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 c₀ for ordering the product plus a cost of c₁ per 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 goodwill and future business; this cost is ƒ 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 ƒ = $25. 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.

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P(satisfying demand) = F(q*) = d-c₁+f d-e+f