Weis's paint store uses a $(Q,R)$ inventory system to control its stock levels. For a particularly popular white latex paint, historical data show that the distribution of monthly demand is approximately normal, with mean $28$ and standard deviation $8.$ Replenishment lead time for this paint is about $14$ weeks. Each can of paint costs the store $$6.$ Although excess demands are back ordered, the store owner estimates that unfilled demands cost about $$10$ each in bookkeeping and loss$-$of$-$goodwill costs. Fixed costs of replenishment are $$15$ per order, and holding costs are based on a $30$ percent annual rate of interest.

a$.$ What are the optimal lot sizes and reorder points for this brand of paint?

b$.$ What is the optimal safety stock for this paint?

After taking a production seminar, Al Weiss, the owner of Weiss's paint store, decides that his stock$-$out cost of $$10$ may not be very accurate and switches to a service level model. He decides to set his lot size by the EOQ formula and determines his reorder point so that there is no stock$-$out in $90$ percent of the order cycles.

a$.$ Find the resulting $(Q,R)$ values.

b$.$ Suppose that, unfortunately, he really wanted to satisfy $90$ percent of his demands $($that is$,$ achieve a $90$ percent fill rate$).$ What fill rate did he actually achieve from the policy determined in part $($a$)?$

Answer problem $14$ & $15!$