Matakuliah
Tahun
: K0442-Metode Kuantitatif
: 2009
Decision Analysis
Pertemuan 2
Material Outline
• Problem Formulation
• 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
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 Diagram
• 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
d1
Decisions
4
d2
d3
s2
4
0
1
5
s3
-2
3
-1
-3
Example
• 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
Example
• 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
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; 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
Example
• 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
Decision Making with Probabilities
• 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.
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 (di )   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
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
Model A
Model B
Model C
$10,000
$ 8,000
$ 6,000
$15,000
$18,000
$16,000
$14,000
$12,000
$21,000
Example: Burger Prince
• 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.
Example: Burger Prince
• Decision Tree
2
s1
s2
s3
.4
.2
10,000
15,000
.4
d1
1
Payoffs
14,000
d2
3
d3
s1
s2
s3
.4
.2
8,000
18,000
.4
12,000
4
s1
.4
s2
.2
s3
.4
6,000
16,000
21,000
Example: Burger Prince

Expected Value For Each Decision
d1
EMV = .4(10,000) + .2(15,000) + .4(14,000)
= $12,600
2
Model A
1
Model B
Model C
d2
EMV = .4(8,000) + .2(18,000) + .4(12,000)
= $11,600
3
d3
EMV = .4(6,000) + .2(16,000) + .4(21,000)
= $14,000
4
Choose the model with largest EV, Model C.
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.
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).
Example: Burger Prince
• 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
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.
Example: Burger Prince
• Risk Profile for the Model C Decision Alternative
.50
Probability
.40
.30
.20
.10
5
10
15
20
25
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 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 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.
– 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.
Expected Value of Sample Information
• EVSI Calculation
– Step 1:
Determine the optimal decision and its expected return for
the possible outcomes of the sample or survey 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.
Example: Burger Prince
• 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?
Example: Burger Prince
• Influence Diagram
Legend:
Decision
Chance
Consequence
Market
Survey
Market
Survey
Results
Avg. Number
of Customers
Per Hour
Restaurant
Size
Profit
Example: Burger Prince
• Posterior Probabilities
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
Example: Burger Prince
• 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
Example: Burger Prince
s1 (.148)
• Decision Tree (top half)
4
d1
s2 (.185)
s3 (.667)
$10,000
$15,000
$14,000
s1 (.148)
d2
2
I1
(.54)
5
d3
6
1
$8,000
s2 (.185)
$18,000
s3 (.667)
$12,000
s1 (.148)
$6,000
s2 (.185)
$16,000
s3 (.667)
$21,000
Example: Burger Prince
• Decision Tree (bottom half)
1
s1 (.696)
I2
(.46)
d1
d2
3
7
s2 (.217)
s3 (.087)
8
s1 (.696)
s2 (.217)
s3 (.087)
$10,000
$15,000
$14,000
$8,000
$18,000
$12,000
d3
9
s1 (.696)
s2 (.217)
s3 (.087)
$6,000
$16,000
$21,000
Example: Burger Prince
d1
$17,855
d2
2
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
Example: Burger Prince
• 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.
Example: Burger Prince
• Efficiency of Sample Information
The efficiency of the survey:
EVSI/EVPI = ($900.88)/($2000) = .4504