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Introduction. Varieties of the problem of packing of rectangular parallelepipeds have a wide practical application in various fields of activity, for example, in the optimal filling of containers, in the design and layout of a wide variety of technological objects and systems, in the creation of backup copies on removable media, in the optimization of storage, protection and transportation of goods, in additive manufacturing, etc. This work continues research on this topic and is devoted to the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds of different sizes into a rectangular parallelepiped of minimum volume. It presents a mathematical model for this packing problem and a parallel algorithm for solving it. This algorithm is based on the reduction of the original problem to an unconditional optimization problem using penalty functions, which is solved by the multistart method, in which r-algorithm is used to find local minima from the set of generated starting points. The purpose. Construction of a mathematical model and development of an algorithm for solving the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume. Results. A mathematical model and a parallel algorithm for balanced sparse packing of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume are presented. The algorithm is based on reducing the problem with the help of penalty functions to an unconditional nondifferentiable optimization problem, for finding the solution of which multistart method is used in combination with r-algorithm for finding local minima. The results of numerical experiments are given. Conclusions. The application of the algorithm described in the work based on non-smooth optimization methods allows to improve the results of balanced sparse packing of rectangular parallelepipeds in an acceptable time. Numerical experiments showed effectiveness of the algorithm in practice.

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.

Introduction. When formulating the classical two-stage transportation problem, it is assumed that the product is transported from suppliers to consumers through intermediate points. Intermediary firms and various kinds of storage facilities (warehouses) can act as intermediate points. The article discusses two mathematical models for two-stage transportation problem (linear programming problem and quadratic programming problem) and a fairly universal way to solve them using modern software. It uses the description of the problem in the modeling language AMPL (A Mathematical Programming Language) and depends on which of the known programs is chosen to solve the problem of linear or quadratic programming. The purpose of the article is to propose the use of AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems, to formulate a mathematical model of a quadratic programming two-stage transportation problem and to investigate its properties. Results. The properties of two variants of a two-stage transportation problem are described: a linear programming problem and a quadratic programming problem. An AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems is given. The results of the calculation using Gurobi program for a linear programming two-stage transportation problem, which has many solutions, are presented and analyzed. A quadratic programming two-stage transportation problem was formulated and conditions were found under which it has unique solution. Conclusions. The developed AMPL-code for a linear programming two-stage transportation problem and its modification for a quadratic programming two-stage transportation problem can be used to solve various logistics transportation problems using modern software for solving mathematical programming problems. The developed AMPL code can be easily adapted to take into account the lower and upper bounds for the quantity of products transported from suppliers to intermediate points and from intermediate points to consumers.

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.