A Bayesian Analysis of the Style Goods Inventory Problem

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Abstract
A style goods item has a finite selling period during which the sales rate varies in a seasonal and, to some extent, predictable fashion. There are only a limited number of opportunities to purchase or manufacture the style goods item, and the cost, in general, will depend on the time at which the item is obtained. The unit revenue achieved from sales of the item also varies during the selling season, and, in particular, reaches an appreciably lower terminal salvage value. Previous work on this class of problem has assumed one of the following: (a) There is a known deterministic sales rate during the season. (b) The sales are generated by a stochastic process having exactly known probabilities. Both of these formulations of the problem overlook what we believe to be an essential feature—that initially we have a great uncertainty concerning the sales potential of the item but have an opportunity to develop a better forecast as actual sales become known. The model discussed in this paper treats the sales potential of the item as a subjective random variable whose distribution is changed adaptively (through the use of Bayes' Rule) as the sales history unfolds. We examine several methods for encoding the prior knowledge that lead to a computationally feasible dynamic programming solution to the problem. An interesting feature of this approach is that the dynamic programming state represents both the unsold inventory of the item and the current state of knowledge concerning the sales potential. Numerical examples are discussed, showing the importance of formulating the problem, in a way that includes a state of knowledge that adaptively improves as demand becomes known. Finally, computational shortcuts that make problems of a larger scale computationally feasible are discussed.

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