limited finance test 2.5 hour

3QA3/2DA3

Management Science for

Lecture 06

C01: July 7, 2021

C02: July 8, 2021

Instructors:

Seyyed Hossein Alavi Zdravko Dimitrov

Decision Analysis

Decision Analysis

3

 We live in a world surrounded by uncertainties and risks.

 We have to make a decision, without knowing for sure which

events will occur in the future.

 We’d better consider all decision alternatives before we make a

decision.

 Decision analysis is an analytic and systematic approach to the study

of decision making.

Steps of Decision Making

4

1. Clearly define the problem

2. List ALL possible alternatives

3. Identify all possible outcomes for each alternative

4. Identify the payoff for each alternative and outcome combination

5. Use a decision modeling technique to choose an alternative

Types of Decision Making

5

 Under Certainty

 We know the consequence of all alternatives.

 Under Uncertainty

 We do not know the probability of each outcome.

 Under Risk

 We know the probability of each outcome.

Decision Making Under Uncertainty

6

 The manager can list the possible future outcomes but cannot

estimate the probability that a specific outcome will occur.

 Five different decision-making criteria can be used to make a decision:

1. Maximax

2. Maximin

3. Realism

4. Equally likely

5. Minimax regret

Drilling Example

7

 A drilling company has a land. The manager would like to consider
two alternatives:
1. Drill the land for oil

2. Sell the land

 The possible outcomes (state of nature) are:
1. Oil

2. Dry

 The payoffs are:
1. If the company finds oil: profit of $700

2. If the company cannot find oil: cost of $100

3. If the company sells the land: profit of $90

Drilling Example

8

 Payoff Table:

Alternatives

Outcomes (States of nature)

Oil Dry

Drill for oil $700 -$100

Sell the land $90 $90

1-Maximax

9

 For each alternative, find the maximum payoff over all possible outcomes. Next,

find the maximum of them. Choose the alternative whose maximum payoff gives

this maximum.

 It is choosing the alternative with the best payoff, if the best outcome happens

(optimistic!)

Alternatives

Outcomes

(States of nature) Max

Oil Dry

Drill for oil $700 -$100 700

Sell the land $90 $90 90

Max

2-Maximin

10

 For each alternative, find the minimum payoff over all possible outcomes. Next,

find the maximum of them. Chose the alternative whose minimum payoff gives

this maximum.

 It is choosing the alternative with best payoff, if the worst outcome happens

(pessimistic!)

Alternatives

Outcomes

(States of nature) Min

Oil Dry

Drill for oil $700 -$100 -100

Sell the land $90 $90 90
Max

3-Criterion of Realism

11

 Decision makers should not be optimists or pessimists. They should be realistic.

 Define Coefficient of Realism: 0 ≤ ? ≤ 1
 ? = 1 means the decision maker is totally optimistic.(Maximax)

 ? = 0 means the decision maker is totally pessimistic.(Maximin)

 Calculate the realism payoff for each alternative and select the alternative with the

highest realism payoff.

Realism Payoff = ? × ??? ?????? + 1 − ? × ??? ??????

3-Criterion of Realism

12

 Suppose ? = 0.55

 Alternative 1: 0.55 × 700 + 0.45 × −100 = 340

 Alternative 2: 0.55 × 90 + 0.45 × 90 = 90

Alternatives

Outcomes

(States of nature) Criterion

for Realism
Oil Dry

Drill for oil $700 -$100 340

Sell the land $90 $90 90

Max

3-Criterion of Realism

13

 Suppose ? = 0.15

 Alternative 1: 0.15 × 700 + 0.85 × −100 = 20

 Alternative 2: 0.15 × 90 + 0.15 × 90 = 90

Alternatives

Outcomes

(States of nature) Criterion

for Realism
Oil Dry

Drill for oil $700 -$100 20

Sell the land $90 $90 90
Max

4- Equally Likely (Laplace)

14

 Calculate the average payoff for each alternative and select the alternative with

the highest average payoff.

Average Payoff =
σ ???????

?
where ? is the number of outcomes for the alternative

4- Equally Likely (Laplace)

15

 Alternative 1:
700+(−100)

2
= 300

 Alternative 2:
90+90

2
= 90

Alternatives

Outcomes

(States of nature) Laplace

Oil Dry

Drill for oil $700 -$100 300

Sell the land $90 $90 90

Max

5- Minimax Regret

16

 Opportunity loss or regret

 Is the amount lost by not picking the best alternative.

 Minimax regret finds the alternative that minimizes the maximum regret for

each alternative.

 First find the regrets for all outcome-alternative combinations by:

best payoff for outcome − actual payoff

 Next, find the maximum regret for alternatives and choose the minimum

one among them.

5- Minimax Regret

17

Min
Alternatives

Regret of Outcomes Max

regretOil Dry

Drill for oil 700-700 = 0 90-(-100) = 190 190

Sell the land 700-90 = 610 90-90 = 0 610

Alternatives

Outcomes

(States of nature) Laplace

Oil Dry

Drill for oil $700 -$100 300

Sell the land $90 $90 90

Payoff Table

Opportunity Loss Table (Regret = best payoff for outcome − actual payoff)

Example 2

18

Alternatives

Outcomes

High

Demand

Moderate

Demand

Low

Demand

Build Large Plant 200,000 100,000 -120,000

Build Small Plant 90,000 50,000 -20,000

Do Nothing 0 0 0

1- Maximax

19

Alternatives

Outcomes

MaxHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 200,000

Build Small Plant 90,000 50,000 -20,000 90,000

Do

Nothing
0 0 0 0

2- Maximin

20

Alternatives

Outcomes

MinHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 -120,000

Build Small Plant 90,000 50,000 -20,000 -20,000

Do

Nothing
0 0 0 0

3- Criterion of Realism, ? = 0.45

21

Alternatives

Outcomes

RealismHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 24,000

Build Small Plant 90,000 50,000 -20,000 29,500

Do

Nothing
0 0 0 0

Alternative 1: 0.45 × 200,000 + 0.55 × −120,000 = 24,000

Alternative 2: 0.45 × 90,000 + 0.55 × −20,000 = 29,500

4- Equally Likely (Laplace)

