Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
ISEN 636: Large-Scale Stochastic
Optimization
Topic: Introduction
***
L. Ntaimo
Industrial & Systems Engineering
Texas A&M University
Sept 4, 2009
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Outline
Probability Spaces
Applications
Decision-Making Models
Linear Programming (LP) Review
From LP to Stochastic Programming
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Preliminaries
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Some Basic Probability First
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ω is an “outcome" of a random experiment (we will use ω̃ to
denote a multivariate random variable vector)
Note: The textbook by Birge & Louveaux uses ξ and ξ.
Ω is the set of all possible outcomes (sample space)
A is collection of random outcomes (events) of Ω
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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Probability Spaces
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For each A ∈ A there is a probability measure (or
distribution) P that tells the probability with which A ∈ A
occurs
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0 ≤ P(A) ≤ 1
P(Ω) = 1, P(∅) = 0
T
P(A1 ∪ A2 ) = P(A1 ) + P(A2 ) if A1 A2 = ∅
The triplet (Ω, A, P) is called a probability space
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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Random Variables
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A random variable (rv) ω̃ on a probability space (Ω, A, P) is
a real-valued function ω̃(ω), ω ∈ Ω such that {ω|ω̃(ω) ≤ x}
is an event for all finite x.
For the random variable ω̃ we define its cumulative
distribution by Fω̃ = P(ω̃ ≤ x).
Discrete random variables take on a finite number of values
ω k , k ∈ K with associated probabilities,
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f (ω k ) = P(ω̃ = ω k ) with
P
k ∈K
f (ω k ) = 1
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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Random Variables ...
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Continuous r.v.’s are described by a density function f (ω)
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Probability of ω being in an interval [a, b] is
Z b
P(a ≤ ω̃ ≤ b) =
f (ω̃)d ω̃
Z
a
b
=
dF (ω̃)
a
= F (b) − F (a)
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Contrary to the discrete case, P(ω̃ = x) = 0
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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More
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Expected value of ω̃ is
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P
Discrete case: µ = Eω̃ = kR∈K ω k f (ω k )
R∞
∞
Continuous case: µ = Eω̃ = −∞ ω̃f (ω̃)d ω̃ = −∞ ω̃dF (ω̃)
Variance of ω̃ is Var(ω̃)= E[(ω̃ − µ)2 ]
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Sources of Uncertainty
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Since the real world is uncertain, it becomes imperative to
consider uncertainty in decision-making.
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Example Sources of Uncertainty:
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Market (product, stocks) related.
Financial related.
Technology related.
Competition related.
Weather related (e.g. airline rescheduling).
Catastrophic events (accidents, war, 9/11, etc.).
And many more ...
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Example Applications
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Manufacturing (supply chain planning)
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Transportation (e.g airline industry)
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Telecommunications (e.g. network design)
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Electricity power generation (e.g. power adequacy planning)
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Health care (e.g. patient/resource scheduling)
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Agriculture / forestry (e.g. wildfire emergency response)
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Finance (e.g. portfolio optimization)
And many more ...
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Sequential Decision Models
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Decision Trees
I Graphical representation of the decision process.
I 2 - Decision “event".
I ° - Uncertainty “event".
I → - Time, progressing from left to right.
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Random event is a point at which information is
revealed/provided.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision Trees
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Decision trees help to keep track of interplay between decisions
and random events.
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Decisions that follow “info" can adapt to it.
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Decisions that precede it cannot.
Decision
Observe
Decision
...
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In general, it is “best" to delay the decision as long as possible:
most flexible for adapting to “info".
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There is no advantage to delay if no “info" is anticipated.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Models
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Statistical Decision Theory (SDT)- Wald[1950]...
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Determine best levels of variables that affect the outcome
of an experiment.
x ∈ X , ω ∈ Ω, associated distribution F (ω), and reward
r (x, ω), the basic problem is:
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Max Eω [r (x, ω)|F ] = Max r (x, ω)dF (ω).
(1)
x∈X
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x∈X
ω
Problem (1) is the fundamental form of stochastic
programming. Underlying assumptions lead to major
differences between the fields.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision-Making Models...
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Statistical Decision Theory (SDT)- Wald[1950]...
