Synthesizing Filtering Algorithms in Stochastic Constraint ... · Synthesizing Filtering Algorithms in Stochastic Constraint Programming Brahim Hnich1 Roberto Rossi2 S. Armagan Tarim3

Introduction Constraint Programming Stochastic Constraint Programming Ongoing Research Conclusions

Synthesizing Filtering Algorithms inStochastic Constraint Programming

Brahim Hnich1

Roberto Rossi2

S. Armagan Tarim3

Steven D. Prestwich4

1Faculty of Computer Science, Izmir University of Economics, Izmir, Turkey1LDI, Wageningen UR, the Netherlands

3Dept. of Operations Management, Nottingham University Business School, UK4Cork Constraint Computation Centre, University College Cork, Ireland

XL Annual Conference of the Italian Operational ResearchSociety, AIRO 2009


Decision Making

A Complete Overview

Theoretical Results

Applications

Decision Support Systems

Modeling Frameworks


Decision Making

Decision Making in a Deterministic Setting

DETERMINISTIC

Theoretical Results

Applications


Modeling Frameworks

MIP

CP

LPSimplex Shortest Path

CRM

ERPInventory Control

Production Planning

Transport Scheduling


Decision Making


DETERMINISTIC

Theoretical Results

Applications


Modeling Frameworks

MIP

CP


CRM


Production Planning


CPLEXCPLEX


Decision Making


DETERMINISTIC

Theoretical Results

Applications


Modeling Frameworks

MIP

CP


CRM


Production Planning


OPL STUDIOOPL STUDIO


Decision Making

Decision Making Under Uncertainty: A Pervasive Issue

UNCERTAINTY

Theoretical Results

Applications


Modeling Frameworks

Production Planning

Inventory Control

Land-Crop Allocation

Sustainable Energy Production

Financial Planning

Food Quality Control


Decision Making

Decision Making Under Uncertainty

UNCERTAINTY

Theoretical Results

Applications


Modeling Frameworks

Stochastic Dynamic Programming

Stochastic Programming

Simplex Convex Analysis

CRM


Production Planning



Decision Making


UNCERTAINTY

Theoretical Results

Applications


Modeling Frameworks

Stochastic Dynamic Programming

Stochastic Programming

Simplex Convex Analysis

CRM


Production Planning


STOCHASTIC OPLSTOCHASTIC OPL


An example


0-1 Knapsack Problem

Problem : we have k kinds of items , 1 through k . Each kind ofitem i has

a value ri

a weight wi .

We usually assume that all values and weights arenon-negative. The maximum weight that we can carry in thebag is c.Objective : find a set of objects that provides the maximumvalue and that fits in the given capacity.


An example


0-1 KP: MIP Formulation

Objective:max

∑ki=1 rixi

Constraints:∑ki=1 wixi ≤ c

Decision variables:xi ∈ {0, 1} ∀i ∈ {1, . . . , k}


An example


Static Stochastic Knapsack Problem

Problem : we have k kinds of items and a knapsack of size cinto which to fit them. Each kind of item i has

a deterministic profit ri .

a size Wi , which is not known at the time the decision hasto be made. The decision maker knows the probabilitydistribution of Wi .

A per unit penalty cost p has to be paid for exceeding thecapacity of the knapsack. The probability of not exceedingthe capacity of the knapsack should be greater or equal to agiven threshold θ.Objective : find the knapsack that maximizes the expectedprofit.


An example


SSKP: Stochastic Programming Formulation

Objective:

max

Pki=1 riXi − pE

h

Pki=1 Wi Xi − c

i+ff

Subject to:

Prn

Pki=1 WiXi ≤ c

o

≥ θ

Decision variables:Xi ∈ {0, 1} ∀i ∈ 1, . . . , k

Stochastic variables:Wi → item i weight ∀i ∈ 1, . . . , k

Stage structure:V1 = {X1, . . . , Xk}S1 = {W1, . . . ,Wk}L = [〈V1, S1〉]


Basic Notions

Formal Background

A slightly formal definition

A Constraint Satisfaction Problem (CSP) is a triple 〈V , D, C〉.

V = {v1, . . . , vn} is a set of variables

D is a function mapping each variable vi to a domain D(vi)of values

C is a set of constraints.

A Constraint Optimization Problem (COP) consists of a CSPand objective function f (V ) defined on a subset V of thedecision variables in V . The aim in a COP is to find a feasiblesolution that minimizes (maximizes) the objective function.


Basic Notions

Solution Method

Strategy

Constraint Programming proposes to solve CSPs/COPsby associating with each constraint a filtering algorithm .

