Scenario-based Stochastic Constraint Programming

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Scenario-based Stochastic Constraint Programming

• Suresh Manandhar, Armagan Tarim, University of
  York
• Toby Walsh, 4C, University College Cork

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Motivation

• Constraint programming (CP)
• tool of choice for many domains
• scheduling, routing, assignment,
• .. but perhaps not for cliff diving
• But CP poor at dealing with uncertainty present
  in real world
• past data contain errors
• present data hard to collect
• future data is uncertain

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Stochastic constraint programs

• Introduced at ECAI-2002
• Decision variables
• user sets
• Stochastic variables
• nature sets according to probability distribution
• examples demand, market up/down, arrival time,
• Can be (in)dependent

ECAI 2002 conference dinner
Walsh 2002
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Stochastic constraint programs

• Hard constraints
• over decision vars, or decision and stochastic
  vars
• demandi productionistocki
• not purely over stochastic vars
• relation of (dis)allowed values
• Chance constraints
• not necessarily satisfied in all worlds
• Prob(demandi productionistocki) gt
  0.95

New
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Stochastic constraint programs

• Many different objectives supported
• expectation maximized or minimized
• downside minimized or upside maximized
• spread or variance minimized
• .

New
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One stage

• user makes decisions
• nature then sets stochastic variables

Outcome 1
M11
S13 B12
M10
Outcome 2
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Two stages

• user makes decisions
• nature sets stochastic variables
• user then makes some more decisions

S210 B26
M11
S13 B12
M10
S21 B21
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Multiple stages

• user makes decisions
• nature sets stochastic variables
• user then makes some more decisions

S332 B320
M21
S210 B26
M11
S34 B32
M20
S13 B12
S34 B32
M21
M10
S21 B21
S30 B31
M20
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Financial planning example

• Decision variables
• Si in stocks in ith year
• Bi in bonds in ith year
• Stochastic variables
• Mi 1, market up
• Mi 0, market down
• with 50 probability

S332 B320
M21
S210 B26
M11
S34 B32
M20
S13 B12
S34 B32
M21
M10
S21 B21
S30 B31
M20
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Financial planning example

• Market up
• stocks make 300
• bonds make 100
• Market down
• stocks make -100
• bonds make 0
• All money must be put in bonds or stocks
• Maximize expected profit
• E(SnBn)

S332 B320
M21
S210 B26
M11
S34 B32
M20
S13 B12
S34 B32
M21
M10
S21 B21
S30 B31
M20
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Financial planning example

• Solution now policy
• tree of decisions
• one path for each possible scenario
• Goal is to find policy that maximizes expected
  profit
• EProfit sum(paths)
• prob(path) (SnBn)

S332 B320
M21
S210 B26
M11
S34 B32
M20
S13 B12
S34 B32
M21
M10
S21 B21
S30 B31
M20
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Solving stochastic CPs

• Search method
• and/or search tree ECAI 2002
• stochastic vars gt and nodes
• decisions vars gt or nodes
• Compilation method
• gives regular (deterministic) constraint program

New
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Compiling stochastic CPs

• Introduce new decision vars for each scenario
• replace stochastic vars with their scenario value
• scenarios with common paths share decision vars
• ve
• gives regular (deterministic) CP
• -ve
• exponential space in worst case

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Stochastic OPL

• Compiles down into ILOGs OPL modelling language
• itself compiles down into C Solver code
• Exploits powerful propagators and modelling
  features found in OPL/Solver

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Stochastic OPL

• int n
• var int stocksn
• var int bondsn
• stochastic var 0..1 marketn uniform
• maximize
• expected 4stocksnmarketn
• 2bondsnmarketn
• bondsn(1-marketn)
• subject to
• stocks1bonds15
• forall(i in 1..n-1)
• stocksi1bondsi1
  4stocksimarketi
• 
  2bondsimarketi
• 
  bondsi(1-marketi)

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Stochastic OPL

• Variable declarations
• var int stocksn
• var int bondsn
• stochastic var 0..1 marketn uniform
• stochastic var 100..105 demandn 1,2,3,3,2,1
• stochastic var int incomingCallsn poisson(100)

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Stochastic OPL

• Hard constraints
• forall(i in 1..n-1)
• stocksi1bondsi1
  4stocksimarketi
• 
  2bondsimarketi
• 
  bondsi(1-marketi)
• Chance constraints
• forall(i in 1..n)
• prob(demandi lt productionistocki)
  gt 0.96

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Stochastic OPL

• Objectives
• maximize expected profitn
• subject to
• maximize min profitn
• subject to
• minimize variance profitn
• subject to
• minimize spread profitn
• subject to
• maximize chance profitngt100
• subject to

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Scenario reduction

• To cope with the exponential number of scenarios
• scenario mean
• scenario median
• scenario sample 100 montecarlo
• scenario sample 100 latinhypercube
• scenario top 10
• scenario top 95 percent

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Robust solutions

• Insist on identical decisions in different
  scenarios
• robust solution
• robust production0
• robust production1
• maximize chance
• wealthn gt 2wealth0
• subject to ..
• maximize robustness
• subject to ..

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Conclusions

• Stochastic constraint programs
• stochastic vars
• chance constraints
• several different objectives
• maximized expectation
• minimized downside
• Stochastic OPL
• compiles to (regular) OPL
• exploits power of CP toolkit

Interesting new way to reason with uncertainty?