[Studies in Fuzziness and Soft Computing] Production Engineering and Management under Fuzziness Volume 252 || Manufacturing System Modeling Using Petri Nets

C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 95–124. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

Chapter 6 Manufacturing System Modeling Using Petri Nets

Cengiz Kahraman and Fatih Tüysüz*

Abstract. Petri nets which are used for modeling and analyzing complex systems that can be characterized as synchronous, parallel, simultaneous, distributed, re-source sharing, nondeterministic and/or stochastic form a powerful modeling tool and are widely used today. In this study, fundamental concepts of Petri nets and their extensions are presented. Since the application area of Petri nets is wide, the subject is handled in the view of flexible manufacturing systems. A two stage modeling approach which combines the modeling power of stochastic Petri nets together with fuzzy sets is also presented. A numerical example is given to present how the proposed approach can be applied. We believe that this approach better represents both dimensions of uncertainty, stochastic variability and imprecision, in system modeling.

1 Introduction

Petri nets (PNs) introduced by Petri (1962) are a graphical and mathematical model-ing tool applicable to many systems. PNs can be used for modeling and analyzing complex systems which can be characterized as synchronous, parallel, simultaneous, distributed, resource sharing, nondeterministic and/or stochastic (Murata 1989, Bob-bio 1990, Zhou and Venkatesh 1999, Marsan et al. 1995). These systems require both qualitative and quantitative aspects to be considered for modeling and analysis. Qualitative analysis searches for structural properties like the absence of deadlocks, the absence of overflows or the presence of certain mutual exclusions in case of re-source sharing. Quantitative analysis looks for performance properties such as throughput, average completion times, average queue lengths or utilization rates. Quantitative analysis concerns the evaluation of the efficiency of the modeled sys-tem whereas qualitative analysis concerns the effectiveness of the modeled system.

The complex systems of these types exhibit characteristics which are difficult to describe mathematically using conventional tools like differential equations, Cengiz Kahraman Department of Industrial Engineering, Istanbul Technical University, Macka 34367, Istanbul, Turkey

Fatih Tüysüz T.C. Beykent University, Department of Industrial Engineering, Ayazağa, İstanbul, Turkey

96 C. Kahraman and F. Tüysüz

difference equations (Murata 1989, Jeng 1997b). On the other hand, Petri nets as a mathematical tool provide obtaining state equations describing system behavior, finding algebraic results and developing other mathematical models. With respect to other techniques of graphical system representation like block diagrams or logi-cal trees, Petri nets are particularly more suited to represent in a natural way logi-cal interactions among parts or activities in a system (Bobbio 1990). In modeling point of view, Petri net theory allows the construction of the models amenable both for the effectiveness and efficiency analysis (DiCesare et al. 1993).

By the beginning of the 1980’s, PNs have begun to be used for modeling and analysis of manufacturing systems. Today there is a great and growing literature related to PNs in the area of manufacturing systems including the major topics of modeling, analysis, performance evaluation, scheduling, planning and control. A flexible manufacturing system (FMS) is a discrete-event system and contains a set of versatile machines, an automatic transportation system, a decision-making sys-tem, multiple concurrent flows of job processes that make different products, and often exploits shared resources to reduce the production cost (Zuberek and Kubiak 1994, Jeng 1997a). Due to the graphical nature, ability to describe static and dy-namic system characteristics and system uncertainty, and the presence of mathe-matical analysis techniques, Petri nets as a powerful tool, form an appropriate conceptual infrastructure for the modeling and analysis of FMSs.

In modeling and analysis of systems, one of the most important subjects that must be taken into consideration is the uncertainty due to the reason that input and model parameters are usually in the form of uncertain parameters. Although the most commonly used approach for representing uncertainty in system modeling is the stochastic models that are based on probability theory, the stochastic models are not suitable to describe all dimensions of the uncertainty. Especially the im-precision of data which is for example as a result of the limited precision of meas-uring is not statistical in nature and can not be described by using probability (Viertl and Hareter 2004). Fuzzy set theory is well suitable for modeling and analysis of systems that include uncertainty and imprecision. Due to these reasons, by the beginning of 1990’s one of the major research areas of PN theory has been the fuzzy PNs which is also the main focus of this study.

Since PNs have a wide area of applications, for the rest of the chapter the sub-ject is handled in the view of flexible manufacturing systems. The organization of the chapter is as follows. In Section 2, literature review about PNs and applica-tions are given. In Section 3, formal definition of PNs, their properties and analy-sis are presented. In Section 4, modeling of manufacturing systems with PNs and in Section 5 time concepts and stochasticity in PNs are given. In Section 6, fuzzi-ness in PNs is explained and a new fuzzy PN approach based on stochastic analy-sis is given in detail. Finally the conclusions are presented.

2 Literature Review

In this section literature review about PNs and their extensions mainly in the ap-plication area of manufacturing systems are summarized. For simplicity the litera-ture review is presesented in a tabular form which can be seen in Table 6.1.

Manufacturing System Modeling Using Petri Nets 97

Table 6.1 Literature related to PNs and their extensions

Type of PN Ap-proach

Application Area Contribution

Classical Petri Nets And Extensions

Valette et al. (1982)

classical PNs flexible car produc-tion system modeling

first application of PNs for modeling FMSs

Alla et al. (1985)

colored PNs

modeling the same flexible car produc-tion system of Valette et al. (1982)

possible benefits of using colored PNs over ordinary PNs

Narahari and Viswanadham (1985)

classical PNs manufacturing sys-tem modeling

significance of boundedness, liveness and reversibility in manufacturing sys-tems was presented

Barad and Sip-per (1988)

classical PNs FMS modeling using PNs for comparing manufactur-ing systems on the basis of flexibility

Valavanis (1990)

extended PNs FMS modeling and simulating

introduction of extended PNs

Zhou and DiCesare (1991, 1992, 1993)

classical PNs modeling a FMS with resource sharing

hybrid synthesis approach was pro-posed for sophisticated manufacturing systems

Zhou et al. (1993)

deterministic timed PNs

modeling a FMS cellcomputation of the cycle time of the cell by using deterministic timed PNs.

