Packing non-equal spheres into containers of different shapes

The article reviews a mathematical model of the optimization problem of packing different spheres into a container which can be a cuboid, a sphere, a right circular cylinder, an annular cylinder and a spherical layer. The assumption that sphere radii are variable is exploited. Based on the hypothesis a new way to derive starting point belonging to the feasible region of the problem is offered. Solving the problem is reduced to solving a sequence of mathematical programming problems. A step by step procedure of smooth transition from one local maximal point to another one providing an improvement of the objective function value is suggested. A solution strategy consisting of four stages is proposed. The first stage involves formation of starting points and computation of local minima of the problem. The second stage fulfills continuous transition from one local minimum to another one. The third stage allows to improve solution results due to reduction of the solution space dimension. The forth stage realizes rearrangements of sphere pairs to reduce the objective function value of the problem. A number of numerical results which we compare with benchmark ones are given.