Question $(3)$: $(27$ points$)$

Annual demand for number $2$ pencils at the campus store is normally distributed with mean $1000$ and standard deviation $250.$ The store purchases the pencils for $6$ cents each and sells them for $20$ cents each. There is a two months lead time and the store accountant estimates that the cost of performing the necessary paperwork to initiate and receive an order is $$20.$ He also recommends a $22$ percent annual interest rate for determining the holding cost. The cost of a stockout is the cost of lost profit plus an additional $20$ cents per pencil, where the latter represents the cost of loss of customer goodwill.

a$)$ Find the simultaneous optimal values of $Q$ and $R.$

b$)$ What is the safety stock for this item at the optimal solution?

c$)$ What is the fill rate associated with the optimal solution?