Shiizuka and Suzuki (1994)

colored PNs modeling AGV net-works in FMSs

applicability of PNs in modeling AGV networks

Kiritsis and Porchet (1996)

classical PNs process planning for manufacturing sys-tems

a three stage approach for process planning using PNs was proposed

Jeng (1997b) classical PNs

modeling shared-resource automated manufacturing sys-tems

an efficent PN synthesis approach based on bottom-up and modular composition was presented

Wang and Wu (1998)

colored time ob-ject oriented PNs

modeling automated manufacturing sys-tems

a new class of PNs called colored time object oriented PNs was introduced

Zimmerman et al. (2001)

classical PNs manufacturing sys-tem modeling

a two phase optimization method for the PN models of manufacturing sys-tems was proposed

Lefebvre (2001)

discrete and continuous PNs

estimation of the fir-ing frequencies in discrete and continu-ous PN models in production systems

a method for the estimation of the fir-ing frequencies of timed PN models was presented

Abdallah et al. (2002)

classical PNs scheduling manufac-turing systems

a PN based scheduling algorithm was proposed

Chen and Chen (2003)

colored PNS performance model-ing and evaluation of FMSs

an object oriented method for per-formance evaluation besed on colored PNs was presented


Table 6.1 (continued)

Uzam (2002, 2004)

classical PNs obtaining an optimal deadlock prevention policy for FMSs.

a PN reduction approach was pro-posed

Wang et al. (2005)

hybrid PNs

modeling networked manufacturing sys-tems and control sys-tem architecture

a hybrid PN approach which includes both discrete and continuous system variables was presented

Stochastic Petri Nets And Extensions

Molloy (1982) stochastic PNs stochastic PN theory the first introduction of the stochastic PNs

Marsan et al. (1984)

generalized sto-chastic PNs

stochastic PN theory a new class of stochastic PNs called generalized stochastic PNs were in-troduced

Balbo et al. (1988)


modeling for the so-lution of complex models of system be-havior

a hierarchical modeling approach that combines queueing network models and generalized SPNs was presented

Zhou et al. (1990)

stochastic PNs

performance evalua-tion of deadlock-free manufacturing sys-tems

stochastic PN modeling based on top-down and bottom-up approaches was presented

Al-Jaar and Desrochers (1990)


performance evalua-tion of automated manufacturing sys-tems

advantages of using generalized SPNs for modeling and analysis of complex manufacturing systems

Choi et al. (1993)

determinitic and stochastic PNs

performance evalua-tion of complex sys-tems

a numerical method for transient analysis based on numerical inversion of Laplace-Stieltjes transforms was in-troduced

German (1995)determinitic and stochastic PNs

determinitic and sto-chastic PNs theory

a numerical method for transient and stationary analysis of deterministic and stochastic PNs based on the method of supplementary variables was developed

Koriem and Patnaik (1997)

generalized sto-chastic high-level PNs

performance evalua-tion of parallel and distributed systems

a new class of generalized stochastic PNs was presented

Yan et al. (1998)

extended sto-chastic high-level evaluation PNs

manufacturing sys-tem modeling and analysis

a new class of stochastic PNs was pre-sented

Linderman and Thümmler (1999)

determinitic and stochastic PNs

determinitic and sto-chastic PNs analysis theory

a numerical algorithm for transient analysis of deterministic and stochas-tic PNS was introduced

Zimmerman and Hommel (1999)

colored stochas-tic PNs

manufacturing sys-tem modeling

a new method based on colored PNs for manufacturing system modeling was presented


Table 6.1 (continued)

Buchholz (2004)

stochastic PNs numerical analysis of stochastic PNs

a new approximate solution technique for the numerical analysis of SPNs was presented

Chen et al. (2005)

batch determi-nistic and sto-chastic PNs

modeling and analy-sis of inventory sys-tems and real-life supply chains

a new class of stochastic PNs with batch places and batch tokens was presented

Fuzzy Petri Nets

Valette et al. (1989)

fuzzy-time PNs fuzzy logic based PN theory

a fuzzy PN approach based on the as-sociation of fuzzy enabling duration with the transition which results in attaching a fuzzy firing date to the transition was introduced

Murata (1996) fuzzy-timing high-level PNs

simulation of com-munication protocols

a fuzzy PN approach based on possi-bility theory which includes four fuzzy theoretic functions of time was presented

Yeung et al. (1996)

fuzzy coloured PNs

modeling of a printed circuit board produc-tion system

a new class of fuzzy PNs was pre-sented

Pedrycz ve Camargo (2003)

fuzzy timed PNs fuzzy logic based PN theory

a new version of fuzzy PNs which in-corporates time with the interval and fuzzy set-based models of temporal relationship was introduced

Ding et al. (2005, 2006)

fuzzy timed PNs fuzzy logic based PN theory

a new fuzzy timed PN model in which each transition firing is associated with a fuzzy number was presented

Table 6.1 presents some of the most important studies in PN theory. More re-cent papers including different applications of PNs mainly in the field of manufac-turing are; modeling and analysis of manufacturing systems by using process nets with resources (Jeng 2002), reliability design of industrial plants using PNs (Ber-tolini et al. 2006), scheduling of FMSs with timed PNs (Zuberek and Kubiak 1999, Lee and Korbaa 2006, Kim et al. 2007, Hsu et al. 2008), modeling and scheduling of FMSs with PNs (Zha et al. 2002, Korbaa et al. 2003, Huang et al. 2008), reduced PN models of discrete manufacturing systems (Rangel et al. 2005), property-preserving subnet reductions for designing manufacturing systems with shared resources (Huang et al. 2005), PN approach for error recovery in manufac-turing systems control (Odrey and Mejia 2005), PN modeling of discrete event dynamic systems (Koriem et al. 2004), Modelling and simulation of a bottling plant using hybrid Petri nets (Giua et al. 2005), PN based object-oriented model-ling of hybrid and complex productive systems (Villani et al. 2005, Liu et al. 2005), structuring and composition in PN models (Gomes 2005), hybrid PN and digraph approach for deadlock prevention in automated manufacturing systems (Maione and DiCesare 2005), fuzzy PN modeling of FMSs (Venkateswaran and


