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Scheduling

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					              Planning and Scheduling

                     Stephen F. Smith

                    The Robotics Institute
                  Carnegie Mellon University
                     Pittsburgh PA 15213
                       sfs@cs.cmu.edu
                        412-268-8811



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                         Outline

        • What is Scheduling?
        • Current State of the Art: Constraint-Based
          Scheduling Models
        • Is Scheduling a Solved Problem?




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                What is Scheduling?

        Allocation of resources to activities over time
          so that input demands are met in a timely
          and cost-effective manner
         Most typically, this involves determining a set
          of activity start and end times, together
          with resource assignments, which
            • satisfy all temporal constraints on
              activity execution (following from
              process considerations)
            • satisfy resource capacity constraints,
              and
            • optimize some set of performance
              objectives to the extent possible
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             A Basic Scheduling Problem

             op1        op1
                          2         op1
                                      3
               1

      rel1                                 dd1

                   R1         R2
                                                 i      j     st(i) + p(i) < st(j), where p(i)
                                                              is the processing time of op i

                op2           op2                i      R      j
                  1             2
                                                     st(i) + p(i) < st(j)     st(j) + p(j) < st(i)
        rel 2                        dd2
                                                 rel j < st(i), for each op i of job j
                                                 dd j > st(i) + p(i), for each op i of job j



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              A More Complex Scheduling
                      Problem
     Origin
                          Sea-POE

                Air-POE             Sea-POD    Destination


                                     Air-POD




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                  Scheduling Research:
                   The Last 10 Years

        • Major advances in techniques for solving
          practical problems
            • Constraint solving frameworks
            • Incremental mathematical programming
              models
            • Meta-heuristic search procedures
        • Several significant success stories
        • Commercial enterprises and tools




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          Constraint-Based Scheduling
                     Models
       Components:

         Commitment      Active Data Base        Conflict
          Strategies/   (Current Schedule)       Handling
          Heuristics
                        Constraint Propagation

       Properties:
           • Modeling Generality/Expressiveness
           • Incrementality
           • Compositional
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                    What is a CSP?

        Given a triple {V,D,C}, where
            • V = set of decision variables
            • D = set of domains for variables in V
            • C = set of constraints on the values of
              variables in V
        Find a consistent assignment of values
          to all variables in V



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              A Basic CSP Procedure
        1. [Consistency Enforcement] - Propagate
          constraints to establish the current set vd
          of feasible values for each unassigned
          variable d
        2. If vd = Ø for any variable d , backtrack
        3. If no unassigned variables or no consistent
          assignments for all variables, quit;
          Otherwise
        4. [Variable Ordering] - Select an
          unassigned variable d to assign
        5. [Value Ordering] - Select a value from vd
          to assign to d.
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              Formulating Scheduling
                Problems as CSPs

        “Fixed times” model
           • Find a consistent assignment of start
             times to activities
           • Variables are activity start times
        Disjunctive graph model
           • Post sufficient additional precedence
             constraints between pairs of activities
             to eliminate resource contention
           • Variables are ordering decisions

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         A Simple Job Shop Scheduling
                     CSP
          Variables: start times (stj,i) - Domain: [0,12]

      Job1: O1,1                      O1,2           O1,3
                [0,12]              [0,12]          [0,12]

                              R1                                     R3
      Job2 : O2,1                      O2,2
                [0,12]                [0,12]
                                                          R2
       Job3 :        O3,1                    O3,2                            O3,3
                     [0,12]             [0,12]                              [0,12]


              Oi,j         Oi,k                   Oi,j       Rx          Ok,l
           Sti,j + Duri,j ≤ Sti,k        Sti,j + Duri,j ≤ Stk,l V Stk,l + Durk,l ≤
                                                           Sti,j
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                  Constraint Propagation

        Deductive process of inferring additional
         constraints from existing constraints as
         decisions are made
        Two roles:
            • Early pruning of the search space by
              eliminating infeasible assignments
            • Detection of constraint conflicts




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          Some Constraint Propagation
                Terminology
    K-consistency guarantees that any locally consistent
     instantiation of (K-1) variables is extensible to any
     K-th variable
      Example: 2-consistency (“arc-consistency”)




