misn46 modeling uncertainty 1 f07
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Modeling Uncertainty
Fun with decisions and uncertainty
Sensitivity analysis Preview Monte-Carlo simulation
Brief review of probability distributions and
statistics using Excel
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Warning! Decision and risk
analysis is a radical concept
People, in general, are not comfortable withprobabilistic reasoning
Most people commonly use point estimates for
uncertain quantities and then may carry out a limited1 or 2 variable sensitivity analysis
Everyone will say, too much thinking and planningrequired, dont have time in the real world
but somehow, people have time to revisit the messes theymake with seat of the pants decision making
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Importance and Difficulty of
Uncertainty Modeling
The world is uncertain
Replacing random quantities with averages or singleguesstimates can be dangerous The Flaw of Averages
Allows prediction ofdistribution of results Not just one predicted number or outcome
Sensitivity analysis of outputs to inputs
Which inputs really affect the outputs? Fun with Uncertainty
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Sensitivity Analysis
Sensitivity analysis (SA) a big part of modelingand analysis
SA = What matters in this decision?
which variables might I want to explicit model asuncertain and which ones might I just as well fix to mybest guess of their value?
On which variables should we focus our attention oneither changing their value or predicting their value?
No optimal SA procedure exists SA can help identify Type III errors - solving the
wrong problem
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Some SA Techniques
Scenarios base, pessimistic, optimistic How did we do with scenario planning?
1-way and 2-way data tables and associated
graphs as in the Break Even spreadsheet
Tornado diagrams a one variable at a time technique
Top RankExcel add-in for simple What if?
Risk Analysis or Spreadsheet Simulation direct modeling of uncertainty through probability
distributions
@Risk , CrystalBall sophisticated Excel add-ins
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Tornado Diagrams
Graphical sensitivity analysis technique
Create base, low and high value scenarios foreach input variable
Set all variables at base value Wiggle each variable to its low and high values,
one at a time.
A one-way sensitivity analysis technique
Calculate total profit for each scenario Create tornado diagram - Excel
From Making Hard Decisions by Clemen
GreatThreads-Tornado.xls
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Using Top Rank (see p714-721 in PMS)
Quit Excel (Top Rank acting flaky)
Start | Programs | Palisade Decision Tools | Top Rank 1.5
it will launch Excel and start
Open up your file
GreatThreadsTornado.xls
Select output cell and click Add output cells on TopRank toolbar
Click Step through input cells on TopRank toolbar and specifywhich inputs are to be varied and by how much
Click Run what-if analysis on toolbar
When results come up, click the tornado graph toolbar buttonand select tornado
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Sensitivity Analysis with TopRank
Big bars means
high impact
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Preview of Monte-Carlo Simulation
Simple Excel Simulation example
Revenue for 3 products with fixed prices and
random demands
What is simulation When do you use simulation
Real applications of simulation
Prob distributions the building blocks ofsimulation
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Monte-Carlo Simulation
A modeling approach that allows explicit incorporation ofuncertainty in spreadsheet models
Got its start during the Manhattan project in WWII for modelingnuclear devices
One or more random elements modeled with probabilitydistributions Sample from the input distributions many, many times
Keep track of the values of the outputs for each sampling of the inputs
Analyze the outputs
Often called Risk Analysis Uncertainty is about values of unknown variables
Risk is about consequences of uncertainty
@RISK - Palisade Software
Spreadsheets provide good environment for simulation
Goes beyond expected values and point estimates
Doing simulation involves more than just building models withsoftware must be probability/stats literate to do proper input and output analysis
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Our First Simple Simulation Model
3ProductSimulation-template.xls
What is expected revenue? Deterministic: prices of products A,B,C
Stochastic: demand for products A,B,C
What is the variability in revenue?
