On the generalised stock–cutting problem

In this paper, we formulate the two-stage stock-cutting problem, according to which a set of rectangular pieces of prespecified dimensions are to be cut from an arbitrarily shaped object with arbitrarily shaped holes or defective regions. We show how mathematical morphological operators can be used in order to determine the optimal shifting for a given cutting pattern. It is then proved that the problem of obtaining the optimal cutting pattern is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal NP}$\end{document}-hard and a solution to the unconstrained problem using mathematical programming is proposed. However, for the general problem, good sub-optimal solutions can be obtained using the technique of simulated annealing. Experimental results are also included.