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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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Slides of Mónica Oliveira, MAD
Models and Techniques in Decision
Analysis
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ComplexityUncertainty Problem dominated by
Choice• Decision
trees• Influence
diagrams
Choice• Decision
trees• Influence
diagrams
Evaluating options
• Multicriteria Analysis(MACBETH,EQUITY))
Resource allocation and negotiation
Revising opinion• Bayesian Nets• Event trees• Fault trees
Components decomposition
• Risk Analysis
Slides of Mónica Oliveira, MAD
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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Slides of Mónica Oliveira, MAD
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We want to invest some €’s. We are uncertain about how stock markets and interest rates…
Diário Económico, 17.05.2010http://www.euribor.org/
3 months interest rates
Slides of Mónica Oliveira, MAD
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
Slides of Mónica Oliveira, MAD
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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Slides of Mónica Oliveira, MAD
Influence Diagrams2. 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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One should be ableto identify basicinfluence diagramsand modify/combinethem to matchspecific problems
Basic Influence Diagram: One decision
and one uncertain eventOutcomes
Wild Success
Flop
Alternatives
Savings
Business
Choice Business Result Return
Savings Wild Success 2200
Flop 2200
Business Wild Success 5000
Flop 0
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Slides of Mónica Oliveira, MAD
A case with Imperfect Information
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Forecast
Hits Miami
Misses Miami
Alternatives
Evacuate
Stay
Outcomes
Hits 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 influencediagram results in one optimaldecision for each possibleoutcome of the information source.
Choice Outcome Conseq. risk Conseq. cost
Evacuate Hits Miami Low risk High cost
Misses Miami Low risk High cost
Stay Hits Miami High risk High cost
Misses Miami Low risk Low cost
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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
2nd version
3rd version
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Slides of Mónica Oliveira, MAD
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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Slides of Mónica Oliveira, MAD
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
tree
Decision nodes
Represent decisions
Chance nodes
Represent chance (uncertain) events
Consequences
Consequences are specified at the ends of the branches
The options represented by branches from a decision node must be such that the DM can choose only one option.
Each chance node must havebranches that correspond to a set of mutually exclusive and collectively exhaustiveoutcomes.
When the uncertainty is
resolved, one and only one of the
outcomes occurs.
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Slides of Mónica Oliveira, MAD
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 chanceevent before a decisionmeans that the decision is made conditional on the specific chance outcome having occurred.
• Imperfect information: DM waits for inf. before making a decision.
• The crescent shape indicates that the uncertain eventmay result in any value between two limits.
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Slides of Mónica Oliveira, MAD
Decision Trees
1. Decision Trees are evaluated from left to right2. Only one alternative can be chosen after each
decision node3. 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 sequence6. 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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Slides of Mónica Oliveira, MAD
Decision Trees vs. Influence
DiagramsInfluence Diagrams Decision Trees
Strenghts CompactGood for communication, in particular in the structuring phaseGood overview of large problemsGood for understanding the relevance between uncertainty nodes
Displays details, being good for in-depth understandingFlexible representationBest for assymetric decision problemsAdequate 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
Assess the Cash Flows and probabilities
using the Precision Tree software
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Slides of Mónica Oliveira, MAD
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Laboratory (next week)
Examples from PrecisionTree
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Other concepts
Expected monetary value
Risk profile
Slides of Mónica Oliveira, MAD
The Risk Profile concept• A risk profile is a graph that shows the chances associated with
possible consequences.
• Each risk profile is associated with a strategy, a particular immediate alternative, as well as specific alternatives in future decisions.
0
0,1
0,2
0,3
0,4
0,5
0,6
-100000 -50000 0 50000 100000 150000 200000 250000 300000
Pro
ba
bil
ity
Value
Risk Profile For Oil Diagram of oil_infl.xls
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Slides of Mónica Oliveira, MAD
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.
0
0,2
0,4
0,6
0,8
1
1,2
-100000 -50000 0 50000 100000 150000 200000 250000 300000
Cu
mu
lati
ve
Pro
ba
bil
ity
Value
Cumulative Probability For Oil Diagram of oil_infl.xls
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∑≤
==≤=yii
iYyYyF:
)Pr()Pr()(
Slides of Mónica Oliveira, MAD
The Expected Value concept
The random variable Y has many possible outcomes!
Expected value: “BEST GUESS” for Y, what number
would you give?
Interpretation: If you were able to observe many
outcomes of Y, the calculated average of all the
outcomes would be close to E[Y].
[ ] ∑∑==
===Εn
i
ii
n
i
ii pyyYyY11
*)Pr(*
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Other tools to model uncertainty
Event trees
Fault trees
Slides of Mónica Oliveira, MAD
It is simply a decision tree without any decisions!
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Slides of Mónica Oliveira, MAD
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:
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Source: System Reliability Theory: Models, Statistical Methods, and Applications, M. Rausand, A. HøylandWiley-Interscience (2003)
Slides of Mónica Oliveira, MAD
Example
Applications: Risk analysis of technological systems; Identification of improvements in protection systems and other safety functions
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Slides of Mónica Oliveira, MAD
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 safetybarriers 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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Slides of Mónica Oliveira, MAD
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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Slides of Mónica Oliveira, MAD
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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Slides of Mónica Oliveira, MAD
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.
Source: http://syque.com/quality_tools/toolbook/FTA/fta.htm
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Slides of Mónica Oliveira, MAD
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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