Monitoring manufacturing systems by means of Petri nets with imprecise markings R. Valette∗ J. Cardoso∗∗ D. Dubois∗∗∗ LAAS-CNRS, 7 avenue du Colonel Roche F-31077 Toulouse cedex * LAAS-CNRS and UFSC, Florian´opolis ** LSI-UPS, 118 Route de Narbonne F-31062 Toulouse cedex Abstract—After having described the importance and the complexity of monitoring Flexible Manufacturing Systems, this paper shows the interest of introducing uncertainty and imprecision within Petri net based models. These two concepts are then introduced through a modification of the marking of a Petri net with objects, and through time handling. It is shown how fuzzy firing dates can be computed. Finally, some examples of possible uses are presented.
I.
I NTRODUCTION
Due to its complexity, Flexible Manufacturing System (FMS) Control is commonly decomposed into a hierarchy of the following abstraction levels: planning, scheduling, global coordination and real-time monitoring, sub-systems coordination and local control. Each level operates on a certain model of the manufacturing system. The upper level models are more aggregated but also more global. The decisions made at each level have to be a refinement of those made at the upper levels. At the same time, each level supervises the behavior of the level just below. This is done by checking that the current state of a given level is consistent with the update messages sent by the lower level. The representation of a Flexible Manufacturing System by means of Petri nets, in order to conveniently model resource allocations, implies the use of a kind of high level net. As A.I. techniques are already commonly used for FMS management [9] [8], it seems natural to employ Predicate/Transition nets [11] where each transition is considered as a rule with variables and each place as a set of entities verifying some predicate. These entities are either constants ou tuples of constants. The concept of objects belonging to a class and possessing a set of properties or attributes is also very convenient because the modeling of a shop floor is frequently based on the actual objects (parts, machines, tools, etc.) circulating in it. It is the reason why we have chosen to replace the variables of the Predicate/Transition net by objects rather than by constants i.e. to use Petri Nets with Objects (PNO) [16]. The Petri net structure (underlying condition/event net) is convenient to depict resource allocation mechanisms. A manufacturing policy (or flexible plan) can be defined by associating with each operation a time interval (earliest starting time, latest starting time) and a set of decision rules in order to solve the remaining conflicts in real-time. Firing a transition within this time interval corresponds to making a decision compatible with the shop floor state and with the manufacturing policy . A token that remains in a place associated with the input queue
of a machine (containing the parts waiting for an operation) after the planned time means a violation of this policy [1]. The supervising function is responsible for detecting any abnormal behavior of the physical part of the shop floor. Its requirements are twofold and contradictory: it has to be strict in order to avoid any fault propagation but it has also to be tolerant of human intervention in order to avoid floods of alarms in such situations. A Petri net description of a shop floor allows systematic strict supervision: each event corresponding to normal behavior has to be associated with an enabled transition. On the other hand, an event corresponding to a transition that cannot be fired in the current marking will activate an alarm. However, when the event is known to have a possible human origin, it is better to try to correctly update the shop state rather than produce alarms. In Petri Nets with objects, a token remaining in a place (that represents, for instance, a machining operation) more than the operation duration means that either the machine failed or that the foreseen operation duration was incorrect. The problem is consequently to be tolerant in specific cases and so to decompose the events in three classes: those corresponding to normal operation, the forbidden ones, the acceptable ones. For example, let us consider in a manufacturing shop an automatic guided vehicle stopped between two contacts (positions where the controller can detect it and send commands to it). After a human intervention supposed to solve the trouble either by restarting the vehicle or by dragging it to the maintenance station, the normal event is the arrival at the next contact, the forbidden ones are the arrival at any other contact, but an acceptable event is the arrival at the maintenance station. Another case in which it is necessary to have the notion of acceptable operations which differs from that of normal ones and of forbidden ones, concerns the control of manufacturing schedules. In case of schedule violation, in order to recompute a new schedule it is necessary to know if the incident results from a progressive drift (shift) or a machine failure. In the first case, a trace of all the decisions which have been at the margin of a schedule violation (acceptable decision) is required. As a matter of fact a progressive drift results from a bad (optimistic) evaluation of effective operation durations and rescheduling has to be done with updated durations. In order to make it possible to express the notion of acceptable events, a logic including the notion of imprecision or uncertainty is required. In the case of Petri nets, the requirements are that we should express the fact that the existence of an object modeled by a token is known but that its localization is imprecise (i.e. the location of the token must
be described in terms of a set of places). In other words, it is necessary to represent the fact that a certain sequence is being executed even though the precise step within the sequence is unknown. As it stems from the example given above (supervising the shop floor operation or the schedule), time considerations are essential. The events transforming an imprecise marking into another one are associated with fuzzy dates. So in this paper the notion of fuzzy-time Petri nets will be introduced, and the presence of a given token in a place will correspond to a fuzzy temporal window (fuzzy time interval). It is well known [2] [12] [5] that one way of simplifying a very complex system is to allow some degree of uncertainty in its description. Statements obtained from this simplified system are less precise but their relevance to the original system is fully maintained.
