Chapter 18 sensitivity analysis

Slide Sets to © 2005 by McGraw-Hill,18-1
Developed By:
Dr. Don Smith, P.E.
Department of Industrial
Engineering
Texas A&M University
College Station, Texas
Executive Summary Version
Chapter 18
Formalized Sensitivity
Analysis and
Expected Value
Decisions

LEARNING OBJECTIVESLEARNING OBJECTIVES
1. Sensitivity to
variation
2. Three estimates
3. Expected value
4. Expected value
of cash flows
5. Decision trees

Sct 18.1 Determining Sensitivity toSct 18.1 Determining Sensitivity to
Parameter VariationParameter Variation
A parameter is a variable or factor for which an
estimate or stated value is required to conduct the
analysis at hand.
 Examples:
 P, F, A;
 i, n;
 Future costs, salvages, etc.
 Sensitivity analysis
 Seeks to determine what parameters matter most in an
economic analysis

SensitivitySensitivity
 Sensitivity is concerned with variability
 Variance associated with input parameters
impact the output variable the most
 The MARR as a parameter
 Interest rates and other interest factors tend to be
more stable from project to project
The analyst can limit the range over which these
type of parameters vary

Visualizing the Impact of ParametersVisualizing the Impact of Parameters
 Plot the PW, AW, or ROR vs. input parameters
 Steps
Pre-select the desired input parameters
Select the probable range and increment of
variation for each parameter
Select the measure of worth
Compute the results for each parameter
Graphically display the results by plotting the
parameter vs. the measure of worth

ExampleExample
Low High
$PW
Parameter
Assume a parameter of interest (show on the X-axis). Vary that parameter from some low
value to an assumed high value. Plot the resultant values on the Y-axis.

Sensitivity of PW: Example 18.1Sensitivity of PW: Example 18.1
Year Cflow
0 -$80,000
1 $25,000
2 $23,000
3 $21,000
4 $19,000
5 $17,000
6 $15,000
7 $13,000
8 $11,000
9 $9,000
10 $7,000
MARR PW(i%)
10% $27,831.49
15% $11,510.26
20% -$962.36
25% -$10,711.51
Ex. 18.1 PW vs. MARR
-$15,000.00
-$10,000.00
-$5,000.00
$0.00
$5,000.00
$10,000.00
$15,000.00
$20,000.00
$25,000.00
$30,000.00
0% 10% 20% 30%
MARR
$-PW
One can better visualize the relationship on PW vs. a
selected range of discount rates. As the discount
rate increases from 10% to 25% the resultant PW is
substantially lowered at a rather accelerated rate.

Sensitivity of Several ParametersSensitivity of Several Parameters
 For several parameters with one alternative
Graph the percentage change for each parameter
vs. the measure of worth
One will plot the percent deviation from the most
likely estimate on the x-axis
 This type of plot results in what is termed a
spider plot

Spider Plot: Figure 18-3Spider Plot: Figure 18-3
Those plots with positive slope have a
positive correlation on the output variable:
Those plots with negative slope have a
inverse relationship with the output
variable

Sct 18.2 Formalized Sensitivity AnalysisSct 18.2 Formalized Sensitivity Analysis
Using Three EstimatesUsing Three Estimates
 Given an input parameter of interest
Provide three estimates for that parameter
 A pessimistic estimate, P
 A most likely estimate, ML
 An optimistic estimate, O
 Note: This approach comes from PERT/CPM
analysis and is based upon the beta
distribution

Three Estimates: Example 18.3Three Estimates: Example 18.3
 Three alternatives (A, B, C) with 4 Parameters
First cost, salvage value, AOC, and life
For each parameter we formulate
Parameter
P pessimistic estimate
ML most likely estimate
O optimistic estimate
See Table 18.2 and observe the dominance by
alternative B (cost problem so lower cost is preferred)

Example 18.3 -- SetupExample 18.3 -- Setup
Strategy First Cost SV AOC Life
Alt A.
P -20,000 0 -11,000 3
ML -20,000 0 -9,000 5
O -20,000 0 -5,000 8
Alt. B
P -15,000 500 -4,000 2
ML -15,000 1,000 -3,500 4
O -15,000 2,000 -2,000 7
Alt. C
P -30,000 3,000 -8,000 3
ML -30,000 3,000 -7,000 7
O -30,000 3,000 -3,500 9

Plot for Example 18.3Plot for Example 18.3
Ex. 18.3
$0
$5,000
$10,000
$15,000
$20,000
$25,000
1 2 3 4 5 6 7 8 9
Life - Years
A.Cost
Alt A
Alt B
Alt C
Observe the
dominance by
alternative B over A
and C. A plot like this
clearly shows the
relationships.

