AQ$3.$ Inventory Management $(12$ marks$)$ A supermarket opens $365$ days a year. The daily demand for its store brand of facial tissue averages $80$ boxes. The daily demand is normally distributed with a standard deviation of $6$ boxes. It takes $4$ days for the ordered facial tissue to arrive at the store.

$1.$ If the ordering cost is $$36$:$5$ and inventory carrying cost is $$1$ per box per year, how many boxes of facial tissue should the supermarket manager order each time? $2.$ To ensure the stockout risk to be within $2\%,$ at what level of inventory should the manager places an order?

$3.$ Suppose the lead time becomes random and is normally distributed with a mean of $4$ days and a standard deviation of $0$:$4$ days. If the manager continues to use the reorder point as that obtained in part $2$ question, what is the stockout risk now?