Bhat 2006), modeling, analysis and control of hybrid dynamic systems (Lefebvre et al. 2007), fuzzy PN modeling of intelligent databases (Korpeoglu and Yazici 2007), dynamic fuzzy PNs for course generation (Huang et al. 2008), scheduling by using timed PNs (Kim et al. 2007, Ghaeli et al. 2008) , cyclic scheduling of FMSs by using PNs and genetic algorithm (Hsu et al. 2008), resource-oriented PN for deadlock avoidance in flexible assembly systems (Wu et al. 2008), multipara-digm modeling of hybrid dynamic systems using PNs (Lee et al. 2008), reachabil-ity and state space analysis of PNs (Praveen and Lodaya 2008, Reinhardt 2008, Fronk and Kehden 2009), modeling and scheduling of job shops by timed PNs (Zhang and Gu 2009).

3 Definition, Properties and Analysis of Petri Nets

3.1 Formal Definition of PNs

The formal definition of PNs introduced by Petri (1962) is as follows. A marked Petri net (PN) ),,,,( mOITPZ = is a five-tuple where

1. { } 0,,...,, 21 >= npppP n , is a finite set of places pictured by circles

2. { } 0,,...,, 21 >= stttT n , is a finite set of transitions pictured by bars, with

∅≠∪TP and ∅=∩TP 3. NTPI →×: , is an input function that defines the set of directed arcs from P

to T where { },...2,1,0=N

4. NPTO →×: , is an output function that defines the set of directed arcs from T to P

5. NPm →: , is a marking whose ith component represents the number of tokens in the ith place. An initial marking is denoted by 0m . The tokens are pictured

by dots.

A marked PN and its elements are shown in Figure 6.1. The four-tuple ),,,( OITP

is called a PN structure that defines a directed graph structure. A PN models sys-tem dynamics using tokens and their firing rules. Introducing tokens into places and their flow through transitions make it possible to describe and study the dis-crete-event dynamic behavior of the PN.

Fig. 6.1 A marked PN and its elements


Table 6.2 Interpretations of the PN elements (Zhou and Jeng 1998)

PN Elements Interpretation

Places Resource status, operations and conditions

Transitions Operations, processes, activities and events

Directed Arcs Material, resource, information, and control flow direction

Table 6.3 Manufacturing concepts representation in PN models (Zhou and Jeng 1998)

Manufacturing Concepts PN Modeling

Moving or production lot size

Weight of directed arcs modeling moving

Number of resources, e.g. AGVs, machines.

The number of tokens in places modeling quantity of the corresponding resources

Capacity of a workstation The number of tokens in places modeling of its availability

Work-in-process The number of tokens in places modeling the buffers and operations of all machines

Production volume The number of the tokens in places modeling the counter foror the number of firings of transitions modeling end of a product

Time of an operation, e.g. setup, processing.

Time delays associated with the place or transitions model-ing the operation

Conveyance or transpor-tation time

Time delays associated with the directed arc, place, or tran-sition modeling the conveyance or transportation

System state PN marking (plus the timing information for timed PN)

The interpretations of the PN elements are given in Table 6.2. Table 6.3 pre-sents how the PN models can represent the concepts in manufacturing.

The behavior of a system is described in terms of the system states and their changes. The system state in a PN model is defined as a marking. Everytime a transition fires a new marking occurs which defines the new system state. A state or a marking in a PN is changed according to the execution rules which include enabling and firing rules.

• Enabling Rule: a transition Tt ∈ is enabled if and only if ),()( tpIpm ≥ ,

Pp ∈∀

• Firing Rule: enabled in a marking m, t fires and results in a new marking ,m

which is described as PptpOtpIpmpm ∈∀+−= ),,(),()()(, .

Marking ,m is said to be immediately reachable from marking m. The enabling rule says that if all the input places of transition t have at least the required number of tokens, then t is enabled. The firing rule states that an enabled transition t re-moves ),( tpw tokens from each input place p of t, and adds ),( ptw tokens to


Fig. 6.2 (a) 1t is enabled (b) enabled transition 1t fires

each output place p of t, where ),( ptw is the weight of the arc from t to p. The

execution rule is depicted in Figure 6.2.

3.2 Properties of PNs

PNs have some important fundamental properties which allows to identify the presence or absence of the functional properties of the modeled system. They can be classified into two groups, behavioral properties and structural properties. Be-havioral properties are the ones which depend on the initial marking, whereas structural properties are the ones which do not depend on the initial marking of a PN. The most important properties are explained below.

• Reachability: given a PN ),,,,( 0mOITPZ = , marking m is reachable from

marking m0 if there exists a sequence of transition firings that changes m0 to m. Reachability which is a behavioral property is a fundamental basis for analyz-ing the dynamic properties of any system. In terms of manufacturing systems, this property helps us to analyze whether the system can reach a specific state or perform a functional behavior.

• Boundedness: given a PN ),,,,( 0mOITPZ = and its reachability set R, a place

Pp ∈ is k-bounded if RmBpm ∈∀≤ ,)( . In other words, a PN is said to be k-

bounded if the number of the tokens in each place does not exceed a finite number k for any marking reacheable from m0. A PN is said to be safe if it is 1-bounded. Places are usually used to represent storage areas, tools, pallets and AGVs in manufacturing systems. The concept of boundedness of a PN is used to identify the existence of overflows for these resources.

• Liveness: a transition t is live if at any marking Rm ∈ , there exists a sequence of transitions of which firing reaches a marking enabling t. A PN is live if all its transitions are live. A transition t is dead if at any marking, such that there ex-ists no sequence of transition firings to enable t starting from m. Liveness is an important property for many practical systems since the concept of liveness is related to the absence of deadlocks. The deadlocks occur as a result of inappro-priate resource utilization. In manufacturing systems, these resources can be


machines, robots, AGVs, storage areas or buffers. Therefore, if a PN is live, then there is no deadlock related to such resources.

