The Theory and Practice of Constraint Programming: An Overview Brahim Hnich Faculty of Computer...

The Theory and Practice

of Constraint Programming:

An Overview

Brahim Hnich

Faculty of Computer Science

Izmir University of Economics

Izmir, Turkey

18/04/23 A CP Tutorial: Hnich 2

Quotation

“Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it.”

Eugene C. Freuder, Constraints, April 1997

Caveat

•In this talk:

Constraint programming for combinatorial problems

“Programming” refers to its roots in computer science (programming languages)

Outline

•Modelling

•Constraint propagation

•Search

•Demo: Lot-sizing with stochastic non-stationary demand

A Puzzle

• Place the numbers 1 through 8 in the nodes such that: Each number appears exactly once No connected nodes have consecutive numbers

Modeling

• Each node a decision variable

• {1, …, 8} values in the domain of each variable

• No consecutive numbers a constraint (vi, vj) |vi – vj| > 1

• All values used a clique of not-equals constraints forall i<j. vi ≠ vj,

Modeling

• Each node a decision variable

• {1, …, 8} values in the domain of each variable

• No consecutive numbers a constraint (vi, vj) |vi – vj| > 1

• All values used forall i<j. vi ≠ vj,

Or more compactly, all-different[v1,…,v8]

Heuristic Search

?? 1 8

{1, 2, 3, 4, 5, 6, 7, 8}

Inference/Propagation

?? 1 8

{1, 2, 3, 4, 5, 6, 7, 8}

Inference/Propagation

?? 1 8

{3, 4, 5, 6}

{3, 4, 5, 6} {3, 4, 5, 6}

{3, 4, 5, 6, 7} {2, 3, 4, 5, 6}

An OPL Model

int N=8;struct edge{int x; int y;};{edge} Edges ={<1,2>, <1,3>,…,<7,8>};range Nodes 1..N;range Values 1..N;

var int Solution[Nodes] in Values;

solve{forall(e in Edges)

abs(Solution[e.x] - Solution[e.y]) >1;

alldifferent(Solution); };

The Core of Constraint Computation

Solving

HeuristicSearch

Propagation

Modelling

ANY QUESTIONS?

Modeling

Finite-domain Variables

•To each variable x is associated a set of values called its domain

e.g x Є {1,3,11}

•Each variable must take a value from its domain

•That domain is updated as decisions are made

•A domain may only shrink (no value is ever added)

Constraint Satisfaction Problems

CSP: (X, D, C)o X = {x1, x2,…, xn} variables

o D = {d1, d2,…,dn} domains (finite)

o C = {c1,c2,…,ce } constraints

c ЄC var(c) = {xi, xj,…, xk} scope

rel(c) ⊆ di x dj x .. x dk permitted

tuples

Solution: o assignment satisfying every constraint

o NP-complete task

CSP: Relevance

Artificial Intelligenceo temporal reasoning

Control Theoryo controllers for sensory based robots

Concurencyo process comm. and synchr.

Computer Graphicso geometric coherence

Database Systemso constraint databases

Bioinformaticso sequence alignment

Operations researcho optimization

Real-life applications

Production planningStaff schedulingResource allocationCircuit designOption tradingDNA sequencing...

Many problems can be represented as CSP:

Constraint Programming

CP: provides a platform for solving CSPs proven useful in many real applications

Platform: set of common structures to reuse best known algorithms for propagation & solving

Two stages: modelling solving

Language of Constraints

The usual relational operators (<, =, >, ≤, …)

… including ≠

Linear and nonlinear constraints

Logical connectives (→, ↔, ┐, …)

Set constraints (subset, union, intersection,…)

“Global constraints”: constraints capturing a common substructure (pattern) of combinatorial problems

Map Colouring

Variables: F, N, S

Values: { }

Constraints: N ≠ S ≠ F ≠ N

A solution: F

Word Design Problem

This problem has its roots in Bioinformatics and Coding Theory.

