$[$do not use chatgpt$]$Annual demand for number $2$ pencils at the campus store is normally distributed with mean $1,500$ and standard deviation $235.$ The store purchases the pencils for $6$ cents each and sells them for $20$ cents each. There is a two$-$month lead time from the initiation to the receipt of an order. The store accountant estimates that the cost in employee time for performing the necessary paperwork to initiate and receive an order is $$20$ and recommends a $25$ percent annual interest rate for determining holding cost. The cost of a stock$-$out is the cost of lost profit plus an additional $20$ cents per pencil, which represents the cost of loss of goodwill.

a$.$ Find the optimal value of the reorder point $R$ assuming that the lot size used is the EOQ.

b$.$ Find the simultaneous optimal values of $Q$ and $R.$

c$.$ Compare the average annual holding, setup, and stock$-$out costs of the policies determined in parts $($a$)$ and $($b$).$

d$.$ What is the safety stock for this item at the optimal solution?