• Conservativeness: a PN ),,,,( 0mOITPZ = is conservative with respect to a

vector w if there is a vector ),...,,( 21 nwwww = and niwi ,...,2,1,0 => , such that

Rmmwmw ∈∀= ,0ττ where R is the reachability set. Conservativeness states

that the weighted sum of tokens remains the same for each marking. In real life systems, the number or amount of resources are limited. Since tokens are usu-ally used to represent resources which are constant, then the number of the to-kens in a PN model remains unchanged without regard to the marking change.

• Reversibility and Home state: a PN is said to be reversible if for each marking )( 0mRm ∈ , m0 is reachable from m. Therefore, in a reversible PN it is always

possible to return back to the initial marking or state. This property is important in error recovery in manufacturing systems. If a net contains a deadlock, then the net is not reversible.

The PN properties and their meanings in manufacturing systems are summarized in Table 6.4.

Table 6.4 PN properties and their meanings in manufacturing systems (Zhou and Venkatesh 1999)

PA Properties Meanings in Manufacturing Systems

Reachability A certain state can be reached from the initial conditions

Boundedness No capacity such as buffer, storage and workstation overflow

Safeness Availability of single resource or no request to start an ongoing process

Liveness Freedom from deadlock and guarantee the possibility of a modeled event, operation, process or activity to be ongoing

Conservativeness Conservation of nonconsumable resources such as machines, equipments and AGVs

Reversibility Reinitialization and cyclic behavior

3.3 Analysis of PNs

There are two main analysis methods for PNs which are reachability analysis and invariant analysis. The first method is the fundamental analysis method for PNs and can be applied to all classes of PNs. Although invariant analysis method is powerful, it does not include all the information of a general PN. It can be applied only to special subclasses of PNs or special situations, in this study invariant analysis method is omitted and the reachability analysis method is handled. Fur-ther details about invariant method can be found in Peterson (1981), Murata (1989).

In reachability analysis method, starting from the initial state of the system, it is aimed at deriving all the possible states that the system can reach together with their relationships. The obtained reachability graph enable us to analyze all the


Fig. 6.3 A PN model and its reachability graph

behavioral properties if the number of the states or markings is finite. In reachabil-ity graph the nodes represent markings obtained from initial marking 0m and its

successors, and each arc represents a transition firing which changes one marking to another.

Figure 6.3 shows a finite state PN model and its reachability graph which repre-sents all the possible markings enumerated. It can be easily seen that the system represented by this PN model is bounded, live and reversible.

4 Modeling of Manufacturing Systems Using PNs

Modern manufacturing systems are complex and automation intense systems which include many elements and relations among them. Modeling such systems requires taking into accounts not only the elements but also their relations. The ba-sic relations between the processes and operations can be classified as follows (Zhou and Robbi 1994):

• Sequential: when one operation follows another, the places and transitions rep-resenting them form a sequential relation in PNs.

• Concurrent: when two or more operations are initiated by an event, they form a parallel structure starting with the same transition. Such a situation can be mod-eled as sequentially connected series of places/transitions in which multiple places are marked simultaneously or multiple transitions are enabled at certain markings

• Conflicting: if either of two operations follows an operation, then the transi-tions form two outputs from the same place.


• Cyclic: when a sequence of operations follows one after another and comple-tion of the last one launches the first one, then the cyclic structure between these operations is formed.

• Mutually Exclusive: mutual exclusion between two processes occurs if they cannot be performed at the same time because of the limited usage of a shared resource.

These are the most commonly encountered relations in manufacturing systems. The representation of these basic relations in PN models are given in Figure 6.4.

Fig. 6.4 PN representation examples of basic relations: (a) Sequential (b) Concurrent (c) Conflicting (d) Cyclic (e) Mutually Exclusive

In modeling flexible manufacturing systems by using PNs, some commonly used PN modules can be very helpful in modeling process. These modules can be summarized as follows (Zhou and Jeng 1998):

1. Resource/operation module: it defines a single operation stage that requires a dedicated resource

2. Periodically-maintained resource/operation module: it is the extension of the resource/operation module that requires periodical maintenance.

3. Fault-prone resource/operation module: resource/operation module can be ex-tended to a fault-prone resource by adding a loop to it.

4. Priority module: it describes different routes each of which requires a dedicated resource.

5. Rework module: when a process fails, a part may be required to rework. This module can be modeled by adding a new transition to the resource/operation module which which reroutes the part back.


Fig. 6.5 Some commonly used PN modules in flexible manufacturing (Zhou and Venkatesh 1999): (a) Resource/operation, (b) Periodically-maintained resource/operation, (c) Fault-prone resource/operation, (d) Priority, (e) Rework

Figure 6.5 shows these commonly used PN models for flexible manufacturing systems.

5 Stochastic PNs

Although the concept of time was not included in the original work by C. A. Petri (1962), for many practical applications, the addition of time is a necessity. With-out an explicit notion of time, it is not possible to conduct temporal performance analysis, i.e. to determine production rate, resource utilization. In modeling time critical systems with PNs such as a FMS, timing and activity durations for analyz-ing temporal performance and dynamics of the system should be taken into consideration.

Merlin (1974) and Ramchandani (1973) were the first who independently ex-tended PNs to include time delays in different ways. Curently the two proposed ap-proaches which are named as firing durations and enabling durations are the basis of


representing time in PNs. In timed PNs, although time delays can be associated to any transitions, places or arcs, it is often associated to transitions. The reason for this is that transitions represent events in a model and it is more natural to consider events to take time rather than time to be related to conditions, that is, places (Mu-rata 1989, Zhou and Venkatesh 1999, Bowden 2000, Gharbi ve Ioualalen 2002). For detailed information and comparison of approaches associating time to PNs, see Bowden (2000).

