| Abstract: | We examine vendor‐managed inventory contracts in a (Q, r) inventory system between a supplier and a retailer, in which a stockout penalty is charged to the supplier based on the length of the time period during which stockouts occur at the retailer. Linear and quadratic forms of the time‐dependent stockout penalty are considered. For the deterministic demand case, we find that the quadratic form of time‐dependent stockout penalty is equivalent to a proportional stockout penalty per unit short per unit time. For the stochastic demand case, we provide the exact cost expressions for the supplier and the retailer with a linear time‐dependent stockout penalty. We also discuss how the stochastic model can be extended to the case with a quadratic time‐dependent stockout penalty when there is at most one outstanding replenishment order at any point of time. We provide several interesting computational results. |
