MISn46 Modeling Uncertainty 1 F07

8/8/2019 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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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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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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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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Components of a Simulation ModelComponents of a Simulation Model


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Components of a Simulation ModelComponents of a Simulation Model

MISn46 Modeling Uncertainty 1 F07

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