Knowledge that Transforms

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.

Abstracts of the key papers presented at the Third Annual International Meeting of the Institute of Management Sciences, October 18–19, Los Angeles.

In France, the investment operations in basic industries, notably in the railroad, coal, electricity, and gas sectors, are submitted for approval of the government and the parliament. The authors of these programs have thus to determine the most economical solution of problems and then to enlighten and persuade the authorities in charge of approving operations. Towards the end of 1954 and the beginning of 1955, in the course of undertaking these tasks we found ourselves independently faced with remarkably similar problems.

In this paper, it is shown that the “caterer” problem, a problem in mathematical economics and logistics which has been discussed by Jacobs, Gaddum, Hoffman and Sokolowsky, and Prager, can be reduced to the problem of determining the maximum of the linear form, subject to a series of “triangular” constraints in the form of inequalities, under an assumption concerning the non-accumulation of dirty laundry. This maximization problem is solved explicitly, using the functional equation technique of dynamic programming.

This is to describe a proposal for comparing assumptions about the risks of war with our capabilities to handle those risks. The proposed method offers a concise, shorthand way to bring into focus the key parts of the defense planning picture. It is not a short cut to answers but it is a shorter, clearer way to present what the answers are at any one time.

In a recent paper, [Bellman, Richard. 1956. On the theory of dynamic programming—a warehousing problem. Management Sci. 2 (3, April) 272.], Bellman uses dynamic programming to establish a computational algorithm for the solution of the “warehouse” problem. The present paper also employs the dynamic-programming approach and shows that the structure of the solution can be determined analytically, with numerical results easily obtained via recursive formulas.

An accelerating increase in linear programming applications to industrial problems has made it virtually impossible to keep abreast of them, not only because of their number (and diversity) but also because of the conditions under which many are carried out. Industrial (and governmental) secrecy is often present. Other conditions also bar access to ascertainment and assessment of the pattern of applications. Lack of a tradition for publication is one. Failure to ascertain the general significance of particular findings is another, as is discouragement arising from the fact that similar applications have previously been published by others. Immediate remedies are not available for these difficulties. Presumably conventions such as this will help, over a period of time, by encouraging informal contacts between interested persons. A talk on “industrial applications of linear programming” must be altered to suit these circumstances. In place of a survey or evaluation of industrial studies, two broad issues which are relevant to all such applications will be discussed. These are, (1) use of linear programming models as guides to data collection and (2) analysis (and prognosis) of fruitful areas of additional research, especially those which appear to have been opened by industrial applications.

Much research has been undertaken in the past several years on the uses of Leontief input-output matrices; applications have been made ranging from industrial growth models to models of interregional dynamics to estimates of structural changes in the economy. This paper discusses a general linear programming formulation of Leontief type relationships. Since we may construct models for a single firm analogous to Leontief macro-economic systems, our results are applicable to a wide variety of situations.