A filtering algorithm removes from decision variabledomains values that cannot belong to any solution of theCSP/COP.

Constraint Propagation is the process that repeatedlycalls filtering algorithms until no new deduction can bemade.

Constraint Solving interleaves filtering algorithms and asearch procedure (for instance a backtracking algorithm).


Basic Notions

An Example

P0

P1

x1 ∈ {1}

Sample COP: 0-1 KPV = {x1, . . . , x3}D(xi) = {0, 1} ∀i ∈ {1, . . . , 3}C = {8x1 + 5x2 + 4x3 ≤ 10}f (x1, . . . , x3) = 8x1 + 15x2 + 10x3

Filtered domains at P1

D(x1) = {1} D(x2) = {0} D(x3) = {0} D(z) = {8}


Basic Notions

Global Constraints

In constraint programming is common to find constraints over anon-predefined number of variables

alldifferent

element

cumulative

...

These constraints are called global constraints

they can be used in a variety of situations

they are associated with powerful filtering strategies

new custom global constraints can be defined


Basic Notions

Global Constraints

Filtering algorithms

detect inconsistencies in a proactive fashion

speed up the search

provided that the time spent in filtering is less then the timesaved in terms of search efforts.A challenging research topic is the design of efficient filteringstrategies.


Basic Notions

A slightly formal definition

Stochastic Constraint Satisfaction ProblemA Stochastic Constraint Satisfaction Problem (SCSP) is a7-tuple

〈V , S, D, P, C, θ, L〉.

V = {v1, . . . , vn} is a set of decision variables

S = {s1, . . . , sn} is a set of stochastic variables

D is a function mapping each variable to a domain of potential values

P is a function mapping each variable in S to a probability distribution for itsassociated domain

C is a set of (chance)-constraints, possibly involving stochastic variables

θh is a threshold probability associated to chance-constraint h

L = [〈V1, S1〉, . . . , 〈Vi , Si〉, . . . , 〈Vm, Sm〉] is a list of decision stages.

By considering an objective function f (V , S) we obtain a SCOP.


Basic Notions

An Example

Sample SCOP: SSKP

V = {x1, . . . , x3}

D(xi) = {0, 1} ∀i ∈ {1, . . . , 3}

S = {w1, . . . , w3}

D(w1) = {5(0.5), 8(0.5)}, D(w2) ={3(0.5), 9(0.5)}, D(w3) = {15(0.5), 4(0.5)}

C = {Pr(w1x1 + w2x2 + w3x3 ≤ 10) ≥ 0.2}

L = [〈V , S〉]

f (x1, . . . , x3) =

8x1 + 15x2 + 10x3 − 2E max[0,

∑3i=1 wixi − 20

]


Basic Notions

An Example

Sample SCOP: DSKP

V = {x1, . . . , x3}

D(xi) = {0, 1} ∀i ∈ {1, . . . , 3}

S = {w1, . . . , w3}

D(w1) = {5(0.5), 8(0.5)}, D(w2) ={3(0.5), 9(0.5)}, D(w3) = {15(0.5), 4(0.5)}

C = {Pr(w1x1 + w2x2 + w3x3 ≤ 10) ≥ 0.2}

L = [〈{x1}, {w1}〉, 〈{x2}, {w2}〉, 〈{x3}, {w3}〉]

f (x1, . . . , x3) =

E[8x1 + 15x2 + 10x3] − 2E max[0,

∑3i=1 wixi − 20

]


Stochastic OPL

Stochastic OPL

A language specifically introduced by Tarim et al. (Tarim et al.,2006) for modeling decision problems under uncertainty .It captures several high level concepts that facilitate theprocess of modeling uncertainty:

stochastic variables (independent or conditionaldistributions)

several probabilistic measures for the objective function(expectation, variance, etc.)

chance-constraints

decision stages

...


