ConstraintBased Scheduling in the Real World

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Constraint-Based Scheduling in the Real World

• Mark Boddy
• Adventium Labs
• Mark.Boddy_at_adventiumlabs.org

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Finite-Capacity Scheduling

• Specify activities linked to operations
  (transfers, tool changes, delivery dates,
  production runs, etc.).
• This specification includes
• Activity start and end times
• Resource usage, including raw materials, power,
  equipment needed, labor required

Finite capacity schedules can be used to
determine predicted inventory levels, campaign
completion/product delivery dates, economic
performance, unit operating modes, and resource
bottlenecks.
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Another view of the problem
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Complex Operational Domains
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Common Features

• Continuous dynamics
• Continuous time
• Metric resources (power, weight, process heat)
• Consumable resources (energy, volume/capacity,
  fuel)
• Continuous action effects
• Rate is a free variable
• Objective functions expressed in terms of
  continuous parameters.
• Actions
• Concurrent actions (power generation and
  consumption)
• Action choice
• State (action sequences, state resources,
  action preconditions, )

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Scheduling The Current State of the Art

• Current state of the art Issues
• 60 of manufacturing facilities have no automated
  scheduling capability.
• 30 of airline flights dont go as planned, on a
  good day.
• 10 reduction in waiting time for trucking
  industry would allow 6 of all trucks to be taken
  off the highways altogether.
• Most current scheduling done by unsophisticated
  means - spreadsheets or manually.

• Gaps to be closed by an Integrated Solution
• 1) Schedule awareness and editing, then
• 2) Autogeneration of feasible schedules, then
• 3) Autogeneration of optimal schedules.

Most manufacturing enterprises do not have an
effective solution to address Step 1!
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Value of Scheduling for Manufacturing

• Reduce Costs
• Reduce disruption frequency and severity (what-if
  scenarios, detailed maintenance schedules)
• Improve parts/raw material ordering (better
  tracking of on-hand inventory)
• Reduce Work-In-Process inventory (reduced
  lot-rot, more timely vendor ordering, better
  synchronization of production to delivery dates)
• Reduce changeover costs (cleaning, new catalyst,
  temperature changes, ).
• Enhance Capital Investment Planning
• prioritize and quantify payoff for capital
  investment
• Increase Effective Capacity
• Identify and remove bottlenecks
• Increase agility (improve Available to Promise)

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What is Scheduling?

• Determining when and how to accomplish some set
  of tasks.
• The classic job shop
• A set J of jobs to be run, each with
• a set of tasks to run in sequence, each on
  some unique element of
• a set M of machines
• Common problem statements
• Find a feasible schedule.
• Minimize makespan.
• Given a set of deadlines, minimize tardiness.

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Possible Complications (a brief sample)

• Choice of resource
• Other resource constraints
• (Global) capacity constraints
• Inter-activity constraints
• Consumable resources
• Complex temporal constraints (latency,
  preemption, hierarchical activity relationships).
• System dynamics (flow rates, chemical
  composition, )
• Activity generation
• Reasoning about state

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Effective Domain Modeling is Hard

• Example large transport pipelines
• Pipeline volume must be modelled explicitly (not
  like shipping and receiving).
• Currently model is to have slugs of material with
  associated volume, etc., plus a position in the
  pipeline.
• Material movement in the pipeline requires both
  input and output flows.
• Rate is independent of individual movements

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How schedules are used is a factor as well

• Schedules are generated to satisfy existing
  plans.
• Schedules are generated in a historical context.
• Schedules are usually not constructed completely
  automatically.
• Schedules frequently require input from multiple
  parties.
• Schedules are used in operations.
• Updating, as events occur.
• Rescheduling, as conflicts are detected.
• Post mortem examinations.
• (Schedules are persistent artifacts.)
• Schedules are large.
• Optimization is important, but hard to formalize
  properly.

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Manufacturing IT Architecture
Capital investment, Finance, Long-term supply
and demand forecasts
More than 6 months
Rough-cut capacity planning, Orders
forecasts, Long-leadtime vendor orders
1 month to 1 year
MRP-II
2 weeks to 3 months
Transportation and Warehouse Management
1 day to 1 month
Manufacturing Execution
1 hour to 1 day
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Scheduling as a Constraint Satisfaction Problem

• CSPs are specified as
• A set V of variables
• A set C of constraints, each constraint a
  relation specifying tuples of permissible values
  for some subset of V.
• The objective is to find a feasible (alt.,
  optimal) complete assignment for V, consistent
  with C.

