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Basic elements about decision
trees and influence diagrams
Bibiography:
P. Goodwin & G. Wright (2003) Decision Analysis for Management
Judgement, John Wiley and Sons (chapter 6)
R. T. Clemen (1999) Making Hard Decisions: An Introduction to
Decision Analysis, Duxbury (chapter 3)
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Models and Techniques in Decision
Analysis
Uncertainty
Revising opinion
• Bayesian Nets
• Event trees
• Fault trees
Components
decomposition
• Risk Analysis
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Problem dominated by
Complexity
Evaluating
options
Choice
• Decision
trees
• Influence
diagrams
• Multicriteria
Analysis
(MACBETH,
EQUITY))
Resource
allocation and
negotiation
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Concepts
Influence diagrams
Decision trees
Complementary concepts:
Expected monetary value
Risk profile and cumulative risk profile
Other tools to model uncertainty: fault trees
and event trees
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We want to invest some €’s. We are uncertain
about how stock markets and interest rates…
3 months interest rates
Diário Económico, 17.05.2010
http://www.euribor.org/
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Structuring uncertainty within
problems
• Logical and time structure between decisions
• Logical structure (dependent) between uncertain
events
• Time structure of the sequence of uncertain
events, related with a sequence of decisions
• Representations using key concepts:
Influence diagrams
Decision trees
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Influence diagrams and decision
trees
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Influence Diagrams
1. Elements are represented by:
(rectangles) represent
decisions (and
alternatives)
(ovals) represent
uncertain events (and
outcomes) (chance
events)
(and calculation) nodes
– represent
consequences (and
calculations)
Nodes are put together in a graph, connected by ARCS.
Arcs represent relationships (relevance or sequence) between nodes:
Predecessor node successor node
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Influence Diagrams
2. Logical relationships are represented by: arrows
Sequence
Relevance
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Calculation
nodes
Consequence
nodes
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Building an Influence Diagram
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Basic Influence Diagram: One decision
and one uncertain event
One should be able
to identify basic
influence diagrams
and modify/combine
them to match
specific problems
Outcomes
Wild Success
Flop
Alternatives
Savings
Choice
Business Result
Return
Business
Savings
Wild Success
2200
Flop
2200
Wild Success
5000
Flop
0
Business
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A case with Imperfect Information
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Outcomes
Forecast
Hits Miami
Hits Miami
Misses Miami
Misses Miami
Imperfect Information:
• Involves one decision and two
uncertain events at the time of the
Decision Analysis.
• One uncertain event is known at the
time that the immediate decision
is made.
• Solving the influence
diagram results in one optimal
decision for each possible
outcome of the information source.
Alternatives
Evacuate
Stay
Choice
Outcome
Conseq. risk
Conseq.
cost
Evacuate
Hits Miami
Low risk
High cost
Misses Miami
Low risk
High cost
Hits Miami
High risk
High cost
Misses Miami
Low risk
Low cost
Stay
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But if there is missing information: The
case for sequential decisions...
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More on sequential decisions
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Developing financial models while
accounting for uncertainty…
1st version
3rd version
2nd version
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A DECISION TREE represents all of the
possible paths that the DM might follow
through time, including all possible
decision alternatives and outcomes of
chance events
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A simple Decision Tree
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Decision tree and the objectives
hierarchy
Outcomes measured in multiple dimensions…
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Representing elements in a decision
The options represented by
tree
branches from a decision
Decision nodes
Represent decisions
Chance nodes
Represent chance
(uncertain)
events
node must be such that
the DM can choose only one
option.
Each chance node must have
branches that correspond to a
set of mutually exclusive and
collectively exhaustive
outcomes.
Consequences
Consequences
are specified at
the ends of the
branches
When the
uncertainty is
resolved, one and
only one of the
outcomes occurs.
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Reading decision trees…
• If a chance node is to the
right of a decision node, the
decision must be made in
anticipation of the chance
event.
• Conversely, placing a chance
event before a decision
means that the decision is
made conditional on the
specific chance outcome
having occurred.
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• Imperfect information: DM
waits for inf. before making a
decision.
• The crescent shape indicates
that the uncertain event
may result in any value
between two limits.
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Decision Trees
1. Decision Trees are evaluated from left to right
2. Only one alternative can be chosen after each
decision node
3. Outcome from a chance event needs to be
complete, i.e. not more than one outcome can
happen at the same time and one outcome will
happen
4. Decision Trees represent all possible future
scenarios
5. Think of nodes as occurring in time sequence
6. If for chance nodes the order is not important, then
use the easiest interpretation
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Again the hurricane example… with
imperfect information
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Decision Trees vs. Influence
Diagrams
Influence Diagrams
Decision Trees
Strenghts
Compact
Good for
communication, in
particular in the
structuring phase
Good overview of large
problems
Good for understanding
the relevance between
uncertainty nodes
Displays details, being
good for in-depth
understanding
Flexible representation
Best for assymetric
decision problems
Adequate for performing
sensitivity analysis
Weaknesses
Details suppressed
Becomes very messy for
large problems
Complementary use of decision trees
and influence diagrams!
