A finite-horizon inventory system with partial backorders and inventory holdback

We consider a periodic-review base stock inventory system with partial backorders. At the beginning of each period t of a T period problem, an order of size qt is placed for delivery one period later. As the stochastic demand is realized, as much as possible of it is filled immediately from the inventory on-hand. If the realized demand exceeds the inventory on-hand, up to qt-kt units of excess demand are backordered to be filled from the pipeline inventory or future orders. The on-order quantity kt denotes the reservation quantity held back for use in subsequent periods. The case kt=qt yields the full lost sales model while the case kt=−∞ yields the full backorder model.