%0 Journal Article
%T Packing Different Cuboids with Rotations and Spheres into a Cuboid
%A Y. G. Stoyan
%A A. M. Chugay
%J Advances in Decision Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/571743
%X 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 ( -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. 1. Introduction At present, scientific research concerning mathematical modeling of optimization packing problems of 3D geometrical objects is intensively performed. The great interest to the given class of problems is motivated by the need of wide use of the problems both in scientific researches and applications in different branches of industry. Therefore, when tackling these classes of problems, development of fundamental bases and tools for mathematical and computer modeling is very important. A state-of-the-art review of bin packing techniques is considered in [1]. It should be noted that there are many publications devoted to packing of cuboids which can be rotated only through 90¡ã around of all coordinate axes. Currently, 3D packing problems of cuboids, for which translations and continuous rotations are allowed, and spheres are poorly investigated. The problem of packing nonoriented polyhedrons can be applied in CAD system for rapid prototyping which uses selective laser sintering process of a special powder [2]. Besides, the problem is applied in nanotechnologies for 3D modeling, visual and quality analysis of structural characteristics and mechanical properties of various composite, firm, liquid, glassy materials, granulated media, and biological systems [3¨C6]. Packing of spheres and cuboids is used to model heterogeneous and porous material morphologies such as concrete, sand, coal, porous explosives, and solid rocket propellants [7]. Also, the problem of packing nonoriented cuboids is applied to car design [8]. Various optimization search algorithms for solving 3D layout problems are considered in paper [9]. In the paper, the authors notice that simulated annealing algorithms and genetic algorithms are stochastic methods that are used in a wide variety of 3D problems. A lot of authors traditionally use either spheres or
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