Chapter 6
Decision Trees
and
Influence Diagrams
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Introduction
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Decision problems are multi-stage in character
when the choice of a given option may result in
circumstances which will require yet another
decision to be made.
The decisions made at the different points in time
are interconnected.
Influence diagrams offer an alternative way of
structuring a complex decision problem and some
analysts find that people relate to them much
more easily.
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Constructing a decision tree:
An initial tree...
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A new decision tree for the foodprocessor problem
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Determining the optimal policy
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A decision tree consists of a set of policies.
A policy is a plan of action stating which
option is to be chosen at each decision node
that might be reached under that policy.
For simplicity, assume that monetary return
is the only attribute which is relevant to the
decision
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Assume that, because the company is
involved in a large number of projects, it is
neutral to the risk involved in this
development and therefore the expected
monetary value (EMV) criterion is appropriate.
The technique for determining the optimal
policy in a decision tree is known as the
rollback method.
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Rolling back the tree
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The decision tree suggests the best policy
based on the information which is available
at the time it is constructed.
By the time the engineer knows whether or
not the gas-powered design is successful
his perception of the problem may have
changed and he would then, of course, be
advised to review the decision.
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Decision trees and utility: The
engineer’s utility function
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Applying rollback to utilities
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If the engineer had wished to include
other attributes besides money in his
decision model then multi-attribute utilities
would have appeared at the ends of the
tree.
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Decision trees involving continuous
probability distributions
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In some problems the number of possible
outcomes may be very large or even infinite.
Variables could be represented by
continuous probability distributions, but how
can we incorporate such distributions into
our decision tree format?
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One obvious solution is to use a discrete
probability distribution as an approximation.
The Extended Pearson-Tukey (EP-T)
approximation, proposed by Keefer and
Bodily.
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The value in the distribution which has a 95%
chance of being exceeded. This value is allocated
a probability of 0.185.
The value in the distribution which has a 50%
chance of being exceeded. This value is allocated
a probability of 0.63.
The value in the distribution which has only a 5%
chance of being exceeded. This value is also
allocated a probability of 0.185.
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The extended Pearson-Tukey
approximation method
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The EP-T approximation does have its
limitations.
It would be inappropriate to use it where
the continuous probability distribution has
more than one peak (or mode).
The approximation would probably not be a
good one if the shape of the continuous
distribution was very asymmetric.
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In some decision problems a subsequent
decision depends upon the achievement of
a particular level of a variable.
Some successful applications in Pages 152
and 153.
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Assessment of decision structure
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Imagine that you are a businessman and
you are considering making electronic
calculators. Your factory can be equipped to
manufacture them and you recognize that
other companies have profited from
producing them. However, equipping the
factory for production will be very expensive
and you have seen the price of calculators
dropping steadily. What should you do?
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Eliciting decision structure: One
representation of the calculator problem
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Towards a correct representation of
the calculator problem?
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Structuring is therefore a major problem in
decision analysis, for if the structuring is
wrong then it is a necessary consequence
that assessments of utilities and
probabilities may be inappropriate and the
expected utility computations may be
invalid.
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Phases of a decision analysis
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What determines the decision analyst's
provisional representation of the decision
problem? Generally, it will be based upon
past experience with similar classes of
decision problems and, to a significant
extent, intuition.
Problem representation is an art rather than
a science.
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Eliciting decision tree representations
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Influence diagrams
designed to summarize the dependencies that are
seen to exist among events and acts within a
decision.
influence diagrams can be converted to trees.
The advantage of starting with influence diagrams
is that their graphic representation is more
appealing to the intuition of decision makers who
may be unfamiliar with decision technologies.
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Definitions used in influence diagrams
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Definitions used in influence diagrams
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One step-by-step procedure for turning an
influence diagram into a decision tree
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(1) Identify a node with no arrows pointing into it.
(2) If there is a choice between a decision node and
an event node, choose the decision node.
(3) Place the node at the beginning of the tree and
'remove' the node from the influence diagram.
(4) For the now-reduced diagram, choose another
node with no arrows pointing into it. If there is a
choice a decision node should be chosen.
(5) Place this node next in the tree and 'remove7 it
from the influence diagram.
(6) Repeat the above procedure until all the nodes
have been removed from the influence diagram.
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Tree derived from influence diagram
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