22

Alternatives

Outcomes

LaplaceHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 60,000

Build Small Plant 90,000 50,000 -20,000 40,000

Do

Nothing
0 0 0 0

Alternative 1:
200,000+100,000+(−120,000)

3
= 60,000

Alternative 2:
90,000+50,000+(−20,000)

3
= 40,000

5- Minimax Regret

23

A
lt

e
r
n

a
ti

v
e

s

Regret of Outcomes

Max

High

Demand

Moderate

Demand

Low

Demand

Build

Large

Plant

200,000-200,000 = 0
100,000-100,000=

0

0-(-120,000)=

120,000
120,000

Build

Small

Plant

200,000-90,000=

110,000

100,000-0,000=

50,000

0-(-20,000) =

20,000
110,000

Do

Nothing 200,000-0= 200,000
100,000-0=

100,000
0-0=0 200,000

Using Excel

24

 All formulas can be implemented in Excel.

 There is a prepared Excel sheet on Avenue:
Solver-Decision Analysis Under Uncertainty.xlsx

Decision Making Under Risk

25

 When the decision maker has some idea about the

probabilities of outcomes.

 We use probabilities of outcomes to find the best

alternative.

1. Expected Monetary Value (EMV)

2. Expected Opportunity Loss (EOL)

3. Expected Value of Perfect Information (EVPI)

Expected Monetary Value (EMV)

26

 EMV is the weighted average of all possible payoffs,

where weights are probabilities of outcomes

 EMV for each alternative is

෍

????????

??????????? × ??????

 We would like to maximize the EMV

 Select the alternative with largest EMV

Expected Monetary Value (EMV)

27

Alternatives

Outcomes

EMVHigh

Demand

Moderate

Demand
Low Demand

1- Build Large

Plant
200,000 100,000 -120,000 86,000

2- Build Small

Plant
90,000 50,000 -20,000 40,000

3- Do

Nothing
0 0 0 0

Probability 0.3 0.5 0.2

EMV 1 = 200,000 × 0.3 + 100,000 × 0.5 − 120,000 × 0.2 = 86,000

EMV 2 = 90,000 × 0.3 + 50,000 × 0.5 − 20,000 × 0.2 = 48,000

Expected Opportunity Loss (EOL)

28

 Opportunity loss (Regret) is the difference between the optimal payoff
and the actual payoff.

 EOL is the weighted (with probabilities) average of all regrets.

 EOL is the expected cost of not picking the best solution.

 EOL for each alternative is

෍

????????

??????????? × ??????

 We would like to minimize the EOL

 Select the alternative with smallest EOL

Expected Opportunity Loss (EOL)

29

EOL 1 = 200,000 − 200,000 × 0.3 + 100,000 − 100,000 × 0.5 + (0 + 120,000) × 0.2 = 24,000

EOL 2 = (200,000-90,000) × 0.3 + 100,000 − 50,000 × 0.5 + (0 + 20,000) × 0.2 = 62,000

EOL 3 = 200,000 − 0 × 0.3 + 100,000 − 0 × 0.5 + (0 − 0) × 0.2 = 110,000

Alternatives

Regret of Outcomes

EOLHigh

Demand

Moderate

Demand
Low Demand

Build Large

Plant
200,000-200,000 100,000-100,000 0-(-120,000) 24,000

Build Small

Plant
200,000-90,000 100,000-0,000 0-(-20,000) 62,000

Do

Nothing 200,000-0 100,000-0 0-0 110,000

Probability 0.3 0.5 0.2

EMV vs EOL

30

 In EMV, we choose the alternative with maximum EMV.

 In EOL, we choose the alternative with minimum EOL.

 In fact, these two always suggest the same alternative:

The alternative suggested by Max EMV

is the same as

the alternative suggested by Min EOL

Expected Value of Perfect Information (EVPI)

31

 A company claims that they can predict what happens in the future with
certainty.

 They have the perfect information, and they want to sell it to us!

 We want to know the monetary value of this information.

 If we have perfect information, we select alternative with the best payoff.

 Expected Value with Perfect Information (EVwPI)

 Weighted average of payoffs with certainty

෍

????????

??????????? × ???? ??????

Expected Value of Perfect Information (EVPI)

32

 EVwPI is the average payoff when we have the perfect information.

 Max EMV (Min EOL) is what we can do without any information!
(expected value without perfect information)

 There for EVwPI minus Max EMV is the maximum worth of
perfect information:

EVPI = EVwPI – Max EMV

 We must not pay more than EVPI for the information.
 EVPI is the upper bound for the money that we should spend for

the information.

EVPI = Min EOL

33

Alternatives

Outcomes

EMV EOLHigh

Demand

Moderate

Demand

Low

Demand

1- Build Large

Plant
200,000 100,000 -120,000 86,000 24,000

2- Build Small

Plant
90,000 50,000 -20,000 40,000 62,000

3- Do

Nothing
0 0 0 0 110,000

Probability 0.3 0.5 0.2

EVwPI = 200,000 × 0.3 + 100,000 × 0.5 + 0 × 0.2 = 110,000

EVPI= EVwPI- Max EMV = 110,000 − 86000 = 24,000

EVPI = Min EOL = 24,000

Expected Value of Perfect Information (EVPI)

Expected Value of Perfect Information (EVPI)

34

 We have:

EVPI = EVwPI – Max EMV

EVPI = Min EOL

 Therefore:

EVwPI = Max EMV + Min EOL

 For any decision alternative:

EVwPI = EMV + EOL

35

Alternatives

Outcomes

EMV EOL EVwPI
High

Demand

Moderate

Demand

Low

Demand

1- Build

Large Plant
200,000 100,000 -120,000 86,000 24,000 110,000

2- Build

Small Plant
90,000 50,000 -20,000 40,000 62,000 110,000

3- Do

Nothing
0 0 0 0 110,000 110,000

EVwPI = EMV + EOL

Expected Value of Perfect Information (EVPI)

Expected Value of Perfect Information (EVPI)

36

EVwPI Max payoff with Perfect Information

EOL

EOL
EOL

EOL

EVPI

EMV
EMV

EMV
EMV

Alternative 1 Alternative 2 Alternative 3 Alternative 4

EVwPI = EMV + EOL

Example 3

37

 We would like to decide on the order quantities for snowboards
over next 5 months.

