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Name: Le Duc Thinh ID: BABAIU14255 QUANTITATIVE METHODS FOR BUSINESS – ASSIGNMENT: DECISION TREE 3–35: The physicians in Problem 3–34 have been approached by a market research firm that offers to perform a study of the market at a fee of $5,000. The market researchers claim their experience enables them to use Bayes’ theorem to make the following statements of probability:

Probability Probability Probability Probability Probability Probability

of of of of of of

a favorable market given a favorable study = 0.82 an unfavorable market given a favorable study = 0.18 a favorable market given an unfavorable study = 0.11 an unfavorable market given an unfavorable study = 0.89 a favorable research study = 0.55 an unfavorable research study = 0.45

(a) Develop a new decision tree for the medical professionals to reflect the options now open with the market study. (b) Use the EMV approach to recommend a strategy. (c) What is the expected value of sample information? How much might the physicians be willing to pay for a market study? (d) Calculate the efficiency of this sample information. Solution: (b) – Given favorable survey results, we have: EMV(node 2) = EMV(construct | positive survey) = (0.82)*($95,000) + (0.18)*(–$45,000) = $69,800 The EMV of the situation that the clinic is not constructed in this case is – $5,000. Thus, if the survey results are favorable, they should construct the clinic. – Given negative survey results, we have: EMV(node 3) = EMV(construct | negative survey) = (0.11)*($95,000) + (0.89)*(–$45,000) = –$29,600 The EMV of the situation that the clinic is not constructed in this case is – $5,000. Thus, if the survey results are negative, they should not construct the clinic. – The expected value of conducting the market survey is: EMV(node 1) = EMV(conduct survey) = (0.55)*($69,800) + (0.45)*(–$5,000) = $36,140

– If the market survey is not conducted, we have: EMV(node 4) = (0.5)*($100,000) + (0.5)*(–$40,000) = $30,000 The EMV of the situation that the clinic is not constructed in this case is $0. Thus, constructing the clinic is the best choice, given that the market survey is not performed. Therefore survey).

the survey should be used since EMV (conduct survey) > EMV(no

If it is favorable, construct the clinic. If it is unfavorable, don’t

construct the clinic. (c) We have: EVSI = (EV with sample information + cost) – (EV without sample information) = ($36,140 + $5,000) – $30,000 = $11,140 Thus, the physicians would pay up to $11,140 for the survey. (d) We have the decision table as following: DECISION ALTERNATIVE Construct clinic Do nothing With perfect information Probability

STATE OF NATURE FAVORABLE MARKET UNFAVORABLE MARKET ($) ($) 100,000 0 100,000 0.5

–40,000 0 0 0.5

We have: EMV(construct clinic) = (0.5)*($100,000) + (0.5)*(–$40,000) = $30,000 EMV(do nothing) = (0.5)*($0) + (0.5)*($0) = $0 Best alternative for favorable and unfavorable market is constructing the clinic with a payoff of $100,000 and not construct the clinic with a payoff of $0, respectively. We have: EV(with perfect information) = (0.5)*($100,000) + (0.5)*($0) = $50,000 EVPI = EV(with perfect information) – Maximum EMV(without perfect information) = $50,000 – $30,000 = $20,000 Efficiency of sample information =

EVSI ×100 EVPI

=

$ 11,140 × 100 $ 20,000

55.7% Thus, the market survey is only 55.7% as efficient as perfect information. 3–41: A financial advisor has recommended two possible mutual funds for investment: Fund A

=

and Fund B. The return that will be achieved by each of these depends on whether the economy is good, fair, or poor. A payoff table has been constructed to illustrate this situation: (a) Draw the decision tree to represent this situation. (b) Perform the necessary calculations to determine which of the two mutual funds is better. Which one should you choose to maximize the expected value? (c) Suppose there is question about the return of Fund A in a good economy. It could be higher or lower than $10,000. What value for this would cause a person to be indifferent between Fund A and Fund B (i.e., the EMVs would be the same)? INVESTMENT Fund A Fund B Probability

