Stochastic Constraint Programming and Global Stochastic … · 2020-07-07 · Stochastic Constraint...

Stochastic Constraint Programming andGlobal Stochastic Constraints

A modelling and solution framework for effective decisionmaking under uncertainty

Dr. Roberto Rossi

The University of Edinburgh Business SchoolThe University of Edinburgh

[email protected]

Friday, November the 29th, 2013

1

What is Constraint Programming (CP)

A subfield of Artificial Intelligence

It represents an efficient means for finding solutions tocombinatorial problems in areas such as:

◮ timetabling

◮ scheduling

◮ design & configuration

◮ . . .

Closely related to a number of disciplines:

◮ logic programming & inference

◮ graph theory

◮ combinatorial optimization

◮ . . .

2

Who cares about Constraints

◮ IBM acquired ILOG in 2009, a leading vendor of constrainttechnology.

◮ CISCO acquired the ECLiPSe constraint logic programmingsystem.

◮ St Andrews Minion solver is used to schedule the CB1000Nanoproteomic Analysis Systemdx.doi.org/10.1007/978-3-642-04244-7_8.

◮ In Scotland there is a fairly large academic community: thereare constraint programers in Edinburgh, Glasgow, St Andrewsand Dundee.

3

dx.doi.org/10.1007/978-3-642-04244-7_8

Constraints

Constraints represent a natural means of knowledge

representation.

Examples

“x+y=30”

“The helicopter can carry one passenger”

University timetabling:

◮ “no student can attend two lectures at once”

◮ “George Square lecture theatre has a capacity of 400students”

◮ “art history lectures require a slide projector”

4

Constraints

Constraint programming offers many expressive constraints thatcan be used to build a constraint model, e.g.

allDifferent(x1, . . . , xn)

Global constraint catalog

The catalogue presents a list of 354 global constraints issued fromthe literature in constraint programming and from popularconstraint systems.

http://www.emn.fr/z-info/sdemasse/gccat/

5

http://www.emn.fr/z-info/sdemasse/gccat/

Solving problems with constraints

Two phases:

1. Describe the problem to be solved as a constraint model,a format suitable for input to a constraint solver.

2. Search (automatically) for solutions to the model with aconstraint solver (e.g. IBM ILOG, Choco, ECLiPse,. . . ).

6

The (finite domain) Constraint Satisfaction Problem

Given

◮ a finite set of decision variables

◮ for each decision variable, a finite domain of potential values

◮ a finite set of constraints on the decision variables

Find an assignment of values to variables such that all constraintsare satisfied.

7

The (finite domain) Constraint Satisfaction Problem

Decision variableA decision variable corresponds to a choice that must be made insolving a problem.

DomainsValues in the domain of a decision variable correspond to theoptions for a particular choice.

ConstraintsRepresent relations that identify which assignments of values tovariables are allowed.

8

Constraint solving

Typically interleaves two components

◮ systematic search through a space of partial assignments◮ extend an assignment to a subset of the variables incrementally◮ backtrack (i.e. “undo”) if establish that current partial

assignment cannot be extended to a solution

◮ constraint propagation◮ logical inference based on constraints and current domains◮ aims to reduce domains

9

The Crystal Maze Puzzle

Thanks to Patrick Prosser

Computing ScienceGlasgow University17 Lilybank GardensGlasgow G12 8RZ

10

Crystal MazeGiven the network

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Place numbers 1, . . . , 8 on nodes by making sure that◮ each number appears exactly once◮ consecutive numbers do not appear on adjacent nodes

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Crystal Maze as a CSP

The Crystal Maze puzzle can be modelled as a CSP.

A constraint solver can then make the inferences you justperformed by hand, automatically.

12


Variables

x3

x7

x4

x1

x8

x5

x2

x6

The indexing adopted is irrelevant.

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Domains

x3

x7

x4

x1

x8

x5

x2

x6

All variables have the same domain: {1, 2, 3, 4, 5, 6, 7, 8}.

14


Constraints

x3

x7

x4

x1

x8

x5

x2

x6

◮ All values in the domains should be used:alldifferent(x1, x2, x3, x4, x5, x6, x7, x8)

◮ No consecutive numbers on adjacent nodes:|x1 − x2| > 1|x1 − x3| > 1. . .