 Everything must go during the winter season.

 The profit is $100.
 Unsold snowboards will have a loss of $50.

 Alternatives: Order quantity of 100,200,300,400,500

Outcome

1

Outcome

2

Outcome

3

Outcome

4

Outcome

5

Demand 100 200 300 400 500

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – Payoff Table

38

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000

200 $5,000 $20,000 $20,000 $20,000 $20,000

300 0 $15,000 $30,000 $30,000 $30,000

400 -$5,000 $10,000 $25,000 $40,000 $40,000

500 -$10,000 $5,000 $20,000 $35,000 $50,000

Probability 0.2 0.3 0.3 0.1 0.1

Payoff (Profit) =$100 × 200 − $50 × 200 = $10,000 (Demand < Order Size) Payoff (Profit) =$100 × 300 = $30,000 (Demand = Order Size) Payoff (Profit) =$100 × 200 = $20,000 (Demand > Order Size)

Example 3 – EMV

39

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

EMV

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000

200 $5,000 $20,000 $20,000 $20,000 $20,000 $17,000

300 0 $15,000 $30,000 $30,000 $30,000 $19,500

400 -$5,000 $10,000 $25,000 $40,000 $40,000 $17,500

500 -$10,000 $5,000 $20,000 $35,000 $50,000 $14,000

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – EOL Table

40

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demands (regret of outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

EOL

100 200 300 400 500

100 0 $10,000 $20,000 $30,000 $40,000 $16,000

200 $5,000 0 $10,000 $20,000 $30,000 $9,000

300 $10,000 $5,000 0 $10,000 $20,000 $6,500

400 $15,000 $10,000 $5,000 0 $10,000 $8,500

500 $20,000 $15,000 $10,000 $5,000 0 $12,000

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – EVPI

41

EVwPI = 10,000 × 0.2 + 20,000 × 0.3 + 30,000 × 0.3 +
40,000 × 0.1 + 50,000 × 0.1 = 26,000

or

EVwPI = EMV + EOL = 10,000 + 16,000 = 26,000

EVPI= EVwPI- Max EMV = 26,000 − 19,500 = 6,500

EVPI = Min EOL = 6,500

➢A perfect forecast of demand is worth no more than

$6,500.

Example 3 – EVPI

42

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome

1

Outcome

2
Outcome 3 Outcome 4 Outcome 5

EMV EOL

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $16,000

200 $5,000 $20,000 $20,000 $20,000 $20,000 $17,000 $9,000

300 0 $15,000 $30,000 $30,000 $30,000 $19,500 $6,500

400 -$5,000 $10,000 $25,000 $40,000 $40,000 $17,500 $8,500

500 -$10,000 $5,000 $20,000 $35,000 $50,000 $14,000 $12,000

Probability 0.2 0.3 0.3 0.1 0.1

EVwPI

Using Excel

43

 All formulas can be implemented in Excel.

 There is a prepared Excel sheet on Avenue:

Solver-Decision Analysis Under Risk-One Stage.xlt

Practice Problems

 CHAPTER 8

 Discussion Questions 4, 5

 Problems 14, 15, 19, 20

44

limited finance test 2.5 hour

3QA3/2DA3

Management Science for

Lecture 06

C01: July 7, 2021

C02: July 8, 2021

Instructors:

Seyyed Hossein Alavi Zdravko Dimitrov

Decision Analysis

Decision Analysis

3

 We live in a world surrounded by uncertainties and risks.

 We have to make a decision, without knowing for sure which

events will occur in the future.

 We’d better consider all decision alternatives before we make a

decision.

 Decision analysis is an analytic and systematic approach to the study

of decision making.

Steps of Decision Making

4

1. Clearly define the problem

2. List ALL possible alternatives

3. Identify all possible outcomes for each alternative

4. Identify the payoff for each alternative and outcome combination

5. Use a decision modeling technique to choose an alternative

Types of Decision Making

5

 Under Certainty

 We know the consequence of all alternatives.

 Under Uncertainty

 We do not know the probability of each outcome.

 Under Risk

 We know the probability of each outcome.

Decision Making Under Uncertainty

6

 The manager can list the possible future outcomes but cannot

estimate the probability that a specific outcome will occur.

 Five different decision-making criteria can be used to make a decision:

1. Maximax

2. Maximin

3. Realism

4. Equally likely

5. Minimax regret

Drilling Example

7

 A drilling company has a land. The manager would like to consider
two alternatives:
1. Drill the land for oil

2. Sell the land

 The possible outcomes (state of nature) are:
1. Oil

2. Dry

 The payoffs are:
1. If the company finds oil: profit of $700

2. If the company cannot find oil: cost of $100

3. If the company sells the land: profit of $90

Drilling Example

8

 Payoff Table:

Alternatives

Outcomes (States of nature)

Oil Dry

Drill for oil $700 -$100

Sell the land $90 $90

1-Maximax

9

 For each alternative, find the maximum payoff over all possible outcomes. Next,

find the maximum of them. Choose the alternative whose maximum payoff gives

this maximum.

 It is choosing the alternative with the best payoff, if the best outcome happens

(optimistic!)

