A company advertises a $95$ per cent cycle service level for all stocked items. Based on historical data the company estimates an item has a Normally distributed demand

with mean $200$ units a week and standard deviation of $20$ units. The company works with a reliable supplier that guarantees a lead time of four weeks. The reorder and

handling cost is $$80,$ holding cost is $$5$ per unit per week.

$($a$)$ What reorder level should they use?

$($b$)$ If the lead time is unstable and follows a normal distribution with mean of four weeks and variance of one week, what is the ordering policy?

$($c$)$ If shortage cost is considered and it is $$250$ per unit per week, illustrate how to find the optimal reorder level and order quantity.