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Suppose that a grocery store buys milk for $2.20 and sells it for $3.70. If the milk gets old then the grocery store can sell their unsold milk back to their wholesaler for $2.70 (so the grocery store loses $1.00 on each gallon that it has to sell back to the wholesaler). Suppose that the demand for milk is normally distributed with a mean of 2900 gallons per week and a standard deviation of 250 gallons per week. The grocery store needs to decide how much milk to order. They decide that they want to order an amount of milk such that their expected profit on the last gallon sold is 0. Therefore, if p denotes the probability of selling a gallon of milk (notice that the probability of selling a gallon of milk is related to the number of gallons of milk because the demand for milk is normally distributed) then the expected profit is p (1.50) + (1-p) (-1.00). If we set this equal to 0 and solve for p then we get p = .40. Therefore, if the grocery store’s goal is to maximize its expected profit then they should order an amount of milk such that the probability of selling that amount of milk (or more) is 40%. How many gallons of milk should the store order?
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