Alternatives

Outcomes

(States of nature) Max

Oil Dry

Drill for oil $700 -$100 700

Sell the land $90 $90 90

Max

2-Maximin

10

 For each alternative, find the minimum payoff over all possible outcomes. Next,

find the maximum of them. Chose the alternative whose minimum payoff gives

this maximum.

 It is choosing the alternative with best payoff, if the worst outcome happens

(pessimistic!)

Alternatives

Outcomes

(States of nature) Min

Oil Dry

Drill for oil $700 -$100 -100

Sell the land $90 $90 90
Max

3-Criterion of Realism

11

 Decision makers should not be optimists or pessimists. They should be realistic.

 Define Coefficient of Realism: 0 ≤ ? ≤ 1
 ? = 1 means the decision maker is totally optimistic.(Maximax)

 ? = 0 means the decision maker is totally pessimistic.(Maximin)

 Calculate the realism payoff for each alternative and select the alternative with the

highest realism payoff.

Realism Payoff = ? × ??? ?????? + 1 − ? × ??? ??????

3-Criterion of Realism

12

 Suppose ? = 0.55

 Alternative 1: 0.55 × 700 + 0.45 × −100 = 340

 Alternative 2: 0.55 × 90 + 0.45 × 90 = 90

Alternatives

Outcomes

(States of nature) Criterion

for Realism
Oil Dry

Drill for oil $700 -$100 340

Sell the land $90 $90 90

Max

3-Criterion of Realism

13

 Suppose ? = 0.15

 Alternative 1: 0.15 × 700 + 0.85 × −100 = 20

 Alternative 2: 0.15 × 90 + 0.15 × 90 = 90

Alternatives

Outcomes

(States of nature) Criterion

for Realism
Oil Dry

Drill for oil $700 -$100 20

Sell the land $90 $90 90
Max

4- Equally Likely (Laplace)

14

 Calculate the average payoff for each alternative and select the alternative with

the highest average payoff.

Average Payoff =
σ ???????

?
where ? is the number of outcomes for the alternative

4- Equally Likely (Laplace)

15

 Alternative 1:
700+(−100)

2
= 300

 Alternative 2:
90+90

2
= 90

Alternatives

Outcomes

(States of nature) Laplace

Oil Dry

Drill for oil $700 -$100 300

Sell the land $90 $90 90

Max

5- Minimax Regret

16

 Opportunity loss or regret

 Is the amount lost by not picking the best alternative.

 Minimax regret finds the alternative that minimizes the maximum regret for

each alternative.

 First find the regrets for all outcome-alternative combinations by:

best payoff for outcome − actual payoff

 Next, find the maximum regret for alternatives and choose the minimum

one among them.

5- Minimax Regret

17

Min
Alternatives

Regret of Outcomes Max

regretOil Dry

Drill for oil 700-700 = 0 90-(-100) = 190 190

Sell the land 700-90 = 610 90-90 = 0 610

Alternatives

Outcomes

(States of nature) Laplace

Oil Dry

Drill for oil $700 -$100 300

Sell the land $90 $90 90

Payoff Table

Opportunity Loss Table (Regret = best payoff for outcome − actual payoff)

Example 2

18

Alternatives

Outcomes

High

Demand

Moderate

Demand

Low

Demand

Build Large Plant 200,000 100,000 -120,000

Build Small Plant 90,000 50,000 -20,000

Do Nothing 0 0 0

1- Maximax

19

Alternatives

Outcomes

MaxHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 200,000

Build Small Plant 90,000 50,000 -20,000 90,000

Do

Nothing
0 0 0 0

2- Maximin

20

Alternatives

Outcomes

MinHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 -120,000

Build Small Plant 90,000 50,000 -20,000 -20,000

Do

Nothing
0 0 0 0

3- Criterion of Realism, ? = 0.45

21

Alternatives

Outcomes

RealismHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 24,000

Build Small Plant 90,000 50,000 -20,000 29,500

Do

Nothing
0 0 0 0

Alternative 1: 0.45 × 200,000 + 0.55 × −120,000 = 24,000

Alternative 2: 0.45 × 90,000 + 0.55 × −20,000 = 29,500

4- Equally Likely (Laplace)

22

Alternatives

Outcomes

LaplaceHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 60,000

Build Small Plant 90,000 50,000 -20,000 40,000

Do

Nothing
0 0 0 0

Alternative 1:
200,000+100,000+(−120,000)

3
= 60,000

Alternative 2:
90,000+50,000+(−20,000)

3
= 40,000

5- Minimax Regret

23

A
lt

e
r
n

a
ti

v
e

s

Regret of Outcomes

Max

High

Demand

Moderate

Demand

Low

Demand

Build

Large

Plant

200,000-200,000 = 0
100,000-100,000=

0

0-(-120,000)=

120,000
120,000

Build

Small

Plant

200,000-90,000=

110,000

100,000-0,000=

50,000

0-(-20,000) =

20,000
110,000

Do

Nothing 200,000-0= 200,000
100,000-0=

100,000
0-0=0 200,000

Using Excel

24

 All formulas can be implemented in Excel.

 There is a prepared Excel sheet on Avenue:
Solver-Decision Analysis Under Uncertainty.xlsx

Decision Making Under Risk

25

 When the decision maker has some idea about the

probabilities of outcomes.

 We use probabilities of outcomes to find the best

alternative.

1. Expected Monetary Value (EMV)

2. Expected Opportunity Loss (EOL)

3. Expected Value of Perfect Information (EVPI)

Expected Monetary Value (EMV)

26

 EMV is the weighted average of all possible payoffs,

where weights are probabilities of outcomes

 EMV for each alternative is

෍

????????

??????????? × ??????

