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Chapter 16
Decision Analysis
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Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution
without
the prior written
reserved.
No reproduction
or distribution without
Frederick
S. HillierEducation.
∎ Gerald J. Lieberman
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the prior written consent of McGraw-Hill Education.
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Introduction
The focus of previous chapters:
• Decision-making when consequences of
alternative decisions are known with a
reasonable degree of certainty.
Decision analysis:
• Addresses decision-making in the face of great
uncertainty.
• Provides framework and methodology.
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16.1 A Prototype Example
Goferbroke Company owns land that may
contain oil.
• Geologist reports 25% chance of oil.
• Another company offers to purchase land for
$90k.
• Goferbroke option: drill for oil at cost of $100k.
• Potential gross profit $800k (net $700k).
• Potential net loss of $100k if land is dry.
• Need to decide whether to drill or sell.
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A Prototype Example
Figure 16.1 Prospective profits for the Goferbroke Company
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16.2 Decision Making Without
Experimentation
Decision maker must choose an alternative.
• From a set of feasible decision alternatives.
State of nature.
• Factors in place at the time of the decision that
affect the outcome.
Payoff table shows payoff for each
combination of decision alternative and state
of nature.
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Decision Making Without Experimentation
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Analogy with game theory.
• Decision maker is player 1.
• Chooses one of the decision alternatives.
• Nature is player 2.
• Chooses one of the possible states of nature.
• Each combination of decision and state of nature
results in a payoff.
• Payoff table should be used to find an optimal
alternative for the decision maker.
• According to an appropriate criterion.
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Decision Making Without Experimentation
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Differences from game theory.
• Nature is not rational or self-promoting.
• Decision maker likely has information about
relative likelihood of possible states of nature.
• Probability distribution: prior distribution.
• Probabilities: prior probabilities.
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Decision Making Without Experimentation
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Formulation of the prototype example.
TABLE 16.2 Payoff table for the decision analysis formulation of
the first Goferbroke Co. problem
Alternative
State of
Nature: Oil
State of
Nature: Dry
1. Drill for oil
2. Sell the land
700
90
−100
90
Prior probability
0.25
0.75
The maximin payoff criterion:
• Extremely conservative in nature.
• Assumes nature is a malevolent opponent.
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Decision Making Without Experimentation
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The maximum likelihood criterion:
• Identify the most likely state of nature.
• Choose the decision alternative with the
maximum payoff for this state of nature.
• In the example: the decision would be to sell,
since the most likely state of nature is dry.
• Does not permit gambling on a low-probability,
big payoff.
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Decision Making Without Experimentation
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Bayes’ decision rule:
• Commonly used.
• Using the best available estimates of the
probabilities of the states of nature, calculate the
payoff value for each decision alternative.
• Choose the alternative with the maximum
expected payoff value.
• Alternative selected: drill for oil.
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Decision Making Without Experimentation
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Sensitivity analysis with Bayes’ decision rule.
• Assume prior probability of oil, p, is between 15
and 35 percent.
• Figure 16.1 shows plot of expected payoff versus p.
Crossover point.
• Point at which decision shifts from one
alternative to another.
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Decision Making Without Experimentation
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FIGURE 16.1
Graphical display of how
the expected payoff for
each decision alternative
changes when the prior
probability of oil changes
for the first Goferbroke
Co. problem.
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16.3 Decision Making With
Experimentation
Additional testing (experimentation).
• Frequently used to improve preliminary
probability estimates.
• Improved estimates: posterior probabilities.
Continuing with oil drilling example.
• Seismic survey can refine the probability.
• Cost of survey is $30,000.
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Decision Making With Experimentation
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Possible survey findings:
• USS: unfavorable seismic soundings.
• Indicates oil is unlikely.
• FSS: favorable seismic soundings.
• Indicates oil is likely.
Based on past experience with seismic soundings:
P ( USS State = Oil ) = 0.4,
so P(FSS State = Oil) = 1 − 0.4 = 0.6.
P ( USS State = Dry ) = 0.8,
so P(FSS State = Dry) = 1 − 0.8 = 0.2.
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n = number of possible states of nature;
P ( State = state i ) = prior probability that true state of nature is
state i, for i = 1, 2, . . . , n;
Finding = finding from experimentation (a random variable);
Finding j = one possible value of finding;
P ( State = state i Finding = finding j ) = posterior probabilty that true state of nature
is state i, given that Finding = finding j, for
i = 1, 2,..., n.
• Bayes’ theorem.
P ( State = state i Finding = finding j ) =
P ( Finding = finding j State = state i ) P (State = state i )
 P ( Finding = finding j State = state k ) P (State = state k )
n
k =1
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Decision Making With Experimentation
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• If seismic survey finding is USS:
P ( State = Oil Finding = USS ) =
0.4 ( 0.25 )
0.4 ( 0.25 ) + 0.8 ( 0.75 )
P ( State = Dry Finding = USS ) = 1 −
=
1
,
7
=
1
,
2
1 6
= .
7 7
• If finding is FSS:
P ( State = Oil Finding = FSS ) =
0.6 ( 0.25 ) + 0.2 ( 0.75 )
P ( State = Dry Finding = FSS) = 1 −
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0.6 ( 0.25 )
1 1
= .