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Emphasis is on:
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Changes in F to some updated distribution F̂x that depends
on a partial choice of x and some observations of ω.
Assumes that this part of analysis dominates any solution
procedure, as when X is a small finite set that can easily be
enumerated.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision-Making Models...
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Decision Analysis (DA)- Raiffa [1968]...
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Particular part of SDT.
Emphasis is on:
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Acquiring information about possible outcomes.
Evaluating the utility associated with possible outcomes.
Defining a limited set of possible outcomes usually in the
form of a decision tree.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision-Making Models...
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Dynamic Programming (DP) and Markov Decision Processes
(MDP) - e.g. Bellman [1957], Ross [1983]...
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Search for optimal actions to take at discrete points in time.
Actions influenced by random outcomes and carry one from
some state at some stage t to another state at stage t + 1.
Emphasizes identifying finite or at least, low-dimensional
state and actions spaces in assuming some Markovian
structure (actions and outcomes depend on current state).
Backward recursion resulting in an optimal decision
associated with each state.
For infinite horizon problems use discounting; finding
stationary policy.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision-Making Models...
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Optimal Stochastic Control
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Models often similar to stochastic programming models.
Problem dimensions are lower.
Emphasizes control rules.
More restrictive constraint assumptions.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision-Making Models...
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Stochastic Programming (SP)
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Generally assumes that difficulties in finding the form of r and changes
in F as a function of actions are small compared to finding expectations
with known F and optimal x with all other info known.
Emphasis is on finding a solution after a suitable problem statement in
the form (1) has been found.
Basically generalizations of deterministic mathematical programs in
which uncontrollable data are not known with certainty.
Note: The "certainty" assumption in linear programming (LP) is violated!
Stochastic linear programming (SLP) deals with linear programs with
random data (course focus).
Stochastic mixed-integer programming (SMIP) deals with mixed integer
programs with random data.
Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Decision-Making Models...
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Stochastic Programming ...
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Key features:
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Many decision variables with many potential values.
Discrete time periods for decisions.
Use of expectation (and other risk measures) functionals for
objectives.
Known or partially known probability distributions.
The relative importance of these features contrasts with the
other models.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
LP Review: You know this already!
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
LP Review
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
LP Review
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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Consider the following linear program (LP):
Min c > x
s.t. Ax ≥ b,
(2a)
(2b)
Tx ≥ r ,
x ≥ 0,
(2c)
(2d)
where,
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x ∈ <n+1 is the decision variable vector
c ∈ <n1 is the cost vector
A ∈ <m1 ×n1 is the constraint matrix
b ∈ <m1 is the RHS vector.
T ∈ <m2 ×n1 is the technology matrix
r ∈ <m2 is the RHS vector.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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In LP we deal with the following concepts:
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Feasibility
Optimality
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Both these concepts are clear. In fact, sensitivity analysis in LP
deals with these two concepts.
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But suppose (T , r ) contains random variables (T̃ , r̃ ). What
should we do?
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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Assumptions:
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Real value of (T , r ) is not known
Uncertainty if expressed by probability distribution (e.g.
"scenarios"):
P(T̃ , r̃ ) : P(T ω , r ω ) = pω , ω ∈ Ω.
This can also be expressed as:
P(T̃ , r̃ ) : P(T s , r s ) = ps , s = 1, ..., S,
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where, S = |Ω|
Probability distribution known (available data, experts,
forecasting/prediction models, etc.)
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
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Decision-making under uncertainty using SP:
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Find x “here-and-now" without knowing the real value of
(T , r ), but knowing its probability distribution
Tx ≥ r can be interpreted as a goal constraint to be
specified more precisely
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches
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Expected Value Solution
I Replace T̃ x ≥ r̃ with E[T̃ ]x ≥ E[r̃ ] = T̄ x ≥ r̄ , where,
P
P
T̄ = s ps T s and r̄ = s ps r s . Then the problem can be
given as follows:
Min c > x
s.t. Ax ≥ b
T̄ x ≥ r̄
x ≥ 0.
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(3)
Advantage: Simple model - deterministic LP.
Disadvantage: Risk is not taken care of since T s x ≥ r s for
some scenarios only!
What else can you do here?