Problem: find as large a set S of strings (words) of length 8 over the alphabet W = { A,C,G,T } with the following properties:

1. Each word in S has 4 symbols from { C,G };

Word Design Problem

Problem: find as large a set S of strings (words) of length 8 over the alphabet W = { A,C,G,T } with the following properties:

2. Each pair of distinct words in S differ in at least 4 positions; and

Word Design Problem

Problem: find as large a set S of strings (words) of length 8 over

the alphabet W = { A,C,G,T } with the following properties:

2. Each pair of distinct words in S differ in at least 4 positions; and

3. Each pair of words x and y in S (where x and y may be identical) are such that xR and yC differ in at least 4 positions. o (x1,…,x8)R = ( x8,…,x1 ) is the reverse of ( x1,…,x8 )

o (y1,…,y8)C is the Watson-Crick complement of ( y1,…,y8 ), i.e. the word where each A is replaced by a T and vice versa and each C is replaced by a G and vice versa.

A Solution

S= { AAGCCGTT, TACGCGAT}

Each word in S has 4 symbols from { C,G };

A Solution

Each pair of distinct words in S differ

in at least 4 positions

A Solution

SR= { TTGCCGAA, TAGCGCAT}

SC= { TTCGGCAA, ATGCGCTA}

Each pair of words x and y in S

(where x and y may be identical) are such that

xR and yC differ in at least 4 positions

A Matrix Model

M 1 2 3 4 5 6 7 8

1{ A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T }

m{ A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T } { A,C,G,T }

Each row represents a word in S

M[i,j]: a decision variable with domain { A,C,G,T }

A Matrix Model

M 1 2 3 4 5 6 7 8

1. Expressing that each word in S has 4 symbols from { C,G }

A Matrix Model

M 1 2 3 4 5 6 7 8

For each row r

sum (p in 1..8) //channelling constraints

(M[r,p]=C or M[r,p]=G) = 4

A Matrix Model

M 1 2 3 4 5 6 7 8

1. …2. Each pair of distinct words in S differ in at least

4 positions

For each distinct rows r1 and r2

sum(p in 1..8) (M[r1,p] ≠ M[r2,p]) >= 4

A Matrix Model

MC 1 2 3 4 5 6 7 8

1. …2. …3. xR and yC differ in at least 4 positions.

Introduce a “compliment matrix”

Each pair //channelling constraints

<M[i,j], MC[i,j]> in {<C,G>, <G,C>, <A,T>, <T,A>}

A Matrix Model

MC 1 2 3 4 5 6 7 8

1. …2. …3. xR and yC differ in at least 4 positions.

For each rows r1 and r2

sum(p in 1..8) (M[r1,9-p] ≠ MC[r2,p]) >= 4

ANY QUESTIONS?

Constraint Propagation

•General principle

•Consistency

•Filtering on simple constraints

•Filtering on global constraints

•Conclusion

General Principle

•Each type of constraint relies on its own specific filtering algorithm to filter out (locally) inconsistent variable assignment

•The communication between the different filtering algorithms takes place through the domains of the variables

•It makes it easy to have different types of constraints work together

General Principle

?? 1 8

{1, 2, 3, 4, 5, 6, 7, 8}

≠ ≠

≠≠

≠ ≠

≠≠

Constraint network

General Principle

•This propagation necessarily terminates

•The results is the same regardless of the order in which we consider the constraints

Triggering the constraints

•A constraint is woken up by one of its variables.•domain event: anytime the domain changes(i.e. some value has been removed)

•range event: only when the minimum or maximum value in the domain has changed (e.g. for x ≤ y constraint)

•value event: only when the domain is reduced to a single value (e.g. for x≠y constraint)

•Consistency

•Conclusion

Consistency

•We reason locally about a constraint, removing

inconsistent values from domains

•We can often characterize the level of consistency achieved by a filtering algorithm

Generalized Arc Consistency

•Given a constraint C on the variables X A support for Xi = vj on C is a partial assignment containing Xi = vj that satisfies C.

