Comments based on material originally prepared for the Seminar on Theory and Research in Modern Management. Cooperative Center for Educational Administration, Columbia University.
The following type of problem has arisen in various contexts in applications of management science: to determine a program of future levels of activity (e.g., production or employment) subject to known future requirements, which will minimize the total cost of the projected program over the entire planning interval of time. Mathematical methods have been developed for treating variants of this problem, and the purpose of this paper is to present an outline of methods appropriate to problems which share a certain common mathematical form.
The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Académie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7–8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.—A. Charnes, Northwestern Technological Institute and The Transportation Center.
The quantitative aspects of business management were left almost entirely in the hands of accountants prior to the advent of “scientific management” some seventy-five years ago. Although this term has fallen into disuse, the emphasis upon quantitative bases for decision-making and control has increased as the management problem has become more complex. Needs for additional data and for more comprehensive analysis have brought changes in accounting to provide more frequent and more detailed information. New systems for processing data have been developed; in some cases these appear to have replaced or to have amended the traditional notions about accounting. Actually, none of the new methods really alters the basic situation: no one approach to managerial measurements is completely effective in all circumstances. What is really needed is a combination of techniques; traditional and novel methods, census and sampling techniques, general- and special-purpose analyses all contribute to the measurement objective. We should strive to combine all available quantitative techniques in such ways as will provide optimum service to meet the needs of management.
This paper studies the planning problem faced by a machine shop required to produce many different items so as to meet a rigid delivery schedule, remain within capacity limitations, and at the same time minimize the use of premium-cost overtime labor. It differs from alternative approaches to this well-known problem by allowing for setup cost indivisibilities. As an approximation, the following linear programming model is suggested: Let an activity be defined as a sequence of the inputs required to satisfy the delivery requirements for a single item over time. The input coefficients for each such activity may then be constructed so as to allow for all setup costs incurred when the activity is operated at the level of unity or at zero. It is then shown that in any solution to this problem, all activity levels will turn out to be either unity or zero, except for those related to a group of items which, in number, must be equal to or less than the original number of capacity constraints. This result means that the linear programming solution should provide a good approximation whenever the number of items being manufactured is large in comparison with the number of capacity constraints.