Modeling, reasoning, and application of fuzzy Petri net

 

Artificial Intelligence Review (2022) 55:6567–6605 https://doi.org/10.1007/s10462-022-10161-0

Modeling, reasoning, reasoning, and application of fuzzy Petri net model: a survey 1

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Wei Jiang  · Kai-Qing Zhou   · Arezoo Sarkheyli-Hägele  · Azlan Mohd Zain Accepted: 28 February 2022 / Published online: 11 March 2022 © The Author(s), under exclusive licence to Springer N Nature ature B.V. B.V. 2022

Abstract A fuzzy Petri net (FPN) is a powerful tool to model and analyze knowledge-based systems containing vague information. This paper systematically reviews recent developments of the FPN model from the following three perspectives: knowledge representation using FPN, reasoning mechanisms mechanisms using an FPN framework, framework, and the latest industrial applications using FPN. In addition, some specific modeling and reasoning approaches to FPN to solve the ‘state-explosion problem’ are illustrated. Furthermore, detailed analysis of the discussed aspects are shown to reveal reveal some interesting findings, as well as their developmental history. Finally, we present conclusions and suggestions for future research directions. Keywords  Fuzzy Petri net · Knowledge Knowledge representation representation · Modeling · Reasoning · Industrial application

1 Introduction Knowledge-based systems (KBSs) or expert systems are intelligent computer systems that rely on captured knowledge to perform complex tasks and programs, usually by domain experts (Valavanis et al. 1994 1994). ). After analyzing basic KBS modules (including the rules, frame, and logic), Chandrasekaran (1988 (1988)) proposed a generic framework for a knowledge system that consists of a series of building blocks with each block owning a function aimed at solving a specific problem. A KBS helps retain the knowledge scattere scattered d in the t he memories of individual domain experts and helps to find solutions by mimicking the reasoning pro1992). ). In the past cess of domain experts (Handelman et al. 1990; 1990; Mockler and Dologite 1992 few decades, KBSs have been employed to implement modeling and reasoning functions to solve different problems in various areas, such as fault diagnosis (Frank 1990), 1990), inference 2000), classification (Zanni et al. 2006 2006), ), and (Chen and Yang 2013), 2013), verification (Koriem 2000), biological modeling (Chaplain 2011). 2011). *    

Kai-Qing Zhou  [email protected]

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College of Information Science and Engineering, Jishou Univer University, sity, Jishou 416000, Hunan, China

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Internet of Things and People Researc Research hC Center, enter, Department of Computer Science and Media Technology, Malmö University, 20506 Malmö, Sweden

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UTM Big Data Centre, Universiti Teknologi Malay Malaysia, sia, 80310 Skudai, Johor, Malay Malaysia sia

 

 

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However, as the sizes of KBSs grow, it becomes difficult to obtain individual and precise knowledge from the experience of the related experts when dealing with complex problems. This also indicates that the difficulty and complexity of KBS modeling and the 1989;; Ha et al. corresponding inference function have increased sharply (Cardoso et al 1989 2007). To handle this, a series of new methods have arisen to represent knowledge and 2007). execute automatic reasoning: fuzzy production rules (FPRs), semantic networks, G-nets, Petri nets (PNs), and fuzzy Petri nets (FPNs). the mentioned methods, FPR ,and have applied fulfill b, cFPN ; Wai andbeen Liu successfully 2009). 2009). FPR has beentowidely theAmong KBS requirements (Liu et al. 2013a, 2013a used to develop expert systems to represent, capture, and store fuzzy knowledge. Zhang et al. (1991 (1991)) proposed fuzzy production rules based on expert systems. They developed a procedure with the Micro-PROLOG language to assist physicians in solving diseases in the field of pneumoconiosis. Based on FPR and PN correspondence, Lipp (1983 ( 1983)) and Looney 1988)) proposed that the extended PN formalism, namely, FPN, combines classical PN (1988 with fuzzy theory to represent the FPR. FPN is a kind of high-level Petri Petri net (HLPN) to deal with fuzzy information. Compared to other modeling methodologies, FPN has some unique features. It provides a formal and general graphical model (Hu et al. 2011) 2011) that can be visualized. Being a formal mathematical model, it allows for the verification of the properties of the modeled systems by utilizing the characteristics of Petri nets and supports knowledge reasoning in KBSs (Liu et al. 2015a,, b; Li et al. 2000 2000). ). Finally, Finally, FPNs can capture the dynamic nature of fuzzy rule-based 2015a reasoning with the evolution of markings and also express the dynamic behavior in algebraic forms (Lee et al. 1998). 1998). Because of the above advantages, FPNs are widely employed to represent FPRs and to model KBSs in a variety of industrial areas, such as robotic systems (Wai and Lin 2009, 2009, 2013), 2013), manufacturing systems (Mhalla et al. 2013), 2013), traffic engineering (Cheng and Yang 2009 2009;; Lee et al. 2006), 2006), and fault diagnosis (Liu et al. 2013a, 2013a, b, c; Zhang et al. 2017a, 2017a, b). Although FPNs and their applications have achieved fruitful results, further utilization of FPNs will be limited due to the increasing complexity of KBSs. This is because the existing FPNs cannot adapt to the changing antecedent propositions in complex KBSs. The main disadvantage of using an FPN for modeling and reasoning consists of the following two aspects. 1.

2.

FPNs lack lack a learning mechanism to adapt to the growth of KBSs. Accurate values values of FPN parameters (weight, threshold, and certainty factors) are increasingly difficult to obtain. As with a PN, it is a huge challenge to solve solve the state space explosion explosion problem of largescale FPNs using existing analytic approaches and reasoning mechanisms.

2016)) to summarize and This paper extended our previous review study (Zhou and Zain 2016 analyze the latest research results regarding modeling and reasoning methods using FPN and the related typical industrial applications. Related articles in journals (indexed by SCI) and conferences (indexed by CPCI and Scopus) between 2014 and 2020 are gathered and reviewed. revie wed. Compared with our previous paper, the specific objectives of this manuscript are as follows: 1.

To explore the modeling and reasoning of KBSs using FPN FPN under the general backbackground and the state-explosion background.

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To summarize and review current typical applications applications of FPN from from 2014 to 2020 to reveal the different development lines of FPN. To discuss the potential application domains of FPN in the future.

Based on the mentioned objectives, the rest of this paper is organized as follows. Basic notions of PN and FPN and the related correspondences cor respondences are presented in Sect. 2. Section 3  reviews HLPN formalisms and recalls the development of the FPN family. Section 4 summarizes and analyzes existing methods for using FPN for sizeable KBS modeling. Section 5  discusses the reasoning mechanisms using FPN. Section 6  reviews and discusses FPN applications in various areas from 2014 to 2020. Section 7 concludes and points to potential future research directions.

2 Petri net an and d fuzzy Petri net 2.1 PN and elementary net system The Petri net (PN) model, proposed by C. A. Petri in this thesis in 1962, is a graphical mathematical formalism used to model, analyze, and describe the control and information flows in discrete events or distributed systems with asynchronous and concurrent activi2008). ). The PN model provides an ties (Narahari and Viswanadham 1985; 1985; Petri and Reisig 2008 intuitive way to understand systems with dynamic behaviors (Heiner 2011). 2011). According to related review papers (Murata 1989; 1989; Zurawski and Zhou 1994 1994), ), a PN could be defined as follows.

Definition 1  PN is a 6-tuple:PN   = {P, T ;F , K , W , M 0 } , where 1. 2.

P  = { p1 , p2 , p3 , , pn } is a finite set of places. The places are represented by circles; T   = {t 1 , t 2 , t 3 , , t m } is a finite set of transitions. The transitions are represented as ⋯

⋯

rectangles; 3.

F   ⊆  ( P

× T ) ∪ (T   × P) is a set of flow relations or arcs. The relation (or arc) is used to

illustrate the flow relationship of a place to a transition or a transition to a place; K   = {   , 2, 2, , , n} i; 4. capacity of eachwhich placeindicates 2, , , n} is 5. W   = {   , 2,  is the a weight function, how many existing resources will be consumed or how many new resources will be created in the firing process; and 6.  M 0 is the initial marking. ⋯

⋯

The elementary net system (EN_system) is the most fundamental model of the PN family, in which the places are considered to be conditions represented by B. The transitions 1987). ). In genare considered to be events represented by E (Thiagarajan 1986; 1986; Graubmann 1987 eral, according to related review papers (Rozenberg 1987; 1987; Pelz 1989 1989), ), the EN_system formalism is defined as: _system  = { B,  E ;F , c} , where Definition 2  An EN_system EN_system is a 4-tuple:  EN _system

1. 2. 3.

 B is  E  is  is

a finite set of places; a finite set of transitions; F   ⊆  ( P   × T ) ∪ (T   ×   P) is a finite set of flow relations or arcs;

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4.

c   ∈   B is

an  EN _system _system case.

In every state of the EN_system, the status of conditions in B can be divided into two classes. If  M (s)  =   1 , then these conditions are true; otherwise,  M (s)  =   0 and other conditions are false. Based on this finding, a subset of B (replaced by C) is used to represent all “true” conditions.

2.2 FPN Although the PN can simulate and analyze the systems that own clearly defined states and transitions, it is difficult to endow them with uncertainty and vague information. Looney 1988)) proposed a rough notion for an FPN to execute approximate reasoning. Since then, (1988 the basic framework of FPN and related applications have been discussed frequently by 1994a,, b; Liu et al. 2013a 2013a,, b, c; Zhou et al. scholars (Chen et al. 1990; 1990; Yeung and Tsang 1994a 2015a 2015a;; Liu et al. 2017a, 2017a, b) to support the structural organization of fuzzy information in KBSs, to provide visualization of diagnosis processing and to design efficient fuzzy inference algorithms. Formally, an FPN formalism structure is defined as:

Definition 3  FPN is an 8-tuple: FPN   = {P

,

T   M   I  O W   CF } , ,

,

,

,

,

,

where

P  = { p1 , p2 , p3 , , pn } is a finite set of places; T   = {t 1 , t 2 , , t n } is a finite set of transitions;  M   = (m1 , m2 , , mn )T  is a vector of fuzzy markings, mi   ∈ [0,   ] is the truth tr uth degree of  pi (i  =   , 2, , n) , and the initial truth t ruth degree d egree of the vector is denoted by 0; 4.  I   ∶   P ×  T  → {0,   } is the × m input matrix, in which  I ( pi  t  j ) records whether a directed  j  =  1, 2, , m  m)) , where arc from  pi to t  j exists (i  =  1, 2, , n; j = 1   if ther theree is a dir direc ecte ted d arc arc fr from om pi to t  j ;  I ( pi , t  j ) =  ; and   0   if ther theree is no nott a dir direct ected ed arc arc fr from om pi to t  j . 5. O  ∶   P × T  → {0,   } is the n × m output matrix. Here, O( pi  t  j ) records whether a directed  j  =  1, 2, … , m  m)) , where arc from t  j to  pi exists (i  =  1, 2, … , n; j =

1. 2. 3.

⋯

⋯

⋯

⋯

,

⋯

⋯



,

1   if ther theree is a di dirrecte ected d arc arc fr from om t  j to pi ;

O( pi , t  j ) =

0   if there there is no nott a dir direct ected ed arc arc fr from om t  top .



 j

i

6. w(i  j)  j) is the weight of the arc from  pi to t  j; 7.    ∶    → (0,   ] ,  j is the threshold of  j ; and 8. CF  is  is the belief strength, where CF ij   ∈ [0,   ] is the support strength of the arc from t  j to  pi and exists as (i  =  1, 2, … , n; j =  j  =  1, 2, … , m  m)) ,

2.3 Correspondences among EN EN_system, _system, PN, and FPN Based on the existing results, the correspondences among EN_system, PN, and FPN are shown in Table 1 (Zhou et al. 2015a 2015a;; Zhou and Zain 2016; 2016; Liu et al. 2018a 2018a,, b). Table 1  further indicates the main characteristics of FPN. Compared with EN_system and PN, the largest modification of FPN is to obscure the value of each parameter. Hence, the FPN could be considered a back-extensive formalism of PN without destroying the inner-dynamic structure to model and execute the inference or computation task for systems with uncertainty. It is also why the intrinsic graphical description method and

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Modeling, reasoning, and application of fuzzy Petri net model:… Table 1 Corresponds between EN_system, PN and FPN  EN_system  EN_syst em

Formalism Weight

,

K (s)

 

= 1

i =1

 

 

Marking Support strength

M  s

 

K (s)

 

W (i,   j) =  1

 

Threshold

FPN 

EN _system  = { B, E ;F , c}PN   = {P, T ;F , K , W ,  M 0 }FPN   = {P T   M   I  O W 

 

Capacity

PN

i =1

 1

 

M  s

 

K (s)

 

W (i,   j) =  1

   

( )= CF ( j,   i) =  1

=  1

 1

,

,

,

,

 CF } ,

= 1

W (i, j  j)) ∈ [0, 1]

 

i   ∈

 

( )= CF ( j,   i) =  1

,

 

[0, 1]

M  s

 

( ) ∈ [0, 1] CF ( j,   i) ∈ (0, 1]

the algebra analytic technique of PN can also be employed by FPN to deal with fuzzy information.

3 High-level FPNs and their lines of development The problems of using FPN in industrial fields and some constraints on the real systems (such as time delays and negation issues) should be considered to ensure that the obtained model fully reflects all the core properties of the corresponding system (Ribaric and Basic 1998; 2013;; Amin and Shebl 2014; 2014; Liu et al. 2016 2016;; Yang and Li 2018). 2018). There1998; Wu et al. 2013 fore,, a series of HLFPNs, such as fuzzy colored Petri nets (FCPNs) (Scarpelli et al. 1996), fore 1996), fuzzy time Petri nets (FTPNs) (Berthomieu and Diaz 1991), 1991), fuzzy stochastic Petri nets (FSPNs) (Molloy 1982), 1982), and intuitionistic fuzzy Petri nets (IFPNs) (Shen et al. 2009 2009)) have been proposed by researchers and practitioners to address problems resulting from different constraints.