 We would like to maximize the EMV

 Select the alternative with largest EMV

Expected Monetary Value (EMV)

27

Alternatives

Outcomes

EMVHigh

Demand

Moderate

Demand
Low Demand

1- Build Large

Plant
200,000 100,000 -120,000 86,000

2- Build Small

Plant
90,000 50,000 -20,000 40,000

3- Do

Nothing
0 0 0 0

Probability 0.3 0.5 0.2

EMV 1 = 200,000 × 0.3 + 100,000 × 0.5 − 120,000 × 0.2 = 86,000

EMV 2 = 90,000 × 0.3 + 50,000 × 0.5 − 20,000 × 0.2 = 48,000

Expected Opportunity Loss (EOL)

28

 Opportunity loss (Regret) is the difference between the optimal payoff
and the actual payoff.

 EOL is the weighted (with probabilities) average of all regrets.

 EOL is the expected cost of not picking the best solution.

 EOL for each alternative is

෍

????????

??????????? × ??????

 We would like to minimize the EOL

 Select the alternative with smallest EOL

Expected Opportunity Loss (EOL)

29

EOL 1 = 200,000 − 200,000 × 0.3 + 100,000 − 100,000 × 0.5 + (0 + 120,000) × 0.2 = 24,000

EOL 2 = (200,000-90,000) × 0.3 + 100,000 − 50,000 × 0.5 + (0 + 20,000) × 0.2 = 62,000

EOL 3 = 200,000 − 0 × 0.3 + 100,000 − 0 × 0.5 + (0 − 0) × 0.2 = 110,000

Alternatives

Regret of Outcomes

EOLHigh

Demand

Moderate

Demand
Low Demand

Build Large

Plant
200,000-200,000 100,000-100,000 0-(-120,000) 24,000

Build Small

Plant
200,000-90,000 100,000-0,000 0-(-20,000) 62,000

Do

Nothing 200,000-0 100,000-0 0-0 110,000

Probability 0.3 0.5 0.2

EMV vs EOL

30

 In EMV, we choose the alternative with maximum EMV.

 In EOL, we choose the alternative with minimum EOL.

 In fact, these two always suggest the same alternative:

The alternative suggested by Max EMV

is the same as

the alternative suggested by Min EOL

Expected Value of Perfect Information (EVPI)

31

 A company claims that they can predict what happens in the future with
certainty.

 They have the perfect information, and they want to sell it to us!

 We want to know the monetary value of this information.

 If we have perfect information, we select alternative with the best payoff.

 Expected Value with Perfect Information (EVwPI)

 Weighted average of payoffs with certainty

෍

????????

??????????? × ???? ??????

Expected Value of Perfect Information (EVPI)

32

 EVwPI is the average payoff when we have the perfect information.

 Max EMV (Min EOL) is what we can do without any information!
(expected value without perfect information)

 There for EVwPI minus Max EMV is the maximum worth of
perfect information:

EVPI = EVwPI – Max EMV

 We must not pay more than EVPI for the information.
 EVPI is the upper bound for the money that we should spend for

the information.

EVPI = Min EOL

33

Alternatives

Outcomes

EMV EOLHigh

Demand

Moderate

Demand

Low

Demand

1- Build Large

Plant
200,000 100,000 -120,000 86,000 24,000

2- Build Small

Plant
90,000 50,000 -20,000 40,000 62,000

3- Do

Nothing
0 0 0 0 110,000

Probability 0.3 0.5 0.2

EVwPI = 200,000 × 0.3 + 100,000 × 0.5 + 0 × 0.2 = 110,000

EVPI= EVwPI- Max EMV = 110,000 − 86000 = 24,000

EVPI = Min EOL = 24,000

Expected Value of Perfect Information (EVPI)

Expected Value of Perfect Information (EVPI)

34

 We have:

EVPI = EVwPI – Max EMV

EVPI = Min EOL

 Therefore:

EVwPI = Max EMV + Min EOL

 For any decision alternative:

EVwPI = EMV + EOL

35

Alternatives

Outcomes

EMV EOL EVwPI
High

Demand

Moderate

Demand

Low

Demand

1- Build

Large Plant
200,000 100,000 -120,000 86,000 24,000 110,000

2- Build

Small Plant
90,000 50,000 -20,000 40,000 62,000 110,000

3- Do

Nothing
0 0 0 0 110,000 110,000

EVwPI = EMV + EOL

Expected Value of Perfect Information (EVPI)

Expected Value of Perfect Information (EVPI)

36

EVwPI Max payoff with Perfect Information

EOL

EOL
EOL

EOL

EVPI

EMV
EMV

EMV
EMV

Alternative 1 Alternative 2 Alternative 3 Alternative 4

EVwPI = EMV + EOL

Example 3

37

 We would like to decide on the order quantities for snowboards
over next 5 months.

 Everything must go during the winter season.

 The profit is $100.
 Unsold snowboards will have a loss of $50.

 Alternatives: Order quantity of 100,200,300,400,500

Outcome

1

Outcome

2

Outcome

3

Outcome

4

Outcome

5

Demand 100 200 300 400 500

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – Payoff Table

38

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000

200 $5,000 $20,000 $20,000 $20,000 $20,000

300 0 $15,000 $30,000 $30,000 $30,000

400 -$5,000 $10,000 $25,000 $40,000 $40,000

500 -$10,000 $5,000 $20,000 $35,000 $50,000

Probability 0.2 0.3 0.3 0.1 0.1

Payoff (Profit) =$100 × 200 − $50 × 200 = $10,000 (Demand < Order Size) Payoff (Profit) =$100 × 300 = $30,000 (Demand = Order Size) Payoff (Profit) =$100 × 200 = $20,000 (Demand > Order Size)

Example 3 – EMV

39

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

EMV

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000

200 $5,000 $20,000 $20,000 $20,000 $20,000 $17,000

300 0 $15,000 $30,000 $30,000 $30,000 $19,500

400 -$5,000 $10,000 $25,000 $40,000 $40,000 $17,500

500 -$10,000 $5,000 $20,000 $35,000 $50,000 $14,000

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – EOL Table

40

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demands (regret of outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

EOL

100 200 300 400 500

100 0 $10,000 $20,000 $30,000 $40,000 $16,000

200 $5,000 0 $10,000 $20,000 $30,000 $9,000

300 $10,000 $5,000 0 $10,000 $20,000 $6,500

400 $15,000 $10,000 $5,000 0 $10,000 $8,500

500 $20,000 $15,000 $10,000 $5,000 0 $12,000

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – EVPI

41

EVwPI = 10,000 × 0.2 + 20,000 × 0.3 + 30,000 × 0.3 +
40,000 × 0.1 + 50,000 × 0.1 = 26,000

or

EVwPI = EMV + EOL = 10,000 + 16,000 = 26,000

EVPI= EVwPI- Max EMV = 26,000 − 19,500 = 6,500

EVPI = Min EOL = 6,500

➢A perfect forecast of demand is worth no more than

$6,500.

