$(18$ points$)$

Consider the following demand scenario:

The variable production cost is $$\frac{45}{?}$ unit and the fixed cost is $$125,000.$ The product is sold to

end customers for $$\frac{145}{?}$ unit during the season and any unsold units are sold for $$\frac{20}{?}$ unit after the

season. In a buy back scenario, the manufacturer will buy back units at $$\frac{55}{u}$nit. 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$).$