Slides prepared by JOHN LOUCKS St. Edward’s University

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© 2006 Thomson South-Western. All Rights Reserved.

Slidesprepared by

JOHN LOUCKS

St. Edward’sUniversity

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Chapter 4Decision Analysis

Problem Formulation Decision Making without Probabilities Decision Making with Probabilities Risk Analysis and Sensitivity Analysis Decision Analysis with Sample Information Computing Branch Probabilities

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Problem Formulation The first step in the decision analysis process

is problem formulation. We begin with a verbal statement of the

problem. Then we identify:• the decision alternatives• the states of nature (uncertain future

events)• the payoff (consequences) associated with

each specific combination of:• decision alternative• state of nature

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Burger Prince Restaurant is considering

opening a new restaurant on Main Street.

The company has three differentbuilding designs (A, B, and C), eachwith a different seating capacity.

Burger Prince estimates that theaverage number of customers

arrivingper hour will be 40, 60, or 80.

Example: Burger Prince Restaurant

Problem Formulation

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Problem Formulation

d1 = use building design Ad2 = use building design Bd3 = use building design C

Decision Alternatives

States of Natures1 = an average of 40 customers arriving per hour

s2 = an average of 60 customers arriving per hours3 = an average of 80 customers arriving per hour

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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.

Problem Formulation Payoff

Table

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Payoff Table (Payoffs are Profit Per Week)

$ 6,000 $16,000 $21,000$ 8,000 $18,000 $12,000$10,000 $15,000 $14,000

Average Number of Customers Per

Hours1 = 40 s2 = 60 s3 = 80

Design ADesign BDesign C

Problem Formulation

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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.

Influence Diagram

Problem Formulation

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RestaurantDesign Profit

AverageNumber ofCustomersPer Hour

Influence Diagram

Problem Formulation

Design A (d1)Design B (d2)Design C (d3)

DecisionAlternatives

40 customers per hour (s1)60 customers per hour (s2)80 customers per hour (s3)

States of Nature

ConsequenceProfit

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

A decision tree has two types of nodes:• round nodes correspond to chance

events• square nodes correspond to decisions

Branches leaving a round node represent the different states of nature; branches leaving a square node represent the different decision alternatives.

At the end of a limb of the tree is the payoff attained from the series of branches making up the limb.

Problem Formulation Decision

Tree

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1

10,00015,000 14,000 8,000 18,000 12,000 6,000 16,000 21,000

2

3

4

Problem Formulation

40 customers per hour (s1)

Design A (d1)

Design B (d2)

Design C (d3)

Decision Tree

60 customers per hour (s2)80 customers per hour (s3)



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Decision Making without Probabilities

Three commonly used criteria for decisionmaking when probability information regarding thelikelihood of the states of nature is unavailable are:

• the optimistic (maximax) approach• the conservative (maximin) approach• the minimax regret approach.

Criteria for Decision Making

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

The decision with the overall largest payoff is chosen.

If the payoff table is in terms of costs, the decision with the overall lowest cost will be chosen (hence, a minimin approach).

Optimistic (Maximax) Approach


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The decision that has the largest single value in the payoff table is chosen.

Maximax

payoff

Maximaxdecisio

n

Optimistic (Maximax) Approach


States of Nature Decision (Customers Per Hour)Alternative 40 s1 60 s2 80 s3

Design A d1 10,000 15,000 14,000Design B d2 8,000 18,000 12,000Design C d3 6,000 16,000 21,000

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

For each decision the minimum payoff is listed.

The decision corresponding to the maximum of these minimum payoffs is selected.

If payoffs are in terms of costs, the maximum costs will be determined for each decision and then the decision corresponding to the minimum of these maximum costs will be selected. (Hence, a minimax approach)

Decision Making without Probabilities Conservative (Maximin)

Approach

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

Conservative (Maximin) Approach


Maximinpayoff

Maximindecisio

n Decision MinimumAlternative PayoffDesign A d1 10,000Design B d2 8,000Design C d3 6,000

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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 corresponding to the minimum of the maximum regrets is chosen.

Decision Making without Probabilities Minimax Regret Approach

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First compute a regret table by subtracting each payoff in a column from the largest payoff in that column. The resulting regret table is:


States of Nature Decision (Customers Per Hour)Alternative 40 s1 60 s2 80 s3

Design A d1 0 3,000 7,000Design B d2 2,000 0 9,000Design C d3 4,000 2,000 0

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For each decision list the maximum regret. Choose the decision with the minimum of these values.