Example 3 – EVPI

42

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome

1

Outcome

2
Outcome 3 Outcome 4 Outcome 5

EMV EOL

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $16,000

200 $5,000 $20,000 $20,000 $20,000 $20,000 $17,000 $9,000

300 0 $15,000 $30,000 $30,000 $30,000 $19,500 $6,500

400 -$5,000 $10,000 $25,000 $40,000 $40,000 $17,500 $8,500

500 -$10,000 $5,000 $20,000 $35,000 $50,000 $14,000 $12,000

Probability 0.2 0.3 0.3 0.1 0.1

EVwPI

Using Excel

43

 All formulas can be implemented in Excel.

 There is a prepared Excel sheet on Avenue:

Solver-Decision Analysis Under Risk-One Stage.xlt

Practice Problems

 CHAPTER 8

 Discussion Questions 4, 5

 Problems 14, 15, 19, 20

44

limited finance test 2.5 hour

3QA3/2DA3

Management Science for

Lecture 06

C01: July 7, 2021

C02: July 8, 2021

Instructors:

Seyyed Hossein Alavi Zdravko Dimitrov

Decision Analysis

Decision Analysis

3

 We live in a world surrounded by uncertainties and risks.

 We have to make a decision, without knowing for sure which

events will occur in the future.

 We’d better consider all decision alternatives before we make a

decision.

 Decision analysis is an analytic and systematic approach to the study

of decision making.

Steps of Decision Making

4

1. Clearly define the problem

2. List ALL possible alternatives

3. Identify all possible outcomes for each alternative

4. Identify the payoff for each alternative and outcome combination

5. Use a decision modeling technique to choose an alternative

Types of Decision Making

5

 Under Certainty

 We know the consequence of all alternatives.

 Under Uncertainty

 We do not know the probability of each outcome.

 Under Risk

 We know the probability of each outcome.

Decision Making Under Uncertainty

6

 The manager can list the possible future outcomes but cannot

estimate the probability that a specific outcome will occur.

 Five different decision-making criteria can be used to make a decision:

1. Maximax

2. Maximin

3. Realism

4. Equally likely

5. Minimax regret

Drilling Example

7

 A drilling company has a land. The manager would like to consider
two alternatives:
1. Drill the land for oil

2. Sell the land

 The possible outcomes (state of nature) are:
1. Oil

2. Dry

 The payoffs are:
1. If the company finds oil: profit of $700

2. If the company cannot find oil: cost of $100

3. If the company sells the land: profit of $90

Drilling Example

8

 Payoff Table:

Alternatives

Outcomes (States of nature)

Oil Dry

Drill for oil $700 -$100

Sell the land $90 $90

1-Maximax

9

 For each alternative, find the maximum payoff over all possible outcomes. Next,

find the maximum of them. Choose the alternative whose maximum payoff gives

this maximum.

 It is choosing the alternative with the best payoff, if the best outcome happens

(optimistic!)

Alternatives

Outcomes

(States of nature) Max

Oil Dry

Drill for oil $700 -$100 700

Sell the land $90 $90 90

Max

2-Maximin

10

 For each alternative, find the minimum payoff over all possible outcomes. Next,

find the maximum of them. Chose the alternative whose minimum payoff gives

this maximum.

 It is choosing the alternative with best payoff, if the worst outcome happens

(pessimistic!)

Alternatives

Outcomes

(States of nature) Min

Oil Dry

Drill for oil $700 -$100 -100

Sell the land $90 $90 90
Max

3-Criterion of Realism

11

 Decision makers should not be optimists or pessimists. They should be realistic.

 Define Coefficient of Realism: 0 ≤ ? ≤ 1
 ? = 1 means the decision maker is totally optimistic.(Maximax)

 ? = 0 means the decision maker is totally pessimistic.(Maximin)

 Calculate the realism payoff for each alternative and select the alternative with the

highest realism payoff.

Realism Payoff = ? × ??? ?????? + 1 − ? × ??? ??????

3-Criterion of Realism

12

 Suppose ? = 0.55

 Alternative 1: 0.55 × 700 + 0.45 × −100 = 340

 Alternative 2: 0.55 × 90 + 0.45 × 90 = 90

Alternatives

Outcomes

(States of nature) Criterion

for Realism
Oil Dry

Drill for oil $700 -$100 340

Sell the land $90 $90 90

Max

3-Criterion of Realism

13

 Suppose ? = 0.15

 Alternative 1: 0.15 × 700 + 0.85 × −100 = 20

 Alternative 2: 0.15 × 90 + 0.15 × 90 = 90

Alternatives

Outcomes

(States of nature) Criterion

for Realism
Oil Dry

Drill for oil $700 -$100 20

Sell the land $90 $90 90
Max

4- Equally Likely (Laplace)

14

 Calculate the average payoff for each alternative and select the alternative with

the highest average payoff.

Average Payoff =
σ ???????

?
where ? is the number of outcomes for the alternative

4- Equally Likely (Laplace)

15

 Alternative 1:
700+(−100)

2
= 300

 Alternative 2:
90+90

2
= 90

Alternatives

Outcomes

(States of nature) Laplace

Oil Dry

Drill for oil $700 -$100 300

Sell the land $90 $90 90

Max

5- Minimax Regret

16

 Opportunity loss or regret

 Is the amount lost by not picking the best alternative.