    Complexity: Enforcing K-consistency is (in general)
      exponential in K
        • Forward Checking: partial arc-consistency only
          involving constraints between an instantiated
                         variable and a non-instantiated
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          one
       Temporal Constraint Propagation
       through Precedence Constraints
        Assume dui,j = 3 for all Oi,j
           • Before propagation:
                  O1,1           O1,2      O1,3
                  [0,12]       [0,12]     [0,12]

           • Forward propagation

                   O1,1           O1,2      O1,3
                  [0,12]        [3,12]    [6,12]
           • Backward propagation

                   O1,1            O1,2     O1,3
                  [0,6]         [0,9]     [6,12]


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       Capacity Constraint Propagation

        Observation: Enforcing consistency with
         respect to capacity constraints is more
         difficult due to the disjunctive nature of
         these constraints
        Forward Checking:
                   O1,1   Scheduled to start at time 6
                        [6,6]


                      R1

                                Before propagation: [6,12]
                      O2,1      After propagation:  [9,12]

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           Pruning Operation Ordering
                  Alternatives
       Example: Erschler’s dominance conditions




        Conclusion: Oi cannot precede Oj
        In general: For any unordered pair of operations {Oi, Oj},
        we have four possible cases:
                  1. LSTi < EFTj and LSTj ≥ EFTi: Oi is before Oj
                  2. LSTj < EFTi and LSTi ≥ EFTj : Oj is before Oi
                  3. LSTi < EFTj and LSTj < EFTi : inconsistency
                  4. LSTi ≥ EFTj and LSTj ≥ EFTi: both options remain open
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                            Edge Finding

        • S - a set of operations competing for resource R
        • O - an operation not in S also requiring R
    ((LFT(S) - EST(S) < Dur(O) + Dur(S))
                                                   EST(O) ≥ EST(S) + Dur(S)
      (LFT(S) - EST(O) < Dur(O) + EST(O))

                                     OP k


                                     OP j


                                     OP i


                               10           20         30

                   S = {OP ,OP }; O = OP
                         i   j         k         Start TimekOP ≥ 25
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             More Complex Temporal
                   Constraints

        “Simple Temporal Problem” (STP)
          [Dechter91]
            • Edge-weighted graph of time points
              expressing constraints of the form:
             atpjtpib
            • Assuming no disjunction, allows incorporation
              of
                • Temporal relations:
                    • finish-to-start <0, ∞> (precedence)
                    • start-to-finish <t1,t2> (duration)
                    • Start-to-start <0,0> (same-start)
                    • ...
                • Metric bounds: offsets from time origin
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            • Efficiently solved via all-pairs shortest path
        Constraint-Posting Scheduling
                   Models
         • Conduct search in the space of
           ordering decisions
             • variables - Ordering(i,j,R) for
               operations i and j contending for
               resource R
             • values - i before j, j before i
         • Constraint posting and propagation in
           the underlying temporal constraint
           network (time points and distances)



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               Search Heuristics
         (Variable and Value Ordering)
      • Slack/Temporal Flexibility
         • Choose pair of activities with least
           sequencing flexibility
         • Post sequencing constraint that leaves the
           most slack
      • Resource Demand/Contention
         • Identify bottleneck resource
         • Schedule (or sequence) those activities
           contributing most to demand
      • Minimal critical sets
         • Generalization to multi-capacity resources
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                  Search Control

        • Backtracking-based search
        • Least-Discrepancy Search
        • Iterative Re-starting with randomized
          heuristics
        • Local search - Tabu, GAs, etc.




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               The Broader Picture
    Constraint posting provides a framework for
     integrating planning and scheduling
      • contemporary temporal planners operate with
        analogous representational assumptions
      • E.g., IXTET, HSTS/RAX, COMIREM, …
      • “Constraint-Based Interval Planning” [D. Smith
        00]
    Constraint posting is a relatively unexplored approach
       to scheduling with several advantages
         • more flexible solutions
         • simple heuristics can yield high performance
           solution techniques under a wide variety of
           problem constraints
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              Technological Strengths

        • Scalability
        • Modeling flexibility
        • Optimization
        • Configurable


        So, Is scheduling a solved problem?