2 2
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Decision Making With Experimentation
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FIGURE 16.2
Probability tree
diagram for the full
Goferbroke Co.
problem showing all
the probabilities
leading to the
calculation of each
posterior of the state of
nature given the
finding of the seismic
survey.
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Decision Making With Experimentation
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TABLE 16.5 The optimal policy with experimentation, under
Bayes’ decision rule, for the full Goferbroke Co. problem
Finding from
Seismic Survey
Optimal
Alternative
USS
FSS
Sell the land
Drill for oil
Expected Payoff
Expected Payoff
Excluding Cost of Survey Including Cost of Survey
90
300
60
270
Is it worth it to undertake the cost of the
survey?
• Need to determine potential value of the information.
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Decision Making With Experimentation
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Expected value of perfect information:
• Provides an upper bound on the potential value
of the experiment.
• If upper bound is less than experiment cost:
• Forgo the experiment.
• If upper bound is higher than experiment cost:
• Calculate the actual improvement in the expected
payoff.
• Compare this improvement with experiment cost.
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Decision Making With Experimentation
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TABLE 16.6 Expected payoff with perfect information for the full
Goferborke Co. problem
Alternative
State of Nature: Oil
State of Nature: Dry
1. Drill for oil
2. Sell the land
700
90
−100
90
Maximum payoff
Prior probability
700
0.25
90
0.75
Expected payoff with perfect information = 0.25(700) + 0.75(90) = 242.5
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Decision Making With Experimentation
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Expected value of experimentation (EVE):
• The difference between the expected payoff with
experimentation and the expected payoff without
experimentation.
• For the Goforbroke Co.
EVE = 153 − 100 = 53.
• Since this exceeds the experiment cost, the
experiment should be done.
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16.4 Decision Trees
Functions:
• Visually displaying a problem.
• Organizing computational work.
• Especially helpful when a sequence of decisions
must be made.
Constructing the decision tree:
• Should a seismic survey be conducted before a
decision is chosen?
• Which action (drill for oil or sell land) should be
chosen?
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Decision Trees
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Nodes (forks).
• Junction points in the tree.
Branches.
• Lines in the decision tree.
Decision node.
• Indicated by a square.
• Indicates decision needs to be made at that
point.
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Decision Trees
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Event node (chance node).
• Indicates random event occurring at that point.
• Note expected payoff over its decision node.
Indicate chosen alternative.
• Insert a double dash as a barrier through each
rejected branch.
Backward induction procedure.
• Leads to optimal policy.
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Decision Trees
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FIGURE 16.4
The decision tree
(before including
any numbers) for
the full Goferbroke
Co. problem.
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Decision Trees
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FIGURE 16.5
The decision tree in
Fig. 16.4 after
adding both the
probabilities of
random events and
the payoffs.
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Decision Trees
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FIGURE 16.6
The final decision
tree that records the
analysis for the full
Goferbroke Co.
problem when using
monetary payoffs.
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16.5 Utility Theory
Monetary values may not always correctly
represent consequences of taking an action.
Utility functions for money U(M).
• People may have an increasing or decreasing
marginal utility for money.
Decision maker is indifferent between two
courses of action if they have the same utility.
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Utility Theory
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Equivalent lottery method for determining
utilities.
• See Page 676 in the text.
Applying utility theory to the full Goforbroke
Co. problem.
• Identify the utilities for all the possible monetary
payoffs.
• Shown in Table 16.7 on next slide.
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Utility Theory
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TABLE 16.7 Utilities for the full Goferbroke Co. problem
Monetary Payoff
Utility
−130
−100
60
90
670
700
0
0.05
0.30
0.333
0.97
1
Exponential utility function:
• Another approach for estimating U(M).
• Involves an individual’s risk tolerance, R.
U (M ) = 1 − e
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16.6 The Practical Application of Decision
Analysis
Real applications involve many more
decisions and states of nature:
• Than were considered in the prototype example.
• Decision trees would become large and unwieldy.
Several software packages are available.
Algebraic techniques being developed and
incorporated into computer solvers.
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The Practical Application of Decision
Analysis
Other graphical techniques:
• Tornado charts.
• Influence diagrams.
Decision conferencing:
• Technique for group decision making.
• Group of people work with a facilitator.
• Analyst builds and solves models on the spot.
• Consulting firm may be used if company does not
have analyst trained in OR techniques.
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16.7 Multiple Criteria Decision Analysis,
Including Goal Programming
• Thus far have assumed that there is just a single
criterion for measuring the performance of the
outcome.
• Frequently need to consider a number of
different criteria simultaneously instead.
• Multiple criteria decision analysis (MCDA) is
designed for doing this.
• Goal programming is one important MCDA
technique.
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Multiple Criteria Decision Analysis,
Including Goal Programming
The basic approach of goal programming:
• For each criterion, establish a specific numerical
goal for the objective.
• Formulate an objective function for each
objective.
• Establish weights on the relative importance of
the objectives.
• Seek a solution that minimizes the weighted sum
of deviations of the objective functions from their
respective goals.
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16.8 Conclusions
Decision analysis is an important technique
when facing decisions with considerable
uncertainty.
Decision analysis involves:
• Enumerating potential alternatives.
• Identifying payoffs for all possible outcomes.
• Quantifying probabilities for random events.
Decision trees and utility theory are tools for
decision analysis.
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End of Main Content
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