Mathematical model and efficient algorithms for object packing problem

The article is devoted to mathematical models and practical algorithms for solving the cutting and packing (C&P) problem. We review and further enhance the main tool of our studies - phi-functions. Those are constructed here for 2D and 3D objects (unlike other standard tools, such as No-Fit Polygons, which are restricted to the 2D geometry). We also demonstrate that in many realistic cases the Phi-functions can be described by quite simple formulas without radicals and other complications. Lastly, a general Solution strategy using the phi-functions is Outlined and illustrated by several 2D and 3D examples. (C) 2010 Elsevier B.V. All rights reserved.