$2.[40$ marks$]$ Suppose that a cellphone manufacturer sells its cellphone to supply the U$.$S$.$ market, which is served from a warehouse located in St$.$ Louis. Daily demand at the St$.$ Louis warehouse is normally distributed, with a mean of $1,000$ and a standard deviation of $500.$ The ordering lead time is $4$ days and shipping costs $$1$ per unit. Each cellphone costs $$100,$ and the warehouse incurs an annual holding cost rate of $10\%.$

$($a$)$ The warehouse follows a continuous review policy to manage inventory and aims for a cycle service level $($CSL$)$ of $98\%.$ Given the minimum lot sizes, the warehouse would order $7,000$ cellphones each time $($on average, once every week$).$ What is the safety stock that satisfies the required CSL$?$ What is the expected annual safety stock holding cost and annual cycle stock holding cost?

$($b$)$ Suppose that the warehouse changes its replenishment policy from a continuous review to a periodic review, and the review period is $5$ days. What is the order$-$up$-$to level that achieves a CSL of $98\%?$