Burger Prince

7/29/2019 Burger Prince

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CONTEMPORARYMANAGEMENT SCIENCEWITH SPREADSHEETS

ANDERSON SWEENEY WILLIAMS

SLIDES PREPARED BY

JOHN LOUCKS

1999 South-Western College Publishing


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Chapter 9Decision Analysis

Structuring the Decision Problem

Decision Making Without Probabilities

Decision Making with Probabilities

Expected Value of Perfect Information Decision Analysis with Sample Information

Developing a Decision Strategy

Expected Value of Sample Information


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Slide

Structuring the Decision Problem

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 aremutually exclusive and collectively exhaustive.

For each decision alternative and state of nature,

there is an outcome. These outcomes are often represented in a matrix

called a payoff table.


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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 squarenodes correspond to the decision alternatives.

The branches leaving each round node represent thedifferent states of nature while the branches leavingeach square node represent the different decisionalternatives.

At the end of each limb of a tree are the payoffsattained from the series of branches making up thatlimb.


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Decision Making Without Probabilities

If the decision maker does not know with certaintywhich state of nature will occur, then he is said to bedoing decision making under uncertainty.

Three commonly used criteria for decision makingunder uncertainty when probability informationregarding the likelihood of the states of nature isunavailable 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 ischosen.

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 aconservative decision maker.

For each decision the minimum payoff is listed andthen the decision corresponding to the maximum of

these minimum payoffs is selected. (Hence, theminimum possible payoff is maximized.)

If the payoff was in terms of costs, the maximumcosts would be determined for each decision and

then the decision corresponding to the minimum ofthese maximum costs is selected. (Hence, themaximum possible cost is minimized.)


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Minimax Regret Approach

The minimax regret approach requires theconstruction of a regret table or an opportunity losstable.

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 foreach 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 withthe following payoff table representing profits:

States of Nature

s1 s2 s3

d1 4 4 -2

Decisions d2 0 3 -1d3 1 5 -3


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Example

Optimistic Approach

An optimistic decision maker would use theoptimistic approach. All we really need to do is tochoose the decision that has the largest single value inthe 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 forOptimistic Approach

A B C D E F

1

2

3 De cision Ma x im um Re com m ende d

4 Alternative s1 s2 s3 Payoff Decision

5 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

Spreadsheet for Optimistic Approach

A B C D E F

1

2

3 De cision M a x im u m Re com m e n de d

4 Alternative s1 s2 s3 Pa yoff De cision

5 d1 4 4 -2 4

6 d2 0 3 -1 3

7 d3 1 5 -3 5 d3

89 5

Sta te of Nature

Bes t Pay off

PAYOFF TABLE


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Example

Conservative Approach

A conservative decision maker would use theconservative approach. List the minimum payoff foreach decision. Choose the decision with the maximumof these minimum payoffs.

MinimumDecision 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 F

1

2

3 De cision Minim um Re com m ende d

4 Alternative s1 s2 s3 Payoff Decision

5 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 Pa yoff

PAYOFF TABLE


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Example

Spreadsheet for Conservative Approach

A B C D E F

1

2

3 De cision M inim u m Re com m e n de d

4 Alternative s1 s2 s3 Pa yoff De cision

5 d1 4 4 -2 -2

6 d2 0 3 -1 -1 d2

7 d3 1 5 -3 -3

89 -1

Sta te of Nature

Be st Pa yoff

PAYOFF TABLE


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Example

Minimax Regret Approach

For the minimax regret approach, first compute aregret 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 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 1d2 4 2 0

d3 3 0 2


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Example

Minimax Regret Approach (continued)

For each decision list the maximum regret. Choosethe 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 Approach

A B C D E F

1

2 Decision

3 Al tern. s1 s2 s3

4 d1 4 4 -2

5 d2 0 3 -16 d3 1 5 -3

7

8

9 Decision Max imum Recommended

10 Altern. s1 s2 s3 Regret Decision

11 d1=MA X($B$4:$B$6)-B4 =MA X($C$4:$C$6)-C4 =MAX( $D$4:$D$6)-D4

=MAX(B11:D11) =IF(E11=$E$14,A11,"")12 d2 =MA X($B$4:$B$6)-B5 =MA X($C$4:$C$6)-C5 =MAX( $D$4:$D$6)-D5 =MAX(B12:D12) =IF(E12=$E$14,A12,"")

