Doctorante : Thèse encadrée par : * Touria BENRAHHOU ...Cyclic Scheduling 1) Basic Cyclic...
Transcript of Doctorante : Thèse encadrée par : * Touria BENRAHHOU ...Cyclic Scheduling 1) Basic Cyclic...
Doctorante :* Touria BENRAHHOU
Thèse encadrée par : * Laurent HOUSSIN
* LAAS-CNRS; Université de Toulouse, France.{ben-rahhou, houssin}@laas.fr13ème Congrès des Doctorants 2012.
SummarySummary
I. Cyclic SchedulingI. Cyclic Scheduling
II. Heap of pieces theory
III. Application to the cyclic scheduling
IV. Conclusion and perspectives
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Cyclic SchedulingCyclic Scheduling
1) Basic Cyclic Scheduling (BCS)
Organize the implementation of tasks repeated many times,
• Realization of n jobs.
• A job "i" consists of ni elementary tasks.
• The jth task of the job "i" is denoted (i,j).j j ( j)
With precedence constraints between tasks,
• Precedence Relationship : (i,1) ‹ (i,2) ‹ … ‹ (i,ni).3
Cyclic SchedulingCyclic Scheduling
2) Cyclic Scheduling (C.S.) with resource constraints
Constraints on the use and availability of resources required by tasks.
• (i j) uses the machine m without interruption for a period p• (i,j) uses the machine mij , without interruption for a period pij.
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Cyclic SchedulingCyclic Scheduling
3) Jobs does not interwine
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Cyclic SchedulingCyclic Scheduling
Several possible scheduling by the criterion to optimizeSeveral possible scheduling by the criterion to optimize
Minimizing the asymptotic cycle time.g y p y
Asymptotic cycle time:Asymptotic cycle time:
The average cycle time
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Cyclic SchedulingCyclic Scheduling
4) Some classical methods of performance ) pevaluation of cyclic job-shop:
Petri nets based methods, especially timed event graphs.
G h h Graph theory.
(max +) - linear system (max,+) - linear system.
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Cyclic SchedulingCyclic Scheduling
5) Example of cyclic job-shop:
4 elementary tasksy 2 jobs 2 machines
Task (i,j) mij pij
a1 (1,1) 2 61 ( , )
a2 (1,2) 1 4
b1 (2,1) 1 7
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b2 (2,2) 2 3
Cyclic SchedulingCyclic Scheduling
5) Example of cyclic job-shop:Task (i,j) mij pij
a1 (1,1) 2 6
a2 (1,2) 1 4
b1 (2,1) 1 7
b (2 2) 2 3m1
b2 (2,2) 2 3
Cycle time = 11m2
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m1 b1 a2 b1 a2 b1 a2
m2 a1 b2 a1 b2 a1 b2
SummarySummary
I. Cyclic SchedulingI. Cyclic Scheduling
II.Heap of pieces theory
III. Application to the cyclic scheduling
IV. Conclusion and perspectives
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Heap of pieces theoryHeap of pieces theory
Graphic représentation
An analytical modeling in (max,+) algebra.
First application to the cyclic scheduling:
S. Gaubert and J. Mairesse.
" Modeling and analysis of timed petri nets using heap of pieces." Modeling and analysis of timed petri nets using heap of pieces.
IEEE Trans. on Automatic Control, 44(4), Apr. 1999.
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Heap of pieces theoryHeap of pieces theory
1) (max,+) Algebra
The set
Two operators:
• The sum corresponds to the max operator,• The product denotes the usual sum• The product denotes the usual sum.
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Heap of pieces theoryHeap of pieces theory
(max +) matrix algebra:(max,+) matrix algebra:
Square matrices with coefficient in Square matrices with coefficient in
For matrices, additions and products give:, p g
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Heap of pieces theoryHeap of pieces theory
Calculus example:p
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Heap of pieces theoryHeap of pieces theory
Cyclicity:
X an irreducible matrix
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Heap of pieces theoryHeap of pieces theory
2) Heap model definition
A heap model is composed by H = (T,R,R,l,U)
•T a finite set of pieces,T a finite set of pieces,
•R a finite set of slots,
• R :T P(R) gives the subset of slots occupied by a piece,
• l :T ₓ R gives the lower contour of a piece,
• U :T ₓ R gives the upper contour of a piece.U : ₓ g ves t e uppe co tou o a p ece.
