Solving the Two-Stage Cutting Stock Problem

I N T R O D U C T I O N In a number of industries, such as paper and film, there is an important class of problems OVER THE past 20 years, a substantial amount that falls in between the well known one and of work has been done on one and two-dimentwo-dimensional problems. These are referred sional cutting stock problems. A number of to here as two-stage cutting stock problems early authors [1, 6-8] recognized that the roll because the material is processed through two trim problem in the paper industry could be distinct cutting operations. It differs from the formulated as a linear programming model, multi-stage cutting problems discussed by GilThe pioneering computational work by Gilmore and Gomory because the first and second more and Gomory [2,33 on delayed pattern stage cuts are parallel to one another. This generation techniques made it practical to find .pr°blem has not been discussed in the literalinear programming solutions to large oneture even though it occurs in a number of indimensional problems. Follow-up research by dustries. This two-stage cutting may be necessPierce [9] focused on the use of this algorithm ary because of the nature of the production in the paper industry to solve the roll trim process and the type of equipment available problem. Gilmore and Gomory [4] subseor to give flexibility so that orders that vary quently extended their work to higher dimenin some characteristic other than width can be sional problems such as the cutting of reccombined across the width of the primary protangles from rectangles, a common problem in duction process. Regardless of the reason, prothe glass industry. They overcame the difficulty duction rolls are first slit into master roll of generating patterns in higher dimensions by widths at the first stage and the master roll limiting themselves to patterns with guillotine widths are then slit to the ordered sizes at the cuts (once a cut is begun it must be continued second stage. Limitations in the master roll through to the other side of the material in sizes that can be processed at the second stage a straight line). As a result the problem of make it necessary to impose additional restricgenerating a higher dimensional cutting pattions on the pattern generating process relative tern can be handled by sequentially solving a to the well known one-dimensional roll trim number of one-dimensional problems. Their problem. multi-stage approach in the two-dimensional case focuses first on cutting patterns for strips G E N E R A L PROBLEM of known width and length and then on ways CONSIDERATIONS these strips can be combined to form twodimensional patterns. Consider the production process sketched in