Transcript Chapter 04

Chapter 9 - Decision Analysis - Part II
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Decision Making under Uncertainty with Probabilities
Decision Tree
Expected Value of Perfect Information
Bayes’ Theorem and Posterior Probabilities
Expected Value of Sample Information
Developing a Decision Strategy
Efficiency of Sample Information
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Decision Making Under Uncertainty
With Probabilities
Expected Value Approach
– The decision maker generally will have some information
about the relative likelihood of the possible states of
nature. These are referred to as the prior probabilities.
– 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.
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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:
EV(d i ) 
where:
n
 P(s j )Vij
j 1
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
3 - Chap 09
Example: Burger Prince
Burger Prince Restaurant is planning to open a new restaurant on
Main Street. It has three different models under consideration, each
with a different seating capacity.
Three decisions:
d1: Model A: Small capacity
d2: Model B: Medium capacity, and
d3: Model C: Large capacity
Burger Prince estimates that the average number of customers per
hour will be 80, 100, or 120.
Three states of nature:
s1 = 80
s2 = 100, and
s3 = 120
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Example: Burger Prince
The profit payoff table for the three models is as follows:
PAYOFF TABLE
Decision
Alternative
d1: Model A
d2: Model B
d3: Model C
State of Nature
Average Number of Customers Per Hr
s1 = 80
s2 = 100
s3 = 120
$10,000
$15,000
$14,000
$8,000
$18,000
$12,000
$6,000
$16,000
$21,000
Burger Prince has estimated that the prior probabilities of
S1, S2 and S3 are 0.4, 0.2 and 0.4 respectively.
Question: Using the expected value approach, which model
should Burger Prince choose in order to maximize the profit
payoff?
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Example: Burger Prince
Expected Value Approach
PAYOFF TABLE
Decision
Alternative
d1: Model A
d2: Model B
d3: Model C
Probability
State of Nature
Average Number of Customers Per Hr
s1 = 80
s2 = 100
s3 = 120
$10,000
$15,000
$14,000
$8,000
$18,000
$12,000
$6,000
$16,000
$21,000
0.4
0.2
0.4
Maximum Expected Value
Expected
Value
$12,600
$11,600
$14,000
Recommended
Decision
d3: Model C
$14,000
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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 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.
7 - Chap 09
Example: Burger Prince
Payoffs
• Decision Tree
2
d1
1
.4
.2
.4
10,000
15,000
14,000
d2
d3
s1
s2
s3
3
4
s1
s2
s3
s1
s2
s3
.4
.2
.4
.4
.2
.4
8,000
18,000
12,000
6,000
16,000
21,000
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Steps to Conduct Decision Tree Analysis
1.
2.
3.
4.
5.
6.
Define the problem.
Structure or draw the decision tree.
Assign probabilities to the states of nature.
Estimate payoffs for each possible combination of alternatives
and states of nature.
Solve the problem by computing expected monetary value
(EMV) for each state of nature node.
Choose the best decision alternative at each decision node
and eliminate the other decisions.
Example: Burger Prince

Expected Value For Each Decision
d1
EMV = .4(10,000) + .2(15,000) + .4(14,000)
2
= $12,600
Model A
1
Model B
Model C
d2
EMV = .4(8,000) + .2(18,000) + .4(12,000)
3
= $11,600
d3
EMV = .4(6,000) + .2(16,000) + .4(21,000)
4
= $14,000
Choose the model with the largest EMV, Model C.
10 - Chap 09
Decision Making Under Certainty
Expected Value of Perfect Information (EVPI)
• 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.
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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 (EPPI).
– Step 3:
Subtract the EV of the optimal decision from the amount
determined in step (2).
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Expected Value of Perfect Information
Example: Burger Prince
Expected Value of Perfect Information
PAYOFF TABLE
Decision
Alternative
d1: Model A
d2: Model B
d3: Model C
Probability
State of Nature
Average Number of Customers Per Hr
s1 = 80
s2 = 100
s3 = 120
$10,000
$15,000
$14,000
$8,000
$18,000
$12,000
$6,000
$16,000
$21,000
0.4
0.2
0.4
Maximum Expected Value
Maximum Payoff
$10,000
$18,000
$21,000
Expected
Value
$12,600
$11,600
$14,000
Recommended
Decision
d3: Model C
$14,000
EPPI
$16,000
EVPI
$2,000
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Expected Value of Perfect Information
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
14 - Chap 09
Expected Value of Sample Information
Example: Burger Prince
Sample Information
Burger Prince must decide whether or not to purchase a marketing
survey from Stanton Marketing for $500. The results of the survey
are "favorable (F)" or "unfavorable (U)". The conditional
probabilities are:
P(favorable | 80 customers per hour) = P(F|S1) = 0.2
P(favorable | 100 customers per hour) = P(F|S2) = 0.5
P(favorable | 120 customers per hour) = P(F|S3) = 0.9
Should Burger Prince have the survey performed by Stanton
Marketing?
