Probabilistic Operations Research Modelsjpbrooks/oldclasses/2006fall691/bnfo691-2.pdfProbabilistic Operations Research Models ... Hillier, FS and Lieberman, GJ. Introduction to Operations

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Operations Research ModelsAxioms of Probability

Markov ChainsSimulation

Probabilistic Operations Research Models

Paul Brooks Jill Hardin

Department of Statistical Sciences and Operations Research

Virginia Commonwealth University

BNFO 691 December 5, 2006

Paul Brooks, Jill Hardin

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Outline

1 Operations Research Models2 Axioms of Probability

DefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom Variables

3 Markov ChainsMarkov PropertyBlood Types II

4 SimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event Simulation


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Operations Research Models

Operations Research

Deterministic OR

Continuous

Variables

DiscreteVariables

Probabilistic OR

Discrete Time Continuous Time

Models

FunctionsLinear Linear Nonlinear Continuous

SpaceNonlinearFunctions Functions Functions

DiscreteSpace

ContinuousSpace

DiscreteSpace


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Linear Programming

Operations Research

Deterministic OR

Continuous

Variables

DiscreteVariables

Probabilistic OR


Models

FunctionsLinear Nonlinear

FunctionsLinear

FunctionsNonlinearFunctions Space

Discrete ContinuousSpace

ContinuousSpaceSpace

Discrete


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Integer Programming

Operations Research

Deterministic OR

Continuous

Variables

DiscreteVariables

Probabilistic OR


Models


FunctionsLinear




Discrete


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Markov Chains

Operations Research

Deterministic OR

Continuous

Variables

DiscreteVariables

Probabilistic OR


Models


FunctionsLinear




Discrete


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Discrete-Event Simulation

Operations Research

Deterministic OR

Continuous

Variables

DiscreteVariables

Probabilistic OR


Models


FunctionsLinear




Discrete


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Discrete vs. Continuous Models

Discrete

means “space between”

countable, e.g., integers, binary numbers

attributes, variables, time, space

Continuous

uncountable, e.g., real numbers, intervals of real numbers

attributes, variables, time, space


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Linear vs. Nonlinear Models

Linear

additivity - every function is the sum of the individualcontributions of activities

proportionality - the contribution of an activity to a functionis proportional to the level of the activity.

Hillier, FS and Lieberman, GJ. Introduction to Operations Research, 6th edition. McGraw-Hill, 1995.

Nonlinear

Additivity or proportionality (or both) are violated


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Probabilistic vs. Deterministic Models

ProbabilisticProbability is used to model behaviors that are uncertain orunknown

DeterministicRandomness is not considered; systems are assumed to betotally determined. Sensitivity analysis can help incorporateuncertainty into models.


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Outline






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What is Probability?

Simply speaking probabilities are numbers between 0and 1 that reflect the chances of “something”happening.Synonymous with chance, likelihood, odds.Has different interpretations.


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Cards and Dice

What is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?

These are called events

Are you sure? What assumptions did you make?

Were they correct?

How can I correct these assumptions?

How can I determine a more accurate probability?


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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Cards and Dice




Were they correct?




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Outline






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Interpretations of Probability

Classical/AnalyticalTheoretically determined probabilities

Probability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2

Advantagesprobabilities are accurateno experimentation requiredobjective

Disadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)


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Relative Frequency/EmpiricalObserved proportion of successful events

10 cards selected, 6 of them black → probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years → probability of having at least 60 inchesof snow this year is 0.95.

Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjective

Disadvantage: situation must be replicable


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Personal/Subjective“What do you think are the odds?”

What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?

Relies on expert information (definition of “expert” is fluid).Advantage:

always applicable - everybody has an opinionuseful in risk analysis

Disadvantage: difficult (sometimes impossible?) todetermine accuracy.


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Outline






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Probability Rules

1. Probability is always between 0 and 1.probability of an event E is written P(E)

2. If event E cannot occur then P(E) = 0.E = “Jill will grow to be 6 feet tall”. P(E) = 0.

3. If an event is certain, then P(E) = 1.E = “Class will end before midnight.” P(E) = 1.

4. The sum of the probabilities of all possible outcomes is 1.


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Probability Rules






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Probability Rules






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Probability Rules






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Probability Rules






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Probability Rules






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Probability Rules






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Probability Rules






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Probability Rules

5. For two events A and B:P(A or B) = P(A) + P(B) − P(A and B)P(A and B) = P(A) + P(B) − P(A or B)


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Probability Rules

6. For two events A and BP(B occurs given that A occurs)= P(B|A) = P(A and B)/P(A)⇒ P(A and B) = P(B|A)P(A)

If P(B|A) = P(B) and P(A|B) = P(A) then A and B are saidto be independent.

