A motorcycle dealer must determine how many next year\'s model should be ordered
ID: 3072739 • Letter: A
Question
A motorcycle dealer must determine how many next year's model should be ordered in August. Each car costs the dealer $10,000. The demand for the dealer's next year models has the following probability: d pd) F(d) 20.30.30 25 .15 .45 30 .15 .60 35 .20 80 40 .20 1 Each motorcycle is sold for $15,000. If the demand exceeds the number the dealer ordered, the dealer must reorder at a cost of $12,000 per unit. If the demand falls short, the dealer may dispose of excess units in an end-of-model-year sale for $9,500 per unit (a) How many new models should be ordered to maximize the expected profit? (b) Calculate the expected profit when the dealer orders the optimal quantity.Explanation / Answer
a)
Cost (C)=10000, Selling price (S)=15000
goodwill cost(V)= 12000-10000=2000 Salvage value (s)= 15000-9500=5500
calculate critical ratio (C3) = C2/(C1 + C2).
C1= over stockinf cost, C2 = under stocking cost
C1 = C - V= 8000
C2 = S - C + s = 10500
Then C3 = 10500 / (10500+8000) = 0.54
From Cumulative distribution, 0.57 lies between 0.45 and 0.60. i.e. it lies between the demand 25 to 30 and hence the new models to be ordered is 30.
b)
Let Q = item ordered and D = item demanded
Sell of model
The payoff matrix is obtained by using the formaula obtained as 9500D-4500Q when Q >= D and 13000D-8000Q when Q < D.
Now create payoff matrix for Q and D
It is found that the optimal quantity should be 30 units. Hence the expected profit for optimal quantity ordered is
= (20000*0.30)+(85000*0.15)+(150000*0.15)+(197500*0.20)+(245000*0.20)
= 6000+12750+22500+39500+49000
=129750
Therefore the expected profite will be $129750 if the dealer order the optimal quantity of 30 units.
Payoff for Q >= D Q < D Cost of model -10000Q -10000QSell of model
15000D 15000D Goodwiil cost - -2000(D-Q) Salvage cost 5500 (Q-D) - Total Payoff 9500D-4500Q 13000D-8000QRelated Questions
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