15

Solving the Crystal Maze Puzzle

But how would a constraint solver reason about it?

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Which nodes are the hardest to label?

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Which nodes are the hardest to label?

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Which are the least constraining values to use?

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Recall that domains are {1, 2, 3, 4, 5, 6, 7, 8}.

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Which are the least constraining values to use?

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Recall that domains are {1, 2, 3, 4, 5, 6, 7, 8}.

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Exploit symmetry!

It would be unnecessary to consider the following symmetricassignment.

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Constraint propagation

Exploit constraint propagation to eliminate infeasible values fromdomains associated with other nodes (i.e. decision variables).

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Exploit constraint propagation to eliminate infeasible values fromdomains associated with other nodes (i.e. decision variables).

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Values 1 and 8 appear exactly once.

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Enforce adjacency restrictions.

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Symmetry again. . .

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Symmetry. . .

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SearchGuess, but be prepared to backtrack. . .

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Solution.

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Generally, it doesn’t go as smooth as this.

Search will often involve backtracking.

54

Search

Search is often carried out by using “standard” depth-first search(DFS) procedures.

At each node of the search tree two heuristics operate:

◮ variable selection: what variable do we branch on (e.g.smallest domain, most constrained, . . . )?

◮ value selection: what value in the domain do we try first(e.g. increasing, decreasing, . . . )?

In general, there is no easy way to decide what strategies are goodfor a given problem!

55

Propagation

Propagation operates by exploiting dedicated filtering algorithmsthat enforce some degree of “consistency” by removing provablyinfeasible value from the domains of the constrained variables.

Possible “levels” of consistency (from weaker to stronger) are:

◮ arc consistency

◮ bound consistency

◮ range consistency

◮ . . .

◮ hyper-arc consistency

56

Propagation

Arc consistency

Arc consistency is the simplest (and weakest) possible form ofconsistency that can be enforced; it is enforced by checking eachpair of variables against a given constraint that constrains them —as we did in the Crystal Maze example!

57

Propagation

Arc consistency: example

¹

¹ ¹

x1 x2

x3

81,2,3<

81,2< 81,2<

58

Propagation

Bound consistency

Bound consistency operates by pruning edges of domains, a typicalexample involves linear inequalities.

59

Propagation

Bound consistency: example

Variables & Domains:

x1 ∈ {1, . . . , 3}

x2 ∈ {5, . . . , 9}

x3 ∈ {1, . . . , 2}

Constraints:x1 + x2 + x3 ≥ 15

Reasoning:x1 ≥ 15− x2 − x3

where xi denotes the maximum value in the domain of xi.

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Propagation

Bound consistency: example

Variables & Domains:

x1 ∈ {1, . . . , 3}

x2 ∈ {5, . . . , 9}

x3 ∈ {1, . . . , 2}

Constraints:x1 + x2 + x3 ≥ 15

Reasoning:

x1 ≥ 15− 9− 2 = 4 /∈ {1, . . . , 3} inconsistent!

where xi denotes the maximum value in the domain of xi.

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Propagation

Hyper-arc consistency

This is the strongest possible level of consistency.

Informally speaking, it guarantees that it is always possible toextend the current partial assignment to a solution viabacktrack-free search.

If the problem of interest is NP-hard, enforcing hyper-arcconsistency on it is also NP-hard.

We shall demonstrate how it is possible to enforce hyper-arcconsistency for the alldifferent(x1, . . . , xn) constraint, whichensures that x1, . . . , xn take all different values.

Jean-Charles Regin: A Filtering Algorithm for Constraints ofDifference in CSPs. AAAI 1994: 362-367

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Propagation

Hyper-arc consistency: example

¹

¹ ¹

x1 x2

x3

81,2,3<

81,2< 81,2<

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Propagation


Reformulate the problem as that of finding a Maximum Matchingin a Bipartite Graph — known to be solvable in polynomial time.

x1

x2

x3

1

2

3

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Propagation


Find a maximum matching. If the cardinality of the maximummatching is less than the number of variables, then there is nofeasible assignment; hence we remove all values from domains.

x1

x2

x3

1

2

3

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Propagation


In this case there is a maximum matching of size 3, therefore thereis at least one feasible assignment (displayed).

x1

x2

x3

1

2

3

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Propagation


However, to enforce hyper-arc consistency, in principle we shouldbe able to identify all possible maximum matchings of size 3 — sothat we can retain every values in decision variable domains, whichbelongs at least to a maximum matching.

