Problem 9: Consider the following demand scenario Consider the following demand

Question

Problem 9: Consider the following demand scenario Consider the following demand scenario uantit 2000 2100 2200 2300 2400 2500 2600 2700 Probability 8% 15% 30% 5% Suppose the manufacturer produces at a cost of $20/unit. The distributor sells to end customers for S50/unit during season and unsold units are sold for $10/unit after season. a) What is the system optimal production quantity and expected profit under global optimization? b) Suppose the manufacturer is make-to-order (i.e., the distributor must order before it receives demand from end customers) () Suppose the manufacturer sells to the distributor at $40/unit, how much should the distributor order? What is the expected profit for the manufacturer? What is the expected profit for distributor? (ii) Find an option contract such that both the manufacturer and distributor enjoy a higher expected profit than (b)(i). What is the expected profit for the manufacturer and the distributor? c) Suppose the manufacturer is make-to-stock. (i.e., the manufacturer must decide how much to produce before the distributor sees the demand and places an order.) (i) Using the same wholesale price contract as part (b)(), calculate the production level of the manufacturer. What are the expected profits for the manufacturer and for the distributor? Compare your results with part (b)(i) (ii) Find a cost sharing contract such that both the manufacturer and distributor enjoy a higher expected profit than that in (c)(i) and calculate their expected profits

Explanation / Answer

a) Under global optimization, Underage cost, Cu = selling price to end customer - manufacturing cost = 50 - 20 = 30

Overage cost, Co = Manufacturing cost - Salvage value = 20 - 10 = 10

Critical ratio = Cu/(Cu+Co) = 30/(30+10) = 0.75

Refer the cumulative probability distribution table as under

Lookup cumulative probability greater than critical ratio 0.75, the corresponding quantity is 2500

Optimal production quantity = 2500

Expected shortage = (2600-2500)*0.10 + (2700-2500)*0.05 = 20

Expected inventory = (2500-2000)*0.03+(2500-2100)*.08+(2500-2200)*.15+(2500-2300)*.3+(2500-2400)*.17+(2500-2500)*.12 = 169

Average demand = 2000*0.03+2100*0.08+2200*0.15+2300*0.3+2400*0.17+2500*0.12+2600*0.1+2700*0.05 = 2351

Expected Sales = Average demand - expected shortage = 2351 - 20 = 2331

Expected profit = Expected Sales*Cu - Expected inventory *Co = 2331*30 - 169*10 = $ 68,240

b) (i) For distributor, Underage cost, Cu = 50 - 40 = 10

Overage cost, Co = 40 - 10 = 30

Critical ratio = Cu/(Cu+Co) = 10/(10+30) = 0.25

Optimal order quantity for distributor corresponding to cumulative probability 0.25 or higher is 2200

Optimal quantity distributor should order = 2200

Expected profit for manufacturer = 2200*(40-20) = 44000

Expected shortage for distributor = (2300-2200)*0.3+(2400-2200)*0.17+(2500-2200)*0.12+(2600-2200)*0.10+(2700-2200)*0.05 = 165

Expected sales = Average demand - Expected shortage = 2351-165 = 2186

Expected inventory = 2200-2186 = 14

Expected profit for distributor = 2186*10 - 14*30 = $ 21440

Quantity Probability Cumulative probability 2000 3% 3% 2100 8% 11% 2200 15% 26% 2300 30% 56% 2400 17% 73% 2500 12% 85% 2600 10% 95% 2700 5% 100%

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