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This paper deals with the packing problem of circles and non-convex polygons, which can be both translated and rotated into a strip with prohibited regions. Using the F-function technique, a mathematical model of the problem is constructed and its characteristics are investigated. Based on the characteristics, a solution approach to the problem is offered. The approach includes the following methods: an optimization method by groups of variables to construct starting points, a modification of the Zoutendijk feasible direction method to search for local minima and a special non-exhaustive search of local minima to find an approximation to a global minimum. A number of numerical results are given. The numerical results are compared with the best known ones.

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A mathematical model is constructed and a method is developed for the solution of the packing problem for n-dimensional parallelepipeds with the feasibility of changing their orthogonal orientation in an n-dimensional parallelepiped. To search for an approximation to the global minimum, a combination of the sequentially-single placement method and a modification of the decremental neighborhood method is used. The offered approach contributes to the improvement of the results of packing oriented n-dimensional parallelepipeds.

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The paper considers a packing optimization problem of different spheres and cuboids into a cuboid of the minimal height. Translations and continuous rotations of cuboids are allowed. In the paper, we offer a way of construction of special functions (Φ-functions) describing how rotations can be dealt with. These functions permit us to construct the mathematical model of the problem as a classical mathematical programming problem. Basic characteristics of the mathematical model are investigated. When solving the problem, the characteristics allow us to apply a number of original and state-of-the-art efficient methods of local and global optimization. Numerical examples of packing from 20 to 300 geometric objects are given.