CHAPTER
5
Sensitivity Analysis
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Overview of Chapter 5
In this lecture, we will cover sensitivity analysis and
several tools for performing sensitivity analysis.
 A modeling approach to sensitivity analysis
• Identification and structuring of problems
 Dominance among alternatives
 Probability assessment
 One-way and two-way sensitivity analysis
 Sensitivity graphs and tornado diagrams
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Before We Start…
...keep in mind:
 There is no single “optimal” sensitivity-analysis procedure.
 Model building is an art, and since sensitivity analysis is a
part of modeling, it is also an art.
 We need to understand the overall philosophy of model
building and where sensitivity analysis fits into this
philosophy.
• We need principles, not rules.
 There are many tools for doing sensitivity analysis
• E.g., Excel, PrecisionTree
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Why Do Sensitivity Analysis?
 To determine the potential impact of each input
variable in the model by varying the input values
 To provide guidance for the development of a
requisite decision model
 To gain valuable insights into the decision situation
Sensitivity analysis is a critical part of the decisionanalysis cycle. It answers the question, “What makes
a difference in this decision?”
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
A Modeling Approach
 We will use an approach to sensitivity analysis built
on the idea of requisite decision models.
• A requisite decision model is one whose form and
content are just sufficient to solve a particular problem.
 Requisite decision modeling determines which
issues to address and which to ignore.
• Address those in the model—they matter
• Ignore those left out of the model—they don’t matter
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Problem Identification and Structure
 First ask, “Are we solving the right problem?”
 Different questions will usually lead to different
sensitivity analyses—and different decisions.
 Avoid Type III errors (asking the wrong questions):
• Is the problem at hand the fundamental problem?
• What do you really want as the outcome?
• Is the problem properly specified?
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Problem Identification and Structure
There are multiple ways to represent problems.
Sensitivity analysis helps in deciding which to use.
 Different problem contexts
• For example, should an oil drilling problem be stated as
a regulatory problem or a competitive problem?
 Different graphical representations
• Decision tree or influence diagram? Both?
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
House Hunting Example
We will now illustrate the various steps in sensitivity
analysis by using the House Hunting example from
the textbook.
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House Hunting Example
The steps will cover in this example are:
1. Graphical model
2. Model as set of equations
3. Calculations
• Each alternative
• Upper and lower bounds
• Comparison of alternatives
4. One-way sensitivity analysis
5. Two-way sensitivity analysis
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
House Hunting Example
Example as an influence diagram. Determine the values of the rounded
rectangles (chance nodes). Some may be constants.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
House Hunting Example
The influence diagram model can be shown as a set of equations derived from
the decision tree. This will allow for calculations.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
House Hunting Example
Find or determine the values for the equation and solve.
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Problem Identification and Structure
Before running the sensitivity analysis, estimate the
values of variables in the equations that might change
 Some of these will consist of values with upper and
lower bounds
• E.g., total rent will change if you stay in the apartment
shorter or longer than expected.
 Some values will be specific events unlikely to
happen
• E.g., replacing the roof on a house three years old
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
House Hunting Example
Upper and lower bounds for House Hunting Example
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Problem Identification and Structure
With above steps completed, you now have all the
information you need to run a sensitivity analysis
 Critical that sensitivity analysis accurately reports
the effect of changing values across the alternatives
• Can be done using cell referencing in Excel
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
House Hunting Example
Final sensitivity
analysis using cell
referencing to
guarantee that
changes in the input
values are
appropriately
reflected across the
alternatives.
(*) are the variables
used in the final
analysis
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
One-Way Sensitivity Analysis
 Vary one and only one input variable at a time
 Tells whether the individual variables in the
requisite model are influential
• If so, keep in the model
 Frequently represented graphically
• Sensitivity graphs and tornado diagrams
• Visual representations can provide intuitive and
immediate insights into a problem
 Can also be represented as a table
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Sensitivity Graphs and Tornado Diagrams
 Sensitivity graphs
• Input values on the x-axis and the calculated consequence
values on the y-axis
• Show the impact of one variable across alternatives
 Tornado diagrams
• Horizontal bar chart each bar represents one variable length
of each bar represents the change in the consequence value
as that one variable swings from its lower bound to its upper
bound
• Shows the relative impact of each variable on one alternative
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Sensitivity Graphs or Tornado Diagrams?
 Which is better?
