### How many elvis wigs should be ordered

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Pony Express Creations Inc. (www.pony-ex.com) is a manufacturer of party hats, primarily for the Halloween season. (80 percent of their yearly sales occur over a six-week period.) One of their popular products is the Elvis wig, complete with sideburns and metallic glasses. The Elvis wig is produced in China, so Pony Express must make a single order well in advance of the upcoming season. Ryan, the owner of Pony Express, expects demand to be 25,000 and the following is his entire demand forecast:

 Q Prob (D = Q) F(Q) L(Q) 5,000 0.0183 0.0183 20,000 10,000 0.0733 0.0916 15,092 15,000 0.1465 0.2381 10,550 20,000 0.1954 0.4335 6,740 25,000 0.1954 0.6289 3,908 30,000 0.1563 0.7852 2,052 35,000 0.1042 0.8894 978 40,000 0.0595 0.9489 425 45,000 0.0298 0.9787 170 50,000 0.0132 0.9919 63 55,000 0.0053 0.9972 22 60,000 0.0019 0.9991 8 65,000 0.0006 0.9997 4 70,000 0.0002 0.9999 2 75,000 0.0001 1.0000 2

The Elvis wig retails for \$25, but Pony Express's wholesale price is \$ 1 2. Their production cost is \$6. Leftover inventory can be sold to discounters for \$2.50.

a. Suppose Pony Express orders 40,000 Elvis wigs. What is the chance they have to liquidate 1 0,000 or more wigs with a discounter?

b. What order quantity maximizes Pony Express's expected profit?

c. If Pony Express wants to have a 90 percent fill rate, then how many Elvis wigs should be ordered?

d. If Pony Express orders the quantity chosen in part c, what is Pony Express's actual fill rate?

e. If Pony Express wants to have a 90 percent in-stock probability, then how many Elvis wigs should be ordered?

f. If Pony Express orders 50,000 units, then how many wigs can they expect to have to liquidate with discounters?

g. If Pony Express insists on a 1 00 percent in-stock probability for its customers, then what is its expected profit?

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