GOOD ECONOMY $10,000 $6,000 0.2

STATE OF NATURE FAIR ECONOMY $2,000 $4,000 0.3

POOR ECONOMY –$5,000 0 0.5

(b) We have: EMV(fund A) = (0.2)*($10,000) + (0.3)*($2,000) + (0.5)*(–$5,000) = $100 EMV(fund B) = (0.2)*($6,000) + (0.3)*($4,000) + (0.5)*($0) = $2,400 EMV(no investment) = $0 Hence, investing in fund B is the best choice since it maximizes the expected value. (c) Let X (dollars) be the return of fund A in a good economy. We are indifferent when the EMV of fund A is the same as the EMV of fund B, which is $2,400. We have: EMV(fund A) = EMV(fund B) (0.2)*(X) + (0.3)*($2,000) + (0.5)*(–$5,000) = $2,400 (0.2)*X = $4,300 => X = $21,500 Therefore, the return of fund A in a good economy would have to be $21,500 for the two funds to be equally desirable based on the expected values. 3–42: Jim Sellers is thinking about producing a new type of electric razor for men. If the market were favorable, he would get a return of $100,000, but if the market for this new type of razor were unfavorable, he would lose $60,000. Since Ron Bush is a good friend of Jim Sellers, Jim is considering the possibility of using Bush Marketing Research to gather additional information

about the market for the razor. Ron has suggested that Jim either use a survey or a pilot study to test the market. The survey would be a sophisticated questionnaire administered to a test market. It will cost $5,000. Another alternative is to run a pilot study. This would involve producing a limited number of the new razors and trying to sell them in two cities that are typical of American cities. The pilot study is more accurate but is also more expensive. It will cost $20,000. Ron Bush has suggested that it would be a good idea for Jim to conduct either the survey or the pilot before Jim makes the decision concerning whether to produce the new razor. But Jim is not sure if the value of the survey or the pilot is worth the cost. Jim estimates that the probability of a successful market without performing a survey or pilot study is 0.5. Furthermore, the probability of a favorable survey result given a favorable market for razors is 0.7, and the probability of a favorable survey result given an unsuccessful market for razors is 0.2. In addition, the probability of an unfavorable pilot study given an unfavorable market is 0.9, and the probability of an unsuccessful pilot study result given a favorable market for razors is 0.2 (a) Draw the decision tree for this problem without the probability values. (b) Compute the revised probabilities needed to complete the decision, and place these values in the decision tree. (c) What is the best decision for Jim? Use EMV as the decision criterion.

(b) Let FM, UM, FS, US, FP and UP respectively be the favorable market, unfavorable market, favorable survey, unfavorable survey, favorable pilot study and unfavorable pilot study. We have:

P(FM) = 0.5

P(UM) = 0.5

P(FS | FM) = 0.7

P(FS | UM) = 0.2

P(US | FM) = 1 – P(FS | FM) = 0.3 P(US | UM) = 1 – P(FS | UM) = 0.8

P(UP | UM) = 0.9

P(FP | UM) = 1 – P(UP | UM) = 0.1

P(UP | FM) = 0.2

P(FP | FM) = 1 – P(UP | FM) = 0.8

Now, we can compute the revised probabilities needed to complete the decision in this situation:

P(FM | FS)

P(UM | FS)

=

FS|UM ) × P(UM ) P( FS∨FM )× P( FM )+ P ¿ P( FS∨FM )× P( FM ) ¿

=

( 0.7 ) ×(0.5) ( 0.7 ) × ( 0.5 ) + ( 0.2 ) ×(0.5)

= 0.78

= 1 – P(FM | FS) = 0.22

P(FM | US)

P(UM | US)

=

US |UM ) × P(UM ) P(US∨FM )× P( FM )+P ¿ P(US∨FM )× P(FM ) ¿

=

( 0.3 ) ×(0.5) ( 0.3 ) × ( 0.5 ) + ( 0.8 ) ×(0.5)

= 0.27

= 1 – P(FM | US) = 0.73

P(FM | FP)

P(UM | FP)

=

FP |UM ) × P(UM ) P( FP∨FM )× P( FM )+ P ¿ P( FP∨FM )× P (FM ) ¿

=

( 0.8 ) ×(0.5) ( 0.8 ) × ( 0.5 ) + ( 0.1 ) ×(0.5)

= 0.89

= 1 – P(FM | FP) = 0.11

P(FM | UP)

P(UM | UP)

=

UP |UM ) × P(UM ) P(UP∨FM ) × P(FM )+ P ¿ P(UP∨FM )× P( FM ) ¿

=

( 0.2 ) ×(0.5) ( 0.2 ) × ( 0.5 )+ ( 0.9 ) ×(0.5)

= 1 – P(FM | UP) = 0.82

P(FS | FM)

=

FM |US ) × P(US) P( FM ∨FS)× P( FS)+ P ¿ P( FM ∨FS)× P( FS) ¿

= 0.18

0.78 × P(FS) 0.78× P( FS)+ 0.27 ×(1−P ( FS ))

