scheduling of petri nets as a multi-objective shortest path problem

Scheduling of Petri nets as a Multi-objective Shortest Path Problem

Uno Wikborg* and Tae-Eog Lee

KAIST Industrial and Systems Engineering Department

CASE – August 21st 2012 – Seoul, Korea

System Integration & Modeling Lab.

Problem definition

• Find a firing sequence with shortest makespan which brings a Petri net with place delays from one given marking to another given marking

• Does not consider maximum residency time of tokens

• Going from the marking [1,0,0,0,0] to [0,0,0,0,0] given delay times for each place in the Petri net below

2

1

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

0

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5


Related literature

• Dawande M, Geismar HN, Sethi SP, C S. Sequencing and scheduling in robotic cells; recent developments. Journal of Scheduling. 2005; 8(5): p. 387-426.

• Jung C. An Efficient Mixed Integer Programming Model Based on Timed Petri Nets for Diverse Complex Cluster Tool Scheduling Problems. 2010

• Daellenbach HG, De Kluyver CA. Note on Multiple Objective Dynamic Pro-gramming. The Journal of the Operational Research Society. 1980 July; 31(7): p. 591-594.

• Wikborg U, Lee TE. Non-Cyclic Scheduling for Timed Discrete Event Systems with Application to Single-Armed Cluster Tools using Pareto-Optimal Opti-mization. Accepted to be published. IEEE Transactions on Automation Sci-ence and Engineering.

3


Motivation

• Petri nets are well suited for modelling cluster tools for semiconductor production

• Wafer recipes tend to change more often

• Cyclic scheduling is not efficient when the steady state is short

4


Example 1: A Single Objective Approach (1)

• Problem description: Which firing sequence uses the shortest time to change the marking from [1,0,0,0,0] to [0,0,0,0,0]?

5

1

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Place p1 p2 p3 p4 p5Delay 0 s 2 s 2 s 1 s 2 s



• The two enabled transitions give us two possible next marking.

6

0

1

0

1 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t51

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5 0

0

1

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

t1

t2



• The tokens have to wait at p2 and p3 for the delay period.

• The resulting state can therefore be defined by the marking, a time stamp for each token and a time stamp for when the last transition was fired.

• For the case when t1 is fired: marking [0,1,1,0,0], tokens {2},{3}, fire time 0

7

0

1

0

1 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Place p1 p2 p3 p4 p5Delay 0 s 2 s 2 s 1 s 2 s



• This gives us a tree like transition graph

• The last fired transition is used as the objective

• The optimal solution can be found by traversing the graph

• Tree like structures tend to grow too fast when the problem size increases

8

[1,0,0,0,0]{0}0

[0,1,1,0,0]{2},{2}

0

[0,0,0,1,0]{1}0

[0,0,1,0,1]{2},{4}

2

[0,0,0,0,0]∅4

[0,0,1,0,1]{3},{3}

1

[0,0,0,0,0]∅3

t1 t2

t3 t5

t4 t4


A Multi-objective Approach

• Tree structure was caused by many state variables

• The time stamps can instead be considered objective functions

• The marking then defines the state

• A multi-objective shortest path problem in the untimed transition graph

• Note that the number of objectives vary from state to state and that the length of the edges depend on the previous path

9

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

[1,0,0,0,0]{0}0

[0,1,1,0,0]{2},{2}

0

[0,0,0,1,0]{1}0

[0,0,1,0,1]{2},{4}

2

[0,0,0,0,0]∅4

[0,0,1,0,1]{3},{3}

1

[0,0,0,0,0]∅3

t1t1 t2t2

t3t3 t5t5

t4t4 t4


Label correction algorithm

• Based on bottom-up-dynamic programming

• Keep the best solutions for each state

• A set of non-dominated solutions is kept for each marking

• A hash map is used with the marking as the key and a set of all non-dominated paths to the marking as the value

• Terminate when no better solution to the goal marking can be found

10


Example 2: Multi-objective Solution

Path End State Last Firing Time Token Time Stamps∅ [1,0,0,0,0] 0 [ , , , , ]∅∅∅∅∅

[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

11

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring∅

1

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

12

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4


1

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

13

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4


1

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore

t2




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

14

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

1

0

1 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to exploret2




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

15

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

1

0

1 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to exploret2




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

16

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

1

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

17

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

1

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

18

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

1

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

19

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

1 1p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

20

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

1 1p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

21

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

1 1p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

22

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

1 1p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 4 [ , , , , ]∅∅∅∅∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

23

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

1 1p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

24

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

1 1p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore




[0,1,1,0,0] 0 [ ,{2},{2}, , ]∅ ∅∅[0,0,0,1,0] 0 [ , , ,{1}, ]∅∅∅ ∅[0,0,1,0,1] 2 [ , ,{2}, ,{4}]∅∅ ∅[0,0,1,0,1] 1 [ , ,{3}, ,{3}]∅∅ ∅[0,0,0,0,0] 3 [ , , , , ]∅∅∅∅∅

25

[1,0,0,0,0]

[0,1,1,0,0] [0,0,0,1,0]

[0,0,1,0,1]

[0,0,0,0,0]

t1 t2

t3 t5

t4

Currently exploring

0

0

0

0 0p1

p2

p3

p4

p5

t1

t2

t3

t4

t5

Paths to explore


Theoretical properties

26

Theorem 2. There exists an optimal pathand such that for any:

A solution can be found which only includes non-dominated sub-problems.

Theorem 3.

The values of the objectives for all non-dominated solutions for a given problem falls within a range of two times the largest place delay.

Theorem 1. The sequencing problem of a timed Petri net can be transformed into an equivalent multi-objective shortest path problem.


Non-cyclic scheduling of Cluster Tools for Semiconductor Production

• Well suited for non-cyclic scheduling

• Single armed cluster tool with three processing chambers and 500 wafers

• Dual armed cluster tool with two double and one single processing module and a changing recipe

27

500 0 0 0 0 0 0 0

1 1 1

1

p1 p2 p3 p5

p4 p7

p6 p8 p9 p11

p10

p12

t1 t2 t3 t4 t5 t6 t7 t8

25 0 0 0 0 0 0 0

2 2 12

p1 p2 p3 p5 p6 p8 p9 p11t1 t2 t3 t4 t5 t6 t7 t8

1

0 0 0 0 0 0 0 0

0

25

25

p7p4 p10p12 p13

p14 p15 p16 p17 p18 p19 P20 p21

p22

t9 t10 t11 t12 t13 t14 t15 t16

t17


Problem variations

• Single and dual armed cluster tools

• Multiple recipes – frequent changes

• Cleaning processes

• Changing between steady states

• Start up and close down of cluster tools

• Non-identical robot movement times28


Future work

• Study how different parameters effect the efficiency

• Apply the method to more problems

• Improve the direction of the search

• Extend the method to handle additional requirements

29


Questions?

?• For further inquires please contact [email protected]

30

scheduling of petri nets as a multi-objective shortest path problem

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