Xavier makes STEM learning project kits for children using electronic components. Each project kit costs $4.50 for Xavier and he sells the kits to a retailer at $10.00 per kit. The retailer sells each kit for $19.95. The monthly demand for the project kits is normally distributed with a mean of 400 kits and a standard deviation of 50. The retailer places a single order from Xavier at the beginning of the month. Any unsold project kits at the end of the month is sold at a discount price of $7 by the retailer. (Any project kits that did not sell at full price sell at this price).
a. If the retailer is acting independently, how many project kits should they order?
b. If the retailer is acting independently, what is their expected profit if they order an optimal quantity? What is the expected profit for Xavier?
c. Xavier offers an option to the retailer: Retailer would get refunded for any project kit that does not sell during the month. As before, retailer will discount the unsold project kits at $7 per kit and Xavier will refund $2 per project kit in addition. Under this plan, how many project kits will Retailer order?
d. Under the refund option, what is the expected profit of the retailer? What is the expected profit for Xavier?