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]

The network flow problem was first considered by Ford and Fulkerson[1] who introduced the basic concepts of flow, cut, etc. used here and provided the main tool, the maximum-flow minimum-cut theorem. Ford and Fulkerson wrote about the flow between two special points, the source and the sink. Mayeda [2] then took up the multi-terminal problem, where flows are considered between all pairs of nodes in a network, and Chien [3] discussed the synthesis of such a network. In this paper we continue with the multi-terminal problem, giving results on realizability, analysis, and synthesis. Although this paper is self-contained we will use throughout many of the notions of [1, 2, 3].We consider connected networks consisting of nodes N and branches *

The problem of obtaining the best integer solution to a linear program comes up in several contexts. The connection with combinatorial problems is given by Dantzig in [l], the connection with problems involving economies of scale is given by Markowitz and Manne [3 ] in a paper which also contains an interesting example of the effect of discrete variables on a scheduling problem. Also Dreyfus [4] has discussed the role played by the requirement of discreteness of variables in limiting the range of problems amenable to linear programming techniques.It is the purpose of this note to outline a finite algorithm for obtaining integer solutions to linear programs. The algorithm has been programmed successfully on an E101 computer and used to run off the integer solution to small (seven or less variables) linear programs completely automatically.The algorithm closely resembles the procedures already used by Dantzig, Fulkerson and Johnson [2], and Markowitz and Manne [3] to obtain solutions to discrete variable programming problems. Their procedure is essentially this. Given the linear program, first maximize the objective function using the simplex method, then examine the solution. If the solution is not in integers, ingenuity is used to formulate a new constraint that can be shown to be satisfied by the still unknown integer solution but not by the noninteger solution already attained. This additional constraint is added to the original ones, the solution already attained becomes nonfeasible, and a new maximum satisfying the new constraint is sought. This process is repeated until an integer maximum is obtained, or until some argument shows that a nearby integer point is optimal. What has been needed to transform this procedure into an algorithm is a systematic method for generating

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.