Shelf space in the grocery business is a valuable asset. Every good grocery store 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. Let us now consider the shelf space allocation problem at a local 7-11 store. This particular 7-11 store has decided to make a policy that every product will receive enough shelf space to ensure that 99% of customers will find their first choice product is available (i.e., the fill rate is 99%). Consider the Coca Cola on the shelf as an example. Soft drink like cola has being a stable product, the normal distribution is a good assumption to represent demand. Average daily demand for Coca Cola in the following image is 90 bottles at this store and the daily standard deviation is 30 bottles. Bottles of Coca cola shown below can be stacked 20 deep per facing. (A facing is the width on a shelf required to display one item of a product. See the picture below for an illustration. Thus, 20 deep per facing means that there are 19 bottles of juices in each facing behind the first bottle that you see in the picture.)

Deliveries from 7-11s central distribution center occur 2 days after a store manager submits an order while the store manager reviews inventory once every day. Based on the information provided above, answer the following questions (5 parts, please show all calculations for each part). Please note that z-table and loss function tables for standard normal distribution have been included in this exam. Also, assume there is no backroom storage for this type of bottled Coca Cola.

(HINT: Think carefully what determines the amount of shelf space (i.e., facings) in this setting? Is it S or something else?)

a. (18 points) How many facings should 7-11 allocate to Coca cola in this store?

b. (18 points) How many facings are needed to achieve a 98.75% in-stock probability (or service level) instead?

c. (12 points) Suppose 7-11 allocates 7 facings to Coca Cola in this type. On average, how many bottles of Coca cola are on the shelf at the end of any day just before closing? (Assume deliveries are made after-hours, i.e., after the store is closed)

d.(12 points) Although 7-11 does not want to incur the cost of holding excessive inventory, it does want to leave customers with the impression that it is well stocked. Hence, 7-11 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 the employees allocate 4 facings to Coca cola of this type. What is the chance that at the end of the day there will be an empty facing, i.e., a facing without any Coca Cola left?

e. (20 points) The 7-11 store under consideration also sells Pepsi Cola. The daily demand for Pepsi also follows a normal distribution with an average of 40 bottles and a standard deviation of 20 bottles. The lead-time for replenishment and the review period are identical to those of Coca Cola. Bottles of Pepsi Cola are also stacked 20 deep per facing. The research by the store manager has further found that there is perfect substitutability between the two brands of cola. This implies that when one of the two brands is out of stock, all of the customers with unmet demand will buy the second brand if it is in stock. Given this, how many total facings should 7-11 allocate to Coca Cola and Pepsi Cola together? Assume that 7-11 still wants a combined (i.e. considering both brands together) 99% fill rate and that demand correlation between Pepsi Cola and Coca Cola is zero.