A variable Xi is generalized arc consistent (GAC) on C iff every value in D(Xi) has support on C.

C is GAC iff each constrained variable is GAC on C.

Bounds Consistency

•Given a constraint C on the variables X A bound support on C is a support where the interval [min(Xi); max(Xi)] is substituted for the domain of each constrained variable Xi.

A variable Xi is bound consistent (BC) on C if min(Xi) and max(Xi) have bound support on C.

C is BC iff all constrained variables are BC on C.

Consistency: BC example

•Consider 4x + 3y - 2z = 10

•with Dx = Dy = Dz = {0, 1, . . . , 9}

•Maintaining BC reduces domains toDx = {0, 1, . . . , 7},Dy = Dz = {0, 1, . . . , 9}

•BC is triggered on range event

Consistency: GAC example

•Consider 4x + 3y - 2z = 10

•with Dx = Dy = Dz = {0, 1, . . . , 9}

•Maintaining GAC reduces domains toDx = {0, 1, . . . , 7},Dy = {0, 2, 4, 6, 8},Dz = {0, 1, . . . , 9}

•GAC is triggered on domain event

•Consistency

•Conclusion

Disequalities: Filtering Algorithm

•Consider x≠y with Dx = Dy = {0, 1, . . . , 9}

•Only once one of the variables is fixed may we remove any value from the domain of the other:Dx = {3} → Dy = {0, . . . , 2, 4, . . . , 9}

•achieves GAC

•triggered on value event

Inequalities: Filtering Algorithm

•Consider x<y with Dx = Dy = {0, 1, 2, 3}

•Only once one of the variables’ bound is changed may we remove any value from the domain of the other:

Dx = {0, 1, 2} Dy = {1, 2, 3}

•achieves BC (which is equivalent to GAC because of monotonicity)

•triggered on range event

•Consistency

•Conclusion

Example: all-different

3 binary constraints,they are GAC,

no pruning

Var: F, N, S; Val: { }; Ctrs: N ≠ S ≠ F ≠ N

≠ ≠

•Using binary disequalities, this inconsistency goes undetected

no pruning

≠ ≠

We can do something simple:

1. Count the number of variables, n

2. Count the number of values in the union of their domains, m

3. If n > m then no solution can possibly be found3 binary constraints,

they are GAC,no pruning

≠ ≠

We can do something simple:

1. Count the number of variables, n

2. Count the number of values in the union of their domains, m

3. If n > m then no solution can possibly be found

• ...still fooled by

Dx = Dy = Dz = {a, b}, Dw = {c, d}3 binary constraints,they are GAC,

no pruning

≠ ≠

no pruning

1 ternary constraint, not GAC,GAC pruning empty domain

no solution!!

logically

equivalent

≠ ≠

all-different

All-different: A Filtering Algorithm

•Build the corresponding bipartite graph

721 3 54 6

x2x1 x3 x5x4 x6

• ∃ solution iff ∃ matching covering all the variables

721 3 54 6

x2x1 x3 x5x4 x6

• ∃ solution iff ∃ matching covering all the variables

•filteringfind alternating cycles and pathsremove inconsistent values (useless edges) [X4=2]fix variables (vital edges) [X4=4]

721 3 54 6

x2x1 x3 x5x4 x6

• ∃ solution iff ∃ matching covering all the variables •starting from the current matching at each call makes the algorithm incremental•achieves GAC

721 3 54 6

x2x1 x3 x5x4 x6

Cardinality: A Filtering Algorithm

•distribute([c1, . . . , cm],[v1, . . . , vm],[x1, . . . ,xn]),ci [li, ui]∈

•transformed into a network flow problem

• ∃ solution iff feasible flow∃

[0,1] [l1,u1]

[lm,um]

•Compute a maximum flow

•Build the residual graph

•Find its strongly connected components

•Remove zero-flow arcs between components

•achieves GAC

Lexicographic Ordering: A Filtering Algorithm

• A new family of global constraints

• Linear time complexity

• Ensures that a pair of vectors of variables are lexicographically ordered.

