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

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Carnegie Mellon 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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Carnegie Mellon A Basic Scheduling Problem

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Carnegie Mellon A More Complex Scheduling Problem Origin Air-POE Sea-POE Sea-POD Air-POD Destination

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Carnegie Mellon 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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Carnegie Mellon Constraint-Based Scheduling Models Properties: Modeling Generality/Expressiveness Incrementality Compositional Active Data Base (Current Schedule) Constraint Propagation Commitment Strategies/ Heuristics Conflict Handling Components:

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Carnegie Mellon 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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Carnegie Mellon A Basic CSP Procedure 1. [Consistency Enforcement] - Propagate constraints to establish the current set v d of feasible values for each unassigned variable d 2. If v d = 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 v d to assign to d. 6. Go to step 1

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Carnegie Mellon 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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Carnegie Mellon A Simple Job Shop Scheduling CSP Variables: start times (st j,i ) - Domain: [0,12] O 1,1 Job 1 : [0,12] R1R1 R2R2 R3R3 Job 3 : Job 2 : O 1,3 O 1,2 O 2,2 O 2,1 O 3,3 O 3,2 O 3,1 St i,j + Dur i,j ≤ St i,k O i,j O i,k O i,j O k,l RxRx St i,j + Dur i,j ≤ St k,l V St k,l + Dur k,l ≤ St i,j

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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon Temporal Constraint Propagation through Precedence Constraints Assume du i,j = 3 for all O i,j Before propagation: Forward propagation Backward propagation O 1,1 [0,12] O 1,3 O 1,2 O 1,1 [0,12][3,12] [6,12] O 1,3 O 1,2 O 1,1 [0,6][0,9] [6,12] O 1,3 O 1,2

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Carnegie Mellon Capacity Constraint Propagation Observation: Enforcing consistency with respect to capacity constraints is more difficult due to the disjunctive nature of these constraints Forward Checking: O 1,1 R1R1 O 2,1 Before propagation: [6,12] After propagation: [9,12] Scheduled to start at time 6 [6,6]

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Carnegie Mellon Pruning Operation Ordering Alternatives Example: Erschler’s dominance conditions Conclusion: O i cannot precede O j In general: For any unordered pair of operations {O i, O j }, we have four possible cases: 1. LST i < EFT j and LST j ≥ EFT i : O i is before O j 2. LST j < EFT i and LST i ≥ EFT j : O j is before O i 3. LST i < EFT j and LST j < EFT i : inconsistency 4. LST i ≥ EFT j and LST j ≥ EFT i : both options remain open

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Carnegie Mellon Edge Finding S - a set of operations competing for resource R O - an operation not in S also requiring R 10 20 30 OP i j k 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 ≥ 25 k k i j

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Carnegie Mellon More Complex Temporal Constraints “Simple Temporal Problem” (STP) [Dechter91] Edge-weighted graph of time points expressing constraints of the form: a tp j tp i b Assuming no disjunction, allows incorporation of Temporal relations: finish-to-start (precedence) start-to-finish (duration) Start-to-start (same-start)... Metric bounds: offsets from time origin Efficiently solved via all-pairs shortest path

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

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Carnegie Mellon 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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Carnegie Mellon Technological Strengths Scalability Modeling flexibility Optimization Configurable So, Is scheduling a solved problem?

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Carnegie Mellon 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 OP 1,1 OP 1,2 OP 1,3 OP 2,1 OP 2,2 R1R1 R2R2 rd 1 dd 1 dd 2 rd 2 i j st(i) + p(i) ≤ st(j), where p(i) is the processing time of op i ij R 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 OP1,2 OP2,1 R2 R1

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Carnegie Mellon 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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Carnegie Mellon Managing Change “Scheduling” is really an ongoing process of responding to change Predictable, Stable Environment Optimized plans Unpredictable, Dynamic Environment Robust response Manufacturing Crisis Action Planning Project Management

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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon Delayed-Commitment Scheduling Procedures Identify a contention peak and post a leveling constraint Carnegie Mellon Activity 2 R1R1 Activity 1 Activity 2 R1R1 Activity 1 Advantages Retain flexibility implied by problem constraints (time and capacity) Can establish conditions for guaranteed executability

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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon Morley’s GM Paint Shop System Dispatcher Paint Booth 1 Paint Booth 2 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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Carnegie Mellon 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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Carnegie Mellon Adaptive Systems: “Routing Wasps” in the Factory Machine 1 Machine 2 Machine N...... ST2ST2 P(route|S T,Ø T ) = _________ S T 2 + Ø T 2 Response Thresholds: Ø A, Ø B, Ø C,... A A B B B C C Stimulus : S B R-Wasp Agent N R-Wasp Agent 2 R-Wasp Agent 1 Jobs

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Carnegie Mellon Updating Response Thresholds Ø T = Ø T – ∆ 1 if next job is same type as current job Ø T = Ø T + ∆ 2 if next job is a different type Ø T = Ø T – ∆ 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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Carnegie Mellon 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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Carnegie Mellon Integrating Planning & Scheduling Mixed-Initiative Model Waterfall Model PlanSchedule Planner Scheduler Planner Scheduler Schedule Plan “Planning & scheduling are rarely separable”

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Carnegie Mellon 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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Carnegie Mellon JFACC Planner/Scheduler Plan Server Constraint Based Scheduler (CMU) HTN Planner (SRI) PLANS SCHEDULES 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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Carnegie Mellon 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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Carnegie Mellon 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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Carnegie Mellon Use of Relaxed Models to Identify Resource Capacity Shortfalls

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Carnegie Mellon 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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Carnegie Mellon Parameterizable Search Procedures I 1,305 I 2,305 I 1,437... 305th AMW AssignMission: C141, [t1,t2] Configuration 60th AW 62nd AW 437th AMW... I 2,437 I 1,60... I 1,62... I 1,60 Gen Resources Gen Intervals Eval Criteria... Feasible - Search Configurations Feasible - Delay - Over-Allocate - <Gen RequestedRes, Gen OverInts, Eval MinOverUsage > Bump – Alternative-MDS - < Gen AlternRes, Gen Intervals, Eval MinCompletion > Composite Relaxations - …

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Carnegie Mellon Generate Relaxation Options

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Carnegie Mellon 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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Carnegie Mellon 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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