the non-smooth penaltyfunction for finding local minima of the problem are given. The investigatedproblems belong to the class of NP-hard problems. Mathematical models arerepresented by multi-extremal nonlinear programming problems. To find thebest feasible solution, the multistart method is used in combination with Shor's ralgorithm. For this, the problem is reduced to the unconditional optimizationproblem using penalty functions in the form of maximum functions, and nonsmooth optimization methods based on the use of software implementations of ralgorithm are used to find local minima from a set of starting points. Mathematical models and sequential and parallel algorithms under consideration can beused to develop software tools for solving problems of finding a balanced sparsepacking of spherical and cubic objects into spherical and cubic containers. Thematerial is presented in three sections. The first section presents a mathematicalmodel and algorithms for solving the problem of finding a balanced sparse packing of balls of different radii into a spherical container. The sequential and parallel algorithms for finding the best feasible problem solution are described. Section 2 provides a mathematical model and algorithms for solving the problem offinding a balanced sparse packing of balls of different radii into a cubic container. The sequential and parallel algorithms for finding the best feasible problemsolution are described. Section 3 presents a mathematical model of the problemof finding a balanced sparse packing of cubes into a cubic container. A description ofthe non-smooth penalty function for finding local minima of the problem is given.

Introduction. The identification of structural and technological disproportions that affect crisis phenomena in the economy and the analysis of ways to eliminate them require a wide application of quantitative research methods, in particular, mathematical modeling. “Input-Output” tables of Leontief turned out to be quite a convenient tool for analyzing these economic issues. In Leontief-type models, the matrix of technical coefficients (matrix of direct costs) is assumed to be known and calculated on the basis of statistical information from the “input-output” tables. M.V. Mykhalevych formulated the “inverse” problem: how to determine those structural and technological changes that would reduce the cost of production and thereby increase the incomes of end consumers and make the economy more dynamic. Or, in other words, how to choose or adjust technical coefficients to improve the properties of the economic process. This work is devoted to two optimization problems built on the basis of models of this type. The purpose of the article is to optimize the interdisciplinary planning of structural and technological changes. Results. Inverse models of the Leontief type for optimization of structural and technological transformations in economic systems are considered. These models are formulated in terms of nonlinear programming problems and include two objective functions for maximization: total consumer incomes and the “income growth–production growth” multiplier. Algorithms and software for solving these problems are presented. Numerical optimization procedures are based on Shor's r-algorithm. Conclusions. The use of inverse models of the Leontief type will allow choosing promising directions of structural and technological transformations in both the macro- and microeconomy. The proposed mathematical apparatus based on non-smooth optimization algorithms proved to be a sufficiently effective tool for solving appropriate optimization problems in practice. Keywords: structural and technological changes, inter-industry balance, Leontief model, “input-output” matrix, inverse Leontief-type models, non-smooth optimization algorithms, software.