3.1 Classical HLFPNs To consider colored markings, Lin and Hwang (1996 ( 1996)) developed a fuzzy colored PN model (FCPN) to verify a KBS and to classify the t he fuzziness of consistency consistency.. Lee and Seong ((2004 2004)) utilized a novel FCPN model to analyze and model nuclear power plants and execute inference engine decision-making. Chiang et al. (2006 (2006)) proposed a queueing-colored Petri nets (QCPNs) model with a fuzzy factor to check the wafer fabrication performance evaluation.  Jorge et et al. (1996 (1996)) proposed an HLFPN by combining fuzzy time Petri nets with G-nets for analysis of real-time distributed systems. Slobodan et al. (1999 ( 1999)) modified the original PN model by combining FTPN with fuzzy temporal relations to implement knowledge representation and reasoning tasks. Pedrycz and Camargo (2003 (2003)) proposed a modified FTPN for adding the time factor to the transitions and places. Liu et al. (2011 ( 2011)) modified FTPN to realize a chemical abnormality monitoring function from two angles: assigning a time factor for each transition and associating a degree of reliability with each place. Yang and Li (2018 (2018)) proposed a dynamic time fuzzy Petri nets (DTFPNs) modeling approach using dynamic delay time analysis. Wu (1999 ( 1999)) proposed a fuzzy-timing colored Petri net (FTCPN) to comprehensively comprehensively measure the continuous time scale by using the time scale as a linear-ordered commutative To consider the strength ofmonoid. fuzzy sets in dealing with uncertain information, researchers developed the concept of a fuzzy stochastic Petri net (FSPN) by combining SPNs and

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fuzzy sets. Zhang and Yao (2015 ( 2015)) analyzed multistate systems utilizing FSPN and introduced a systemic approach to investigate investigate the adaptability and reliability of FSPN. Liu et al. (2018a, 2018a, b) examined both the randomness and fuzziness of biological systems. After that, they modeled and analyzed the biological systems with uncertain kinetic parameters using FSPN. Shafiekhani et al. (2018 (2018)) examined the dynamic properties of a tumor in the tumorimmune order. They captured the dynamic behavior of the tumor-immune system involving uncertain kinetic parameters using a FSPN model and calculated the steady-state response of the system. 2009)) proposed an intuitionistic FPN (IFPN) and related reasoning algoShen et al. ((2009 rithms. In the framework of IFPN, the credibility degree of a rule and the threshold of transition are represented by intuitionistic fuzzy numbers (with operators including membership, non-membership, and hesitancy). Liu et al. (2015a (2015a,, b) proposed a new type of FPN model based on intuitionistic fuzzy sets and ordered weighted averaging operators to address the knowledge representation of uncertainty in expert systems and the incomplete reasoning results. Based on this new framework, framework, a max algebra-based reasoning algorithm is proposed to automatically implement intuitionistic, fuzzy reasoning. This method was applied to the fault diagnosis problem of aircraft generators. Meng et al. (2017 ( 2017)) proposed a hybrid reasoning algorithm by combining forward reasoning with backward reasoning based on the intuitionistic FPN model. Zhang et al. (2018 ( 2018)) employed the IFPN model to forecast the influence of the uncertainty of alarm information in fault diagnosis in power grid fault systems. Scholars have proposed various HLFPNs by using neural network theory. In 1994, Looney proposed an FPN to adjust the threshold value parameter, but the other two types of parameters still need to be adjusted by experts. Li and Lara-Rosano (2000 (2000)) proposed an adaptive FPN (AFPN) model to amend the certainty factors from transition to place in the FPN model. The model not only performs knowledge reasoning, reasoning, but it also enables the whole net learning process to be decomposed into several simple learning processes, which reduces the complexity of the learning algorithm. It is, in addition, a learning algorithm presented as an NN. Pedrycz and Gomide (1994 (1994)) proposed an adjustable FPN to model the reasoning by combining neural network learning algorithms. However, the model was only suitable for handling the logical relationships ’OR’ and ’And’. Shen et al. (2010 (2010)) developed supervised and unsupervised learning algorithms for machine learning PN (MLPN) models to make them wholly trained and to remedy the difficulties encountered by artificial neural networks (ANNs). Vahidipour et al. (2015 (2015)) proposed an adaptive PN with learning automata (APN-LA) to resolve the transitions’ conflicts. Shojafar et al. (2015 (2015)) proposed an adaptive stochastic stochastic Petri nets (ASPNs) model with a fuzzy factor to enhance the consumption time for tasks in economic grids. gr ids. This model is based on learning automata approaches that predict the system’s next state and previous state based on current status.

3.2 Development thread of HLFPNs The abovementioned studies indicate that HLFPNs are often formulated by combining the general FPN model with other HLPNs to enhance the descriptive ability of the original FPN. Therefore, these novel models have modified the corresponding large-size FPN models to handle specific problems, expanded the research domain of FPN theory, and improved the FPN model’s self-learning. On the other hand, FPN is also a kind of backward-extension HLPN to handle vague information without destroying the intrinsic

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Modeling, reasoning, and application of fuzzy Petri net model:…

Fig. 1 Developme Development nt of the FPN model family

advantages and inner structural properties of PNs. The developmental lineage of the FPN family famil y is depicted in Fig. 1.

4 Modeling sizeable KBS u using sing FPN FPRs have been widely employed to represent the captured fuzzy information in a KBS by using an ‘IF–THEN’ form. Broadly speaking, the FPN model is also constructed to execute approximate reasoning based on descriptive graphics and parallel mathematical analysis. The first challenge of modeling sizeable KBS using FPN is to investigate the hidden associations between an FPN and FPRs.

4.1 Modeling FPR using FPN FPR is typical for representing uncertain, fuzzy knowledge in the KBS (Negoita 1985 1985;; 2002). ). In a KBS, each FPR could be expressed as a fuzzy ‘IFChen 1988; 1988; Song and Lee 2002 THEN’ form. The FPR could be divided into two parts: the antecedent and the consequent. Each part is represented by fuzzy sets (Chen 1996; 1996; Awan and Awais 2011). 2011). Moreover, a composite FPR is a kind of FPR that contains either the ‘AND’ or ‘OR’ connectors (Bal2006; Ting et al. 2008). 2008). The general FPR formalazinski et al. 2002; 2002; Novák and Lehmke 2006; ism is described as follows.

Definition 4  A general general FPR formalism formalism rule is presented as: ()   t en   (

 

,

 w) , where ,

1.  D is the antecedent part of an FPR with  D  = { D ,  D , ,  Dn }; 2. Q is the consequent part of an FPR with Q  = {Q11 , Q22 , , Qn }; 3.  is the truth degree of each module of the antecedent part,    ∈ [0, 1]; ⋯

⋯

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4.

CF   ∈ [0,   ] is

the belief strength of the t he FPR, which is the credibility implementation of

the rule after; 5. i   ∈ [0,   ] is the threshold of the rule; and 6. wi   ∈ [0, 1] is the weight of each precondition. According to the definitions above, the inner correspondence between FPN and FPR is listed in Table 2. Table 3  shows that there is a strict correspondence relationship between an FPN and an FPR. In particular, the antecedent and consequent parts of each FPR could be illustrated as locations in an FPN model, and the three types of parameters also exhibit point-to-point correspondence. Hence, each FPR could be represented as an FPN, and a KBS with multiple FPRs can be depicted as a complex FPN model (Zhou et al. 2019 2019). ). Based on the above-mentioned relationships, it is easy to see that there exists a correspondence between FPR and FPN in the modeling process. Three typical FPN models corresponding to three types of FPRs, the simple rule, ’AND’ rule, and ’OR’ rule, are illustrated in Figs. 2, 3 and 4. Type 1: Simple rule and corresponding FPN model Table 2 Correspondence between FPN and FPR

Parameter

FPN  

Formalism

FPN   = (P T   I  O  M  ()CF  )  If DW    then then ,

Capacity function Weight function

 

 

,

= 1

,

,

i   ∈

[0, 1]

M (s)

∈ [0, 1]

,

,

D  =

 

W (i, j  j)) ∈  [ 0, 1]

 

 

Support strength

K (s)

 

Threshold function Marking

FPR

 

CF ( j,   i) ∈ (0, 1]

 

{ D1 ,  D2 ,

Q   (CF   w) ,

⋯

,

,  Dn }

W   ∈  [0,  [ 0, 1]   ∈

 

,

[0, 1]

null   CF   ∈ [0, 1]

Table 3 Comparison between reasoning by graphical method and by algebraic form

Reasoning algorithm by graphical method

Reasoning algorithm by algebraic method

Advantages

Disadvantages

(1) Easy to execute an inference

(1) The reachability and adjacent

operation intuitively and visually (2) Easy to recognize and find inference path

places set rapidly increase while the structure of the FPN changes (2) In the reachability tree, it is difficult to mark the inner ‘AND or OR’ relationship among places in the FPN

(3) Easy to transform all relationships among places into matrices (4) Easy to implement an efficient parallel inference operation

(5) The dimensions of matrices and vectors increased with the growth of the scale of the FPN model (6) Lack of intuitive methods

Fig. 2 FPN model of simple rule

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Modeling, reasoning, and application of fuzzy Petri net model:… Fig. 3 FPN model of ‘AND’ rule

Fig. 4 FPN model of ‘OR’ rule

 If D()   then Q   (CF , , w  =  1 ) Type 2: ‘AND’ rule and corresponding FPN model  If D(1 ) AND D(2 ) AND

then en Q (CF ,  ,  AND D(n ) th

⋯

n 

wi   =   1)

i=1

Type 3: ‘OR’ rrule ule and corresponding FPN model  If D(1 ) OR D(2 ) OR OR D(n ) th then en Q (CF i , i , wi   =  1 ) ⋯

4.2 Modeling KBS using FPN Knowledge-based systems (KBSs) are expert systems that can solve complex problems, such as modeling and verification (Koriem 2000), 2000), fault diagnosis or analysis (ParedesFrigolett and Flavio 2016; 2016; Garcia-Crespo et al. 2011), 2011), and knowledge reasoning and representation (Fay 2000). 2000). Due to the correspondence relationship between an FPN and a KBS, various modeling algorithms algorit hms of a KBS using an FPN have been widely discussed in the existing literature. Yeung and Tsang (1994b ( 1994b)) proposed an FPN model to capture more FPRs KBSmore that maps FPRs with different threshold values in a proposition into an FPNintothe reflect information in the FPR. Chen ( 2002) (2002 ) proposed a weighted FPN (WFPN) and a weighted fuzzy reasoning algorithm for rule-based systems. In this

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WFPN, the truth degrees, weight values of the propositions, and certainty factors of the rules are all represented by fuzzy numbers. Shen (2006 ( 2006)) proposed a high-level fuzzy Petri net (HLFPN) to represent the FPRs in a KBS to implement fuzzy reasoning. Ha et al. (2007 (2007)) proposed a generalized fuzzy Petri net (GFPN). The GFPN model has two knowledge representation parameters, namely, input and output weights. The two kinds of weights replace the previous local weight and certainty factors of a rule used in the GFPN framework. Suraj (2012b (2012b)) proposed a matrix representation of parametrized FPN (PFPN), which can quickly quickly implement parallel firing firi ng of independent transitions in a reasoning process using matrix equations or inequalities. Liu et al. (2013a (2013a)) proposed an active adaptive fuzzy Petri net (DAFPN) for knowledge representation and reasoning and developed a max-algebra-based parallel reasoning algorithm to implement the reasoning process automatically. Liu et al. (2013c ( 2013c)) proposed a new knowledge acquisition and reasoning approach based on DAFPN and fuzzy evidential reasoning (FER). In the modified DAFPN, situations in which various rules shared the same parameters were systematically analyzed. Furthermore, the corresponding solving strategies are also discussed. Liu et al. (2015a (2015a,, b) enhanced Suraj’s work (2013 (2013)) to propose a new FPN based on intuitionistic fuzzy sets (IFSs) and to order weighted averaging operators. The model aimed to execute the various types of uncertain knowledge information in expert systems and presented a max algebra-based reasoning algorithm to automatically implement fuzzy reasoning, thereby improving the traditional FPN. However, there are many circumstances in which it is inappropriate to use fuzzy sets or IFSs to represent experts’ knowledge. Liu et al. (2017b ( 2017b)) proposed a linguistic Petri Nets model by combining the theory of FPN with the concept of interval clouds and the hybrid averaging operator. This model is called cloud reasoning Petri nets (CRPNs) and it is used for knowledge representation and reasoning. Li et al. (2018a 2018a,, b) employed linguistic interval 2-tuples to acquire and represent default knowledge in the interval-valued intuitionistic FPN. Yue et al. (2019 ( 2019)) proposed a self-learning interval type-2 FPN (SLIT-2FPN) for knowledge representation and reasoning. The methods introduced an interval number ordered by the weighted averaging operator to enhance the knowledge reasoning capabilities of SLIT-2FPN and extended TOPSIS to determine the parameters of each fuzzy membership function. Xu et al. (2019a ( 2019a,, b) proposed a new type of FPN, which is called picture fuzzy Petri nets (PFPN), to represent and acquire imprecise, uncertain expert knowledge. knowledge. The existing FPN formalisms lack the self-learning ability to improve the accuracy of a relevant parameter value in the FPN model. It is not easy to accurately represent fuzzy information. Tsang et al. (1999 (1999)) proposed an FPN with learning ability by combining it with an artificial neural network (ANN). Similarly, Similarly, Shen et al. (2010 (2010)) developed two kinds of machine learning PN (MLPN) models based on ANNs to enhance the self-learning of the Petri net (PN) model by supervised and unsupervised learning algorithms. Wang et al. (2014a 2014a,, b) employed an efficient genetic particle swarm optimization (GPSO) learning algorithm to execute a given FPN model’s self-learning function. However, GPSO, the proposed learning algorithm, is not suitable for more complex and large-scale FPN models. Accordingly, Jiang et al. (2019 (2019)) proposed a hybrid GA-SFLA algorithm by combining the advantages of GA and SFLA to enhance the accuracy of each type of FPN parameter. The discussed literature above indicates that the t he knowledge knowledge representation using an FPN could be separated into two phases. From the 1990s to 2015, most methods described fuzzy information by internally modifying the FPN model, as illustrated by WFPN (Chen 2002), 2002), GFPN (Shen 2006 2006), ), and DAFPN (Liu et al. 2013a, 2013a, b, c). However, since 2015, a group of hybrid frameworks combining FPN and other theories were introduced to combine vague

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2017a,, b), SLIT-2FPN (Yue et al. knowledge with other factors, such as CRPN (Liu et al. 2017a 2019 2019), ), and PFPN (Xu et al. 2019a, 2019a, b). Meanwhile, the self-learning ability of FPN has also been widely investigated to improve the accuracy of knowledge representation and reasoning.