0.7

=

=> P(FS)

= 0.45, P(US) = 0.55

P(FP | FM)

=

FM |UP ) × P(UP ) P( FM ∨FP)× P( FP)+ P ¿ P( FM ∨FP)× P (FP) ¿

0.8

=

0.89 × P( FP) 0.89× P ( FP)+0.18 ×(1−P ( FP ))

=> P(FP)

= 0.45, P(UP) = 0.55

(c) *Use survey: – Given favorable survey results, we have: EMV(node 3) = (0.78)*($95,000) + (0.22)*(–$65,000) = $59,800 The EMV of the situation that the razor is not produced in this case is – $5,000. Thus, if the survey results are favorable, they should produce the razor. – Given negative survey results, we have: EMV(node 4) = (0.27)*($95,000) + (0.73)*(–$65,000) = –$21,800 The EMV of the situation that the razor is not produced in this case is – $5,000. Thus, if the survey results are unfavorable, they should not produce the razor. – The expected value of conducting the market survey is: EMV(node 1) = EMV(conduct survey) = (0.45)*($59,800) + (0.55)*(–$5,000) = $24,160 *Use pilot study: – Given favorable study results, we have: EMV(node 5) = (0.89)*($80,000) + (0.11)*(–$80,000) = $62,400 The EMV of the situation that the razor is not produced in this case is – $20,000. Thus, if the study results are favorable, they should produce the razor. – If the study results is negative, we have: EMV(node 6) = (0.18)*($80,000) + (0.82)*(–$80,000) = –$51,200 The EMV of the situation that the razor is not produced in this case is – $20,000. Thus, if the study results are unfavorable, they should not produce the razor. – The expected value of conducting the market survey is: EMV(node 2) = EMV(conduct pilot study)

= (0.45)*($62,400) + (0.55)*(–$20,000) = $17,080 *Not conduct anything: EMV(node 7) = (0.5)*($100,000) + (0.5)*(–$60,000) = $20,000 The EMV of the situation that the razor is not produced in this case is $0. Thus, producing the razor is the best choice, given that the market survey and pilot study is not performed. => So, since EMV(conduct survey) > EMV(not conduct anything) > EMV(conduct pilot study), the survey should be used.

If it is favorable, produce the razor. If it is

unfavorable, do not produce the razor.

View more...
Probability Probability Probability Probability Probability Probability

of of of of of of

a favorable market given a favorable study = 0.82 an unfavorable market given a favorable study = 0.18 a favorable market given an unfavorable study = 0.11 an unfavorable market given an unfavorable study = 0.89 a favorable research study = 0.55 an unfavorable research study = 0.45

(a) Develop a new decision tree for the medical professionals to reflect the options now open with the market study. (b) Use the EMV approach to recommend a strategy. (c) What is the expected value of sample information? How much might the physicians be willing to pay for a market study? (d) Calculate the efficiency of this sample information. Solution: (b) – Given favorable survey results, we have: EMV(node 2) = EMV(construct | positive survey) = (0.82)*($95,000) + (0.18)*(–$45,000) = $69,800 The EMV of the situation that the clinic is not constructed in this case is – $5,000. Thus, if the survey results are favorable, they should construct the clinic. – Given negative survey results, we have: EMV(node 3) = EMV(construct | negative survey) = (0.11)*($95,000) + (0.89)*(–$45,000) = –$29,600 The EMV of the situation that the clinic is not constructed in this case is – $5,000. Thus, if the survey results are negative, they should not construct the clinic. – The expected value of conducting the market survey is: EMV(node 1) = EMV(conduct survey) = (0.55)*($69,800) + (0.45)*(–$5,000) = $36,140

– If the market survey is not conducted, we have: EMV(node 4) = (0.5)*($100,000) + (0.5)*(–$40,000) = $30,000 The EMV of the situation that the clinic is not constructed in this case is $0. Thus, constructing the clinic is the best choice, given that the market survey is not performed. Therefore survey).

the survey should be used since EMV (conduct survey) > EMV(no

If it is favorable, construct the clinic. If it is unfavorable, don’t

construct the clinic. (c) We have: EVSI = (EV with sample information + cost) – (EV without sample information) = ($36,140 + $5,000) – $30,000 = $11,140 Thus, the physicians would pay up to $11,140 for the survey. (d) We have the decision table as following: DECISION ALTERNATIVE Construct clinic Do nothing With perfect information Probability

