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Engineering Economics in Canada
Chapter 12
Dealing with Risk: Probability Analysis
12.1 Introduction to Uncertainty and Risk
• We all encounter situations where we don’t
know for sure what events will happen in the
future.
• Only until the event occurs, we know the exact
outcome.
• We always talk about the “chance” that the
certain event may take place, which is
mathematically described by probability theory.
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What will be covered in this chapter?
• Decision making schemes using the knowledge
of the probability (chances) of uncertainties.
• decision trees, for decomposing a problem into
its decision alternatives and uncertain events.
• decision criteria for evaluating alternatives in the
decision tree obtained.
• a brief introduction of Monte Carlo simulation,
for analyzing complex problems.
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Short introduction of probability theory
• Uncertainty is characterized by unknown outcomes.
• Unknown outcomes can be uniquely represented
by a random variable.
• A random variable can take on a number of
possible values. Only one of these values will
eventually occur.
• A function describes the likelihood of each value of
a random variable: probability distribution function
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Probability Distribution Function (PDF)
• Consider a random variable X (capitalized) that can
take on m discrete outcomes x1, x2, …, xm
(lowercase): discrete random variable
• If these outcomes are mutually exclusive and
collectively exhaustive, a probability distribution
function p(x) is a set of numerical measures p(xi)
such that:
m
0  p( xi )  1 for i  1,..., m and  p(xi )  1
i 1
• Pr(X= xi) = p(xi). Intuitively, the higher p(xi), the
more likely it is that xi will occur.
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Example 12.1
• You are testing three solder joints on a printed
circuit board. Every solder can either be open
(O) or closed (C).
• determine the PDF for the random variable X,
the number of open joints in three tested joints.
• The probability that a single tested joint will be
open is 20%.
Solution: X can take on four possible values:
x1 = 0, x2 = 1, x3 = 2, and x4 = 3.
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Table 12.1
Test
Sequence
(O,O,O)
(O,O,C)
(O,C,O)
(C,O,O)
(O,C,C)
(C,C,O)
(C,O,C)
(C,C,C)
Number of
“Opens”
3
2
2
2
1
1
1
0
Probability
0.008 = 0.2 × 0.2 × 0.2
0.032 = 0.2 × 0.2 × 0.8
0.032 = 0.2 × 0.8 × 0.2
0.032 = 0.8 × 0.2 × 0.2
0.128 = 0.2 × 0.8 × 0.8
0.128 = 0.8 × 0.8 × 0.2
0.128 = 0.8 × 0.2 × 0.8
0.512 = 0.8 × 0.8 × 0.8
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Example 12.1…
• the PDF for X, the number of “open” joints in the three tests is:
Pr(X = 0) = p(x1) = 0.512
Pr(X = 1) = p(x2) = 0.384
Pr(X = 2) = p(x3) = 0.096
Pr(X = 3) = p(x4) = 0.008
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Cumulative Distribution Functions (CDF)
• The cumulative distribution function for a
discrete random variable X is defined as
follows:
P ( x )  Pr( X  x )   p( xi )
xi  x
• For Example 12.1, the CDF for the number of
open joints is:
Pr(X ≤ 0) = P(x1) = 0.512
Pr(X ≤ 1 ) = P(x2) = 0.896
Pr(X ≤ 2 ) = P(x3) = 0.992
Pr(X ≤ 3 ) = P(x4) = 1.000
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Mean and Variance
• Two summary statistics useful in describing a
random variable X is its expected value, or
mean, E(X), and variance, Var(X).
• If X can take values x1, x2, …, xm , then its
mean is:
m
E ( X )   xi p( xi )
i 1
• And its variance is:
m
Var ( X )   p( xi )( xi  E ( X ))2
i 1
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Mean and Variance
• The expected value of a random variable is
much like the centre of mass for an object.
– The expected value is simply the centre of
the probability “mass.”
• The variance measures the degree of spread
or dispersion of a random variable about the
mean.
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12.4 Decision Trees
• Formal methods can help by:
– providing a means of decomposing a problem
and structuring it clearly.
– suggesting a variety of decision criteria to
help with the process of selecting a preferred
course of action.
• decision trees, a graphical means of structuring
a decision-making situation where uncertainties
can be characterized by probability distributions.
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Decision Trees
• A decision tree is a graphical representation of
the logical structure of a decision problem in
terms of
– the sequence of decisions to be made and
– outcomes of chance events.
• It provides a mechanism to decompose a large
and complex problem into a sequence of small
and essential components.
• It clarifies the options a decision maker has
and provides a framework with which to deal
with the risk involved.
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Example 12.2
• Edwin Electronics (EE) has a factory for assembling
TVs. EE outsources the TV screen to a supplier, but are
considering bringing screen production in-house.
• Uncertainty in demand for the company’s TVs has an
important bearing on the decision.
– If the future demand is low, outsourcing seems to be
the reasonable option in order to save production
costs.
– On the other hand, if the demand is high, it may be
worthwhile to produce the screens on-site due to
economies of scale.
• EE’s engineers represented their decision in a graphical
manner with a decision tree.
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Example 12.2…
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Components of a Decision Tree
•
•
There are four main components in a decision tree:
1. A decision node represents a decision to be made
by the decision maker. It is denoted by a square.
2. A chance node represents an event whose outcome
is uncertain. It is denoted by a circle.
3. The branches of a tree are the lines connecting
nodes from left to right, depicting the sequence of
possible decisions and chance events.
4. Finally, the leaves indicate the values, or payoffs,
associated with each terminal (rightmost) branch of
the decision tree.
A decision tree grows from left to right and usually
begins with a decision node.
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12.5 Decision Criteria
• Once a complete decision tree is structured, an
analyst is in a better position to select a
preferred alternative from a set of possible
choices.
