Anderson
Sweeney
Williams
QUANTITATIVE
METHODS FOR
BUSINESS 8e
Slides Prepared by JOHN LOUCKS
© 2001 South-Western College Publishing/Thomson Learning
Slide 1
Chapter 4
Decision Analysis, Part B




Expected Value of Perfect Information
Decision Analysis with Sample Information
Developing a Decision Strategy
Expected Value of Sample Information
Slide 2
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
Slide 3
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.
Slide 4
Example: Burger Prince

Decision Tree
Payoffs
2
d1
d3
.4
.2
10,000
15,000
.4
14,000
d2
1
s1
s2
s3
3
s1
s2
s3
.4
.2
8,000
18,000
.4
12,000
4
s1
s2
s3
.4
6,000
.2
16,000
.4
21,000
Slide 5
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.
Slide 6
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.
Slide 7
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).
Slide 8
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
Slide 9
Example: Burger Prince

Spreadsheet for Expected Value of Perfect Information
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
F
Recommended
Decision
d3 = Model C
EVPI
2000
Slide 10
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.
Slide 11
Example: Burger Prince
Risk Profile for the Model C Decision Alternative
.50
Probability

.40
.30
.20
.10
5
10
15
20
25
Slide 12
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.
Slide 13
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.
Slide 14
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.
Slide 15
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.
Slide 16
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).
Slide 17
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.
Slide 18
Example: Burger Prince

Sample Information
Burger Prince must decide whether or not to
purchase a marketing survey from Stanton Marketing
for $1,000. The results of the survey are "favorable" or
"unfavorable". The conditional probabilities are:
P(favorable | 80 customers per hour) = .2
P(favorable | 100 customers per hour) = .5
P(favorable | 120 customers per hour) = .9
Should Burger Prince have the survey performed
by Stanton Marketing?
Slide 19
Example: Burger Prince

Influence Diagram
Legend:
Decision
Chance
Consequence
Market
Survey
Market
Survey
Results
Avg. Number
of Customers
Per Hour
Restaurant
Size
Profit
Slide 20
Example: Burger Prince

Posterior Probabilities
Favorable
State
80
100
120
Prior
.4
.2
.4
Conditional
.2
.5
.9
Total
Joint
.08
.10
.36
.54
Posterior
.148
.185
.667
1.000
P(favorable) = .54
Slide 21
Example: Burger Prince

Posterior Probabilities
Unfavorable
State
80
100
120
Prior
.4
.2
.4
Conditional
.8
.5
.1
Total
Joint
.32
.10
.04
.46
Posterior
.696
.217
.087
1.000
P(unfavorable) = .46
Slide 22
Example: Burger Prince

Formula Spreadsheet for Posterior Probabilities
A
B
Market Research Favorable
Prior
State of Nature Probabilities
s1 = 80
0.4
s2 = 100
0.2
s3 = 120
0.4
C
D
1
2
Conditional
Joint
3
Probabilities
Probabilities
4
0.2
=B4*C4
5
0.5
=B5*C5
6
0.9
=B6*C6
7
P(Favorable) = =SUM(D4:D6)
8
9 Market Research Unfavorable
10
Prior
Conditional
Joint
11 State of Nature Probabilities Probabilities
Probabilities
12
s1 = 80
0.4
0.8
=B12*C12
13
s2 = 100
0.2
0.5
=B13*C13
14
s3 = 120
0.4
0.1
=B14*C14
15
P(Unfavorable) = =SUM(D12:D14)
E
Posterior
Probabilities
=D4/$D$7
=D5/$D$7
=D6/$D$7
Posterior
Probabilities
=D12/$D$15
=D13/$D$15
=D14/$D$15
Slide 23
Example: Burger Prince

Solution Spreadsheet for Posterior Probabilities
A
B
Market Research Favorable
Prior
State of Nature Probabilities
s1 = 80
0.4
s2 = 100
0.2
s3 = 120
0.4
C
D
1
2
Conditional
Joint
3
Probabilities Probabilities
4
0.2
0.08
5
0.5
0.10
6
0.9
0.36
7
P(Favorable) =
0.54
8
9 Market Research Unfavorable
10
Prior
Conditional
Joint
11 State of Nature Probabilities Probabilities Probabilities
12
s1 = 80
0.4
0.8
0.32
13
s2 = 100
0.2
0.5
0.10
14
s3 = 120
0.4
0.1
0.04
15
P(Favorable) =
0.46
E
Posterior
Probabilities
0.148
0.185
0.667
Posterior
Probabilities
0.696
0.217
0.087
Slide 24
Example: Burger Prince

Decision Tree (top half)
s1 (.148)
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
Slide 25
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
Slide 26
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
Slide 27
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.
Slide 28
Example: Burger Prince

Efficiency of Sample Information
The efficiency of the survey:
EVSI/EVPI = ($900.88)/($2000) = .4504
Slide 29
The End of Chapter 4, Part B
Slide 30