Problem: Newsboy Model

Betty Cooker runs a bakery in San Francisco that specializes in her famous Black Forest cakes. These cakes come with four kinds of frostings-Vanilla, Chocolate, Raspberry and Devilicious. She estimates that the daily demand for each type of cake is independent and normally distributed with mean 50 and standard deviation 20. Each customer wants to buy exactly one cake. Customers who favor a particular type of frosting will not buy any other if their preferred frosting is out of stock. Every day in the morning, Betty Cooker and her team of bakers prepare a fresh batch of the cakes for sale that day. Her costs to bake and top each cake are $5. Each cake sells for $15. Betty's Bakery prides itself on its fresh assortment, so cakes not sold by the end of that day are given away to a soup kitchen for the homeless.

a) Suppose Betty wants to bake enough cakes so that she can be 97.5% sure that she can satisfy the demand for all of her customers. How many cakes with Devilicious frosting should she prepare daily in the morning? What is the associated expected profit?

b) Suppose Betty wants to bake enough cakes so that she can satisfy 97.5% of her customers' demand. How many cakes with Devilicious frosting should she prepare daily in the morning? What is the associated expected profit?

c) How many cakes with Devilicious frosting should Betty prepare daily in the morning if her objective is to maximize expected profit? What is the associated expected profit?