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Constraint-based Scheduling

Claude Le Pape

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Outline

• Introduction• Scheduling constraints• Non-preemptive scheduling

– Temporal constraints

– Resource constraints

– Problem-solving examples

• Preemptive scheduling• Conclusion (applications)

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Scheduling problems

"Scheduling is the allocation of resources over time to perform a collection of tasks" [Baker 74]

– Pure scheduling problems: assignment of start and end times to activities (each activity requires given resources with given capacities)

– Pure resource allocation problems: assignment of resources to activities (the start and end times of each activity are given)

– Joint scheduling and resource allocation problems: assignment of resources and start and end times to activities

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Scheduling problems

• Transportation– Traffic scheduling and control (aircrafts, buses, trains, trucks)

– Loading and unloading (aircrafts, ships, trucks)

– Crew rostering

• Production and maintenance

• Manpower planning and timetabling (shifts, courses, exams)

• Project or mission scheduling

• Network planning and routing

• Computer or printer job scheduling

• Mixture planning, rack configuration, textile cutting, ...

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Variety of scheduling problems

• Activities

• Resources

• Constraints

• Optimization criteria

• Problem size

• Time available to make a decision

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Activities

• Interval (block) activities

• Splittable activities (with interruption cost?)

A

time

A

time

A A A

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Resources

• Unary resources– one person

– one machine

• Discrete resources– a group of people with the same capabilities

• State resources– an oven with different possible temperatures

• Energetic resources– a limited number of human-days each week

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Constraints

• Temporal constraints– Fixed or variable durations

– Precedences

– Minimal and maximal delays

• Resource constraints– Fixed capacity

– Variable capacity (time versus capacity tradeoffs)

– Variable capacity over time

• Specific constraints

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Optimization criteria

• No optimization criterion• A well-defined criterion

– Project makespan

– Number of activities performed within given delays

– Maximal or average tardiness or earliness

– Peak or average resource utilization

• A combination of well-defined criteria• Preferences (soft constraints)• Optimization versus robustness

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Problem sizenumber

of resources

numberof activities

1000

100

10

11 10 100 1000 10000

unary resourcesdiscrete resourcesseveral types of resources

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Problem sizenumber

of resources

numberof activities

1000

100

10

11 10 100 1000 10000

production and maintenancenetwork routing and traffic controlother applications

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Suitable response time

• Reactive train traffic control: two seconds• Predictive production and maintenance

scheduling: minutes to hours• Reactive production and maintenance

scheduling: seconds• Predictive timetabling: minutes to hours• Reactive computer or printer job scheduling:

small fraction of the average job duration

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Constraint programming

Problemdefinition

Decision-making(and retracting)

Constraintpropagation

Problem specificationor partial solution interms of constraints

Initial constraintsDynamic changes

Deduced constraintsContradictions

New constraints(decisions)

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Constraint programming

• Explicit problem definition

• Separation between problem definition and problem solving

• Systematic deduction of the consequences of made decisions (constraint propagation)

Partial constraint propagation development of heuristic search algorithms to generate and optimize solutions

• Incremental constraint propagation (with fixpoint semantics)

• Localized definition of the constraint propagation process (each constraint propagates independently of other constraints)

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Incrementality principle (1)

Masonry (7)

Carpentry (3)

Roofing (1)

Windows (1) Facade (2) Garden (1) Painting (2)

Moving (1)

Plumbing (8) Ceilings (3)

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Incrementality principle (2)

Masonry

Plumbing

Carpentry

R.

Ceilings

Facade

W.

G.

Paint.

M.

0 5 10 15 20

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Incrementality principle (3)

• "Plumber" "Roofer"

• Solution: Order "Plumbing" and "Roofing"

• Heuristic choice

For example, "Plumbing" before "Roofing"

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Incrementality principle (4)

Masonry

Plumbing

Carpentry

R.

Ceilings

Facade

W.

G.

Paint.

0 5 10 15 20

M.

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Incrementality principle (5)

Masonry

Plumbing

Carpentry

R.

Ceilings

Facade

W.

G.

Paint.

0 5 10 15 20

M.

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Incrementality principle (6)

Masonry

Plumbing

Carpentry

R.

Ceilings

Facade

W.

G.

Paint.

M.

0 5 10 15 20

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Incrementality principle (7)

Masonry

Plumbing

Carpentry

R.

Ceilings

Facade

W.

G.

Paint.

0 5 10 15 20

M.

