Large-Scale Stochastic Optimization: Decision-Making Under Uncertainty - Prof. Lewis Ntaim, Study notes of Systems Engineering

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ISEN 689

Large-Scale Stochastic Optimization

Introduction

Dr. Lewis Ntaimo

Sept 4, 2007

Outline

◮

Decision-Making in an Uncertain World

Applications

Decision Stages

Decision Trees

◮

Decision-Making Models

◮

Preliminaries

Probability Spaces

Random Variables

Linear Programming

Stochastic Programming Applications

◮

Example Applications

Manufacturing (supply chain planning)

Transportation (e.g airline industry)

Telecommunications (e.g. network design)

adequacy planning)Electricity power generation (e.g. power

Health care (e.g. patient/resource scheduling)

response)Agriculture / forestry (e.g. wildfire emergency

Finance (e.g. portfolio optimization)

◮

And many more ...

Sequential Decision Models

◮

Decision Trees

Graphical representation of the decision process

• Decision “event”

• Uncertainty “event”

• Time, progressing from left to right

revealed/providedRandom event is a point at which information is

Decision-Making Models

◮

Statistical Decision Theory (SDT)- Wald[1950]...

outcome of an experimentDetermine best levels of variables that affect the

x

X, ω

, associated distribution

F

ω

) , and

reward

r ( x, ω

, the basic problem is:

x^ max ∈ X

E

ω (^) [ r ( x, ω

F

(^) ] = max

Z

ω

r ( x, ω

dF

ω

)

major differences between the fields.programming. Underlying assumptions lead toProblem (1) is the fundamental form of stochastic

Decision-Making Models ...

◮

Decision Analysis (DA)- Raiffa [1968]...

Particular part of SDT

 Emphasis is on:

outcomesAcquiring information about possible



outcomesEvaluating the utility associated with possible



usually in the form of a decision treeDefining a limited set of possible outcomes

Decision-Making Models ...

◮

Optimal Stochastic Control

modelsModels often similar to stochastic programming

Problem dimensions are lower

Emphasizes control rules

More restrictive constraint assumptions

Decision-Making Models ...

◮

Stochastic Programming (SP)

data are not known with certaintymathematical programs in which uncontrollableBasically generalizations of deterministic

Note:

The ”certainty” assumption in linear

programming (LP) is violated!

linear programs with random data (Stochastic linear programming (SLP) deals with

course focus

datadeals with mixed integer programs with randomStochastic mixed-integer programming (SMIP)

Preliminaries

◮

Some Basic Probability First

ω

is an “outcome” of a random experiment (we

will use

˜ω

to denote a multivariate random

Note: The textbook by Birge & Louveaux usesvariable vector)

ξ

and

ξ .

is the set of all possible outcomes (sample

space)

A

is collection of random outcomes (events) of

Preliminaries

◮

Probability Spaces

For each

A

∈ A

there is a probability measure (or

distribution)

P

that tells the probability with which

A

∈ A

occurs



P

A



P

, P



P

A

1

∪

A

2 ) =

P

A

1 ) +

P

A

2 )

if

A

1

T

A^

2

=

The triplet (

A

, P

(^) ) is called a

probability space

Preliminaries

◮

Random Variables ...

Continuous rv’s are described by a

density

function

f (^) ( ω

)



Probability of

ω

being in an interval

[

a, b

]

is

P

a

˜ω

≤ b ) = Z b

a

f (^) (˜ ω

) d ˜ω

Z

b

a

dF

ω

)

F

b )

−

F

a )



Contrary to the discrete case,

P

ω

x

) = 0

Preliminaries

◮

More

Expected value of

˜ω

is



Discrete case:

μ

E

˜ω

= P k ∈ K ω k

f^ (^) ( ω

k )^



μ Continuous case:

E

˜ω

=

R

∞

−∞

˜ωf

ω

) d

˜ω

R

−∞∞

˜ωdF

ω

)

Variance of

˜ω

is Var(

˜ω

)=

E

[(˜

ω − μ ) 2 ]

Preliminaries

◮

In LP we deal with the following concepts:

Feasibility

Optimality

◮

Both these concepts are clear. In fact,

sensitivity

analysis

in LP deals with these two concepts.

◮

But suppose

T, r

contains random variables

T ,˜

˜r ).

What should we do?

Preliminaries

◮

Assumptions:

Real value of

( T, r

)

is not known

”scenarios”):Uncertainty if expressed by probability distribution (e.g.

P (^) (

˜

T ,

(^) ˜r ) = (

T (^) ω

, r^

ω ) =^

p ω (^) , ω

∈

Ω

.

This can also be expressed as:

P (^) ( T ,˜

˜r ) = (

T (^) s , r

s ) =

p s , s

= 1

, ..., S,

where,

S

=

| Ω

|

Probability distribution known (data, experts, etc)