Knowledge that Transforms

Some stochastic models are analyzed which describe the time fluctuations of the inventory levels of companies which are in the rental business. The stochastic process which describes the fluctuations is shown to be applicable to a wide variety of physical situations. Examples are given, and analyses are made of several types of profit functions which are applicable to rental situations.

In many optimization problems the objective function may depend on a random set of coefficients that have some known distribution. For example, a profit function may depend on certain market conditions which can at best be estimated by a set of random variables with some given distribution. An important question to be answered in such problems is: What is the expected value of the maximum profit? On the answer to this question may hinge certain very important decisions that an organization may have to make. It may also provide good estimates of future profits. While the problem of determining the expected value of the maximum may be very difficult at times, a number of related problems, some quite important in their own right, may be considerably easier to solve and also may provide bounds for the expected value of the maximum. In this work, one upper and three lower bounds are given in terms of the solutions of related problems. In addition a convexity property for a class of parametric nonlinear programming problems is obtained. This property is also used in deriving some of the bounds.

I have detected the following minor errors in my article, “Operating Characteristics of Opportunistic Replacement and Inspection Policies," Management Science, Vol. 10, No. 1 (October 1963), pp. 85–98, that might possibly cause some confusion. The first five “t's” appearing in the Preliminaries (p. 87), the first five “t's” in the Preliminaries (p. 88) and the first five “t's” in the Preliminaries (p. 90) should all be “X's”.

Ludwig von Bertalanffy, in 1951, and Kenneth Boulding, in 1956, wrote articles which have provided a modern foundation for general systems theory [Bertalanffy, L. von. 1951. General system theory: A new approach to unity of science. Human Biol. (December) 303–361; Boulding, K. 1956. General systems theory: The skeleton of science. Management Sci. (April) 197–208.]. We build on that foundation in applying general systems theory to management. The general theory is reviewed for the reader. Next, it is applied as a theory for business, and an illustrative model of the systems concept is developed to show the business application. Finally, the systems concept is related to the traditional functions of a business, i.e., planning, organizing, control, and communications.

This article is based on an idea proposed by C. W. Carroll for transforming a mathematical programming problem into a sequence of unconstrained minimization problems. It describes the theoretical validation of Carroll's proposal for the convex programming problem. A number of important new results are derived that were not originally envisaged: The method generates primal-feasible and dual-feasible points, the primal objective is monotonically decreased, and a subproblem of the original programming problem is solved with each unconstrained minimization. Briefly surveyed is computational experience with a newly developed algorithm that makes the technique competitive with known methodology. (A subsequent article describing the computational algorithm is in preparation.)

The use of multiple performance criteria which account for both the static and dynamic aspects of marketing systems is proposed as a necessary step toward scientific marketing analysis. Currently, over-all evaluations using multiple performance measures are difficult to make and even more difficult to interpret meaningfully. Fisher's technique of discriminant functions is utilized to estimate the mathematical relationship between the relevant marketing performance measures which best “matches” subjective classifications agreed upon by a number of business executives. An application of the technique involving the evaluation of a consumer product's performance in various market areas is discussed. The problems of bias and multicollinearity, their relationship to the conceptual approaches, and their effect on the use of the performance measure are also discussed using the particular problems encountered in the application to illustrate the various points.

An exploration of the margin of error entailed in using a “one-point move” algorithm for solving a class of fixed-charge problems. The algorithm is of interest both from the viewpoint of numerical analysis and also from the analogy with market mechanisms. Despite the presence of economies-of-scale, there is the possibility of operating a decentralized system. This is a two-price system; it implies discriminatory and also marginal cost pricing.

A simplex-type method for finding a local maximum of [Formula: see text] subject to [Formula: see text] and [Formula: see text] is proposed. At a local maximum, the objective function (1), can be expressed, in terms of the non-basic variables λ 0 , as [Formula: see text] and the vector of partial derivatives of (13), with respect to the non-basic variables may be written, [Formula: see text] This allows calculation of the maximum values of the non-basic variables, increased one at a time, consistent with ∇Z ≧ 0. A “cutting plane” a ** λ′ ≧ 1 is then defined which excludes the local optimum, and many lower values (but no higher values) of (1). The form of the square matrix C is immaterial.

In this paper we consider the dynamic inventory problem in which the demand distribution possesses a density belonging to either the exponential or range family of densities and having an unknown parameter. An a priori density is chosen for the unknown parameter. Using a Bayesian estimation scheme, inequalities are obtained for the optimal purchase policies as the amount of demand information varies. In addition, asymptotic expansions for the optimal policies are found as the number of observations of the demand becomes large. This paper extends the results of Scarf, [8].

The static deterministic inventory model has been extended to the problem of multi-purpose equipment scheduling. The cycle time which will minimize the sum of inventory, backorder, and setup charges is determined.