question 1)

A producer sells a certain type of LED lights to LPS, a local hardware store, at $12 each. The marginal production cost for the producer is $1 per item. LPS prices the lights at $24 and expects demand to be normally distributed with a mean of 20,000 and a standard deviation of 5,000. LPS places a single order with the producer. Currently, LPS discounts any unsold lights at the end of two months down to $3, and any lights that did not sell at full price sell at this price.

1. How many lights should LPS order?
2. What is LPS's expected profit?
3. How many lights does LPS expect to sell at a discount?
4. What is the expected under stock?
5. What is the producer's expected profit?
6. What is the supply chain's profit?

For question 1, a plan under discussion is that the producer buys back any number of unsold lights for $8 per item.

1. Under this plan how many items will LPS order?
2. What is the expected profit for LPS?
3. How many lights are expected to be unsold?
4. What is the expected under stock?
5. What is the expected profit for the producer?
6. What is the supply chain's profit?
7. What should the producer do?

Please show all of your work. Thank You.