Minimaxregret

Minimaxdecisio

n

Decision MaximumAlternative RegretDesign A d1 7,000Design B d2 9,000Design C d3 4,000

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Decision Making with Probabilities

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( ) ( ) ( ) ( ) 1N

j Nj

P s P s P s P s

Once we have defined the decision alternatives and states of nature for the chance events, we focus on determining probabilities for the states of nature. The classical, relative frequency, or subjective method of assigning probabilities may be used. Because only one of the N states of nature can occur, the probabilities must satisfy two conditions:

P(sj) > 0 for all states of nature

Assigning Probabilities

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Decision Making with Probabilities

We use the expected value approach to identify the best or recommended decision alternative. The expected value of each decision alternative is calculated (explained on the next slide). The decision alternative yielding the best expected value is chosen.

Expected Value Approach

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EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1

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

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

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

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Expected Value Approach Calculate the expected value (EV) for

each decision. The decision tree on the next slide can assist

in this calculation. Here d1, d2, d3 represent the decision

alternatives of Designs A, B, and C. And s1, s2, s3 represent the states of nature of

40, 60, and 80 customers per hour. The decision alternative with the greatest EV

is the optimal decision.

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1

10,00015,000 14,000 8,000 18,000 12,000 6,000 16,000 21,000

2

3

4

Design A (d1)

Design B (d2)

Design C (d3)

Decision Tree 40 customers (s1) P(s1) = .4

60 customers (s2) P(s2) = .280 customers (s3) P(s3) = .440 customers (s1) P(s1) = .460 customers (s2) P(s2) = .280 customers (s3) P(s3) = .440 customers (s1) P(s1) = .460 customers (s2) P(s2) = .280 customers (s3) P(s3) = .4

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EV(d1) = .4(10,000) + .2(15,000) + .4(14,000) = $12,600

EV(d2) = .4(8,000) + .2(18,000) + .4(12,000) = $11,600

EV(d3) = .4(6,000) + .2(16,000) + .4(21,000) = $14,000

Design A d1

Design B d2

Design C d3

2

1

4

Choose the decision alternative with the largest EV:Design C

3

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Expected Value of Perfect Information Frequently information is available which can

improve the probability estimates for the states of nature.

The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur.

The EVPI provides an upper bound on the expected value of any sample or survey information.

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Expected value of perfect information is defined as

Expected Value of Perfect Information

EVPI = |EVwPI – EVwoPI|

EVPI = expected value of perfect information EVwPI = expected value with perfect information about the states of natureEVwoPI = expected value without perfect

information about the states of nature

where:

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Expected Value of Perfect Information EVPI Calculation

• Step 1:Determine the optimal return

corresponding to each state of nature.• Step 2:

Compute the expected value of these optimal returns.

• Step 3:Subtract the EV of the optimal

decision from the amount determined in step (2).

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Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

Expected Value of Perfect Information EVPI Calculation

EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000

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Risk Analysis Risk analysis helps the decision maker

recognize the difference between:• the expected value of a decision alternative,

and• the payoff that might actually occur

The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.

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Risk Analysis Risk Profile for Decision Alternative

d3

.10

.20

.30

.40

.50

5 10 15 20 25

Prob

abilit

y

Profit ($ thousands)

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Sensitivity Analysis Sensitivity analysis can be used to determine

how changes to the following inputs affect the recommended decision alternative:• probabilities for the states of nature• values of the payoffs

If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value.

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Sensitivity Analysis

EV(d1) .35 .30 .35.333 .333 .333 .40 .30 .30 .50 .25 .25 .60 .20 .20 .25 .50 .25 .20 .60 .20 .15 .70 .15

12,40012,65412,20011,50010,80014,00014,80015,600

P(s1)

Resolving Using Different Values for the

Probabilities of the States of Nature

12,90012,98712,70012,25011,80013,50013,80014,100

14,25014,31913,50012,25011,00014,75015,00015,250

P(s2) P(s3) EV(d2) EV(d3)d3

d3

d3

d1 and d3

d1

d3

d3

d2

Optimal d

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Decision Analysis With Sample Information

Knowledge of sample (survey) information can be used to revise the probability estimates for the states of nature.

Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities.

With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem.

The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.

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Burger Prince must decide whether to purchase a

marketing survey from Stanton Marketing for$1,000. The results of the survey are

"favorable“or "unfavorable". The branch probabilitiescorresponding to all the chance nodesare listed on the next slide.


Example: Burger Prince

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Branch Probabilities

P(80 customers per hour | unfavorable) = .087P(60 customers per hour | unfavorable) = .217P(40 customers per hour | unfavorable) = .696

P(80 customers per hour | favorable) = .667P(60 customers per hour | favorable) = .185P(40 customers per hour | favorable) = .148

P(Unfavorable market survey) = .46P(Favorable market survey) = .54

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Restaurant Design(seating capacity) Profit

Avg. Numberof Customers

Per Hour

MarketSurveyResults

MarketSurvey

DecisionChanceConsequence


Influence Diagram

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• A decision strategy is a sequence of decisions and chance outcomes.

• The sequence of decisions chosen depends on the yet to be determined outcomes of chance events.

• The EVwSI is calculated by making a backward pass through the decision tree.


