Quadratic Inventory Cost Approximations and the Aggregation of Individual Products

The purpose of this paper is to place into focus a whole class of problems that have not yet received the attention they deserve. The problem studied, although seemingly narrow in scope, reflects a much broader class of problems relating aggregate and disaggregate cost functions. It raises questions relating to the efficacy of constructing aggregate cost functions that adequately represent the "true" costs and, at the same time, are easy to manipulate in decision models. It also questions whether seemingly optimal aggregate decisions will lead to serious suboptimalities when they are broken down to the level of disaggregation necessary for implementation. The questions are raised in the context of the aggregate inventory cost function of HMMS as well as their suggested procedure for using disaggregate data to estimate the parameters of the aggregate cost function. It is found that, under conditions of varying demand, the parameters of the aggregate cost function ordinarily thought to be independent of demand are in fact highly related to demand. Approximate cost functions designed to hold in certain ranges of the decision variables may fit poorly in other ranges and may therefore be unreasonable for developing decision rules for respond-to rather wide variations in demand.