4.3 Decomposition strategies for la large-scale rge-scale FPN With the rapidly increasing KBS sizes, the number of FPRs in a rule-based system also sharply increased. As a result, the number of nodes also increased in the correspond1996). ). This phenomenon is called the state explosion ing FPN model (Godefroid et al. 1996 (Valmari 1998 1998;; Milinković et al. 2013). 2013). The state explosion problem severely hinders further applications of the resulting FPN model. The influences of the state-explosion problem in industrial process-based FPNs can be summarized along the following two lines: In the forward inference process, the number of required parameters is raised in the FPN model, which results in the increase in the dimensions of the matrices and vectors involved in the associated inference algorithm. Accordingly, the time complexity of the algorithm will also increase. Hence, the accuracy of reasoning results becomes more challenging with the increasing size si ze of FPN (Liu et al. 2017a 2017a). ). In the backward inference process, the advantages of FPN model, inverse reasoning and algebraic operations are combined. The algorithm separates the part of the rules related to the problem from the knowledge base system, thereby simplifying the reasoning process and meeting real-time requirements. In a decomposition process, whether the decomposed FPN submodel destroys the inference path integrity is the first consid2013b;; Chun and Bien 1993). 1993). eration (Hu et al. 2011; 2011; Liu et al. 2013b To address the state st ate explosion problem, many researchers have have developed techniques such as state space reductions, storage size reductions, parallel and distributed computation methods, randomized techniques, and heuristics in order to decompose largescale FPN models (Clarke and Grumberg 1987 1987;; Clarke et al. 2001; 2001; Demri et al. 2006; 2006; Camilli 2012). 2012). Garg et al. (1991 ( 1991)) proposed a novel type of FPN utilizing three reduction rules to control the scale of FPN and to check the consistency of KBS effectiveness. Lee and Favre (1985 (1985)) proposed a hierarchical reduction method to analyze the existing origin Petri net model. Zaitsev (2004 ( 2004)) proposed a decomposition algorithm of PNs, which can divide a large-scale PN into a series of small-scale PNs using the properties of similar dynamics or structure. Nishi and Matsumoto (2015 ( 2015)) proposed a decomposition approach of PNs to derive a near-optimal solution of deadlock-free and noncyclic scheduling of dual-armed cluster tools to reduce the computational complexity. Zhou et al. (2015a (2015a)) analyzed the dynamic properties of the FPN model, such as reachability, boundedness, safeness, liveness, and fairness. The authors provided theoretical evidence for the design-related decomposition decompositi on algorithm. algorit hm. However, However, as a backward-extended highlevel network topology, FPN can consider fuzzy factors without destroying the original PN structure. Therefore, the PN model’s original decomposition algorithm can be used for reference reference in the decomposition process of the t he FPN model. To reduce the number of FPN nodes, Zhou et al. (2015b ( 2015b)) proposed a decomposition algorithm of FPN by combining the backward search stage and forward strategy methods. The proposed algorithm can divide a large-scale FPN model into a sub-FPN model series using an index function and incidence matrix. All marking values could be obtained simultaneously in an inference process but without enumerating all possible

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paths. Zhou et al. (2018b ( 2018b)) proposed a bidirectional reasoning algorithm by combining a forward forwar d reasoning mechanism with wit h a backward reasoning algorithm to reduce the operational matrices’ dimensions and simplify the reasoning process by removing irrelevant places and transitioning large-scale FPNs to effectively overcome the state explosion problems. Zhou et al. (2019 (2019)) developed an equivalent FPN model for generating corresponding large-scale KBSs by utilizing four primary operations: separation, simplifying, generating, and merging. By executing inference operations, the authors proved that it could completely protect inner-reasoning paths among places and transitions in a large-scale FPN.

5 Reasoning algorithms using FPN As mentioned in Sect. 2, FPN is a kind of backward-extended HLFPN designed to handle fuzzy, vague, and uncertain information without destroying the inner structure of traditional PN. This means that FPN can also describe the KBS by graphical methods (e.g., reachability trees) and algebraic methods (e.g., incidence matrices). Therefore, reasoning algorithms using FPN also have developed from two viewpoints: reasoning algorithms based on the graphical method and reasoning algorithms based on the t he algebraic method.

5.1 Reasoning algorithm b by y gra graphical phical method The primary thinking behind the t he reasoning algorithm is to obtain the final inference re results sults from the corresponding reachability tree of the FPN model. Chen et al. (1990 (1990)) introduced the reachability tree by using an immediately reachable set (IRS) and reachable set (RS). They then generated the equivalent FPN for different types of FPR and presented a forward reasoning algorithm. Kasirolvalad et al. (2004 (2004)) first constructed an AND/OR graph to execute fuzzy inference operations. Similarly, Yang et al. (2002) 2002) created a backward tree inference method. To improve the fuzzy inference ability, Chen (2000 (2000)) proposed a backward reasoning algorithm based on their previous works. The proposed algorithm could automatically estimate the degree of the truth of any proposition. Chen (2002 ( 2002)) proposed a weighted FPN model (WFPN) by adding weight to the FPN model and developed a weighted fuzzy reasoning algorithm using WFPN that automatically performs weighted fuzzy reason reasoning. ing. Yeung and Tsang (1994b (1994b)) established reachability sets and adjacent places and presented an enhanced fuzzy reasoning algorithm based on Chen’s previous work. Like Chen’s work, this work also ignored the function of three types of FPN parameters. To improve the capability of weighted FPN, Yeung and Tsang (1998 (1998)) developed a multilevel weighted fuzzy reasoning algorithm (MLWFRA). In this algorithm, the weight values are thought to improve the accuracy of reasoning results by using a new FPR evaluation method (FPREM). However However,, the certainty factors and the t he weight values of the propositions were restricted and represented by real values between [0,1]. To solve these drawbacks, Manoj et al. (1998 ( 1998)) proposed a novel hierarchical FPN for data abstraction to generate the FPN for a system at different levels of abstraction and reduce the work of determining whether the location is reachable. At the same time, the corresponding reasoning algorithm regarding the hierarchical FPN is also introduced. Amin and Shebl (2014 (2014)) proposed adaptive fuzzy higher-order PNs (AFHOPNs) to control the arc’s weight changes during the fuzzy inference process. The AFHOPN model

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has learning ability as a neural network (NN) and can be used for knowledge representation and reasoning. Moreover, a back-propagation (BP) learning algorithm is used in this AFHOPN model to ensure the convergence convergence of the weights using the transition firing rules. The traditional reachability tree used in the reasoning algorithm can adequately represent the flow relationship between place and transition in the FPN model, but the ’And’ or ’Or’ relationship among FPNs is challenging to deal with. Hence, Zhou et al. 2018a,, b) proposed a modified reachability tree based on and/or graphs to automati(2018a cally generate a corresponding reachability tree model for a large-scale FPN model. It also enhanced the application range of this kind of reasoning algorithm. From the above discussion, discussion, it can be seen that t hat the data structure used by this type of algorithm is complex, and the inference process is still performed using a step-by-step step-by-step method of judgment. The implementation is quite complicated and does not make optimal use of the parallel reasoning capabilities of an FPN.

5.2 Reasoning algo algorithm rithm by the algebrai algebraicc method Although the reachable tree algorithm using FPN have the unique advantages of a description of the flow relationship between the places and the transitions, it cannot fully realize the parallel processing capability of the FPN model. Further, logical relationships such as AND/OR graphs are ignored. Moreover, the need to enumerate all possible paths in the inference process results in relatively low reasoning efficiency. In response, another popular approach approach of reasoning using the FPN model is to combine algebraic forms (matrix-operation) or vectors to execute fuzzy inference operations in a parallel way. On the one hand, the algebra-format method makes the data structure simple and easy for computer processing. Gao et al. (2003 ( 2003)) proposed fuzzy reasoning Petri nets (FRPNs) to execute a fuzzy reasoning algorithm using max-algebra forms automatically.. Fryc et al. (2004 cally ( 2004)) proposed an algebraic representation formal using FPN for performing fuzzy reasoning procedures. Suraj (2012a ( 2012a)) proposed a generalized fuzzy Petri nets (GFPNs) model for knowledge representation and reasoning in decision support systems. This model extends the existing FPNs by introducing two operators: t-norms and s-norms, instead of min and max operators. Suraj (2012b (2012b)) proposed a parameterized fuzzy Petri nets (PFPNs) model by introducing two parameterized families of sums and algebraic product operations to replace previous operator operators. s. The experimental results show that the presented model is more flexible than the GFPN model is via an application in the domain of train traffic control systems. On the other hand, the backward reasoning method can reduce the space complexity in a KBS so that the complexity of large-scale KBSs can be reduced and the KBS can be turned into a more straightforward straightforward system. Scarpelli and Gomide (1993 (1993)) developed a backward reasoning algorithm based on the HFPN model for extracting information from knowledge bases that contain fuzzy production rules. Amit and Lakhmi (2005 (2005)) summarized two underlying reasoning mechanisms, forward reasoning models and backward reasoning models, and introduced a machine learning strategy in fuzzy pattern recognition from noisy training instances using FPN. Zhou et al. (2015b (2015b)) proposed ainclude decomposition algorithm of bidirectional thinking FPN. Thea algorithms a backward search stage and a forward strategy,based whichoncan divide large-scale FPN model into a series of sub-FPN models using an index function and an incidence

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matrix. Furthermore, this bidirectional thinking is also used to reduce the inference 2018b). ). Meng algorithm’s complexity for a large-scale FPN model by Zhou et al. ( 2018b et al. (2016 (2016)) also built an IFPN model to carry out reasoning by matrix operation.

5.3 Analysis of reasoning algorithms As mentioned above, the main reasoning strategies using FPN could be classified into two types. On the one hand, reasoning utilizes the graphical description ability of an FPN, and on the other, reasoning utilizes the algebraic analysis ability of an FPN. In addition, the reasoning mechanisms deployed can be divided into three approaches: forward thinking, backward backward thinking, and bidirectional bidi rectional thinking.

5.3.1 Characteristics of two rreasoning easoning stra strategies tegies The characteristics of these two reasoning strategies are summar summarized ized below. below. The reachability tree is a powerful visual tool to reveal the inner relationshi relationships ps among places of FPN and is frequently used to graphically describe the inference process. Reasoning using reachability trees was widely discussed and applied in the twentieth century to provide a clear path for each inference chain. The core goal of reachability treebased algorithms is to implement the inference process using the graphical description of an FPN. In the inference process, the primary step is to generate the reachability tree of an FPN based on RSs and IRSs and implement different reasoning strategies within the reachability tree. On the other hand, the reasoning algorithm using algebraic forms takes advantage of the parallel operation of FPN formalism. The core thinking of this kind of algorithm is to transform the inference process into matrix operations in order to achieve multiple places to transition transformations at the same time. Table 3  shows the comparisons between these two reasoning mechanisms.

5.3.2 Reasoning mechanisms Based on the summarization of two reasoning strategies, there are three basic reasoning mechanisms: forward inference, backward inference, and bidirectional inference. The advantages and disadvantages of the three reasoning mechanisms are summarized in Table 4. Table 4  shows the main advantages of the above three reasoning mechanisms. The graphical ability of the FPN is fully exploited by the first mechanism to reveal the visual reasoning process based on the inner connections among places. The second mechanism is widely applied to execute parallel operations based on algebraic forms, such as vectors and matrices. The bidirectional inference strategy is proposed to simplify the reasoning process for a complex FPN model by combining the highlights of the first two approaches.

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6 Typical applications using FPN between 2014 and 2020 As a potential graphical tool for modeling and reasoning in a rule-based system or KBSs, FPNs have have been extensively applied in various industrial areas. The typical applications of an FPN from 2014 to 2020 are reviewed and summarized to reveal the latest trends in this section.

6.1 Successful applications of FPN In the past few years (2014–2020), FPNs have been applied in various industrial domains and have gained some remarkable results, including fault diagnosis, risk assessment, manufacturing systems, health care and biological systems, automatic control systems, and so on. In the following, the papers about the FPN application field and highlight(s) are summarized in the Appendix.