STATE OF NATURE FAVORABLE MARKET UNFAVORABLE MARKET ($) ($) 100,000 0 100,000 0.5

–40,000 0 0 0.5

We have: EMV(construct clinic) = (0.5)*($100,000) + (0.5)*(–$40,000) = $30,000 EMV(do nothing) = (0.5)*($0) + (0.5)*($0) = $0 Best alternative for favorable and unfavorable market is constructing the clinic with a payoff of $100,000 and not construct the clinic with a payoff of $0, respectively. We have: EV(with perfect information) = (0.5)*($100,000) + (0.5)*($0) = $50,000 EVPI = EV(with perfect information) – Maximum EMV(without perfect information) = $50,000 – $30,000 = $20,000 Efficiency of sample information =

EVSI ×100 EVPI

=

$ 11,140 × 100 $ 20,000

55.7% Thus, the market survey is only 55.7% as efficient as perfect information. 3–41: A financial advisor has recommended two possible mutual funds for investment: Fund A

=

and Fund B. The return that will be achieved by each of these depends on whether the economy is good, fair, or poor. A payoff table has been constructed to illustrate this situation: (a) Draw the decision tree to represent this situation. (b) Perform the necessary calculations to determine which of the two mutual funds is better. Which one should you choose to maximize the expected value? (c) Suppose there is question about the return of Fund A in a good economy. It could be higher or lower than $10,000. What value for this would cause a person to be indifferent between Fund A and Fund B (i.e., the EMVs would be the same)? INVESTMENT Fund A Fund B Probability

GOOD ECONOMY $10,000 $6,000 0.2

STATE OF NATURE FAIR ECONOMY $2,000 $4,000 0.3

POOR ECONOMY –$5,000 0 0.5

(b) We have: EMV(fund A) = (0.2)*($10,000) + (0.3)*($2,000) + (0.5)*(–$5,000) = $100 EMV(fund B) = (0.2)*($6,000) + (0.3)*($4,000) + (0.5)*($0) = $2,400 EMV(no investment) = $0 Hence, investing in fund B is the best choice since it maximizes the expected value. (c) Let X (dollars) be the return of fund A in a good economy. We are indifferent when the EMV of fund A is the same as the EMV of fund B, which is $2,400. We have: EMV(fund A) = EMV(fund B) (0.2)*(X) + (0.3)*($2,000) + (0.5)*(–$5,000) = $2,400 (0.2)*X = $4,300 => X = $21,500 Therefore, the return of fund A in a good economy would have to be $21,500 for the two funds to be equally desirable based on the expected values. 3–42: Jim Sellers is thinking about producing a new type of electric razor for men. If the market were favorable, he would get a return of $100,000, but if the market for this new type of razor were unfavorable, he would lose $60,000. Since Ron Bush is a good friend of Jim Sellers, Jim is considering the possibility of using Bush Marketing Research to gather additional information

about the market for the razor. Ron has suggested that Jim either use a survey or a pilot study to test the market. The survey would be a sophisticated questionnaire administered to a test market. It will cost $5,000. Another alternative is to run a pilot study. This would involve producing a limited number of the new razors and trying to sell them in two cities that are typical of American cities. The pilot study is more accurate but is also more expensive. It will cost $20,000. Ron Bush has suggested that it would be a good idea for Jim to conduct either the survey or the pilot before Jim makes the decision concerning whether to produce the new razor. But Jim is not sure if the value of the survey or the pilot is worth the cost. Jim estimates that the probability of a successful market without performing a survey or pilot study is 0.5. Furthermore, the probability of a favorable survey result given a favorable market for razors is 0.7, and the probability of a favorable survey result given an unsuccessful market for razors is 0.2. In addition, the probability of an unfavorable pilot study given an unfavorable market is 0.9, and the probability of an unsuccessful pilot study result given a favorable market for razors is 0.2 (a) Draw the decision tree for this problem without the probability values. (b) Compute the revised probabilities needed to complete the decision, and place these values in the decision tree. (c) What is the best decision for Jim? Use EMV as the decision criterion.

