Scheduling with Constraint Programming

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Scheduling with Constraint Programming

• February 24/25, 2000

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Today

• Optimization
• Scheduling
• Assessment

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Review

• Constraint programming is a framework for
  integrating three families of algorithms
• Propagation algorithms
• Branching algorithms
• Exploration algorithms

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S E N D M O R E M O N E Y
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 4
E ? 4

S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 5
E ? 5
S ? 9 E ? 5 N ? 6 D ? 7 M ? 1 O ? 0 R
? 8 Y ? 2
S ? 9 E ? 6..7 N ? 7..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E ? 6
E 6


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Using OPL Syntax
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by increasing
dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ? Run
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Today

• Optimization
• Scheduling
• Assessment

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Optimization

• Modeling define optimization function
• Propagation algorithms identify propagation
  algorithms for optimization function
• Branching algorithms identify branching
  algorithms that lead to good solutions early
• Exploration algorithms extend existing
  exploration algorithms to achieve optimization

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Optimization Example

• SEND MOST MONEY

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SEND MOST MONEY

• Assign distinct digits to the letters
• S, E, N, D, M, O, T, Y
• such that
• S E N D
• M O S T
• M O N E Y holds and
• M O N E Y is maximal.

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Modeling

• Formalize the problem as a constraint
  optimization problem
• Number of variables n
• Constraints c1,,cm ? ?n
• Optimization constraints d1,,dm ? ?n ?2?n Given
  a solution a, and an optimization constraint di ,
  the constraint di(a)? ?n contains only those
  assignments b for which b is better than a.

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A Model for MONEY

• number of variables 8
• constraints
• c1 (S,E,N,D,M,O,T,Y)? ?8 0 ? S,,Y ? 9
• c2 (S,E,N,D,M,O,T,Y)? ?8
• 1000S 100E 10N D
• 1000M 100O 10S T
• 10000M 1000O 100N 10E Y

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A Model for MONEY (continued)

• more constraints
• c3 (S,E,N,D,M,O,T,Y)? ?8 S ? 0
• c4 (S,E,N,D,M,O,T,Y)? ?8 M ? 0
• c5 (S,E,N,D,M,O,T,Y)? ?8 SY all
  different
• optimization constraint
  
• d (s,e,n,d,m,o,t,y) ?
• (S,E,N,D,M,O,T,Y)? ?8
• 10000m 1000o 100n 10e y
• lt 10000M 1000O 100N 10E Y

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Propagation Algorithms

• Identify a propagation algorithm to implement the
  optimization constraints
• Example SEND MOST MONEY
• d (s,e,n,d,m,o,t,y) ?
• (S,E,N,D,M,O,T,Y))? ?8??8
• 10000m 1000o 100n 10e y
• lt 10000M 1000O 100N 10E Y
• Given a solution a, choose propagation algorithm
  for d(a).

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Branching Algorithms

• Identify a branching algorithm that finds good
  solutions early.
• Example SEND MOST MONEY
• Idea Naïve enumeration in the order M, O,
  N, E, Y.
• Try highest values first.

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Exploration Algorithms

• Modify exploration such that for each solution a,
  corresponding optimization constraints are added.
• Two strategies
• branch-and-bound continue as in original
  exploration
• restart optimization after each solution, start
  from the root.

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Using OPL Syntax
enum Letter M,O,N,E,Y,S,D,T var int lLetter
in 0..9 minimize lM10000 lO1000
lN100 lE10 lY subject to
alldifferent(l) lS ltgt 0 lM ltgt 0
lS1000 lE100 lN10 lD
lM1000 lO100 lR10
lE lM10000 lO1000 lN100
lE10 lY search forall(i in
Letter) tryall(v in 0..9 ordered by
decreasing v) li v ?All
Solutions Execution ? Run
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Experiments with MONEY
MONEY Search
MONEY Optimization
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Today

• Optimization
• Scheduling
• Assessment

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Scheduling

• Scheduling Problems
• Propagation Algorithms for Resource Constraints
• Branching Algorithms
• Other Constraints
• Exploration Algorithms
• Literature

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Scheduling Problems

• Assign starting times (and sometimes durations)
  to tasks, subject to
• resource constraints,
• precedence constraints,
• idiosyncratic and other constraints, and
• an optimization function, typically to minimize
  overall schedule duration.