6.1.1 Fault diagnosis using FPN 2014)) developed an automatic reasoning algorithm using FPN to judge the Zhang et al. ((2014 transformers’ legal fault cause for The method two main building an IF–THEN str ucture structure and power findingsystems. the t he transformer’s transformer’ s faultinvolves reason using matrixsteps: iteration. The authors indicate that the proposed algorithm makes the diagnosis process clearer and faster. Additionally, the case study verifies the correction and feasibility of the proposed approach. He et al. (2014a (2014a,, b) proposed a novel fault diagnosis method using adaptive fuzzy Petri nets (AFPNs) to implement inference for a complex power system. The proposed AFPN can achieve more accurate results by keeping the descriptive advantages of FPN and enhancing the self-learning ability of FPN. Wang et al. (2014a (2014a,, b) developed a novel hierarchical fault diagnosis approach using a FPN. The method employed a directional weighted fuzzy Petri net (DWFPN) using a fault recorder and WAMS to analyze the smart power grid. The case study shows shows that the t he proposed method can effectively solve solve the defects defe cts and obtain correct cor rect diagnosis results in the presence of incomplete inf information. ormation. Sun et al. (2015 (2015)) proposed a time sequence fuzzy Petri net (TSFPN) to build a fault diagnosis model for a power system. The proposed method considers the influence of incomplete and uncertain alarm information on fault diagnosis. It also employs protective relays relays and circuit breakers’ breakers’ time constraint characteristics characteristics to eliminate conflicting time sequence information. Wang et al. (2015 ( 2015)) developed a fault diagnosis algorithm based on FPN to address uncertain power grid elements. The discussed algorithm utilizes the features of FPN to execute the logical inference on the base of an associate tree for finding located in point of failure completely. Zhang et al. (2016 (2016)) employed the FPN model to implement a fault diagnosis of power systems. In this strategy, strategy, the FPN is used to cope with temporal constraints and fuzzy inforinformation, and the temporal attributes are assigned to the propositions in PNs to ensure that the obtained structured model can adapt to new protection schemes and topology changes. The proposed approach is suitable for online fault diagnosis in large-scale power systems compared with three existing fault diagnosis methods. To decrease the complexity of diagnosis processing, Wang et al. (2016 (2016)) proposed a fault recognition method using fuzzy fault Petri nets (FFPNs) to simplify the process of fault propagation in a high-speed train traction system. The proposed method has better validity and practicability in obtaining fault

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recognition through forward and backward reasoning algorithms using associated failures between equipment. Cheng et al. (2017 (2017)) proposed a novel approach to implement fault diagnosis for a power grid system. A hierarchy-weighted fuzzy Petri net (WFPN) is constructed by considering the related time association character (TAC) to improve the diagnostic accuracy. Specifically, the TAC consistency check is used to correct the operation probability of the protection and circuit breaker that does not meet the time correlation consistency consistency.. This effectively reduces the failure probability of nonfaulty components and greatly improves the accuracy of fault diagnosis. Sun et al. (2017 ( 2017)) proposed a fuzzy fault Petri net (FFPN) to address the fault propagation characteristics in power systems. The established FFPN model aims to describe the system failure occurrence, evaluate the severity of the fault and transmission path, and provide real data maintenance. Zhang et al. (2018 (2018)) proposed intuitionistic FPNs (IFPNs) to represent the certainty and uncertainty of alarm information in a power system. The reasoning mechanism using IFPNs is used to calculate the certainty and uncertainty of electrical device fault events and evaluate the robustness of real-time diagnosis in complicated circumstances where multiple components operate incorrectly. Li et al. (2018a ( 2018a,, b) proposed a fuzzy neural Petri nets (FNPNs) method to improve a resistance strain weighing sensor’s operating stability and eliminate vague factors in the fault diagnosis process. The proposed approach establishes a fault diagnosis model by FNPN to adjust the t he weights of models and gain the order of faults for each each part using PN characteristics. Liu et al. (2018a (2018a,, b) proposed a modified online fault diagnosis algorithm by using generalized generalized mutual exclusion constraints (GMECs) with integer linear programming (ILP) and the FPN model. The algorithm further furt her addresses the ILP problem of the POPN model with a remarkably increased DES diagnosis ability ability.. Xu et al. (2019a (2019a,, b) proposed a novel fault diagnosis model based on temporally constrained fuzzy Petri nets (TCFPNs) to address the uncertainty and temporal constraints of alarms for power system fault diagnosis. Similarly, Tan et al. (2019 ( 2019)) proposed intuitionistic fuzzy Petri nets (IFPNs) based on time series matching to improve the reliability of power (2019)) both focused on improving grid fault diagnosis. Xu et al. (2019a (2019a,, b) and Tan et al. (2019 the accuracy of the diagnosis results of power systems by considering uncertain information and alarm information. By use of an example, the accuracy of the fault diagnosis of the proposed method is demonstrated. Wang et al. (2020 (2020)) proposed an adaptive neural fuzzy Petri network (NFPN) algorithm by combining BPNN and FPN to execute fault diagnosis for a complex system with a three-phase asynchronous motor. The established NFPN model-based motor system using the Gaussian function is employed to replace the traditional transition confidence in the NFPN model and to enhance the flexibility of NFPN by utilizing the t he backpropagation backpropagation features of BPNN. Jiang et al. (2020 ( 2020)) proposed a weighted fuzzy neural Petri net (WFNPN) to create a microgrid fault diagnosis model. On the one hand, the parameters of the WFNPN are trained using a fuzzy neural network (FNN) to enhance the calculation accuracy. On the other hand, WFNPN is used to reduce the computational complexity of the diagnosis process.

6.1.2 Risk assessment using FPN Bharathi et al. (2014 (2014)) developed a FPN based forecast model to predict the post-implementation risks in small and medium enterprises (SMEs). In the same year, Bharathi (2014 ( 2014)) utilized the FPN model to assess the risks inherent in the implementation phase of SMEs.

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These two projects aim to develop the prediction and assessment risk tools of ERP in SMEs, to further enhance the accuracy of the risk assessment technique and to expand the areas to which risk assessment tools can be applied. Pramod et al. (2014 ( 2014)) utilized the FPN model to assess the risk associated with the post-implementation phase or the usage phase of enterprise resource planning (ERP) in small and medium enterprises. The method provided an effective and intuition means of representing and quantifying risks in the different phases of ERP adoption for SMEs to enhance the level of risk assessment. To expand the previous research results in 2014, Bharathi et al. (2017 (2017)) addressed the problems of risk identification and assessment during the extension phase. The authors utilized the FPN model because the risks are categorized and further graded for decision support to SME stakeholders. Chiang (2015 (2015)) proposed an ECG-based mental stress assessment system by combining fuzzy theory and an associated PN. This system detects mental stress and reduces subjective errors accurately by using SVD, MEPA, and APN. Moreover, this system provides diagnostic information immediately by producing a mental stress assessment model. Meng et al. (2015 (2015)) proposed a local adaptive fuzzy Petri net (LAFPN) to predict the t he shortage of water risk values in Beijing. The LAFPN could adjust the local weights according to the t he real situation and respond to chang changes es quickly to enhance the improvement of the accurate prediction of the risk value. Guo et al. (2016 (2016)) developed a comprehensive risk evaluation method using the FPN model for long-distance oil and gas transportation pipelines. The proposed model describes the propagation path of the fault and the systems’ characteristics. Meanwhile, the model also provides some decision support for the risk management of oil and gas pipelines to reduce the risk. Kuchárik and Balogh (2016 (2016)) developed a new form of evaluation approach by combining the FPN and Trans Place Sim tools to assess the risk of prioritizing the teaching process. Zhou et al. (2017 (2017)) proposed weighted FPNs (WFPNs) and a corresponding security risk inference method using matrix operations to evaluate the importance of risk factors and their different relationships in chemical industry areas. Majma et al. (2017 ( 2017)) proposed a hierarchical hierarc hical fuzzy colored Petri-net (FCPN) to monitor device software behavior to reduce the failure risk of medical devices. Compared with the simple PN and nonhierarchical FCPN, the HFCPN has less runtime for device runtime verification. Zakharov et al. (2018 (2018)) proposed a modified FPN to represent and evaluate the fuzzy production system’s system’s ecological risk. r isk. Compared with the conventional approaches, approaches, the proposed method directly defines and solves the optimization problems by determining the complex frequency distribution of the original parameters or an unknown frequency disdeveloped a fuzzy reasoning algorithm using the FPN model tribution. Chang et al. (2018 ( 2018)) developed to assess the risk r isk degree of the t he deep-water deep-water drilling riser system, in which the analytic hierarchy process (AHP) and entropy method (EM) are adopted to improve the accuracy of evaluation. Yao et al. (2018 (2018)) proposed a modified FPN to assess the safety safety risk in air traffic control. While focusing on the huge number of nodes and strong subjectivity of change weights in the assessment process, the risk level threshold, the AHP and the utilized reverse rev erse search method are used in this proposed modified FPN to simplify the complicated evaluation eva luation process to improve improve the accuracy of the assessment result. Shi et al. (2019 (2019)) proposed a novel failure mode and effect analysis (FMEA) method by hybrid FPN and a fuzzy evidential reasoning strategy to improve the risk assessment of traditional FMEA and determine the failure risk priority modes in FMEA. The proposed risk assessment method produces more reliable risk ranking results and aims for applications related to more fields to enhance the safety and reliability of products, processes, and services. Li et al. (2019 ( 2019)) proposed a new FMEA method by combining linguistic term

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sets (PLTSs) and FPNs to assess the risk failure of reciprocating compressor failure in a marine ship system. Sachan and Donchak (2020 ( 2020)) developed a generalized stochastic Petri Net (GSPN) with fuzzy parameters to evaluate infrastructure asset management policy and enhance the reliability and availability availability of infrastructural assets. Zhou and Reniers (2020 (2020)) proposed a weighted fuzzy Petri net (WFPN) with inhibitor arcs to model relationships between risk factors and establish the risk assessment structure considering veto factors. The WFPN model can properly address the risk assessment problem considering veto factors and provide useful and reasonable results. Zhang et al. (2020) 2020) proposed an interval intuitionistic integrated cloud Petri net (IIICPN) model. The improvements improv ements of the IIICPN can be summarized summari zed into two parts. On the t he one hand, intervalvalued intuitionistic uncertain linguistic numbers have been converted into IIICs. On the other hand, related definition and reasoning rules are introduced into the IIICPN. Finally, Finally, a case study of subway fire risk assessment is used to verify the correctness of the proposed model. The experiments reveal that the proposed IIICPN can overcome the defects of FPNs and trapezium clouds.

6.1.3 Manufacturing ssystems ystems using FPN Baruwa and Piera ((2014 2014)) proposed a timed colored PN (TCPN) approach to evaluate schedule flexible manufacturing systems (FMSs). Compared with traditional methods, the presented method can be parallelized on several processors with distributed memory to increase the amount of memory available. available. This enhances its capabilities to return solutions for large-sized problems. Li et al. (2014 ( 2014)) proposed a colored Petri net and a hybrid metaheuristic by combining simulated annealing (SA) and tabu search (TS) to optimize the plan schedule of manufacturing systems and thereby provided a theoretical basis for more complicated industrial cases. Mhalla and Benrejeb (2015 (2015)) proposed a monitoring-maintenance method based on FPN for maintenance (FPNM) using P-time PN and fault trees to model manufacturing systems with time constraints and analyze the behavior of diagnosis operations. Basak and Albayrak (2015 (2015)) proposed object-oriented Petri nets (OOPNs) to model and analyze the shop floor scheduling problem of flexible automotive manufacturing systems (FAMSs). Case analysis illustrates that the performance evaluation problem of timed PN can be transformed into a simple linear programming problem using the proposed OOPN model. Yu et al. (2016 (2016)) proposed object-oriented Petri nets (OOPNs) and π-calculus approaches. On the one hand, OOPNs are applied to visualize the initial structure and system behaviors behaviors of reconfigurable manufacturing manufacturing systems (RMSs). On tthe he other hand, π-calculus is used to describe the reconfiguration of RMSs. Similarly, Kahloul et al. (2016 (2016)) employed reconfigurable object Petri nets (ROPNs) to model, simulate, and analyze RMSs. Shah et al. (2017 (2017)) proposed a colored Petri net (CPN) with a fuzzy factor to reduce the total number of elementary circuits of flexible manufacturing systems (FMSs) and decrease the complexity of the system. Although the presented framework framework does not predict tool life monitoring, it can keep track of the entire machining time during a cycle. Saren et al. (2017 (2017)) proposed a hierarchical colored Petri net (HCPN) to resolve conflict situations in manufacturing processes. The proposed HCPN plays an essential role in the cell-based overall ove rall performance on arrival time and part processing time in the hierarchical model and provides useful information about system behavior for this manufacturing system.

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Saren et al. (2018 (2018)) extended their previous work to construct a delivery date model using HCPN approaches. In this approach, three main modules of the FMS system, namely EDD, EXACT, and OVER, are defined as three sub-models. Compared with other CPNs, this HCPN model easily analyzes the behaviors of the complex FMS. Fei et al. ( 2018) 2018) developed ‘switching the machine on/off’ status approaches using FPN to reduce the idle period in manufacturing systems. Simulation experiments validate that machines’ idle status can be effectively transformed into an energy-saving status to realize energy-efficient production by utilizing the proposed approach. approach. Wang et al. (2019 (2019)) proposed a dynamic adaptive fuzzy reasoning Petri net (DAFRPN) to infer or decide on machine statues for reducing idle times by considering energy-saving aspects: real-time buffer levels and production rates in manufacturing systems. Experimental analysis shows that the proposed model can effectively anal analyze yze in parallel, assembly and disassembly structures of manufacturing systems. Chen et al. (2019 (2019)) built an FPN model to analyze the fault diagnosis of manufacturing systems by executing a bidirectional reasoning algorithm. The experimental results verify the reliability and accuracy of the fault diagnosis results. Kaid et al. (2020 (2020)) proposed a different deadlock control strategy for automated manufacturing systems (AMSs) based on PNs and NNs. The strategy includes three steps: to obtain the controlled system (deadlock-free), to merge all obtained monitors into a single monitor by CPN (colored Petri net), and to detect and treat faults by combining neural networks with CPN. Al-Ahmari et al. (2020 (2020)) proposed a two-step robust deadlock control strategy to analyze AMSs with unreliable resources. A live (deadlock-free) controlled system is derived by using strict minimal siphon control, and then deadlock control is solved by CPNs. The proposed approach inherits the advantages of PNs and NNs to address the deadlock problem in AMSs and to detect and treat failures. Nabi and Aized (2020 (2020)) developed a compact, editable, and deadlock-free deadlock-free flexible manufacturing system model based on CPNs to fully represent discrete event dynamical system characteristics and build a precise and stable model of a discrete event dynamical system.

