Q13.1 * ( Teddy Bower ) Teddy Bower sources a parka from an Asian supplier for $
ID: 465220 • Letter: Q
Question
Q13.1 * ( Teddy Bower ) Teddy Bower sources a parka from an Asian supplier for $10 each and sells them to customers for $22 each. Leftover parkas at the end of the season have no salvage value. (Recall Q12.6.) The demand forecast is normally distributed with mean 2,100 and standard deviation 1,200. Now suppose Teddy Bower found a reliable vendor in the UnitedStates that can produce parkas very quickly but at a higher price than Teddy Bower’s Asian supplier. Hence, in addition to parkas from Asia, Teddy Bower can buy an unlimited quantity of additional parkas from this American vendor at $15 each after demand is known.
a. Suppose Teddy Bower orders 1,500 parkas from the Asian supplier. What is the probability that Teddy Bower will order from the American supplier once demand is known?
b. Again assume that Teddy Bower orders 1,500 parkas from the Asian supplier. What is the American supplier’s expected demand; that is, how many parkas should the American supplier expect that Teddy Bower will order?
c. Given the opportunity to order from the American supplier at $15 per parka, what order quantity from its Asian supplier now maximizes Teddy Bower’s expected profit?
d. Given the order quantity evaluated in part c, what is Teddy Bower’s expected profit?
e. If Teddy Bower didn’t order any parkas from the Asian supplier, then what would Teddy Bower’s expected profit be?
Q12.6 (Teddy Bower Parkas ) Teddy Bower is an outdoor clothing and accessories chain that purchases a line of parkas at $10 each from its Asian supplier, TeddySports. Unfortunately, at the time of order placement, demand is still uncertain. Teddy Bower forecasts that its demand is normally distributed with mean of 2,100 and standard deviation of 1,200. Teddy Bower sells these parkas at $22 each. Unsold parkas have little salvage value; Teddy Bower simply gives them away to a charity.
a. What is the probability this parka turns out to be a “dog,” defined as a product that sells less than half of the forecast?
b. How many parkas should Teddy Bower buy from TeddySports to maximize expected profit?
c. If Teddy Bower wishes to ensure a 98.5 percent in-stock probability, how many parkas should it order?
For parts d and e, assume Teddy Bower orders 3,000 parkas.
d. Evaluate Teddy Bower’s expected profit.
e. Evaluate Teddy Bower’s stockout probability
Cachon, Gerard; Terwiesch, Christian. Matching Supply with Demand: An Introduction to Operations Management (Page 267). McGraw-Hill Education. Kindle Edition
Explanation / Answer
Q12.6
Mean Demand, µ = 2,100
Standard deviation, = 1,200
Price value, P = $22
Cost value, C = $10
Salvage value, S = $0
a) What is the probability this parka turns out to be a "dog", defined as a product that sells less than half of the forecast?
The parka sells less than half of the forecast, then Q = 2100/2 = 1050 or fewer units.
The probability that the parka turns out to be a “dog” = P(z <= Q)
The Z-score for the Q = 1050 is;
z = (Q – µ)/ = (1050 – 2100)/1200 = 0.875
From Standard Normal Distribution Table: P(z = 0.875) = 0.1907
P = 0.1907 implies there is a 19.07% probability that the parka will be a dog.
b) How many parkas should Teddy Bower buy from TeddySports to maximize expected profit?
Calculate Underage and Overage cost:
Cu = P – C = 22 – 10 = $12
C0 = C – S = 10 – 0 = $10
Critical Ratio = Cu /( Co + Cu) = 12/(10+12) = 0.5455
Probability of critical ratio is 0.5455.
z-score for the p = 0.5455 = 0.1142
The order quantity for z = 0.1142 is given as follows:
Q = µ + z = 2100 + 0.1142 x 1200 = 2237 units
Teddy Bower should buy 2220 parkes from TeddySports to maximize expected profit.
c) If Teddy Bower wishes to ensure a 98.5% in-stock probability, how many parkas should Teddy Bower's order?
To hit the target in-stock probability of 98.5%, we need to find the z-statistic such that z = 0.9850. We see from the Standard Normal Distribution Function,
z for p = 0.9850 = 2.17
The order quantity for z = 2.7 is given as follows:
Q = µ + z = 2100 + 2.7 x 1200 = 4704 units
Teddy Bower's should order4704 parkas to ensure a 98.5% in-stock probability.
d) Evaluate Teddy Bower's expected profit.
Q = 3000
Z-score is obtained as follows:
z = (Q – µ)/ = (3000 – 2100)/1200 = 0.75
Now look up expected lost sales with the Standard Normal distribution in the Standard Normal Loss Function Table:
L(0.75) = 0.1312 .
Convert that lost sales into the expected lost sales with the L(z ) = L(z) x = 1200 x 0.1312 = 157.4
Expected actual demand distribution:
Sales = expected demand expected lost sales = 2100 157.4 = 1942.6.
Expected left over inventory = Order quantity – Expected Sale = 3000 1942.6 = 1057.4.
Finally,
Expected profit = Cu x expected sales Co x expected expected left over
Expected profit = 12 x 1942.6 10 x 1057.4 = $12, 737
e) Evaluate Teddy Bower's stock-out probability
In-stock Probability = P(z = 0.75) = 0.7733
Out-stock Probability = 1 – In-stock Probability = 1 – 0.7733 = 0.2267
13.1
a. Suppose Teddy Bower orders 1,500 parkas from the Asian supplier. What is the probability that Teddy Bower will order from the American supplier once demand is known?
Determine P(demand > 1500) = 1 – P(demand <=Q) = 1 – F(Q)
Z score for Q = 1500 is
z = (Q – µ)/ = (1500 – 2100)/1200 = 0.5
Probability for z score of -0.5 = 0.3085
1 – F(Q) = 1 – 0.3085 = 0.69
P(demand > 1500) = 69%
probability that Teddy Bower will order from the American supplier once demand is known is 69%.
b. Again assume that Teddy Bower orders 1,500 parkas from the Asian supplier. What is the American supplier’s expected demand; that is, how many parkas should the American supplier expect that Teddy Bower will order?
Expected Shortage with Q = 1500
Z = -0.5
Probability of expected loss
L(-0.5) = 0.6978
Expected demand = x L(-0.5) = 1200 x 0.6978 = 837.4
Americal supplier expect that Teddy Bower will order 837.4 parkas
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