Doctorante : Thèse encadrée par : * Touria BENRAHHOU ...Cyclic Scheduling 1) Basic Cyclic Scheduling (BCS) Organize the implementation of tasks repeated many times, • Realization

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

2

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


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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3) Jobs does not interwine

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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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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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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

8

b2 (2,2) 2 3


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


II.Heap of pieces theory



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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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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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(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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Calculus example:p

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Cyclicity:

X an irreducible matrix

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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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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


Each piece "a" is represented by its matrix as:Each piece a is represented by its matrix as:

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Example: 10

9

8 (d)

7

(b)6 ( )5

(c)4

3

2

1 (a)

S1 S2 S3

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Example (suite): 10

9

8 (d)

7

(b)6 ( )5

(c)4

3

2

1 (a)

S1 S2 S3

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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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Example (suite): 10

9

8 (d)

7

(b)6 ( )5

(c)4

3

2

1 (a)

S1 S2 S3

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SummarySummary





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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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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

18

17

b1 b1

16

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

25

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


Tasks matrices:11

a2 a210

b2 b29

8

7

Consider:

7

b1 b1

6

a1 a1

5

4

3

2

1

m1 m2 J1 J2

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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

30

29

b b

28

27

b1 b1a1 a1

26

25

24

23

22

a2 a221

b bb2 b220

19

18

b1 b1

17

a1 a1

16

15

14

13

12

11

a2 a210

b2 b29

88

7

b1 b1

6

a1 a1

5

4

3

2

29

1

m1 m2 J1 J2


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





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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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Documents

Doctorante : Thèse encadrée par : * Touria BENRAHHOU ...Cyclic Scheduling 1) Basic Cyclic Scheduling (BCS) Organize the implementation of tasks repeated many times, • Realization