(b) Let FM, UM, FS, US, FP and UP respectively be the favorable market, unfavorable market, favorable survey, unfavorable survey, favorable pilot study and unfavorable pilot study. We have:

P(FM) = 0.5

P(UM) = 0.5

P(FS | FM) = 0.7

P(FS | UM) = 0.2

P(US | FM) = 1 – P(FS | FM) = 0.3 P(US | UM) = 1 – P(FS | UM) = 0.8

P(UP | UM) = 0.9

P(FP | UM) = 1 – P(UP | UM) = 0.1

P(UP | FM) = 0.2

P(FP | FM) = 1 – P(UP | FM) = 0.8

Now, we can compute the revised probabilities needed to complete the decision in this situation:

P(FM | FS)

P(UM | FS)

=

FS|UM ) × P(UM ) P( FS∨FM )× P( FM )+ P ¿ P( FS∨FM )× P( FM ) ¿

=

( 0.7 ) ×(0.5) ( 0.7 ) × ( 0.5 ) + ( 0.2 ) ×(0.5)

= 0.78

= 1 – P(FM | FS) = 0.22

P(FM | US)

P(UM | US)

=

US |UM ) × P(UM ) P(US∨FM )× P( FM )+P ¿ P(US∨FM )× P(FM ) ¿

=

( 0.3 ) ×(0.5) ( 0.3 ) × ( 0.5 ) + ( 0.8 ) ×(0.5)

= 0.27

= 1 – P(FM | US) = 0.73

P(FM | FP)

P(UM | FP)

=

FP |UM ) × P(UM ) P( FP∨FM )× P( FM )+ P ¿ P( FP∨FM )× P (FM ) ¿

=

( 0.8 ) ×(0.5) ( 0.8 ) × ( 0.5 ) + ( 0.1 ) ×(0.5)

= 0.89

= 1 – P(FM | FP) = 0.11

P(FM | UP)

P(UM | UP)

=

UP |UM ) × P(UM ) P(UP∨FM ) × P(FM )+ P ¿ P(UP∨FM )× P( FM ) ¿

=

( 0.2 ) ×(0.5) ( 0.2 ) × ( 0.5 )+ ( 0.9 ) ×(0.5)

= 1 – P(FM | UP) = 0.82

P(FS | FM)

=

FM |US ) × P(US) P( FM ∨FS)× P( FS)+ P ¿ P( FM ∨FS)× P( FS) ¿

= 0.18

0.78 × P(FS) 0.78× P( FS)+ 0.27 ×(1−P ( FS ))

0.7

=

=> P(FS)

= 0.45, P(US) = 0.55

P(FP | FM)

=

FM |UP ) × P(UP ) P( FM ∨FP)× P( FP)+ P ¿ P( FM ∨FP)× P (FP) ¿

0.8

=

0.89 × P( FP) 0.89× P ( FP)+0.18 ×(1−P ( FP ))

=> P(FP)

= 0.45, P(UP) = 0.55

(c) *Use survey: – Given favorable survey results, we have: EMV(node 3) = (0.78)*($95,000) + (0.22)*(–$65,000) = $59,800 The EMV of the situation that the razor is not produced in this case is – $5,000. Thus, if the survey results are favorable, they should produce the razor. – Given negative survey results, we have: EMV(node 4) = (0.27)*($95,000) + (0.73)*(–$65,000) = –$21,800 The EMV of the situation that the razor is not produced in this case is – $5,000. Thus, if the survey results are unfavorable, they should not produce the razor. – The expected value of conducting the market survey is: EMV(node 1) = EMV(conduct survey) = (0.45)*($59,800) + (0.55)*(–$5,000) = $24,160 *Use pilot study: – Given favorable study results, we have: EMV(node 5) = (0.89)*($80,000) + (0.11)*(–$80,000) = $62,400 The EMV of the situation that the razor is not produced in this case is – $20,000. Thus, if the study results are favorable, they should produce the razor. – If the study results is negative, we have: EMV(node 6) = (0.18)*($80,000) + (0.82)*(–$80,000) = –$51,200 The EMV of the situation that the razor is not produced in this case is – $20,000. Thus, if the study results are unfavorable, they should not produce the razor. – The expected value of conducting the market survey is: EMV(node 2) = EMV(conduct pilot study)

= (0.45)*($62,400) + (0.55)*(–$20,000) = $17,080 *Not conduct anything: EMV(node 7) = (0.5)*($100,000) + (0.5)*(–$60,000) = $20,000 The EMV of the situation that the razor is not produced in this case is $0. Thus, producing the razor is the best choice, given that the market survey and pilot study is not performed. => So, since EMV(conduct survey) > EMV(not conduct anything) > EMV(conduct pilot study), the survey should be used.

If it is favorable, produce the razor. If it is

unfavorable, do not produce the razor.

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