Stock cutting to minimize cutting length

In this paper we investigate the following problem: Given two convex P_in, and P_out where P_in is completely contained in P_out, we wish to find a sequence of ‘guillotine cuts’ to cut out P_in from P_out such that the total length of the cutting sequence is minimized. This problem has applications in stock cutting where a particular shape or design (in this case the polygon P_in) needs to be cut out of a given piece of parent material (the polygon P_out) using only guillotine cuts and where it is desired to minimize the cutting sequence length to improve the cutting time required per piece. We first prove some properties of the optimal solution to the problem and then give an approximation scheme for the problem that, given an error range δ, produces a cutting sequence whose total length is atmost δ more than that of the optimal cutting sequence. Then it is shown that this problem has optimal solutions that lie in the algebraic extension of the field that the input data belongs to — hence due to this algebraic nature of the problem, an approximation scheme is the best that can be achieved. Extensions of these results are also studied in the case where the polygons P_in and P_out are non-convex.