A Family of Inventory Models

An inventory control model in which the state of the system is reviewed only at discrete, equally spaced time intervals is studied. A procurement is made if at the review period the inventory position of the system is less than or equal to a number k. The quantity procured is an integral multiple of a number Q. The total expected cost of review, procurement, holding inventory, and stockouts is determined under the assumption that all demands are ultimately met. The cost of a stockout is taken to consist of a fixed cost per unit out of stock plus a variable cost which is proportional to the time out of stock. Specific expressions for all the cost expressions are found for the case of (1) Poisson demands and fixed lead times, and (2) Poisson demands and gamma lead times. It is shown that many of the inventory models discussed in the literature are special cases of the model here described.