An algorithm for the determination of optimal cutting patterns

Abstract

This paper presents a new mathematical programming formulation for the problem of determining the optimal manner in which several product rolls of given sizes are to be cut out of raw rolls of one or more standard types. The objective is to perform this task so as to maximize the profit taking account of the revenue from the sales, the costs of the original rolls, the costs of changing the cutting pattern and the costs of disposal of the trim. A mixed integer linear programming (MILP) model is proposed which is solved to global optimality using standard techniques. A number of example problems, including an industrial case study, are presented to illustrate the efficiency and applicability of the proposed model. Scope and purpose One-dimensional cutting stock (trim loss) problems arise when production items must be physically divided into pieces with a diversity of sizes in one dimension (e.g. when slitting master rolls of paper into narrower width rolls). Such problems occur when there are no economies of scale associated with the production of the larger raw (master) rolls. In general, the objectives in solving such problems are to [5]: • minimize trim loss; • avoid production over-runs and/or; • avoid unnecessary slitter setups. The above problem is particularly important in the paper converting industry when a set of paper rolls need to be cut from raw paper rolls. Since the width of a product is fully independent of the width of the raw paper a highly combinatorial problem arises. In general, the cutting process always produces inevitable trim-loss which has to be burned or processed in some waste treatment plant. Trim-loss problems in the paper industry have, in recent years, mainly been solved using heuristic rules. The practical problem formulation has, therefore, in most cases been restricted by the fact that the solution methods ought to be able to handle the entire problem. Consequently, only a suboptimal solution to the original problem has been obtained and very often this rather significant economic problem has been left to a manual stage. This work presents a novel algorithm for efficiently determining optimal cutting patterns in the paper converting process. A mixed-integer linear programming model is proposed which is solved to global optimality using available computer tools. A number of example problems including an industrial case study are presented to illustrate the applicability of the proposed algorithm.