 Minimax regret finds the alternative that minimizes the maximum regret for

each alternative.

 First find the regrets for all outcome-alternative combinations by:

best payoff for outcome − actual payoff

 Next, find the maximum regret for alternatives and choose the minimum

one among them.

5- Minimax Regret

17

Min
Alternatives

Regret of Outcomes Max

regretOil Dry

Drill for oil 700-700 = 0 90-(-100) = 190 190

Sell the land 700-90 = 610 90-90 = 0 610

Alternatives

Outcomes

(States of nature) Laplace

Oil Dry

Drill for oil $700 -$100 300

Sell the land $90 $90 90

Payoff Table

Opportunity Loss Table (Regret = best payoff for outcome − actual payoff)

Example 2

18

Alternatives

Outcomes

High

Demand

Moderate

Demand

Low

Demand

Build Large Plant 200,000 100,000 -120,000

Build Small Plant 90,000 50,000 -20,000

Do Nothing 0 0 0

1- Maximax

19

Alternatives

Outcomes

MaxHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 200,000

Build Small Plant 90,000 50,000 -20,000 90,000

Do

Nothing
0 0 0 0

2- Maximin

20

Alternatives

Outcomes

MinHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 -120,000

Build Small Plant 90,000 50,000 -20,000 -20,000

Do

Nothing
0 0 0 0

3- Criterion of Realism, ? = 0.45

21

Alternatives

Outcomes

RealismHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 24,000

Build Small Plant 90,000 50,000 -20,000 29,500

Do

Nothing
0 0 0 0

Alternative 1: 0.45 × 200,000 + 0.55 × −120,000 = 24,000

Alternative 2: 0.45 × 90,000 + 0.55 × −20,000 = 29,500

4- Equally Likely (Laplace)

22

Alternatives

Outcomes

LaplaceHigh

Demand

Moderate

Demand
Low Demand

Build Large Plant 200,000 100,000 -120,000 60,000

Build Small Plant 90,000 50,000 -20,000 40,000

Do

Nothing
0 0 0 0

Alternative 1:
200,000+100,000+(−120,000)

3
= 60,000

Alternative 2:
90,000+50,000+(−20,000)

3
= 40,000

5- Minimax Regret

23

A
lt

e
r
n

a
ti

v
e

s

Regret of Outcomes

Max

High

Demand

Moderate

Demand

Low

Demand

Build

Large

Plant

200,000-200,000 = 0
100,000-100,000=

0

0-(-120,000)=

120,000
120,000

Build

Small

Plant

200,000-90,000=

110,000

100,000-0,000=

50,000

0-(-20,000) =

20,000
110,000

Do

Nothing 200,000-0= 200,000
100,000-0=

100,000
0-0=0 200,000

Using Excel

24

 All formulas can be implemented in Excel.

 There is a prepared Excel sheet on Avenue:
Solver-Decision Analysis Under Uncertainty.xlsx

Decision Making Under Risk

25

 When the decision maker has some idea about the

probabilities of outcomes.

 We use probabilities of outcomes to find the best

alternative.

1. Expected Monetary Value (EMV)

2. Expected Opportunity Loss (EOL)

3. Expected Value of Perfect Information (EVPI)

Expected Monetary Value (EMV)

26

 EMV is the weighted average of all possible payoffs,

where weights are probabilities of outcomes

 EMV for each alternative is

෍

????????

??????????? × ??????

 We would like to maximize the EMV

 Select the alternative with largest EMV

Expected Monetary Value (EMV)

27

Alternatives

Outcomes

EMVHigh

Demand

Moderate

Demand
Low Demand

1- Build Large

Plant
200,000 100,000 -120,000 86,000

2- Build Small

Plant
90,000 50,000 -20,000 40,000

3- Do

Nothing
0 0 0 0

Probability 0.3 0.5 0.2

EMV 1 = 200,000 × 0.3 + 100,000 × 0.5 − 120,000 × 0.2 = 86,000

EMV 2 = 90,000 × 0.3 + 50,000 × 0.5 − 20,000 × 0.2 = 48,000

Expected Opportunity Loss (EOL)

28

 Opportunity loss (Regret) is the difference between the optimal payoff
and the actual payoff.

 EOL is the weighted (with probabilities) average of all regrets.

 EOL is the expected cost of not picking the best solution.

 EOL for each alternative is

෍

????????

??????????? × ??????

 We would like to minimize the EOL

 Select the alternative with smallest EOL

Expected Opportunity Loss (EOL)

29

EOL 1 = 200,000 − 200,000 × 0.3 + 100,000 − 100,000 × 0.5 + (0 + 120,000) × 0.2 = 24,000

EOL 2 = (200,000-90,000) × 0.3 + 100,000 − 50,000 × 0.5 + (0 + 20,000) × 0.2 = 62,000

EOL 3 = 200,000 − 0 × 0.3 + 100,000 − 0 × 0.5 + (0 − 0) × 0.2 = 110,000

Alternatives

Regret of Outcomes

EOLHigh

Demand

Moderate

Demand
Low Demand

Build Large

Plant
200,000-200,000 100,000-100,000 0-(-120,000) 24,000

Build Small

Plant
200,000-90,000 100,000-0,000 0-(-20,000) 62,000

Do

Nothing 200,000-0 100,000-0 0-0 110,000

Probability 0.3 0.5 0.2

EMV vs EOL

30

 In EMV, we choose the alternative with maximum EMV.

 In EOL, we choose the alternative with minimum EOL.

 In fact, these two always suggest the same alternative:

The alternative suggested by Max EMV

is the same as

the alternative suggested by Min EOL

Expected Value of Perfect Information (EVPI)

31

 A company claims that they can predict what happens in the future with
certainty.