• This section deals with several commonly used
decision criteria for situations that involve
uncertainty.
– Expected Value
– Dominance
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12.5.1 Expected Value
•
One criterion for selecting among risky alternatives is
expected value, EV. With a decision tree, this is carried
out as follows:
1. Structure the problem: Develop a decision tree.
2. Rollback: moving from right to left on the tree:
– At each chance node, compute the expected value
of the possible outcomes.
– At each decision node, select the option with the
best expected value. This becomes the value
associated with the decision node. Mark option(s)
not selected with a double-slash (//) on the
corresponding branch.
– Continue until the leftmost node is reached.
3. Conclusion: The expected value associated with the
final node is the expected value of the overall decision.
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Example 12.3
• Carry out a decision tree analysis given the figure below.
• What decision should they make based on expected value?
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Example 12.7…
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Example 12.7…
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12.5.2 Dominance
• The expected value criterion is straightforward,
but is only a summary measure.
– It does not consider the dispersion of the
outcomes associated with a decision.
• information from the probability distributions
allows a decision maker to use dominance
concepts to screen out less preferred
alternatives, or to pick the best of several
alternatives.
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Mean-Variance Dominance
• Suppose that an engineer is attempting to
select between two projects X and Y where the
outcomes are monetary (i.e., more is better).
Alternative X is said to have mean-variance
dominance over alternative Y:
(a) If EV(X) ≥ EV(Y) and Var(X) < Var(Y), or
(b) If EV(X) > EV(Y) and Var(X) ≤ Var(Y).
• an alternative is said to be mean-variance
efficient if no other alternative has both a
higher mean and a lower variance.
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Example 12.9
Prod’n
Demand Pattern
Strategy
1
2
3
Mean Variance
1
420
310
600
401
12 169
2
280
340
630
380
16 300
3
500
290
425
380
8775
4
600
275
390
395
19 812
5
415
300
590
392.5 12 231
• Strategies 1 and 3 remain.
• A choice between the two will require management to
assess its willingness to trade-off mean profits with
variability in profits.
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Outcome Dominance
• Outcome dominance of alternative X over
alternative Y can occur in one of two ways.
– the worst outcome for alternative X is at
least as good as the best outcome for
alternative Y.
– when one alternative is at least as preferred
to another for each outcome, and is better
for at least one outcome.
• Outcome dominance can be useful in
screening out alternatives that are clearly
worse than others
• Though it straightforward to apply, it may not
remove many alternatives.
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Stochastic Dominance
• Example 12.10: Suppose that the probability distribution
functions (risk profiles) of the outcomes for the two decision
alternatives that EE is considering are as shown in Figure
12.10
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Stochastic Dominance…
• A look at the cumulative distribution functions, (cumulative
risk profiles) for the two alternatives provides further insight.
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Stochastic Dominance…
• The cumulative risk profile for the outsource
decision either overlaps with or lies to the left
and above of the cumulative risk profile of the
produce decision
– for all outcomes, the probability that the
outsource decision gives a lower profit per
unit is equal to or greater than the
corresponding probability for the produce
decision.
• The produce strategy is said to dominate the
outsource strategy according to (first-order)
stochastic dominance.
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Dominance
• Stochastic dominance and outcome
dominance can be used to screen alternatives,
but they are often not able to provide a
definitive best alternative.
• Despite this limitation, cumulative risk profiles
can be very useful in making statements such
as:
– “Alternative A is more likely to produce a
profit in excess of $1 000 000 than
alternative B, or
– “Project C is more likely to suffer a loss than
project D”.
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12.6 Monte Carlo Simulation
• Monte Carlo simulation can be very useful
when analyzing complex problems
characterized by multiple sources of risk.
• Each decision strategy is evaluated by
repeatedly randomly sampling branches of the
decision tree and then constructing risk profiles
for the relevant performance measures.
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12.6.2 Probability Distribution Estimation
• Monte Carlo simulation attempts to construct
the probability distribution of an outcome
performance measure of a project (e.g.,
present worth) by repeatedly sampling from the
input random variable probability distributions.
• Once the probability distribution is estimated,
summary statistics associated with a decision
strategy can be used to gain insight into the
possible performance level for the project.
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12.6.3 The Monte Carlo Simulation
Approach
•
A Monte Carlo simulation model is constructed via a five
step process:
1. Analytical model: Identify input random variables
that affect the outcome performance measure of the
project. Develop the equation(s) necessary to
compute the outcome performance measure from a
particular realization of the input random variables.
2. Probability distributions: Establish an appropriate
probability distribution for each input random
variable.
3. Random sampling: Sample a value for each input
random variable from its associated probability
distribution.
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The Monte Carlo Simulation Approach…
4. Repeat sampling: Continue sampling until a sufficient
sample size is obtained for the sampled outcomes (the
computed performance measure)
5. Summary: Summarize the frequency distribution of the
sample outcomes using a histogram. Summary
statistics, like the range of possible outcomes and
expected value can also be calculated from the sample
outcomes.
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12.7 Application Issues
• Decision trees and Monte Carlo simulation are
powerful tools for analyzing decision making
situations that involve risk.
– Use of probability distributions permits the
engineer to get an overall picture of risk.
• A drawback of these methods is that specifying
the probability distribution of outcomes can be
challenging, and at times, highly subjective.
• Despite this drawback, decision trees and
Monte Carlo simulation are widely used.
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Summary
•
•
•
•
•
Introduction to Uncertainty and Risk
Basic Concepts of Probability
Random Variables and Probability Distributions
Structuring Decisions with Decision Trees
Decision Criteria
– Expected Value
– Dominance
• Monte Carlo Simulation
– Dealing with Complexity
– Probability Distribution Estimation
– The Monte Carlo Simulation Approach
• Application Issues
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