B. Imprecise marking. Traditionally, the marking of a Petri net is defined as the mapping M of the set of places P to the set N of natural numbers. In order to have a place in relation to any element of O∗ , we introduce a virtual place φ containing all the defined tuples of objects which are not in a place of the net and we define Pφ as P ∪ {φ}. When no tuple appears in a place with an arity greater than one, the marking can be defined as: M : O∗ x Pφ −→ {0, 1}
(1)
If M (o∗ , p) = 1, the tuple of objects o∗ is in the place p. When ∃ p1 , p2 , o∗ such that M (o∗ , p1 ) = M (o∗ , p2 ) = 1
II.
I MPRECISION WITHIN P ETRI N ETS .
A. Petri Nets with Objects. Let us now introduce the concept of Petri Net with Objects which is a class of Predicate/Transition nets as mentioned in the introduction. Its chief feature is that tokens are tuples of instances of classes or sub-classes of objects which lead to the association of classes or sub-classes with the variables attached to the arcs and with the places. The objects’ attributes may be involved in predicates associated with the transitions that act as extra firing conditions. They can also be modified by the execution of an action when a transition is applied (fired) to the corresponding objects. As the tokens are tuples of objects and not tuples of constants, it is possible to define a concept of ubiquity. Ubiquity appears when a given object belongs to more than one place or when a given place contains a given objet with an arity greater than one [SIB 85]. In our case, tokens are tuples of physical objects such as parts, machines or tools, and places are the predicates meaning that some objects are in a given state. For example, we might name two instances of the class P art parts p1 and p2 . Each instance represents a distinct object. As a given object cannot be in two different states, the Petri Net with objects describing the shop floor has to be without ubiquity. For example, object p1 cannot be found at a machine and be in inventory at the same time. In the next section, we shall show that ubiquity will be interpreted as imprecision about the object localization. Another way of defining non ubiquity is to say that a given object is available in a unique instance (exemplary) and is visible at a unique place. It has to be pointed out that the situation where an object appears in two different tuples contained in two different places corresponds also to ubiquity. In our case, ubiquity represents, in fact, the imprecision associated with the object localization. In this way, a marking where the object p1 is in the place that depicts the machining operation and in the place that depicts the inventory shows that we know that p1 is in one of the two sites, but we do not know in which one.
there is an ubiquity situation, ubiquity that we shall interpret as imprecision. When an abnormal event occurs, no part, tool or automated guided vehicle can suddenly disappear from the manufacturing workshop but it can happen that we are not sure of its localization. This will be expressed by imprecision attached to the occupancy of a token at a place. In order to represent this imprecision, we introduce a possibility distribution πo∗ [6] assigned to each tuple o∗ ∈ O∗ and defined on the set Pφ . o∗ ∈ O ∗
πo∗ : Pφ −→ {0, 1}
(2)
The set of all these functions describes the marking in a way which resembles (1). A fuzzy marking is defined by a set of possibility distributions πo∗ such that o∗ ∈ O ∗
πo∗ : Pφ −→ [0, 1]
(3)
Clearly, the possibility distribution assigned to an element of a tuple is the same as that of the tuple. If a given element appears in more than one tuple for a place, then its possibility will be the maximum value over the corresponding tuples. Similarly, we can also introduce possibility distributions πo of the objects which will be computed from that of the tuples: πo (p) = max{πo∗ (p), o ∈ o∗ } ∀p ∈ P We can then make the following statements: - πo (p) = 1 represents the fact that p is a possible localization of o, - πo (p) = 0 expresses the certainty that o is not present in place p. - It can happen that πo (p) = 1 and πo (p0 ) = 1 for p 6= p0 at a given time, i.e. object o may be in place p or place p0 . When we are certain that a tuple o∗ exists in the shop (πo∗ (φ) = 0) and when πo∗ (p) = 1 for a unique p ∈ P , then we are certain that the tuple o∗ belongs to p.
