Shelf space in the grocery business is a valuable asset. Every good supermarket spends a significant amount of effort attempting to determine the optimal shelf space allocation across products. Many factors are relevant to this decision: the profitability of each product, the size of each product, the demand characteristics of each product, and so forth. Southern Fresh has decided to make a policy that every product will receive enough shelf space to ensure that 98.75 percent of customers will find that their first choice is available. Consider Hot Bull corn chips, a local favorite. Average daily demand for this product is 55, with a standard deviation of 30. Bags of Hot Bull can be stacked 20 deep per facing. (A facing is the width on a shelf required to display one item of a product.) Deliveries from Southern Fresh's central warehouse occur two days after a store manager submits an order. (Actually, in most stores, orders are generated by a centralized computer system that is linked to its point-of-sales data. But even these orders are received two days after they are transmitted.)

a.How many facings should Southern Fresh allocate to Hot Bull corn chips?

b.How many facings are needed to achieve a 98.75 percent in-stock probability?

c. Suppose Southern Fresh allocates 11 facings to Hot Bull corn chips. On average, how many bags of Hot Bull are on the shelf at the end of the day?

d. Although Southern Fresh does not want to incur the cost of holding inventory, it does want to leave customers with the impression that it is well stocked. Hence, Southern Fresh employees continually roam the aisles of the store to adjust the presentation of the product. In particular, they shift product around so that there is an item in each facing whenever possible. Suppose Southern Fresh allocates 11 facings to Hot Bull corn chips. What is the probability that at the end of the day there will be an empty facing, that is, a facing without any product?