Decision Analysis
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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
Slide 1
Structuring the Decision Problem
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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.
For each decision alternative and state of nature,
there is an outcome.
These outcomes are often represented in a matrix
called a payoff table.
Slide 2
Decision Trees
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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.
Slide 3
Decision Making Without Probabilities
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If the decision maker does not know with certainty
which state of nature will occur, then he is said to be
doing decision making under uncertainty.
Three commonly used criteria for decision making
under uncertainty when probability information
regarding the likelihood of the states of nature is
unavailable are:
the optimistic approach
• the conservative approach
• the minimax regret approach.
Slide 4
Optimistic Approach
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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.
Slide 5
Conservative Approach
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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.)
Slide 6
Minimax Regret Approach
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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 chosen is the one corresponding to the
minimum of the maximum regrets.
Slide 7
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
Decisions d2
d3
4
0
1
4
3
5
-2
-1
-3
Slide 8
Example
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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
Slide 9
Example
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Formula Spreadsheet for Optimistic Approach
A
1
B
C
D
E
F
M a x im um
Re com m e nde d
P A Y OFF TA B LE
2
3
De cision
S ta te of Na ture
4
Alte rna tive
s1
s2
s3
P a yoff
De cision
5
d1
4
4
-2
= M A X(B 5:D5)
= IF(E 5= $E $9,A 5,"")
6
d2
0
3
-1
= M A X(B 6:D6)
= IF(E 6= $E $9,A 6,"")
7
d3
1
5
-3
= M A X(B 7:D7)
= IF(E 7= $E $9,A 7,"")
8
9
B es t P ay off
= M A X(E 5:E 7)
Slide 10
Example
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Spreadsheet for Optimistic Approach
A
1
B
C
D
E
F
M a x im u m
Re co m m e n d e d
De cisio n
P A Y O F F TA B LE
2
3
De cisio n
S ta te o f Na tu re
4
Alte rn a tive
s1
s2
s3
P a yo ff
5
d1
4
4
-2
4
6
d2
0
3
-1
3
7
d3
1
5
-3
5
d3
8
9
B es t P ay off
5
Slide 11
Example
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Conservative Approach
A 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
Slide 12
Example
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Formula Spreadsheet for Conservative Approach
A
1
B
C
D
E
F
M inim um
Re com m e nde d
P A Y OFF TA B LE
2
3
De cision
S ta te of Na ture
4
Alte rna tive
s1
s2
s3
P a yoff
De cision
5
d1
4
4
-2
= M IN(B 5:D5)
= IF(E 5= $E $9,A 5,"")
6
d2
0
3
-1
= M IN(B 6:D6)
= IF(E 6= $E $9,A 6,"")
7
d3
1
5
-3
= M IN(B 7:D7)
= IF(E 7= $E $9,A 7,"")
8
9
Be st P a yoff
= M A X(E 5:E 7)
Slide 13
Example
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Spreadsheet for Conservative Approach
A
1
B
C
D
E
F
M in im u m
Re co m m e n d e d
P A Y O F F TA B LE
2
3
De cisio n
S ta te o f Na tu re
4
Alte rn a tive
s1
s2
s3
P a yo ff
5
d1
4
4
-2
-2
6
d2
0
3
-1
-1
7
d3
1
5
-3
-3
De cisio n
d2
8
9
Be st P a yo ff
-1
Slide 14
Example
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Minimax Regret Approach
For the minimax regret approach, first compute a
regret table by subtracting each payoff in a column
from the largest payoff in that column. In this
example, in the first column subtract 4, 0, and 1 from
4; in the second column, subtract 4, 3, and 5 from 5;
etc. The resulting regret table is:
s1 s2 s3
d1
d2
d3
0
4
3
1
2
0
1
0
2
Slide 15
Example
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Minimax Regret Approach (continued)
For each decision list the maximum regret. Choose
the decision with the minimum of these values.
choose d1
Decision
d1
d2
d3
Maximum Regret
1
minimum
4
3
Slide 16
Example
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Formula Spreadsheet for Minimax Regret Approach
A
1
B
C
D
E
F
Ma x imum
Re comme nde d
Re gre t
De cision
PAYOFF TABLE
2 De cision
Sta te of Na ture
3
Alte rn.
s1
s2
s3
4
d1
4
4
-2
5
d2
0
3
-1
6
d3
1
5
-3
7
8
OPPORTUNITY LOSS TABLE
9 De cision
Sta te of Na ture
10
Alte rn.
s1
s2
s3
11
d1
=MA X($B$4:$B$6)-B4
=MA X($C$4:$C$6)-C4
=MA X($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$6)-C5
=MA X($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$6)-C6
=MA X($D$4:$D$6)-D6
=MAX(B13:D13) =IF(E13=$E$14,A13,"")
14
Minima x Re gre t Va lue
=MIN(E11:E13)
Slide 17
Example
Spreadsheet for Minimax Regret Approach
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1
P A Y O F F TA B L E
2
D e c i si o n
S ta te o f N a tu r e
3
A l te r n a ti v e
s1
s2
s3
4
d1
4
4
-2
5
d2
0
3
-1
6
d3
1
5
-3
7
8
O P P O R TU N ITY L O S S TA B L E
9
D e c i si o n
10
A l te r n a ti v e
s1
s2
11
d1
0
12
d2
13
d3
14
S ta te o f N a tu r e
M a x im u m
Re co m m e n d e d
s3
R e g re t
D e c i si o n
1
1
1
d1
4
2
0
4
3
0
2
3
