Decision Making in Operations Management

7/27/2019 Decision Making in Operations Management

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Slide 2005 Thomson/South-Western

Chapter 13Decision Analysis

Problem Formulation Decision Making without

Probabilities

Decision Making with Probabilities Risk Analysis and Sensitivity

Analysis

Decision Analysis with SampleInformation

Computing Branch Probabilities


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Problem Formulation

A decision problem is characterized bydecision alternatives, states of nature, andresulting payoffs.

The decision alternatives are the different

possible strategies the decision maker canemploy.

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


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Influence Diagrams

An influence diagram is a graphical deviceshowing the relationships among thedecisions, the chance events, and theconsequences.

Squares or rectangles depict decision nodes. Circles or ovals depict chance nodes.

Diamonds depict consequence nodes.

Lines or arcs connecting the nodes show thedirection of influence.


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Payoff Tables

The consequence resulting from a specificcombination of a decision alternative and astate of nature is a payoff.

A table showing payoffs for all combinations

of decision alternatives and states of nature isa payoff table.

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


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Decision Trees

A decision tree is a chronologicalrepresentation of the decision problem.

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


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The branches leaving each round noderepresent the different states of naturewhile the branches leaving each

square node represent the differentdecision alternatives.

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


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Decision Making without Probabilities

Three commonly used criteria fordecision making when probabilityinformation regarding the likelihoodof the states of nature is unavailable

are:

the optimistic approachthe conservative approachthe minimax regret approach.



Optimistic Approach

The optimistic approach would be usedby an optimistic decision maker.

The decision with the largest possiblepayoff is chosen.

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



Conservative Approach

The conservative approach would be used by a

conservative decision maker.

For each decision the minimum payoff is listed andthen the decision corresponding to the maximumof 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 andthen the decision corresponding to the minimum

of these maximum costs is selected. (Hence, themaximum possible cost is minimized.)



Minimax Regret Approach

The minimax regret approach requires the

construction of a regret table or an opportunityloss table.

This is done by calculating for each state of naturethe 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.



Example

Consider the following problem with three decision

alternatives and three states of nature with thefollowing payoff table representing profits:

States of Nature

s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1

d3 1 5 -3



Example: Optimistic Approach

An optimistic decision maker would use the

optimistic (maximax) approach. We choose thedecision that has the largest single value in thepayoff table.

MaximumDecision Payoff

d1 4

d2 3

d3 5

Maximaxpayoff

Maximaxdecision



Example: Optimistic Approach

Solution Spreadsheet

A B C D E F

1

2

3 Decision Maximum Recommended

4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 4

6 d2 0 3 -1 3

7 d3 1 5 -3 5 d3

8

9 5

State of Nature

Best Payoff

PAYOFF TABLE


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15Slide 2005 Thomson/South-Western

Example: Conservative Approach


A B C D E F

1

2

3 Decision Minimum Recommended

4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 -2

6 d2 0 3 -1 -1 d2

7 d3 1 5 -3 -3

8

9 -1

State of Nature

Best Payoff

PAYOFF TABLE


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For the minimax regret approach, first compute a

regret table by subtracting each payoff in a columnfrom the largest payoff in that column. In thisexample, in the first column subtract 4, 0, and 1 from4; etc. The resulting regret table is:

s1 s2 s3

d1 0 1 1

d2 4 2 0

d3 3 0 2

Example: Minimax Regret Approach


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For each decision list the maximum regret.

Choose the decision with the minimum of thesevalues.

Maximum

Decision Regretd1 1

d2 4

d3 3


Minimaxdecision

Minimaxregret


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A B C D E F

1

2 Decision

3 Alternative s1 s2 s3

4 d1 4 4 -2

5 d2 0 3 -1

6 d3 1 5 -3

7

8

9 Decision Maximum Recommended

10 Alternative s1 s2 s3 Regret Decision11 d1 0 1 1 1 d1

12 d2 4 2 0 4

13 d3 3 0 2 3

14 1Minimax Regret Value

State of Nature

PAYOFF TABLE

State of Nature

OPPORTUNITY LOSS TABLE



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Decision Making with Probabilities

Expected Value Approach

If probabilistic information regarding the statesof nature is available, one may use the expectedvalue (EV) approach.

