Consider the following demand scenario:

Quantity 20 21 22 23 24 25 26

Probability 5% 10% 15% 30% 17% 13% 10%

The manufacturer incurs $29/unit towards the cost of manufacturing a product. The distributor sells the product to end-users (customers) for $69/unit. The salvage value of unsold items at the end of the selling season is $10/unit. In case of product shortage, each unmet demand receives a coupon for $5. Both the manufacturer and the distributor give these coupons, but whenever the shortage for a scenario is 4 or less. (E.g., if a scenario has a shortage of 5 (demand 25, stock 20), no coupon is given)

1. What are the system's optimal production quantity and expected profit under the global optimizer?

2. Assume that both the manufacturer and the distributors are deciding in favor of their individual interests. Suppose the manufacturer sells to the distributor at $54/unit; how much will the distributor order? What is the expected profit for the manufacturer and the distributor?

3. How much additional profit do both the manufacturer and the distributor earn if they collaborate instead of focusing on their individual profits?