In $2017$ Oppo released BDP$-103,$ a high$-$quality Blu$-$ray Disc Player. Oppo sold the product through Best Buy. The selling price of BDP$-103$ on Best Buy was $$500,$ and it costed Oppo $$300$ to produce one unit of BDP$-103.$ The per unit wholesale price Oppo charged Best Buy was $$400.$ Customer demand D for BDP$-103$ was uniformly distributed as D $$ Unif$[500,1000]($here$,$ for simplicity, we assume D can take non$-$integer values$).$ Best Buy must order BDP$-103$ before the start of the selling season. Because of the lag time, production and logistics constraints, the product is not replenishable during selling season. For each unsold product, Best Buy can salvage $$100.$

$($a$)$ What is the optimal ordering quantity of Best Buy? Under this optimal quantity, what is the expected profit of Best Buy? What is the expected profit of Oppo?

$($b$)$ Suppose Best Buy selects the ordering quantity to maximize the total supply chain profit $($i$.$e$.,$ sum of the profits of Best Buy and Oppo$).$ What would be the optimal ordering quantity then?

$($c$)$ Going back to the setting of $($a$),$ suppose Oppo guarantees to buy back each unsold unit from Best Buy for x dollars. For what value of x is the quantity ordered by Best Buy $($to maximize its own profit$)$ equal to that found in $($b$)?$

$($d$)$ Under the setting of $($c$),$ what is the profit of Best Buy and Oppo? Is Oppo better$-$off making such an offer?