The most popular extension of PNs which are widely used in the application field of manufacturing systems is stochastic PNs. A stochastic PN (SPN) is a Petri net where each transition is associated with an exponentially distributed random variable that expresses the delay from the enabling to the firing of the transition. Due to the memoryless property of the exponential distribution of firing delays, Molloy (1982) showed that the reachability graph of a bounded SPN is isomorphic to a finite Markov chain. Queueing networks and Markov chains provide flexible, powerful and easy to use tools for modeling and analysis of complex manufactur-ing systems and are widely used (Al-Jaar and Descrochers 1990). In SPN models, we can explicitly describe the causal relation of uncertain events by using places, transitions, and arcs. Therefore, using SPNs, we can construct the model of a FMS more easily than using the other models. SPNs combine the modeling power of PNs and the analytical tractability of Markov processes for the purpose of per-formance analysis (Molloy 1982).

An ordinary continuous-time stochastic PN is a PN with a set of positive, finite and exponentially distributed firing rates ),...,( 1 mλλ=Λ , possibly marking de-

pendent, associated with all its transitions. An enabled transition can fire after an exponentially distributed time delay with parameter λ/1 elapses.

Assume that every transition in a PN is associated with an exponentially dis-tributed random delay from the enabling to the firing of the transition. Then the firing time of each transition can be characterized as a firing rate.

A stochastic PN ),,,,,( 0 Λ= mOITPZ is a six-tuple where

1. { } 0,,...,, 21 >= npppP n and is a finite set of places

2. { } 0,,...,, 21 >= stttT s and is a finite set of transitions with ∅≠∪TP and

∅=∩TP 3. NTPI →×: and is an input function that defines the set of directed arcs from

P to T where { },...2,1,0=N

4. NPTO →×: and is an output function that defines the set of directed arcs from T to P

5. NPmi →: and is a marking whose ith component represents the number of

tokens in the ith place. An initial marking is denoted by 0m .

6. +→Λ RT: and is a firing function whose ith component represents the firing

rate of the ith transition where iλ denotes the firing rate of it and +R is the set

of all positive real numbers.


In a SPN, when a transition is enabled at marking m, the tokens remain in its input places during the firing time delay. At the end of the firing time, tokens are removed from its input places and deposited in its output places. The number of tokens in the flow depends on the input and output functions.

Live and bounded SPNs are isomorphic to continuous-time Markov chains due to the memoryless property of exponential distribution (Molloy 1982). This im-portant property allows for the analysis of SPNs and the derivation of the some important performance measures. The states of the Markov chain are the markings in the reachabillity graph, and the state transition rates are the exponential firing rates of the transitions in the SPN. By solving a system of linear equations repre-senting the Markov chain, performance measures can be computed. The perform-ance analysis of a SPN can be summarized as follows (Murata 1989, Marsan et al. 1995, Zhou ve Venkatesh 1999):

After generating the reachability graph )( 0mR , the Markov process is obtained

by assigning each arc with the rate of the corresponding transition. The steady-state probability distribution ),...,,( 10 qπππ=Π of a SPN is obtained by solving

the following linear system

0=ΠA , ∑−

=

=1

0

1q

iiπ (6.1)

where ( )qqijaA

×= is the transition rate matrix. For 1,...,1,0 −= qi , A’s ith row

elements, i.e. ija , 1,...,1,0 −= qj are determined as follows:

1. if ij ≠ , ija is the sum of all outgoing arcs from state im to jm

2. Since any row elements in A satisfies ∑−

=

=1

0

0q

jija , then ∑

−

≠

−=1q

ijijii aa , where

iia represents the sum of firing rates of transitions enabled at im , i.e. transition

rates leaving state im .

From the steady-state distribution Π and transition firing rates Λ , the required performance indices of the system modeled by the SPN can be obtained. For example,

• The probability of a particular condition: Let B the subset of )( 0mR satisfying

a particular condition. Then the required probability is ∑∈

=Bm

i

i

BP π)( . For ex-

ample, if the particular condition is that a machine produces a product, then P(B) is the machine utilization.

• The expected vaalue of the number of tokens: let Bij be the subset of )( 0mR

such that at each marking in Bij, the number of tokens in place pi is j. Then the expected value of the number of tokens in a k-bounded place pi is


[ ] ∑=

=k

jiji BjPpmE

1

)()( . For example, if pi represents the availabity of one kind

of product, then [ ])( ipmE is the average inventory of the product.

• The mean number of firings in unit time: let Bj be the subset of )( 0mR in

which a given transition tj is enabled. Then the mean number of firings of tj in

unit time is ∑∈

=ji Bm

jiiijf λπ , where jiλ is the firing rate of tj at marking mi.

• The mean system throughput or production rate: assume that a product is pro-duced each time a transition in a subset T’ fires. Then the average throughput

rate is ∑∈

='Tt

j

j

fg .

The limitation of the SPN is that the number of states of the associated Markov chain grows very fast as the complexity of the SPN model increases (Marsan et al. 1984, Marsan et al. 1995). Marsan et al. (1984) introduced the generalized SPNs to reduce the complexity of solving a SPN model in which the number of reach-able markings is smaller than that in a topologically identical SPN. A generalized SPN is basically a SPN with transitions that are either timed (to describe the exe-cution of time consuming activities) or immediate (to describe some logical be-havior of the model). Timed transitions behave as in SPNs, whereas the immediate transitions have an infinite firing rate and fire in zero time. Although generalized SPNs provide a smaller reachability set and reduce the complexity of analyzing SPNs, they both give similar results. Also, the development of software packages which are currently available both for academic and commercial purposes makes the analysis of SPN models quite easier. More information and different modeling examples of both SPNs and generalized SPNs can be found in Marsan et al. (1995).

6 Fuzzy PNs and a New Fuzzy PN Approach

Although both PN theory and fuzzy set theory were introduced in 1960’s, the re-searches for the use of these together began much later, in early 1990’s. Petri (1987) presented some criticism related to timed and stochastic PNs about the conceptualization of time and chance. In his latter study Petri (1996) presented many axioms, among which the axioms of measurement and control related to time and nets, and emphasized mainly on uncertainty. These studies turned the at-tention on fuzzy set theory and fuzzy logic (Zadeh 1965, Zadeh 1973) which have been applied successfully in modeling and designing many real world systems in environments of uncertainty and imprecision.

