PI. BISCUITS A restaurant sells biscuits for $1.5 each from 8:00am to 3:00pm. The biscuits are ordered from a supplier at the start of each day and delivered before the store opens. The supplier charges $1 per biscuit and there is no fixed ordering cost. If biscuits spend a night in storage they lose most of their taste and so if at 3:00pm some biscuits are left, the restaurant sells them off at $.50 per biscuit. At that price, all "leftover" biscuits are sold (there is always enough demand at that price). The restaurant has estimated that if a customer demands a biscuit before 3:00pm but the restaurant has run out, besides losing the sale, this costs the restaurant $2.00 in loss of future sales. Moreover, demand for biscuits before 3:00pm varies, with the probability of each demand scenario given in the following table. Demand before 3:00pm (biscuits) 40 41 42 43 Probability (%) 10 15 25 25 15 10 44 45 The table suggests that for 10% of days the demand before 3:00pm is 40 biscuits, for 15% of days the demand before 3:00pm is 41 biscuits, etc. The restaurant currently orders 42 biscuits every day. The goal of the problem is to provide the restaurant with a recommendation about the number of biscuits to order on each day. (a) What is the expected number of biscuits demanded before 3:00pm on a given day? What is the expected number of sales before 3:00pm given the current order size? How about the expected number of lost sales (units of demand that cannot be fulfilled due to lack of inventory) before 3:00pm? What is the expected number of biscuits left-over that need to be sold at a deep discount? (b) With its current order quantity of 42, what is the probability the restaurant fulfills all pre-3:00pm demand? How about the probability that a randomly chosen customer arriving before 3:00pm has their demand fulfilled? (c) Suppose that on a particular day 41 biscuits are demanded before 3:00pm. How much profit did the restaurant lose on that day by ordering the 42th biscuit? (This is the per-unit regret of over- ordering.) Now suppose that on a different day 43 biscuits are demanded before 3:00pm. How much profit did the restaurant lose on that day by not ordering a 43th biscuit? (This is the per-unit regret of under-ordering.) Is the restaurant ordering too many or too few biscuits? Why? (d) How many biscuits would you recommend the restaurant orders? (e) Suppose that, instead of the probability table given above, you were given the Excel spreadsheet BiscuitData.xls., which contains data on total demand for biscuits prior to 3:00pm over 40 days. Based on that data, how many biscuits would you recommend the restaurant orders? [Hint: Can you transform the data into a table like the one above?]