Problem $4\mathrm{.}11($Managing Blood Inventory$)$ A hospital purchases blood from a local

blood$-$donation organization and uses it for patients during surgeries and emergency

procedures. The hospital pays $$175$ for each unit of blood purchased. Orders must be

placed first thing in the morning, and any blood not used by the end of the day must be

discarded. There is no salvage value or cost to discard a unit of blood. If the hospital

needs more blood on a given day than they purchased that morning, they must place

an emergency order; blood ordered this way costs $$420$ instead of $$175.$ The number

of units of blood that the hospital uses on a given day is normally distributed, with a

mean of $150$ and a standard deviation of $40.$

$($a$)$ Interpret this problem as a newsvendor problem. What are the holding and

stockout costs, h and p$?$

$($b$)$ What is the optimal number of units of blood for the hospital to purchase in the

morning? $($Fractional answers are OK$.)$

$($c$)$ On what fraction of days will the hospital need to order at least one emergency

unit of blood?