13 d3 =MA X($B$4:$B$6)-B6 =MA X($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 Nature

PAYOFF TABLE

State of Nature

OPPORTUNITY LOSS TABLE


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19Slide

Example

Spreadsheet for Minimax Regret Approach1

2 D e c isio n

3 A l te r n a ti v e s 1 s 2 s 3

4 d 1 4 4 -2

5 d 2 0 3 -1

6 d 3 1 5 -3

7

8

9 De c ision M a x im um Re com m e nd e d

10 A l te rna ti ve s 1 s 2 s 3 R e g re t D e c isio n

11 d 1 0 1 1 1 d 112 d 2 4 2 0 4

13 d 3 3 0 2 3

14 1M in im a x R e g re t V a lu e

S ta te o f N a tu re

P A Y O F F TA B LE

S ta te o f N a tu re

O P P O R TU N ITY LO S S TA B LE


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Decision Making with Probabilities

Expected Value Approach

If probabilistic information regarding he states ofnature is available, one may use the expectedvalue (EV) approach.

Here the expected return for each decision is

calculated by summing the products of the payoffunder each state of nature and the probability ofthe respective state of nature occurring.

The decision yielding the best expected return is

chosen.


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21Slide

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 sjVij = the payoff corresponding to decisionalternative di and state of nature sj

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 contemplating

opening a new restaurant on Main Street. It has threedifferent models, each with a different seatingcapacity. Burger Prince estimates that the averagenumber 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

d1 = Model A $10,000 $15,000 $14,000d2 = Model B $ 8,000 $18,000 $12,000

d3 = Model C $ 6,000 $16,000 $21,000


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Expected Value Approach

Calculate the expected value for each decision. Thedecision tree on the next slide can assist in thiscalculation. Here d1, d2, d3 represent the decisionalternatives 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

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

2

3

4


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Expected Value For Each Decision

Choose the model with largest EMV -- Model C.

3

4

d1

d2

d3

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

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

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

Model A

Model B

Model C

2

1


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Formula Spreadsheet for Expected Value Approach

A B C D E F

1

2

3 Decision Expected Recommended

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

5 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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Spreadsheet for Expected Value Approach

A B C D E F

1

2

3 Decision Expected Recommended

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

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

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

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

8 Probabil ity 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 canimprove the probability estimates for the states ofnature.

The expected value of perfect information (EVPI) isthe increase in the expected profit that would result ifone knew with certainty which state of nature wouldoccur.

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


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EVPI Calculation

Step 1:

Determine the optimal return corresponding toeach state of nature.

Step 2:

Compute the expected value of these optimalreturns.

Step 3:

Subtract the EV of the optimal decision from theamount determined in step (2).


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Calculate the expected value for the optimumpayoff 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 for Expected Value of Perfect Information

A B C D E F

1

2

3 D e cisio n Ex p e c te d R e co m m e n d ed

4 A lte r na tiv e s 1 = 80 s 2 = 100 s 3 = 120 V a lu e De cisio n5 d1 = M odel A 10,000 15,000 14,000 12600

6 d2 = M odel B 8,000 18,000 12,000 11600

7 d3 = M odel C 6,000 16,000 21,000 14000 d 3 = M o d e l C

8 P ro b a b il i ty 0.4 0.2 0.4

9 14000

10

11 EV w P I EV P I

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

S ta te of Na ture

M a x i m u m Ex p e c te d Va l u e

P A Y O F F TA B L E

M a x im um P a yoff


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Decision Analysis With Sample Information

Knowledge of sample or survey information can be

used to revise the probability estimates for the states ofnature.

Prior to obtaining this information, the probabilityestimates for the states of nature are called prior

probabilities. With knowledge of conditional probabilities for the

outcomes or indicators of the sample or surveyinformation, these prior probabilities can be revised by

employing Bayes' Theorem. The outcomes of this analysis are called posterior

probabilities.


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Posterior Probabilities

Posterior Probabilities Calculation

Step 1:

For each state of nature, multiply the priorprobability by its conditional probability for theindicator -- this gives the joint probabilities for the

states and indicator. Step 2:

Sum these joint probabilities over all states -- thisgives the marginal probability for the indicator.

