(24 points total) Solve the question on paper, then take a photo and upload the document to SuCourse+. SUGGESTED TIME: 40 MINUTES Consider a supply chain composed of a manufacturer that produces a product and a retailer that sells this product to consumers. The manufacturer operates make-to-order and the retailer operates procure-to-stock. First, the retailer determines his order quantity from the manufacturer. Next, the manufacturer produces the retailer's order quantity and delivers it to the retailer. Finally, consumer demand is realized according to the following distribution Probability 0.15 0.12 0.13 0.14 0.14 0.14 0.78 Demand 900 1000 1100 1200 1350 1400 1500 The manufacturer's unit cost of production is $20. The fixed cost of production is zero. The retailer sells the product to end customers for $80. The salvage value for the product is $0. The firms operate under a wholesale price contract with a wholesale price of $60 per unit. (a) (6 points) Find the retailer's optimal order quantity. Then, calculate each firm's expected profits if the retailer orders this quantity. (b) (6 points) Find the system-optimal (i.e., supply-chain optimal) production quantity and calculate the resulting expected system-wide (i.e., supply chain) profit if the retailer orders this quantity. () (7 points) Assume now that the manufacturer will pay the retailer $30 for each unsold item (i.e., a buyback contract). Otherwise, the setting is the same as the original problem, including the wholesale price of $60. Find the retailer's optimal order quantity and calculate each firm's expected profit if the retailer uses this order quantity. (d) (5 points) Compare the total supply chain profits under parts a, b and c above. How do you explain the difference