Consider the following demand scenario: QuantityProbability$4,00010\%4,40012\%4,80018\%5,00017\%5,60013\%6,00011\%6,4008\%6,8006\%7,2005\%$ The variable production cost is $$45/$unit and the fixed cost is $$125,000.$ The product is sold to end customers for $$145/$unit during the season and any unsold units are sold for $$20/$unit after the season. In a buy back scenario, the manufacturer will buy back units at $$55/$unit$.$ Also, in a payback scenario, the retailer will pay $$15$ for each unit it does not purchase. In both the payback and buy back scenarios, the manufacturer sells the product for $$75\mathrm{.}00.$ a$)$ What is the system optimal production quantity and expected profit under global optimization? b$)$ What is the system optimal production quantity and expected profit under a payback scenario? What is the profit for the retailer and for the manufacturer? c$)$ What is the system optimal production quantity and expected profit under a buyback scenario? What is the profit for the retailer and for the manufacturer? d$)$ Provide the results of your analyses supporting your answers to parts a$)$ through c$).$