$1.$ Consider an item whose inventory is controlled by a periodic review, base$-$stock policy.$$ Suppose the review period is $1$ week $(7$ days$),$ the replenishment lead$-$time is $4$ days, and the daily demand is normally distributed with mean $100$ and standard deviation $40.$ The cost of placing the order is $$200,$ and the unit cost of the product is $$50.$ The annual interest rate is $20\%$ in this industry.

a$)$ What is the basestock level $($order$-$up$-$to level$)$ S to assure that the coverage probability is $0\mathrm{.}95?$ What is the average inventory level? What is the annual total inventory$-$related cost? How many times of stock$-$out do you expect per year? $($Assume $52$ weeks per year$).$