6.1.4 Health care and biological system usi using ng FPN Tao et al. (2014 (2014)) employed FPNs to replace the knowledge model of agents to handle uncertainty information and analyze and verify agents’ communication with each other in collaborative health care diagnosis systems. Compared to the traditional FPN formalism, there are two main differences. On the one hand, tokens are put on arcs instead of places. On the other hand, places are permitted the capability to control the propagation of tokens. These changes make FPN more suitable for logical inference and more flexible in applications where knowledge is frequently updated. Chen et al. (2014 (2014)) proposed a rule-based decision-making diagnosis system using FPN to evaluate arteriovenous shunt (AVS) stenosis for long-term hemodialysis treatment. The study employed human diagnosis work to present a fuzzy inference algorithm for a rule-based system, and the degree range was subdivided into nine scales to analyze the occlusion levels. Hamed (2015 (2015)) proposed an adaptive FPN (AFPN) reasoning algorithm for rule-based systems. The proposed algorithm predicts potential risk for esophageal esophageal cancer based on the serum concentrations of C-reactive protein (CRP) and albumin as a set of input variables. Moreover, Moreov er, this approach combined FPN reasoning and the capability of an adaptive learning mechanism to make qualitative inferences of a biological system. Chiang (2015 ( 2015)) developed a rule-based reasoning model by combining fuzzy computing with associative Petri

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net (APN) for electrocardiogram (ECG)-based mental stress assessment. The approach combines SVD, MEPA, and the APN to build a mental stress assessment model that can address signal noise and instability and precisely identify the degree of mental stress. Chiang and Pao (2016 (2016)) proposed a hybrid inference system combining fuzzy theory and the APN model to implement early diagnosis and prediction of Alzheimer’s disease. The case study shows that the proposed model could effectively detect Alzheimer’s disease and use it as an objective tool for clinical diagnosis assistance. Pennisi et al. (2016 ( 2016)) proposed colored PNs (CPNs) to model the cellular scale humoral immune system. The proposed model has a suitable level of granularity in the description of cell behavior. It permits the analysis of the biological network’s fundamental properties under study, such as reachability,, liveness, boundedness, place invariants, circuits, and t-invariants. ity Hamed (2017 (2017)) developed a quantitative framework of biological systems using FPN. Liu et al. (2017a (2017a,, b) proposed a fuzzy continuous PN (FCPN) that combined CPN with fuzzy logic to model biological systems with uncertain kinetic parameters. The tw two o studies show that FPN is a reasonable model and very successful for the information representation of intuitionistic fuzzy frameworks. Chiang et al. (2018 (2018)) proposed a diagnosis model of sleep quality based on fast Fourier transform and the FPN to explore the impact of various presleep activities on subsequent sleep quality. Experiments demonstrate that these variables are adequate for assessing the impact of different bedtime activities on sleep quality. Liu et al. (2018a (2018a,, b) systematically reviewed the existing modeling methods of uncertainty with biological systems using FPNs and classified them into three categories: basic FPNs, fuzzy quantitative Petri nets, and Petri nets with fuzzy kinetic parameters. A fuzzy continuous Petri net (FCPN) is proposed by combining CPNs and fuzzy logic by viewing some kinetic parameters as missing or inaccurate. Simulation experiments show that the FCPN algorithm can be used to model and analyze biological systems with uncertain kinetic parameters. Y Yang ang et al. (2018 ( 2018)) established a biological system model based on CPN to represent the flow of genetic information involving transcription and translation in the DNA molecular biological system. The modeling method provides an intuitive and clear description of the detailed state changes in the process of protein synthesis and establishes a theoretical foundation for intuitively modeling and analyzing other essential biological processes. Rosdi et al. (2019 (2019)) proposed a classification method using FPN to evaluate the speech intelligibility of children with speech impairments. This method can improve the discrimination ability of automatic speech detectors detectors for children with speech impairments. Liu et al. 2019)) developed a framework to simulate a biological system with uncertainties by com(2019 bining continuous PNs with fuzzy theory. This method provides a suitable hybrid uncertain/certain approach to modeling and analyzing biological systems with uncertainties.

6.1.5 Control system using FPN Weng et al. (2014 (2014)) developed a parallel railroad crossing control system using TPNs. The proposed model demonstrated how to verify the level of parallel railroads from the models to critical scenarios for crossing traffic systems. Additionally, the proposed model can quickly quickl y amend some operations to handle some exceptiona exceptionall cases, such as inter interrupting rupting traffic lights’ regularity to allow emergency cars to be prioritized. Tian et al. (2014 (2014)) proposed a fuzzy hybrid Petri net (FHPN) to control the traffic signal system. This method not only computes the best green light duration of each stage in a period, but it also determines a reasonable and alterable stage sequence. He et al. (2014a ( 2014a,, b) proposed an airplane fault

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diagnosis model for an environmental control system (ECS) using weighted FPN (WFPN). The failure diagnosis method uses a reverse inference control strategy by considering the false alarm rate and underreporting under reporting rate to improve improve fault locating. Shen et al. (2015 (2015)) proposed an extended HLFPN for smartphones to analyze, develop, and identify human actions, such as regular action, exercising, and falling. A smartphone could execute fall detection using this algorithm on either side of the thigh pocket without a specific figure on a particular area. Lin and Li (2015 2015)) proposed a dynamic Petri fuzzy cerebellar (DPFC) model by integrating a cerebellar model articulation controller (CMAC) to simulate the magnetic levitation system (MLS) and a two-axis linear piezoelectric ceramic motor (LPCM) drive system. In the DPFC model, the online algorithm for learning the controller’s parameters is obtained by using the Lyapunov function to guarantee the system’s stability. Experimental results show that the proposed control scheme has a perfect tracking response to analyze, develop, develop, and identify human actions, such as regular actions, exercising, and falling. Hajduk and Wojtowicz (2016 (2016)) proposed an FPN with Lukasiewicz norms to model and analyze the costs of implementing control systems. Letia and Kilyen (2016 ( 2016)) proposed a hybrid control system (HCS) model to describe discrete event and discrete-time features. The model combined the enhanced TPN with fuzzy logic rules (FLETPNs) to describe the concurrent, synchronous, and asynchronous behavior of HCS. Bibi et al. (2017 (2017)) proposed a Petri Type 2 fuzzy neural network (PT2FNN) approximator to realize adaptive control of uncertain nonlinear systems. The PT2FNN is obtained by integrating the Petri layer into the T2FNN and replacing the K–M iterative algorithm with adaptive modulation. Compared to T1FNN and T2FNN, the PT2FNN approach has better performance. Kim and Yang (2017 (2017)) developed a self-navigating robot system using FPN to simulate, evaluate, and interconnect the navigational algorithm and solve the unknown maze problem. Hansen et al. (2018 (2018)) utilized an FPN model to analyze soccer ball recognition and distance prediction. Ammour et al. (2018 (2018)) proposed a partially observed stochastic PN (POSPN) to model a system with sensors for predicting the faults of discrete event systems. Sun et al. ( 2018) 2018) proposed a fused fuzzy Petri net (FFPN) with a shared control method to solve the control problem of a brain actuated discrete event system by combining automatic robot control (AC) with brain actuated control (BCI). The simulation test builds a software model using the FFPN to adjust the control parameter and achieve achieve the required control. To extend his previous work, Bibi et al. (2019 (2019)) employed PT2FNN to approximate 2019)) proadaptive control for a class of single-input single-output nonlinear systems. Le ((2019 posed a self-organizing recurrent interval type-2 Petri fuzzy controller (SORIT2PFC) to control time-varying delay systems by hybridizing an interval type-2 fuzzy network, recurrent network, and FPN. Moreover, a self-organizing algorithm is also proposed to optimize the construction of the SORIT2PFC network, network, improve system performance, and reduce the computational burden. Sha et al. (2019 ( 2019)) proposed a cooperator fault detection system using a FPN to organize several fault detectors in a smart factory that has a complex production process. Bibi et al. (2020 (2020)) proposed a hybrid structure by combining type 2 fuzzy logic and PN (PT2FNN) as a new approximator to reduce uncertainties with optimal cost and time consumption.

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6.1.6 Other typical applications using FPN Shen and Chen (2015 (2015)) employed a high-level fuzzy Petri net (HLFPN) to complete music segmentation and karaoke system correction. The proposed model can implement an automatic calibration system for Chinese karaoke lyrics and obtain song lyrics by GPRs and HLFPN. Cheng et al. (2015 (2015)) proposed an automatic web service composition method based on a fuzzy predicate Petri net (FPPN) to model the Horn clause set and the T-invariT-invariant technique to determine the existence of composite services fulfilling the user input and output requirements. Suraj (2015 (2015)) proposed a modified generalized FPN (GFPN) to model the rain traffic control system. The GFPN model combined the graphical power of Petri nets and fuzzy logic to model rule-based expert knowledge knowledge in a decision support system. Suraj et al. (2016 (2016)) proposed a flexible generalized fuzzy Petri net (FGFP-net) by GFPN to optimize decision support systems. The proposed approach can be used to control design, knowledge representation and reasoning in decision support systems. Xia et al. (2016) 2016) developed a human–machine (HM) system model using a fuzzy inference Petri net (FIPN) to realize adaptive task allocation (ATA) between human operators and machines. The FIPN constitutes a new modeling tool suitable for systems that combine (Mamdanitype) FIS and discrete logical switching. Fat et al. (2016 ( 2016)) built a stock market price prediction strategy using fuzzy logic enhanced TPN (FLETPN). On the one hand, the GA algorithm is used to optimize the system parameters to effectively assess the change in stock market prices. On the other hand, the model can also identify asynchronous events influencing the market. Nie et al. (2017 (2017)) built a personalized learning model using FPN to analyze students’ learning behaviors, characteristics, and mastery of knowledge level in education systems. To improve the precision of individualized education, students’ level of knowledge and learning speed are considered essential factors. In the implementation, this is combined with the outer edge of the student’s knowledge state. The simulation results show that the proposed method can dynamically plan the optimal learning path for students and compares well with other predominant personalized education methods. Shen et al. (2017 (2017)) proposed a shot boundary detection (SBD) hybrid method by combining HLFPNs and keypoint matching to address the membership function of SBD in social media to adapt the transition variation of each video. On the one hand, removing all possible false shots is used to improve the precision of SBD on regular transition. On the other hand, the hybrid approach approac h of the HLFPN model and keypoint matching can efficiently increase the precision of SBD by removing false shots and can be applied to various types of videos. To further improve the overall performance of HM systems, Zhang et al. (2017a ( 2017a,, b) proposed a fuzzy inference PN (FIPN) to model the continuous and discrete components of an HM hybrid system. The model is uses two methods to predict operators’ performance using WangMendel (WM) fuzzy modeling and optimizes the membership function parameters of the fuzzy OFS model using an artificial bee colony (ABC) algorithm. Shen et al. (2018 (2018)) developed a financial investment system model as a HLFPN to decide the stock prediction system’s investors, enhance profits and assist investors so that the desired investment targets are met. Furthermore, fuzzy neural reasoning or technical analysis is used to decide on investments. Yang et al. (2018 ( 2018)) proposed novel dynamic timed fuzzy Petri net (DTFPN) approaches based on dynamic time delay analysis (e-DTA) to overcome the limitation of the TFPN field region. Compared to traditional methods, DTFPNs have have a strong ability to describe dynamic correlations among process variables.

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Yang et al. (2019 (2019)) employed an HLFPN model to detect a threshold value in shot boundary detection. Compared with other methods, the proposed model can reduce the time in the video stream process and improper shot changes caused by motions. Zhang et al. (2019 (2019)) proposed a random FPN with a time delay to model behavior reconstruction systems and control the user behavior character. Using this formalism, the behavioral flow of different user groups is constructed by the system behavior reconstruction model. Shen (2019 2019)) proposed a high-level fuzzy Petri net (HLFPN) with an abstract syntax tree to detect code plagiarism operations. The experimental results reveal that the proposed method makes a better determination of code plagiarism detection. Cheng et al. (2019 ( 2019)) developed a fuzzy spatiotemporal ontology model using FPN to construct fuzzy spatiotemporal ontologies. The fuzzy spatiotemporal data represented by FPN models are directly upgraded to semantic web contents by using this approach so that the existing fuzzy spatiotemporal data resources in fuzzy Petri net models can be used in the context of the semantic web. Samantra et al. (2020 (2020)) proposed an FPN to optimize the ad hoc network for discovering an optimal route between the source and destination by considering different factors, such as node battery power, power, node mobility, and path length. Shahidinejad et al. (2020 ( 2020)) proposed an elastic controller using CPNs and a queueing system to manage cloud infrastructures automatically. Compared with other approaches, the proposed method can significantly outperform in terms of the elasticity value, response time, and CPU utilization.

6.2 Bibliometric analysis Based on the analysis in Sect. 6.1, 6.1, 93 articles are classified into various application fields as illustrated in Fig. 5. Figure 5  shows the extensive use of FPNs in traditional industrial fields, including fault diagnosis and manufacturing manufacturing systems. Therefore, Therefore, these areas have yielded significant achievements by combining HLPN and HLFPN. There have also been academic attempts to improve the accuracy of fault diagnosis result in the fault diagnosis process by combining it with other HLFPNs and as also in the manufacturing engineering sector. Compared with the traditional applications, FPN has been widely applied in various complex systems, such as health care and biological systems and risk assessment. In the health care and biological sector, FPN is mainly used for disease assessment and diagnosis in medical engineering and the performance improvement of medical equipment. In the risk assessment sector, the main focus has been on the risk evaluation method and the accuracy of risk value estimation by combining HLFPN and other theories.

