Decision analysis

1Slide© 2005 Thomson/South-Western
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

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.

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.

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.

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.

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.

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.

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.

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.)

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.

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 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 the
decision that has the largest single value in the
payoff table.
Maximum
Decision Payoff
d1 4
d2 3
d3 5
Maximax
payoff
Maximax
decision

Example: Optimistic Approach
 Solution Spreadsheet
A B C D E F
1
2
3 Decision Maximum Recommended
4 Alternative s1 s2 s3 Payoff Decision
5 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

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.
Minimum
Decision Payoff
d1 -2
d2 -1
d3 -3
Maximin
decision
Maximin
payoff

Example: Conservative Approach
A B C D E F
1
2
3 Decision Minimum Recommended
4 Alternative s1 s2 s3 Payoff Decision
5 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

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:
s1 s2 s3
d1 0 1 1
d2 4 2 0
d3 3 0 2
Example: Minimax Regret Approach

For each decision list the maximum regret.
Choose the decision with the minimum of these
values.
Maximum
Decision Regret
d1 1
d2 4
d3 3
Minimax
decision
Minimax
regret

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 Decision
11 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

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.

 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 decision
alternative di and state of nature sj
Expected Value of a Decision Alternative
EV( ) ( )d P s Vi j ij
j
N
1
EV( ) ( )d P s Vi j ij
j
N
1

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.

Payoff Table
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A $10,000 $15,000 $14,000
Model B $ 8,000 $18,000 $12,000
Model C $ 6,000 $16,000 $21,000

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.

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

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

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
Expected Value Approach

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.

 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).

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

 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

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.

Risk Profile
 Model C Decision Alternative
.10
.20
.30
.40
.50
5 10 15 20 25
Probability

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.

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.

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.

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.

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.

 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).

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.

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
Should Burger Prince have the survey performed
by Stanton Marketing?
Sample Information

Influence Diagram
Restaurant
Size
Profit
Avg. Number
of Customers
Per Hour
Market
Survey
Results
Market
Survey
Decision
Chance
Consequence

Favorable
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
Posterior Probabilities

Unfavorable
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

A B C D E
1
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 Probabilities
12 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) =

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

 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

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
Decision Tree

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.

Efficiency of Sample Information
The efficiency of the survey:
EVSI/EVPI = ($900.88)/($2000) = .4504

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.

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 | Ij)
n
1i
iij
iij
j
ji
)S(P)S|I(P
)S(P)S|I(P
)I(P
)IS(P

Home Work
 Problem 13-10
 Problem 13-21
 Due Date: Nov 11, 2008

Decision analysis

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