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Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-1
Developed By:
Dr. Don Smith, P.E.
Department of Industrial Engineering
Texas A&M University
College Station, Texas
Executive Summary Version
Chapter 19
More on Variation and Decision Making
Under Risk
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-2
LEARNING OBJECTIVESLEARNING OBJECTIVES
1. Certainty and risk
2. Variables and distributions
3. Random samples
4. Average and dispersion
5. Monte Carlo simulation
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-3
Sct 19.1 Interpretation of Certainty, Risk, Sct 19.1 Interpretation of Certainty, Risk, and Uncertaintyand Uncertainty
Certainty – Everything know for sure; not present in the real world of estimation, but can be ‘assumed’
Risk – a decision making situation where all of the outcomes are know and the associated probabilities are defined
Uncertainty – One has two or more observable values but the probabilities associated with the values are unknown Observable values – states of nature
See Example 19.1 – about risk
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-4
Types of Decision MakingTypes of Decision Making Decision Making under Certainty
Process of making a decision where all of the input parameters are known or assumed to be known
Outcomes – known Termed a deterministic analysis Parameters are estimated with certainty
Decision Making under Risk Inputs are viewed as uncertain, and element of chance is
considered Variation is present and must be accounted for Probabilities are assigned or estimated Involves the notion of random variables
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-5
Two Ways to Consider Risk in Two Ways to Consider Risk in Decision MakingDecision Making
Expected Value (EV) analysisApplies the notion of expected value (Chapter 18)Calculation of EV of a given outcomeSelection of the outcome with the most
advantageous outcome
Simulation AnalysisForm of generating artificial data from assumed
probability distributionsRelies on the use of random variables and the laws
associated with the algebra of random variables
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-6
Sct 19.2 Elements Important to Decision Sct 19.2 Elements Important to Decision Making Under RiskMaking Under Risk
The concept of a random variableA decision rule that assigns an outcome to a
sample spaceDiscrete variable or Continuous variable
Discrete variable – finite number of outcomes possible Continuous variable – infinite number of outcomes
ProbabilityNumber between 0 and 1Expresses the “chance” in decimal form that a
random variable will take on any specific value
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-7
Types of Random VariablesTypes of Random Variables
Continuous
Discrete
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-8
Distributions - Continuous VariablesDistributions - Continuous Variables Probability Distribution (pdf)
A function that describes how probability is distributed over the different values of a variable
P(Xi) = probability that X = Xi
Cumulative Distribution (cdf) Accumulation of probability over
all values of a variable up to and including a specified value
F(Xi) = sum of all probabilities through the value Xi
= P(X Xi)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-9
Three Common Random VariablesThree Common Random Variables
Uniform – equally likely outcomes
Triangular
Normal
Study
Example 19.3
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-10
Discrete Density and Cumulative ExampleDiscrete Density and Cumulative Example
pdf cdf
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-11
Sct 19.3 Random SamplesSct 19.3 Random Samples Random Sample
A random sample of size n is the selection in a random fashion with an assumed or known probability distribution such that the values of the variable have the same chance of occurring in the sample as they appear in the population
Basis for Monte Carlo Simulation Can sample from:
Discrete distributions … orContinuous distributions
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-12
Sampling from a Continuous DistributionSampling from a Continuous Distribution
Form the cumulative distribution in closed form from the pdf
Generate a uniform random number on the interval {0 – 1}, called U(0,1)
Locate U(0,1) point on y-axis
Map across to intersect the cdf function
Map down to read the outcome (variable value) on x-axis
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-13
Sct 19.4 Expected Value and Standard Sct 19.4 Expected Value and Standard DeviationDeviation
Two important parameters of a given random variable:Mean -
Measure of central tendency
Standard Deviation - Measure of variability or spread
Two Concepts to work withinPopulationSample from a population
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-14
Sample
Population vs SamplePopulation vs Sample
Population - population mean 2 - population
variance - population
standard deviation Often sample from a
population in order to make estimates
X
2s
s
Sample mean
Sample variance
Sample standard deviation
These values, properly sampled, attempt to estimate their population counterparts
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-15
Important RelationshipsImportant Relationships
Population Mean Distribution
E(x) =
Sample
Measure of the central tendency of the population If one samples from a population the hope is that
sample mean is an unbiased estimator of the true, but unknown, population mean
( )i iX P X
i i iX f X
n n
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-16
Variance and Standard DeviationVariance and Standard Deviation
Notes relating to variance and standard deviation properties
Illustration of variances for discrete and continuous distributions
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-17
Population vs. SamplePopulation vs. Sample
Variance of a population
Standard deviation of a population:
N2
ii=1
(x )
N
2 Var(X)X X
N2
i2 i=1
(x )
N
Variance of a sample
Standard deviation of a sample
2
2 2
1 1iX n
s Xn n
S is termed an unbiased estimator of the population standard deviation
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-18
Combining the Average and Standard Combining the Average and Standard DeviationDeviation
Determine the percentage or fraction of the sample that is within ±1, ±2, ±3 standard deviations of the sample mean . . .
