Question

9.3. Jean Clark is the manager of the Midtown Saveway Grocery Store. She now needs to replenish her supply of strawberries. Her regular supplier can provide as many cases as she wants. However, because these strawberries already are very ripe, she will need to sell them tomorrow and then discard any that remain unsold. Jean estimates that she will be able to sell 10, 11, 12, or 13 cases tomorrow. She can purchase the strawberries for $3 per case and sell them for $8 per case. Jean now needs to decide how many cases to purchase.

Jean has checked the store’s records on daily sales of strawberries. On this basis, she estimates that the prior probabilities are 0.2, 0.4, 0.3, and 0.1 for being able to sell 10, 11, 12, and 13 cases of strawberries tomorrow.

4. How many cases should be purchased if she uses the maximum likelihood criterion?

5. How many cases should be purchased according to Bayes’ decision rule?

6. Jean thinks she has the prior probabilities just about right for selling 10 cases and selling 13 cases, but is uncertain about how to split the prior probabilities for 11 cases and 12 cases. Reapply Bayes’ decision rule when the prior probabilities of 11 and 12 cases are (i) 0.2 and 0.5, (ii) 0.3 and 0.4, and (iii) 0.5 and 0.2.

Answer 1

a)

Decision alternatives: Number of cases of strawberry to purchase

States of nature: Demand in number of cases

Payoff = MIN(D,Q)*p-Q*c, where D is the demand, Q is the qty purchased, p is selling price and c is the purchase cost.

Resulting Payoff table is following:

	Demand
Purchase Qty	10	11	12	13
10	50	50	50	50
11	47	55	55	55
12	44	52	60	60
13	41	49	57	65

Calculation example: Payoff for (Q=12, D=13) = MIN(12,13)*8-12*3 = 60

b) Using Maximax approach, decision alternatives whose maximum payoff is the maximum of all alternatives' maximum payoff is selected.

The maximum of the maximum payoff of all alternatives is 65 for Q=13. Therefore, best alternative as per Maximax approach is to purchase 13 cases.

c) Using Maximin approach, decision alternatives whose minimum payoff is the maximum of all alternatives' minimum payoff is selected.

The maximum of the minimum payoff of all alternatives is 50 for Q=10. Therefore, best alternative as per Maximin approach is to purchase 10 cases.

d) Using Maximum likelihood approach, decision alternatives whose average payoff is the maximum of all alternatives' average payoff is selected.

Average payoff of order qty 10 cases = 50

Average payoff of order qty 11 cases = (47+55+55+55)/4 = 53

Average payoff of order qty 12 cases = (44+52+60+60)/4 = 54

Average payoff of order qty 13 cases = (41+49+57+65)/4 = 53

The maximum of the average payoff of all alternatives is 54 for Q=12. Therefore, best alternative as per Maximax approach is to purchase 12 cases.

e) Using Bayes' decision rule, Expected Monetary Value (EMV) is computed as the SUMPRODUCT of payoff and corresponding probabilities

EMV of order qty of 10 cases = .2*50+.4*50+.3*50+.1*50 = 50

EMV of order qty of 11 cases = .2*47+.4*55+.3*55+.1*55 = 53.4

EMV of order qty of 12 cases = .2*44+.4*52+.3*60+.1*60 = 53.6

EMV of order qty of 13 cases = .2*41+.4*49+.3*57+.1*65 = 51.4

Highest EMV is 53.6 of order qty of 12 cases

Therefore, best decision is to order 12 cases

f) this part of the question is incomplete.. prior probabilities of 11 and 12 cases are cropped out.