Great Bows$($GB$)$ is a manufacturer of high$-$end bows and sells their bows through Downtown Sporting Goods$($DSG$)$ retail store. Their top$-$selling bow is called $$Sterling$.$ Demand for Sterling is assumed to follow a Normal distribution with a mean of $100$ and standard deviation of $35.$ DSG sells the Sterling for $$700,$ during the hunting season. Once the hunting season is over DSG sells the unsold Sterlings to a discount sporting goods store for $$300.$ GB$$s unit manufacturing cost is $$200.$ GB is considering charging DSG either $$400,$ or $$450,$ or $$500$ or $$550$per unit.

a$)$ What unit price should GB charge to maximize its expected profit and how many units would DSG order at this price? $($Please round up all order sizes$).$ Hint: Think how many units DSG will buy at each price.

b$)$ If GB charges the price you came up with in part $($a$)$ above and DSG adopts a service level of $0\mathrm{.}90$

i$)$ What is DSG$$s optimal order size?

ii$)$ What is DSG$$s implied Goodwill cost?