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.

An important class of mathematical programming problems is the scheduling of manufacturing and transportation systems. In many cases, the independent variables which describe the manufacturing system are interrelated in a highly nonlinear manner. The majority of the system variables are normally required to represent transportation and allocation. These variables appear linearly and must satisfy a system of equalities which is very large if considered as a single matrix. With such systems it is usually possible to select a relatively small number of the system variables so that when these selected (decision or coupling) variables are held fixed the complete nonlinear system can be partitioned into a number of relatively small independent linear sub-problems. An iterative method for the solution of such problems has been presented [Rosen, J. B. 1963. Convex partition programming. R. L. Graves, P. Wolfe, eds. Recent Advances in Mathematical Programming. McGraw Hill, 159–176.]. The method starts with initial values for the decision variables and solves the separate linear subproblems. The optimal solution to each subproblem is then used to determine that the complete system optimum has been found or to find improved values of the decision variables. This procedure is continued until the complete system optimum has been obtained. Application of the Partition Programming method to a typical large manufacturing-transportation system will be described, including computational experience. This will illustrate that the method can be successfully used for systems which do not satisfy the mathematical requirements which insure convergence of the method to a global optimum. The economic interpretation of an optimal solution will be discussed, showing how the complete system shadow prices are obtained from those of the individual subproblems.

Reviews the book "A Tentative Set of Broad Accounting Principles for Business Enterprises," by Robert T. Sprouse and Maurice Moonitz.

The optimal composition of cattle feed, which can be formulated as a linear programming problem in the case of certainty, is considered when compositions of inputs vary. In the corresponding linear programming formulation the coefficients of the constraints are not constant but can be considered as stochastic. Reformulating the constraints as chance constraints, a nonlinear programming problem results. For an illustrative example this problem is solved using one of Zoutendijk's methods of feasible directions.

T HIS paper examines the effects of debt management on aggregate expenditure during I9 53-58. The Treasury in this period lengthened the debt in recession and allowed it to shorten somewhat in prosperity (Table i), the opposite of the anti-cyclical policy advocated by some economists. Treasury policy was defended on the grounds that it did not unduly intensify recessions and that offerings of longterm securities in prosperity provided undesirable competition with new issues of private, state, and local government securities and increased interest costs.1 Debt management for purposes of this paper