In continuous review models with a fixed delivery lag T, the state of the system is conveniently described by the net inventory position = (inventory on hand) plus (outstanding orders), in spite of most cost components depending on the actual inventory on hand. To relate these two inventory concepts one observes that the distribution of the inventory on hand at time t + T is determined by the inventory position at time t. This explains the standard convention of charging the expected costs incurred in [Sn + T1, Sn+1 + T) to the decision made at time Sn, where Sn denotes the nth decision epoch. This paper derives simple expressions for the expected costs in [Sn + T1, Sn + 1 + T) as a function of the inventory position just after decision epoch Sn.
Federgruen, Awi, and Zvi Schechner. "Cost formulas for continuous review inventory models with fixed delivery lags." Operations Research 31, no. 5 (1983): 957-965.
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