0 1 4 2

2 9 8 7

Motivation: Symmetry

Symmetry: transformation of an entity that preserves the properties of the entity

Example:

Motivation: Symmetry

• Frequently occurs Combinatorial problems like

covering arrayso Rows and columns can be permuted

Messy real world problems like nurse rostering

o Nurses with same skills can be swapped

• Tough for IP Very active research area

within CP Some effective techniques

have been developed

Motivation

• An important class of symmetries in CP matrices of decision variables rows/columns represent indistinguishable

objects, hence symmetric

• Rows and columns can be permuted without affecting satisfiability

• Encountered frequently

• Schedule games between n teams over n-1 weeks• Each week is divided into n/2 periods• Each period has 2 slots: home and away• Find a schedule such that

every team plays exactly once a week

every team plays against every other team

every team plays at most twice in the same period over the tournament

Example: Sports Scheduling

Example: Sport Scheduling

Period3

Period4

Period2

Period1

0 vs 72 vs 72 vs 60 vs 41 vs 63 vs 54 vs 5

0 vs 5

1 vs 4

3 vs 7

Week 5

3 vs 4

0 vs 6

1 vs 5

Week 6

1 vs 31 vs 22 vs 54 vs 66 vs 7

5 vs 65 vs 70 vs 31 vs 72 vs 3

2 vs 4 3 vs 64 vs 70 vs 20 vs 1

Week 7Week 4Week 3Week 2Week1

• We need a table of meetings!

Period3

Period4

Period2

Period1

0 vs 72 vs 72 vs 60 vs 41 vs 63 vs 54 vs 5

0 vs 5

1 vs 4

3 vs 7

Week 5

3 vs 4

0 vs 6

1 vs 5

Week 6

1 vs 31 vs 22 vs 54 vs 66 vs 7

5 vs 65 vs 70 vs 31 vs 72 vs 3

2 vs 4 3 vs 64 vs 70 vs 20 vs 1

Week 7Week 4Week 3Week 2Week1

• Weeks are indistinguishable• Periods are indistinguishable

• We need a table of meetings!

0 vs 72 vs 72 vs 60 vs 41 vs 63 vs 54 vs 5Period 3

0 vs 5

1 vs 4

3 vs 7

Week 5

3 vs 4

0 vs 6

1 vs 5

Week 6

1 vs 31 vs 22 vs 54 vs 66 vs 7Period 4

5 vs 65 vs 70 vs 31 vs 72 vs 3Period 2

2 vs 4 3 vs 64 vs 70 vs 20 vs 1Period 1

Week 7Week 4Week 3Week 2Week 1

0 vs 72 vs 72 vs 6 0 vs 41 vs 6 3 vs 54 vs 5Period 3

0 vs 5

1 vs 4

3 vs 7

Week 5

3 vs 4

0 vs 6

1 vs 5

Week 6

1 vs 31 vs 22 vs 5 4 vs 66 vs 7Period 4

5 vs 65 vs 70 vs 3 1 vs 72 vs 3Period 2

2 vs 4 3 vs 64 vs 7 0 vs 20 vs 1Period 1

Week 7Week 4Week 3Week 2Week 1

Example: Bin Packing

• Consider 2 identical bins:

Example

• Consider 2 identical bins:

• We must pack 6 items: 3 41 652

Example

• Here is one solution:

Example

• Here is another:

Example

• Is there any fundamental difference?