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Propagation


However, to enforce hyper-arc consistency, in principle we shouldbe able to identify all possible maximum matchings of size 3 — sothat we can retain every values in decision variable domains, whichbelongs at least to a maximum matching.

Fortunately, to find out if a values in a decision variable domainbelongs at least to a maximum matching there is no need toenumerate all maximum matchings.

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Propagation


To understand why, we can look at the extended graph used by theFord-Fulkerson algorithm — which can be used to determine amaximum matching — and focus on its strongly connectedcomponents (SCC).

s

x1

x2

x3

1

2

3

t

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Propagation


A directed graph is said to be strongly connected, if every vertex isreachable from every other vertex.

The strongly connected components of an arbitrary directed graphform a partition into subgraphs that are strongly connected; thesecan be computed in linear time.

s

x1

x2

x3

1

2

3

t

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Propagation


Observation: by flipping every edge in a strongly connectedcomponent we obtain a new maximum matching.

s

x1

x2

x3

1

2

3

t

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Propagation


Observation: by flipping every edge in a strongly connectedcomponent we obtain a new maximum matching.

s

x1

x2

x3

1

2

3

t

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Propagation


This means that all edges that connect nodes within a stronglyconnected component also belong to a maximum matching; hencethe respective values in decision variable domains should beretained.

s

x1

x2

x3

1

2

3

t

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Propagation


Conversely, all edges that connect nodes belonging to differentstrongly connected component and that do not belong to thecurrent matching, can not belong to any maximum matching (notethat strongly connected components are all saturated); hence therespective values in decision variable domains should be removed.

s

x1

x2

x3

1

2

3

t

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Propagation


¹

¹ ¹

x1 x2

x3

81<

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Decision making under uncertainty

Can we extend constraint programming so that it can be applied toproblems of decision making under uncertainty?

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Stochastic Constraint Programming

European Conference on Artificial Intelligence, 2002 Constraints, 2006

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Stochastic Constraint Programming (Walsh, 2002)

A Stochastic Constraint Satisfaction Problem (SCSP) is a 7-tuple

〈V, S,D,P,C, θ, L〉

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

2. D is a function mapping each decision variable in V to a domain of potentialvalues

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

4. P is a function mapping each random variable in S to a probability distribution

5. C is a set of (chance) constraints, possibly involving random variables

6. θh is a threshold probability associated with chance-constraint h

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

by considering an objective function f(V, S) we obtain aStochastic Constraint Optimization Problem (SCOP).

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An SCSP

◮ V1 = {x1},V2 = {x2}

◮ D(x1) = {1, . . . , 4},D(x2) = {3, . . . , 6}

◮ S1 = {s1},S2 = {s2}

◮ P (s1) = {(0.5)5, (0.5)4},P (s2) = {(0.5)3, (0.5)4}

◮ C =

{

c1 : Pr{s1x1 + s2x2 ≥ 30} ≥ 0.75c2 : Pr{s2x1 = 12} ≥ 0.5;

}

◮ L = [〈V1, S1〉, 〈V2, S2〉]

78


A solution to the previous SCSP

s1 = 5

x1 = 3

s1 = 4

s2 = 4

s2 = 3

s2 = 4

s2 = 3

0.25

0.25

0.25

0.25

x2 = 4

x2 = 6

1

2

V1

S1

Scenarioprobability CV

2S

2c

1: 5á3 + 4á4 ³ 30

c2: 4á3 = 12

c1: 4á3 + 4á6 ³ 30

c2: 4á3 = 12

c1: 5á3 + 3á4 < 30

c2: 3á3 12

c1: 4á3 + 3á6 ³ 30

c2: 3á3 12

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An SCOP: Static Stochastic Knapsack

◮ V = {x1, . . . , x3}

◮ D(xi) = {0, 1} ∀i ∈ {1, . . . , 3}

◮ S = {w1, . . . , w3}

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

◮ C = {Pr(w1x1 + w2x2 + w3x3 ≤ C) ≥ θ}

◮ L = [〈V, S〉]

◮ f(x1, . . . , x3) =

E

[

p1x1 + p2x2 + p3x3 − smax(

0,∑3

i=1wixi − 20)]

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An SCOP: Dynamic Stochastic Knapsack

◮ V = {x1, . . . , x3}

◮ D(xi) = {0, 1} ∀i ∈ {1, . . . , 3}

◮ S = {w1, . . . , w3}

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

◮ C = {Pr(w1x1 + w2x2 + w3x3 ≤ C) ≥ θ}

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

◮ f(x1, . . . , x3) =

E

[

p1x1 + p2x2 + p3x3 − smax(

0,∑3

i=1wixi − 20)]

81

Stochastic OPL (Tarim et al., 2006)

A language specifically introduced for modeling decision

problems under uncertainty.