There are several approaches that combine fuzzy sets and PN theories, differing not only in the fuzzy tools used but also in the elements of the nets that are fuzzi-fied. A PN structure is a four tuple consisting of places, transitions, tokens and arcs, and theoretically each of these can be fuzzified (Srinivasan and Gracanin


1993). Today there are some fuzzy PN approaches proposed by some reasearchers which are differently named, but there is not a commonly agreed standard ap-proach. The most common point of these approaches is that they usually focus on the fuzzification of time in PNs. Here we give a brief overview of the important fuzzy PN approaches.

Valette et. al. (1989) were the first who presented a brief description of fuzzy time PNs and oulined a procedure for computing fuzzy markings and fuzzy firing dates, in context of imprecise markings for modeling real-time control and moni-toring manufacturing systems. This approach is an extension of the crisp time in-terval of Merlin (1974)’s time PN to a fuzzy time interval since to each transition a fuzzy date or interval is assigned.

Another important fuzzy PN approach was introduced by Murata (1996) and called as fuzzy timing high level PNs. This approach defines four fuzzy set theo-retic functions of time called fuzzy timestamp, fuzzy enabling time, fuzzy occur-ence time and fuzzy delay. These fuzzy time functions are defined by triangular or trapezoidal possibility distributions.

Although these approaches mainly use simple fuzzy operations, the computa-tion procedure still can be time consuming and sometimes impractical in case of large complex systems. And also as mentioned before there is not still an agreed fuzzy approach for PNs and these approaches need further modifications and de-velopments. For example, Wu (1999) claims that, from algebraic point of view, the fuzzy occurence time of Murata (1996)’s approach is not closed which implies that the fuzzy timestamps of the produced tokens no longer have the initial forms (i.e. the trapezoidal or triangularvpossibility distributions). It seems that there is still a long way for the development and applications of PN theory with fuzzy logic. For some other approaches and applications of fuzzy set theory to PNs, the interested reader can see the researches of Yeung et.al. (1996), Wu (1999), Pedrycz and Camargo (2003), Ding et. al. (2005) and Ding et. al. (2006).

Fuzzy PNs and stochastic PNs have been separately used in modeling and analysis of discrete-event dynamic systems, such as FMSs. Analysis and design of such complex systems often involve two kinds of uncertainty: randomness and fuzziness. Randomness models stochastic variability which refers to describing the behavior of the parameters by using probability distribution functions. Fuzziness models measurement imprecision due to linguistic structure or incomplete infor-mation. Although the dominating concept to describe uncertainty in modeling is stochastic models which are based on probability, probabilistic models are not suitable to describe all kinds of uncertainty, but only randomness. Especially the imprecision of data which is for example as a result of the limited precision of measuring is not statistical in nature and can not be described by using probability (Viertl and Hareter 2004). To be able to better represent uncertainty, both stochas-tic or probabilistic variability and imprecision, we present an approach which we name as stochastic PNs with fuzzy parameters for modeling FMSs. This approach is based on the stochastic modeling and analysis of PNs by using fuzzy parame-ters. We believe that this new approach makes significant contribution to both sto-chastic PN and fuzzy PN literature.


The memoryless property of exponential distribution of firing delays is very important since live and bounded SPNs are isomorphic to continuous-time Markov chains due to the memoryless property of exponential distribution (Molloy 1982). In our approach, we describe the exponentially distributed transition firing rates as triangular fuzzy numbers. Since our aim is to take into consideration both randomness and fuzziness, the memoryless property for fuzzy exponential function must be satisfied in order to perform the stochastic analysis of fuzzy parameters.

The exponential )(λE has density

, 0( ; )

0,

xe xf x

otherwise

λλλ−⎧ ≥

= ⎨⎩

(6.2)

The mean and variance of )(λE is λ/1 and 2/1 λ , respectively.

The probability statement of the memoryless property of crisp exponential is

[ ] [ ]ττ ≥=≥+≥ XPtXtXP (6.3)

If we substitute λ~ for λ in Eq. (6.2) we obtain the fuzzy exponential, )~

(λE .

If μ~ denotes the mean, we find its α -cuts as

)}(~

{~

0

αλλλμ λ ∈= ∫∞

− dxex x (6.4)

for all α . However, each integral in the above equation equals λ/1 . Hence

λμ ~/1~ = . If 2~σ is the fuzzy variance, then we write down an equation to find its

α -cuts and we obtain 22 ~/1~ λσ = . It can be seen that the fuzzy mean (variance) is

the fuzzification of the crisp mean (variance). The conditional probability of fuzzy event A given a fuzzy event B is de-fined by Zadeh (1968) as

)(~

)(~

)(~

BP

BAPBAP

⋅= , 0)(~ >BP (6.5)

By using the fuzzy exponential α -cuts of the fuzzy conditional probability, relat-ing to the left side of Equation (3)

[ ] )}(~

{)( αλλ

λ

λ

ατλ

τ

λ

∈=≥+≥

∫

∫∞

−

∞

+

−

dxe

dxe

tXtXP

t

x

t

x

(6.6)


for [ ]1,0∈α . Now the quotient of the integrals in Eq. (6.6) equals, after evalua-

tion, λτ−e , so

[ ] )}(~

{)( αλλλαττ

λ ∈=≥+≥ ∫∞

− dxetXtXP x (6.7)

which equals [ ] )(~ ατ≥XP . Hence, Equation (6.3) holds for the fuzzy exponential

which shows the memoryless property. Our approach is a two stage modeling approach. The first stage is same as the

conventional SPN modeling approach. The only difference is that the steady-state distributions are obtained parametrically by using Equation (6.1) and no numeric results are calculated. In other words, each steady-state probability, iπ , is de-

scribed in terms of transition firing rates, as a function of iλ . Up to the second

stage, the system is a crisp one and describing the stochastic nature of the system. At this stage we represent the transition firing rates, iλ , as triangular fuzzy

numbers which may depend on the opinions of experts. After replacing the fuzzy numeric values of transition firing rates, by using the fuzzy calculation theory we obtain the α -cuts of the fuzzy steady-state probabilities of the system. As Buck-ley (2005) states that whenever we use interval arithmetic with α -cuts to compute the functions of fuzzy variables, we may get something larger than that obtained by using the extension principle. To be able to find the α -cuts of the fuzzy-steady probabilities we solve an optimization problem that makes the solution feasible.