Example

• Consider a matrix model:

1 2 3 4 5 6

A 1 0 1 0 1 0

B 0 1 0 1 0 1

1 2 3 4 5 6

A 0 1 0 1 0 1

B 1 0 1 0 1 0

Example

NB: ‘1’ means place this item in this bin:

1 2 3 4 5 6

A 1 0 1 0 1 0

B 0 1 0 1 0 1

1 2 3 4 5 6

A 0 1 0 1 0 1

B 1 0 1 0 1 0

Example

If we insist that row A lex row B,

we remove a) from the solution set.

1 2 3 4 5 6

A 1 0 1 0 1 0

B 0 1 0 1 0 1

1 2 3 4 5 6

A 0 1 0 1 0 1

B 1 0 1 0 1 0

Example

• Notice that items 3 and 4 are identical.

1 2 3 4 5 6

A 0 1 0 1 0 1

B 1 0 1 0 1 0

1 2 3 4 5 6

A 0 1 1 0 0 1

B 1 0 0 1 1 0

Example

• Notice that items 3 and 4 are identical.

1 2 3 4 5 6

A 0 1 0 1 0 1

B 1 0 1 0 1 0

1 2 3 4 5 6

A 0 1 1 0 0 1

B 1 0 0 1 1 0

If we insist that col 3 lex col 4,

we remove c) from the solution set.

• Main Goal Eliminate row and column symmetries effectively and efficiently.

• Aims: Investigate types of ordering constraints to break row and column

symmetries.

Devise global constraints to easily pose and efficiently solve the ordering constraints.

Examine the effectiveness of the ordering constraint

we focus on lexicographic ordering constraints

How GACLex Works

• Consider the following example.

• We have two vectors of decision variables:

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

How GACLex Works

• Consider the following example.

• We have two vectors of decision variables:

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• We want to enforce GAC on: x lex y.

A Tale of Two Pointers

• We use two pointers, α and β, to avoid repeatedly traversing the vectors.

• We index the vectors as follows:

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

Most Significant Index

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• α: index such that all variables at more significant indices are ground and equal.

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• β: most significant index from which the two vectors’ tails necessarily violate the constraint.

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• β: If tails never violate the constraint:

Pointer Initialisation

• Needs one traversal of the vectors (linear).

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

Pointer Initialisation

• Needs one traversal of the vectors (linear).

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

Failure

• Inconsistent if β α.

How GACLex Works

• We maintain α and β as assignments made.

How GACLex Works

• When β = α + 1 we enforce bounds consistency on: xα < yα

How GACLex Works

The variable at the αth element of each vector.

How GACLex Works

• When β > α + 1 we enforce bounds consistency

on: xα yα

How GACLex Works

• Key: we reduce GAC on vectors to BC on binary constraints.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• 0, 1 removed from yα.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• 0, 1 removed from yα.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• Update α.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• Update α.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• 3, 4 removed from xα.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• 3, 4 removed from xα.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• Update α.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• Update α.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• 4, 5 removed from xα, 0, 1 removed from yα.

How GACLex Works

0 1 2 3 4

x {2} {1,3,4} {1,2,3,4,5} {1,2} {3,4,5}

y {0,1,2} {1} {0,1,2,3,4} {0,1} {0,1,2}

• 4, 5 removed from xα, 0, 1 removed from yα.

Complexity

• Initialisation: O(n)

• Propagation:• We enforce bounds consistency between at most n pairs of

variables: xα < yα or xα yα.

• Cost: b.

• Overall cost: O(nb).

• Amortised cost: O(b)

Results: BIBD

Time(s)

1 10 100 1000 10000

GACLex

Decomposition takesAbout 9 times longer on each of these instances.

•Consistency

•Conclusion

Conclusion

•Constraint propagation is the glue to combine efficient

filtering algorithms for common substructuresMatching theory, network flow theory, automata theory, computational geometry, …, encapsulated in constraints

•Characterization of level of consistency

•Aim for incrementality

•Amount/frequency of filtering vs processing time

ANY QUESTIONS?