It captures several high level concepts that facilitate the processof modeling uncertainty:

◮ stochastic variables (independent or conditional distributions)

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

◮ chance-constraints

◮ decision stages

◮ ...

82


Static Stochastic Knapsack

int N = 3;

int c = 10;

int p = 2;

float θ = 0.2

range 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*z

subject 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;

};

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Dynamic Stochastic Knapsack

int N = 3;

int c = 10;

int p = 2;

float θ = 0.2

range 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*z

subject 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;

};

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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 deterministicequivalent CSP.

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Static Stochastic Knapsack — deterministic equivalent model

int nbWorlds = 8;

range Worlds 1..nbWorlds;

int nbItems = 3;

range Items 1..nbItems;

float s = 0.2

float W[Worlds,Items] =

[[5,3,15],[5,3,4],[5,9,15],[5,9,4],[8,3,15],[8,3,4],[8,9,15],[8,9,4]];

float Pr[Worlds] = [0.125,0.125,0.125,0.125,0.125,0.125,0.125,0.125];

float p[Items] = [8,15,10];

float C = 10;

var float+ z[Worlds];

var int+ w[Worlds] in 0..1;

var int+ x[Items] in 0..1;

maximize sum(i in Items) x[i]*r[i] - p*(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])-C;

forall(j in Worlds) (sum(i in Items)W[j,i]*x[i] <= C) => w[i]==1;

sum(j in Worlds) Pr[j]*w[j] >= 0.2;

};

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Advantages

◮ Seamless modeling under uncertainty

◮ Stochastic OPL not necessarily linked to CP

Drawbacks

◮ Size of the compiled model

◮ Constraint propagation not fully supported

86

Decision making under uncertainty

Can we reuse existing propagation algorithms, e.g. alldifferent, in astochastic setting?

87

Global stochastic constraints & Global chance constraints

International Conference on Principles and Practice ofConstraint Programming, 2008 Artificial Intelligence, 2012

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Global stochastic constraints & Global chance constraintsA global stochastic constraint (Rossi et al., 2008) represents arelation among a non-predefined number of decision and random

variables.

It implements dedicated filtering algorithms based on

◮ feasibility reasoning

◮ optimality reasoning

Stochastic Programming

Pr{

∑ki=1 wixi ≤ C

}

≥ θ

xi → decision variablewi → random variable or constant parameterC random variable or constant parameterθ → satisfaction probability

Global Chance ConstraintstochLinIneq(x1, . . . , xk, w1, . . . , wk, C, θ);

89

Filtering algorithms for global chance constraints: an

example

x1=0x2=0

x3=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 Tree

X

X

X

X

90


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

90


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


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

90


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


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

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

90


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


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}

90


example

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

90


example

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}

90


example

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}

90


example

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}

90


example

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}

90


example

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

90


example

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

90


example

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

90


example

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}

90


example

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

90


example

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

90


example

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

90

Global chance constraints vs Scenario-based approachRandom SCSPs in (Hnich et al., 2012)

91


What are the advantages of modelling problems of decision makingunder uncertainty via global stochastic constraints?

