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Planning and Scheduling

Stephen F. Smith

The Robotics InstituteCarnegie Mellon University

Pittsburgh PA 15213sfs@cs.cmu.edu412-268-8811

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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

op11

R2R1

rel1 dd1

rel2 dd2

op12 op1

3

op12 op2

2 i jR

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

st(i) + p(i) < st(j) st(j) + p(j) < st(i)

rel < st(i), for each op i of job jj

dd > st(i) + p(i), for each op i of job jj

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A More Complex Scheduling Problem

Origin

Air-POE

Sea-POE

Sea-POD

Air-POD

Destination

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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

Properties:•Modeling Generality/Expressiveness•Incrementality•Compositional

Active Data Base(Current Schedule)

Constraint Propagation

CommitmentStrategies/Heuristics

ConflictHandling

Components:

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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.

6. Go to step 1

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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]

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

[0,12] [0,12]

[0,12] [0,12] [0,12]

R1

R2

R3

Job3 :

Job2 :

O1,3O1,2

O2,2O2,1

O3,3O3,2O3,1

Sti,j + Duri,j ≤ Sti,k

Oi,j Oi,kOi,j Ok,lRx

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 one

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Temporal Constraint Propagation through

Precedence ConstraintsAssume dui,j = 3 for all Oi,j

• Before propagation:

• Forward propagation

• Backward propagation

O1,1[0,12] [0,12] [0,12]

O1,3O1,2

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

O1,3O1,2

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

O1,3O1,2

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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

R1

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

Scheduled to start at time 6[6,6]

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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 : inconsistency4. 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

10 20 30

OPi

OPj

OPk

EST(O) ≥ EST(S) + Dur(S)((LFT(S) - EST(S) < Dur(O) + Dur(S)) (LFT(S) - EST(O) < Dur(O) + EST(O))

S = {OP ,OP }; O = OP Start Time OP ≥ 25kki j

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More Complex Temporal Constraints

“Simple Temporal Problem” (STP) [Dechter91]• Edge-weighted graph of time points expressing

constraints of the form: atpjtpib• 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• Efficiently solved via all-pairs shortest path

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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

OP1,1 OP1,2 OP1,3

OP2,1 OP2,2

R1 R2rd1 dd1

dd2rd2

i j

st(i) + p(i) ≤ st(j), where p(i)is the processing time of op i

i jR

st(i) + p(i) ≤ st(j) V st(j) + p(j) ≤ st(i)

rd(j) ≤ st(i) for each op i of job j

Minimize ∑ |c(j) - dd(j)|

OP1,1 OP2,2 OP1,3

OP1,2OP2,1

R2

R1

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What’s Missing from the Classical View of

SchedulingPractical 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

• Predictable, Stable Environment• Optimized plans

• Unpredictable, Dynamic Environment• Robust response

Manufacturing

Crisis ActionPlanning

ProjectManagement

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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

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Activity 2

R1Activity 1

Activity 2

R1Activity 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

Dispatcher

PaintBooth

1

PaintBooth

2

Bid

Bid

Announcement(new truck)

Bid parameters:

- same color as last truck

- space in queue

- empty queue

“If bid for same color then award else if empty booth then award 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

Machine1

Machine2

MachineN

...

ST2

P(route|ST,ØT) = _________ ST

2 + ØT2

Response Thresholds:ØA, ØB, ØC, ...

AA

B

B

B

C

C

Stimulus:SB

R-WaspAgentN

R-WaspAgent2

R-WaspAgent1

Jobs

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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

Mixed-Initiative Model

Waterfall Model

Plan Schedule

Planner

Scheduler

Planner

SchedulerSchedule

Plan

“Planning & scheduling are rarely separable”

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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

ConstraintBased

Scheduler(CMU)

HTNPlanner

(SRI)

PLANSSCHEDULES

ANNOTATIONS

TRIGGERS

Experimental• Simple, low-cost info. exchanges

yield• Marked reduction in comp.

time• Comparable plan/schedule

quality• More complex models can improve

performance further

Technological• Interleaved generation &

repair of plans/schedules• Distributed architecture

to support remote collaboration

SRI International

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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 satisfied

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Parameterizable Search Procedures

I1,305 I2,305 I1,437...

305thAMW

AssignMission:C141, [t1,t2]

Configuration

60thAW

62ndAW

437thAMW

...I2,437 I1,60 ... I1,62 ...I1,60

GenResources

GenIntervals

EvalCriteria

...

Feasible - <GenResources, GenIntervals, EvalMinCompletion>

Search ConfigurationsFeasible - <GenRequestedRes,GenIntervals,EvalMinCompletion >

Delay - <GenRequestedRes, GenDelayInts, EvalMinTardiness>

Over-Allocate - <GenRequestedRes, GenOverInts,

EvalMinOverUsage>

Bump – <GenRequestedRes, GenBumpInts, EvalMinDisruption>

Alternative-MDS - < GenAlternRes, GenIntervals,

EvalMinCompletion>

Composite Relaxations - …

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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