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Probabilistic methods in operations research
GPEM - UPF
José Niño MoraJosé Niño Mora
April 6, 2000April 6, 2000
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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
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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
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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)
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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
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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, …)
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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
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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.
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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. .
......
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Application: Expected benefit
Have Have
E
FCostESalesEPSalesE
FCostSalesEPSalesE
FCostPSalesEE
000.70
30.000-7,5080.000-700.000
][][][
] [][
])([]Benefit[
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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
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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[
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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
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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!
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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
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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?
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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)]?
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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?
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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
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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
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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
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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.