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                           What is Scheduling (Again)?
                Classic view:
                   • Scheduling is a puzzle solving activity-
                                    • Given problem constraints and objective criterion,
                                      figure out how to best tile the capacity over time
                                      surface with operations
                           • Research agenda - specify new puzzles
                             and/or provide new best solutions
                                                                      i       j
            OP1,1           OP1,2         OP1,3
                                                             st(i) + p(i) ≤ st(j), where p(i)
    rd1                                                     is the processing time of op i                      OP2,1 OP1,2
                      R1             R2             dd1                                                 R1
                                                                  i       R       j

                                                          st(i) + p(i) ≤ st(j) V st(j) + p(j) ≤ st(i)   R2   OP1,1    OP2,2   OP1,3
                    OP2,1           OP2,2


                                                           rd(j) ≤ st(i) for each op i of job j
          rd2                                     dd2

                                                             Minimize ∑ |c(j) - dd(j)|

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            What’s Missing from the
          Classical View of Scheduling
       Practical problems can rarely be formulated as
         static optimization tasks
          • Ongoing iterative process
          • Situated in a larger problem-solving
            context
          • Dynamic, unpredictable environment




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                   Managing Change

        “Scheduling” is really an ongoing process of
          responding to change


              Manufacturing

               Project Crisis Action
             Management Planning


          • Unpredictable, Dynamic     • Predictable, Stable
              Environment                   Environment
          • Robust response            • Optimized plans
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      Approaches to Managing Change

        • Build schedules that retain flexibility
        • Produce schedules that promote localized
          recovery
        • Incremental re-scheduling techniques
          (e.g., that consider “continuity” as an
          objective criteria)
        • Self-scheduling control systems




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          Incremental Schedule Repair

        Several competing approaches to maintaining
          solution stability
            • Minimally disruptive schedule revision
              (temporal delay, resource area, etc.)
            • Priority-based change
            • Regeneration with preference for same
              decisions
        Little understanding of how these techniques
          stack up against each other
        Even less understanding of how to trade
         stability concerns off against (re)optimization
         needs
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          Delayed-Commitment
          Scheduling Procedures
       Identify a contention peak and post a leveling
       constraint
              Activity 2                             Activity 2
  R1        Activity 1
                                  R1   Activity 1




        Advantages
         • Retain flexibility implied by problem
           constraints (time and capacity)
         • Can establish conditions for guaranteed
           executability


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            Building Robust Schedules

        Some open questions:
            • Extended conditions for
              “Dispatchability”
            • Robustness versus optimization
            • Use of knowledge about domain
              uncertainties
            • Local search with robust
              representations




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             Self-Scheduling Systems

         • Distribute decision-making among
           individual entities (machines, tools,
           parts, operators; manufacturers,
           suppliers)
         • Specify local behaviors and protocols
           for interaction
         • Robust, emergent global behavior




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       Morley’s GM Paint Shop System


                                           Paint
                            Bid            Booth   Bid
                                             1
                                                    parameters:
            Dispatcher     Announcement            - same color
                            (new truck)             as last truck
                                                   - space in
                                           Paint    queue
                                  Bid      Booth   - empty
 “If bid for same color then award
     else if empty booth then award          2      queue
             else if queue space then award”


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                          Tradeoffs

      Advantages:
          •   Complexity reduction
          •   Simple, configurable software systems
          •   Robust to component failures
          •   More stable computational load
      Problems:
          • No understanding of global optima (or how to
            achieve global behavior that attends to specific
            performance goals)
          • Prediction only at aggregate level (can become
            unstable)


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              Adaptive Systems:
        “Routing Wasps” in the Factory
        Machine
          1       R-Wasp                         ST2
                             P(route|ST,ØT) = _________
                  Agent1
                                                ST2 + ØT2
        Machine
          2
           .
           .
                   R-Wasp
                   Agent2
                                    B         C
           .                         A       Jobs   B
                  R-Wasp       A
        Machine   AgentN                        C       Stimulus:
          N                              B                  SB

                   Response Thresholds:
                       ØA, ØB, ØC, ...
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        Updating Response Thresholds

       ØT = ØT – ∆1 if next job is same type as current
         job
       ØT = ØT + ∆2 if next job is a different type
       ØT = ØT – ∆3 if the machine is currently idle



       • Routing framework can be seen as an
         adaptive variant of Morley’s bidding rule
       • Experimental results showing significant
         performance improvement
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           Some Open Issues in Multi-
               Agent Scheduling

        • Self-scheduling approaches do not preclude
          the use of advance schedules
            • How to incorporate?
        • Opportunistic optimization
        • Cooperative, distributed scheduling is a fact of
          life in many domains (geographic constraints,
          autonomous business entities, etc.)
            • How to negotiate and compromise?
            • Can self-interest be compatible with global
              performance objectives?


