Consider a two-echelon supply chain with a manufacturer and a distributor. The table shows the demand distribution.

Quantity	25	26	27	28	29	30	31
Probability	0.03	0.08	0.23	0.05	0.14	0.31	0.16

The manufacturer incurs $30/unit towards the cost of manufacturing. The distributor sells the product to end-users (customers) for $70/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. This coupon enables parties to retain customers; otherwise, customers may switch to other distributors. Manufacturers also give coupons of $5/unit whenever the manufacturer fails to fulfill the distributors orders.

1. What will be the system optimal production quantity and expected profit under global optimizer?

2. Assume that the partners are deciding in favor of their interests. Suppose the manufacturer sells to the distributor at $55/unit. How much will the distributor order? What is the expected profit for the manufacturer and distributor?

3. Compare the profits of partners (when silos and collaborate) using a diagram depicting profit lines. Explain briefly what the profit lines infer. How much additional profits, the partners can earn if they collaborate?

Please show all steps, formulas, calculations and explanations.