Stochastic Programming models, algorithms and applications

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Stochastic Programming models, algorithms and
applications
E-mailChandra.Poojari_at_brunel.ac.uk Webpage
www.brunel.ac.uk/mastcap
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Outline

• The Centre for analysis of Risk and optimisation
  modelling applications(CARISMA), Brunel
  university.
• Introduction to Stochastic programming.
• Application areas of Stochastic programming.
• Modelling paradigms in Stochastic programming.
• Solution techniques in Stochastic programming.
• An integrated environment for modelling and
  processing Stochastic programming models.

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The Centre for the Analysis of Risk and
Optimisation Modelling Applications http//carisma
.brunel.ac.uk
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People in CARISMA
Director Prof Gautam Mitra Deputy Director-
Prof Christos Ionnadies Research Lecturers-
Dr Paresh Date Dr
Fabio Spagnolia Dr Chandra
Poojari Research Associates-
Mr. Frank Ellison Mr George
Birbilis Dr Patrick Valente
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Mission of CARISMA

• The mission of CARISMA is to be a centre of
  excellence
• recognised for its research and scholarship in
  the following
•  

• the analysis of risk,
• optimisation modelling,
• the combined paradigm of risk and return
  quantification.
• Industry Focus
• Finance Industry - Bank, Insurance, Pension Funds
• Large Corporates - FTSE 100, Multinationals,
  EUROTOP
• Public Sector/Utilities, Environment, Food,
  Agriculture,
• Health

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Mathematical programming components

• Modelling systemsA computer language for
  describing large-scale optimisation or
  mathematical programming problems.
• MPL
• AMPL
• Solvers
• CPLEX
• OSL
• FortMP
• FortSP (stochastic programming solver)

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Need for planning under uncertainty
Consider the following situations
1. A single car breaks down on a freeway and
hundreds of motorist are
caught in a horrific
traffic jam
2. A single circuit breaker tripped in a storm
causes domino effect plunging a wide area into
darkness
3. Every year death and destruction due to
natural calamities like earthquakes, flood, and
hurricanes.
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Decision making under uncertainty
Hamlet
Shakespeare a Stochastic programmer ?
To be
Not to be
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Decision making under uncertainty
Shakespeare a Stochastic programmer ?
Merchant of Venice
My ventures are not in one bottom trusted/nor to
one place, nor is my whole estate/upon the
fortune of this present.
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Linear programming to Stochastic programming
A general deterministic linear program
Min
Is the solution sensitive to changes in parameter
values ?
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Extensions to LP

• Decisions not infinitely divisible
• Non-linear relationship
• Multi-time period decisions
• Probabilistic relationship

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Extensions to LP
discrete choice
LP
MIP
probabilistic modelling
probabilistic modelling
discrete choice
SLP
SIP
Multi-time period
Multi-time period
discrete choice
MSIP
MSLP
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Applications Supply Chain planning
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Applications Finance
Binomial tree of stock prices
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Applications Telecom
plant
plant
plant
A network connection
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What are Stochastic programs
A stochastic linear program
Min
Modelling techniques

• Chance constrained program
• Recourse program
• Quadratic program.

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Chance constrained model
Min
Properties
is log-concave, and A and c are fixed
1.Convex optimisation if
2.Quadratic programming if
is multivariate normal
3. Properties unknown other wise, but in general
Non-linear programming over non-convex set
otherwise.
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Two-stage SP with linear recourse
random event Probability Second-stage
cost Technical matrix Recourse
matrix Right-hand side First-stage
decisions Second-stage decisions.
Min
Subject to
let
Min
Subject to
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Recourse model using scenarios
Properties
1. Piece-wise Convex with non-linear objective
function
2. Requires multi-dimensional summation.
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Solving a two-stage Stochastic Linear Program
Master
Scenario sub-problem
Scenario sub-problem
Scenario sub-problems
Master problem
x
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Algorithms The deterministic equivalent
1. Simplex Worst-case complexity is
exponential. 2. Interior point method
Polynomial time complexity.
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Algorithms Stage-Decomposition
Benders decomposition
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Algorithms Scenario-Decomposition
Scenario sub-problems
Augmented Lagrangian decomposition
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Stochastic Programming Integrated environment
(SPInE)
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Subsystems in SPInE

• Scenario generator
• Modelling system
• Solver system
• Report generation

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SPInEs menu commands
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View of the scenario tree
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SMPS generation
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Solver control
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Computation of Value-at-Risk
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The Airlines model
Properties of the STORM model

• Two-stage air-freight scheduling model
• Has 1000 scenarios
• The Deterministic equivalent has
• 528,185 rows, 1,259,121 columns and
  3,341,696 non-zeroes.

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The Telecom model
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The Supply chain model
Properties of the PLTEXP model

• Five-stage capacity planning model
• Has 1296 scenarios
• The Deterministic equivalent has
• 539,198 rows, 1,410,236 columns and
  2,867,137 non-zeroes.

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The Finance model
Properties of the ALM model

• Four-stage asset liability model
• Has 1296 scenarios
• The Deterministic equivalent has
• 539,198 rows, 1,410,236 columns and
  2,867,137 non-zeroes.

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Speed-up of the decomposition algorithm
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Speed-up of the decomposition algorithm
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New developments in SPInE

• Application specific generator
• Lagrangean based column generator for SIP
• Stochastic decomposition
• Sampling techniques

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Conclusions

• Stochastic programming arises in many practical
  contexts.
• Alternative modelling techniques.
• Structure exploitation based algorithms.
• High performance and grid computing.
• Integrated optimisation tools.

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Thank You
http//carisma.brunel.ac.uk/
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