The procedure to compute the fuzzy-steady state probabilities is as follows:

Stage 1 1.1. Model the system using a PN and associate exponential time delays with

transitions. 1.2. Generate the reachability graph. Assign each arc with the rate of the corre-

sponding transition. Label all states or markings. 1.3. By using Equation (6.1) find the steady-state probabilities parametrically,

in terms of transition firing rates.

Stage 2 2.1. Place the transition firing rates described as triangular fuzzy numbers in

parametric steady-state probabilities obtained in Step (1.3). 2.2. Compute the fuzzy steady-state probabilities by using Equations (6.4)-(6.9)

in terms α -cuts. 2.3. For each fuzzy steady-state probability, iπ , find o=α values which gives

the largest possible interval. It should be noted that for 1=α the obtained result is the steady-state distribution of the crisp SPN.

2.4. For each iπ , the maximum and minimum value ( o=α value) must be in

the interval [ ]1,0 . If o=α for each iπ does satisfy this, the result is feasible. If

any of the obtained fuzzy probability does not satisfy this, apply the optimization in the next step.


2.5. As mentioned before, interval arithmetic with α -cuts method is mainly based on max and min operators which may produce larger intervals. Theoreti-cally, o=α cut of a fuzzy number gives the largest possible interval of values. Since we want to calculate the fuzzy probabilities the largest possible interval is restricted to the interval [ ]1,0 . The problem is finding the min α -cut that satisfies this condition which can be found solving the following optimization problem:

Assume that the α -cut representation for the fuzzy steady-state probability is

[ ])(),( απαππ +−= iii , where ni ,...,2,1= and n is the number of the states. Then

the structure of the problem is

α=)(ZMin

s.t.

)()(

10

0)(

1)(

απαπααπ

απ

+−

−

+

≤

≤≤≥

≤

ii

i

i

In the next section a numerical illustration of the proposed approach is given.

7 A Numerical Example for the Proposed Fuzzy Approach

Our proposed approach is applied to a FM cell which was selected from (Zhou and Venkatesh 1999). This FM cell is illustrated in Figure 6.6.

Fig. 6.6 The illustration of the flexible manufacturing cell

Our FM cell consists of two machines (M1 and M2), each of which is served by a dedicated robot (R1 and R2) for loading and unloading, as shown in Figure 6.6. An incoming conveyor carries pallets with raw materials one by one, from which R1 loads M1. An outgoing conveyor takes the finished product, to which R2 unloads M2. There is a buffer with capacity of two intermediate parts between two machines. The system produces a specific type of final parts. Each raw workpiece


fixtured with one of three available pallets is processed by M1 and then M2. A pallet with a finished product is automatically defixtured, then fixtured with raw material, and finally returns to the incoming conveyor. Now suppose that

1. M1 performs faster M2 does, however subject to failures when it is processing a part. On the average, M1 takes two time units to break down, and a quarter time unit to be repaired. Thus, its average failure and repair rates (1/time unit) are 0.5 and 4 respectively. M2 and the two robots are failure-free.

2. R1’s loading speed is 40 per unit time. The average rate for M1’s processing plus R1’s loading is 5 per unit time.

3. The average rate for M2’s processing plus the related R2’s loading and unload-ing is 4 per unit time.

4. All the time delays associated with the above operations are exponential. The problem is to find the average utilization of M2, assuming that only one pallet is available.

The SPN model of the system is given in Figure 6.7. Table 6.5 gives the expla-nation and interpretation of the PN elements used in the model.

Fig. 6.7 The stochastic PN model of the flexible manufacturing cell

If two pallets were available, the PN model of the system would contain two tokens in place 1p .

As it can be seen in Table 6.5, the redundant information in each marking on the token values in places 987 ,, ppp has been eliminated since 7p holds the same

number of tokens as 5p does and each of the other two always has one token.

The reachability graph and the Markov chain of the modeled system is given in Figure 6.8.


Table 6.5 Places, transitions and their firing rates used in the model

Places Interpretation

p1 Pallets with workpieces available

p2 M1 in process

p3 Intermediate parts available for processing at M2

p4 M1 in repair

p5 M1 available

p6 Conveyor slots available

p7 R1 available (redundant from the analysis viewpoint)

p8 M2 available (redundant from the analysis viewpoint)

p9 R2 available (redundant from the analysis viewpoint)

Transitions Interpretation Firing Rates

t1 R1 loads a part to M1 401 =λ

t2 M1 machines and R1 unloads a part 52 =λ

t3 R1 loads/unloads and M2 machines a part 43 =λ

t4 M1 breaks down 5.04 =λ

t5 M1 is repaired 5.05 =λ

Fig. 6.8 The reachability graph and Markov chain of the system

By using Equation (6.1) we obtain the following system of equations:

1

0

00

00

0

00

),,,(

3210

55

33

4242

11

3210

=+++

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

−−−

ππππλλ

λλλλλλ

λλ

ππππ


The solution of the above system gives the steady-state probabilities parametri-cally, in terms of transition firing rates, as follows:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Π

λλλλλλλλλλλλλλλλ

ππππ

/

/

/

/

431

521

531

532

3

2

1

0

T (6.8)

where 431521531532 λλλλλλλλλλλλλ +++= .

After obtaining the steady-state probabilities in terms of transition firing rates, in the second stage transition firing rates are represented as triangular fuzzy num-bers. Assume that the fuzzy number values of each transition firing rate are as in Table 6.6.