C. Marking Computation. We have just defined imprecise markings for a Petri net, now we are going to show how, each time an event occurs, a new marking can be computed. Classically, events are taken into account in a Petri net model by firing transitions. As explained already, an imprecise marking is a way of describing a set of all the possible markings at a given time, knowing that the actual state of the system corresponds to one and only one of these markings. The marking computation procedures executed after each event have to be consistent with this point of view. Let us consider, first, the simplest case: the occurrence of a precisely known event involving objects (or a tuple of object) whose localization is precisely known. This corresponds to the classical firing of a transition. It is the case of a normal operation of the system. Let us considerer now the case where the event is a deduction rather than the reception of a message. The actual state of the system may have changed or not. The corresponding transition will not be fired but rather pseudo-fired by adding the tokens in its output places without removing them from the input places. In doing so, imprecision is augmented in such a way that the new imprecise marking covers the possible former states and the possible new one. The last case concerns events, precisely known, corresponding to transitions which are enabled by tokens whose localization is not precisely known. Before firing the transition (normal firing), a new computation of the possibility distribution of these tokens is required in order to reach a precise localization. An algorithm is proposed in [4]. The basis of this computation is the calculation of a firing sequence leading from the last precisely known localization of the tokens to a new one which is such that the concerned transition is enabled. In doing so, the marking representing the actual state of the system will be consistent, i.e. will be an element of the set of reachable markings. An example concerning a FMS transport system is described in [4]. When a recomputation of the localization of an automated vehicle is done, the list of the free and occupied sections of the transport system is automatically updated.
continuous parameter [13] [15] [14], either for performance evaluation [10], or for formal verification [3]. There are two main approaches: Timed Petri Nets [15] and Time Petri Nets [13]. The classical example of Time Petri Nets has been defined by Merlin [13]. An interval [a, b] is associated with each transition, the enabling duration has to be greater than a and less than b. It is clearly a way of defining imprecise enabling duration. The aim of this paper being the introduction of timing consideration as well as imprecision and uncertainty, the Time Petri Net [13] is a solution. Instead of an interval [a, b] used to represent an imprecise enabling duration, we simply use a fuzzy interval [a1 , a2 , b1 , b2 ] meaning that the normal value for the duration is an element of [a2 , b2 ] while an element of [a1 , b1 ] is not excluded. Another difference with Merlin’s approach is that we work with Petri nets with objects. The issue is to determine if a transition will be fired and to compute the firing time, for a given tuple of objects. The enabling duration is therefore associated with a transition and a tuple of objects. This duration may be the result of a computation valuing the attributes of these objects. B. Fuzzy date, fuzzy interval. A date a has only one value, which may be ill-known. Therefore, the fuzzy set of the possible values of an ill-known date is a disjunctive set [17]. A time interval [a, b] is a period of time between two dates a and b. A time interval [a, b] for which a and b are fuzzy dates is consequently defined by two fuzzy sets, A and B representing these dates. The set of dates after A and before B is a conjunctive set. Let l be the length of the interval [a, b]. L denotes the fuzzy set which restricts the possible values of l, then: L = B A where denotes the extended substraction. If A and B are depicted by trapezoidal functions (see figure 6) represented by the quadruples (a, a∗ , a∗ , a) and (b, b∗ , b∗ , b) respectively, then L is depicted by the quadruple [7]: (l, l∗ , l∗ , l) = (b − a, b∗ − a∗ , b∗ − a∗ , b − a)
III.
F UZZY- TIME P ETRI N ETS .
As stated above, imprecision has to be increased when a possible state change is deduced and decreased when a message is received from the actual system. Generally, the reasoning about possible state changes is based on temporal considerations such as: after the normal duration of the operation, the event end-of-operation occurs. The aim of this section is to explicitly introduce time at the level of the Petri net with Objects as a framework for handling fuzzy marking computation. A. Time and Petri Net. A Petri net describes causality (an event is a consequence of another one), decision and independence. It is a way of describing a partial order among a set of events, i.e. introducing some implicit timing considerations. Much work has been done in order to make the time explicit as a quantifiable and
Let us consider the figure 1.a. A and B describe the fuzzy dates a and b. A fuzzy date can be associated with the firing of a transition for a tuple of tokens. If for a given token o∗ , a is associated with t1 and b with t2 in figure 1.b, then L = B A describes the fuzzy enabling duration of transition t2 . When the date a is associated with an event which certainly occurred, occurs or will occur, then the core of the fuzzy set A is not empty. So, if the core of A is empty, it implies that the event will not necessarily occur. Let us consider the Petri net in figure 2.a, for a given tuple of objects in p1 , either transition t2 or (exclusive or) transition t3 may fire. The core of one the fuzzy set B ∪ C is not empty. However if the sequence a; b is the normal operation and c describes a possible abnormal event, then the core of C will be empty (see figure 2.b).
