Knowledge that Transforms

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.

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.

A problem of primary significance to a variety of industries is the suppression of trim losses in cutting rolls of paper, textiles, cellophane, metallic foil, or other material, for the execution of business orders. This problem is amenable to solution by the application of mathematical tools. We shall illustrate the general treatment of such problems by demonstrating a numerical example.

The word “leadership” has been widely used. Political orators, business executives, social workers, and scholars employ it in speech and writing. Yet, there is widespread disagreement as to its meaning. Among social scientists, the theoretical formulations of the leadership concept have continued to shift, focusing first upon one aspect and then upon another. Much still needs to be done to develop a basic, systematic theory. The time seems ripe for attempting a careful statement of a frame of reference which may serve to make available research more meaningful, and which may guide future research and practice. Specifically, such a frame of reference can perform the useful function of pointing to the variables which need to be measured. It can help us to state hypotheses concerning the key variables underlying leadership effectiveness. It can also provide meaningful objectives for the development of more adequate leaders.

This paper is the result of a survey made during the summer of 1956. It is a progress report on applications of linear programming by a number of oil companies. Examples are presented of applications to a variety of problems arising in the areas of Drilling and Production, Manufacturing, and Marketing and Distribution. The examples were selected to illustrate both the power and the limitations of present linear programming methods when applied to actual problems.