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Example Building a Bridge
Show Problem
Animate Solution
Gantt Chart

• Resource constraints
• Example a1 and a2 use excavator, cannot overlap
  in time
• Precedence constraints
• Example p1 requires a3, a3 must end before p1
  starts

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Modeling

• Indices
• Task begin,a1,a2,a3,a4,a5,a6,p1,...,end
• Constants duration of tasks
• duration begin0, a14, a22,...
• Variables represent each task with a finite
  domain variable representing its starting time.
• Activity at in Task(durationt)
• at.start is finite domain variable
  representing the starting time of at

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Precedence Constraints

• For each two tasks t1, t2, where t1 must precede
  t2, introduce a constraint
• at1.start at1.duration ? at2.start
• In OPL, just write
• at1 precedes at2

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Resource Constraints

• For each two tasks t1, t2 that require a unary
  resource r, we have the constraint
• at1.start at1.duration ?
• at2.start
• \/ at2.start at2.duration ?
• at1.start
• But many constraints and weak propagation.
• Thus, introduce global resource constraints.

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Scheduling

• Scheduling Problems
• Propagation Algorithms for Resource Constraints
• Branching Algorithms
• Other Constraints
• Exploration Algorithms
• Literature

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Global Resource Constraints in OPL

• Declare resource
• UnaryResource excavator
• Require resource
• aa1 requires excavator
• aa2 requires excavator...
• All requires constraints on a resource together
  form a global resource constraint.

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Propagation Disjunctive Resource

• For all tasks t1, t2 using the same resource
• at1.start at1.duration ?
• at2.start
• \/ at2.start at2.duration ?
• at1.start
• Weakest but most efficient form of propagation
  for unary resources.

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Propagation Edge finding

• For a given task t and set of tasks S, all
  sharing the same unary resource , find out
  whether t can occur
• before all tasks in S,
• after all tasks in S,
• between two tasks in S,
• and infer corresponding basic constraints.

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Propagation Task Intervals

• Notation est(t) earliest starting time
• lct(t) latest completion time
• For two tasks t1 and t2 where est(t1)?est(t2) and
  lct(t1) ? lct(t2), the task interval I(t1,t2) is
  defined
• I(t1,t2) t est(t1) ? est(t) and
• lct(t) ? lct(t2)
• Perform edge finding on all task intervals.

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Task Intervals Example
A
B
C
D
I(A,B) A,B,C I(C,D) B,C,D
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Which Edge Finder?

• As usual, trade-off between run-time of
  propagation algorithm and strength of
  propagation. Edge finding can be expensive
  depending on what tasks are considered.
• Recent edge finders have complexity n2 for one
  propagation step, where n is the number of tasks
  using the resource.
• Edge finding based on task intervals can be
  stronger than these, but typically have
  complexity n3 .

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Specifying Propagation Algorithms

• OPL fixes propagation for unary resources to an
  (undisclosed) edge-finding algorithm.
• For discrete (non-unary) resources, the user can
  choose between default, disjunctive and
  edgeFinder when declaring a resource
• DiscreteResource
• crane(3) using edgeFinder

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Demo Propagation for Unary Resources
Bridge
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Scheduling

• Scheduling Problems
• Propagation Algorithms for Resource Constraints
• Branching Algorithms
• Other Constraints
• Exploration Algorithms
• Literature

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Branching Algorithms Serializers

• Simple enumeration techniques such as first-fail
  are hopeless for scheduling.
• Use unary resources to guide branching.
• For two tasks t1, t2 sharing the same resource,
  use the constraints
• t1.start t1.duration ? t2.start
• and
• t2.start t2.duration ? t1.start
• for branching.

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Which Tasks To Serialize?

• Resource-oriented serialization Serialize all
  tasks of one resource completely, before tasks of
  another resource are serialized.
• Most-used-resource serialization
• Global slack
• Local slack
• Task-oriented serialization Choose two suitable
  tasks at a time, regardless of resources.
• Slack-based task serialization (see later)

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Most-used-resource Serialization

• Most-used-resource serialization serialize
  first the resource that is used the most.
• Let T be a set of tasks running on resource r.
• demand(T) ?t?T t.duration
• Let S be the set of all tasks using resource r.
• demand(r) demand(S),
• Serialize the resource r with maximal demand(r)
  first.

