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We address a one-dimensional cutting stock problem where, in addition to trim-loss minimization, cutting patterns must be sequenced so that no more than s different part types are in production at any time. We propose a new integer linear programming formulation whose constraints grow quadratically with the number of distinct part types and whose linear relaxation can be solved by a standard column generation procedure. The formulation allowed us to solve problems with 20 part types for which an optimal solution was unknown.

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This paper presents a decision support tool for solving a cutting and reuse problem arising in a European plant devoted to the production of gear belts. In this production, rectangular pieces of rubberised nylon are cut using machines employing parallel blades, so as to obtain rectangular components of identical height and (possibly) different width. A component is then used to produce a set of belts with the same girth; but, if necessary, the girth required can also be obtained by sewing together two components. The major objectives of optimisation are: trim loss minimisation, quality control, workload equalisation, setup minimisation. The problem, a particular one-dimensional cutting stock with both cutting and reuse decision variables, has been formulated in terms of integer linear programming and then efficiently solved by applying standard packages within a column generation scheme. A significant improvement of performance has been obtained in terms of both economic savings and product quality. This has convinced the management to implement the model in the plant operation.

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