Chapter 4
Decision Trees and
Utility Theory
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to:
• Develop accurate and useful
decision trees
• Revise probability estimates using
Bayesian Analysis
• Understand the importance and
use of utility theory in decision
making
• Use computers to solve more
complex decision problems
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter Outline
4.1 Introduction
4.2 Decision Trees
4.3 How Probability Values Are
Estimated by Bayesian Analysis
4.4 Utility Theory
4.5 Sensitivity Analysis
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Introduction
Decision trees enable one to look
at decisions:
• with many alternatives and
states of nature
• which must be made in
sequence
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Decision Trees
A graphical representation where:
a decision node from which one
of several alternatives may be
chosen
 a state-of-nature node out of
which one state of nature will
occur
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Thompson’s Decision
Tree Fig. 4.1
A State
of Nature
Node
1
Favorable Market
Unfavorable Market
A
Decision
Node
Favorable Market
Construct
Small
Plant2
Unfavorable Market
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Five Steps to
Decision Tree Analysis
1. Define the problem
2. Structure or draw the decision
tree
3. Assign probabilities to the states
of nature
4. Estimate payoffs for each
possible combination of
alternatives and states of nature
5. Solve the problem by computing
expected monetary values
(EMVs) for each state of nature
node.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-7
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Thompson’s Decision
Tree Fig. 4.2
A State of
Nature
Node
1
EMV
=$10,000
A
Decision
Node Construct
Small Plant 2
EMV
=$40,000
Favorable
Market (0.5) $200,000
Unfavorable
Market (0.5) -$180,000
Favorable
Market (0.5) $100,000
Unfavorable
Market (0.5) -$20,000
0
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-8
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Decision Tree Analysis on
R&D Projects
Define problem
Test solution
Discovery of a new,
unpatentable process
Traditional decision tree with
expected net present values
(ENPV) as outcomes
Collected both probability
and monetary values:
technical success, significant
market, commercial success
Traditional decision tree
analysis
Analyzed risks of the process
Analyze results
ENPV was $3.2 million
Implement results
Decision made to investigate
further. Field testing resulted
in cancellation
Develop model
Acquire data
Develop solution
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Thompson’s Decision
Tree -Fig. 4.3
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-10
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Thompson’s Decision
Tree -Fig. 4.4
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-11
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Thompson Decision Tree
Problem Using QM for
Windows
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-12
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Thompson Decision
Tree Problem using
Excel
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-13
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Expected Value of
Sample Information
Expected
value of best
EVSI =
decision with
Expected value
sample
of best decision
information,
without sample
assuming no
information
cost to gather
it
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-14
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Estimating Probability
Values by Bayesian
Analysis
• Management experience or
intuition
• History
• Existing data
• Need to be able to revise
probabilities based upon new
data
Bayes Theorem
Prior
probabilities
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
New data
4-15
Posterior
probabilities
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Table 4.1
Market Survey Reliability in Predicting Actual
States of Nature
Actual States of Nature
Result of Survey
Favorable
Market (FM)
Unfavorable
Market (UM)
Positive (predicts
favorable market
for product)
Negative (predicts
unfavorable
market for
product)
P(survey positive|FM)
= 0.70
P(survey positive|UM)
= 0.20
P(survey
negative|FM) = 0.30
P(survey negative|UM)
= 0.80
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-16
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Table 4.2
Probability Revisions Given a Positive Survey
Conditional
Probability
State
P(Survey
of
positive|State
Nature of Nature
FM
UM
0.70
0.20
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
* 0.50
* 0.50
4-17
0.35
0.35 = 0.78
0.45
0.10
0.10 = 0.22
0.45
0.45
1.00
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Table 4.3
Probability Revisions Given a
Negative Survey
Conditional
Probability
State P(Survey
of
negative|State
Nature of Nature)
FM
0.30
* 0.50
0.15
0.15= 0.27
0.55
UM
0.80
* 0.50
0.40
0.40 = 0.73
0.55
1.00
0.55
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-18
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Utility Theory
$2,000,000
$0
Heads
(0.5)
Tails
(0.5)
$5,000,000
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-19
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Utility Assessment
• Utility assessment assigns the
worst outcome a utility of 0, and
the best outcome, a utility of 1.
• A standard gamble is used to
determine utility values.
• When you are indifferent, the
utility values are equal.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-20
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Standard Gamble for
Utility Assessment - Fig. 4.6
(p)
Best outcome
Utility = 1
(1-p)
Worst outcome
Utility = 0
Other outcome
Utility = ??
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-21
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Figure 4.7
p= 0.80 $10,000
U($10,000) = 1.0
(1-p)= 0.20 0
U(0)=0
$5,000
U($5,000)=p
=0.80
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-22
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Utility
Utility Curve for Jane
Dickson Fig. 4.8
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
$-
$2,000
$4,000
$6,000
$8,000
$10,000
Monetary Value
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-23
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Utility
Preferences for Risk
Fig. 4.9
Monetary Outcome
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-24
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Decision Facing Mark
Simkin
Fig. 4.10
Tack lands
point up (0.45)
$10,000
Tack lands
point down (0.55)
-$10,000
Mark does not play the game
0
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-25
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Utility Curve for Mark
Simkin
Fig. 4.11
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-$20,000
-$10,000
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
$0
$10,000
4-26
$20,000
$30,000
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Using Expected
Utilities in Decision
Making - Fig. 4.12
Utility
Tack lands
point up (0.45)
0.30
Tack lands
point down (0.55)
0.05
Don’t play
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-27
0.15
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Calculations for
Thompson Lumber
Sensitivity Analysis
EMV(node 1) = ($106,400)p + ( 1 - p )($2,000)
= $104,000 p + 2,400
Equating the EMV(node 1) to the EMV of not
conducting the survey, we have
$104,000 p + $2,400 = $40,000
$104,000 p = $37,000
or
p=
$37,000
= 0.36
$104,000
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4-28
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458