Probabilistic methods in operations research

GPEM - UPF

José Niño MoraJosé Niño Mora

April 6, 2000April 6, 2000

Outline

More about the courseMore about the course Elements of probabilistic modelsElements of probabilistic models Idealized probability distributionsIdealized probability distributions Multivariate distributionsMultivariate distributions Conditional probabilitiesConditional probabilities The buildings of uncertainty: Functions of The buildings of uncertainty: Functions of

random variablesrandom variables Simulation / OptimizationSimulation / Optimization

Course objectives

Given a complex business Given a complex business decision making decision making problem under uncertaintyproblem under uncertainty, learn how to: , learn how to:

1. Build a probabilistic model1. Build a probabilistic model 2. Solve the model (analysis/simulation)2. Solve the model (analysis/simulation) 3. Interpret the solution in terms of original 3. Interpret the solution in terms of original

problemproblem

Course features

Emphasis:Emphasis: NOTNOT on abstract analysison abstract analysis But on:But on: Modeling, Analysis/Simulation and Modeling, Analysis/Simulation and

Solution in the setting of CONCRETE Solution in the setting of CONCRETE planning problemsplanning problems

YET: Need to learn fundamental methods YET: Need to learn fundamental methods and modeling techniquesand modeling techniques

Also: Will solve/simulate models with Also: Will solve/simulate models with computer (Excel)computer (Excel)

Course overview (revised)

1. Review of probability1. Review of probability 2. Decision trees2. Decision trees 3. Dynamic programming3. Dynamic programming 4. Queueing (Business process flows) systems4. Queueing (Business process flows) systems 5. Simulation5. Simulation Methods Methods illustrated through applicationsillustrated through applications

Course web page

Look at: Look at:

http://www.econ.upf.es/~ninomora/pmor.htmhttp://www.econ.upf.es/~ninomora/pmor.htm Contains:Contains:

class presentations, Excel spreadsheetsclass presentations, Excel spreadsheets Links to useful resources (probability, OR, …)Links to useful resources (probability, OR, …)

About grading ...

Final exam: 66%Final exam: 66% Problem sets (biweekly): 17%Problem sets (biweekly): 17% Course project: 17%Course project: 17% Class participation: for boundary gradesClass participation: for boundary grades

Resources for probability review & for spreadsheet modeling

In course web page, look at:In course web page, look at: Links: Probability Links: Probability Ex: Ex: The layman’s guide to probability theoryThe layman’s guide to probability theory

Look also at Bibliography:Look also at Bibliography: Ex: Feller: Ex: Feller: An introduction to prob. TheoryAn introduction to prob. Theory

For spreadsheet modeling: will useFor spreadsheet modeling: will use Insight.xla (Business Analysis Software). Sam Insight.xla (Business Analysis Software). Sam

L. Savage.L. Savage.

References

Course transparenciesCourse transparencies Copies from books/articlesCopies from books/articles

Anupindi et al. (1999). Managing Business Anupindi et al. (1999). Managing Business Process Flows. Prentice HallProcess Flows. Prentice Hall..

D.E. Bell et al. (1995). Decision making under D.E. Bell et al. (1995). Decision making under uncertainty. Course Technologyuncertainty. Course Technology. .

......

Ex: Uncertain benefits

Introducing new product in marketIntroducing new product in market ¿Benefit?¿Benefit? Depends on:Depends on:

Sales (in units)Sales (in units) Price/unitPrice/unit Cost/unitCost/unit (production, marketing, sales, ...)(production, marketing, sales, ...) Fixed costsFixed costs (overhead, publicidad) = E30.000(overhead, publicidad) = E30.000

Benefit = Benefit = Sales Sales ** (Price (Price-- Cost_unit) Cost_unit) -- Fixed costs Fixed costs

Market scenarios New market: UncertaintyNew market: Uncertainty Scenarios: high or low volume (50%)Scenarios: high or low volume (50%)

Scenario: cost/unitScenario: cost/unit

Low volume High volume Mean volumeProbability 50% 50%Units 60000 100000 80000Price/unit(E) 10 8 9

Market Scenarios

Low More likely High Mean costProbab. 25% 50% 25%Cost/u.(E) 6 7,5 9 7,5

Cost/unit Scenarios

The building blocks of uncertainty

1. Uncertain numbers: Random numbers1. Uncertain numbers: Random numbers 2. Averages: Diversification2. Averages: Diversification 3. Important classes of random numbers: 3. Important classes of random numbers:

Idealized distributionsIdealized distributions 4. Functions of random numbers: 4. Functions of random numbers:

uncertainty managementuncertainty management

Exponential distribution

Models time between events, e.g., teleph. Models time between events, e.g., teleph. Calls, or product orders:Calls, or product orders:

