Contenu connexe Similaire à Chapter 18 sensitivity analysis (20) Plus de Bich Lien Pham (20) Chapter 18 sensitivity analysis1. 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
2. Slide Sets to © 2005 by McGraw-Hill,18-2
LEARNING OBJECTIVESLEARNING OBJECTIVES
1. Sensitivity to
variation
2. Three estimates
3. Expected value
4. Expected value
of cash flows
5. Decision trees
3. Slide Sets to © 2005 by McGraw-Hill,18-3
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
4. Slide Sets to © 2005 by McGraw-Hill,18-4
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
5. Slide Sets to © 2005 by McGraw-Hill,18-5
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
6. Slide Sets to © 2005 by McGraw-Hill,18-6
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.
7. Slide Sets to © 2005 by McGraw-Hill,18-7
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.
8. Slide Sets to © 2005 by McGraw-Hill,18-8
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
9. Slide Sets to © 2005 by McGraw-Hill,18-9
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
10. Slide Sets to © 2005 by McGraw-Hill,18-10
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
11. Slide Sets to © 2005 by McGraw-Hill,18-11
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)
12. Slide Sets to © 2005 by McGraw-Hill,18-12
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
13. Slide Sets to © 2005 by McGraw-Hill,18-13
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.
14. Slide Sets to © 2005 by McGraw-Hill,18-14
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
15. Slide Sets to © 2005 by McGraw-Hill,18-15
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
16. Slide Sets to © 2005 by McGraw-Hill,18-16
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
17. Slide Sets to © 2005 by McGraw-Hill,18-17
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)
18. Slide Sets to © 2005 by McGraw-Hill,18-18
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
19. Slide Sets to © 2005 by McGraw-Hill,18-19
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
20. Slide Sets to © 2005 by McGraw-Hill,18-20
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
21. Slide Sets to © 2005 by McGraw-Hill,18-21
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= ∑
22. Slide Sets to © 2005 by McGraw-Hill,18-22
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
23. Slide Sets to © 2005 by McGraw-Hill,18-23
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
24. Slide Sets to © 2005 by McGraw-Hill,18-24
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
25. Slide Sets to © 2005 by McGraw-Hill,18-25
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.
26. Slide Sets to © 2005 by McGraw-Hill,18-26
Chapter 18Chapter 18
End of SetEnd of Set
Notes de l'éditeur At this point:
1. Introduce yourself - your students are likely to want to know something about your qualifications and interests - overall, where you are coming from.
2. Have students introduce themselves. Ask why they are taking this class. If you are fortunate enough to have a Polaroid camera, take pictures of each student for later posting on a class “board” so both they and you get to know each other.
3. Discuss both choice of textbook and development of syllabus.
4. If you are expecting students to work in teams, at east introduce the choice of team members. If at all possible, have students participate in a team building or team study exercise. It works wonders. Most student have been told to work in teams in prior classes, but have never examined exactly what a team is and how it works. One hour spent in a team building/examination exercise saves many hours and avoids many problems later on.