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Apply sensitivity analysis: Still poor model of
decision-making under uncertainty [See Higle and Wallace,
2003].
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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‘Fat’ Solution
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Replace T̃ x ≥ r̃ with T s x ≥ r s , s = 1, ..., S. Then the
problem can be given as follows:
Min c > x
s.t. Ax ≥ b
T s x ≥ r s , s = 1, ..., S
x ≥ 0.
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(4)
Advantage: Deterministic LP.
Disadvantage: This is a conservative, expensive, restrictive
model: often no feasible solution exists. Yields overly
conservative solutions driven by extreme events, no matter
how rare!
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Scenario Analysis
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Solve
Min c > x s
s.t. Ax ≥ b
T sx s ≥ r s
x s ≥ 0.
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(5)
for every scenario (T s , r s ), s = 1, ..., S.
Get solutions x s , s = 1, ..., S. Find an overall solution
“based on the scenario solutions".
Advantage: Each scenario solution is a deterministic LP;
improvement over the expected value approach; very
popular approach.
Disadvantage: How do you find an overall solution?
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Chance (Probabilistic) Constraints
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Replace T̃ x ≥ r̃ with P{T̃ x ≥ r̃ } ≥ α, for some specified
reliability level α ∈ (0.5, 1). Then the problem can be given
as follows:
Min c > x
s.t. Ax ≥ b
P{T̃ x ≥ r̃ } ≥ α,
x ≥ 0.
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(6)
Advantage: Risk is taken care of explicitly (1 − α is maximal
acceptable risk)
Disadvantage: Difficult to compute; discrete distributions
may lead to MIP model; in general, possibly nonconvex
model.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Two-Stage Recourse Model
I Introduce explicitly corrective actions. Replace T̃ x ≥ r̃ with
T̃ x + Wy ≥ r̃ ,
<n+2
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where y ∈
is the decision vector of a second-stage LP
problem.
The value of y depends on the realization of (T̃ , r̃ )
Penalize corrective actions, called recourse actions in SP.
Minimize total expected costs.
Decision
x
Stage 1
Observe
Uncertainty
Decision
y = ys
Stage 2
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Two-Stage SLP with Recourse Model (discrete distribution):
Min c > x +
S
X
ps q > y s
(7a)
s=1
s.t. Ax
≥ b,
s
s
T x + Wy ≥ r s ,
x ≥ 0, y s ≥ 0, s = 1, ..., S,
(7b)
(7c)
(7d)
with q unit recourse costs.
Objective = c > x + expected recourse costs.
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Two-Stage SP with Recourse Model ...
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Advantage: Risk is taken care of explicitly (expected
recourse costs); large-scale LP model.
Disadvantage: model may be too large to solve, e.g. 10
independent random variables, 6 realizations each
⇒ S = 610 ≈ 60, 000, 000!
SP dimensions: matrix A is (m1 × n1 ), W is (m2 × n2 ), SP
has
(n1 + n2 S) decision variables
(m1 + m2 S) constraints.
Large-scale model ⇒ DECOMPOSITION approach to
solve!
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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A general two-stage SP with recourse model can be written as
follows:
Min c > x + Eω̃ [f (x, ω̃)]
s.t. Ax ≥ b
x ≥ 0,
(8)
where for any realization ω of ω̃ (defined on a probability space
(Ω, A, P)) we have
Min q(ω)> y
s.t. W (ω)y ≥ r (ω) − T (ω)x
y ≥ 0.
(9)
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Some standard SP terminology:
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Function E[f (x, ω̃)]: Expected recourse function
Matrix W (ω): Recourse matrix
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Matrix W (ω) = W fixed: Fixed recourse
Matrix W (ω) random: Random recourse
Matrix W = [I, −I]: Simple recourse
If (f (x, ω̃)) < ∞ w.p.1 ∀x ∈ <n1 : Complete recourse
If (f (x, ω̃)) < ∞ w.p.1 ∀x ∈ X , X = {x ∈ <n1 |Ax ≥ b}
Relatively complete recourse
We will talk more about these later
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Probability Spaces Applications Decision-Making Models Linear Programming (LP) Review From LP to Stochastic Programming
Approaches...
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Next ...
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Stochastic programming modeling examples
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