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© 2001 South-Western College Publishing/Thomson Learning

Anderson Sweeney Williams

Anderson Sweeney Williams

Slides Prepared by JOHN LOUCKS

QUANTITATIVE METHODS FORBUSINESS 8e

QUANTITATIVE METHODS FORBUSINESS 8e

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Chapter 4Decision Analysis, Part A

Problem Formulation Decision Making without Probabilities Decision Making with Probabilities

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Problem Formulation

A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.

The decision alternatives are the different possible strategies the decision maker can employ.

The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

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Influence Diagram

An influence diagram is a graphical device showing the relationships among the decisions, the chance events, and the consequences.

Squares or rectangles depict decision nodes. Circles or ovals depict chance nodes. Diamonds depict consequence nodes. Lines or arcs connecting the nodes show the

direction of influence.

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Payoff Tables

The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff.

A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table.

Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.

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Decision Trees

A decision tree is a chronological representation of the decision problem.

Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives.

The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.

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Decision Making without Probabilities

Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: • the optimistic approach• the conservative approach• the minimax regret approach.

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Optimistic Approach

The optimistic approach would be used by an optimistic decision maker.

The decision with the largest possible payoff is chosen.

If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

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Conservative Approach

The conservative approach would be used by a conservative decision maker.

For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.)

If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)

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Minimax Regret Approach

The minimax regret approach requires the construction of a regret table or an opportunity loss table.

This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature.

Then, using this regret table, the maximum regret for each possible decision is listed.

The decision chosen is the one corresponding to the minimum of the maximum regrets.

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Example

Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits:

States of Nature s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1

d3 1 5 -3

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Example

Optimistic Approach An optimistic decision maker would use the

optimistic approach. All we really need to do is to choose the decision that has the largest single value in the payoff table. This largest value is 5, and hence the optimal decision is d3.

Maximum Decision Payoff

d1 4

d2 3

choose d3 d3 5 maximum

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Example

Formula Spreadsheet for Optimistic Approach

A B C D E F123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MAX(B5:D5) =IF(E5=$E$9,A5,"")6 d2 0 3 -1 =MAX(B6:D6) =IF(E6=$E$9,A6,"")7 d3 1 5 -3 =MAX(B7:D7) =IF(E7=$E$9,A7,"")89 =MAX(E5:E7)

State of Nature

Best Payoff

PAYOFF TABLE

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Example

Solution Spreadsheet for Optimistic Approach

A B C D E F123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 46 d2 0 3 -1 37 d3 1 5 -3 5 d389 5

State of Nature

Best Payoff

PAYOFF TABLE

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Example

Conservative ApproachA conservative decision maker would use

the conservative approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.

Minimum Decision Payoff

d1 -2

choose d2 d2 -1 maximum

d3 -3

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Example

Formula Spreadsheet for Conservative Approach

A B C D E F123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MIN(B5:D5) =IF(E5=$E$9,A5,"")6 d2 0 3 -1 =MIN(B6:D6) =IF(E6=$E$9,A6,"")7 d3 1 5 -3 =MIN(B7:D7) =IF(E7=$E$9,A7,"")89 =MAX(E5:E7)

State of Nature

Best Payoff

PAYOFF TABLE

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Example

Solution Spreadsheet for Conservative Approach

A B C D E F123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 -26 d2 0 3 -1 -1 d27 d3 1 5 -3 -389 -1

State of Nature

Best Payoff

PAYOFF TABLE

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Example

Minimax Regret ApproachFor the minimax regret approach, first

compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from 4; in the second column, subtract 4, 3, and 5 from 5; etc. The resulting regret table is:

s1 s2 s3

d1 0 1 1

d2 4 2 0

d3 3 0 2

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Example

Minimax Regret Approach (continued)For each decision list the maximum regret.

Choose the decision with the minimum of these values.

Decision Maximum Regret choose d1 d1 1

minimum d2 4

d3 3

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Example

Formula Spreadsheet for Minimax Regret ApproachA B C D E F

12 Decision 3 Altern. s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Altern. s1 s2 s3 Regret Decision11 d1 =MAX($B$4:$B$6)-B4 =MAX($C$4:$C$6)-C4 =MAX($D$4:$D$6)-D4 =MAX(B11:D11) =IF(E11=$E$14,A11,"")12 d2 =MAX($B$4:$B$6)-B5 =MAX($C$4:$C$6)-C5 =MAX($D$4:$D$6)-D5 =MAX(B12:D12) =IF(E12=$E$14,A12,"")13 d3 =MAX($B$4:$B$6)-B6 =MAX($C$4:$C$6)-C6 =MAX($D$4:$D$6)-D6 =MAX(B13:D13) =IF(E13=$E$14,A13,"")14 =MIN(E11:E13)Minimax Regret Value

State of NaturePAYOFF TABLE

State of NatureOPPORTUNITY LOSS TABLE

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Example

Solution Spreadsheet for Minimax Regret Approach1

2 Decision 3 Alternative s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Alternative s1 s2 s3 Regret Decision11 d1 0 1 1 1 d112 d2 4 2 0 413 d3 3 0 2 314 1Minimax Regret Value

State of NaturePAYOFF TABLE

State of NatureOPPORTUNITY LOSS TABLE

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Decision Making with Probabilities

Expected Value Approach• If probabilistic information regarding he

states of nature is available, one may use the expected value (EV) approach.

• Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring.

• The decision yielding the best expected return is chosen.

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Expected Value of a Decision Alternative

The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative.

The expected value (EV) of decision alternative di is defined as:

where: N = the number of states of nature P(sj ) = the probability of state of

nature sj

Vij = the payoff corresponding to decision alternative di and state of nature sj

EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1

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Example: Burger Prince

Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table for the three models is as follows:

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000

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Example: Burger Prince

Expected Value ApproachCalculate the expected value for each

decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3 represent the decision alternatives of models A, B, C, and s1, s2, s3 represent the states of nature of 80, 100, and 120.

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Example: Burger Prince

Decision Tree

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.2

.4

.4

.4

.2

.4

.4

.2

.4

d1

d2

d3

s1

s1

s1

s2

s3

s2

s2

s3

s3

Payoffs

10,000

15,000

14,000

8,000

18,000

12,000

6,000

16,000

21,000

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33

44

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Example: Burger Prince

Expected Value For Each Decision

Choose the model with largest EV, Model C.

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d1

d2

d3

EMV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000

Model A

Model B

Model C

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Example: Burger Prince

Formula Spreadsheet for Expected Value Approach

A B C D E F123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 Model A 10,000 15,000 14,000 =$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"")6 Model B 8,000 18,000 12,000 =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"")7 Model C 6,000 16,000 21,000 =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"")8 Probability 0.4 0.2 0.49 =MAX(E5:E7)

State of Nature

Maximum Expected Value

PAYOFF TABLE

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Solution Spreadsheet for Expected Value Approach

Example: Burger Prince

A B C D E F123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 Model A 10,000 15,000 14,000 126006 Model B 8,000 18,000 12,000 116007 Model C 6,000 16,000 21,000 14000 Model C8 Probability 0.4 0.2 0.49 14000

State of Nature

Maximum Expected Value

PAYOFF TABLE

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The End of Chapter 4, Part A