Density function:Density function: Distribución:Distribución:

0,)( tetf t)(ExpX

0,}{ tetXP t

2

1]Var[

1]E[

X

X

Relation Exponential-Poisson

Suppose time between consecutive calls isSuppose time between consecutive calls is

Then, number of calls ocurring in [0, t) es: Then, number of calls ocurring in [0, t) es:

Hence,Hence,

)(ExpX

)( tPY

0,!/)(}{ jjtejYP t

Uniform distribution

Uniform distr. between a and b (a < b):Uniform distr. between a and b (a < b):

Density function:Density function: Distribution:Distribution:

),( baUX bxa

abxf

,

1)(

12)(

]Var[

2/)(]E[

,}P{

2abX

baX

bxaabax

xX

Uniform distribution (cont)

The RAND() Excel function: The RAND() Excel function:

Usefulness of in simulation:Usefulness of in simulation:

Ex: Ex:

U(0,1)RAND() )1,0(UU

XUF

xXxF

)( Then,

}.P{)(Let 1

XU

UF

exFExpX x

)1log()(

Then,

1)();(

1

Geometric distribution

Models no. of independent trials until first Models no. of independent trials until first success, with success prob. success, with success prob. pp

2

1

/)1(]Var[

/1]E[

1,)1(}P{

)(

ppX

pX

jppjX

pGXj

Multivariate distributions

Main example: Main example: Multivariate Normal:Multivariate Normal:

jiijjjj

jijjiiij

jj

XXXXE

XE

NXX

,:Note

) and bet. e(covarianc )])([(

mean) (marginal ][

),,(),(

2

2221

1211

2121 Σμ

Multivariate distr. (cont)

Given by Joint Distribution:Given by Joint Distribution:

or by Joint Density: Ex (Normal)or by Joint Density: Ex (Normal)

},{),( 221121 xXxXPxxF

2/)(

2121),( xxQKexxf

Covariance/correlation

Are Are measures of Linear Dependencemeasures of Linear Dependence between two r.v.:between two r.v.:

1),(1-

:Note

),Cov(),(

:nCorrelatio

)])([(),Cov(

:Covariance

21

21

2121

221121

XX

XXXX

XXEXX

Dependence/Independence of r.v.

If thenIf then If thenIf then If then NO linear relationIf then NO linear relation Def: Two r.v. are INDEPENDENT ifDef: Two r.v. are INDEPENDENT if

Ej: Two independent exponentials:Ej: Two independent exponentials:

1),( 21 XX )0( ,12 KKXX,1),( 21 XX )0(,12 KKXX

,0),( 21 XX

}P{}P{},P{ 22112211 xXxXxXxX

)1)(1(},{ 2211

2211

xx eexXxXP

Conditional expectation/probability

Conditional probabilitiy: probability of a Conditional probabilitiy: probability of a success given another success occurs:success given another success occurs:

Conditional expectation:Conditional expectation:

}P{},P{

}|P{

}P{},P{

}|P{

11

22111122

11

22111122

xXxXxX

xXxX

xXxXxX

xXxX

]|E[ ],|E[ 112112 xXXxXX

Conditional prob./exp. and Independence Suppose are independent r.v.Suppose are independent r.v. Then, Then,

A useful identity:A useful identity:

21 , XX

][][][

][][][

][]|[

}{}|{

2121

2121

2112

221122

XVarXVarXXVar

XEXEXXE

XExXXE

xXPxXxXP

]E[]|E[E 212 XXX

Application: Expected benefit

Have Have

E

FCostESalesEPSalesE

FCostSalesEPSalesE

FCostPSalesEE

000.70

30.000-7,5080.000-700.000

][][][

] [][

])([]Benefit[

Ex: conditional prob./exp.

Cars enter a gas station with interarrival Cars enter a gas station with interarrival timestimes

Each car brings an Each car brings an independent independent number of number of people distributed as : people distributed as :

¿Distribution/mean of the number ¿Distribution/mean of the number YY of of peoplepeople arriving in time interval arriving in time interval [0, t)?[0, t)?

)(Exp

1},{)( 1 jjZPjp

Ex: conditional prob./exp.

