Tom owns a small firm that manufactures “Tom Sunglasses.” He has the opportunity
ID: 366794 • Letter: T
Question
Tom owns a small firm that manufactures “Tom Sunglasses.” He has the opportunity to sell a particular seasonal model to Land’s End. Tom offers Land’s End two purchasing options: Option 1. Tom offers a price of $55 for each unit, but returns are no longer accepted. In this case, Land’s End throws out unsold units at the end of the season. Option 2. Tom offers to set his price at $65 and agrees to credit Land’s End $53 for each unit Land’s End returns to Tom at the end of the season. This season’s demand for this model will be normally distributed with mean of 200 and standard deviation of 125. Land’s End will sell those sunglasses for $110 each.
(1) How much would Land’s End buy if they choose Option 1?
(2) How much would Land’s End buy if they choose Option 2? What is the probability that Land’s End will return sunglasses to Tom at the end of the season?
Explanation / Answer
Mean =200
Std deviation = 125
OPTION 1
Price p= $110
Cost c =$55
Salvage value v=0
cost of ordering one fewer unit than required, Cu = Price – Cost
Underage cost Cu= $110 - $55= $55
cost of ordering one more unit than required, Co = Cost – Salvage value
Overage cost Co= $55- $0 = $55
to minimize the expected total cost choose Q such that we do not have lost sales (i.e. demand is Q or lower) with a probability that equals to the critical ratio
F(Q) = Cu/(Co + Cu) which is the critical ratio
=55/(55+55)
=0.5
Look up critical ratio in the Standard Normal Distribution Function Table
And we get z=0
Convert the z-statistic into an order quantity:
Q= mean + Z*std deviation
=200+ 0*125 = 200
OPTION 2
Price p= $110
Cost c =$65
Salvage value v=$53
Underage cost Cu= $110 - $65= $45
Overage cost Co= $65- $53 = $12
Critical ratio = Cu/(Co + Cu)
=45/(12+45)
=0.7894
Look up critical ratio in the Standard Normal Distribution Function Table
And we get z=0.81
Convert the z-statistic into an order quantity:
Q= mean + Z*std deviation
=200+ 0.81*125 = 301.25 =302(approx)
Expected Lost sales = std deviation * L(z) = 125* 0.1181 = 14.7625
Expected Sales = mean – Expected Lost Sales = 200 – 14.76 = 185.24
Expected Left Over Inventory = Q – Expected Sales = 302 – 185 = 117 units
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