Search

•Variable selection heuristics

•Value selection heuristics

•Conclusion

General principle

alldifferent(Solution); };

General principle

solve{… }; search {…}

General principle

alldifferent(Solution); }; search {…}

With a good search strategy

We can quickly find good solution

Searching for a good solution

•For any interesting problem, propagation alone is not enough

•We typically proceed by tree search

Solving CSP by Search

Search tree: root: empty node one variable per level sucessors of a node: every value of the next level var

Searching for a good solution

•Two main decisions to control searchchoose a variablechoose a value from its domain

Search

•Conclusion

Variable Selection Heuristics

•Has an impact on tree topology

•static vs dynamic ordering

•smallest-domain-firstfirst-fail principle: “To succeed, try first where you are most likely to fail.”

•regret: favour variable with greatest difference in cost between two best values in domain

Search

•Conclusion

Value Selection Heuristics

•Not as critical

•Very problem-dependent

•Alternative for large domains: domain splitting

Search

•Conclusion

Conclusion

•A lot of control over the search strategy

•Many heuristics developed, some of them generic

ANY QUESTIONS?

Lot-sizing under demand uncertainty

Demand Uncertainty in Supply Chain Networks

ProductionInventories

When to order?

How much to order?

Demand

Uncertainty

When to order?

How much to order?

Demand

Uncertainty

When to order?

How much to order?

work in collaboration

with Bell Labs Ireland

Determining the optimal inventory control policy parameters is key to profitability for any company involved in distribution and/or production of goods

We developed a CP model to find the optimal dynamic (R,S) inventory policy parameters such that

• the expected cost is minimized;

• demand is stochastic, non-stationary; and

• a minimum service level is required

Results

• State-of-the-art improvement for the stochastic non-stationary formulation of the lot-sizing problem

• Real-world instances can be solved in few seconds

• The strategy could be extended to deal with Capacity constraints Lead time uncertainty

(Demo)

Experimental results

Seasonal Demands

100000

24 26 28 30 32 34 36 38 40 42 44 46 48 50

periods

s ILOG - CP model

CPLEX - MIP model

Choco - CP model dyn red

ANY QUESTIONS?

Successful CP

•(Machine) Scheduling

(whole book on constraint-based scheduling)

•Sports Scheduling (e.g. NFL)

•Rostering

•Allocation (e.g. terminal gates to aircrafts)

•Transportation (e.g. VRP, airline crew rotation)

•Even pure problems like Maximum Clique

•Production planning (Lot-sizing)

Finding out more

•Talks: CP, CPAIOR, IJCAI, AAAI, ECAI,

INFORMS, CORS

•Papers: Lecture Notes in Computer Science (Springer), Constraints (Kluwer), AI journals, OR journals

•Software: CHIP; ECLiPSe; ECLAIR; FaCiLe;

ILOG OPL, Solver; SISCtus Prolog, Choco,. . .

Finding out more

Books:

•Apt, K., Principles of Constraint Programming, Cambridge University Press, 2003.•Baptiste, P., Le Pape, C., Nuijten, W.. Constraint-Based Scheduling, Kluwer Academic Publishers,2001.•Hooker, J., Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, John Wiley & Sons, 2000.•Marriott, K., Stuckey, P.J., Programming with Constraints: An Introduction, MIT Press, 1998.•Constraint and Integer Programming: Toward a Unified Methodology, edited by M. Milano, Kluwer Academic Publishers, 2003.•Tsang, E., Foundations of Constraint Satisfaction, Academic Press, 1993.•Van Hentenryck, P., The OPL Optimization Programming Language, MIT Press, 1999.

Acknowledgements

•Some parts of this tutorial are adapted material from tutorials given by:

Gilles PesantPedro MesseguerChris Beck

•Most parts of the tutorial is work done in collaboration with:

Alan Frisch, Ian miguel, Zeynep Kiziltan, Toby Walsh, Armagan Tarim, Roberto Rossi, Steven Prestwich

The Theory and Practice of Constraint Programming: An Overview Brahim Hnich Faculty of Computer...

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