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Advantages to a CSP approach

• Flexible, declarative representation.
• Lots of previous and current work on solution
  methods.
• General solution techniques
• Static structural analysis
• Propagation
• Search
• Requirements and scheduling decisions can be
  represented as constraints.

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Scheduling as Search Using a Dynamic CSP

• CSP Variables
• Activity start and end times
• Activity resource assignments
• Constraints
• Temporal constraints (duration, ordering, release
  times, deadlines, )
• Resource constraints (permissible assignments,
  usage requirements, state information, ...)
• System dynamics (rate limitations, allowable
  state changes, )
• Search over
• Activity generation
• Resource assignments
• Activity orderings, start and end times

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Hybrid Systems

• Hybrid systems have both continuous and discrete
  components, as opposed to
• Combinatorial problems, such as knapsack, TSP, or
  jobshop
• Continuous problems, captured as sets of
  mathematical equations and inequalities.
• Combinatorial, continuous, and hybrid constraint
  problems can all be framed in terms of either
  satisfiability (CSP) or optimality (COP).
• Scheduling is a hybrid constraint problem.

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Solving Hybrid Systems

• MI(N)LP
• Discrete choices represented using integer
  variables.
• Continuous variables represented using equations
  and inequalities
• Typical solution methods involve iterating some
  form of relaxation to a purely continuous model,
  followed by forcing one or more integer variables
  to an integral value.
• Hybrid CLP/MP approaches
• The problem is represented explicitly in two
  separate models, one a conventional mathematical
  program, the other a discrete CSP.
• Variants
• Conditional math programming (Hooker, Grossman)
• Variable correspondence (Heipcke)
• Cooperating solvers (Ilog Solver)

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Constraint Envelope Scheduling

• A least commitment approach
• Accumulate additional constraints (e.g., activity
  orderings), so as to narrow the space of possible
  schedules, until all schedules in the remaining
  set are feasible.
• Other people doing similar things
• Nicola Muscettola et al. HSTS/Europa
• Malik Ghallab IxTeT
• Steve Smith DITOPS

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Constraint Envelope Scheduling

• Search variables
• Unary resources lt A before B, B before A gt
• Capacity resources lt A before B, B before A, A
  overlaps B gt

A1
A2
A3
A1
A2
A3
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Constraint Envelope Scheduling

• Advantages
• No premature commitment
• Flexible search strategies
• Natural representation for decisions (do it
  before lunch)
• Natural interleaving of search and propagation
• Disadvantages
• (Somewhat) cumbersome to implement.
• Intermediate results are less intuitive to users
  (more difficult to present, anyway).
• Efficiency concerns (a matter of degree)

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Constraint Envelope Scheduling Solver

• A cooperating solver approach
• Mathematical model is purely continuous, heavily
  optimized for speed and scaling.
• All interesting disjunction is captured in the
  discrete model.
• Standard architecture permits swapping of
  continuous solvers.
• Requirements
• Global consistency determination
• Incremental (or very fast) add and delete
• Culprit identification
• Integrated continuous solvers
• Simple Temporal Problem
• Linear solver
• Nonlinear solver

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CES Solver Architecture
1. Continuous constraints added as result of
discrete decisions
Discrete Constraint Engine
ContinuousConstraint Engine
3. Discrete constraints added as result of
continuous decisions
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Three Different Continuous Solvers

• The Simple Temporal Problem constraints of the
  form P2 - P1 lt D.
• Inference supported (dis)proving the
  consistency of a specified temporal relation
  between activity begin and end points.
• Implementation shortest-path search in a graph.
• Up to 60,000 nodes (so far).
• Incremental consistency checking and propagation.
• Linear solver currently using Ilog CPLX
• Full, commercial linear solver.
• Solution times are a few milliseconds
• Irreducible Infeasible Sets
• Nonlinear solver locally developed subdivision
  solver.

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Basic Control Flow for Planning and Scheduling

• Refinement search over discrete decisions
• Activity generation
• Discrete choices within resulting CSP (resource
  choice)
• Continuous feasibility checking
• Constraint reduction
• Propagation
• Linear solve
• Subdivision search over full nonlinear model.
• Objective function used as a heuristic during
  search, with a final optimizing solve on full set
  of continuous constraints.

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Less and Less Commitment

• For the past 13 years, our approach has been to
  shove as much into the continuous model as
  possible, without slowing it down.
• 50,000 temporal constraints for SAFEbus avionics
  scheduling problem
• 18,000 continuous constraints (2700 quadratic) on
  14,000 variables for scheduling an entire
  petroleum refinery
• Why?
• Reduces backtracking due to premature commitment
  in discrete search
• Reduces size of discrete search, as well
• Resulting flexibility is useful for schedule
  updates and for optimization.
• Reduced need for explicit bookkeeping (we dont
  keep resource timelines for consumable resources).

