The cutting-stock problem is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut from given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. This same difficulty persists when only an approximate solution is being sought by linear programming. In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the problem. The technique enables one to compute always with a matrix which has no more columns than it has rows. OME linear programming problems arising from combinatorial prob-< lems become intractable because of the large number of variables involved. Usually each variable represents some activity, and the difficulty is that there are too many possible competing activities satisfying the combinatorial restrictions of the problem. An example of this is the cutting-stock problem described below in a form similar to that used by EISEMANN. [1]

In this paper, the methods for stock cutting outlined in an earlier paper in this Journal [Opns Res 9, 849–859 (1961)] are extended and adapted to the specific full-scale paper trim problem. The paper describes a new and faster knapsack method, experiments, and formulation changes. The experiments include ones used to evaluate speed-up devices and to explore a connection with integer programming. Other experiments give waste as a function of stock length, examine the effect of multiple stock lengths on waste, and the effect of a cutting knife limitation. The formulation changes discussed are (i) limitation on the number of cutting knives available, (n) balancing of multiple machine usage when orders are being filled from more than one machine, and (m) introduction of a rational objective function when customers' orders are not for fixed amounts, but rather for a range of amounts. The methods developed are also applicable to a variety of cutting problems outside of the paper industry.

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We consider a machine with a single real variable x that describes its state. Jobs J1, …, JN are to be sequenced on the machine. Each job requires a starting state A, and leaves a final state Bi. This means that Ji can be started only when x = Ai and, at the completion of the job, x = Bi. There is a cost, which may represent time or money, etc., for changing the machine state x so that the next job may start. The problem is to find the minimal cost sequence for the N jobs. This problem is a special case of the traveling salesman problem. We give a solution requiring only 0(N2) simple steps. A solution is also provided for the bottleneck form of this traveling salesman problem under special cost assumptions. This solution permits a characterization of those directed graphs of a special class which possess Hamiltonian circuits.