track-mounted cranes move materials & equipment
TRANSCRIPT
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Optimal Crane SchedulingSamid Hoda, John Hooker
Latife Genc Kaya, Ben Peterson
Carnegie Mellon University
1/16EWO - 13 November 2007
Iiro Harjunkoski
ABB Corporate Research
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Problem
• Track-mounted cranes move materials & equipment.
– We focus on longitudinal moves (left-right).– Cranes must not cross paths.
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Example crane trajectories
time
longitude
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• Schedule cranes to transfer material between locations.– 50-300 jobs with 1-8 tasks each.– Origin/destination and stop times for
each task.– Time windows and priorities.– Precedence constraints.
• Goal:– Follow production schedule as
closely as possible.
Problem
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• Three problem levels.– Assign jobs to cranes.
– Sequence jobs on each crane.
– Plan crane trajectories.• Time granularity ≈ 10 sec over ∼24 hrs
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Crane 1
14
5
10
7
9 11
15
1217
Crane 2
38
6
2 16
13
1814
Crane 1
Crane 2
Problem
…171512111097541Crane 1
…181614138632Crane 2
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…
…1097541Crane 1
…8632Crane 2
…109746Crane 1
85312Crane 2
Algorithm 1: Two-phase search
• For 2-crane problems.• Phase 1
– Assign and sequence with local search.
• Phase 2– Compute optimal trajectory with dynamic programming.
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Computes penalty
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• Basic DP recursion:
Transitions that satisfy constraints
Position
Loading/unloading time
Segment
Algorithm 1: Two-phase search
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State Space Reduction
• Only certain trajectories must be considered.– Canonical trajectory for the left crane:
Wait as long as possible
Move as soon as possible
Follow leftmost trajectory that never moves backward (away from destination)
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Depart from canonical trajectory
At each moment, follow canonical trajectory or right
crane’s trajectory, whichever is further to the left
Left crane Right crane
State Space Reduction
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– Properties of the minimal trajectory:
• The left crane never stops en route unless it is adjacent to the right crane.
• The left crane never moves backward unless it is adjacent to the right crane.
State Space Reduction
Depart from canonical trajectory
Left crane Right crane
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• Theorem: Some optimal pair of trajectories are minimal with respect to each other.
State Space Reduction
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• Represent processing-time states as an interval of states.– Exploit the fact that cranes are processing (and
therefore at rest) most of the time.– Store these states in a 2-dimensional circular queue
data structure that persists through time.
State Space Reduction
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c15c14c13c12c11
c21
c31
c41Compute these costs when a task pair for which both tasks are at their stop locations appears in the state space.
cij = cost-to-go when the left crane has been processing i time units and the right crane has been processing j time units.
State Space Reduction
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c21c25c24c23c22
c31c32
c41c42
c11c15c14c13c12Compute these costs in the next period. No need to update existing costs.
Data structure is a 2-dimensional circular queue.
cij = cost-to-go when the left crane has been processing i time units and the right crane has been processing j time units.
State Space Reduction
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c32c31c35c34c33
c42c41c43
c12c11c15c14c13
c22c21c25c24c23And similarly in the next period.
cij = cost-to-go when the left crane has been processing i time units and the right crane has been processing j time units.
State Space Reduction
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c43c42c41c45c44
c13c12c11c15c14
c23c22c21c25c24
c33c32c31c35c34These costs do not correspond to separate states.
After this point the table is no longer needed and memory can be released.
cij = cost-to-go when the left crane has been processing i time units and the right crane has been processing j time units.
State Space Reduction
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• New data structure reduced computation time an order of magnitude.– Computation time is sensitive to width of time
windows.– Can now solve 60-job problem with wide time
windows (∼1 hour)• Wide windows are necessary for feasible solution.
Computational Results
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Effect of New Data Structure
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Solution of 20-job Problem
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State Space Size
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Solution of 40-job Problem
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State Space Size
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Solution of 60-job Problem
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State Space Size
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Computation Time for one DP
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crane 1new job:1,2,none
crane 2new job:2,3,none
craneyield:1,2
TrimStates
TrimStates
TrimStates
SimulationTime
State Space
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Algorithm 2: Inexact DP
• Crane simulation.– Splits at decision
points.
• State trimming:– Eliminates inferior
states
• Applies to > 2 cranes.
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WORKING
WAITING
MOVING
YIELDING
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Algorithm 2: Inexact DP
• Discrete-event simulation– Continuous time– 4 crane actions– State transition only when an action is completed
• Cranes will work and move until:– Time to pick up new job.
• Path splits for each job choice.
– Cranes interfere.• Path splits for each
yielding choice.
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C1 C2 C3 …
S
D
SD
S
D
• Exact state trimming by domination.– State D dominates state S if
• Obj fcn value of D ≤ Obj fcn value of S
• {jobs completed in D} ⊂ {jobs completed in S}• For all cranes c,
– c working on same job in D & S or c is waiting in D, and
– progress on c’s job in D ≥ progress on c’s job in S
• Inexact state trimming by size limit– Drop states with higher obj fcn values
Algorithm 2: Inexact DP
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0
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1000
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000 34000
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Uncut States
Cut States
- Schedules hundreds of tasks per minuteunder typical conditions
- State explosions caused by specific schedule features
2-crane, 72-job schedule test
- Runtime is essentially linear in number of jobs.
Computational Results
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• Inexact DP algorithm:– Meets runtime goals.– Can accommodate multiples cranes.
• State space size:– Still sensitive to time window width.
– Inexact pruning required for ∼1 hour windows.
• Unfinished business:– Compare solution quality with Algorithm 1.– Solve additional problems.
Computational Results