Table 6.6 The fuzzified transition firing rates and their −α cut representations

Fuzzy λ value −α cut representation

)50/40/30(1 =λ [ ]ααλ 1050;10301 −+=

)6/5/4(2 =λ [ ]ααλ −+= 6;42

)5/4/3(3 =λ [ ]ααλ −+= 5;33

)6.0/5.0/4.0(4 =λ [ ]ααλ 1.06.0;1.04.04 −+=

)6.0/5.0/4.0(5 =λ [ ]ααλ 1.06.0;1.04.05 −+=

By placing the fuzzy values of Table 6.6 in the previously obtained parametric steady-state probability representations, Equation (6.8), and applying fuzzy mathematics, the following −α cut representations of fuzzy steady-state prob-abilities are obtained.

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++++−+−

+−+−+++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++++−+−

+−+−+++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++++−+−

+−+−+++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++++−+−

+−+−+++=

8.1241101.321.3

1508516;

4986.2757.501.3

363310)(

8.1241101.321.3

1809617;

4986.2757.501.3

484011)(

8.1241101.321.3

1508516;

4986.2757.501.3

363310)(

8.1241101.321.3

186.97.11.0;

4986.2757.501.3

8.441.11.0)(

23

23

23

23

3

23

23

23

23

2

23

23

23

23

1

23

23

23

23

0

αααααα

αααααααπ

αααααα

αααααααπ

αααααα

αααααααπ

αααααα

αααααααπ


0

0,2

0,4

0,6

0,8

1

1,2

0,01

0

0,01

3

0,01

7

0,02

2

0,02

8

0,03

7

0,04

8

0,06

3

0,08

3

0,10

9

0,14

4

0

0,2

0,4

0,6

0,8

1

1,2

0,07

2

0,09

7

0,12

9

0,17

0

0,22

5

0,29

6

0,39

0

0,51

5

0,68

0

0,90

2

1,20

2

(a) (b)

0

0,2

0,4

0,6

0,8

1

1,2

0,09

6

0,12

7

0,16

6

0,21

8

0,28

4

0,37

0

0,48

3

0,63

2

0,82

8

1,09

0

1,44

2

0

0,2

0,4

0,6

0,8

1

1,2

0,07

2

0,09

7

0,12

9

0,17

0

0,22

5

0,29

6

0,39

0

0,51

5

0,68

0

0,90

2

1,20

2

(c) (d)

Fig. 6.9 The graphical representation of the fuzzy steady-state probabilities, (a) 0π , (b)

1π , (c) 2π , (d) 3π

The graphics of the fuzzy steady-state probabilities are given in Figures 6.9 (a)-(d).

Although for each [ ])(),( απαππ +−= iii , the maximum and minimum value

( o=α value) must be in the interval [ ]1,0 , it can be seen that ++21 ,ππ and +

3π do

not satisfy this condition. So we must apply optimization to find the min α -cut

that satisfies this condition. Since 31 ππ = and 0)( ≥− απ i , the structure of the op-

timization problem can be reduced to the following:

α=)(ZMin

s.t.

0)(),(),(

10

1)(

1)(

1)(

110

2

1

0

≥

≤≤≤

≤

≤

+++

+

+

+

απαπαπααπ

απ

απ

αα


By making the necessary simplifications the problem is obtained as follows:

α=)(ZMin

s.t.

10

02.552061.151.4

02.251951.161.4

08.1066.1194.302.3

23

23

23

≤≤≤+−−−

≤+−−−

≤−−−−

ααααααα

ααα

The solution for the problem can be found by using software packages such as MATLAB or a spread sheet like Excel. The result of the optimization problem is 0.263. This α value is the one that makes the fuzzy steady-state probabilities fea-sible. The final fuzzy steady-state probabilities are presented in Table 6.7.

Table 6.7 The final fuzzy steady-state probabilities

0=α cut 1=α cut

0π [ ]1.0;014.0 0.037

1π [ ]826.0;106.0 0.296

2π [ ]1;138.0 0.37

3π [ ]826.0;106.0 0.296

Note that 0=α cut value represents the largest interval of probability whereas 1=α cut value represents the crisp SPN probability.

M1’s utilization is determined by the probability that M1 is machining a raw workpiece. This corresponds to the marking 1m at which 2p is marked, or state

probability 1π . Therefore, the expected M1’s utilization is 1π which is

)826.0/296.0/106.0( .

8 Conclusions

There is a growing literature and interest in Petri nets theory, and in the scope of this chapter we presented a review of the important concepts of Petri nets and their application mainly in the area of flexible manufacturing systems. Complex sys-tems such as flexible manufacturing systems exhibit characteristics which are dif-ficult to describe mathematically using conventional tools and methods. Although there are many methods and tools used for modeling and analysis of FMSs such as queueing networks, Markov chains and simulation, Petri nets as a mathematical tool provide obtaining state equations describing system behavior, finding


algebraic results and developing other mathematical models. With respect to other techniques of graphical system representation like block diagrams or logical trees, Petri nets are particularly more suited to represent in a natural way logical interac-tions among parts or activities in a system. In modeling point of view, Petri net theory allows the construction of the models amenable both for the effectiveness and efficiency analysis. Due to the graphical nature, ability to describe static and dynamic system characteristics and system uncertainty, and the presence of mathematical analysis techniques, Petri nets form an appropriate conceptual infra-structure for modeling and analysis of flexible manufacturing systems.

One of the most important points in modeling real world systems is to be able to represent the uncertainty in the system model. Analysis and design of complex systems often involve two kinds of uncertainty: randomness and fuzziness. Ran-domness refers to describing the behavior of the parameters by using probability distribution functions. Fuzziness models measurement imprecision due to linguis-tic structure or incomplete information. Although the dominating concept to describe uncertainty in modeling is stochastic models which are based on prob-ability, probabilistic models are not suitable to describe all kinds of uncertainty, but only randomness. We believe that to be able to better represent both dimen-sions of uncertainty, models that combine stochasticity and fuzziness should be used together. For this reason, the approach which was explained in detail in this study plays a significant role for the use of fuzzy set theory together with stochas-tic PNs especially in manufacturing system modeling.

For further research, the application of the proposed approach in different areas rather than FMSs and sensitivity analysis for the fuzzy parameters can be considered.

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[Studies in Fuzziness and Soft Computing] Production Engineering and Management under Fuzziness Volume 252 || Manufacturing System Modeling Using Petri Nets