Thr r t Sim l ti l
if rm m i tri ti
r t ri i
A $ 0 20 0
B $1 0 0 0C $ 00 0 10
Tot l
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A few simulation applications
T Rowe Price 529 Simulator
NFL play calling
http://www.sciencedaily.com/releases/2006/04/060420232621.htm
http://www.pigskinrevolution.com/index.html
@Risk case studies
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Building a Spreadsheet Based Simulation Model
(1)Builddeterministic
model
Inputs OutputsFormulas
(2) Chooseinputs to model
as random
Inputs
Stochastic or Uncertain or Random Inputs
Deterministic Inputs
(3) Model uncertain inputs with probability distributions
Discrete Probability Distribution of Demand
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
100 150 200 250 300
Demand
Probability
Uniform Normal PoissonExponential Empirical
Many more
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Building a Spreadsheet Based Simulation Model
(4) Recalculate
spreadsheet many times 2 options
(4.2) Use spreadsheet simulation add-in
such as @Risk or Crystal Ball (Ex 11.2)
(4.1) Manually, through formulas and
either many rows or VBA(Ex 11.1)
Running the model
@Risk
www.palisade.com Crystal Ball
www.decisioneering.com
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Probability: The Language of Uncertainty
Distributions: Building Blocks ofSimulation
Random variables Discrete probability distributions
Expected value of a discrete randomvariable
Continuous probability distributions
Using Excels probability and statisticsfunctions
Using the RiskView add-inYou learned about most of the above and more in your
Statistics course. Ill just do a quick refresher as needed on
some concepts well need for this course.
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Random variables (RV) and
probability distributions RV is a variable whose value depends on the outcome of an
uncertain event(s)
Low bid by competing firms, project completion date
Demand for some product or service next year
Number of patients requiring open heart surgery next month at
Hospital H Cost of Drug X in December, 2004
Probability of various outcomes determined by probabilitydistribution associated with the RV
Probability distributions are the shapes of RVs
As modelers, we select appropriate distributions Probability distributions
mathematical functions
Assign numeric probabilities to uncertain events modeled bythe distribution
See Distributions, Simulation and Excel Functions handout that Prof. Doanecreated and that Ive posted on Web.
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Using Distributions for
Simulation
We will model uncertain inputs with probabilitydistributions Need to be able to generate random numbers from various
probability distributions We may fit probability distributions to raw data to
serve as a convenient model of the data
Simulation model outputs will be distributions Need to know how to compute various measures from
distributions
Simulating different scenarios - Need to know how tocompare features of distributions with each other
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Two Types of Distributions
Discrete Distributions
Integer, countable X
EX: # of warranty claims in a day
P(X) is the probability at each point
P(X)may be summed over X values
Continuous Distributions
X defined over an interval
EX: Length of stay for open heart surgery patients
Points have no area
Calculus gives area under curve
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P.D.F. vs. C.D.F
Probability Density Function X axis shows values of X
Y axis shows probability
7 P(X) = 1 if discrete
f(x) = 1 if continuous
Histogram is pdf for data
Cumulative Distribution Function
X axis shows values of X
Y axis shows cumulative probability
0 e F(X)e 1 and is non-decreasing
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Discrete RVs and Probability Distributions
Countable # of outcome values
Each possible outcome has an
associated probability
Di
r
t
r
ilit
Di
tri
ti
fDem
0.00
0.0
0.10
0.1
0.20
0.2
0.
0
0.
5
100 150 200 250 300
Dem
r
ilit
Expected Demand Total Probability
A few discrete distributions
Empirical
Binomial BINOMDIST()
Poisson POISSON()
1
[ ] [ ]n
i i
i
E X x P X x!
! !Expected Value of Discrete RV
DistributionReview.xls
x rob[ x] rob[ < x]
De and robabilit
u ulative
robabilit
. .
. .
. .8
. .9
. .
7 . .
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Cumulative Distribution Function (CDF)
for a Discrete Random Variable
e
exx
i
i
xpxxF )()Pr()(
1)(0 ee xF
The probability a random variable X
takes on a value less than or equal to x.
Properties of the CDF
F(x) is nondecreasing in x
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Distribution Review
Download DistributionReview.xls Lets answer questions on sheet Discrete
Well do Continuous sheet momentarily
Excel has many probability and statisticalrelated functions
Remember, probability distributions are a
type ofmodel for some uncertain quantity
Think of histograms as empirical probability
distribution functions
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Continuous RVs and Probability
Distributions
Infinite # of outcome values
Has a probability distribution
(density) function (pdf), f(x),
We calculate probabilities over
intervals using the cumulative
distribution function (cdf), F(x),which is P X =b
Area under the f(x) curve
from infinity to b
[ ] ( )b
P X b f x dxg
e
[ ] ( )E X xfx dxg
g!
