Knowledge that Transforms

Consider a linear-programming problem in which the “right-hand side” is a random vector whose expected value is known and where the expected value of the objective function is to be minimized. An approximate solution is often found by replacing the “right-hand side” by its expected value and solving the resulting linear programming problem. In this paper conditions are given for the equality of the expected value of the objective function for the optimal solution and the value of the objective function for the approximate solution; bounds on these values are also given. In addition, the relation between this problem and a related problem, where one makes an observation on the “right-hand side” and solves the (nonstochastic) linear programming problem based on this observation, is discussed.

Using an illustration drawn from the area of inventory control, this paper demonstrates how a typical sequential probabilistic model may be formulated in terms of (a) an initial decision rule and (b) a Markov process, and then optimized by means of linear programming. This linear programming technique may turn out to be an efficient alternative to the functional equation approach in the numerical analysis of such problems. Regardless of computational significance, however, it is of interest that there should be such a close relationship between the two traditionally distinct areas of dynamic programming and linear programming.

The relation between productivity, efficiency and size of a technical organization as affected by internally generated and circulated paperwork is analyzed. It is shown that there exists an upper bound to total productive output which is independent of the number of employees; and that as the organization size is increased the efficiency generally first rises and then falls off inversely proportionally to the number of employees.

In Manne [Manne, A. S. 1960. Linear programming and sequential decisions. Management Sci. (April).] has suggested a linear programming formulation for on optimal steady state solution of sequential decision models. In this note we offer a proof which appeals primarily to the properties of linear programs that an optimal solution exists involving only pure strategies.

In the last several years there have been a number of papers discussing optimal policies for the inventory problem. Almost without exception these papers are devoted to the determination of optimal purchasing quantities at a single installation faced with some pattern of demand. It has been customary to make the assumption that when the installation in question requests a shipment of stock, this shipment will be delivered in a fixed or perhaps random length of time, but at any rate with a time lag which is independent of the size of the order placed. There are, however, a number of situations met in practice in which this assumption is not a tenable one. An important example arises when there are several installations, say 1, 2, …, N, with installation 1 receiving stock from 2, with 2 receiving stock from 3, etc. In this example, if an order is placed by installation 1 for stock from installation 2, the length of time for delivery of this stock is determined not only by the natural lead time between these two sites, but also by the availability of stock at the second installation. In this paper we shall consider the problem of determining optimal purchasing quantities in a multi-installation model of this type.

This paper reports the results of an analytic problem arising out of a history. Although some of the context and all of the numbers have been changed to protect proprietary information, the following important facet of the original case history remains invariant in the transformation. Management of the sponsoring firm had planned to make a policy decision based on differences between averages of overlapping distributions. After the influence of the probability distributions was evaluated and explained, management reversed its intended policy decision.

Discussion on two early papers of Professor Kantorovich, from 1939 and 1949, that proved to be remarkable documents in the history of management science, of linear programming, and of economic theory in general. The 1949 paper discusses transportation models for a single commodity and for many commodities (including empty vehicles), and a single-commodity model for a capacitated network, with applications to sections of the Russian railroad network. All problems considered in the 1939 paper reprinted in this issue are what would now be called linear programming problems. The coefficient matrices of the problems labeled “A” and “B” exhibit special structures somewhat similar to that of the transportation problem matrix. Problem “C,” while appearing still to have a somewhat special structure, is in fact equivalent to the general linear programming problem.

In recent years considerable attention has been paid by marketing theorists to the role of advertising and other promotional activities within the marketing policy of the firm and to their optimal setting in the combined operation of the various instruments available to such a policy. These authors have generally assumed efficient spending of money on advertising as far as choice of media and of purchase motivating themes are concerned. It is the object of the present paper to consider in some detail the second of these two problems—the efficient selection of purchase motivating themes for activation through advertising.

The problem of specifying an allowance for defects in a production lot is that of balancing the cost of producing too many items against the risk of not having enough to meet requirements. A model of these costs is here proposed. Sufficient conditions are developed on the probability distribution of defectives for total cost to have a single minimum with respect to the allowance. A sequential algorithm is investigated and shown to produce an optimum allowance if certain further conditions on the probability are met. Next, it is shown for a special class of probability distributions that the above conditions are satisfied. This class is that for which the probability of an item being defective is independent of previous defects in the lot, and includes the binomial distribution. Finally some computational aspects of this algorithm are discussed, and an easily computable starting value is given.

The goal of the article is to show how sometimes the use of contour integrals can help in solving problems of inventory and queueing theories by enabling the target function to be uniquely represented in the analytical form min(0, X). The use of those integrals allows us to rewrite some well-known formulas from queueing theory in a larger setting that enables generalizations. That is our goal in the first part of the paper. In the second part, this idea is applied to a particular problem to determine the inventory that is changing ten months in a year by regular reduction for use in production and by supplies arriving by ships, randomly distributed following the Poisson distribution. Penalties imposed are proportional to the waiting time for a ship to arrive. The mathematical problem that enables to determine the optimum between the costs of waiting for a ship to arrive and the costs of maintaining the inventory is expressed as the residue of a contour integral. An asymptotic formula useful for numerical calculation is derived.