M in im a x R e g re t V a lu e
1
Slide 18
Decision Making with Probabilities
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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.
Slide 19
Expected Value of a Decision Alternative
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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:
N
EV( d i )   P( s j )Vij
j 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
Slide 20
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
d1 = Model A
d2 = Model B
d3 = Model C
$10,000
$ 8,000
$ 6,000
$15,000
$18,000
$16,000
$14,000
$12,000
$21,000
Slide 21
Example: Burger Prince
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Expected Value Approach
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.
Slide 22
Example: Burger Prince
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Decision Tree
Payoffs
2
d1
1
s1 .4
10,000
s2 .2
s3 .4
15,000
14,000
d2
3
d3
s1 .4
s2 .2
s3 .4
8,000
18,000
12,000
4
s1
.4
s2
s3
.2
6,000
16,000
.4
21,000
Slide 23
Example: Burger Prince
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Expected Value For Each Decision
d1
EV = .4(10,000) + .2(15,000) + .4(14,000)
= $12,600
2
Model A
1
Model B
Model C
d2
3
EV = .4(8,000) + .2(18,000) + .4(12,000)
= $11,600
d3
EV = .4(6,000) + .2(16,000) + .4(21,000)
= $14,000
4
Choose the model with largest EMV -- Model C.
Slide 24
Example: Burger Prince
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Formula Spreadsheet for Expected Value Approach
A
B
C
D
E
F
Expected
Recommended
Value
Decision
1 PAYOFF TABLE
2
3
Decision
State of Nature
4 Alternative s1 = 80 s2 = 100 s3 = 120
5
Model A
10,000
15,000
14,000 =$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,"")
0.4
0.2
8 Probability
9
0.4
Maximum Expected Value
=MAX(E5:E7)
Slide 25
Example: Burger Prince
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Spreadsheet for Expected Value Approach
A
1
B
C
D
E
F
Ex pe cte d
Re com m e nded
Va lue
De cision
PAYOFF TABLE
2
3
De cision
4
Alte rna tive
s1 = 80
5
M odel A
10,000
15,000
14,000
12600
6
M odel B
8,000
18,000
12,000
11600
7
M odel C
6,000
16,000
21,000
14000
8
Proba bility
0.4
0.2
0.4
9
Sta te of Na ture
s2 = 100 s3 = 120
M a x im um Ex pe cte d Va lue
M ode l C
14000
Slide 26
Expected Value of Perfect Information
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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.
Slide 27
Expected Value of Perfect 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).
Slide 28
Example: Burger Prince
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Expected Value of Perfect Information
Calculate the expected value for the optimum
payoff for each state of nature and subtract the EV of
the optimal decision.
EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000
Slide 29
Example: Burger Prince
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Spreadsheet for Expected Value of Perfect Information
A
1
B
C
D
E
F
Ex p e c te d
Re co m m e n d ed
D e c i si o n
P A Y O F F TA B L E
2
3
D e c i si o n
4
A l te rn a ti v e
s 1 = 80
s 2 = 100
s 3 = 120
V a lu e
5
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
8
P ro b a b i l i ty
0.4
0.2
0.4
9
S ta te o f N a tu re
d3 = M ode l C
M a x i m u m Ex p e c te d V a l u e
14000
M a x i m u m P a y o ff
EV w P I
EV P I
16000
2000
10
11
12
10,000
18,000
21,000
Slide 30
Decision Analysis With Sample Information
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Knowledge of sample or 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.
Slide 31
Posterior Probabilities
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Posterior Probabilities Calculation
• Step 1:
For each state of nature, multiply the prior
probability by its conditional probability for the
indicator -- this gives the joint probabilities for the
states and indicator.
• 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 gives
the posterior probability distribution.
Slide 32
Expected Value of Sample Information
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The expected value of sample information (EVSI) is
the additional expected profit possible through
knowledge of the sample or survey information.
Slide 33
Expected Value of Sample Information
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EVSI Calculation
• Step 1:
Determine the optimal decision and its expected
return 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.
• Step 3:
Subtract the EV of the optimal decision obtained
without using the sample information from the
amount determined in step (2).
Slide 34
Efficiency of Sample Information
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
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.
Slide 35
Example: Burger Prince
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Sample Information
Burger Prince must decide whether or not to
purchase a marketing survey from Stanton Marketing
for $1,000. The results of the survey are "favorable" or
"unfavorable". The conditional probabilities are:
P(favorable | 80 customers per hour) = .2
P(favorable | 100 customers per hour) = .5
P(favorable | 120 customers per hour) = .9
Should Burger Prince have the survey performed
by Stanton Marketing?
Slide 36
Example: Burger Prince