Here the expected return for each decision iscalculated by summing the products of thepayoff under each state of nature and theprobability of the respective state of natureoccurring.

The decision yielding the best expected return ischosen.


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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 decisionalternative di and state of nature sj

Expected Value of a Decision Alternative

EV( ) ( )d P s V i j ijj

N

1EV( ) ( )d P s V

i j ijj

N

1


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Example: Burger Prince

Burger Prince Restaurant is considering 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 ofcustomers per hour will be 80, 100, or

120. The payoff table for the three

models is on the next slide.


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Payoff Table

Average Number of Customers Per Hour

s1 = 80 s2 = 100 s3 = 120

Model A $10,000 $15,000 $14,000Model B $ 8,000 $18,000 $12,000

Model C $ 6,000 $16,000 $21,000


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Calculate 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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Decision Tree

1

.2

.4

.4

.4

.2

.4

.4

.2

.4

d1

d2

d3

s1

s1

s1

s2s3

s2

s2s3

s3

Payoffs10,000

15,000

14,0008,000

18,000

12,000

6,000

16,000

21,000

2

3

4


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Expected Value for Each Decision

Choose the model with largest EV, Model C.

3

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

2

1

4


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A B C D E F

1

2

3 Decision Expected Recommended

4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision

5 d1 = Model A 10,000 15,000 14,000 12600

6 d2 = Model B 8,000 18,000 12,000 11600

7 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C

8 Probability 0.4 0.2 0.4

9 14000

State of Nature

Maximum Expected Value

PAYOFF TABLE



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Expected Value of Perfect Information

Frequently information is available which can

improve the probability estimates for the states ofnature.

The expected value of perfect information (EVPI) isthe increase in the expected profit that would

result if one knew with certainty which state ofnature would occur.

The EVPI provides an upper bound on theexpected value of any sample or survey

information.


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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 ofthe optimal decision.

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



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Spreadsheet

A B C D E F

1

2

3 Decision Expected Recommended

4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision

5 d1 = Model A 10,000 15,000 14,000 12600

6 d2 = Model B 8,000 18,000 12,000 11600

7 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C

8 Probability 0.4 0.2 0.4

9 14000

10

11 EVwPI EVPI

12 10,000 18,000 21,000 16000 2000

State of Nature

Maximum Expected Value

PAYOFF TABLE

Maximum Payoff



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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 thepossible payoffs for the decision alternative alongwith their associated probabilities.


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Risk Profile

Model C Decision Alternative

.10

.20

.30

.40

.50

5 10 15 20 25

Probab

ility


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Sensitivity Analysis

Sensitivity analysis can be used to determine how

changes to the following inputs affect therecommended decision alternative:

probabilities for the states of nature values of the payoffs

If a small change in the value of one of the inputscauses a change in the recommended decisionalternative, extra effort and care should be taken inestimating the input value.


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Bayes Theorem and Posterior Probabilities

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 priorprobabilities.

With knowledge of conditional probabilities for theoutcomes or indicators of the sample or surveyinformation, these prior probabilities can be revised byemploying Bayes' Theorem.

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


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Branch (Posterior) Probabilities Calculation

Step 1:For each state of nature, multiply the prior

probability by its conditional probability for theindicator -- this gives the joint probabilities for the

states and indicator.


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Branch (Posterior) Probabilities Calculation

Step 2:Sum these joint probabilities over all states -- this

gives the marginal probability for the indicator.

Step 3:For each state, divide its joint probability by the

marginal probability for the indicator -- this givesthe posterior probability distribution.


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Expected Value of Sample Information

The expected value of sample information (EVSI) is

the additional expected profit possible throughknowledge of the sample or survey information.


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EVSI Calculation

Step 1:Determine the optimal decision and its expected

return for the possible outcomes of the sample usingthe posterior probabilities for the states of nature.

Step 2:Compute the expected value of these optimal

returns.

Step 3:Subtract the EV of the optimal decision obtained

without using the sample information from theamount determined in step (2).


