Scarf's State Reduction Method, Flexibility, and a Dependent Demand Inventory Model

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Abstract
We consider a finite horizon inventory model with linear holding, shortage, and ordering costs. The demand random variables are dependent, and average demand is described by an exponential smoothing formula. This model can be formulated as a two-state variable (inventory level, weighted past demands) dynamic program. Using a procedure first developed by Scarf for a Bayesian inventory model, we are able to reformulate the model as a one-state variable dynamic program, resulting in a considerable computational saving over the two-state variable formulation. We also show that this dependent demand model orders less than or equal the amount ordered by a comparable independent demand model. We establish this result under the assumption that demand represents failures that are repaired by the beginning of the next period.

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