Sct 18.3 Economic Variability and TheSct 18.3 Economic Variability and The
Expected ValueExpected Value
 Expected Value
Long-run average based upon occurrence and
probability of occurrence
 Definition of Expected Value
1
( ) ( )
m
i i
i
E X X P X
=
= ∑
Xi = value of the variable X for i from 1 to m different values
P(Xi) = probability that a specific value of X will occur
Subject to:
1
( ) 1.0
m
i
i
P X
=
=∑ See Example 18.4

Sct 18.4 Expected Value Computations forSct 18.4 Expected Value Computations for
AlternativesAlternatives
 Two applications for use of Expected Value
(EV)
1. Prepare information for a more complete analysis
of an economic analysis
2. To evaluate expected utility of a fully formulated
alternative
 Examples 18.5 and 18.6 illustrate this
concept

Sect 18.5 Staged Evaluation ofSect 18.5 Staged Evaluation of
Alternatives Using Decision TreesAlternatives Using Decision Trees
 Some problems involve staged decisions that
occur in sequence
 Define the staged decisions and assign the
respective probabilities to the various defined
outcomes
 Useful tool for modeling such a process
involves Decision Trees
 The objective: Make Risk More Explicit

Decision Tree AttributesDecision Tree Attributes
 More than one stage of alternative selection
 Selection of an alternative at one stage that
leads to another stage
 Expected results from a decision at each
stage
 Probability estimates for each outcome
 Estimates of economic value (cost or
revenue) for each outcome
 Measure of worth as the selection criterion,
e.g. E(PW)

ExamplesExamples
60.0% 0
0 500000
FALSE Chance
500000 500000
40.0% 0
0 500000
Decision
1000000
30.0% 0.3
0 1000000
TRUE Chance
1000000 1000000
60.0% 0.6
0 1000000
10.0% 0.1
0 1000000
Example Tree
Lease
Buy
Good Outcome
Bad Outcome
Good Outcome
Really Bad Outcome
Fair Outcome
Decision
Node
Probability
Node
Marginal
Probabilities
Outcomes

Solving Decision TreesSolving Decision Trees
 Once designed, Decision Tree is solved by folding
back the tree
 First, define all of the decision and the decision
points
 Define the various outcomes given the decision
 Assign the probabilities to the mutually exclusive
outcomes emanating from each decision node
 See Example 18-8 for a comprehensive analysis
illustrating the decision tree approach

Important Points for Decision TreesImportant Points for Decision Trees
 Estimate the probabilities associated with
each outcome
 These probabilities must sum to 1 for each set
of outcomes (branches) that are possible from
a given decision
 Required economic information for each
decision alternative are investments and
estimated cash flows

Starting OutStarting Out
 Assuming the tree logic has been defined…
 Start at the top right of the tree
Determine the PW for each outcome branch
applying the time value of money
Calculate the expected value for each decision
alternative as:
At each decision node, select the best E( decision )
value
Continue moving to the left of the tree back to the
root in order to select the best alternative
Trace the best decision path back through the tree
( ) ( estimate)P(outcome)E decision outcome= ∑

Example 18.8Example 18.8
D 1
M a r k e t
S e ll
D 2
D 3
H ig h
L o w
9
1 4
1 4
0 .2
0 .5
0 .5
1 2
1 6
4
6
- 1
0 . 8
0 .2
0 .4
0 .4
0 .2
0 .8
0 .2
0 .4
0 .4
0 .2
- 3
- 3
6
- 2
2
P W o f
C F B T
( $ m i ll i o n )
1 .0 9
E x p e c t e d v a l u e s f o r e a c h
a lt e r n a t i v e ( c a l c u la t e d )
I n t .
N a t .
I n t .
N a t .
2
4 .2
4 .2
9

Chapter SummaryChapter Summary
The emphasis is on sensitivity to variation in one or more
parameters using a specific measure of worth.
 When two alternatives are compared compute and graph
the measure of worth for different values of the parameter
to determine when each alternative is better.
 When several parameters are expected to vary over a
predictable range, the measure of worth is plotted and
calculated using three estimates for a parameter:
 Most likely
 Pessimistic
 Optimistic

Summary - continuedSummary - continued
The combination of parameter and probability
estimates results in the expected value relations
E(X) = ∑(X)P(X)
This expression is also used to calculate
E(revenue), E(cost), E(cash flow) E(PW), and E(i)
for the entire cash flow sequence of an alternative.
 E9X0 is a measure of central tendency

Summary - continuedSummary - continued
 Decision trees are used to make a series of alternative
selections.
This is a way to explicitly take risk into account.
 It is necessary to make several types of estimates for a
decision tree:
 Outcomes for each possible decision, cash flows, and
probabilities.
 Expected value computations are coupled with those for the
measure of worth to “solve” the tree structure.
 Assist is identifying the best alternative stage-by-stage.

Chapter 18Chapter 18
End of SetEnd of Set

Chapter 18 sensitivity analysis

More Related Content

What's hot (20)

Viewers also liked (19)

Similar to Chapter 18 sensitivity analysis (20)

More from Bich Lien Pham (9)

Recently uploaded (20)

Chapter 18 sensitivity analysis

Editor's Notes