Chapter 13
Decision Analysis
Problem Formulation
 Decision Making without
Probabilities
 Decision Making with Probabilities
 Risk Analysis and Sensitivity
Analysis
 Decision Analysis with Sample
Information
 Computing Branch Probabilities

© 2005 Thomson/South-Western
Slide 1
Problem Formulation



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.
© 2005 Thomson/South-Western
Slide 2
Influence Diagrams





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.
© 2005 Thomson/South-Western
Slide 3
Payoff Tables



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.
© 2005 Thomson/South-Western
Slide 4
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 square nodes
correspond to the decision alternatives.
© 2005 Thomson/South-Western
Slide 5
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.
© 2005 Thomson/South-Western
Slide 6
Decision Making without Probabilities

Three commonly used criteria for
decision making when probability
information regarding the likelihood
of the states of nature is unavailable
are:
•the optimistic approach
•the conservative approach
•the minimax regret approach.
© 2005 Thomson/South-Western
Slide 7
Optimistic Approach
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.

© 2005 Thomson/South-Western
Slide 8
Conservative Approach



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.)
© 2005 Thomson/South-Western
Slide 9
Minimax Regret Approach




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.
© 2005 Thomson/South-Western
Slide 10
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
© 2005 Thomson/South-Western
4
0
1
4
3
5
-2
-1
-3
Slide 11
Example: Optimistic Approach
An optimistic decision maker would use the
optimistic (maximax) approach. We choose the
decision that has the largest single value in the
payoff table.
Maximax
decision
Decision
d1
d2
d3
© 2005 Thomson/South-Western
Maximum
Payoff
4
Maximax
payoff
3
5
Slide 12
Example: Optimistic Approach

1
2
3
4
5
6
7
8
9
Solution Spreadsheet
A
B
PAYOFF TABLE
Decision
Alternative
d1
d2
d3
C
E
F
State of Nature
s1
s2
s3
4
4
-2
0
3
-1
1
5
-3
Maximum
Payoff
4
3
5
Recommended
Decision
Best Payoff
5
© 2005 Thomson/South-Western
D
d3
Slide 13
Example: Conservative Approach
A conservative decision maker would use the
conservative (maximin) approach. List the minimum
payoff for each decision. Choose the decision with
the maximum of these minimum payoffs.
Maximin
decision
Decision
d1
d2
d3
© 2005 Thomson/South-Western
Minimum
Payoff
-2
-1
-3
Maximin
payoff
Slide 14
Example: Conservative Approach

Solution Spreadsheet
A
B
C
D
1 PAYOFF TABLE
2
3
Decision
State of Nature
4 Alternative
s1
s2
s3
5
d1
4
4
-2
6
d2
0
3
-1
7
d3
1
5
-3
8
9
Best Payoff
© 2005 Thomson/South-Western
E
F
Minimum
Payoff
-2
-1
-3
Recommended
Decision
d2
-1
Slide 15
Example: 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; etc. The resulting regret table is:
d1
d2
d3
© 2005 Thomson/South-Western
s1
s2
s3
0
4
3
1
2
0
1
0
2
Slide 16
Example: Minimax Regret Approach
For each decision list the maximum regret.
Choose the decision with the minimum of these
values.
Minimax
decision
Maximum
Decision
Regret
d1
1
d2
4
d3
3
© 2005 Thomson/South-Western
Minimax
regret
Slide 17
Example: Minimax Regret Approach

Solution Spreadsheet
A
B
C
D
PAYOFF TABLE
Decision
State of Nature
Alternative
s1
s2
s3
d1
4
4
-2
d2
0
3
-1
d3
1
5
-3
1
2
3
4
5
6
7
8 OPPORTUNITY LOSS TABLE
9
Decision
State of Nature
10 Alternative
s1
s2
s3
11
d1
0
1
1
12
d2
4
2
0
13
d3
3
0
2
14
Minimax Regret Value
© 2005 Thomson/South-Western
E
Maximum
Regret
1
4
3
1
F
Recommended
Decision
d1
Slide 18
Decision Making with Probabilities

Expected Value Approach
• If probabilistic information regarding the 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.
© 2005 Thomson/South-Western
Slide 19
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:
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
© 2005 Thomson/South-Western
Slide 20
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 of
customers per hour will be 80, 100, or
120. The payoff table for the three
models is on the next slide.
© 2005 Thomson/South-Western
Slide 21
Payoff Table
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A
Model B
Model C
$10,000
$ 8,000
$ 6,000
© 2005 Thomson/South-Western
$15,000
$18,000
$16,000
$14,000
$12,000
$21,000
Slide 22
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.
© 2005 Thomson/South-Western
Slide 23
Decision Tree
d1
1
d2
d3
2
3
4
© 2005 Thomson/South-Western
s1
s2
s3
.4
.2
.4
s1
.4
s2
s3
.2
s1
s2
s3
.4
.4
.2
.4
Payoffs
10,000
15,000
14,000
8,000
18,000
12,000
6,000
16,000
21,000
Slide 24
Expected Value for Each Decision
Model A
1
d1
Model B d2
Model C
EMV = .4(10,000) + .2(15,000) + .4(14,000)
= $12,600
2
EMV = .4(8,000) + .2(18,000) + .4(12,000)
= $11,600
3
d3 EMV = .4(6,000) + .2(16,000) + .4(21,000)
4
= $14,000
Choose the model with largest EV, Model C.
© 2005 Thomson/South-Western
Slide 25
Expected Value Approach

