Mathematical Models and Algorithms for Optimal Packing of Balls and Cubes Into Spherical and Cubic Containers

Abstract

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.