C. Propagation of fuzzy temporal constraints. The purpose of reasoning with imprecision is to be able, in some situations, to diminish this imprecision in order to make some conclusions. For example, let us consider the case described in figure 1 supposing that transition t1 is associated with the event beginning-of-operation and transition t2 with the event endof-operation of a part on a machine. Before the actual date of the operation the planning and scheduling level may provide a fuzzy date A for the firing of t1 for a given part o∗ , a fuzzy date B for t2 and a fuzzy duration L = B A. As soon as the operation actually begins at date τ for the part, A is reevaluated and takes the form of the singleton {τ }, then B can be recomputed as B = {τ } ⊕ L. The interest of using the Petri net formalism is that it represents a partial order between the events and so a complex reasoning is possible. Let us consider some examples. In figure 3, let us suppose that the firing date of t1 for < o1 > is represented by Ao1 (πa,o1 ) and that the firing date of t2 for < o2 > is represented by Bo2 (πb,o2 ) . This may correspond to the duration of operations associated with p1 and p2 assuming that these operations have begun exactly at time zero. The possibility distribution of the presence of < o1 > in place p3 is [Ao1 , +∞) which is the conjunctive set of dates which are possibly after the date a. That of the presence of < o2 > in place p4 is [Bo2 , +∞). As a condition for firing transition t3 is the presence of a token in p3 and p4 , a representation of the firing date c of transition t3 for the tuple < o1 , o2 > is C = ∩ ([Ao1 , +∞), [Bo2 , +∞)). It corresponds to a concept of earliest firing time. This is illustrated in figure 4. Let us now refine the evaluation of the firing date c. Suppose for example that t3 fires with no delay (as soon as possible when enabled). Then it is impossible to have simultaneously a token in p3 and one in p4 . This piece of information allows us to compute something similar to a latest firing time. The necessity of the presence of < o1 > in place p3 is ]Ao1 , ∞) which is the conjunctive set of dates which are necessarily after the date a. That of the presence of < o2 > in place p4 is ]Bo2 , ∞). The necessity of the presence of < o1 > in place p3 and, simultaneously, of < o2 > in p4 is D> = ]Ao1 , +∞) ∩ ]Bo2 , +∞) (see figure 5). It is exactly the set of dates which are necessarily strictly after the firing date of transition t3 . ¯ From the well known formulae [6], Π(A) = 1 − N (A) where Π is the possibility for a set A and N the necessity ¯ we can compute the set of dates for the complementary set A, which are possible before the firing date of transition t3 , D< . Co1 ,o2 can be interpreted as the set of dates possibly after the firing of t3 and clearly the firing date with no delay, Co0 1 ,o2 of t3 is given by the intersection of Co1 ,o2 with D< (see figure 5). D. Fuzzy marking. In section II-B, imprecise and fuzzy markings were defined by (2) and (3), i.e. by a possibility distribution πo∗ attached
to each tuple of objects o∗ and associating with each place a possibility value. Clearly, these possibility distributions are defined at a given time. Now we shall introduce time explicitly. As stated before, we use the definition of Time Petri Nets [13] with the only modification that the enabling durations associated to the transitions are defined by fuzzy intervals. As we are operating with Petri net with objects, the fuzzy enabling duration may be the result of a computation involving the attributes of the tokens enabling the transition. It must be pointed out that our work is not concerned with analysis of net properties. We are concerned with the real-time control and monitoring of manufacturing systems. A fuzzy marking is a way of describing a set of possible markings which all are reachable markings of the underlying Petri net. A given place p contains a token o∗ between the two following events: the firing of one of its input transitions for o∗ , for example firing transition t1 at date a, and the firing of one of its output transition for o∗ , for example firing transition t2 at date b. This is the time interval [a, b], a fuzzy one when at least one of these data is ill-known. As this interval is conjunctive, it is not described by a possibility distribution but it can be represented by a membership function µo∗ ,p associated with each pair (o∗ , p), i.e. with the set of times for which the token o∗ belongs to place p. A way of defining µ is the following one: µo∗ ,p (t) = πo∗ (p) at time t. The computation of fuzzy markings and fuzzy firing dates are not performed for all the times greater than the current time. It would be clumsy and very time-consuming. A fuzzy firing date is only computed when the current time belongs to the support of the fuzzy date. Consequently, the computation is incremental and can be performed in real-time. The computation procedure has been partially described in the preceding section and is the following one: 1) 2) 3)
4)
5) 6)
7) 8)
start a new computation when the current time is equal to the smallest time of the support of a firing date, suppose it is transition t for token o∗ , deduce the time interval possible after this fuzzy date, it describes when o∗ appears in the output places of t, consider each transition following one of the output places and build all the tuples of tokens enabling them at the current time (only tokens whose possibility distribution differs from zero are considered), compute the time interval possibly following the date from which these transitions can be fired, (similar to an earliest firing time in the case of a firing with no delay), compute the time interval necessary strictly after the fuzzy date of the firing of t for o∗ , compute the time interval necessary strictly after the firing of the transitions following the output places of t, (similar to latest firing time in the case of a firing with no delay), deduce the fuzzy firing dates of these transitions in the case of a no delay firing, add the fuzzy delay (if it exists) to each firing date,
9)
compute the new marking of all the output places of t,
This procedure is partially illustrated in figure 5. IV.
