Scheduling cutting machines in a clothing industry

In a clothing industry, clothes are produced by assembling fabric cutouts. Fabric rolls on a line, and is cut by automated machines arranged along the line. A fabric cutout fj requiring time Pj to be cut, is available for cutting on the first machine at a release date r1j, and must be completed, on the same machine, within a deadline D1j. On machine Mi, i = 2, …, m, release dates and deadlines related to the cutout fj are r1j = r1j + Δ(i - 1) and Dij = Dij + Δ(i - 1) respectively, where Δ is a constant factor depending on the speed rate of the fabric. A set-up time occurs on a machine, when switching from one cutout to another. The problem of assigning and scheduling cutting tasks to the minimum number of machines is addressed.