Posterior Probabilities
Favorable Survey Results
State
80
100
120
Prior
.4
.2
.4
Conditional Joint
.2
.08
.5
.10
.9
.36
Total .54
Posterior
.148
.185
.667
1.000
P(favorable) = .54
Slide 37
Example: Burger Prince
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Posterior Probabilities
Unfavorable Survey Results
State
80
100
120
Prior
.4
.2
.4
Conditional
.8
.5
.1
Total
Joint
.32
.10
.04
.46
Posterior
.696
.217
.087
1.000
P(unfavorable) = .46
Slide 38
Example: Burger Prince

Formula Spreadsheet for Posterior Probabilities
A
1
B
C
D
E
P rio r
C o n d it io n a l
Jo in t
P o s t erio r
M a rk e t R e s e a rc h F a vo ra b le
2
3
S t a t e o f N a t u re
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
4
s 1 = 80
0.4
0.2
= B 4 *C 4
= D 4 / $ D$ 7
5
s 2 = 100
0.2
0.5
= B 5 *C 5
= D 5 / $ D$ 7
6
s 3 = 120
0.4
0.9
= B 6 *C 6
= D 6 / $ D$ 7
P (F a vo ra b le ) =
= S U M (D 4 : D 6 )
P rio r
C o n d it io n a l
Jo in t
P o s t erio r
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
7
8
9
M a rk e t R e s e a rc h U n fa vo ra b le
10
1 1 S t a t e o f N a t u re
12
s 1 = 80
0.4
0.8
= B 1 2 *C 1 2
= D 1 2 / $ D$ 1 5
13
s 2 = 100
0.2
0.5
= B 1 3 *C 1 3
= D 1 3 / $ D$ 1 5
14
s 3 = 120
0.4
0.1
= B 1 4 *C 1 4
= D 1 4 / $ D$ 1 5
15
P (U n fa vo ra b le ) = = S U M (D 1 2 : D 1 4 )
Slide 39
Example: Burger Prince

Spreadsheet for Posterior Probabilities
A
1
B
C
D
E
P rio r
C o n d it io n a l
Jo in t
P o s t e rior
M a rk e t R e s e a rc h F a vo ra b le
2
3
S t a t e o f N a t u re
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
4
s 1 = 80
0.4
0.2
0.08
0.148
5
s 2 = 100
0.2
0.5
0.10
0.185
6
s 3 = 120
0.4
0.9
0.36
0.667
P (F a vo ra b le ) =
0.54
P rio r
C o n d it io n a l
Jo in t
P o s t e rior
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
P ro b a b ilit ie s
7
8
9
M a rk e t R e s e a rc h U n fa vo ra b le
10
1 1 S t a t e o f N a t u re
12
s 1 = 80
0.4
0.8
0.32
0.696
13
s 2 = 100
0.2
0.5
0.10
0.217
14
s 3 = 120
0.4
0.1
0.04
0.087
P (F a vo ra b le ) =
0.46
15
Slide 40
Example: Burger Prince

Decision Tree (top half)
s1 (.148)
4
s2 (.185)
s1 (.148)
2
5
s2 (.185)
s3 (.667)
I1
d3
s1 (.148)
(.54)
6
s2 (.185)
s3 (.667)
1
$15,000
s3 (.667)
d1
d2
$10,000
$14,000
$8,000
$18,000
$12,000
$6,000
$16,000
$21,000
Slide 41
Example: Burger Prince

Decision Tree (bottom half)
1
I2
d1
(.46)
7
s1 (.696)
$10,000
s2 (.217)
$15,000
s3 (.087)
s1 (.696)
d2
3
8
d3
9
$14,000
$8,000
s2 (.217)
$18,000
s3 (.087)
$12,000
s1 (.696)
$6,000
s2 (.217)
s3 (.087)
$16,000
$21,000
Slide 42
Example: Burger Prince
d1
$17,855
2
d2
4
EMV = .148(10,000) + .185(15,000)
+ .667(14,000) = $13,593
5
EMV = .148 (8,000) + .185(18,000)
+ .667(12,000) = $12,518
6
EMV = .148(6,000) + .185(16,000)
+.667(21,000) = $17,855
7
EMV = .696(10,000) + .217(15,000)
+.087(14,000)= $11,433
8
EMV = .696(8,000) + .217(18,000)
+ .087(12,000) = $10,554
9
EMV = .696(6,000) + .217(16,000)
+.087(21,000) = $9,475
d3
I1
(.54)
1
d1
I2
(.46)
d2
3
$11,433
d3
Slide 43
Example: Burger Prince

Decision Strategy Assuming the Survey is Undertaken:
• If the outcome of the survey is favorable, choose
Model C.
• If it is unfavorable, choose Model A.
Slide 44
Example: Burger Prince

Question:
Should the survey be undertaken?

Answer:
If the Expected Value with Sample Information
(EVwSI) is greater, after deducting expenses, than
the Expected Value without Sample Information
(EVwoSI), the survey is recommended.
Slide 45
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
Slide 46
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.
Slide 47
Example: Burger Prince

Efficiency of Sample Information
The efficiency of the survey:
EVSI/EVPI = ($900.88)/($2000) = .4504
Slide 48
The End of Chapter 9
Slide 49