CPG Bagels starts the day with a large production run of bagels. Through- out the morning, additional bagels are produced as needed. The last bake is completed at 3 P. M. and the store closes at 8 P. M . It costs approximately $0.20 in materials and labor to make a bagel. The price of a fresh bagel is $0.60. Bagels not sold by the end of the day are sold the next day as "day old" bagels in bags of six, for $0.99 a bag. About two-thirds of the day-old bagels are sold; the remainder are just thrown away. There are many bagel flavors, but for simplicity, concentrate just on the plain bagels. The store manager predicts that demand for plain bagels from3 P. M. until closing is normally distributed with mean of 54 and standard deviation of 21.

a. How many bagels should the store have at 3P. M . to maximize the store's expected profit (from sales between 3 P. M. until closing)?

b. Suppose that the store manager is concerned that stockouts might cause a loss of future business. To explore this idea, the store manager feels that it is appropriate to assign a stockout cost of $5 per bagel that is demanded but not filled. (Customers frequently purchase more than one bagel at a time. This cost is per bagel demanded that is not satisfied rather than per customer that does not receive a complete order.) Given the additional stockout cost, how many bagels should the store have at 3P. M. to maximize the store's expected profit?

c. Suppose the store manager has 101 bagels at 3 P. M. How many bagels should the store manager expect to have at the end of the day?