U SE FOR INTELLIGENT CONTROL AND MONITORING .
The aim of this section is to show that fuzzy time Petri net can be applied within the context of FMS control. The first application concerns monitoring the shop floor. It means that the sequences of events occurring in it are checked in real-time in order to verify that they correspond to a normal behavior which has been previously defined. For example, let us consider a transport system based on automated guided vehicles. The localizations of the vehicles are only known when the vehicles pass by contacts. When a vehicle, moving along a section, does not reach the contact within a given time interval, it means that some trouble occurred. In figure 7.a, the fuzzy date associated with the event arrival at the contact is described. At time a2 , the human operator is called, in order to see what is happening. If he can restart the vehicle, the latter will reach the contact later, and the monitoring system will be reset automatically. It is why the fuzzy date expresses the fact that the arrival at the contact is acceptable until time a3 . However, it may happen that the battery has gone flat and that the vehicle has to be moved to the maintenance station. The fuzzy date of the event arrival at the maintenance station is given in figure 7.b. During the interval [a2 , a3 ] the two events are possible. After a3 , only the arrival at the maintenance station remains acceptable. When one of these two events occurs, markings and firing dates are recomputed and the state representation is reset automatically by the monitoring instead of sounding new alarms. The second appilcation concerns monitoring the manufacturing schedule. It is well known that a rigid schedule cannot be respected in a Flexible Manufacturing system. In fact, incidents and unforeseen events occur frequently. A solution is to permit some flexibility in order to make real-time decisions which are consistent with the schedule, for example, instead of allocating a precise date for the beginning of the operations, we may use an imprecise date. This is the case when the interval defined by the earliest starting time and the due date is greater than the duration of the operation. A drawback of this approach is that when the current time becomes greater than the latest starting time (due date minus operation duration), the system suddenly goes into an abnormal state, re-scheduling is required. No aid is provided by the monitoring module for a diagnosis. If we choose to associate a fuzzy date with the beginning of the operation, then, if the current time belongs to the core, it is the normal operation whereas a fault occurs when the current time is necessarily greater than the fuzzy date. In figure 6, t1 corresponds to a normal operation and t2 to a fault. But two new situations may occur now. The first case corresponds to time t3. It is only an acceptable date for the operation beginning. Re-scheduling is not required but this event can be interpreted as a pessimistic scheduling. Either the machine load, or the required time to machine the part before this operation have been over-estimated. This kind of information may be fed back at the scheduling level, as a kind of learning process, to improve its knowledge about the behavior of the workshop. The other situation corresponds to
the case of time t4 in figure 6. It is the case of an optimistic schedule. When the cause of the schedule violation is a shift rather than a machine failure, the diagnosis is quite difficult. By logging these situations it will be possible to know if the shift results from a saturated machine or from a critical operation sequence concerning a part. Once the machining operation has begun, it corresponds now to a precise wellknown date. Generally, machining durations are on the contrary ill-known because they depend upon the degree of use of the tools. Consequently, the result of the computation of the end-of-operation date is a fuzzy date. In this case also, it is useful to have the notion of normal date, abnormal date or acceptable date. The case of an acceptable date signifies that the evaluation of the operation duration was not very good. Once more this information can be fed back to the scheduling and planning levels. V.