Fig. 5 Number of articles on FPN application in various fields

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 1 3

 

W. Jiang et al.

6592

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  n   o   n   n   n    i   s    t   o   y   o   o    k   a    i    i    i   m   s    t    t    t    t    i   a   r   r   a   a   a    t   r    d   r    t    t    t   o    l   s   g   e   n   n   e   n   p   u   o   g   s    d   e    fi   e   e    i    l    d    d   a   n   e    i   n    t   o   r    t   m   e    l   m   e   a    l   m   e    l   r    h   r   a    fi    i   o   n    l   c   g    t   p   p   p   a     e   a   c   n   s   m   i   m   o   m   i   n    i   c    i    i   r   r    i    h    t    l    l    N    i   e   c   c   e    t    P    P   e   a   p   a    P   m   e    P   p   p    R    R    l    F    i   e    R    h    A    E    E    E    W   P    T    E    C   g   n    i   s   u    t    )   n    4   e    1   m    0   s    2   s    (   e    i   s   s    h    t   a    )   a   r    6    k   a   s    1    i    h    0   r    2    B    f    (  ,   o    )    )    h    )    4    7   n   g    4    1    1   o    )   o    i    1    l    0    )    0    t    5    )    0   a    2    2   a    1    6    (    7    2    1    (   c    B    0    (    1    i    )  .  .    2    0    l    l    l    2  .    0    5    d    (    l   p   a   a    (    2    1   a    0    l  .    (   n   p    t    t  .   a    t    l   a  .    k    A    (   e    d   e    2   e    t    )   s    i    t   a    l    (    i   a    h    h   e   r    i   r    t    t    t   g   e     6    á   o   o   e   a   a   n   g    e    h   r   r   a   n   o    h     l   c    t   a   m   a   u    i   o     b   a   e   u   u   u    h    h    h    h   r    a     T    A    B    P    C   M   G   K   B    Z

 1 3

  m   e    t   e   c   s    i   s   v   y   a    d   e    l   c    l    i   a   c   g   o    i    l    d   o   e   c    M   E

   )    8    )    1    7    0    1    2    (    0  .    2    l    (   a  .    l   e   a    t    t   e   o   v   r   a   a   m   h    j   a    k   a    M   Z

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   )    0    2    )    0    0    2    (    2    0    k    2   a    )    (    )    h    8   s   c   r    0    1    )    2    )   n   e    0    8    0    i    )    9   o    2    1    2   n    9    1    (    (    0    D   e  .  .    2    0    1    l   a    (    2   a    (    0    d    R    l    (    2  .    d   n    t    t   e    l   a  .    l  .   n   a   a    l   n   a   e    t   g   a   g    t   a   e   u   n   e    t   n    h   a   e   o   a   o    i   c    h   a    h    i   a    h    h    C   Y   S    L    S    Z    Z

 

6593

Modeling, reasoning, and application of fuzzy Petri net model:…

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  n    2   e    k    (   o    j   a    i   r   e   a    t   r   r   y   a   a    )   n   e   c    )    i    i   e    6    b    l    4    l    P    1    B   p    1    A   0    d   p    0    d    2   n   n    (  .   n    d    A    (    )   s   a    l    (    2  .   a   a    l   r   a   a     7   a    l   o   w    k    t    l    e    t    h   u     l   e   a   a    t   e    h   s   r     b   u   a   u   a    i    a     T    A    B    L    M   B    Y

  m   e    t   s   y   s   g   n    i   r    t   c   u   a    f   m   u   e   n   a    t   s   m   y   s   e   g    l   n    b    i   a   r   r   u   u    t   g   c   a    fi    f   n   u   o   n   c   e   a    R   M

  s   m   m   e    t   e    t   s   s   s   y    i   n   g    i   s   y   s   y   y   g   n   s   r   r   g   g   g   g   u   u   n   n   n   n   m   i   m   e    t    t    i    i    i   e   r   r   r   r   c   c    t    t   a   a   u   u   u   u   s   s    t    t    t    t    f    f   y   y   c   c   c   c   u   u   s   s   a   n   n    f   a    f   a   a    f    f   a   a   u   g   g   u   u   u   n   n   m   m   n   n    i   n   a   n    i   r   r   a   a    d    d   u   a   u    t   m   m   m   t   m   e   e   c   c   e    t   e    l   e    f   e   a    t   a    l   a    l   a    f    l    b    b    b   m   m   b   u   u    i    i    i    i   o    t   o   n   n   x   x   x   x    t   a   a   e   e   e   e   u   u    l    l    l    l    F    F    M   M   F    A   A   F

   )    6    1    )    0    7    2    1    (    0  .    l    (   a    2    t  .   a    l   e    l    t   u   e   o    h    l    h   a   a    h    K   S

   )    )    0    2    0    2    0    0    2    )    )    )    )    2    (    )    9    7    8    (    9  .    0    1    1    1    )    l    1    2   a    0    0    0    8    0    0    t    d   e    2    2    2    1   z    2    (    2    (    (   e    i    (    0    (  .  .  .    i    2  .    l    l    l    A  .    t   a    t   a   a    l    (  .    t    t   a    l   a   r   a    l    t   n   e   e   m   d   e   a   e   e    h   a    t   g   n   n    i   n   e   n   e    d   e    i    A   r   e   -    b   r    i   a   a   a   e   a    h   a    l    S    S    F    W   C   K   A   N

  s   s   m   e   m   s   s    t   e    t   s   m   m   s   y   y   e   e    t   s   s    t   s   s

 1 3

 

W. Jiang et al.

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   N    P    F   g   n    i   s   u   g   n    i    t    l   e   n   e    d   o   m    i   r    t   m    i   n   m   e   a   o   m   p    i   s    t   e   y    t   m   a   m   s    i    l   s   e    t   y   m    h   s   r   s   a   c   c   y   o    l   e    i    f   s   a   e   n    l   c    i   m   p   g    i   o   a   s    l   g    i   c   t   c   e   e   o    i   o    i    t    l   s   g   e    h    b   o   o   n   y    i    l   s    t   r   e    b   y   a   o    l   o    i    f   g   e    t   a   y    f   c    i    b   z   c    t   s    i   y   s   a   o   n   y    l    f    N   e    i    t   g   a   a   z    i    l   a    P   o    l   y   a    t   y   n   c   a    F    l   r   m   w   z   o   l   o   a   u   e    d   z    i    fl   n   q   c   h   n    d    t   e   b   a   u   g   n   a    i    f   n   n    i   r   r   p    h   a   s   u   o    t    l    d   a    d   e   e    l    t   u   n    l   e    f   n   g    t   u   a   e   a   o   l   a   y    l   n   c    l   s   u    d   a   s   g   e   g   e   a   o    f   e   s    l    N   l    d   o   o   o   n   e   o   v    P   o    d   r    l    i   e   m    h    C   n   o   p   m   e   o   n    t   m   d   e   o   e   o    t    h    t   g   r    l    A   s   o   o   m  a   n   c   e    i   o   N   l   e    t   e    d    t   m   n   e    t   s    l   u   g   e   s    i   r    l    D   d   o   o    i   s   a   o    N   e   h    k   r    i   e   m   n    d   m   o   m   N    l    P    d   o    t   o   e   n   m   e    C    N   m   i    d   g   e   n    P    P   w   a   e   o    N   i   r    F    l   e    F    C    h   a   n    P    d    t   m    F   m   v    t   o   a    C   a   w  e    i   o   a   n   r    t   s    f   p    F   p   e   o   n   l   e   a   a   p   a   s   m   l   a    l   p   o    i   o   a    l   o   v   o   e   n    l   e    d    d    d   e   e   e   r    l    l    l   a    i   v    R   P    i   v    i   v   r   p   e   u   e   u   t   u   e  .  .   m    B   D   B    D   1   2    B    I    D

   N    P    t    ’   s    F   n   g   e   n   g   a    i   s   u   y    f    i   m   r   e   e   v    t   s   y    d    N   s   n    P   s   a    i    F   s   n    A   o   o    i   n   g    t   g   a   n   a    i    i   s    N   m    d   r   u    P    l   o   g    F    f   e    N   n   g    d   n    i    P    i   n    k   o    F    i   y   a   s   z   m   g   m   z   n   -   u   g    i   u   s   n   n    f   m    i   u   o    h    i   e   n    t    l   s    i    h   o    i   e   r    t   s   c    d   o    t   a   e   g   e   n    d    l   r   o   e   a   s    d   m   e    d   s   a   s   e   p   r    i   s    i   n   g   a   s   e   o   e   n    b   r    b   n     o     e   s   o    l   a    l   e   g   a    t    i   u   u   e   r    d   r   r   s    N   a   a   a   a    t    P    h   p   p   e   g    F   r   s   s   o   e   o    i   e    l    l   e    l   o   o   z   e   e    i    h    h   p   p    l    t   v   g    i   o    i   e   o    t   o   v   r   r   e    H    U    D   P    P    D    t   n   e   m   s   s   e   s   s   a   s   s   r   e   e    N   r   c    t    P   n   s    F   a    l   m   c   g   a   e   g    t    t    l   n   n   s    i   n    i   a   s    l   y   e   e   s   e   u   g   m    d   s   o   e   m    i   s   e    h   a    d   p   o   m    t   s   o   s   n   a   s   s   n   y   g   m   e    b   s   o   s   a    i   e   -    t    i    i    t    f    l    )   o   s    d   a   o    G   c   n   y   c   e   e   e   m    i   s    k    t    t   s    C   e   r   e   g    l    t   a    S    i    E   o   a   s   r    (    d   c    t    l    l   c   y   y    i    h   e   n   o   s    t    t   s    i   a   s    i    l   g   u    i    l   a    b    d   a    h    t   m   e   m   m   a    l   o   e   a   e   n   n    d    l    S   e   u    i   e    t   s   r    i   o   e    h    t   u   e   n   q    t   g   s   s   s    d   a    fi   e    b   o   o   m   u   y   y   p   y   v   s   s   s   e   o   p    i   e    i   n    ’    d   m   l   r    t    l   r   n   e    i   r   e   o   a    l    t   a   n   a    i   e   a   e   a    l    h   r    t   c   c    S    i   c    i    t   c   m   a   v   a   a    i   o    h   r   o   s   o    t   s   c   g   r    l   r    i   o   g    i   s    t   e   s   r    t   a    i   o   e    l    b   o    l   e   a   e    l    h    l   e   c   m   p   c    l   e   s   s    t   z   o   s   n   o    i   e    l   r   s    l   u    i    h   p   o    f    A    C    B    H    A   U    A    A    A    E    B   o    )   n    )    )    )    6   o    8    b    b    i    )    1    1    t  ,    0   a  ,    4    )    0   a   a    1   o    4   c    2    7    2    8    i    )    )    )    0    5    5   a    (    7    1    (    1    1    l  .    P    l  .    0    0    2   p    0    l    (    2   p   a    1  .    1    0    1    0    d    0    2    (    (    (   a    2    (    l  .    t    l   a    2    A    (   e    2    )   s    l  .    )    t    (    2   e   a   n   6    (    l  .    t   a    i   a   a   r    d   g   g   g   e     8   s    d    t    1    t    t    i   o   e   e   n   n   n   e    e   n    0   n   m   e    i    h   a    i   a   e   a   2     l    t   e   m    i   o   n   u     b   u    (   a   a   u    h    h    h    h    i    i   a   e    a     T    A    T    C    H   C   C    P    H   L    C   L

 1 3

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   )    9    )    1    9    0    1    2    (    0  .    2    l    (  .    t   a    l   a   e    t    i   e    d   s   u   o    i    R    L

 

Modeling, reasoning, and application of fuzzy Petri net model:…

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W. Jiang et al.

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Other applications using FPNs can also be found in traffic control systems, artificial intelligence, web services, social media, system control, detection and identification, prediction, and classification.

7 Conclusion and potential future work  As a graphical, mathematical model tool, FPN provides a visual method for the reasoning process. Systematic summaries and analysis regarding the modeling and reasoning of a KBS using FPNs have been presented in this survey regarding the following aspects. 1.

2.

3.

4.

A series of comparisons between between PN, EN_system, FPN, FPN, aand nd HLFPNs are shown shown to reveal the various relationships and indicate that an FPN is a backward extension of HLPN from the EN_system model. Therefore, Therefore, FPN can solve fuzzy information by using the inherent graphical description and parallel analysis ability of the PN model. The correspondences between between FPRs and FPNs have been discussed to explore why why FPNs can fully represent the entire information of a large-scale KBS. Furthermore, the modeling techniques using FPNs have have also been reviewed reviewed in detail. Specifically, Specifically, the current research achievements around the decomposition algorithm of FPNs has also been mentioned. A systematic recall of reasoning using PFN has been analyzed from from two angles: inferinference by graphical method and inference algebraic form. Further, the advantage advantagess and disadvantages of these two mechanisms have been discussed. Various industrial applications using FPNs from 2014 to 2020 have have been reviewed reviewed and summarized to reveal the application trends of FPNs. These industrial applications using FPNs further indicate that th at various HLFPNs combined with different considerations consid erations have been used to implement modeling, diagnosis, assessment, and prediction from traditional manufacturing systems to recent biological systems.

Although FPNs have outstanding achievements to their credit over the last 30 years, as highlighted by the literature reviews, reviews, some possible directions for future research are:

1.

2.

3.

The existing existingand models using FPNs are supportive and robust enough are for not knowledge knowledg e representation reasoning. However, fuzzy reasoning algorithms suitable for parallel reasoning and they lack a learning mechanism. In other words, the inefficient self-adaptive capability of the FPN model is still a barrier to obtaining more accurate inference result(s). Therefore, how to enhance the self-learning/adaptive ability of a multioutput FPN is the first potential research hotspot. Industrial applications applications of F FPNs PNs have have received received significant attention in the past few few years, years, such as fault diagnosis, manufacturing systems, modeling systems with uncertainty, health care and biological systems, and risk assessment. However However,, with the rapid development of KBSs, current FPN technologies do not meet the requirements of some specific problems. Hence, how to propose HLFPNs by combining advanced techniques, such as artificial intelligence, machine learning, and deep learning, is i s the second potential research point. The existing FPN models have have limitations limit ations and are inapplicab inapplicable le to representat representation ion and management in the big data era. Therefore, continuously discovering and developing novel FPN models to handle the state-explosion issue in the big data era and real-time

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applications in order to carry out large-scale and complex knowledge representations and reasoning is the third potential research domain.

