2. (16 pts) All Boots is a retailer of boots. It sources a kind of waterproof hu

Question

2. (16 pts) All Boots is a retailer of boots. It sources a kind of waterproof hunting boots from an Asian supplier for $50 each and sells them to customers for $128 each. Leftover boots at the end of season will be sold to an outlet mall at $40 each. Given the S128 retail price, All Boots forecasts the demand distribution as follows: 100 200 300 400 500 600 700 800 900 1000 Pr(D 0.05 0.09 0.12 0.18 0.22 0.13 0.10 0.06 0.03 0.02 F(D 0.05 0.14 0.26 0.44 0.66 0.79 0.89 0.95 0.98 Now suppose All Boots found a reliable vendor in the United States that can produce boots very quickly but at a higher price than All Boots' Asian supplier. Hence, in addition to boots from Asia, All Boots can buy an unlimited quantity of additional boots from this American vendor at $78 each after demand is know.

Explanation / Answer

(a)

Probability of placing a second order = probability that the demand is more than 500 = 1 - 0.66 = 0.34 or 34%

(b)

Expected order quantity
= (600-500)*0.13 + (700-500)*0.10 + (800-500)*0.06 + (900-500)*0.03 + (1000-500)*0.02 = 73 unit

(c)

Cu = Cost of underage (i.e. cost of ordering one unit less than the demand) = $78 - $50 = $28
Co = Cost of overage (i.e. cost of ordering one unit more than the demand) = $50 - $40 = $10

Critical factor = Cu / (Co+Cu) = 28/(10+28) = 0.737

At optimality, the F(D) >= Critical factor and this happens for D=600. So, the optimal order quantity = 600.

(d)

Q=600 D Pr(D) Sold
(Asian) Sold
(American) Leftovers Profit = sold*78
(Asian) Profit= sold*50
(American) Loss
=leftover*(-10) Net profit 100 0.05 100 0 500 7800 0 -5000 2800 200 0.09 200 0 400 15600 0 -4000 11600 300 0.12 300 0 300 23400 0 -3000 20400 400 0.18 400 0 200 31200 0 -2000 29200 500 0.22 500 0 100 39000 0 -1000 38000 600 0.13 600 0 0 46800 0 0 46800 700 0.10 600 100 0 46800 5000 0 51800 800 0.06 600 200 0 46800 10000 0 56800 900 0.03 600 300 0 46800 15000 0 61800 1000 0.02 600 400 0 46800 20000 0 66800 Expected profit = sum of {Pr(D) x Net profit} 35110

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