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Blend scheduling
Rundowns
Blendstocks
Blenders
Products
Shipments
Rundown Activities
Tank Capacities
Unary resources
Tank Capacities
Shipment Activities
Blend Activities
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Evolution of Scheduling a Blend Activity

• 3) CES scheduler with linear solver
• Blender specified
• Blend sequence specified
• Blend recipe floats
• Blend volume specified
• 4) CES scheduler with NL solver
• Blender specified
• Blend sequence specified
• Blend recipe floats
• Blend volume floats

• 1) Timeline scheduler
• Blender specified
• Blend recipe specified
• Blend start and end times specified
• Blend volume specified
• 2) CES scheduler with STP
• Blender specified
• Blend sequence specified
• Blend recipe specified
• Blend volume specified

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Least Commitment for a SOFIA-Like Domain

• Constraints
• Time
• Fuel
• Orientation
• Sum of velocities
• Objective Maximize (prioritized) useful science

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Problem statement

• Specified
• Crude deliveries
• Product liftings (shipments)
• Initial tank contents, volume and qualities
• Constrained
• Product specifications (in terms of qualities)
• Tank volume min/max
• Unit constraints (operating ranges)
• Material balances (Hydrogen, RFG)
• Objective function
• Product giveaway
• Inventory value, by component
• Ending inventory targets

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Solution Must Specify

• Material movements
• Crude charges
• Shipments
• Blends
• Unit modes
• Crude Distillation Unit
• Distillate Hydrotreater (desulphurizer)
• Unit controls
• Split fractions (e.g., CDU)
• Conversion (e.g., desulphurization, platformer)

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Mathematical Nature of the Problem

• Any model with variable properties and flow or
  tank mixing requires bilinear equations, which
  are therefore quadratic.
• Systems of quadratic equations can represent any
  polynomial or rational function, and can
  approximate any nonlinear function to arbitrary
  accuracy.
• Systems of quadratic equations are not equivalent
  to quadratic programs, which have linear
  equations, only the objective is quadratic.
• Quadratic and bilinear systems typically show
  multiple local minima in optimization problems.

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Discrete search

• 3-5 crude charges in a 10-day schedule.
• 22 shipments
• 2-5 gasoline blends
• blend volume is currently a discrete choice

• Summary
• 1) Create a crude-charge activity
• start, end time
• CDU mode
• DHT mode
• 2) Create shipment activities
• 3) Create gasoline blend activities
• 4) Repeat, to end of schedule

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Continuous model

• Activity start and end times (interval model)
• Process unit control settings
• Flow volumes and rates
• Flow qualities
• Tank volumes and qualities as a function of time
  (at activity endpoints)
• Both mass and volume are tracked
• Some unit controls affect Specific Gravity
• Products may be sold by weight, by volume, or
  both.

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Boeing 777 AIMS Scheduling
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SAFEbus Scheduler for Boeing 777 AIMS

• AIMS requirements represent one of the largest,
  most comprehensive sets of constraints ever
  successfully scheduled
• 29,000 items are scheduled, subject to 97,000
  complex metric constraints specified by AIMS
  applications developers
• More than 230,000 decisions, each with between 4
  and 4,000 possible choices were made in
  scheduling
• This corresponds to finding a solution with a
  directed search of 107 elements in a state space
  of 10140000.

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How We Solved It

• Ginsbergs Dynamic Backtracking algorithm
• Published in 1994
• We fixed a bug
• We extended it
• Moral sometimes the path from research to
  product takes months, not years.

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Concluding Remarks

• Theoretical results and algorithms from CSP/COP
  are frequently of great practical use.
• But, they are almost never of use unaltered.
• And they are almost never of use in isolation.

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Scheduling Systems are 95 Boring
Plan
Planning Tool
Ext Sched
Marketing
Archive
Archived Schedule
Config
Published Schedule
Scheduler
PRM
Plant Model
Publish
Plant
Plant Historian
Working Schedule
Data Externally Accessible
Activity Hist
MES
External Data Interchange
Scheduling UI
External Data Interchange
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Areas for Further (Interesting!) Work

• Optimal Scheduling (for the right definition of
  optimal)
• Incorporating probabilities
• Robust schedules vs. responsive schedulers
• Moving up into production planning
• Better integration with action selection and
  states (but see workshops at ICAPS-04, ICAPS-05)