Multi-Item Production and Inventory Management Under Price Uncertainty

Abstract

A model is presented for the derivation and implementation of an optimal linear decision rule for a firm producing and dealing in a number of interacting products, and possessing partial influence on their prices. The behavior of a multi-item production-inventory complex is represented by the dynamics of suitably defined state variables under the influence of decision rules that are linear in the state variables, but otherwise unspecified. The dynamic equations are stochastic owing to the presence of stochastic processes in the forcing terms. The statistical properties of these processes, together with the decision rules, determine the statistics of a functional describing the outcome. The optimum inventory decision is then derived as the best linear transformation on the past of the state variables, such that the mean value of the criterion functional is optimized subject to the system of constraints. A mechanism is also developed such that the optimal linear rule may be implemented. THE FACT THAT several products are produced and sold by a single organization raises certain problems associated with the direct and indirect interaction among the products. These interactions typically due to complementarity and substitutability-have their origin in the possibility of joint and simultaneous claim on the capacity of one or more resources or activities. As economists have well recognized, such interactions render it difficult to represent aggregate cost of production as the sum of individual costs. An entirely different kind of problem is presented when the problem of inventory management is considered in the context of one or another form of competition. In an industry in which there are relatively few firms (with or without product differentiation), each firm is potentially capable of influencing the profits of the others. That is to say, an individual firm will, in general, have some influence on the prices realized for its products, although the magnitude of such influence is not completely predictable. Consequently, a firm's decision to withold or to make available a certain quantity of output has an impact on its own price and profits. The model to be discussed in this paper attempts to solve the principal decision problem how much to make available and what incremental inventories to plan for the current period-for a multiproduct firm operating under conditions of limited competition.