Constraintbased Scheduling ACC01 Tutorial Session

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Constraint-based SchedulingACC01 Tutorial
Session

• Markus Fromherz
• June 2001

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Outline

• Tutorial presentation
• Introduction
• Scheduling
• Constraint programming
• Constraint-based scheduling
• Application example
• Conclusion
• Markus FromherzXerox Palo Alto Research
  Centerhttp//www.parc.com/fromherz

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Outline

• Short presentations
• Simon de Givry (Thales LCR)A constraint-based
  scheduling and planning framework for real-time
  applications
• Stephen F. Smith (CMU, The Robotics
  Institute)Continuous, mixed-initiative
  management of transportation resources
• Mark Boddy (Honeywell Technology Center)On-line
  event dispatch and schedule updating with
  information from operations, using a
  constraint-based scheduling approach

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Introduction

• Scheduling is
• the process of allocating resources to
  activities over time.
• Scheduling problems are everywhere
• job-shop scheduling, product assembly sequencing
• manpower and service scheduling, logistics
  resource allocation and scheduling, fleet
  assignment
• Scheduling is an important component of many
  business processes.

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Introduction

• Constraint-based scheduling
• separates the model from the solving algorithms
• uses powerful, generic solving algorithms
• which enables
• wider variety of constraints
• dynamic model, model composition
• model re-use for other tasks
• Recent use in control
• on-line and real-time constraint-based scheduling
• even for solving times of tens or hundreds of
  milliseconds
• Scheduling as a new tool for reconfigurable
  system control.

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Next

• Tutorial presentation
• Introduction
• Scheduling
• Constraint programming
• Constraint-based scheduling
• Application example
• Conclusion

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Scheduling

• Scheduling in context

• Job
• Task 1
• Task 2

Products
Operations
Operations
Resource
Resource
Resource
Resource
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Scheduling Application example

• Modular print engines
• 3-20 modules,100s of components
• customer-configurable machines
• complex interactions
• hierarchical, distributed control
• CBS enables
• auto-configuration
• plug-and-playplanning andscheduling
• model multi-use

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Scheduling

• Classic machine shop metaphor
• machines, jobs, tasks
• flow shop task order constrained, resource use
  fixed
• job shop task order constrained, resource use
  flexible
• open shop task order and resource use flexible
• precedence and resource constraints
• much work on deterministic algorithms for
  problems with few machines and jobs
• Constraint-based methods proved successful when
• problems are hard
• domain-specific / redundant constraints are
  available
• problems change often
• Real problems have a wide variety of constraints.

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Next

• Tutorial presentation
• Introduction
• Scheduling
• Constraint programming
• Constraint-based scheduling
• Application example
• Conclusion

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Constraint programming Intro

• Scheduling as a constraint satisfaction and
  optimization problem
• variables timing and resource selection of tasks
• constraints precedence, resource constraints,
  etc.
• objective function optimality criterion (e.g.,
  schedule length)
• solution assignment to the resource and timing
  variables
• Roots
• AI search procedures
• CLP modeling, programming, constraint operations
• OR solving for specific constraint systems

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Constraint programming Constraint problems

• Constraint satisfaction problem (CSP)
• n variables xi with domains Di, m constraints
  cj(x1,,xn)
• solution consistent variable assignment
• given n variables xi with domains Di and m
  constraints cj,find a solution ?x1,,xn?
  ?v1,,vn?such that vi ? Di i1,,n cj(v1,,vn) j
  1,,m
• Constrained optimization problem (COP)
• additional objective function h(x1,,xn)
• given n variables xi with domains Di, m
  constraints cj, andobjective function h,find a
  solution ?x1,,xn? ?v1,,vn?with
  minimal h(v1,,vn)subject to vi ? Di i
  1,,n cj(v1,,vn) j 1,,m

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Constraint programming Modeling

• Modeling constraint problems
• domain types
• integer domains l,u
• logical (or Boolean) domains false,true (0,1)
• enumeration (or choice) domains 1,k
• real domains
• set domains
• constraint types
• arithmetic constraints (e.g., 3xy ? z)
• Boolean formulae (e.g., x ? (y ? ?z))
• set constraints (e.g., subset(x,y))

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Constraint programming Modeling

• Modeling constraint problems
• constraint system
• set of primitive constraints together with rules
  for how to combine and derive constraints
• linear or nonlinear over discrete or continuous
  variables
• equality, inequality, disequality (?),
  special-purpose relations
• other constraint systems possible in CP framework
• objective function
• single variable (e.g., time of last task in a
  schedule)
• combinations of objectives (e.g., weighted sum of
  individual objectives)

