A particular product is purchased for $$3\mathrm{.}7$ each. Cost of ordering is $$155$ per order. Holding cost is given as $$1\mathrm{.}4$ per unit per year. Monthly demand for the product follows a normal distribution with mean $395$ and standard deviation $75.$ Order lead time is given as $4$ months. Assume that the cost of back ordering is given as $$14.$

a$)$ Determine the optimal values of order quantity and reorder point $($Q$,$R$).$

b$)$ Determine the average annual total cost of holding, setup and stockout associated with this product.

c$)$ Suppose that stockout cost is replaced with a Type I service objective of $95\%.$ Find optimal values of $($Q$,$R$)$ in this case.

d$)$ Suppose that stockout cost is replaced with a Type II service objective of $90\%.$ Find optimal values of $($Q$,$R$)$ in this case.