That is, knowing that one event will occur doesn’t give us anyinformation about the other.

If A and B are independent, then P(A and B) = P(A)P(B).


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Probability Rules






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Probability Rules






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Probability Rules






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Probability Rules

DefinitionMutually exclusive events are non-overlapping— that is, theycannot happen at the same time.

A = randomly chosen person is maleB = randomly chosen person if femaleThese events are mutually exclusive.

A = randomly chosen person has blue eyesB = randomly chosen person has brown hairThese events are not mutually exclusive.


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Probability Rules

Bayes’ RuleSuppose A1, A2, . . . , Ak are mutually exclusive events sothat

P(Ai) > 0, i = 1, . . . , kP(A1) + P(A2) + · · · + P(Ak ) = 1 (i.e., they are exhaustive).

Let B be another event with P(B) > 0. Then

P(Ai |B) =P(Ai and B)

P(B)=

P(B|Ai)P(Ai)k∑

i=1P(B|Ai)P(Ai)


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Probability Rules

Bayes’ Rule

P(Ai |B) =P(Ai and B)

P(B)=

P(B|Ai)P(Ai)k∑

i=1P(B|Ai)P(Ai)


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Probability Rules

Bayes’ Rule: ExampleSuppose that 5% of all athletes useperformance-enhancing drugs. Suppose further that forthe drug test in use, the false positive rate is 3% and thefalse negative rate is 7%.An athlete is tested, and her results are positive. What isthe probability that she uses drugs?

5%?97%something else?


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Probability Rules

Bayes’ Rule: ExampleWe want to find P(drug use|positive test). What we have are

P(positive test|no drug use) = P(false positive)P(negative test|drug use) = P(false negative)

Bayes’ Rule says that

P(drugs|positive) =P(positive|drugs)P(drugs)

P(positive|drugs)P(drugs) + P(positive|no drugs)P(no drugs)


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Probability Rules

Bayes’ Rule: Example

P(drugs|positive) = P(positive|drugs)P(drugs)P(positive|drugs)P(drugs)+P(positive|no drugs)P(no drugs)

= (0.93)(0.05)(0.93)(0.05)+(0.03)(0.95)

= 0.62

When we first met the athlete, we thought the chance of herbeing a drug user was 5%. We were able to use Bayes’ Rule,along with the test results, to update our “expert” information.


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Outline






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Example: Blood Types

Suppose the distribution of blood types (and genotypes) in apopulation is as follows:

A 40% AA 20%AO 20%

B 12% BB 6%BO 6%

AB 5% AB 5%O 43% OO 43%


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What is the probability of producing a child with type O blood ifyour genotype is


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What is the probability of producing a child with type O blood ifyour genotype isAO?P(OOchild) =P(mate with AO or BO, you contribute O, mate contributes O) +P(mate with OO, you contribute O)= (0.26)(0.5)(0.5) + (0.43)(0.5)= 0.28


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What is the probability of producing a child with type O blood ifyour genotype isOO?P(OOchild) =P(mate with AO or BO, and mate contributes O) + P(mate with OO)= (0.26)(0.5) + 0.43= 0.56


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What is the probability of producing a child with type O blood ifyour genotype isBB?P(OOchild) is zero! (impossible event)


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Outline






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Random Variables

Sometimes we’re more interested in some function ofan outcome, rather than the outcome itself.

If I flip a coin 5 times, how many are heads? This is afunction of the outcomes on five separate flips.How long will it be before Jill and Paul stop talking?

This function of the outcome is called a randomvariable.Observed value is determined by chance.


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Random Variables

Need to know what values are possible— discrete orcontinuous?

Need to know what values are probable— how likely areeach of these values?

Probabilities defined by the probability density function (orprobability mass function)


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pmf’s and pdf’s

Probability Mass Function

For a discrete random variable X , f (x) = P(X = x) for eachpossible value of x .

Probability Density Function

f (x) ≥ 0 for all x

P(a ≤ X ≤ b) = area under f (x) between a and b

Total area under f is 1

P(X = x) = 0


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Probability Density Functions

Uniform

Normal

Triangular

Exponential


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Markov PropertyBlood Types II

Outline






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Markov Property

Definition

A stochastic process with state variable Xt is said to possess theMarkov Property if

P(Xt+1 = it+1|X0 = i0, X1 = i1, . . . , Xt = it) = P(Xt+1 = it+1|Xt = it)

Translation: “The probabilities that a stochastic process movesto a new state depends only on the current state; theprobabilities are independent of all past events.”


