C$.$ Problem Solving.

$(20$ points$)$ A store uses a $($Q$,$ R$)$ inventory system to control its stock levels. For a particular

part, historical data show that the distribution of monthly demand is approximately normal, with

mean $28$ and standard deviation $8.$ Replenishment lead time for this part is about $4$ weeks. The

optimal EOQ for this part has been determined to be equal to $75.$ The owner of the store would

like to maintain that there is no stock$-$out in $95\%$ of the order cycles.

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

$($b$)$ Suppose that he wants to achieve a $90\%$ fill rate. What fill rate did he actually achieve from

the policy determined in part $($a$)?$