Probabilistic methods in operations research GPEM UPF

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Probabilistic methods in operations
researchGPEM - UPF

• José Niño Mora
• April 6, 2000

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Outline

• More about the course
• Elements of probabilistic models
• Idealized probability distributions
• Multivariate distributions
• Conditional probabilities
• The buildings of uncertainty Functions of random
  variables
• Simulation / Optimization

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Course objectives

• Given a complex business decision making problem
  under uncertainty, learn how to
• 1. Build a probabilistic model
• 2. Solve the model (analysis/simulation)
• 3. Interpret the solution in terms of original
  problem

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Course features

• Emphasis NOT on abstract analysis
• But on Modeling, Analysis/Simulation and
  Solution in the setting of CONCRETE planning
  problems
• YET Need to learn fundamental methods and
  modeling techniques
• Also Will solve/simulate models with computer
  (Excel)

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Course overview (revised)

• 1. Review of probability
• 2. Decision trees
• 3. Dynamic programming
• 4. Queueing (Business process flows) systems
• 5. Simulation
• Methods illustrated through applications

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Course web page

• Look at
• http//www.econ.upf.es/ninomora/pmor.htm
• Contains
• class presentations, Excel spreadsheets
• Links to useful resources (probability, OR, )

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About grading ...

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

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Resources for probability review for
spreadsheet modeling

• In course web page, look at
• Links Probability
• Ex The laymans guide to probability theory
• Look also at Bibliography
• Ex Feller An introduction to prob. Theory
• For spreadsheet modeling will use
• Insight.xla (Business Analysis Software). Sam L.
  Savage.

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References

• Course transparencies
• Copies from books/articles
• Anupindi et al. (1999). Managing Business Process
  Flows. Prentice Hall.
• D.E. Bell et al. (1995). Decision making under
  uncertainty. Course Technology.
• ...

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Ex Uncertain benefits

• Introducing new product in market
• Benefit? Depends on
• Sales (in units)
• Price/unit
• Cost/unit (production, marketing, sales, ...)
• Fixed costs (overhead, publicidad) E30.000
• Benefit
• Sales (Price- Cost_unit) - Fixed costs

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Market scenarios

• New market Uncertainty
• Scenarios high or low volume (50)
• Scenario cost/unit

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The building blocks of uncertainty

• 1. Uncertain numbers Random numbers
• 2. Averages Diversification
• 3. Important classes of random numbers Idealized
  distributions
• 4. Functions of random numbers uncertainty
  management

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Exponential distribution

• Models time between events, e.g., teleph. Calls,
  or product orders
• Density function
• Distribución

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Relation Exponential-Poisson

• Suppose time between consecutive calls is
• Then, number of calls ocurring in 0, t) es
• Hence,

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Uniform distribution

• Uniform distr. between a and b (a lt b)
• Density function
• Distribution

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Uniform distribution (cont)

• The RAND() Excel function
• Usefulness of in simulation
• Ex

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Geometric distribution

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

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Multivariate distributions

• Main example Multivariate Normal

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Multivariate distr. (cont)

• Given by Joint Distribution
• or by Joint Density Ex (Normal)

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Covariance/correlation

• Are measures of Linear Dependence between two
  r.v.

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Dependence/Independence of r.v.

• If then
• If then
• If then NO linear relation
• Def Two r.v. are INDEPENDENT if
• Ej Two independent exponentials

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Conditional expectation/probability

• Conditional probabilitiy probability of a
  success given another success occurs
• Conditional expectation

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Conditional prob./exp. and Independence

• Suppose are independent r.v.
• Then,
• A useful identity

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Application Expected benefit

• Have

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Ex conditional prob./exp.

• Cars enter a gas station with interarrival times
• Each car brings an independent number of people
  distributed as
• Distribution/mean of the number Y of people
  arriving in time interval 0, t)?

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Ex conditional prob./exp.

• Know number X of cars arriving in 0, t) is
  Poisson
• Let
  
• Then,

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Ex Conditional expectation

• Have
• So, by previous slide,

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The buildings of uncertainty Functions of random
variables

• Managers routinely input uncertain numbers into
  spreadsheet models
• customer satisfaction
• future demand for a product
• future workload requirements,
• Outputs are functions of random variables
• Tempting plug in best guesses
• Does it work? NO!!
• Instead plug in ALL uncertain inputs!

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Functions of random variables

• If X, Y, Z, are random variables
• and f(x, y, z, ) is a function,
• f(X, Y, Z, ) is a function of r.v.
• Ex linear functions of r.v.
• f(X, Y, Z) 5 X 4 Y - 2 Z
• The output of a probabilistic model is of the
  form f(X, Y, Z, )
• Ex profit(revenues, cost) revenues - cost

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The average of a function of random variables

• Wanted average value of f(X), Ef(X)
• Can just plug in average values? Is it true
• Ef(X)f(EX)?
• NO!! In general, Ef(X) distinct from f(EX) !
• When are they equal?

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Averages of functions of r.v.

• A sobering counterexample
• Consider a drunk, wandering left and right from
  the middle of a highway in heavy traffic.
• Take X drunks left-right position
• f(X) drunks fate (A/D)
• What is f(EX)? What is Ef(X)?

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Averages of functions of r.v.

• We can relate Ef(X) with f(EX) under certain
  conditions
• Jensens inequality if f(x) is convex, then
• Ef(X) gt f(EX)
• So, then can calculate lower bound
• What is the intuition?

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Simulation estimating Ef(X)

• If cannot obtain
  analytically, estimate it with Monte Carlo
  simulation
• Generate sample X1, , Xn
• Estimate is
• How many trials are enough?

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How many trials are enough?

• Markov inequality
• Let Y gt r.v., and a gt 0. Then,
• Useful consequence for simulation

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Optimization under under uncertainty

• Ex Let f(X,a) be the benefit in an inventory
  system, under random demand X, with inventory
  level a
• Wanted max Ef(X, a) over feasible a
• How to do it?
• Analysis Newsboys model
• Parameterized simulation vary a
• Another view Policy optimization

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More references

• Ross, S.M. Stochastic Processes. Wiley, 1983.
• Feller, W. An Introduction to Probability Theory
  and its Applications. Wiley, 1957.
• Savage, S. Insight.xla Business Analysis
  Software, 1998.
• Bernstein, P. Against the Gods The Remarkable
  Story of Risk. Wiley, 1996.