Inventory models with back-order discounts and variable lead time

For inventory problems with stock-out under probabilistic demands, most of the published data assumes that the average shortages are very small and thus are neglected. However, some of the stock-out in practice might be significant and is back ordered and filled as soon as an adequate size of replenishment arrives. Typically, a supplier will institute an emergency expediting order to obtain the item when a shortage occurs and a price discount can always be offered on the stock-out item in order to secure more back orders. In this article, an inventory model with negotiable back orders is first proposed, then another model where lead time is also subject to change is discussed. Numerical examples are included to illustrate the procedures of the solution.