burger prince
TRANSCRIPT
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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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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$9,A7,"")
89 =MAX(E5:E7)
State of Nature
Best Pa yoff
PAYOFF TABLE
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Example
Spreadsheet for Conservative Approach
A B C D E F
1
2
3 De cision M inim u m Re com m e n de d
4 Alternative s1 s2 s3 Pa yoff De cision
5 d1 4 4 -2 -2
6 d2 0 3 -1 -1 d2
7 d3 1 5 -3 -3
89 -1
Sta te of Nature
Be st Pa yoff
PAYOFF TABLE
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Example
Minimax Regret Approach
For the minimax regret approach, first compute aregret table by subtracting each payoff in a columnfrom the largest payoff in that column. In thisexample, in the first column subtract 4, 0, and 1 from
4; in the second column, subtract 4, 3, and 5 from 5;etc. The resulting regret table is:
s1 s2 s3
d1 0 1 1d2 4 2 0
d3 3 0 2
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Example
Minimax Regret Approach (continued)
For each decision list the maximum regret. Choosethe decision with the minimum of these values.
Decision Maximum Regret
choose d1 d1 1 minimum
d2 4
d3 3
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18Slide
Example
Formula Spreadsheet for Minimax Regret Approach
A B C D E F
1
2 Decision
3 Al tern. s1 s2 s3
4 d1 4 4 -2
5 d2 0 3 -16 d3 1 5 -3
7
8
9 Decision Max imum Recommended
10 Altern. s1 s2 s3 Regret Decision
11 d1=MA X($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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20Slide
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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23Slide
Example: Burger Prince
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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24Slide
Example: Burger Prince
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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25Slide
Example: Burger Prince
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)= $14,000
Model A
Model B
Model C
2
1
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26Slide
Example: Burger Prince
Formula Spreadsheet for Expected Value Approach
A B C D E F
1
2
3 Decision Expected Recommended
4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision
5 Model A 10,000 15,000 14,000 =$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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Example: Burger Prince
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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29Slide
Expected Value of Perfect Information
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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Example: Burger Prince
Expected Value of Perfect Information
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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31Slide
Example: Burger Prince
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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32Slide
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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33Slide
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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34Slide
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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Expected Value of Sample Information
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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36Slide
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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37Slide
Example: Burger Prince
Sample Information
Burger Prince must decide whether or not topurchase a marketing survey from Stanton Marketingfor $1,000. The results of the survey are "favorable" or"unfavorable". The conditional probabilities are:
P(favorable | 80 customers per hour) = .2P(favorable | 100 customers per hour) = .5
P(favorable | 120 customers per hour) = .9
Should Burger Prince have the survey performedby Stanton Marketing?
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38Slide
Example: Burger Prince
Posterior Probabilities
Favorable Survey Results
State Prior Conditional Joint Posterior
80 .4 .2 .08 .148
100 .2 .5 .10 .185
120 .4 .9 .36 .667
Total .54 1.000
P(favorable) = .54
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39Slide
Example: Burger Prince
Posterior Probabilities
Unfavorable Survey Results
State Prior Conditional Joint Posterior
80 .4 .8 .32 .696
100 .2 .5 .10 .217
120 .4 .1 .04 .087
Total .46 1.000
P(unfavorable) = .46
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40Slide
Example: Burger Prince
Formula Spreadsheet for Posterior Probabilities
A B C D E
1
2 P rior Condit iona l Jo in t P os terior
3 S t a t e o f N a tu re P r ob a bilit ie s P r ob a bilit ie s P r ob a bilit ie s P r ob a bilit ie s
4 s 1 = 80 0 .4 0 .2 = B 4*C4 = D 4/$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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41Slide
Example: Burger Prince
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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Example: Burger Prince
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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Example: Burger Prince
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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Example: Burger Prince
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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Example: Burger Prince
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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46Slide
Example: Burger Prince
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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47Slide
Example: Burger Prince
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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48Slide
Example: Burger Prince
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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49Slide
Example: Burger Prince
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