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Constraint programming Solving
Problem

• Constraint graph
• variables are nodes, constraints are edges
• unary, binary, or any number of variables per
  constraint
• Solving finite-domain constraint problems
• domain reduction constraint propagation search

Graph
Solution t1 2, t2 10, t3 13, t4 15
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Constraint programming Solving

• Domain reduction
• application of a unary constraint c(x)
• Constraint propagation
• propagation of domain changes between connected
  variables

x gt 3
x ? 0,10
x ? 4,10
(propagation)
(reduction propagation)
x ? 0,10
x ? 0,7
x ? 4,7


x ? y3
x ? y3
x gt 3
x ? y3
y ? 0,10
y ? 3,10
y ? 7,10
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Constraint programming Solving

• Refinement-based search methods
• incremental assignment of values to variables
• backtracking on constraint violation
• set X ?x1,,xn?while there are unassigned
  variables in X select an unassigned variable x
  from X select a value v from the domain of
  x assign x v (and propagate) backtrack (to
  value then variable selection) if a constraint is
  violatedend while
• Components
• choice points variable selection, value
  selection
• heuristics (domain-independent or
  domain-specific)
• backtrack procedure (chronological, backjumping,
  etc.)
• degree of arc-consistency (forward checking, full
  look-ahead, etc.)

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Constraint programming Solving

• Optimization with refinement-based search
• branch-and-bound search
• add new constraint h(x1,,xn) lt h(v1,,vn)
  during searchevery time a new solution ?v1,,vn?
  is found
• binary search
• restart search with progressively narrowerlower
  and upper bounds on h
• iterative deepening
• restart search with an increasing upper limit on
  huntil a solution is found

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Constraint programming Solving

• Repair-based search methods
• changing variable values in a complete assignment
• iteration on constraint violation
• set V ?v1,,vn? as initial solution for
  ?x1,,xn?while V is inconsistent select an
  inconsistent assignment x v from V select a
  new value v for x assign x v in Vend
  while
• Components
• choice points variable selection, value
  selection
• heuristics
• Optimization with repair-based search
• hill climbing
• simulated annealing

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Next

• Tutorial presentation
• Introduction
• Scheduling
• Constraint programming
• Constraint-based scheduling
• Application example
• Conclusion

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Constraint-based scheduling Modeling

• Modeling scheduling problems
• scheduling-specific variables and constraints
• intervals (e.g., start and duration)
• various classes of resources
• hard and soft constraints

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Constraint-based scheduling Modeling
Renewable resources

• Modeling scheduling problems
• resource constraints
• define and constrain the available resourcesby
  restricting multiple uses of a resource
• renewable and consumable resources
• preemptive and non-preemptive tasks
• incremental allocation constraints(e.g.,
  unary_resource(r), allocate(r,t1),
  allocate(r,t2))or global constraints(e.g.,
  cumulative(t1,t2,1,1,1))

unary
volumetric
state
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Constraint-based scheduling Modeling

• Resource allocation examples

unary_resource(r),allocate(r, t1),allocate(r,
t2) ? cumulative(t1,t2,1,1,1)
volumetric_resource(r, 3),allocate(r, t1, 1),
use1allocate(r, t2, 2), use2allocate(r,
t3, 1) use1 ? cumulative(t1,t2,t3,1,2,1,3
)
state_resource(r, 1,2),allocate(r, t1, 1),
state1allocate(r, t2, 2), state2allocate(r,
t3, 2) state2 ? cumulative(t1,t2,1,1,1),c
umulative(t1,t3,1,1,1)
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Constraint-based scheduling Propagation

• Scheduling-specific propagation techniques
• reasoning about resources and intervals
• resource timetables
• maintains a timetable with required and available
  capacity at any time for each resource
• propagates between resources and task times
• edge finding
• reasons about execution order of tasks on a given
  resource
• updates earliest and latest possible execution
  times
• Domain-specific redundant constraints and search
  heuristics can lead to increased propagation and
  smaller search trees.

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Constraint-based scheduling Example

• Job-shop scheduling problem
• given
• set of machines each processes one task at a
  time
• set of jobs each a set of tasks with given
  durations, fixed order, assigned machines
• find schedule (task times) that minimizes the
  makespan
• Model
• tasks interval variables t, durations
  dmachines unary resource variables r
• objective h l
• constraints For every task t 0 ? t ? l. For
  every task t, given duration d t.d d. For
  every job with tasks ?t1,,tn? ti ? ti1 (i
  1,,n1). For every task t, given assigned
  machine r allocate(r, t).