Appendix: Various about FPN application field and highlight(s) 10.. See Tables 5, 6, 7, 8, 9, 10 Acknowledgements  This work was supported by the National Natura Naturall Science Foundation of China (No. 62066016), the Natural Science Foundation of Hunan Province, China (Nos. 2020JJ5458 and 2019JJ40234), the Research Foundation of Education Bureau of Hunan Province, China (Nos. 18B317 and 19C1535), the  Jishou University Graduate Researc Research h and Innovation Project (No. JDY19046) JDY19046),, and the open project of the collaborative innovation innovation center of industrial technology of manganese, zinc, vanadium (No. MXF201926).

References Al-Ahmari A, Kaid H, Li Z et al (2020) Strict minimal siphon-based colored Petri net supervisor synthesis for automated manufacturing systems with unreliable resources. IEEE Access 8:22411–22424 Amin M, Shebl D (2014) Reasoning dynamic fuzzy systems based on adaptive fuzzy higher order Petri nets. Inf Sci 286:161–172 Amit K, Lakhmi J (2005) Distributed reasoning by fuzzy Petri nets: a review. In: Cognitive engineering advanced information and knowledge processing. Springer, Springer, London, pp 85–122 Ammour R, Leclercq E, Sanlaville E et al (2018) Faults prognosis using partially observed stochastic Petri-nets: an incremental approach. Discrete Event Dyn Syst 28:247–267 Awan MSK, Awais MM (2011) Predicting weather events using fuzzy rule-based system. J Appl Soft Comput 11(1):56–63 Balazinski M, Czogala E, Jemielniak K et al (2002) Tool condition monitoring using artificial intelligence methods. Eng Appl Artif Intell 15(1):73–80 Baruwa OT, Piera MA (2014) Anytime heuristic heur istic search for scheduling flexible manufacturing systems: a timed colored Petri net approach. Int J Adv Manuf Technol 75(1–4):123–137 Basak O, Albayrak YE (2015) Petri net-based decision system modeling in real-time scheduling and control of flexible automotive manufacturing systems. Comput Ind Eng 86:116–126 8 6:116–126 Berthomieu B, Diaz M (1991) Modeling and verification of time dependent systems using time Petri nets. IEEE Trans Softw Eng 17(3):259–273 Bharathi V (2014) A fuzzy Petri-net model for predicting the post-implementation risks of ERP in small and medium enterprises. Int Rev Compute Softw 9(11):1852–1860 Bharathi SV, Raman R, Pramod D (2014) A FPN based risk assessment model for ERP implementation in small and medium enterprises. Middle-East J Sci Res 19(6):747–759 Bharathi SV, SV, Pramod D, Ramakrishnan R (2017) Risks assessment using fuzzy Petri nets for ERP extension in small and medium enterprises. enterpr ises. Inf Resour Manage J 30(4):1–23 Bibi Y, Bouhali O, Bouktir T (2017) A novel Petri type 2 fuzzy neural networks approximator for adaptive control of uncertain nonlinear systems. IET Control Theory Appl 11(17):3130–3136 Bibi Y, Bouhali O, Bouktir T (2019) Robust direct adaptive controller for a class of uncertain nonlinear systems using petri type 2 fuzzy neural networks (PT2FNN) as a new approximator. J Electr Syst 15(2):181–196 Bibi Y, Seghiri M, Bouhali O et al (2020) Petri type 2 fuzzy neural networks (PT2FNN) for identification and control of dynamic systems—a new structure and a comparative study. In: Proceedings of the 4th international conference on electrical engineering and control applications constantine, Algeria, pp 319–330 Camilli M (2012) Petri nets state space analysis in the cloud. In: International conference on software engineering, pp 1638–1640 Cardoso J, Valette R, Dubois D (1989) Petri nets with uncertain markings. In: Proceedings of the 10th international conference on applications and theory of Petri nets, pp 64–78 Chandrasekaran B (1988) Generic tasks as building blocks for knowledge-based systems: the diagnosis and routine design examples. Knowl Eng Rev 3(3):183–210

 1 3

 

Modeling, reasoning, and application of fuzzy Petri net model:…

6599

Chang YJ, Wang XF, Chen GM et al (2018) Comprehensive risk assessment of deepwater drilling riser using fuzzy Petri net model. Process Saf Environ Prot 117:483–497 Chaplain MAJ (2011) Multiscale mathematical modelling in biology and medicine. IMA J Appl Math 76(3):371–388 Chen SM (1988) A new approach to handling fuzzy decision-making problems. IEEE Trans Syst Man Cybern 18(6):1012–1016 Chen SM (1996) A fuzzy reasoning approach for rule-based systems based on fuzzy logics. IEEE Trans Syst Man Cybern B 26(5):769–778 Chen SM (2000) Fuzzy backw b ackward ard reasoning using fuzzy Petri nets. IEEE Trans Cybern B 30(6):846–856 30(6):846–8 56 Chen SM (2002) Weighted fuzzy reasoning using weighted fuzzy Petri nets. IEEE Trans Knowl Data Eng 14(2):386–397 Chen CH, Yang SY (2013) A knowledge-based cooperative differential evolution for neural fuzzy infer infer-ence systems. Soft Comput 17(5):883–895 Chen SM, Ke JS, Chang JF (1990 (1990)) Knowledge representation using fuzzy Petri nets. IEEE Trans Knowl Data Eng 2(3):311–319 Chen YF, Li ZW, ZW, Barkaoui K et al (2014) New Petri net structure and its application to optimal supervisory control: interval inhibitor inhib itor arcs. IEEE Trans Syst Man Cybern 44(10):1384–1400 Chen ZY, Xu B, Lu N et al (2019) A fault diagnosis method for flexible manufacturing tool system based on fuzzy Petri net. In: International conference on intelligent informatics and biomedical sciences, pp 374–379 Cheng YH, Yang LA (2009) A fuzzy Petri nets approach for railway traffic control in case of abnormality: evidence from Taiwan Taiwan railway system. Expert Syst Appl 36(4):8040–8048 Cheng J, Liu C, Zhou MC et al (2015) Automatic composition of semantic web services based on fuzzy predicate Petri nets. IEEE Trans Autom Sci Eng 12(2):680–689 Cheng X, Lin X, Zhu C et al (2017) Power system fault analysis based on hierarchical fuzzy petri net considering time association character character.. Trans China Electrotech Soc 32(14):229–237 Cheng H, Yan L, Ma ZM et al (2019) Fuzzy spatio-temporal ontologies and formal construction based on fuzzy Petri nets. Comput Intell 35(1):204–239 Chiang HS (2015) ECG-based mental stress assessment using fuzzy computing and associative petri net. Med Biol Eng 35(6):833–844 Chiang HS, Pao SC (2016) An EEG-based fuzzy probability model for early diagnosis of Alzheimer’s disease. Environ Syst Decis 40(5):125 Chiang TC, Huang AC, Fu LC (2006) Modeling, scheduling, and performance evalua evaluation tion for wafer ffabricaabrication: a queueing-colored Petri-net and GA-based approach. IEEE Trans Autom Sci Eng 3(3):330–338 Chiang HS, Chen MY, Wu ZM (2018) Applying fuzzy petri nets for evaluating the impact of bedtime behaviors on sleep quality. Granul Comput 3:321–332 Chun MG, Bien Z (1993) Fuzzy Petri net representation and reasoning methods for rule-based decisionmaking systems. IEICE Trans Fundam Electron Commun Comput Sci 76(6):974–983 Clarke EM, Grumberg O (1987) Avoiding the state explosion problem in temporal logic model checking. Proc ACM Symp Princ Distrib Comput 68(3):163–172 Clarke EM, Grumberg O, Jha S et al (2001) Progress on the state explosion problem in model checking algorithms. Springer, Berlin, pp 176–194 Demri S, Laroussinie F, Schnoebelen P (2006) A parametric analysis of the state-explosion problem in model checking. J Comput Syst Sci 72(4):547–575 Fat R, Mic L, Kilyen AO et al (2016) Model and method for the stock market forecast. In: IEEE international conference on automation, quality and testing, robotics, pp 1–5 Fay A (2000) A fuzzy knowledge-based system for railway traffic control. Eng Appl Artif Intell 13(6):719–729 Fei ZC, Li SQ, Chang Q et al (2018) Fuzzy petri net based intelligent machine operation of energy efficient manufacturing system. In: Proceedings of the 14th IEEE conference conference on automation science and engineering, pp 1593–1598 Frank PM (1990) Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy: a survey and some new results. Automatica 26(3):459–474 Fryc B, Pancerz K, Peters JF et al (2004) On fuzzy reasoning using matrix representation of extended fuzzy Petri nets. Fundam Inform 60(1–4):143–157 Gao MM, Zhou MC, Huang XG et al (2003) Fuzzy reasoning Petri nets. IEEE Trans Syst Man Cybern A 33(3):314–324 Garcia-Crespo A, Lopez-Cuadrado JL, Colomo-Palacios R et al (2011) Sem-Fit: a semantic based expert system to provide recommendations in the tourism domain. Expert Syst Appl Int J 38(10):13310–13319

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6600

W. Jiang et al.

Garg ML, Ahson SI, Gupta PV (1991) A fuzzy Petri net for knowledge representation and reasoning. Inf Process Lett 39(3):165–171 Godefroid P, van Leeuwen J, Hartmanis J et al (1996) Partial-order methods for the verification of concurrent systems: an approach to the state-explosion problem. Springer, Berlin Berlin Graubmann P (1987) The construction of EN systems from a given trace behavior. In: Proceedings of the 8th European workshop and theory of Petri nets, pp 133–153 Guo Y, Meng X, Wang D et al (2016) Comprehensive risk evaluation of long-distance oil and gas transportation pipelines using a fuzzy Petri net model. J Nat Gas Sci Eng 33:18–29 Ha MH, Li Y, Wang XF (2007) Fuzzy knowledge representation and reasoning using a generalized fuzzy petri net and a similarity measure. Soft Comput 11(4):323–327 Hajduk Z, Wojtowicz J (2016) Hardware implementation of fuzzy Petri nets with Lukasiewicz norms for modelling of control systems. Asian conference on intelligent information and database systems. Springer, Berlin, pp 449–458 Hamed RI (2015) Esophageal cancer prediction based on qualitative fea features tures using adaptive fuzzy reasoning method. J King Saud Univ Comput Inf Sci Arch 27(2):129–139 Hamed RI (2017) Quantitative modelling of gene networks of biological systems using fuzzy Petri nets and fuzzy sets. J King Saud Univ Sci 30(1):112–119 Handelman DA, Lane SH, Gelfand JJ (1990) Integrating neural networks and knowledge-based systems for intelligent robotic control. IEEE Control Syst Mag 10(3):77–87 Hansen P, Franco P, Kim SY (2018) Soccer ball recognition and distance prediction using fuzzy Petri nets. In: Proceedings of 2018 201 8 IEEE international conference on information reuse and integration, pp 315–322 He C, Wang L, Yan J et al (2014a) Environment control system fault diagnosis expert system. In: Proceedings of the first symposium on aviation maintenance and management-volume, Springer, Springer, Berlin, pp 453–462 He ZY, Yang JW, Zeng QF et al (2014b) Fault section estimation for power systems based on adaptive fuzzy Petri nets. Int J Comput Intell Syst 7(6):605–614 Heiner M (2011) Preface: Petri nets for systems and synthetic biology. Nat Comput 10(3):987–992 Hu H, Li ZW, Al-ahmari A (2011) Reversed fuzzy Petri nets and their application for fault diagnosis. Comput Ind Eng 60(4):505–510  Jiang W, Zhou KQ, Mo LP (2019 (2019)) Parameter optimi optimization zation strategy of fuzzy petri net utiliz utilizing ing hybrid GA-SFLA algorithm. Simulation tools and techniques. Springer, Cham, pp 416–426 416– 426  Jiang T, Du C, Guo S et al ((2020) 2020) Microgr Microgrid id fault diagno diagnosis sis m model odel based on weig weighted hted ffuzzy uzzy n neural eural Pet Petri ri net. In: Proceedings of the IEEE 4th information technology, networking, electronic and automation control conference, pp 2361–2365  Jorge CAF, Perkusich Perk usich A A,, Chan Chang g BSK (1996) Faults and ttiming iming analysis in rea real-time l-time distri distributed buted systems systems:: a fuzzy time Petri-net-based approach. Fuzzy Sets Syst 83(2):143–168 Kahloul L, Bourekkache S, Djouani K (2016) Designing reconfigurable manufacturing systems using reconfigurable object Petri nets. Int J Comput Integr Manuf, pp 1–18 Kaid H, Al-Ahmari A, Nasr EA et al (2020) Petri net model based on neural network for deadlock control and fault detection and treatment in automated manufacturing systems. IEEE Access 8:103219–103235 Kasirolvalad Z, Motlagh MRJ, Shadmani MA (2004) An intelligent modular modelling approach for quality control of CNC machines product using adaptive fuzzy Petri nets. In: IEEE control, automation, robotics and vision conference control, pp 1342–1347 Kim SY, Yang YL (2017) A self-navigating robot using fuzzy Petri nets. Robot Auton Syst 101(2):153–165 Koriem SM (2000) A fuzzy Petri net tool for modeling and verification of knowledge-based systems. Comput J 14(3):206–223 Kuchárik M, Balogh Z (2016) Evaluation of fuzzy Petri nets with the tool trans place sim. In: Proceedings of the IEEE international conference on application of information and communication technologies, pp 1–5 Le TL (2019) Self-organizing recurrent interval type-2 Petri fuzzy design for time-varying delay systems. IEEE Access 7:10505–10514 Lee KH, Favre J (1985) Hierarchical reduction method for analysis and decomposition of Petri nets. IEEE Trans Syst Man Cybern 15(2):272–280 Lee SJ, Seong PH (2004) Development of automated operating procedure system using fuzzy colored petri nets for nuclear power plants. Ann Nucl Energy 31(8):849–869 Lee J, Liu K, Chiang W (1998) Fuzzy Petri nets for modeling rule-based reasoning. Int J Artif Intell Tools 7(4):463–485

