cinnamon-raisin bagels market research

Every weekday morning, University Deli orders and receives a shipment of fresh bagels. Included in this shipment of bagels are cinnamon-raisin bagels. You have heard rumors that more than half of the time, University Deli runs out of cinnamon-raisin bagels by mid-morning. Suppose that University Deli pays $.20 for each cinnamon-raisin bagels that they order, and suppose that University Deli sells the bagels for $.65 each. Assume that any unsold bagels are thrown away at the end of the day.
a) Suppose that daily demand for cinnamon-raisin bagels is equally likely to be any number from one to twenty. Further, suppose that if a customer wants a cinnamon-raisin bagel, and if University Deli has sold out of cinnamon-raisin bagels, then the customer does not buy any other kind of bagel. How many cinnamon-raisin bagels should University Deli order every day to maximize its expected profit?
b) Based on your answer to part (a), does it make sense that University Deli runs out of cinnamon-raisin bagels more than half of the time? If University Deli follows your recommendation of optimal order quantity in part (a), what percentage of the time should they run out of cinnamon-raisin bagels?
c) Suppose that you do some market research and find out that there is a chance that if University Deli has sold out of cinnamon-raisin bagels, a customer wanting a cinnamon-raisin bagel might buy a granola bar from a vending machine at University Deli instead. For simplicity, suppose that there are always granola bars in the vending machines, and suppose that University Deli makes a profit of $.40 on each granola bar. Suppose that there is a 75% chance that a customer wanting a cinnamon-raisin bagel will buy a granola bar if cinnamon-raisin bagels are sold out. Assuming that University Deli is adjusting its optimal number of cinnamon-raisin bagels to this market research and ordering an optimal number of cinnamon-raisin bagels according to this market research, what is the optimal order quantity of cinnamon-raisin bagels?
d) Assuming that University Deli has adjusted its optimal number of cinnamon-raisin bagels to your market research and now orders the optimal number of cinnamon-raisin bagels as you recommend in part (c), what percentage of the time will they run out of cinnamon-raisin bagels?

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