$(30$ points$).$ Michelle's Shop is well known for stocking famous foods. One item it sells is a popular olive oil. The olive oil costs Michelle $$10$ a bottle and requires a $3-$month lead time for replenishment of stock. Michelle uses a $25\%$ annual interest rate to compute holding costs. It estimates that if a customer requests the olive oil when she is out of stock, the loss$-$of$-$goodwill cost is $$20$ a bottle. Bookkeeping expenses for placing an order amount to about $$60.$ During the $3-$month replenishment leadtime, Michelle estimates that she sells an average of $120$ bottles, but there is substantial variation from one $3-$month period to the next. She estimates that the standard deviation of demand during each $3-$month period is $25.$ Assume that the demand is described by a normal distribution. How should Michelle control the replenishment of the olive oil?

$($a$)(15$ pts$)$ Compute $Q^{(0)}$ and $R^{(0)}($for the initial iteration$).$

$($b$)(15$ pts$)$ Find $Q^{(1)}$ and $R^{(1)}($for iteration $1).$