2. The following excerpt from a Wall Street Journal article on Burger King serve
ID: 369854 • Letter: 2
Question
2. The following excerpt from a Wall Street Journal article on Burger King serves as a motivation to the problem:
Burger King intends to bring smiles to the faces 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’ Burger King increase its market share. Nearly, all of the Burger King’s 7,000 U.S. stores are participating in the “toy Story” promotion… Nevertheless, meeting customer 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 customer demand is going to be,” says Richard Taylor, Burger King’s director of youth and family marketing. “This is simply a case of customer 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 version of Burger King’s situation:
Burger Prince (BP), a fast food franchiser, has a total of 120 franchised restaurants in the United Stores served by a single distribution center. In order to bring smiles to the faces of parents and children clients this holiday season, BP is offering a Toy giveaway with every Kids Meal. All 120 restaurants participate in this promotional program. At the time the order must be placed with the factories in Asia, demand (in units of toys) for each restaurant is forecasted by a Normal distribution (assume that demands are independent at each location). Currently, BP characterizes the restaurant in terms of their sales in 3 categories: Low, Medium and High. The number of restaurants in each category and the Average and Standard Deviation of demand for toys during this promotional season are:
Category
Number of Stores
Average Demand per Store (Toy)
Standard Deviation of Demand per Store (Toy)
Low
30
250
75
Medium
40
1000
200
High
50
2000
600
Suppose 6 months in advance of the promotion, BP must make an order for each single restaurant. Furthermore, BP desires to have an 85% probability of a restaurant being in stock at the end of the promotion season.
(5 Points)
a)Find the order quantity for each store in the each category.
(5 Points)
b)What is the implied unit cost of underage if the unit cost of a toy is $0.5?
(5 Points)
c)Suppose BP decides to make a single order for all 120 restaurants. The order will be delivered and stored at the distribution center and then is dispatched to each restaurant during the holiday season as needed. Find the order quantity in this case if an 85% probability of no stock out at the warehouse is desired. What is the average number of leftover (unsold) toys in this case?
(5 Points)
d)If BP ordered the total quantity ordered in (a) but kept the entire order at the distribution center and delivered to each restaurant only as needed, then what is the probability of no stock out at the distribution center at the end of the season? What is the average number of leftover (unsold) toys in this case? What can you comment on the policies found in (c) and (d) regarding the tradeoff between service level and average leftover units and cost of leftovers?
Category
Number of Stores
Average Demand per Store (Toy)
Standard Deviation of Demand per Store (Toy)
Low
30
250
75
Medium
40
1000
200
High
50
2000
600
Explanation / Answer
a) z value for 85% probability = NORMSINV(0.85) = 1.0364
Order qantity for each store in Small category = + z =250 + 1.0364*75 = 328
Order qantity for each store in Large category = + z =1000 + 1.0364*200 = 1207
Order qantity for each store in Largest category = + z =2000 + 1.0364*600 = 2622
b) Critical value = Cu/(Cu+Co) = 0.85
Overage cost, Co = 0.5 (cost of a gift)
Solving the critical value equation for Cu
Cu*(1-0.85) = 0.85*Co
Cu = 0.85*0.5/(1-0.85) = 2.83
Implied cost of underage = $ 2.83
c) Total average demand of 120 stores = 30*250+40*1000+50*2000 = 147,500 units
Standard deviation of demand of 120 stores = (30*752 + 40*2002 + 50*6002) = 4446 units
Optimal order quantity for 85% in-stock probability = 147500 + 1.0364*4446 (1.04 is the z value as determined earlier)
Optimal order quantity = 152,108 units
d) Total quantity ordered in part (a), Q = 328*30+1207*40+2622*50 = 189,220 units
z value = (189200 - 147500)/4446 = 9.38
Corresponding to z value as calculated above, in-stock probability (service level) - NORMSDIST(9.38) = 1 or 100%
Therefore, probability of no stockout at the distribution center at the end of the period = 0% (approx)
From standard normal table, corresonding to z=1, Lost sales, L(z) = 0
Expected lost sales, L = *L(z) = 0*4446 = 20
Expected sales, S = µ - L = 147500 - 0 = 147500
Expected leftover (unsold) inventory = Q - S = 189220 - 147500 = 41,720
Cost of leftover inventory = 41720*0.5 = $ 20,860
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