State reduction in a dependent demand inventory model given by a time series

Introduction In practice we are very often confronted with dynamic inventory problems of dependent demand. However, until now not much work has been done on this subject. Usually, stochastic models assume that demand distributions of inventory in different periods are independent. Recently more attention has been given to dependent demand inventory models e.g. Blinder (1982), and Miller (1986). Usually the exponential smoothing forecasting model is considered. In this paper we consider a more general model for a dependent demand which is given by a time series. The time series will be given by a Stochastic Trend Component and a Seasonal Component. A Polynomial Trend Component and a Trading Day Effect will be neglected in the considered time series model. Consideration of dependent demand leads to addition of a second state variable pertaining to demand in the dynamic programming formulation. The first state is as usual the inventory level. Obviously, taking into account the dependent demand. greatly increases the computational complexity of dynamic programming algorithms. Therefore the main attempt is to reduce the two-state variable dynamic programming problem to a one-state variable problem under assumed demand and cost structure. The method of Scarf (1960) and Karlin (1960) which reduces a two-state variable Bayesian inventory model to a one-state variable model when demand is from exponential and range families can be used to solve the problem. Auzory (1985) has extended this result to the cases where demand is uniform and Weibull. Miller (1986) uses the same concept for the demand described by the exponential smoothing formula. The same approach will be used in this paper for the single item periodic review inventory problem with possible returns of a part of demand in the next period.