Stochastic OPL

Stochastic OPL

SSKPint N = 3;int c = 10;int p = 2;float θ = 0.2range Object [1..3];int value[Object] = [8,15,10];stoch int weight[Object] = [<5(0.5),8(0.5)>,

<3(0.5),9(0.5)>,<15(0.5),4(0.5)>];var int+ X[Object] in 0..1;stages = [<X,weight>];var int+ z;

maximize sum(i in Object) X[i]*value[i] - p*zsubject to{z = max(0,expected(sum(i in Object) X[i]*weight[i] - c));prob(sum(i in Object) X[i]*weight[i] - c ≤ 0) ≥ θ;};


Stochastic OPL

Stochastic OPL

DSKPint N = 3;int c = 10;int p = 2;float θ = 0.2range Object [1..3];int value[Object] = [8,15,10];stoch int weight[Object] = [<5(0.5),8(0.5)>,

<3(0.5),9(0.5)>,<15(0.5),4(0.5)>];var int+ X[Object] in 0..1;stages = [<X[1],weight[1]>,<X[2],weight[2]>,<X[3],weight[3]>];var int+ z;

maximize sum(i in Object) X[i]*value[i] - p*zsubject to{z = max(0,expected(sum(i in Object) X[i]*weight[i] - c));prob(sum(i in Object) X[i]*weight[i] - c ≤ 0) ≥ θ;};


Solution Methods

Scenario-based Compilation

By using the approach discussed in

S. A. Tarim, S. Manandhar and T. Walsh,Stochastic Constraint Programming: A Scenario-Based Approach,Constraints, Vol.11, pp.53-80, 2006

it is possible to compile any SCSP/SCOP down to a determinis-tic equivalent CSP.


Solution Methods


Solver

stoch myrand[onestage]=...;int nbItems=...;float c = ...;float q = ...;range Items 1..nbItems;range onestage 1..1;float W[Items,onestage]^myrand = ...;float r[Items] = ...;dvar float+ z;dvar int x[Items] in 0..1;

maximizesum(i in Items) x[i]*r[i] - expected(c*z)

subject to{z >= sum(i in Items) W[i]*x[i] - q;prob(sum(i in Items) W[i]*x[i] <= q) >= 0.6;};

Solution

Stochastic Constraint Program Stochastic OPL Model

Deterministic equivalent model

int nbWorlds=...;range Worlds 1..nbWorlds;int nbItems=...;range Items 1..nbItems;float c = ...;float W[Worlds,Items] =...;float Pr[Worlds]=...;float r[Items] = ...;float q = ...;var float+ z[Worlds];var int+ x[Items] in 0..1;

maximize ((sum(i in Items)x[i]*r[i])- c*(sum(j in Worlds)Pr[j]*z[j]))subject to{ forall(j in Worlds) z[j]>=(sum(i in Items)W[j,i]*x[i])-q; sum(j in Worlds) Pr[j]*(sum(i in Items)W[j,i]*x[i] <= q) >= 0.2;};

Compiler


Solution Methods


SSKP: Compiled Deterministic Equivalent CSP


Solution Methods


Advantages

Seamless Modeling under Uncertainty!

Stochastic OPL not necessarily linked to CP

DrawbacksSize of the compiled model

Constraint Propagation not fully supported


Solution Methods

An Alternative Approach to Seamless Stochastic Optimizati on

Constraint Programming Solversupporting Global Chance-Constraints

stoch myrand[onestage]=...;int nbItems=...;float c = ...;float q = ...;range Items 1..nbItems;range onestage 1..1;float W[Items,onestage]^myrand = ...;float r[Items] = ...;dvar float+ z;dvar int x[Items] in 0..1;

maximizesum(i in Items) x[i]*r[i] - expected(c*z)

subject to{z >= sum(i in Items) W[i]*x[i] - q;prob(sum(i in Items) W[i]*x[i] <= q) >= 0.6;};

Solution

Stochastic Constraint Program Stochastic OPL Model

Filtering Algorithms forGlobal Chance-Constraints


Global Chance-Constraints

Filtering in SCSPs

Also in Stochastic Constraint Programming (SCP) we have

constraints

filtering algorithms

In contrast to CP, in SCP constraints divide into

hard constraints

chance-constraints

Global Chance-ConstraintsPerhaps the most interesting aspect of SCP is that the conceptof global constraint can be also adopted in a stochasticenvironment, thus leading to

Global Chance-Constraints (Rossi et al., 2008)



Filtering in SCSPs

Stochastic Programming Model

Pr{∑k

i=1 WiXi ≤ c}≥ θ

Global Chance-ConstraintstochLinIneq(x,W,Pr,q,0.2);



Filtering in SCSPs

SSKP: Compiled Deterministic Equivalent CSP withGlobal Chance-Constraints



Filtering in SCSPs

Stochastic Constraint Programming


represent relations among a non-predefined number ofdecision and random variablesimplement dedicated filtering algorithms based on

feasibility reasoningoptimality reasoning

Global Chance-Constraints performing optimality reasoning arecalled Optimization-Oriented Global Chance-Constraints(Rossi et al., 2008).