Step 3:

For each state, divide its joint probability by themarginal probability for the indicator -- this givesthe posterior probability distribution.


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

Step 2:

Compute the expected value of these optimalreturns.

Step 3:

Subtract the EV of the optimal decision obtainedwithout 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.


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Sample Information

Burger Prince must decide whether or not topurchase 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) = .2P(favorable | 100 customers per hour) = .5

P(favorable | 120 customers per hour) = .9

Should Burger Prince have the survey performedby Stanton Marketing?


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Favorable Survey Results

State Prior Conditional Joint Posterior

80 .4 .2 .08 .148

100 .2 .5 .10 .185

120 .4 .9 .36 .667

Total .54 1.000

P(favorable) = .54


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Unfavorable Survey Results

State Prior Conditional Joint Posterior

80 .4 .8 .32 .696

100 .2 .5 .10 .217

120 .4 .1 .04 .087

Total .46 1.000

P(unfavorable) = .46


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Formula Spreadsheet for Posterior Probabilities

A B C D E

1

2 P rior Condit iona l Jo in t P os terior

3 S t a t e o f N a tu re P r ob a bilit ie s P r ob a bilit ie s P r ob a bilit ie s P r ob a bilit ie s

4 s 1 = 80 0 .4 0 .2 = B 4*C4 = D 4/$D$7

5 s 2 = 100 0 .2 0 .5 = B 5*C5 = D 5/$D$76 s 3 = 120 0 .4 0 .9 = B 6*C6 = D 6/$D$7

7 = S UM (D4:D6)

8

9

10 P rior Condit iona l Jo in t P os terior

11 S t a t e o f N a tu re P r ob a bilit ie s P r ob a bilit ie s P r ob a bilit ie s P r ob a bilit ie s

12 s 1 = 80 0 .4 0 .8 = B 12*C12 = D12/$D$15

13 s 2 = 100 0 .2 0 .5 = B 13*C13 = D13/$D$15

14 s 3 = 120 0 .4 0 .1 = B 14*C14 = D14/$D$15

15 = S UM (D12:D14)

M arket Res earch Favorable

P (Favorable) =

M arke t R es earc h U nfavorable

P(Unfavorable) =


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Spreadsheet for Posterior Probabilities

A B C D E

1

2 P rio r C on d it io na l Jo in t P os te rior

3 S t a t e o f N a t u re P ro b a b ili ti es P ro b a b il it ie s P ro b a b il it ie s P ro b a b il it ie s

4 s 1 = 80 0 .4 0 .2 0 .0 8 0 .1 48

5 s 2 = 10 0 0 .2 0 .5 0 .1 0 0 .1 856 s 3 = 12 0 0 .4 0 .9 0 .3 6 0 .6 67

7 0 .54

8

9

10 P rio r C on d it io na l Jo in t P os te rior

11 S t a t e o f N a t u re P ro b a b ili ti es P ro b a b il it ie s P ro b a b il it ie s P ro b a b il it ie s

12 s 1 = 80 0 .4 0 .8 0 .3 2 0 .6 96

13 s 2 = 10 0 0 .2 0 .5 0 .1 0 0 .2 17

14 s 3 = 12 0 0 .4 0 .1 0 .0 4 0 .0 87

15 0 .46

M arke t Res earch Fa vorab le

P (Fa vorab le) =

M arke t Res earch Un favorab le

P (Fa vorab le) =


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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


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Decision Tree (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


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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) = $13,593

EMV = .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

l


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Decision Strategy Assuming the Survey is Undertaken:

If the outcome of the survey is favorable, chooseModel C.

If it is unfavorable, choose Model A.

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Question:

Should the survey be undertaken?

Answer:

If the Expected Value with Sample Information(EVwSI) is greater, after deducting expenses, thanthe Expected Value without Sample Information(EVwoSI), the survey is recommended.

E l B P i


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Expected Value with Sample Information (EVwSI)

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

Expected Value of Sample Information (EVSI)

EVSI = EVwSI - EVwoSI

assuming maximization

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

E l B P i


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Conclusion

EVSI = $900.88

Since the EVSI is less than the cost of the survey ($1000),the survey should not be purchased.

E l B P i


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Efficiency of Sample Information

The efficiency of the survey:

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

Th E d f Ch t 9


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The End of Chapter 9

Burger Prince

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