Solution Spreadsheet
A
B
C
D
1 PAYOFF TABLE
2
3 Decision
State of Nature
4 Alternative s1 = 80 s2 = 100 s3 = 120
5 d1 = Model A 10,000 15,000 14,000
6 d2 = Model B 8,000 18,000 12,000
7 d3 = Model C 6,000 16,000 21,000
8 Probability
0.4
0.2
0.4
9
Maximum Expected Value
© 2005 Thomson/South-Western
E
F
Expected
Value
12600
11600
14000
Recommended
Decision
d3 = Model C
14000
Slide 26
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.
© 2005 Thomson/South-Western
Slide 27
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).
© 2005 Thomson/South-Western
Slide 28
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
© 2005 Thomson/South-Western
Slide 29
Expected Value of Perfect Information

Spreadsheet
A
B
C
D
E
1 PAYOFF TABLE
2
3 Decision
State of Nature
Expected
4 Alternative s1 = 80 s2 = 100 s3 = 120
Value
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
8 Probability
0.4
0.2
0.4
9
Maximum Expected Value
14000
10
11
Maximum Payoff
EVwPI
12
10,000 18,000 21,000
16000
© 2005 Thomson/South-Western
F
Recommended
Decision
d3 = Model C
EVPI
2000
Slide 30
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.
© 2005 Thomson/South-Western
Slide 31
Risk Profile

Model C Decision Alternative
Probability
.50
.40
.30
.20
.10
5
© 2005 Thomson/South-Western
10
15
20
25
Slide 32
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.
© 2005 Thomson/South-Western
Slide 33
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 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.
© 2005 Thomson/South-Western
Slide 34
Computing Branch Probabilities

Branch (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.
© 2005 Thomson/South-Western
Slide 35
Computing Branch Probabilities

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 gives
the posterior probability distribution.
© 2005 Thomson/South-Western
Slide 36
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.
© 2005 Thomson/South-Western
Slide 37
Expected Value of Sample Information

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).
© 2005 Thomson/South-Western
Slide 38
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.
© 2005 Thomson/South-Western
Slide 39
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?
© 2005 Thomson/South-Western
Slide 40
Influence Diagram
Decision
Chance
Consequence
Market
Survey
Market
Survey
Results
Avg. Number
of Customers
Per Hour
Restaurant
Size
Profit
© 2005 Thomson/South-Western
Slide 41
Posterior Probabilities
Favorable
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
© 2005 Thomson/South-Western
Slide 42
Posterior Probabilities
Unfavorable
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
© 2005 Thomson/South-Western
Slide 43
Posterior Probabilities

Solution Spreadsheet
A
B
Market Research Favorable
Prior
State of Nature Probabilities
s1 = 80
0.4
s2 = 100
0.2
s3 = 120
0.4
C
1
2
Conditional
3
Probabilities
4
0.2
5
0.5
6
0.9
7
P(Favorable) =
8
9 Market Research Unfavorable
10
Prior
Conditional
11 State of Nature Probabilities Probabilities
12
s1 = 80
0.4
0.8
13
s2 = 100
0.2
0.5
14
s3 = 120
0.4
0.1
15
P(Favorable) =
© 2005 Thomson/South-Western
D
E
Joint
Probabilities
0.08
0.10
0.36
0.54
Posterior
Probabilities
0.148
0.185
0.667
Joint
Probabilities
0.32
0.10
0.04
0.46
Posterior
Probabilities
0.696
0.217
0.087
Slide 44
Decision Tree

Top Half
s1 (.148)
d1
2
I1
(.54)
d2
4
5
d3
1
© 2005 Thomson/South-Western
s2 (.185)
$15,000
s3 (.667)
$14,000
s1 (.148)
$8,000
s2 (.185)
$18,000
s3 (.667)
s1 (.148)
6
$10,000
s2 (.185)
s3 (.667)
$12,000
$6,000
$16,000
$21,000
Slide 45
Decision Tree

Bottom Half
1
I2
(.46)
d1
3
d2
7
8
d3
9
s1 (.696)
$10,000
s2 (.217)
$15,000
s3 (.087)
s1 (.696)
s2 (.217)
s3 (.087)
$14,000
$8,000
$18,000
$12,000
$6,000
s1 (.696)
s2 (.217)
s3 (.087) $16,000
$21,000
© 2005 Thomson/South-Western
Slide 46
Decision Tree
$17,855
I1
(.54)
2
d1
d2
d3
1
I2
(.46)
d1
3
$11,433
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
© 2005 Thomson/South-Western
Slide 47
Expected Value of Sample Information
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.
© 2005 Thomson/South-Western
Slide 48
Efficiency of Sample Information
The efficiency of the survey:
EVSI/EVPI = ($900.88)/($2000) = .4504
© 2005 Thomson/South-Western
Slide 49
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.
© 2005 Thomson/South-Western
Slide 50
Bayes’ theory
Si: State of Nature (i = 1 ~ n)
P(Si): Prior Probability
Ij: Professional Information (Experiment)( j = 1 ~ n)
P(Ij | Si): Conditional Probability
P(Ij  Si) = P(Si  Ij): Joint Probability
P(Si | Ij): Posterior Probability
P(Si  I j )
P(I j | Si )P(Si )
P(Si | Ij) 

n
P( I j )
 P(I j | Si )P(Si )
© 2005 Thomson/South-Western
i 1
Slide 51
Home Work
 Problem
13-10
 Problem
13-21
 Due
Date: Nov 11, 2008
© 2005 Thomson/South-Western
Slide 52