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DPL Software
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Assess the Cash Flows and probabilities
using the Precision Tree software
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Laboratory (next week)
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Examples from PrecisionTree
Other concepts
Expected monetary value
Risk profile
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The Risk Profile concept
• A risk profile is a graph that shows the chances associated with
possible consequences.
Risk Profile For Oil Diagram of oil_infl.xls
0,6
Probability
0,5
0,4
0,3
0,2
0,1
0
-100000
-50000
0
50000
100000
150000
200000
250000
300000
Value
• Each risk profile is associated with a strategy, a particular
immediate alternative, as well as specific alternatives in future
decisions.
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The Cumulative Risk Profile concept
• In this format, the vertical axis is the chance that the payoff is less
than or equal to the corresponding value on the horizontal axis.
• It results from adding up, or accumulating the chances of the
individual payoffs Along the horizontal axis we can read the
chance that the payoff will be less than or equal to that specific
value.
Cumulative Probability For Oil Diagram of oil_infl.xls
Cumulative Probability
1,2
1
F ( y ) = Pr(Y ≤ y ) = ∑ Pr(Y = i )
0,8
0,6
i:i ≤ y
0,4
0,2
0
-100000
-50000
0
50000
100000
Value
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150000
200000
250000
300000
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The Expected Value concept
The random variable Y has many possible outcomes!
Expected value: “BEST GUESS” for Y, what number
would you give?
n
n
i =1
i =1
Ε[Y ] = ∑ yi * Pr(Y = yi ) = ∑ yi * pi
Interpretation: If you were able to observe many
outcomes of Y, the calculated average of all the
outcomes would be close to E[Y].
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Other tools to model uncertainty
Event trees
Fault trees
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It is simply a decision tree without any
decisions!
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What is Event Tree Analysis?
• An accidental event is defined as the first significant
deviation from a normal situation that may lead to unwanted
consequences (e.g., gas leak, falling object, start of fire)
• It may lead to many different consequences. The potential
consequences may be illustrated by a consequence spectrum:
Source: System Reliability Theory: Models, Statistical Methods, and Applications, M. Rausand,
A. HøylandWiley-Interscience (2003)
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Example
Applications: Risk analysis of technological systems; Identification of improvements
in protection systems and other safety functions
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What is Event Tree Analysis?
• An event tree analysis (ETA) is an inductive procedure that
shows all possible outcomes resulting from an accidental
(initiating) event, taking into account whether installed safety
barriers are functioning or not, and additional events and
factors.
• By studying all relevant accidental events (that have been
identified by a preliminary hazard analysis, or some other
technique), the ETA can be used to identify all potential
accident scenarios and sequences in a complex system.
• Design and procedural weaknesses can be identified, and
probabilities of the various outcomes from an accidental event
can be determined.
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A fault tree begins with an initial system problem, and then
represent all the corrective actions or systems events that can
be taken to correct the default.
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Fault Tree Analysis
• Technique for reliability and safety analysis The failure of an item
in a system is often caused by the failure of other items, for example
where a vehicle's braking failure is caused by water in the brake
cylinders, which may in turn be caused by failure of the cylinder
seals.
• Fault Tree Analysis provides a method of breaking down these
chains of failures, with a key addition for identifying combinations
of faults that cause other faults.
• Combinations of faults come in two main types: (a) where several
items must fail together to cause another item to fail (an 'and'
combination), and (b) where only one of a number of possible faults
need happen to cause another item to fail (an 'or'' combination).
These combinations work as gates by preventing the failure event to
happen if specific conditions are met.
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Fault Tree analysis in problem solving
Logical And and Or in Fault Tree analysis
Source: http://syque.com/quality_tools/toolbook/FTA/fta.htm
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Example
A company president recognized that its personnel
evaluation system was not effective at motivating its
employees, and charged the personnel department
with improving it. As a part of the initial analysis of
the existing system, they use FTA to identify the
different ways that the evaluation system can fail
and lead to lack of motivation.
Identified failure areas were investigated further, and
the new system based on a correction of these
failures. As a result, motivation increased
significantly.
Slides ofSource:
Mónica http://syque.com/quality_tools/toolbook/FTA/fta.htm
Oliveira, MAD
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Fault Trees are particularly
useful to…
• Use when the effect of a failure is known, to find
how this might be caused by combinations of other
failures.
• Use when designing a solution, to identify ways it
may fail and consequently find ways of making the
solution more robust.
• Use to identify risks in a system, and consequently
identify risk reduction measures.
• Use to find failures which can cause the failure of all
parts of a 'fault-tolerant' system.
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