Filtering in SCSPs

“Synthesizing Filtering Algorithms for Global Chance-Constraints” (Hnich et al., 2009)



Filtering Algorithms for GCCs: An example

x1={0,1}x2={0,1}

x3={0,1}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5

Solution TreeSearch Tree




x1={0,1}x2={0,1}

x3={0,1}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={0,1} x2={0,1} x3={0}




x1={0,1}x2={0,1}

x3={0,1}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={0,1} x2={0,1} x3={0}

x1={0,1} x2={0,1} x3={0,1}




x1={0,1}x2={0,1}

x3={0,1}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={0,1} x2={0,1} x3={0}

x1={0,1} x2={0,1} x3={0,1}

x1={0,1} x2={0,1} x3={0}

x1={0,1} x2={0,1} x3={0,1}

x1={0,1} x2={0,1} x3={0}

x1={0,1} x2={0,1} x3={0,1}

x1={0,1} x2={0,1} x3={0}

x1={0,1} x2={0,1} x3={0,1}




x1={1}x2={0,1}

x3={0}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={1} x2={0,1} x3={0}

x1=1




x1={1}x2={0,1}

x3={0}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={1} x2={0,1} x3={0}

x1=1

x1={1} x2={0,1} x3={0}




x1={1}x2={0,1}

x3={0}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={1} x2={0,1} x3={0}

x1=1

x1={1} x2={0,1} x3={0}

x1={1} x2={0} x3={0}




x1={1}x2={0,1}

x3={0}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={1} x2={0,1} x3={0}

x1=1

x1={1} x2={0,1} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}




x1={1}x2={0}

x3={0}

w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

Pr(w1x1+w2x2+w3x3²10)>0.5


x1={1} x2={0,1} x3={0}

x1=1

x1={1} x2={0,1} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}

x1={1} x2={0} x3={0}




x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


Pr(w1x1+w2x2+w3x3²10)³0.5




x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


x1={} x2={1} x3={1}

Pr(w1x1+w2x2+w3x3²10)³0.5




x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

Pr(w1x1+w2x2+w3x3²10)³0.5




x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


Pr(w1x1+w2x2+w3x3²10)³0.5

x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={0} x2={1} x3={0}




x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={0} x2={1} x3={0}

x1={0} x2={1} x3={0}

x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


Pr(w1x1+w2x2+w3x3²10)³0.5




x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={0} x2={1} x3={0}

x1={0} x2={1} x3={0}

x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={} x2={1} x3={1}

x1={} x2={1} x3={1}

Pr(w1x1+w2x2+w3x3²10)³0.5




x3={0,1}

x3=1

x3=1

x3=1

x2=1

x2=1

x1={0,1}w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={0} x2={1} x3={0}

x1={0} x2={1} x3={0}

x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={} x2={1} x3={1}

x1={} x2={1} x3={1}

Pr(w1x1+w2x2+w3x3²10)³0.5




x3=0

x3=1

x3=1

x3=1

x2=1

x2=1

x1=0w1=5

w1=8

w2=3

w2=9

w2=3

w2=9

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4

w3=15

w3=4


x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={0} x2={1} x3={0}

x1={0} x2={1} x3={0}

x1={} x2={1} x3={1}

x1={0} x2={1} x3={1}

x1={} x2={1} x3={1}

x1={} x2={1} x3={1}

Pr(w1x1+w2x2+w3x3²10)³0.5


Filtering Algorithms

B. Hnich, R. Rossi, S. A. Tarim and S.Prestwich,Synthesizing Filtering Algorithmsfor Global Chance-Constraints ,15th International Conference onPrinciples and Practice of ConstraintProgramming (CP-09) Lisbon,Portugal, September 21-24, 2009

ContributionA generic approach for constraint reasoning underuncertainty .Works with any existing propagation algorithm!




DrawbackOnly implemented for linear inequalities/equalities:stochLinIneq(x,W,Pr,q,0.2);

i.e. SSKP → Pr(w1x1 + w2x2 + w3x3 ≤ 10) ≥ 0.2




Future workConsidering more global constraints :allDifferent()NValue()Cumulative()...


Integrated Development Environment


Final remarks

Summary

We discussed a Framework for Modeling DecisionProblems under Uncertainty

Stochastic Constraint ProgrammingGlobal Chance-constraintsStochastic OPL


Questions

Questions

Documents

Synthesizing Filtering Algorithms in Stochastic Constraint ... · Synthesizing Filtering Algorithms in Stochastic Constraint Programming Brahim Hnich1 Roberto Rossi2 S. Armagan Tarim3