Base stock inventory policies for a multi-item demand process

We consider a retail inventory system in which customer orders arrive at random and each order specifies a list of items. The inclusion or otherwise of an item on an order is independent of what other items may or may not be included, subject to at least one item being listed. There is a fixed cost associated with the immediate dispatch of items included on an order and in stock and an additional fixed cost for dispatching, after a delay, an item included on the order but not in stock. There is, therefore, a cost advantage in having items available for immediate dispatch. In addition, time-weighted holding and shortage costs are incurred for each item. The stock of each item is controlled using a base stock system and the objective is to find the item base stocks that jointly minimize the total cost of the system. We replace the customer order arrival process with an equivalent process that permits an order to list no items and use this modified process to develop a solution procedure that is simple to describe and computationally undemanding. We illustrate the procedure through a wide range of numerical examples.