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Constraint-based scheduling Off/on-line

• Off-line/predictive scheduling
• receive and post constraints in one step
• then solve for all tasks to find task times
• On-line/reactive scheduling
• receive and post constraints incrementally(only
  for known tasks)
• find and execute schedule concurrently
• reschedule if constraints change
• few generic schedulers for on-line use

receive problem
post constraints
solve problem
release for execution
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Constraint-based scheduling Inside

• Deployed generic real-time, embedded schedulers
• Xeroxs print-engine paper-path scheduler
• details in next section
• NASAs Remote Agent Planner and Scheduler
  (RAX-PS)
• part of Deep Space One
• constraint solver with customized search
  strategies
• on-line constraint management
• JPLs Aspen planner and scheduler
• for NASA spacecraft
• on-line constraint management
• real-time response by using iterative repair
  techniques

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Constraint-based scheduling Complexity

• Complexity
• most real scheduling problems are NP-hard
• research focus finding generic heuristics and
  enabling programmers to guide algorithms with
  domain-specific knowledge (glass-box approach)
• Possible solutions
• simplify by relaxing certain assumptions, e.g.,
  by allowing to preempt tasks
• dedicated complexity analysis, e.g., separating
  typical from average and worst-case scenarios
• adaptive constraint solving

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Constraint-based scheduling MPC

• Model-predictive control
• similarities to CBS
• model, objectives, and constraints stated
  explicitly as a COP
• incremental nature of processing incoming
  requests
• optimization of decisions with respect to a
  horizon of future events
• reactive to changes in assumptions, system state
• differences from CBS
• order of references points and control inputs
  fixed
• sequence of solutions is the values of the same
  control task at different times
• common requirement from constraint programming
  view
• full optimization with minimal commitment approach

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Next

• Tutorial presentation
• Introduction
• Scheduling
• Constraint programming
• Constraint-based scheduling
• Application example
• Conclusion

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Application example Print-engine scheduling

• System-level task
• on a given machine,Mark engine with 1 feeder and
  2 finishers
• for desired document(s),10 pages, 50 copies,
  duplex, color, face-up
• plan and schedule required operations.Feed at
  0s, mark at 3s, invert at 4s, ...
• Low-level controlexecutes andmonitors
  theoperations.

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Application example Components

• Scheduler components
• machine module models
• system controller
• constraint solver
• Automatic composition,flexible control
  throughconstraint modelsand algorithms.

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Application example Modeling

• Components as transducers
• work units, transformation, resource allocations
• feature, timing constraints on combinations of
  inputs and outputs
• compositional, no function in structure
• Describing structure, constraints,and interfaces.

Transport component
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Application example Modeling

• Transport component in CDL
• domain-specific concurrent constraint language
• for describing component and module capabilities
• constraint systems integer ( interval),
  resource, feature
• Component Transport(int length, int speed)
  EntryPort in ExitPort out IntVariable
  t_out, d FeatureVariable s UnaryResource
  space Capability Move(IntVariable t_in)
  in.Input(s, t_in) out.Output(s, t_out)
  s.size.width lt 200 t_in d
  t_out d length/speed
  space.Allocate(t_in, d) // Capability
  Move // Component Transport

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Application example Capabilities

• Machine capabilities
• composing capabilities by unifying output to
  input events, propagating feature and timing
  constraints
• result
• completespecification ofthe output sheet
• timing andresourceconstraints

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Application example Control

• Auto-configuration
• modules pass up their module models
• module models are composed to machine model
• Run-time control
• scheduler identifies sheet plans for stream of
  sheets
• scheduler schedules sheet plans incrementally,
  optimally
• control commands and instantiated reference times
  are sent to module controllers for execution

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Application example Control

• Sample schedule for ten sheets
• six simplex sheets, two duplex sheets,two
  simplex sheets

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Finally

• Tutorial presentation
• Introduction
• Scheduling
• Constraint programming
• Constraint-based scheduling
• Application example
• Conclusion

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Conclusion CBS

• Constraint-based scheduling
• provides a set of technologies for constructing
  high-level control software
• separates scheduling algorithms from
  system-specific information
• facilitates developing correct and flexible
  control software for complex dynamical systems

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Conclusion Future

• Future work
• further integration of CLP, AI, and OR techniques
• more research on constraint-based techniques
  inreal-time, on-line environments
• more work on making constraint-based scheduling
  accessible
• usable modeling languages
• associated analysis and transformation tools
• Constraint-based scheduling is becoming an
  important part of advanced control software.