Know: number Know: number XX of cars arriving in [0, t) is of cars arriving in [0, t) is Poisson:Poisson:

Let Let Then,Then,

)( tPX iZ i car in passengers ofnumber

X

iiZY

1

X

ii XZ

XYY

1|EE

]|E[E]E[

Ex: Conditional expectation

HaveHave

So, by previous slide,So, by previous slide,

][

]|[

|]|[

1

1

ZjE

jXZE

jXZEjXYE

j

ii

j

ii

][][][

]][[

]]|[[][

11

1

ZtEZEXE

ZXEE

XYEEYE

The buildings of uncertainty: Functions of random variables Managers routinely input uncertain numbers into Managers routinely input uncertain numbers into

spreadsheet models: spreadsheet models: customer satisfactioncustomer satisfaction future demand for a productfuture demand for a product future workload requirements, …future workload requirements, …

Outputs are: functions of random variablesOutputs are: functions of random variables Tempting: plug in “best guesses”Tempting: plug in “best guesses” Does it work? NO!!Does it work? NO!! Instead: plug in ALL uncertain inputs!Instead: plug in ALL uncertain inputs!

Functions of random variables

If X, Y, Z, … are random variablesIf X, Y, Z, … are random variables

and f(x, y, z, …) is a function,and f(x, y, z, …) is a function, f(X, Y, Z, …) is a function of r.v.f(X, Y, Z, …) is a function of r.v. Ex: linear functions of r.v.:Ex: linear functions of r.v.:

f(X, Y, Z) = 5 X + 4 Y - 2 Zf(X, Y, Z) = 5 X + 4 Y - 2 Z The output of a probabilistic model is of the The output of a probabilistic model is of the

form f(X, Y, Z, …) form f(X, Y, Z, …) Ex: profit(revenues, cost) = revenues - costEx: profit(revenues, cost) = revenues - cost

The average of a function of random variables Wanted: average value of f(X), E[f(X)]Wanted: average value of f(X), E[f(X)] Can just plug in average values? Is it trueCan just plug in average values? Is it true

E[f(X)]=f(E[X])?E[f(X)]=f(E[X])? NO!! In general, E[f(X)] distinct from f(E[X]) !NO!! In general, E[f(X)] distinct from f(E[X]) !

When are they equal?When are they equal?

Averages of functions of r.v.

A sobering counterexample:A sobering counterexample: Consider a drunk, wandering left and right Consider a drunk, wandering left and right

from the middle of a highway in heavy from the middle of a highway in heavy traffic.traffic.

Take: X = drunk’s left-right position; Take: X = drunk’s left-right position;

f(X) = drunk’s fate (A/D)f(X) = drunk’s fate (A/D) What is f(E[X])? What is E[f(X)]?What is f(E[X])? What is E[f(X)]?

Averages of functions of r.v.

We can relate E[f(X)] with f(E[X]) under We can relate E[f(X)] with f(E[X]) under certain conditions: certain conditions:

Jensen’s inequality: if Jensen’s inequality: if f(x)f(x) is is convexconvex, then, then

E[f(X)] >= f(E[X])E[f(X)] >= f(E[X])

So, then can calculate So, then can calculate lower boundlower bound What is the intuition?What is the intuition?

Simulation: estimating E[f(X)]

If cannot obtain analytically, If cannot obtain analytically, estimate it with Monte Carlo simulationestimate it with Monte Carlo simulation

Generate sample X1, …, XnGenerate sample X1, …, Xn Estimate is: Estimate is:

How many trials are enough?How many trials are enough?

n

jjn Xf

n 1)(

1̂

)]([ XfE

How many trials are enough?

Markov inequality:Markov inequality: Let Y >= r.v., and a > 0. Then,Let Y >= r.v., and a > 0. Then,

Useful consequence for simulation: Useful consequence for simulation:

aYE

aYP][

}{

][],[ if

1}|{|

2

2

XVarXEk

kXP

Optimization under under uncertainty Ex: LetEx: Let f(X,a) f(X,a) be the benefit in an inventory be the benefit in an inventory

system, under random demand X, with system, under random demand X, with inventory level inventory level aa

Wanted: Wanted: max E[f(X, a)] max E[f(X, a)] over feasibleover feasible a a How to do it?How to do it? Analysis: Newsboy’s modelAnalysis: Newsboy’s model Parameterized simulation: vary Parameterized simulation: vary aa Another view: Another view: Policy optimizationPolicy optimization

More references

Ross, S.M. Stochastic Processes. Wiley, 1983.Ross, S.M. Stochastic Processes. Wiley, 1983. Feller, W. An Introduction to Probability Feller, W. An Introduction to Probability

Theory and its Applications. Wiley, 1957.Theory and its Applications. Wiley, 1957. Savage, S. Insight.xla: Business Analysis Savage, S. Insight.xla: Business Analysis

Software, 1998. Software, 1998. Bernstein, P. Against the Gods: The Bernstein, P. Against the Gods: The

Remarkable Story of Risk. Wiley, 1996.Remarkable Story of Risk. Wiley, 1996.