155. A store will order q gallons of a certain 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 c0 for ordering the product plus a cost of c1 per gallon purchased. The pergallon 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 goodwill 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 Then determine the value of F(q*) if d$35, c0

$25, c1  $15, e  $5, and f  $25. Hint: Let x denote a particular value of X. Develop an expression for profit when x q and another expression for profit when x q. Now write an integral expression for expected profit (as a function of q) and differentiate.