Consider the following excerpts from a Wall Street Journal article on Burger King (Beatty, 1996):
Burger King intends to bring smiles to the faces of millions of parents and children this holiday season with its "Toy Story" promotion. But it has some of them up in arms because local restaurants are running out of the popular toys . . . Every Kids Meal sold every day of the year comes with a giveaway, a program that has been in place for about six years and has helped Grand Metropolitan PLC's Burger King increase its market share. Nearly all of Burger King's 7,000 U.S. stores are participating in the "Toy Story" promotion . . . Nevertheless, meeting consumer demand still remains a conundrum for the giants. That is partly because individual Burger King restaurant owners make their tricky forecasts six months before such promotions begin. "It's asking you to pull out a crystal ball and predict exactly what consumer demand is going to be," says Richard Taylor, Burger King's director of youth and family marketing. "This is simply a case of consumer demand outstripping supply." The long lead times are necessary because the toys are produced overseas to take advantage of lower costs . . . Burger King managers in Houston and Atlanta say the freebies are running out there, too . . . But Burger King, which ordered nearly 50 million of the small plastic dolls, is "nowhere near running out of toys on a national level."
Let's consider a simplified analysis of Burger King's situation. Consider a region with 200 restaurants served by a single distribution center. At the time the order must be placed with the factories in Asia, demand (units of toys) for the promotion at each restaurant is forecasted to be gamma distributed with mean 2,251 and standard deviation 1,600. A discrete version of that gamma distribution is provided in the following table, along with a graph of the density function:
|
Q
|
F(Q)
|
L(Q)
|
|
0
|
0.0000
|
2,251.3
|
|
500
|
0.1312
|
1,751.3
|
|
1,000
|
0.3101
|
1,316.9
|
|
1,500
|
0.4728
|
972.0
|
|
2,000
|
0.6062
|
708.4
|
|
2,500
|
0.7104
|
511.5
|
|
3,000
|
0.7893
|
366.6
|
|
3,500
|
0.8480
|
261.3
|
|
4,000
|
0.8911
|
185.3
|
|
4,500
|
0.9224
|
130.9
|
|
5,000
|
0.9449
|
92.1
|
|
5,500
|
0.9611
|
64.5
|
|
6,000
|
0.9726
|
45.1
|
|
6,500
|
0.9807
|
31.4
|
|
7,000
|
0.9865
|
21.7
|
|
7,500
|
0.9906
|
15.0
|
|
8,000
|
0.9934
|
10.2
|
|
8,500
|
0.9954
|
6.9
|
|
9,000
|
0.9968
|
4.6
|
|
9,500
|
0.9978
|
3.0
|
|
10,000
|
0.9985
|
1.9
|
|
10,500
|
0.9989
|
1.2
|
|
11,000
|
0.9993
|
0.6
|
|
11,500
|
0.9995
|
0.3
|
|
12,000
|
1.0000
|
0.0
|
Suppose, six months in advance of the promotion, Burger King must make a single order for each restaurant. Furthermore, Burger King wants to have an in-stock probability of at least 85 percent.
a. Given those requirements, how many toys must each restaurant order?
b. How many toys should Burger King expect to have at the end of the promotion? Now suppose Burger King makes a single order for all 200 restaurants. The order will be delivered to the distribution center and each restaurant will receive deliveries from that stockpile as needed. If demands were independent across all restaurants, total demand would be 200 × 2,251 = 450,200 with a standard deviation of √(200) x 1,600 = 22,627. But it is unlikely that demands will be independent across restaurants. In other words, it is likely that there is positive correlation. Nevertheless, based on historical data, Burger King estimates the coefficient of variation for the total will be half of what it is for individual stores. As a result, a normal distribution will work for the total demand forecast.
c. How many toys must Burger King order for the distribution center to have an 85 percent in-stock probability?
d. If the quantity in part c is ordered, then how many units should Burger King expect to have at the end of the promotion?
e. If Burger King ordered the quantity evaluated in part a (i.e., the amount such that each restaurant would have its own inventory and generate an 85 percent in-stock probability) but kept that entire quantity at the distribution center and delivered to each restaurant only as needed, then what would the DC's in-stock probability be?