Uniform f(x) Exponential f(x) Normal f(x)
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Excel Add-In, Part of Palisade Decision Tools Suite
Live distribution viewing, Huge number of distributions Online Help has background info on distributions
Start | Palisade Decision Tools | RiskView 4.5
Can also launch from within Excel from the Palisade DecisionTools toolbar (which is visible if any of the Palisade tools are
running, e.g. @Risk)
RiskView
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A few useful distributionsDistribution Illu stration Characteristics
Normal The familiar bell-shaped curve.
Symmetric, with a peak in the
middle and gradually tapering tails.
Pro: Familiar, well-known.
Con: Extreme outcomes possible.
Truncated normal Same as normal but with limits to
prevent extreme cases from arising.
Pro: No wild outcomes.
Con: More complicated.
Triangular Has a central peak and clearly-
defined end points (lowest, most
likely, highest). Can be skewed.
Pro: Easy to understand.
Con: No extremes can occur.
General R e v e n u e fr! " # $ $ e t % & le
0 .00
0 . ' 0
0 .(
0
0 .)
0
0 .4 0
5 0 1 0 0 2 0 0 5 0 0 1 0 0 0
R e v e n u e
P
r
0
1
2
1
ilit
3
Define any k categories and make
sure the probabilities sum to 1.
Pro: Easy to understand.
Con: Need to create categories.
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The Normal Distribution
Two parameters: Mean, standard deviation
Symmetric
Standard normal distribution has mean=0, std dev=1
Normally distributed data with any mean andstandard deviation can be converted to a N(0,1) bystandardizing
X~N(Q,W) Z~N(0,1)X
ZQ
W
!
Excel has a number of functions related to the normal distribution:
NORMDIST(), NORMINV()
NORMSDIST(), NORMSINV()
Lets review handout Excel Functions for Working with Normal
Distributions and do the Continuous tab in DistributionReview.xls
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Descriptive Statistics in Excel
Data Analysis Tool-Pak AVERAGE(), STDEV(),MEDIAN()
FREQUENCY()
PERCENTILE()
RANK(),
PERCENTRANK()
MIN(), MAX()
StatReview.xls
2 ways to create histograms Data Analysis Tool-Pak
Default bins
User specified bins
FREQUENCY() array function
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The very special Uniform
Random Variable (r.v.)
If X~Uniform(0,1) Then E X =1/2
(expected value)
X is equally likely to take any
value between 0 and 1
Probability X =x] = xExcels RAND() function
r.v. has a distribution of type Uniform with a min=0 and max=1
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Uniform Random Numbers for
Simulation
Building blocks of simulation
Modeling randomness
Basis for generating random variables
Normal, exponential, Poisson, triangular, etc.
Need reliable stream of Uniform(0,1) RVs
Excels RAND() function
How do computers generate randomnumbers?
All examples in RandNum_Isken.xls. Letsopen it.
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Uniform Distribution Function
How could youuse a U(0,1) number to
create a random number between a and
b? Lets do it in RandNum_Isken.xls
Question
Implication of shape ofdistribution?
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Using U(0,1)s to generate
other random variables
3
5
6
Demand distrib tion
Cumul ti r ob Demand
0.00 100
0.30 150
0.50 200
0. 0 250
0. 5 300
1
1
1
A B C
Simulation
Replication Random # Demand
1 0.1 100
Find U(0,1) random
number in cumulative
distribution of random
variable you want togenerate.
Return value of random
variable.
Walton example
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Using U(0,1)s to
generate Normal
random variables
Random # =122.57
NORMINV(.1747,160,40)=122.57
NORMDIST(122.57,160,40,TRUE)=.1747
CDF for N(160,40)
Simulation
Replication Random # Demand
1 0.1747 122.57
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Generating Random Numbers
Excels Data Analysis Tool-Pak
Excel RAND() along with transformations
Not possible for all distributions
@Risk functions
@Risk has myriad of functions for generating random
numbers from a wide variety of distributions
The file ProbabilityDistributions.xls (Downloads
section of course web) illustrates generating various
random variables
www.random.org
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Some of the broadly applicable
insights...
