Problem 2: Movie Merchandise

Walt Disney studios will release a new kids' movie on the July 4^th weekend. They are planning to sell an action figure based on the movie's protagonist. All sales will occur during a three-week period around the movie's release date. However, to have enough time to produce and distribute the action figures, they need to place an order to the supplier by the end of this month.

Walt Disney estimates that the total demand for the action figure over the three-week selling period is Normally distributed with a mean of 1 million units and a standard deviation of 0.4 million units. Each action figure will cost them $20 in production and transportation, and will be sold to customers at a price of $60. Any leftover inventory at the end of the selling period will be sold off at a clearance price of $10.

1. What is the underage cost of an action figure in dollars? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

2. What is the overage cost of an action figure in dollars? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

3. What is the optimal number of action figures (in millions) that Walt Disney should order to the supplier to maximize profit?

NOTE: Please enter your answer in millions of units with three decimal places. For example, if your answer is 7.5 million units, please enter 7.500.

4. Suppose your calculated underage cost in Question 8 is u, overage cost in Question 9 is o, and the optimal order quantity in Question 10 is Q*. If Walt Disney orders Q* units, what is the probability that there will NOT be a shortage of action figures by the end of the selling period?