Consider a simple supply chain system for artificial Christmas trees with one manufacturer and one retailer. The retailer is selling the trees over the holiday season and is facing a random demand. The retailer must decide one ordering quantity of the trees long before the holiday season, and we assume that no further replenishment is available during the holiday season. Assume that the total demand over the holiday season for this retailer is forecasted to be 800, or 1000, or 1200, with equal probability. For the manufacturer, to produce the trees, there exists a fixed machine setup cost of $8000, and a per-unit production cost of $40. The unit wholesale price between the manufacturer and the retailer is $200, and the retailing price to the end customers is $300 per unit. Every piece of an unsold tree at the end of the holiday season has a salvage value of $30 at the secondary market. (Assuming all the decision makers in this question are risk-neutral.)

1) Suppose the manufacturer and the retailer act based on their own interest, then what is the optimal order quantity of the retailer? What is the expected profit of the retailer and the manufacturer? Denote them as Rret and Rman, respectively.

2) Supposed both the manufacturer and the retailer is run by the same cooperation, and the decision-maker wants to maximize the expected total profit of the manufacturer and the retailer, then what is the optimal production/order quantity? What is the expected total profit of the retailer and the manufacturer?

3) Now supposed the manufacturer and the retailer still act based on their own interest. Can you introduced a buy-back contract between these two companies, so that under this contract, the expected total profit of the retailer and the manufacturer and both exceeding the expected total profit you computer in part 1 (Rret and Rman)? How does the total expected profit under this case compare with part 2?