(CPG Bagels) CPG Bagels starts the day with a large production run of bagels. Throughout 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 from 3 PM. until closing is normally distributed with mean of 54 and standard deviation of 21.
a. How many bagels should the store have at 3 P.M. to maximize the store's expected profit (from Sales between 3 P.M. until closing)? (Hint: Assume day-old bagels are sold for $0.99/6 = $0.165 each; i.e., don't worry about the fact that day-old bagels are sold in bags of six.)
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 3 P.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?