Operations Management Help

Chapter 9 - Practice Problem Solutions

9-1

9.1 a) Max(A1) = 6, Max(A2) = 4, Max(A3) = 8. Maximax = 8 with alternative A3.

b) Min(A1) = 2, Min(A2) = 3, Min(A3) = 1. Maximin = 3 with alternative A2.

9.2 a) Max(A1) = 30, Max(A2) = 31, Max(A3) = 22, Max(A4) = 29. Maximax = 31 with A2.

b) Min(A1) = 20, Min(A2) = 14, Min(A3) = 22, Min(A4) = 21. Maximin = 22 with A3.

9.3 a)

State of Nature

Alternative Sell 10 cases Sell 11 cases Sell 12 cases Sell 13 cases

Buy 10 cases $50 $50 $50 $50

Buy 11 cases $47 $55 $55 $55

Buy 12 cases $44 $52 $60 $60

Buy 13 cases $41 $49 $57 $65

Prior Probability 0.2 0.4 0.3 0.1

b) Max(Buy 10) = $50, Max(Buy 11) = $55, Max(Buy 12) = $60, Max(Buy 13) = $65.

Maximax = $65 with buying 13 cases.

c) Min(Buy 10) = $50, Min(Buy 11) = $47, Min(Buy 12) = $44, Min(Buy 13) = $41.

Maximin = $50 with buying 10 cases.

d) The most likely state of nature is to sell 11 cases. Under this state, she should buy 11 cases

with a payoff of $55.

e)

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A B C D E F G H

Purchase Pri ce $3 Sel li ng Price $8

Payoff T able Sel l Sel l Sel l Sel l Expected

Al ternati ve 10 11 12 13 Payoff

Buy 10 $50 $50 $50 $50 $50.00 Buy 11 $47 $55 $55 $55 $53.40 Buy 12 $44 $52 $60 $60 $53.60

Buy 13 $41 $49 $57 $65 $51.40

Prior Probabil ity 0.2 0.4 0.3 0.1

State of Nature

Jean should buy 12 cases. The maximum expected payoff is $53.60.

Chapter 9 - Practice Problem Solutions

9-2

f) (i) 0.2 and 0.5

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7 8 9 10

11 12

A B C D E F G H

Purchase Pri ce $3 Sel li ng Price $8

Payoff T able Sel l Sel l Sel l Sel l Expected

Al ternati ve 10 11 12 13 Payoff

Buy 10 $50 $50 $50 $50 $50.00 Buy 11 $47 $55 $55 $55 $53.40 Buy 12 $44 $52 $60 $60 $55.20

Buy 13 $41 $49 $57 $65 $53.00

Prior Probabil ity 0.2 0.2 0.5 0.1

State of Nature

Jean should purchase 12 cases. The maximum expected payoff is $55.20.

(ii) 0.3 and 0.4

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11 12

A B C D E F G H

Purchase Pri ce $3 Sel li ng Price $8

Payoff T able Sel l Sel l Sel l Sel l Expected

Al ternati ve 10 11 12 13 Payoff

Buy 10 $50 $50 $50 $50 $50.00 Buy 11 $47 $55 $55 $55 $53.40 Buy 12 $44 $52 $60 $60 $54.40

Buy 13 $41 $49 $57 $65 $52.20

Prior Probabil ity 0.2 0.3 0.4 0.1

State of Nature

Jean should purchase 12 cases. The maximum expected payoff is $54.40.

Chapter 9 - Practice Problem Solutions

9-3

(iii) 0.5 and 0.2

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7 8 9 10

11 12

A B C D E F G H

Purchase Pri ce $3 Sel li ng Price $8

Payoff T able Sel l Sel l Sel l Sel l Expected

Al ternati ve 10 11 12 13 Payoff

Buy 10 $50 $50 $50 $50 $50.00 Buy 11 $47 $55 $55 $55 $53.40 Buy 12 $44 $52 $60 $60 $52.80

Buy 13 $41 $49 $57 $65 $50.60

Prior Probabil ity 0.2 0.5 0.2 0.1

State of Nature

Jean should purchase 11 cases. The maximum expected payoff is $53.40.

9.4 a) Max(Conservative) = $30 million

Max(Speculative) = $40 million

Max(Countercyclical) = $15 million

Maximax = $40 million with the speculative investment

b) Min(Conservative) = –$10 million

Min(Speculative) = –$30 million

Min(Countercyclical) = –$10 million

Maximin = –$10 million with either the conservative of countercyclical investment.

c) The stable economy is the most likely state of nature.

The speculative investment has the maximum payoff for this state ($10 million).

d) The countercyclical investment has the maximum expected payoff of $5 million.

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A B C D E F

Payoff Table ($mil li on) Expected Al ternati ve State of Nature (Economy) Payoff

(Investment) Improvi ng Stable Worsening ($mi ll ion)

Conservative 30 5 -10 1.5 Specul ati ve 40 10 -30 -3

Countercycli cal -10 0 15 5

Prior Probabil ity 0.1 0.5 0.4

9.7 a) Max(A1) = 80, Max(A2) = 50, Max(A3) = 60.