In terms of probability…
Virtually all of the sample values will fall within the ±3s range of the sample mean
See example 19.6
, t = 1,2,3X ts
P( )X ts X X ts
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-19
Continuous Random VariablesContinuous Random Variables Expected value:
Variance:
For uniform pdf in Example 19.3:
( )( )R
Xf x dxE X 2 2( ) ( ) [ ( )]
R
Var x X f X dX E X
R represents the defined range of the variable in question.
10
2 15
R
2 2 3 15 210
10.2 $10 X $15
5
E(X)= (0.2)Xdx=0.1X 0.1(225 100) $12.50
0.2( ) (0.2) (12.5) (12.5)
3
=0.06667(3375-1000)-156.25=2.08
2.08 $1.44
( )
R
X
Var X X dX X
f x
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-20
Sct 19.5 Monte Carlo Sampling and Sct 19.5 Monte Carlo Sampling and Simulation AnalysisSimulation Analysis
Simulation involves the generation of artificial data from a modeled system
Monte Carlo SamplingThe generation of samples of size n for selected
parameters of formulated alternativesThe sampled parameters are expected to vary
according to a stated probability distribution (assumed)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-21
Key Assumption - IndependenceKey Assumption - Independence For a given problem:
All parameters are assumed to be independentOne variable’s distribution in no way impacts any
other variable’s distributionTermed:
Property of independent random variables
The modeling approach follows a 7-step process (page 678)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-22
SimulationSimulation
Monte Carlo Sampling is a traditional approach (method) for generating pseudo-random numbers (RN) to sample from a prescribed probability distribution.
Pseudo-random refers to the fact that a digital computer can generate approximately random numbers due to fixed word size and round off problems.
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-23
Sampling ProcessSampling Process
Requires: The cdf of the assumed pdf; A uniform random number generator; Application of the inverse transform approach.
Why require the cdf? The y-axis of a cdf is scaled from 0 to 1. That is the same as the range of U(0,1). Facilitates mapping a RN to achieve the outcome value on
the x-axis. The U(0,1) selects a X-value from the cdf.
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-24
The Need for the cdfThe Need for the cdf
x-axis: Outcome values
Cumulative probability on the y-axis
Generate a U(0,1) random number: Locate that value on the y-axis: Map across to the cdf then map down to the x-axis to obtain the outcome
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-25
Summary of the Modeling StepsSummary of the Modeling Steps Formulate the economic analysis:
The alternatives – if more than one;Define which parameters are “constants” and
which are to be viewed as random variables.For the random variables, assign the appropriate
pdf:Discrete and or continuous.
Apply Monte Carlo sampling – a sample size of “n” where it is suggested that n = 30.
Compute the measure of worth (PW, AW, . . )Evaluate and draw conclusions.
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-26
Examples to StudyExamples to Study
Examples 19.7 and 19.8 offer detailed analyses of a simulated economic analysis
Also, refer to the Additional Examples at the end of the chapter Example 19.10 – applying the normal distribution
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-27
Chapter SummaryChapter Summary To perform decision making under risk implies
that some parameters of an engineering alternative are treated as random variables.
Assumptions about the shape of the variable's probability distribution are used to explain how the estimates of parameter values may vary.
Additionally, measures such as the expected value and standard deviation describe the characteristic shape of the distribution.
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-28
Summary - continuedSummary - continued
In this chapter, we learned several of the simple, but useful, discrete and continuous population distributions used in engineering economy -uniform and triangular - as well as specifying our own distribution or assuming the normal distribution.
It is important to note that a sound background in applied statistics is vital to the complete understanding of the simulation process
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved19-29
Chapter 19Chapter 19End of SetEnd of Set