Explicit incorporation and quantification of risks and uncertainties isoften important
Be wary of clairvoyant analysts!
Several methods for trying to incorporate uncertainty in analysis
Quantification of risk is difficult and subject to common humandecision biases
Humans have hard time with uncertainty
Its important to guard against decision biases
Awareness is half the battle
Its OK to say I DONT KNOW
Not all information is worth the cost or equally valid Obtaining data for some of these modeling approaches can be
difficult
probability estimation can be tough
historical data may or may not exist
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What is Simulation?What is Simulation?
AA simulationsimulation is a computer model thatis a computer model thatattempts to imitate the behavior of a realattempts to imitate the behavior of a realsystem or activity.system or activity.
Simulations helps to quantify relationshipsSimulations helps to quantify relationships
among variables that are to complex toamong variables that are to complex toanalyze mathematically.analyze mathematically.
If the simulations predictions differ fromIf the simulations predictions differ fromwhat really happens, refine the model in awhat really happens, refine the model in a
systematic way until its predictions are insystematic way until its predictions are inclose enough agreement with reality.close enough agreement with reality.
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What is Simulation?What is Simulation?
In general, consider simulation whenIn general, consider simulation when
-- The system is complexThe system is complex-- Uncertainty exists in the variablesUncertainty exists in the variables
-- Real experiments are impossible or costlyReal experiments are impossible or costly
--T
he processes are repetitiveT
he processes are repetitive-- Stakeholders cant agree on policyStakeholders cant agree on policy
When Do We Simulate?When Do We Simulate?
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What is Simulation?What is Simulation?
Conversely, we are less inclined to simulateConversely, we are less inclined to simulate
whenwhen
-- The system is simpleThe system is simple
-- Variables are stable or nonstochasticVariables are stable or nonstochastic
-- Real experiments are cheap andReal experiments are cheap and
nondisruptivenondisruptive
-- The event will only happen onceThe event will only happen once
-- Stakeholders agree on policyStakeholders agree on policy
When Do We Simulate?When Do We Simulate?
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What is Simulation?What is Simulation?
In aIn a deterministicdeterministicmodel, variables cantmodel, variables cant
vary.vary.
Simulation lets key variablesSimulation lets key variables changechange inin
random but specified ways.random but specified ways.
Simulation helps us understand theSimulation helps us understand the rangerange ofof
possible outcomes and their probabilities.possible outcomes and their probabilities.
Simulation allowsSimulation allows sensitivity analysissensitivity analysis..
Advantages of SimulationAdvantages of Simulation
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Simulation is useful because itSimulation is useful because it
-- Is less disruptive than real experimentsIs less disruptive than real experiments
-- Forces us to state our assumptions clearlyForces us to state our assumptions clearly
-- Helps us visualize the implications of ourHelps us visualize the implications of ourassumptionsassumptions
-- Reveals system interdependenciesReveals system interdependencies
-- Quantifies risk by showing probabilities ofQuantifies risk by showing probabilities ofeventsevents
-- Helps us see a range of possible outcomesHelps us see a range of possible outcomes
-- Promotes constructive dialogue amongPromotes constructive dialogue among
stakeholdersstakeholders
Advantages of SimulationAdvantages of Simulation
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RiskAssessmentRiskAssessment
Risk assessmentRisk assessmentmeans thinking about ameans thinking about a
range of outcomesrange of outcomes and theirand theirprobabilitiesprobabilities..
Variation is inevitable.Variation is inevitable.
Knowing the 95% range of possible valuesKnowing the 95% range of possible valuesfor the decision variable as well as the mostfor the decision variable as well as the most
likely valuelikely value QQ, is the point of risk, is the point of risk
assessment.assessment.
Risk assessment is useful when the modelRisk assessment is useful when the model
is complex.is complex.
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What is Simulation?What is Simulation?
Components of a Simulation ModelComponents of a Simulation Model
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What is Simulation?What is Simulation?
Components of a Simulation ModelComponents of a Simulation Model