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Incrementality principle (8)

Constraint propagation consists in

incrementally

updating characteristics of a partial problem solution

when an additional constraint is added

These characteristics may be:

– indicators of contradictions

– domains of variables (relational propagation)

– the overall set of constraints (logic/algebraic propagation)

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Incrementality principle (9)

Particular case: arc-consistency

A constraint propagation technique enforces arc-consistency if and only if when propagation stops the following statement holds:

for every constraint c(v1 ... vn)

for every variable vi

for every value vali in the domain of vi

there are values val1 ... vali1 vali1 ... valn

in the domains of v1 ... vi1 vi1 ... vn

such that val1 ... vali1 vali vali1 ... valn satisfy c

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Incrementality principle (10)

Non-monotonic constraint propagation consists in

incrementally

updating characteristics of a partial problem solution

when a constraint is added or retracted

Non-monotonic constraint propagation implies the cancellation (hence the identification) of the consequences of the retracted constraints

Non-monotonic constraint propagation is not used as much as "monotonic" constraint propagation (for complexity reasons)

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Locality principle (1)

• Each constraint (or constraint type) "includes" all the information necessary to enable its propagation and, in particular, to determine whether it is satisfied or not as soon as all its variables are instantiated

• The constraint propagation methods associated with a constraint (or constraint type) are a priori independent of the methods associated with other constraints

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Locality principle (2)

• Example: temporal constraints and resource constraints– User: "Plumber" "Roofer"

– User: "Moving" must end before 20

– Temporal constraints: "Plumbing" must end before 17

– Disjunctive resource constraint: "Plumbing" must precede "Roofing"

– Temporal constraints: "Moving" cannot end before 19

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Constraint programming

• Artificial Intelligence– Declarative programming (problem definition)

– Generic inference mechanisms (constraint propagation)

– Generic search techniques (problem solving)

• Operations Research– Linear programming is "declarative"

– Efficient algorithms for deducing solution characteristics

– Efficient search techniques

• Software Engineering– Incremental and local processes

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Constraint programming

Precision Flexibility

Efficiciency Extensibility

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Precision

• Explicit representation of the overall set of constraints

• Guarantee of constraint satisfaction

Softwarepackage

Specificdevelopment

Use of a constraintprogramming tool

Guarantee of constraint satisfaction BUTlimitation to the constraints considered in the package.

No limitation BUTguarantee of constraint satisfaction to build totally.

Guarantee of satisfaction of the predefined constraints.Possibility of defining specific constraints.

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Precision (example)

• An automatic cart places the batches of products on the waiting areas ( )

• Batches are also evacuated through the waiting areas

• A waiting area contains at most one batch at a time

• There are only two waiting areas per machine

Machine Machine Machine

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Efficiency

• Generation of a "good" solution in a reasonable amount of time

• Response time versus solution quality tradeoffs which correspond to the needs of the users

Softwarepackage

Specificdevelopment

Use of a constraintprogramming tool

CPU time not controlled (although generally correct).Solution quality not controlled.

Compromises made by the development team.Heavy (often omitted) code optimization task.

Compromises made by the development team.Highly optimized predefined constraints and basic mechanisms.

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• Scheduling of 48n activities on 17n resources of 5 different types (2 to 7 resources per activity)

Algorithm 1 Algorithm 2 Algorithm 3 Combination

Efficiency (example)

solution cost

CPU.001 1 1000

solution cost

CPU.001 1 1000

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Flexibility

• Using (combining) different methods to solve different problems (or subproblems)

• Incrementally modifying a problem

Softwarepackage

Specificdevelopment

Use of a constraintprogramming tool

Some flexibility may exist BUTlimited to cases considered by the software designer.

Points of flexibility chosen with respect to actual needs.More or less complex implementation process.

Exploitation of the separation between "problem definition,""constraint propagation," and "decision-making."

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Flexibility (example)

Importance ofbottleneck resource

Importance ofbottleneck resource

Quality of a resource-basedproblem decomposition

Quality of an order-basedproblem decomposition

Importance ofbottleneck resource

Quality of an "opportunistic"problem decomposition

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Extensibility

• Extending and modifying an application when the context in which it is used changes

• Adapting an application to a new problem

Softwarepackage

Specificdevelopment

Use of a constraintprogramming tool

Extensibility is limited.Very often no way of extending the system at all.

Extensible as the source code is available BUThigh extension cost (depending on the source code).

Exploitation of the separation between "problem definition,""constraint propagation," and "decision-making."

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Extensibility (example)

• Allocation of locomotives to trainsConstraints Diesel

Constraints Electric

Constraints TGV

• "Electric to Diesel": immediate adaptation (one additional specific constraint)

• "Electric to TGV": 3 man-months

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Limits

• Not a miracleNP-hard problems remain NP-hard

• Not an immediate solutionConstraint programming is a method for which tools are available

Developing and testing software remains necessary

• Not a universally useful methodMany problems do not necessitate constraint programming (e.g., critical path computation) or are such that their resolution cannot benefit from the use of constraint-based techniques