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Efficiency of Sample Information

Efficiency of sample information is the ratio of EVSI

to EVPI. As the EVPI provides an upper bound for the EVSI,

efficiency is always a number between 0 and 1.

l f


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Burger Prince must decide whether or not to

purchase a marketing survey from Stanton Marketingfor $1,000. The results of the survey are "favorable" or"unfavorable". The conditional probabilities are:

P(favorable | 80 customers per hour) = .2



Should Burger Prince have the survey performed

by Stanton Marketing?

Sample Information

fl D


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Influence Diagram

RestaurantSize Profit

Avg. Numberof Customers

Per HourMarketSurveyResults

MarketSurvey

DecisionChanceConsequence

P i P b bili i


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Favorable

State Prior Conditional Joint Posterior

80 .4 .2 .08 .148

100 .2 .5 .10 .185120 .4 .9 .36 .667

Total .54 1.000

P(favorable) = .54

Posterior Probabilities

P i P b bili i


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Unfavorable

State Prior Conditional Joint Posterior

80 .4 .8 .32 .696

100 .2 .5 .10 .217120 .4 .1 .04 .087

Total .46 1.000

P(unfavorable) = .46


P i P b bili i


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A B C D E1

2 Prior Conditional Joint Posterior

3 State of Nature Probabilities Probabilities Probabilities Probabilities

4 s1 = 80 0.4 0.2 0.08 0.148

5 s2 = 100 0.2 0.5 0.10 0.185

6 s3 = 120 0.4 0.9 0.36 0.667

7 0.54

8

9

10 Prior Conditional Joint Posterior

11 State of Nature Probabilities Probabilities Probabilities Probabilities12 s1 = 80 0.4 0.8 0.32 0.696

13 s2 = 100 0.2 0.5 0.10 0.217

14 s3 = 120 0.4 0.1 0.04 0.087

15 0.46

Market Research Favorable

P(Favorable) =

Market Research Unfavorable

P(Favorable) =


D i i T


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Decision Tree

Top Half

s1 (.148)

s1 (.148)

s1 (.148)s2 (.185)

s2 (.185)

s2 (.185)

s3 (.667)

s3 (.667)

s3 (.667)

$10,000

$15,000

$14,000

$8,000

$18,000

$12,000

$6,000$16,000

$21,000

I1

(.54)

d1

d2

d3

2

4

5

6

1

D i i T


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Bottom Half

s1 (.696)

s1 (.696)

s1 (.696)

s2 (.217)

s2 (.217)

s2 (.217)

s3 (.087)

s3 (.087)

s3 (.087)

$10,000

$15,000

$18,000

$14,000$8,000

$12,000

$6,000$16,000

$21,000

I2(.46) d1

d2

d3

7

9

83

1

Decision Tree

D i i T


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I2

(.46)

d1

d2

d3

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

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

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

I1(.54)

d1

d2

d3

EMV = .148(10,000) + .185(15,000)

+ .667(14,000) = $13,593EMV = .148 (8,000) + .185(18,000)

+ .667(12,000) = $12,518

EMV = .148(6,000) + .185(16,000)

+.667(21,000) = $17,855

4

5

6

7

8

9

2

3

1

$17,855

$11,433

Decision Tree

E t d V l f S l I f ti


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If the outcome of the survey is "favorable,

choose Model C. If it is unfavorable, choose Model A.

EVSI = .54($17,855) + .46($11,433) - $14,000 = $900.88

Since this is less than the cost of the survey, the

survey should not be purchased.


Effi i f S l I f ti


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Efficiency of Sample Information

The efficiency of the survey:

EVSI/EVPI = ($900.88)/($2000) = .4504


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Bayes Decision Rule:Using the best available estimates of the

probabilities of the respective states of nature

(currently the prior probabilities), calculate the

expected value of the payoff for each of the

possible actions. Choose the action with the

maximum expected payoff.


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Bayes theorySi: State of Nature (i = 1 ~ n)

P(Si): Prior Probability

Ij: Professional Information (Experiment)(j = 1 ~ n)

P(Ij | Si): Conditional ProbabilityP(Ij Si) = P(Si Ij): Joint Probability

P(Si | Ij): Posterior Probability

P(Si | Ij)

n

1iiij

iij

j

ji

)S(P)S|I(P

)S(P)S|I(P

)I(P

)IS(P

Home Work


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Home Work

Problem 13-10

Problem 13-21

Due Date: Nov 11, 2008

Decision Making in Operations Management

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