 They have the perfect information, and they want to sell it to us!

 We want to know the monetary value of this information.

 If we have perfect information, we select alternative with the best payoff.

 Expected Value with Perfect Information (EVwPI)

 Weighted average of payoffs with certainty

෍

????????

??????????? × ???? ??????

Expected Value of Perfect Information (EVPI)

32

 EVwPI is the average payoff when we have the perfect information.

 Max EMV (Min EOL) is what we can do without any information!
(expected value without perfect information)

 There for EVwPI minus Max EMV is the maximum worth of
perfect information:

EVPI = EVwPI – Max EMV

 We must not pay more than EVPI for the information.
 EVPI is the upper bound for the money that we should spend for

the information.

EVPI = Min EOL

33

Alternatives

Outcomes

EMV EOLHigh

Demand

Moderate

Demand

Low

Demand

1- Build Large

Plant
200,000 100,000 -120,000 86,000 24,000

2- Build Small

Plant
90,000 50,000 -20,000 40,000 62,000

3- Do

Nothing
0 0 0 0 110,000

Probability 0.3 0.5 0.2

EVwPI = 200,000 × 0.3 + 100,000 × 0.5 + 0 × 0.2 = 110,000

EVPI= EVwPI- Max EMV = 110,000 − 86000 = 24,000

EVPI = Min EOL = 24,000

Expected Value of Perfect Information (EVPI)

Expected Value of Perfect Information (EVPI)

34

 We have:

EVPI = EVwPI – Max EMV

EVPI = Min EOL

 Therefore:

EVwPI = Max EMV + Min EOL

 For any decision alternative:

EVwPI = EMV + EOL

35

Alternatives

Outcomes

EMV EOL EVwPI
High

Demand

Moderate

Demand

Low

Demand

1- Build

Large Plant
200,000 100,000 -120,000 86,000 24,000 110,000

2- Build

Small Plant
90,000 50,000 -20,000 40,000 62,000 110,000

3- Do

Nothing
0 0 0 0 110,000 110,000

EVwPI = EMV + EOL

Expected Value of Perfect Information (EVPI)

Expected Value of Perfect Information (EVPI)

36

EVwPI Max payoff with Perfect Information

EOL

EOL
EOL

EOL

EVPI

EMV
EMV

EMV
EMV

Alternative 1 Alternative 2 Alternative 3 Alternative 4

EVwPI = EMV + EOL

Example 3

37

 We would like to decide on the order quantities for snowboards
over next 5 months.

 Everything must go during the winter season.

 The profit is $100.
 Unsold snowboards will have a loss of $50.

 Alternatives: Order quantity of 100,200,300,400,500

Outcome

1

Outcome

2

Outcome

3

Outcome

4

Outcome

5

Demand 100 200 300 400 500

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – Payoff Table

38

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000

200 $5,000 $20,000 $20,000 $20,000 $20,000

300 0 $15,000 $30,000 $30,000 $30,000

400 -$5,000 $10,000 $25,000 $40,000 $40,000

500 -$10,000 $5,000 $20,000 $35,000 $50,000

Probability 0.2 0.3 0.3 0.1 0.1

Payoff (Profit) =$100 × 200 − $50 × 200 = $10,000 (Demand < Order Size) Payoff (Profit) =$100 × 300 = $30,000 (Demand = Order Size) Payoff (Profit) =$100 × 200 = $20,000 (Demand > Order Size)

Example 3 – EMV

39

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

EMV

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000

200 $5,000 $20,000 $20,000 $20,000 $20,000 $17,000

300 0 $15,000 $30,000 $30,000 $30,000 $19,500

400 -$5,000 $10,000 $25,000 $40,000 $40,000 $17,500

500 -$10,000 $5,000 $20,000 $35,000 $50,000 $14,000

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – EOL Table

40

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demands (regret of outcomes)

Outcome 1 Outcome 2 Outcome 3 Outcome 4 Outcome 5

EOL

100 200 300 400 500

100 0 $10,000 $20,000 $30,000 $40,000 $16,000

200 $5,000 0 $10,000 $20,000 $30,000 $9,000

300 $10,000 $5,000 0 $10,000 $20,000 $6,500

400 $15,000 $10,000 $5,000 0 $10,000 $8,500

500 $20,000 $15,000 $10,000 $5,000 0 $12,000

Probability 0.2 0.3 0.3 0.1 0.1

Example 3 – EVPI

41

EVwPI = 10,000 × 0.2 + 20,000 × 0.3 + 30,000 × 0.3 +
40,000 × 0.1 + 50,000 × 0.1 = 26,000

or

EVwPI = EMV + EOL = 10,000 + 16,000 = 26,000

EVPI= EVwPI- Max EMV = 26,000 − 19,500 = 6,500

EVPI = Min EOL = 6,500

➢A perfect forecast of demand is worth no more than

$6,500.

Example 3 – EVPI

42

O
rd

e
r

S
iz

e

(A
lt

e
r
n

a
ti

v
e

s) Demand (outcomes)

Outcome

1

Outcome

2
Outcome 3 Outcome 4 Outcome 5

EMV EOL

100 200 300 400 500

100 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $16,000

200 $5,000 $20,000 $20,000 $20,000 $20,000 $17,000 $9,000

300 0 $15,000 $30,000 $30,000 $30,000 $19,500 $6,500

400 -$5,000 $10,000 $25,000 $40,000 $40,000 $17,500 $8,500

500 -$10,000 $5,000 $20,000 $35,000 $50,000 $14,000 $12,000

Probability 0.2 0.3 0.3 0.1 0.1

EVwPI

Using Excel

43

 All formulas can be implemented in Excel.

 There is a prepared Excel sheet on Avenue:

Solver-Decision Analysis Under Risk-One Stage.xlt

Practice Problems

 CHAPTER 8

 Discussion Questions 4, 5

 Problems 14, 15, 19, 20

44