A supermarket sells soy milk bottles at $2.00 each to the customers and buys them at $0.80 each with sixty-day expiration. When expired, they are returned to the supplier for $0.20 each (the supplier pays the supermarket). For the order arrived today, the store manager expects a demand of 1,000 bottles, normally distributed with standard deviation 200. (i) Suppose 1,200 bottles arrived today, what is the pro?t if the demand turns out to be exactly 1,000? (ii) Suppose only one bottle arrived today, what is the pro?t if it is sold and what is the pro?t if it remains unsold? (iii) What is the probability that these two events happen? What is the expected pro?t for ordering that single bottle? (iv) Using marginal analysis, show how to ?nd the optimal order quantity. (v) Would that change if the standard deviation increased? If so, how and why?