Phi-functions for 2D objects formed by line segments and circular arcs

Abstract

We study the cutting and packing (CP) 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 CP 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. Copyright © 2012 N. Chernov et al.