Relationships between Petri nets and constraint graphs: application to manufacturing Catherine Mancel : GFI Consulting and LAAS-CNRS, Toulouse, France Pierre Lopez : LAAS-CNRS, Toulouse, France Nicolas Rivière : LAAS-CNRS, Toulouse, France Robert Valette : LAAS-CNRS, Toulouse, France http://www.laas.fr/~robert
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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Objective • (Time(d)) Petri nets seen as (T)Constraint Satisfaction Problems • TCSP algorithms and Constraint Propagation used in the context of Petri nets • Clear separation between discrete (logic) and continuous (time) • Reverse the classical approach (Linear prog. and then Tree based search) : Discrete constraints are satisfied first • Simultaneously solve Assignment and Scheduling
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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Ordinary Petri net and CSP • Petri net defines precedence relations in a procedural way – They are not explicitly enumerated t11
t21
t31
p3
t12
t22
t41
p4
t11
t21
t31
t12
t22
t41
t3 p0
t1
p1
t2
p2
t4
• Two consistent sets of precedence relations • Avoid interleaving (firing sequence) IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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Linear logic and Petri nets • Equivalence between reachability in Petri nets and the provability of specific sequents in Linear logic (Girard's) • Tokens have to be produced before being consumed • Checked during the proof => precedence relations derived • One proof => one consistent set of precedence relations (A fragment of one process in the Petri net unfolding) • One proof = a sequence of decisions i.e. conflict (tokens and transitions) resolutions IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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An example p2 p1
t2
p3
t3
p4
t1
p1
t21
p81 t11
p2 p5
t6 p9 p8 p5
t4
p6
p7
Conflict t2 t4 : decision Two proof trees Two partial orders
t21 p1
t11
p2 p5
p81
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
t31
p82 t41
t5
p3
t61
p7 p6
p3
t5
1
t31
p82 t41
p4
p83
p4
p83 t61
p7 p6
1
t5
5
Introducing time • If one consistent set of precedence relations has been selected time constraints are introduced in a linear way (AOA-graphs) • The following steps are executed: – Petri net model (static definition of manufacturing system and specification of how resource assignments can be done) – From the current marking to another marking => defines an assignment/scheduling problem – Derive one partial order (heuristic based?) => a set of consistent precedence relations – Take continuous time constraint into account => Activity on arcs graphs (AOA) – Use Linear Programming or Constraint Propagation (arc consistency) on AOA
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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p9
6
p9
p-timed Petri nets => AOA • di minimal sojourn time in place pi • Transition firing nodes => firing dates (variables) • Arc labels (place names) => constraints (duration: x3 >= x2 + d3) p1
t21
p81 p2
t11
p5
p3
t31
p82
t61
p7 p6
t41
x2 p4
t51
p8
p9
x1
d2 d5
3
x3
d3
d4
d8 x4
x6
d7 d6
x5
d9
d8
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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p-time Petri net => AOA • dim minimal diM maximal sojourn time in place pi • Transition firing nodes => firing dates (variables) • Arc labels (places) => constraints (duration) (2 arcs each place x3 >= x2 + d3m and x3 AOA • di firing duration attached to each transition • Each node broken down into 2: x: "begin firing" y: "end firing" • Transition arc (y2 >= x2 + d2), token arc (x3 >= y2)
p1
t21
p81 t11
p2 p5
p3
t31
p82 t41
p4
t61
p7 p6
t51
y2
x2 p9
y1
d3
0
0 p83
0
d2
0
x3
x4
y3 0 0
d6 x6
d4 0 d5 y4 x5 y5
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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t-time Petri nets => AOA (1) • dim minimal, diM maximal enabling duration transition ti • Each node broken down into 2: x: "begin enabling" y: "end firing" • Transition arc (y2 >= x2 + d2), token arc (x3 >= y2) • "Backward arcs" from "end firing" of ti to the end of firing producing the last token (ti becomes enabled when it arrives)
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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y6
t-time Petri nets => AOA (2) p2 p1
t2
p3
t3
p4
t1
t6 p9
p1
t4
0
- d2M 0
t5 before t3
0
d3m
- d4M d5m
d4m
- d5M
0
- d6M x6
d6m
y1
y6
0
- d2M 0
0
y5
t3 before t5
t51
d3m
d4m
p9
p83
- d5M
d6m
0
x6
- d4M d5m
0
t61
- d3M y 3
d2m
0
p4 p7
p6
t41
p7
- d3M y 3
d2m y1
t5
t31
p3
p82
p5
p6
0
p2
t11
p8 p5
t21
p81
0
y6
- d6M
y5
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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Conclusion • Comparing the time extensions of Petri nets / constraints: – p-timed, t-timed and t-time Petri nets are such that derived AOA-graphs are without positive oriented circuits, "logical" reachability entails the existence of at least one sequence verifying all the temporal constraints – t-time Petri nets are not adequate because it is necessary to know the firing sequence (combinatorial explosion) – p-time Petri nets are the more general (possibility of temporal inconsistencies) and AOA graphs have the same structure as the precedence graphs
• From a Petri net model: simultaneous assignment + scheduling • A basis to elaborate hybrid approaches (Petri net + Constraints)
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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p-timed Petri nets => AOA p2 p1
t2
t1
p1
t21 p2
t11
p5
p3
t31
p82 t41
t4
p4 t6 p9
p6
t5
p7 x2
p4
t61
p7 p6
t3
d3 p8 p5
p81
p3
t51
p8
p9
d2
x1
d4
d8
d5
3
x3
d3
x4
x6
d7 d6
x5
d9
d8
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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p-time Petri net => AOA p2 p1
t2
p3
t3
p4
t1
t6 p 9 p8 p5
t21
1
p1
p8 t1
1
p2 p5
p3
t31
p82 t41
t4
t51
t5
p7 - d2M
p4
t61
p7 p6
p6
p9
p83
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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x2
- d3M x3
- d4M d4m d2m x1 x6 - d8M d8m d7m d5m d6m - d7M - d5M x4 x5 - d6M d3m
14
t-timed Petri nets => AOA p2 p1
t2
p3
t3
p4
t1
t6 p9 p8 p5
p1
t21
p81 p2
t11
p4
p82
p5
t61
p7 p6
t41
p6
t5
p7 y2
x2
t31
p3
t4
t51
p9
y3
d3
0
0
0 p83
0
d2
0
y1
x3
0
d6 x6
y6
d4 0 d5 y4 x5 y5
x4
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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t-time Petri nets p2 p1
t2
p3
t3
p4
t1
t6 p9
p1
t4
0
- d2M 0
0
d3m
- d4M d5m
d4m
- d5M
0
- d6M x6
d6m
y6
y1
0
- d2M 0
0
y5
d4m
IFAC World Congress 2002, Barcelona, Spain, 21-26 July 2002
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t5
d3m
p83
0
x6
- d4M d5m
0
t61
- d3M y 3
d2m
0
p4 p7
p6
t41
p7
- d3M y 3
d2m y1
t5
t31
p3
p82
p5
p6
0
p2
t11
p8 p5
t21
p81
- d5M
0
d6m
y6
- d6M
y5 16
p9
