Question $4(15$ points$)$: Newsvendor Model

A retail outlet sells a perishable product for $$13$ per unit. The cost of the product is $$6$ per unit. The product not sold has the salvage value of $$2.$

a$)$ Compute the overage cost and underage cost per unit.

b$)$ If the demand follows the normal distribution with an average of $350,$ and the standard deviation of $110.$ What is the optimal order quantity to maximize the expected profit?

c$)$ What is the optimal ordering quantity if the retailer uses the following discrete distribution for demand?

$\backslash$table$[[$quantity$,100,150,200,250,300,350,400,450,500],[$probability$,0\mathrm{.}05,0\mathrm{.}05,0\mathrm{.}15,0\mathrm{.}15,0\mathrm{.}20,0\mathrm{.}15,0\mathrm{.}15,0\mathrm{.}05,0\mathrm{.}05]]$