C ONCLUSION
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam lobortis facilisis sem. Nullam nec mi et neque pharetra sollicitudin. Praesent imperdiet mi nec ante. Donec ullamcorper, felis non sodales commodo, lectus velit ultrices augue, a dignissim nibh lectus placerat pede. Vivamus nunc nunc, molestie ut, ultricies vel, semper in, velit. Ut porttitor. Praesent in sapien. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Duis fringilla tristique neque. Sed interdum libero ut metus. Pellentesque placerat. Nam rutrum augue a leo. Morbi sed elit sit amet ante lobortis sollicitudin. Praesent blandit blandit mauris. Praesent lectus tellus, aliquet aliquam, luctus a, egestas a, turpis. Mauris lacinia lorem sit amet ipsum. Nunc quis urna dictum turpis accumsan semper. ACKNOWLEDGMENT This work has been partially supported by G.I.P. PROMIP (Midi Pyr´en´ees research group in production systems). The second author would like to acknowledge the financial support of CAPES-Brazil. The authors would also like to thank Paul Freedman for his helpful comments. R EFERENCES [1]
[2]
[3] [4]
[5]
[6] [7] [8]
H. Atabakhche. Utilisation conjointe de l’intelligence artificielle et des r´eseaux de Petri: Application au contrˆole d’ex´ecution d’un plan de fabrication. PhD thesis, Universit´e Paul Sabatier, Toulouse, 1987. R. E. Bellman and L. A. Zadeh. Decision-making in a fuzzy environment. In R. R. Yager, editor, Fuzzy sets and applications, John Wiley and Sons, 1988. B. Berthomieu and M. Menasche. An enumerative approach for analysing time petri nets. Information processing, 41–46, 1983. J. Cardoso and R. Valette. Petri nets with uncertain markings. In 10th Internacional Conference on Applications and Theory of Petri nets, Germany, june 1989. D. Dubois and H. Prade. An introduction to possibilistic and fuzzy logics. In P. Smets, editor, Non-Standard logics for automated reasoning, Academic Press, 1988. D. Dubois and H. Prade. Possibility Theory: an approach to computerized processing of uncertainty. Plenum Press, 1988. D. Dubois and H. Prade. Processing fuzzy temporal knowledge. IEEE Trans. on Systems, Man and Cybernetics, 14, 1989. n 4. J. Erschler and P. Esquirol. Decision-aid in job-shop scheduling: a knowledge based approach. In IEEE International Conference on Robotics and automation, San Francisco, april 1986.
[9]
[10]
[11] [12] [13] [14]
[15]
[16] [17]
E. Bensana et all. An expert system approach to industrial jobshop scheduling. In IEEE International Conference on Robotics and automation, San Francisco, april 1986. P. Freedman and A. Malowany. SAGE: A Decision Support System for the Optimization of Repetitive Workcell Sequencing Problems. Report No TR-CIM-88-16, McGill University, may 1988. H.J. Genrich. Predicate/Transition Nets. Technical Report, Institut fur methodische Grundlagen, may 1987. G.J. Klir and T.A. Folger. Fuzzy sets uncertainty and information. Prentice-Hall, 1988. P. Merlin. A Study of the recoverability of Computer Systems. PhD thesis, University of California, Irvine, 1974. M. Molloy. On the integration of delay and throughput measures in distributed processing models. Report No CSD-810021, University of California, september 1981. C. Ramchandani. Analysis of Asynchronous Concurrent Systems by Timed Petri Nets. project MAC TR 120, Massachussets Institute of Technology, feb 1974. C. Sibertin-Blanc. High-level petri nets with data structures. In Workshop on Applications and Theory of Petri nets, Finland, june 1985. R. R. Yager. On different classes of linguistic variables defined via fuzzy subsets. Kibernetes, 13:103–110, 1984.
Fig. 1.
The fuzzy enabling duration of a transition.
Fig. 2.
Conflicts between two events.
Fig. 3.
Fuzzy temporal constraints.
Fig. 4.
Possibility distribution of date c.
Fig. 5. Refined possibility distribution of date c. a) set of dates after A. b) set of dates after B. c) the latest firing time. d) set of dates before the firing date. e) the earliest firing time. f) the firing date refined.
Fig. 6.
A fuzzy date for the beginning of an operation.
Fig. 7. Representation of fuzzy dates. a) arrival at the contact. b) arrival at the maintenance station.
