Deterministic neural network used in the nesting problem

Abstract

A deterministic neural network, used to solve the optimizing problem of nesting (how to put as many 2-D patterns as possible in a given irregular and finite larger area), is presented. Nesting is one of the most important operations that many industries have to deal with; such industries have to cut raw material into patterns that will be the building blocks of a product. Artificial neural networks have proved to be useful in solving n-p kind of problems where the number of possible solutions grow exponentially as the problem gets bigger. It is well known that any deterministic optimizing network of this type will in general find a local minimum of the problem at hand, close enough to the global minimum to be worth considering. We propose an internal energy function E that translates into mathematical terms the goal and the constraints of the nesting problem. Afterward a free energy F expression for this system, which is valid at high temperatures, is derived. A synchronous continuous system is proposed to find a minimum of this free energy and a solution to the problem is found, continuously decreasing the absolute temperature parameter of this system until it reaches the zero temperature where the minimum of the internal energy and the minimum of the free energy coincides, having as a consequence a solution to the initial nesting problem.