Maximax = $80 thousand when choosing alternative A1.

b) Min(A1) = 25, Min(A2) = 30, Min(A3) = 40.

Maximin = $40 thousand when choosing alternative A3.

Chapter 9 - Practice Problem Solutions

9-4

c) S2 is the most likely outcome. For this state, the maximum payoff of $50 thousand occurs with

alternative A2.

d) Alternative A3 has the highest expected payoff of $48 thousand.

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A B C D E

Payoff Table ($thousand) Expected Payoff

Al ternati ve S1 S2 ($thousand)

A1 80 25 47 A2 30 50 42 A3 60 40 48

Prior Probabil ity 0.4 0.6

State of Nature

e)

Chapter 9 - Practice Problem Solutions

9-5

f) When the prior probability of S1 is 0.2, alternative A2 should be chosen, with an expected

payoff of $46 thousand.

When the prior probability of S1 is 0.6, alternative A1 should be chosen, with an expected

payoff of $58 thousand.

Chapter 9 - Practice Problem Solutions

9-6

g)

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19

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21

22

23

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M N O

Prior Expected

Probabil ity Best Payoff

of S1 Al ternati ve ($thousand)

2 46

0.20 2 46

0.24 2 45.2

0.28 3 45.6

0.32 3 46.4 0.36 3 47.2 0.40 3 48 0.44 1 49.2 0.48 1 51.4 0.52 1 53.6 0.56 1 55.8 0.60 1 58

9.19 a)

State of Nature

Alternative Successful Unsuccessful

Develop new product $1,500,000 –$1,800,000

Don’t develop new product 0 0

Prior Probabilities 0.667 0.333

b) Choosing to develop the product maximizes the expected payoff ($400,000).

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A B C D E

Payoff Table ($mil li ons) Expected

Payoff

Al ternati ve Successful Unsuccessful ($mi ll ions)

Devel op Product 1.5 -1.8 0.4

Don't Develop Product 0 0 0

Prior Probabil ity 0.667 0.333

State of Nature

Chapter 9 - Practice Problem Solutions

9-7

c) With perfect information, Telemore should develop the product if it would be successful, and

don’t if it will be unsuccessful.

EP(perfect information) = (0.667)(1.5) + (0.333)(0) = $1 million.

EVPI = EP(with perfect information) – EP(without more information)

= $1,000,000 – $400,000 = $600,000.

This indicates that consideration should be given to conducting the market survey.

d)

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B C D E F G H

Data: State of Prior Nature Probabil ity Predict Successful Predict Unsuccessful

Successful 0.667 0.8 0.2 Unsuccessful 0.333 0.3 0.7

Posterior Probabilities:

Fi nding P(Findi ng) Successful Unsuccessful Predict Successful 0.633 0.842 0.158

Predict Unsuccessful 0.367 0.364 0.636

P(State | Fi nding) State of Nature

P(Findi ng | State) Fi nding

Chapter 9 - Practice Problem Solutions

9-8

e) They should conduct the survey, and develop the product if the survey predicts the product

will be successful. The expected payoff is $520,000.

9.20 a)

State of Nature

Alternative p=0.05 p=0.25

Screen –$1,500 –$1,500

Don’t screen –$750 –$3,750

Prior Probabilities 0.8 0.2

Chapter 9 - Practice Problem Solutions

9-9

b) Choosing not to screen maximizes the expected payoff. The expected cost is $1,350.

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A B C D E

Payoff Table Expected

Alternative p = 0.05 p = 0.25 Payoff

Screen -$1,500 -$1,500 -$1,500

Don't Screen -$750 -$3,750 -$1,350

Prior Probability 0.8 0.2

State of Nature

c) With perfect information, they would screen if p = 0.25, and don’t screen if p = 0.05.

EP(with perfect information) = (0.8)(–$750) + (0.2)(–$1,500) = –$900

EVPI = EP(with perfect information) – EP(without more information)

= (–$900) – (–$1,350) = $450.

This indicates that consideration should be given to inspecting the single item.

d)

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B C D E F G H

Data: State of Prior Nature Probabil ity Defective Nondefective

p = 0.05 0.8 0.05 0.95 p = 0.25 0.2 0.25 0.75

Posterior Probabilities:

Findi ng P(Findi ng) p = 0.05 p = 0.25 Defective 0.09 0.444 0.556

Nondefective 0.91 0.835 0.165

P(State | Fi nding) State of Nature

P(Findi ng | State) Findi ng

Chapter 9 - Practice Problem Solutions

9-10

e) The optimal policy is not to pre-screen or screen.

f) EVSI = EP(with information ignoring cost of information) – EP(without information)

= (–1392.95 + 125) – (–1350) = (–1267.95) – (–1350) = $82.95.

If the prescreening inspection cost is less than $82.96, then it would be worthwhile to use.

Chapter 9 - Practice Problem Solutions

9-11

9.27 a & b) (Note: this decision tree continues on the next page.)

Chapter 9 - Practice Problem Solutions

9-12

The comptroller should invest in stocks the first year. If there is growth during the first year

then she should invest in stocks again the second year. If there is a recession during the first

year then she should invest in bonds for the second year. The expected payoff is $122.94

million.