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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
Carnegie Mellon
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.
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]
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 one
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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 = {OPi ,OP j}; O = OP k ≥
Start Time OP k 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: 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
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 parameters:
1
- same color as
Dispatcher Announcement last truck
(new truck) - space in queue
- empty queue
Paint
Bid
“If bid for same color then award Booth
else if empty booth then award 2
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
S T2 + 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 & repair • Simple, low-cost info. exchanges yield
of plans/schedules • Marked reduction in comp. time
• Distributed architecture to • Comparable plan/schedule quality
support remote collaboration • More complex models can improve
performance further
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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
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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