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Heap of pieces theoryHeap of pieces theory
Example of a heap of pieces11
10 10
9 (d) 9
8(c)
8 (d)
7 7
6 6 (b)6 6
5
(b)
5(c)4 4
3 3
2 2
1 (a) 1 (a)
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S1 S2 S3 S1 S2 S3
Heap of pieces theoryHeap of pieces theory
Each piece "a" is represented by its matrix as:Each piece a is represented by its matrix as:
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Heap of pieces theoryHeap of pieces theory
Example: 10
9
8 (d)
7
(b)6 ( )5
(c)4
3
2
1 (a)
S1 S2 S3
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Heap of pieces theoryHeap of pieces theory
Example (suite): 10
9
8 (d)
7
(b)6 ( )5
(c)4
3
2
1 (a)
S1 S2 S3
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Heap of pieces theoryHeap of pieces theory
The upper contour of a heap of pieces "w" is given pp p p gby:
Height of a heap of pieces "w" is given by:
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Heap of pieces theoryHeap of pieces theory
Example (suite): 10
9
8 (d)
7
(b)6 ( )5
(c)4
3
2
1 (a)
S1 S2 S3
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SummarySummary
I. Cyclic SchedulingI. Cyclic Scheduling
II. Heap of pieces theory
III. Application to the cyclic scheduling
IV. Conclusion and perspectives
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Application to the cyclic schedulingApplication to the cyclic scheduling
Modeling the cyclic job-shop problem as a heap of pieces
The number of slots of the heap = The sum of the number of machines and jobs.
An elementary task a piece
o depending on the resources used,
o depending on the work to which it belongso depending on the work to which it belongs.
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Application to the cyclic schedulingApplication to the cyclic scheduling
1) Application to the example :Task (i,j) mij pij
a1 (1,1) 2 6
(1 2) 1 4a2 (1,2) 1 4
b1 (2,1) 1 7
b2 (2,2) 2 3
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b2 b219
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b1 b1
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15
14
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Tas de pièce : a1a2b1b2
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11
10
a2 a29
8
7
6
a1 a1
5
4
3
2
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2
1
m1 m2 J1 J2
Application to the cyclic schedulingApplication to the cyclic schedulingb2 b2
Task (i,j) mij pij
a1 (1,1) 2 6
(1 2) 1 4
b1 b1
a2 a2
a2 (1,2) 1 4
b1 (2,1) 1 7
b2 (2,2) 2 3
a1 a1
b2 b2
b1 b1
a2 a2
a1 a1
b2 b2
b1 b1
a2 a2b2 b2
b1 b1a1 a1
a2 a2b2 b2
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a2 a2
a1 a1
b1 b1a1 a1
a2 a2b2 b2
b1 b1a a
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26Tas de pièce : a1a2b1b2 a1b1a2b2
m1 m2 J1 J2 m1 m2 J1 J2
a1 a1
Application to the cyclic schedulingApplication to the cyclic scheduling
Tasks matrices:11
a2 a210
b2 b29
8
7
Consider:
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b1 b1
6
a1 a1
5
4
3
2
1
m1 m2 J1 J2
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Application to the cyclic schedulingApplication to the cyclic scheduling
2) Performance evaluation
Cyclicity
The unique eigenvalue of the matrix corresponds to the cycle time of the scheduling "w".
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Application to the cyclic schedulingApplication to the cyclic scheduling33
Example of performance evaluation:a2 a2
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b2 b231
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29
b b
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b1 b1a1 a1
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25
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23
22
a2 a221
b bb2 b220
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18
b1 b1
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a1 a1
16
15
14
13
12
11
a2 a210
b2 b29
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7
b1 b1
6
a1 a1
5
4
3
2
29
1
m1 m2 J1 J2
Application to the cyclic schedulingApplication to the cyclic scheduling
Complexity of performance evaluation in the case of 1-periodic h d lischeduling :
Calculation of eigenvalue (Karp Algorithm): g ( p g )
Matrix product:
Complexity for k-periodic scheduling.
Calculation of eigenvalue (the size of matrices is the same):
Matrix product:p
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SummarySummary
I. Cyclic SchedulingI. Cyclic Scheduling
II. Heap of pieces theory
III. Application to the cyclic scheduling
IV. Conclusion and perspectives
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ConclusionConclusion
Resolution Method,
A branch and bound procedure,p ,
Multiple work in process Multiple work in process.
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PerspectivesPerspectives
Fi di B tt b d Finding Better bounds,
k-periodic solutions.
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Thank you for your attentionThank you for your attention.
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