constraint programming for timetabling

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP

Over-ConstrainedProblemSoft Constraints

Ill-definedproblemSearch

Inconsistencies

Summary

Constraint Programming for Timetabling

Hana Rudová

Faculty of Informatics, Masaryk UniversityBrno, Czech Republic

http://www.fi.muni.cz/~hanka

ASAP Seminar, 2004

http://www.fi.muni.cz/~hanka

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Purdue University Timetabling Problem

Course demands for individual studentsconflicts among classes of one student minimized

Timetable for large lecture classesFall 2001

manually created: slightly smaller (750 classes),less constrained (about 40 assigned prior search)

Fall 2004830 classes .

= 1500 meetings50 classrooms89,633 course demands for 29,808 studentsabout 350 classes assigned before search

Spring 2005easier data setno freshmen consideration, about 780 classes

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Outline

1 Purdue University Timetabling Problem

2 Constraint Satisfaction Problem

3 Over-Constrained ProblemSoft Constraints

4 Ill-defined problemSearchInconsistencies

5 Summary

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Constraint Satisfaction Problem (CSP)

Constraint satisfaction problem (X , D, C)

finite set of domain variables X = {V1, . . . , Vn}finite set of values (domain) D = D1 ∪ . . . ∪ Dnfinite set of constraints C = {c1, . . . , cm}

relations over subsets of variables

Partial assignment of variables (d1, . . . , dk ), k ≤ n

Complete assignment of variables (d1, . . . , dn)

Solution of CSPcomplete assignment which satisfies all constraints

Constrain & Search constraint propagationTimetabling example

variables: time T and classroom R for each classdomains: possible starting times, possible classroomsconstraints: required classrooms, precedence relationsamong times

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary








Constrain & Search constraint propagation

Timetabling examplevariables: time T and classroom R for each classdomains: possible starting times, possible classroomsconstraints: required classrooms, precedence relationsamong times

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary








Constrain & Search constraint propagationTimetabling example

variables: time T and classroom R for each classdomains: possible starting times, possible classroomsconstraints: required classrooms, precedence relationsamong times

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Global Constraints

serialized

classes of the same instructorat different times

disjoint2

two classes must have differenttime or classrooms ?

cumulative

for approximation (?)– classes change classroom– example: big classrooms

on time variables only

3 4 5 6

2

3

2

1

1

3 4 5 6

2

3

2

1

1

3 4 5 6

2

3

2

1

1

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Over-Constrained Problems

Over-constrained problemthere is no solution with all constraints satisfied

Solution:constraint modeling by hard and soft constraints

hard constraints = everything what must be satisfiedsoft constraints = optimization part

part of the problem which can be unsatisfiedoptimization for preferential requirements

Example of soft constraintstoo many preferential time requirementstwo classes share some students

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Soft Constraints

Weighted constraints

each constraint associated with the weight

example: V1#\ = V2@weight

aim: minimize weighted sum of unsatisfied constraints

Weighted constraints + hard constraints

stronger propagation for hard constraints

optimization for weighted constraints

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Solver for Soft Constraints

Each value of the variable associated with a weightunary soft constraints

Soft constraint propagationunsatisfied c @ weight increases weights of valuesfor variables in c

Evaluation of the solution [V1 = v1, . . . , Vn = vn]:∑∀i

w [Vi , vi ]

Aim: minimize evaluation of the solution

Example: V1#\ = V2@weightV1 or V2 is instantiated to value v

⇒ weight of the value v for second variable Vi

increased by weight w [Vi , v ] + weightpartial forward checking

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Solver for Soft Constraints

Each value of the variable associated with a weightunary soft constraints

Soft constraint propagationunsatisfied c @ weight increases weights of valuesfor variables in c

Evaluation of the solution [V1 = v1, . . . , Vn = vn]:∑∀i

w [Vi , vi ]

Aim: minimize evaluation of the solutionExample: V1#\ = V2@weight

V1 or V2 is instantiated to value v⇒ weight of the value v for second variable Vi

increased by weight w [Vi , v ] + weightpartial forward checking

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Examples of Soft Constraints

Unary soft constrainttimes or locations are preferred or discouraged

Soft serialized constraintclass A should not overlap with Bweight is the number of students in common

Soft cumulative constraintAt most N classes of some department are taught at thesame timeN is a soft limit

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Ill-defined problems

Ill-defined problemmistakes in the problem definition

Mistake = contradiction of hard constraintsExample

constraint propagation ⇒ two teachers requirethe same classroom at the same time

Mistakes must be removedfrom the problem definition to find a solutionSolution: detection of mistakes

during searchduring posting hard constraints

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Tree Search

Complete searchDepth First Search

Incomplete search: cutoff strategyconstrain some of the available resources

Depth Bounded Search

DBS(1)

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Limited Assignment Number Search

LAN(2)

Constraint propagationvalues incompatible with the current partial assignmentare removed from the domains

LAN(3) + constraint propagation

do not waste time on „hard” variablesexplore various parts of the search tree

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Maximal Consistent Assignment

Unassigned variables

second variable V2 unassignedsome constraint(s) on V2 remain(s) unsatisfied

Consistent assignmentsatisfy (at least) all constraints over assigned variables

Maximal consistent assignmentconsistent assignment of the largest cardinality

Locally maximal consistent assignmentassignment which can not be extended tonon-instantiated variable

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Restart for LAN Search

1 restart algorithm with different settingunassigned variables firstuntried values for unassigned values firstsuccessful values for assigned variables

2 automated restart until the number of assigned variablesincreases

3 detection of possible problems over unassigned variables+ hard to solve parts, weak propagation

4 continue with restart with the problem changesproblem redefinition does not introduce any changes in restartstrategy

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

How to Discover Inconsistencies?

Undesirable inconsistenciesMistakes in data input

minimize by user interfacetypes of mistakes

introduced during data entrynaturally included as a part of the problem

Explanationscomputationally expensivenot available in standalone constraint solvers

During searchLAN search

user need some partial solutionuser can change the problem

Before searchmanual processautomated process

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Preconditions for Detection of Inconsistencies

Detail (print) information aboutposting each constraintseach value removal

Post constraints in specific ordermost complex first, most simple last

Complex constraintsdisjoint2, cumulativedetection of a conflict here is not easy

Simple constraintsconstraints over small set of variablesunary constraints over location or time

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Towards Automated Process of Detection

1 Fail during constraint posting2 Last posted constraint cLAST failed3 Variables in cLAST are problem variables4 Tracking of value removals for problem variables5 Value removal was due to constraint cREMOVE

6 Check consistency of cLAST and cREMOVE

In our problem, conflict between cLAST and cREMOVE wasalways source of a mistake.

CP forTimetabling

Hana Rudová

PurdueProblem

Outline

CSP



Inconsistencies

Summary

Conclusion

Over-constrained, ill-defined problems

solver for weighted soft constraints

locally maximal consistent assignment during search

detection of mistakes for standalone constraint solvers

Optimization

multi-criteria optimization

minimal perturbation problem

constraint programming for timetabling

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