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Slack-based Resource Serialization

• Let T be a set of tasks running on resource r.
• supply(T) lct(T) - est(T)
• demand(T) ?t?T t.duration
• slack(T) supply(T) - demand(T)
• Let S be the set of all tasks using resource r.
• slack(r) slack(S), S is set of all task
  running on r.
• Global slack serialization Serialize resource
  with smallest slack first

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Local-Slack Resource Serialization

• Let Ir be all task intervals on r. The local
  slack is defined as min slack(I) I ? Ir
• Local slack serialization Serialize resource
  with smallest local slack first. Use global slack
  for tie-breaking.

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Which Tasks Are Serialized?

• Ideas
• Use edge finding to look for possible first tasks
  (and last tasks)
• Choose tasks according to their est, lst (lct,
  ect).

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Task-oriented Serialization

• Among all tasks using all resources, select a
  pair of tasks according to local/global slack
  criteria and other considerations, regardless
  what resources are serialized already.
• Provides more fine-grained control at the expense
  of runtime for finding candidate task pairs among
  all tasks.

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Programming Branching Algorithms in OPL

• Scheduling-specific try constructs
• tryRankFirst(u,a) activity a comes first
• tryRankLast(u,a) activity a comes last
• rank(u) serialize all tasks using u
• Reflective functions for unary resources
• isRanked(Unary) 1 if the resource is ranked
• nbPossibleFirst(Unary)
• number of activities that can
  come first
• globalSlack(Unary) global slack of u
• localSlack(Unary) local slack of u

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Example Task-oriented Serialization in OPL

• while not isRanked(tool) do
• select(r in Resources not isRanked(toolr))
• select(t in tasksr
• not isRanked(toolr,at)
• tryRankFirst(toolr,at)

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Demo Serialization Algorithms
Bridge
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Scheduling

• Scheduling Problems
• Propagation Algorithms for Resource Constraints
• Branching Algorithms
• Other Constraints
• Exploration Algorithms
• Literature

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Other Scheduling Constraints

• Discrete resources
• DiscreteResource crane(3)
• a requires(2) crane
• a consumes(1) crane
• Reservoirs
• Reservoir plumbing(3,1)
• a requires(2) plumbing
• a consumes(2) plumbing
• a provides(2) plumbing
• a produces(2) plumbing

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Exploration Algorithms

• Usually branch-and-bound using minimization of
  the starting time of an end task
• minimize aend.start subject to ...
• Lower-bounding Prove the non-existence of a
  solution with aend.start ? n, add constraint
  aend.start gt n increase n by ?.
• Upper-bounding Find solution with aend.start
  n, add constraint aend.startltn decrease n by
  ?.

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Literature

• Van Hentenryck The OPL optimization programming
  language, 1999
• Constraint Programming Tutorial of Mozart, 2000
  (www.mozart-oz.org)
• Applegate, Cook A computational study of the
  job-shop scheduling problem, 1991
• Carlier, Pinson An algorithm for solving the
  job-shop scheduling problem, 1989
• Various papers by Laburthe, Caseau, Baptiste, Le
  Pape, Nuijten, see Overview paper

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Today

• Optimization
• Scheduling
• Assessment

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Assessment Dont Use It!

• Dont use constraint programming for
• Problems for which there are known efficient
  algorithms or heuristics. Example Traveling
  salesman.
• Problems for which integer programming works
  well. Example Many discrete assignment problems.
• Problems with weak constraints and a complex
  optimization function. Example Timetabling
  problems.

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Assessment Do Use It!

• Use constraint programming for
• Problems for which integer programming does not
  work (linear models too large).
• Problems for which there are no efficient
  solutions available.
• Problems with tight constraints, where
  propagation can be employed. Example ACC 97/98.
• Problems for which strong branching algorithms
  exist. Example Scheduling with unary resources.

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Myths Debunked

• A fad! Constraint programming has been used
  successfully in a number of application areas,
  most spectacularly in scheduling
• Universal! More failure stories than success
  stories.
• All new! Many ideas come from AI search and
  Operations Research. In particular, the important
  ideas for scheduling come from OR.
• Artificial Intelligence! All quite earthly
  algorithms. Constraint programming systems
  provide framework for different kinds of
  algorithms to interact.

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The Future

• Constraint programming will be added in OR as a
  standard technique for solving combinatorial
  problems, along with local search.
• Constraint programming techniques will be tightly
  integrated with integer programming and local
  search.
• Look at the Workshop on Integration of AI and OR
  Techniques for Combinatorial Optimization
  Problems (www.uni-paderborn.de/CP-AI-OR)