Question

A store will order q gallons of a liquid product to
meet demand during a particular time period.
This product can be dispensed to customers in
any amount desired, so demand during the period
is a continuous random variable X with cdf F(x).
There is a fixed cost c0for ordering the product
plus a cost of c1per gallon purchased. The per-
gallon sale price of the product is d. Liquid left
unsold at the end of the time period has a salvage
value of e per gallon. Finally, if demand exceeds
q, there will be a shortage cost for loss of good-
will and future business; this cost is f per gallon
of unfulfilled demand. Show that the value of q
that maximizes expected profit, denoted by q*,
satisfies P(satisfying demand)=F(q*)=(d-c1+f)/(d-e+f)

Answer 1