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Advances in Operations Research
Volume 2012 (2012), Article ID 346358, 26 pages
Research Article

Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

1Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
2Department of Mathematical Modeling, Institute for Mechanical Engineering Problems of The National Academy of Sciences of Ukraine, Kharkov, Ukraine

Received 21 October 2011; Revised 30 January 2012; Accepted 31 January 2012

Academic Editor: Ching-Jong Liao

Copyright © 2012 N. Chernov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the cutting and packing (C&P) problems in two dimensions by using phi-functions. Our phi-functions describe the layout of given objects; they allow us to construct a mathematical model in which C&P problems become constrained optimization problems. Here we define (for the first time) a complete class of basic phi-functions which allow us to derive phi-functions for all 2D objects that are formed by linear segments and circular arcs. Our phi-functions support translations and rotations of objects. In order to deal with restrictions on minimal or maximal distances between objects, we also propose adjusted phi-functions. Our phi-functions are expressed by simple linear and quadratic formulas without radicals. The use of radical-free phi-functions allows us to increase efficiency of optimization algorithms. We include several model examples.