92

Deterministic multiprocessor scheduling (CSP)

Constraints:

(1) cumulative(s, e, t, c,m)Decision variables:

sk ∈ {rk, . . . , dk}, ∀k ∈ 1, . . . , |K|ek ∈ {rk, . . . , dk}, ∀k ∈ 1, . . . , |K|

93

Stochastic multiprocessor scheduling (SCSP)Uncertain processing time

Constraints:

(1) Pr {cumulative(s, e, t, c,m)} ≥ θ

Decision variables:


Stochastic variables:

tk → processing time of order k ∀k ∈ 1, . . . , |K|Stage structure:

V1 = {s1, s2, . . . , s|K|} S1 = {t1, t2 . . . , t|K|}V2 = {e1, e2, . . . , e|K|} S2 = {}L = [〈V1, S1〉, 〈V2, S2〉]

94

Stochastic multiprocessor scheduling (SCSP)Uncertain capacity requirements

Constraints:


Decision variables:



ck → capacity requirement of order k ∀k ∈ 1, . . . , |K|Stage structure:

V1 = {s1, s2, . . . , s|K|, t1, t2 . . . , t|K|, e1, e2, . . . , e|K|}S1 = {c1, c2, . . . , c|K|}L = [〈V1, S1〉]

95

Stochastic multiprocessor scheduling (SCSP)Uncertain processing time and capacity requirements

Constraints:


Decision variables:



tk → processing time of order k ∀k ∈ 1, . . . , |K|ck → capacity requirement of order k ∀k ∈ 1, . . . , |K|

Stage structure:

V1 = {s1, s2, . . . , s|K|} S1 = {t1, t2 . . . , t|K|, c1, c2, . . . , c|K|}V2 = {e1, e2, . . . , e|K|} S2 = {}L = [〈V1, S1〉, 〈V2, S2〉]

96

Stochastic Plane Landing Scheduling ProblemUncertain plane delay

Constraints:

(1) ti = xi + di ∀i ∈ P1, . . . , P5(2) li = xi + fi + di ∀i ∈ P1, . . . , P5(3) bi = xi + di ∀i ∈ P6, . . . , P14(4) Pr {alldiff(t1, l3, t5, b6, b7, b8)} ≥ θ4(5) Pr {alldiff(l1, t2, l4, b9, b10, b11)} ≥ θ5(6) Pr {alldiff(l2, t3, t4, l5, b12, b13, b14)} ≥ θ6(7) Pr {l1 ≤ t2} ≥ θ7(8) l3 ≤ t5

Decision variables:

x1 ∈ {1, . . . , 12}, x2 ∈ {13, 14, 15, 22, 23, 24},x3 ∈ {1, . . . , 9}, x4 ∈ {7, . . . , 12},x5 ∈ {16, . . . , 24},x6 = 18, x7 = 10, x8 = 2,x9 = 15, x10 = 8, x11 = 2,x12 = 13, x13 = 5, x14 = 2,ti ∈ {1, . . . , 28} ∀i ∈ 1, . . . , 5li,∈ {1, . . . , 40} ∀i ∈ 1, . . . , 5bi,∈ {1, . . . , 22} ∀i ∈ 6, . . . , 14


di → delay of plane i ∀i ∈ 1, . . . , 14Stage structure:

V1 = {x1, x2, . . . , x14}S1 = {d1, d2 . . . , d14}V2 = {t1, . . . , t5, l1, . . . , l5, b6, . . . , b14}S2 = {}L = [〈V1, S1〉, 〈V2, S2〉]

97


Can we scale these propagation methods when random variablesupports are large?

98

Global stochastic constraints & Global chance constraintsNumerical study

A comprehensive numerical study on several problems

◮ Random stochastic CSPS

◮ Stochastic knapsack (SCOP)

◮ Stochastic plane landing scheduling problem

is presented in (Hnich et al., 2012).

In this numerical study we demonstrates the effectiveness of globalchance constraints in modelling and solving problems of decisionmaking under uncertainty.

We also demonstrate that global chance constraints outperformclassical scenario-based deterministic equivalent reformulation suchas the one discussed in (Tarim et al., 2006).

99

(α, ϑ)-solutions and Sampled SCSP

22nd International Joint Conferenceon Artificial Intelligence (IJCAI-11)

100

(α, ϑ)-solutions and Sampled SCSP

s1 = 5

x1 = 3

s1 = 4

s2 = 4

s2 = 3

s2 = 4

s2 = 3

2/3

0

1/3

0

x2 = 4

x2 = 6

1

2

V1

S1

Scenarioprobability CV

2S

2c

1: 5á3 + 4á4 ³ 30

c2: 4á3 = 12

c1: 4á3 + 4á6 ³ 30

c2: 4á3 = 12

101

Questions

102

Stochastic Constraint Programming and Global Stochastic … · 2020-07-07 · Stochastic Constraint...

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