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Markov Property

Definition

A stochastic process with state variable Xt is said to possess theMarkov Property if

P(Xt+1 = it+1|X0 = i0, X1 = i1, . . . , Xt = it) = P(Xt+1 = it+1|Xt = it)

Translation: “The probabilities that a stochastic process movesto a new state depends only on the current state; theprobabilities are independent of all past events.”


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Illustrations of the Markov Property

Let Xt = current position on the board after t rolls


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Let Xt = location of a unit of ingested lead at time t

Tissue Bone

Bodyof

Outside

(Outside of Body)

Disposed

Blood

based on a model in Langkamp, G and Hull, J. Quantitative Reasoning and the Environment. Pearson, 2006.Paul Brooks, Jill Hardin

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Let Xt = location of a unit of ingested lead at time t

Tissue Bone

Bodyof

Outside

(Outside of Body)

Disposed

Blood

based on a model in Langkamp, G and Hull, J. Quantitative Reasoning and the Environment. Pearson, 2006.Paul Brooks, Jill Hardin

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Let Xt = political party of the U.S. Representative fromVirginia’s 3rd district after election t

Republican

DemocraticThirdParty

Party

Party


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Outline






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Transition Matrix

ChildAA AB AO BB BO OO

AA 0.33 0.12 0.55 0 0 0AB 0.16 0.22 0.28 0.06 0.28 0AO 0.16 0.06 0.44 0 0.06 0.28

Par

ent

BB 0 0.33 0 0.12 0.55 0BO 0 0.16 0.16 0.06 0.34 0.28OO 0 0 0.33 0 0.12 0.55


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Blood Types

Let Xt = the blood type of a person in generation t


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Blood Types

Let Xt = the blood type of a person in generation t

AB

AA OO

BO

BBAO


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Transition Probabilities

Theorem

Let P be the transition matrix of a Markov Chain. ThenP(Xt = j |X0 = i) is the ij th entry of P t .

Corollary

The probability that a grandchild has genotype BB given thatthe grandparent has genotype AA is given in the appropriateentry of P2 (in our model).


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Steady-State Behavior

Theorem

If all states are accessible from one another, then

limt→∞

P(Xt = j |X0 = i) = πj

where πj is the j th element of the vector π such that π = πP and∑

jπj = 1

This theorem gives us a means to calculate the steady statedistribution of genotypes. The quantity πj represents theprobability that, after a long time, a descendant has genotype j .It also represents the proportion of descendants that havegenotype j .


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The Nature of Simulation ModelingAn Example of a Discrete-Event Simulation

Outline






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A Flowchart for Simulation Modeling

State objective

Output Data

Yes

Noand design study

a Model

Modelis

Valid?

Analyze

Collect

Data Experiments

Design

Construct/Program

Law and Kelton, Simulation Modeling and Analysis, 3rd Ed., 2000.


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Probabilistic Modeling: Simulation vs. Analysis

Analysis

Advantage: Produces exact values for the characteristics of amodel given varied input. These values can easily be comparedfor determining optimal input values.

Disadvantage: Often requires strict assumptions about thenature of the model for any hope of solving for exact values.

Simulation

Advantage: Flexible in terms of assumptions required for model.

Disadvantage: Produces estimates of characteristics of a model;simulations need to be run for a wide variety of inputs and formany replications in order to derive reasonable estimates.


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Types of Simulation

DefinitionA discrete-event simulation is a continuous-time,discrete-space simulation.

Definition

A Monte Carlo simulation is a static simulation.


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Pitfalls of Simulation

Failure to have a well-definedset of objectives at thebeginning of the simulationstudy

Inappropriate level of modeldetail

Failure to collect good systemdata

Obliviously using simulationsoftware whose complexmacro statements may not bewell documented and may notimplement the desiredmodeling logic

Belief that easy-to-usesimulation packages require alower level of technicalcompetence

Failure to account correctly forsources of randomness in theactual system

Using arbitrary distributions asinput

Making a single replication of aparticular system design

Comparing alternative systemdesigns on the basis of onereplication

Law and Kelton, Simulation Modeling and Analysis, 3rd Ed., 2000.Paul Brooks, Jill Hardin

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Outline






Documents

Probabilistic Operations Research Modelsjpbrooks/oldclasses/2006fall691/bnfo691-2.pdfProbabilistic Operations Research Models ... Hillier, FS and Lieberman, GJ. Introduction to Operations