Schedule Execution using Perturbation Analysis

1
Schedule Executionusing Perturbation Analysis
L. Bongaerts, H. Van Brussel, P.
Valckenaers Mechanical Engineering Dept.,
Katholieke Universiteit Leuven Belgium IEEE Intl
Conference on Robotics and Automation,
1998 1998? 7? 14? ???????? ? ? ?
2
Contents

• Introduction
• Holonic SFC architecture
• Holonic schedule execution strategies
• Concepts
• Schedule represented as a graph
• Calculation of reactive scheduler
• On-line control
• Conclusion

3
Introduction (1/2)

• Disturbances in the shop floor
• rush orders, machine breakdowns, processing time
  variations
• Two approaches to cope with disturbances
• reactive scheduling
• neglects the opportunities to immediately react
  with an intelligent dispatcher and have a second
  line reaction in the reactive scheduler
• heterarchical control
• does not address any optimisation either and
  confines itself to fast heuristic
• The Concept of concurrent scheduling and schedule
  execution
• to combine schedule optimisation and robustness
  against disturbances
• reactive scheduler exploits the available time
  for the optimisation
• on-line shop floor control system react
  immediately to disturbances

4
Introduction (2/2)

• This paper
• describe an algorithm for Schedule Execution that
  implements the concept behind holonic
  manufacturing
• the algorithm is based on a Perturbation Analysis

5
Holonic SFC architecture
6
Holonic Schedule Execution Strategies

• A trade-off between the performance and the
  reactivity to disturbances
• hierarchical control vs. heterarchical control
  architecture
• Hierarchical control
• Heterarchical control
• Hybrid control strategies based on heuristics

7
Concepts (1/2)

• Schedule execution problem
• It is impossible to find a good SE algorithm that
  deals with disturbances autonomously and does not
  have to perform rescheduling itself, if the Gantt
  chart is the only available information of the
  schedule
• Example

• If A1 is delayed, on-line SFC system should
  decide whether to keep
• schedule on w1 or swap C1 and B3

8
Concepts (2/2)

• Solution
• the on-line SFC system need to know how local
  decisions affect the global performance
• Perturbation Analysis
• the global performance (?) is expressed in
  function of local parameters (?e)
• partial derivatives of the global performance to
  the local parameters are calculated
• During on-line control
• a number of local decision alternatives are
  defined and evaluated
• for each alternative, the effect (??) on the
  values ?e is calculated
• The alternative with the best ?? is selected

9
Architecture of a co-operating reactive scheduler
and on-line SFC system
10
Schedule Represented as a Graph (1/5)

• Precondition on the schedule active schedules
• Operation start times

11
Schedule Represented as a Graph (2/5)

• Graph
• every node represents an operation
• every edge represents a precedence constraint
  (technological, schedule decision)

12
Scheduled Represented as a Graph (3/5)

• Graph (contd)
• on each instance of time tcut
• schedule head(Gh) a set of operations that start
  before tcut
• schedule body(Gb) a set of operations that start
  at time tcut or later
• cut(E) a set of edges of the graph that connect
  a node of the schedule head with a node of the
  schedule body
• G Gh ? Gb ? E
• feasible if all edges of E are oriented in the
  direction from Gh to Gb
• a cut is sufficient to define the Gh and Gb , if
  the graph G is connected

Schedule Body Gb
0
A1
C1
B3
B2
A2
C2
B1
A3
Schedule Head Gh
13
Schedule Represented as a Graph (4/5)

• Definition of local decision parameters ?e
• for every edge e (n1, n2) in E, a new
  variable(?e) is defined
• the earliest start time operation n2 could start
  if it would respect the precedence constraint
  represented by the edge e

where Ai1,j1,i2,j2 is the auxiliary time (i.e.
setup or transport time)
14
Scheduled Represented as a Graph (5/5)

• Performance in function of ?e
• extending Gb with the dummy nodes for each ?e
• objective function a function of the start date
  of all operations
• or a function of EST ?e of the cut E and the
  start times of the oper. in Gh

0
A1
C1
B3
?4
?6
?1
B2
A2
C2
?5
?2
B1
A3
?3
15
Calculation of Reactive Scheduler (1/2)

• Example
• the function that expresses how the global global
  performance changes due to changes of the
  resource driven earliest start time for op. A2
• Calculation of the partial derivatives of the
  global performance to the local decision
  parameters ?e

16
Calculation of Reactive Scheduler (2/2)

• Example the nominal values of ?e and ??/??e,
  for tcut3

17
On-line Control (1/3)

• Algorithm three phases
• 1) alternative local decisions are proposed
• the allocation of operations to workstations
• the sequencing of operations on a workstation (a
  swap)
• the sequencing of operations on secondary
  resources
• 2) alternatives are evaluated
• ND a set of done operations
• NBu a set of busy operations
• NP a set of operations it is about to take
  local resource allocation decisions about ( the
  pending operations)
• NB a set of operations contained in the
  schedule body
• 3) the best one is chosen

18
On-line Control (2/3)

• Example delayed operation
• Suppose A1 is delayed half a time unit and could
  only finished at time 1.5
• Alternatives
• keeping the sequence
• swapping operations C1 and B3
• First alternative evaluation
• tcut 3, ND A1, B1, NBu B2, NP C1,
  NB A2, A3, B3, C2
• ?1 3.5, ?4 1.5, ?6 3.5
• linear approx. of the new ? ?0 ??/??1??1
  ??/??4??4 ??/??6??6 ?05
• Second alternative evaluation
• tcut 4, ND A1, B1, NBu B2, NP C1,
  A2, B3, NB A3, C2
• tB3 3, tC1 4, tA2 3, ?2 4, ?3 1, ?4
  4, ?6 6
• linear approx. of the new ? ?0 ??/??6??6
  ?0 ? ?0 1 (correct value)
• Therefore, the on-line SFC system concludes that
  it should select the second alternative and swap
  C1 and B3

19
On-line Control (3/3)

• Other examples
• An operation finishes earlier
• A machine breaks down
• An operation is delayed and the next operation on
  that workstation can be executed on another
  workstation
• An operation cannot be executed due to a machine
  failure, but an operation of another type can
  still be executed
• Evaluation and extensions
• this yields a good overall behavior for the SE
  problem
• this algorithm computationally outperforms a
  similar approach based on simulations by an order
  of magnitude O(Noper)
• this approach effectively controls nervousness to
  stay below reasonable bounds, by keeping a good
  homeostasis of the schedule when few disturbances
  occur

20
Conclusion

• This paper
• presents an algorithm for executing a schedule
  based on a graph representation of a schedule and
  the use of partial derivatives for estimating the
  effect of local decisions to the global goal
• provides a generic concept that combines fast
  reaction to disturbances with optimisation by a
  combination of feedback control (reactive
  scheduling) and linearised feedforward
• This algorithms are currently being implemented
  in the control software for an FAS, based on the
  concepts of holonic manufacturing