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      Integrating Planning & Scheduling

       “Planning & scheduling are rarely separable”
                       Waterfall Model
           Planner
                            Plan               Schedule
          Scheduler



                      Mixed-Initiative Model
                              Plan
            Planner

          Scheduler
                               Schedule
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                    Design Issues

        • Integrated search space versus separable
          sub-spaces
        • Single solver versus interacting solvers
        • Resource-driven versus strategy-driven
        • Loose coupling versus tight coupling




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             JFACC Planner/Scheduler

                                 Plan Server

                                   PLANS            Constraint
               HTN
                                 SCHEDULES           Based
             Planner
              (SRI)              ANNOTATIONS        Scheduler
                                                      (CMU)
                                  TRIGGERS



 Technological                     Experimental
 • Interleaved generation &        • Simple, low-cost info. exchanges
   repair of plans/schedules          yield
 • Distributed architecture to          • Marked reduction in comp. time
   support remote                       • Comparable plan/schedule quality
   collaboration                   • More complex models can improve
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                                 SRI International further
         Some Challenges that Remain

        • Scheduling models that incorporate richer
          models of state
        • Can integrated P & S problems really be
          solved as one big optimization task?
            • The limitations of SAT-style approaches
        • How to achieve tighter interleaving of
          action selection and resource allocation
          processes
        • Managing change in this larger arena


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              Requirements Analysis

        “Scheduling is really a process of getting the
          constraints right”

        Current tools designed around a “Specify and
         Solve” model of user/system interaction
           • Inefficient problem solving cycle
        Mixed-Initiative solution models
           • Incremental solution of relaxed
             problems
           • Iterative adjustment of problem
             constraints, preferences, priorities

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      Use of Relaxed Models to Identify
        Resource Capacity Shortfalls




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          The AMC Barrel Allocator
     Domain: Day-to-Day Management of Airlift &
      Tanker Assets at the USAF Air Mobility
      Command (AMC)
      Technical Capabilities:
          • Efficient generation of airlift and tanker
            schedules
          • Incremental solution change to accommodate
            new missions and changes in resource
            availability over time
          • Flexible control over degree of automation
          • Selective, user-controlled constraint relaxation
            and option generation when constraints cannot
            be
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                          Parameterizable Search
                               Procedures
                              AssignMission:
                               C141, [t1,t2]
                              Configuration
                                                                                        Search Configurations
GenResources                                                              Feasible - <GenRequestedRes,GenIntervals,EvalMinCompletion >
                    305th     437th      60th       62nd                  Delay - <GenRequestedRes, GenDelayInts, EvalMinTardiness>
                    AMW       AMW        AW          AW                   Over-Allocate - <GenRequestedRes, GenOverInts,
GenIntervals                                                                                                       EvalMinOverUsage>
                                                                          Bump – <GenRequestedRes, GenBumpInts, EvalMinDisruption>
        I1,305 I2,305 ... I1,437 I2,437 ... I1,60   I1,60 ... I1,62 ...   Alternative-MDS - < GenAlternRes, GenIntervals,
                                                                                                                  EvalMinCompletion>
                                                                          Composite Relaxations - …
                                    ...
                                 EvalCriteria

   Feasible - <GenResources, GenIntervals, EvalMinCompletion>


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                   Generate
                  Relaxation
                   Options




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           Mixed-Initiative Scheduling
                   Challenges

        • Management of user context across
          decision cycles
        • Explanation of scheduling decisions
            • Why did you do this?
            • Why didn’t you do that?
        • Adjustable autonomy




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           Research Directions for the
                 Next 10 Years

        • Deeper integration of AI and OR techniques
        • Robust schedules and scheduling
        • Global coherence through local interaction
        • Extension to larger-scoped problem-solving
          processes
        • Rapid construction of high performance
          scheduling services




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