• Use both—they are complementary
 First, look to the tornado diagram to determine
which variables impact the specific alternative most
• I.e., relative effect
 Then turn to the sensitivity graphs to understand the
exact nature of the variable’s effect
• I.e., individual effect
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
One-Way Sensitivity Analysis: Sensitivity Graphs
Using the House
Hunting example, this
sensitivity
graph shows the
impact on Net
Annual Cost for
each of the three
properties as the
Tax Rate swings
between
25% and 45%.
Insight: No matter
what the tax rate,
Astoria always costs
less.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
One-Way Sensitivity Analysis: Tornado Diagrams
Tornado diagram of
Net Annual Cost of
Barnard when the
variables in model
swing from their lower
bound to their upper
bound.
Insight: The most
influential variable is
Months Occupied,
followed by Interest
Rate, and Monthly
Rent.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Dominance Considerations
 Sensitivity analysis and dominance are similar, but
not identical.
• Cannot conclude from sensitivity analysis
 Why?
1. Model may have a missing variable.
2. Allowing additional variables to change simultaneously
with variable under analysis can lead to different results.
3. Different objective than the one under analysis can
result in a different rank ordering of alternatives.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Two-Way Sensitivity Analysis
 Allows two variables in the model to change
 Can use sensitivity graphs but involves massive
calculations
• Need to computer, e.g., Excel
• Will show strategy region: area of graph containing
best choice of alternatives
 Alternatively, formulate mathematical expression
using < > ≤ ≥ and solve algebraically
• Boundaries will provide strategy regions
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Sensitivity to Probabilities
Most decision making involves uncertainty, especially
in the real world.
 Sensitivity analysis can take this into account using
rules of probability theory.
• For example, the probability of a situation must be less
than or equal to 100%.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Stock Market Example
An investor has funds available to invest in one of three choices: a highrisk stock, a low-risk stock, or a savings account that pays a sure $500.
If he invests in the stocks, he must pay a brokerage fee of $200.His
payoff for the two stocks depends in part on what happens to the market
as a whole. If the market goes up (as measured, say, by the Standard
and Poor’s 500 Index increasing 8% over the next 12 months), he can
expect to earn $1700 from the high-risk stock and $1200 from the lowrisk stock. Finally, if the stock market goes down (as indicated by the
index decreasing by 3% or more), he will lose $800 with the high-risk
stock but still gain $100 with the low-risk stock. If the market stays at
roughly the same level, his payoffs for the high- and low-risk stocks will
be $300 and $400, respectively.
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Stock Market Example
Decision tree for
stock market vs.
savings decision.
Note that t + v
must be less than
or equal to 1
(100%).
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Stock Market Example
 Convert the model as shown in the decision tree
into algebraic expressions
• Solve and plot
Sounds simple, doesn’t it?
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Stock Market Example
Solve and plot EMV(Savings Account) ≥ EMV(Low-Risk Stock)
Beginning the
analysis of the
stock market
problem
Since t + v ≤ 1, the
only feasible points
are within the large
triangular region.
Savings
account
is best
option.
Low-risk
stock is
best
option.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Stock Market Example
Solve and plot EMV(Low-Risk Stock) ≥ EMV(High-Risk Stock)
Second stage in
analysis of the
stock market
problem
A second
inequality has
been incorporated.
The optimal
strategy is clear
now for all regions
except CDA.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Stock Market Example
Solve and plot EMV(Savings Account) ≥ EMV(High-Risk Stock)
Completed twoway sensitivity
graph for the stock
market
problem
Line ID has split
region CDA.
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Real World Example
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Real World Example
Two-way
sensitivity analysis
for the heart
disease treatment
decision
Source: Macartney et
al. (1984).
© British Medical
Association
used by permission
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Sensitivity Analysis: A Built-In Irony
The best way to view sensitivity analysis is as a
source of guidance in modeling a decision problem.
 If you want the guaranteed ‘right’ or ‘best’ solution,
you will be frustrated by sensitivity analysis.
 But if you want to improve your decision making,
sensitivity analysis is comforting.
 Remember: It’s art!
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.
Summary
In this chapter, we have presented an approach and
several tools for performing sensitivity analysis.
We have considered sensitivity analysis in terms of:
 The identification and structuring of problems
 Dominance among alternatives
 Probability assessment
 One-way and two-way analysis
 Tornado diagrams and sensitivity graphs
© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use.