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?