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Expected Value of Sample Information
Example: Burger Prince
Legend:
Decision
Chance
Consequence
Market
Survey
Results
Avg. Number
of Customers
Per Hour
Market
Survey
Restaurant
Size
Profit
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A complete Tree Diagram
d1
d2
A1 No survey
d3
A2 Survey
F
U
d1
d2
d3
d1
d2
d3
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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
for decision trees.
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Computing Posterior Probabilities
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.
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Example: Burger Prince
Posterior Probabilities
Favorable
State Prior Conditional Joint Posterior
S1
.4
.2
.08
.148
S2
.2
.5
.10
.185
S3
.4
.9
.36
.667
Total .54
1.000
P(F) = .54
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Example: Burger Prince
Posterior Probabilities
Unfavorable
State Prior Conditional Joint Posterior
S1
.4
.8
.32
.696
S2
.2
.5
.10
.217
S3
.4
.1
.04
.087
Total .46
1.000
P(U) = .46
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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.
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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).
23 - Chap 09
Example: Burger Prince
Decision Tree (top half)
s1 (.148)
d1
2
F
(.54)
d2
4
s3 (.667)
s1 (.148)
5
d3
s2 (.185)
s3 (.667)
s1 (.148)
6
1
s2 (.185)
s2 (.185)
s3 (.667)
$10,000
$15,000
$14,000
$8,000
$18,000
$12,000
$6,000
$16,000
$21,000
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Example: Burger Prince
Decision Tree (bottom half)
1
s1 (.696)
U
(.46)
d1
d2
3
7
8
s2 (.217)
s3 (.087)
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
25 - Chap 09
Example: Burger Prince
d1
$17,855
2
F
(.54)
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
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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
1
d1
U
(.46)
d2
3
$11,433
d3
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A complete Tree Diagram
$14,000
d1
d2
A1 No survey
$14,000
EMV = .4(8,000) + .2(18,000) + .4(12,000)
= $11,600
d3
$17,855
A2 Survey
$14,900
EMV = .4(10,000) + .2(15,000) + .4(14,000)
= $12,600
F
(.54)
U
(.46)
EMV = .4(6,000) + .2(16,000) + .4(21,000)
= $14,000
d1
d2
d3
d1
d2
$11,433 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
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
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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 higher than the cost of the survey, the survey should
be purchased.
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Developing a Decision Strategy
Example: Burger Prince
General Recommendation:
If the survey cost is less than the EVSI,
Burger Prince should hire the market research firm to conduct
a survey to study the future market.
– If the survey result is favourable, Burger Prince should
choose model C to obtain an expected payoff of $17,855.
– If the survey result is unfavourable, Burger Prince should
choose model A to obtain an expected payoff of $11,433.
If the survey cost is higher than the EVSI,
Burger Prince should not hire the market research firm to
conduct a survey to study the future market,
Burger Prince should choose model C to obtain an expected
payoff of $14,000.
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Efficiency of Sample Information
• Efficiency of sample information measure the value of the market
research 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.
EVSI
Efficencyof sample information 
* 100%
EVPI
30 - Chap 09
Example: Burger Prince
• Efficiency of Sample Information:
The efficiency of the survey:
EVSI/EVPI = ($900.88)/($2000) = .4504
31 - Chap 09
Chapter Summary
Decision with
PRIOR information
Decision with
PERFECT
information
Decision with
POSTERIOR information
EPSI
EV*
EPPI
EVSI
EVPI
EV* = Expected payoff with PRIOR information
EPSI = Expected payoff with POSTERIOR information
EVSI = Expected value of SAMPLE (POSTERIOR) information
EPPI = Expected payoff with PERFECT information
EVPI = Expected value of PERFECT information
32 - Chap 09