Decision Strategy

• The optimal decision strategy is based on the Expected Value with Sample Information (EVwSI).

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• Step 1:Determine the optimal decision strategy and its expected returns for the possible outcomes of the sample using the posterior probabilities for the states of nature.

• Step 2: Compute the expected value of these optimal returns.

Expected Value with Sample Information (EVwSI)


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I2(.46)

d1

d2

d3

EV = .696(10,000) + .217(15,000) +.087(14,000) = $11,433

EV = .696(8,000) + .217(18,000) + .087(12,000) = $10,554

EV = .696(6,000) + .217(16,000) +.087(21,000) = $9,475

I1(.54)

d1

d2

d3

EV = .148(10,000) + .185(15,000) + .667(14,000) = $13,593

EV = .148 (8,000) + .185(18,000) + .667(12,000) = $12,518

EV = .148(6,000) + .185(16,000) +.667(21,000) = $17,855

4

5

6

7

8

9

2

3

1

$17,855

$11,433


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I2(.46)

d1

d2

d3

$11,433

$10,554

$ 9,475

I1(.54)

d1

d2

d3

$13,593

$12,518

$17,855

4

5

6

7

8

9

2

3

1

$17,855

$11,433


EVwSI = .54(17,855) + .46(11,433)= $14,900.88

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If the outcome of the survey is "favorable”,choose Design C. If the outcome of the survey is “unfavorable”, choose Design A.

EVwSI = .54($17,855) + .46($11,433) = $14,900.88


Expected Value with Sample Information (EVwSI)

Optimal Decision Strategy

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PayoffI1 = .54I1 = .54I1 = .54I2 =.46I2 =.46I2 =.46

$ 6,000$16,000$21,000$10,000$15,000$14,000

.08 .10 .36 .32 .10 .04----------------

1.00

Risk Profile for Optimal Decision Strategy

( )P I sP(I)P(s1|I1) = .148P(s2|I1) = .185P(s3|I1) = .667P(s1|I2) = .696P(s2|I2) = .217P(s3|I2) = .087

P(s|I)

Payoffsfor d3

Payoffsfor d1

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Risk Profile for Optimal Decision Strategy

.10

.20

.30

.40

.50

5 10 15 20 25

Prob

abilit

y

Profit ($ thousands)

.08

.32

.04

.36

.10.10

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Expected Value of Sample Information The expected value of sample information

(EVSI) is the additional expected profit possible through knowledge of the sample or survey information.

EVSI = |EVwSI – EVwoSI|

EVSI = expected value of sample information EVwSI = expected value with sample information about the states of natureEVwoSI = expected value without sample

information about the states of nature

where:

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Expected Value of Sample Information

Subtract the EVwoSI (the value of the optimal decision obtained without using the sample information) from the EVwSI.

EVSI Calculation

EVSI = $14,900.88 - $14,000 = $900.88

ConclusionBecause the EVSI ($900.88) is less than the cost of the survey ($1000.00), the survey should not be purchased.

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Efficiency of Sample Information The efficiency rating (E) of sample

information is the ratio of EVSI to EVPI expressed as a percent.

The efficiency rating (E) of the market survey for Burger Prince Restaurant is:

EVSIE 100EVPI

900.88E 100 45%2000.00

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Bayes’ Theorem can be used to compute branch probabilities for decision trees.

For the computations we need to know:• the initial (prior) probabilities for the states

of nature,• the conditional probabilities for the

outcomes or indicators of the sample information, given each state of nature.

A tabular approach is a convenient method for carrying out the computations.

Computing Branch Probabilities

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Step 1For each state of nature, multiply the prior probabilityby its conditional probability for the indicator. Thisgives the joint probabilities for the states and indicator.

Step 2Sum these joint probabilities over all states. This givesthe marginal probability for the indicator.

Step 3For each state, divide its joint probability by themarginal probability for the indicator. This gives theposterior probability distribution.


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Recall that Burger Prince is considering purchasing

a marketing survey from Stanton Marketing. Theresults of the survey are "favorable“ or

"unfavorable".The conditional probabilities are:

Example: Burger Prince Restaurant

P(favorable |80 customers per hour) = .9P(favorable |60 customers per hour) = .5P(favorable |40 customers per hour) = .2

Compute the branch (posterior) probabilitydistribution.


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State Prior Conditional Joint PosteriorFavorable

.08

.10

.36

.148

.185

.6671.000

406080

.4 .2 .4

.2 .5 .9

Posterior Probability Distribution


P(favorable) = .54

Total .54

.08/.54

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Total .46

State Prior Conditional Joint PosteriorUnfavorable

.32

.10

.04

.696

.217

.0871.000

406080

.4 .2 .4

.8 .5 .1

Posterior Probability Distribution


P(unfavorable) = .46

.32/.46

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End of Chapter 4

Slides prepared by JOHN LOUCKS St. Edward’s University

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