A featured presentation at the Ninth Annual International Meeting of The Institute of Management Sciences, jointly with the Econometric Society, at Ann Arbor, Michigan, September 11, 1962. “A unified science of management.” Is it a matter of faith or of enterprise? A unified science of management conceals the self-reflective paradox. Science is an organized activity. Hence it operates according to some managerial principles. A unified science of management implies a management of science: a science of science, a self-reflective science.

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.

This paper presents a new methodology for determining suboptimum relative location patterns for physical facilities. It presents a computer program governed by an algorithm which determines how relative location patterns should be altered to obtain sequentially the most improved pattern with each change, commands their alteration, evaluates the results of alterations, and identifies the sub-optimum relative location patterns. 3 The computer output yields a block diagramatic layout of the facility areas, and the areas need not be equal. 4 A manufacturing plant layout example is used because the methodology is well illustrated by reference to some kind of example, and this one is commonly known. The methodology itself, however, is general in nature and not restricted to such applications.

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.

A linear programming problem is said to be stochastic if one or more of the coefficients in the objective function or the system of constraints or resource availabilities is known only by its probability distribution. Various approaches are available in this case, which may be classified into three broad types: ‘chance constrained programming’, ‘two-stage programming under uncertainty’ and ‘stochastic linear programming’. For problems of ‘stochastic linear programming’ a distinction is usually made between two related approaches to stochastic programming, the passive and the active approach respectively. In the passive approach to stochastic linear programming the statistical distribution of the optimum value of the objective function is estimated either exactly or approximately by numerical methods and optimum decision rules are based on the different characteristics of the estimated distribution. In the active approach, a new set of decision variables are introduced which indicate the proportions of different resources to be allocated to the various activities. One effect of introducing this set of new decision variables in the active approach is the truncation of the statistical distribution of optimal value of the objective function of the passive approach. In this aspect the active approach is useful in suggesting criteria for changing from one optimum decision rule to another, as the probability distribution of the objective function is specified less and less incompletely for different choices of the set of new decision variables. This paper investigates some mathematical relations between the active and passive approach and derives some inequalities for the case when the elements of the coefficient matrix of the inequalities are random variables. Since an ’approximate solution’ of a stochastic linear programming problem is defined usually by replacing each random element by its expected value and then solving the resulting non-stochastic program, some conditions have been derived under which the expected value of the objective function for the optimal solution will exceed the optimal value of the objective function for ’the approximate solution’. Based on the properties of the distribution of extreme values and other order statistics, some approximate bounds have also been specified for the passive approach, which can easily be extended to the active approach.

This paper presents the results of some digital computer simulation tests of a procedure for the economic planning of lot sizes, work force, and inventories. A dynamic, deterministic, linear programming model was used to obtain approximate solutions to the actual problem which is both dynamic and stochastic. The tests were made with data taken from an actual factory. An alternate procedure, based upon single-item inventory control, was also tested; its results were compared with those obtained from the linear programming model. On the basis of these tests, this linear programming method appears to offer a promising method for the practical economic planning of production activities.