 1 3

 

Modeling, reasoning, and application of fuzzy Petri net model:…

6601

Lee YK, Lee JH, Lee SK et al (2006) Nano manufacturing using fountain pen nano-lithography with active membrane pumping. Key Eng Mater 326–328:425–428 Letia TS, Kilyen AO (2016) Evolutionary synthesis of hybrid controllers. In: Proceedings of the IEEE international conference on intelligent computer communication and processing, pp 133–140 Li X, Lara-Rosano F (2000) Adaptive fuzzy petri nets for dynamic knowledge representation and inference. Expert Syst Appl 19(3):235–241 Li X, Yu W, Lara-Rosano F (2000) Dynamic knowledge inference and learning under adaptive fuzzy Petri net framework. IEEE Trans Syst Man Cybern C 30(4):442–450 Li LL, Li CB, Tang Y (2014) A color petri net-based scheduling model for remanufacturing system with stochastic process routing. In: IEEE international conference on automation science and engineering, pp 474–479 Li H, You JX, Liu HC et al (2018a) Acquiring and sharing tacit t acit knowledge based on interval 2-tuple linguistic assessments and extended fuzzy petri nets. Int J Uncertain Fuzz 26(1):43–65 Li J, Zhu XL, Cheng XZ (2018b) Sensor fault diagnosis based on fuzzy neural Petri net. Complexity. https://doi. org/10. 10.1155/ 1155/2018/ 2018/82615 8261549 49 https://doi.org/ Li WJ, He M, Sun YB et al (2019) A n novel ovel lay layered ered fuzzy Petri nets modelling and reasoning method for process equipment failure risk assessment. J Loss Prev Proc 62:103953 Lin CP, Hwang CN (1996) Modeling and verification of fuzzy knowledge base with fuzzy colored Petri nets. In: Proceedings of the IEEE international conference conference on systems man and cybern, pp 586–590 Lin CM, Li HY (2015) Dynamic petri fuzzy cerebellar model articulation controller design for a magnetic levitation system and a two-axis linear piezoelectric ceramic motor drive system. IEEE Trans Control Syst Technol 23(2):693–699 Lipp HP (1983) The application of a fuzzy Petri net for controlling complex industrial processes. In: Proceedings of the IFAC conference on fuzzy information control, pp 471–477 Liu ZJ, Li HG, Zhou PJ (2011) Towards Towards timed fuzzy Petri net algorithms for chemical abnormality monitoring. Expert Syst Appl 38(8):9724–9728 Liu HC, Lin QL, Mao LX et al (2013a) Dynamic adaptive fuzzy petri nets for knowledge knowledge representation and reasoning. IEEE Trans Syst Man Cybern Syst 43(6):1399–1410 Liu HC, Lin QL, Ren ML et al (2013b) Fault diagnosis and cause analysis using fuzzy evidential reasoning approach and dynamic adaptive fuzzy Petri nets. Comput Ind Eng 66(4):899–908 Liu HC, Liu L, Lin QL et al (2013c) Knowledge acquisition and representation using fuzzy evidential reasoning and dynamic adaptive fuzzy petri nets. IEEE Trans Cybern 43(3):1059–1072 Liu HC, You JX, You XY et al (2015a) Fuzzy Petri nets using intuitionistic fuzzy sets and ordered weighted averaging ave raging operators. IEEE Trans Cybern 46(8):1839–1850 Liu HC, You JX, You XY et al (2015b) Linguistic reasoning Petri nets for knowledge representation and reasoning. IEEE Trans Syst Man Cybern: Syst 46(4):499–511 Liu XR, Gao YW, Wang ZL (2016) Method of power distribution network fault diagnosis based on improved improve d time fuzzy petri net. J Northeast Univ 37(2):1526–1529 Liu HC, Xue L, Li ZW et al (2017a) Linguistic Petri nets based on cloud model theory for know knowledge ledge representation and reasoning. IEEE Trans Knowl Data Eng 30(4):717–728 Liu HC, You JX, Li ZW et al (2017b) Fuzzy Petri nets for knowledge knowledge representation and reasoning: a literature review. review. Eng Appl Artif Intell 60(4):45–56 Liu F, Chen SY, Heiner M et al (2018a) Modeling biological systems with uncertain kinetic data using fuzzy continuous Petri nets. BMC Syst Biol 12(4):42 Liu JF, Zhou ZH, Wang ZS (2018b) Online fault diagnosis in discrete event systems with partially observed Petri nets. Int J Control Autom Syst 16(4):217–224 Liu F, Sun W, Heiner M et al (2019) Hybrid modelling of biological systems using fuzzy continuous Petri nets. Brief Bioinform 22(1):438–450 Looney CG (1988) Fuzzy Petri nets for rule-based decision making. IEEE Trans Syst Man Cybern 18(1):178–183 Majma N, Babamir SM, Monadjemi A (2017) Runtime verification of pacemaker functionality using hierarchical fuzzy colored Petri-nets. J Med Syst 41(2):1–21 Manoj TV, Leena J, Soney RB (1998) Knowledge representation using fuzzy Petri nets-revisited. IEEE Trans Knowl Knowl Data Eng 10(4):666–667 Meng DX, Liu SF, Han L (2015) The application of local adaptive fuzzy Petri net in result prediction of multiple-factor multiple-fac tor issues. In: International confer conference ence on human centered computing, pp 512–521 Meng FX, Lei YJ, Zhang B et al (2016) Intuitionistic fuzzy Petri nets for knowledge representation and reasoning. J Digit Inform ManagE 14:104–113 Meng FX, Lei YJ, Lei Y et al (2017) Hybrid reasoning using intuitionistic fuzzy Petri nets. Acta Electron Sin 45(8):1937–1946

 1 3

 

6602

W. Jiang et al.

Mhalla A, Benrejeb M (2015) A monitoring-maintenance approach based on fuzzy petri nets in manufacturing systems with time constraints. In: Azar A, Vaidyanathan S (eds) Computational intelligence applications in modeling and control. Springer, Cham, pp 205–228 Mhalla A, Jenhani O, Dutilleul SC (2013) Contribution to the monitoring of manufacturing systems with time constraints: application to a surface treatment line. In: Proceedings of the 14th international conferencee on sciences and techniques of automatic control and computer engineering, pp 243–250 ferenc Milinković S, Marković M, Vesković S et al (2013) A fuzzy Petri net model to estimate train delays. Simul Model Pract Theory 33:144–157 Mockler RJ, Dologite DG (1992) Knowl Knowledge-based edge-based systems to support strategic planning decision making. In: Proceedings of the 24th annual Hawaii international conference conference on system sciences, pp 173–180 Molloy MK (1982) Performance analysis using stochastic Petri nets. IEEE Trans Comput 31(9):913–917 Murata T (1989) Petri nets: properties, analysis and applications. Proc IEEE 77(4):541–580 Nabi HZ, Aized T (2020) Performance evaluation of a carousel configured multiple products flexible manufacturing system using Petri net. Oper Manage Res 13:109–129 Narahari Y, Y, Viswanadham N (1985) A Petri net approach to the modelling and analysis of flexible manufacturing systems. Ann Oper Res 3(8):449–472 Negoita CV (1985) Expert systems and fuzzy systems. Proc IEEE 75(10):1438–1439 Nie Y, Y, Su GP, Dai JY et al (2017) Research on the application of fuzzy Petri net in personalized education. In: Proceedings of the 2017 2nd international conferenc conferencee on materials science, machinery and energy engineering, pp 385–390 Nishi T, Matsumoto I (2015) Petri net decomposition approach to deadlock-free and non-cyclic scheduling of dual-armed cluster tools. IEEE Trans Autom Sci Eng 12(1):281–294 Novák Nov ák V, Lehmke S (2006) Logical str structure ucture of fuzzy IFIF-THEN THEN rules. Fuzzy Sets Syst 157(15):2003–2029 Paredes-Frigolett Paredes-Frigole tt H, Flavio AMGL (2016) A novel method for rule extraction in a knowledgeknowledge-based based innovation tutoring system. KnowlKnowl-Based Based Syst 92(15):183–199 Pedrycz W, W, Camargo H (2003) Fuzzy timed Petri nets. Fuzzy Sets Syst 140(2):301–330 Pedrycz W, W, Gomide F (1994) A generalized fuzzy Petri net model. IEEE Trans Fuzzy Syst 2(4):295–301 Pelz E (1989) About the concurrent behaviour of EN systems: definability and closure results. In: Proceedings of an advanced course Bad Honnef, pp 354–373 Pennisi M, Cavalieri S, Motta S et al (2016) A methodological approach for using high-level Petri nets to model the immune system response. BMC Bioinform 17(498):91–105 Petri CA, Reisig W (2008) Petri Net. Scholarpedia 3(1):133–136 Pramod D, Bh Arathi S, Raman R (2014) A fuzzy petri-net model for predicting the post-implementation risks of ERP in small and medium enterprises. Int Rev Comput Softw 9(11):1852–1860 Ribaric S, Basic BD (1998) Fuzzy time Petri net primitives for processing fuzzy temporal knowledge. In: Proceedings of 9th Mediterranean electrotec electrotechnical hnical conference, pp 549–553 Rosdi F, Salim SS, Mustafa MB (2019) An FPN-based classification method for speech intelligibility detection of children with speech impairments. Soft Comput 23(1):2391–2408 Rozenberg Rozenber g G (1987) Behaviour of elementary net systems. Adv Course Petri Nets 254:60–94 Sachan S, Donchak N (2020) Generalized stochastic Petri-net algorithm with fuzzy parameters to evaluate infrastructure asset management policy. In: Proceedings of 2020 IEEE international conference on fuzzy systems, pp 1–8 Samantra A, Panda A, Das SK et al (2020) Fuzzy Petri nets-based intelligent routing protocol for ad hoc network. Design frameworks for wireless networks. Springer, Singapore, pp 417–433 Saren SK, Blaga F, Dzitac S et al (2017) Decision based modeling of a flexible manufacturing cell based on hierarchicall timed colored Petri nets. Proced Comput Sci 122:253–260 hierarchica Saren SK, Blaga F, Vesselenyi T (2018) Implementation of fuzzy system using hierarchical colored Petri nets to model flexible manufacturing cell. IOP Conf Ser Mater Sci Eng 400(4):42–50 Scarpelli H, Gomide F (1993) Fuzzy reasoning and fuzzy Petri nets in manufacturing systems modeling. J Intell Fuzzy Syst 1(3):225–241 Scarpelli H, Gomide F, Y Yager ager RR (1996) A reasoning algorithm for high-level fuzzy petri nets. IEEE Trans Fuzzy Syst 4(3):282–294 Sha W, Li W, Jiao G et al (2019) A co-operative fault detection system with multiple detectors for smart factory based on fuzzy Petri net. In: Proceedings of 2019 international conference on artificial intelligence and advanced manufacturing, pp 778–783 Shafiekhani S, Rahbar S, Akbarian F et al (2018) Fuzzy stochastic Petri net with uncertain kinetic parameters for modeling tumor-immune system. In: Proceedings of the 25th national and 3rd international Iranian conference conference on biomedical engineering, pp 1–5 Shah SA, Bohez ELJ, Shah K et al (2017) Colored Petri net model for significant reduction of invariants in flexible manufacturing systems. Int J Adv Manuf Technol 88:1775–1787

 1 3

 

Modeling, reasoning, and application of fuzzy Petri net model:…

6603

Shahidinejad A, Ghobaei-Arani Ghobaei-Arani M, Esmaeili L (2020) An elastic controller using colored Petri nets in cloud computing environment. Cluster Comput 23:1045–1071 Shen VRL (2006) Knowledge Knowledge representation using high-level fuzzy Petri nets. IEEE Trans Syst Man Cybern A 36(6):1220–1227 Shen VRL (2019) Novel code plagiarism detection based on abstract syntax tree and fuzzy Petri nets. Int J Eng Educ 1(1):46–56 Shen VRL, Chen HC (2015) An automatic calibration system for Chinese karaoke lyrics based on high-level fuzzy Petri nets. In: IEEE international conference on machine learning and cybernetics, pp 544–549 Shen XY, Lei YJ, Li CH (2009) Intuitionistic fuzzy Petri nets model and reasoning algorithm. In: Proceedings of the 6th international conference on fuzzy systems and knowl knowledge edge discovery discovery,, pp 119–122 Shen VRL, Chang YS, Juang TTY (2010) Supervised and unsupervised learning by using Petri nets. IEEE Trans Syst Man Cybern A 40(2):363–375 Shen VRL, Lai HY, Lai AF (2015) The implementation of a smartphone-based fall detection system using a high-level fuzzy Petri net. Appl Soft Comput 26:390–400 Shen RK, Lin YN, Shen VRL et al (2017) Automatic detection of video shot boundary in social media using a hybrid approach of HLFPN and key point matching. IEEE Trans Comput Soc Syst 5(1):210–219 Shen RK, Yang CY, Shen VRL et al (2018) A stock market prediction system based on high-level fuzzy Petri nets. Int J Uncertain Fuzziness Know Knowl-Based l-Based Syst 26(5):771–808 Shi H, Wang L, Li XY et al (2019) A novel method for failure mode and effects analysis using fuzzy evidential reasoning and fuzzy Petri nets. J Ambient Intell Hum Comput 11:2381–2395 Shojafar M, Pooranian Z, Meybodi MR (2015) ALATO: ALATO: an efficient intelligent algorithm for time optimization in an economic grid based on adaptive stochastic Pe Petri tri net. J Intell Manuf 26:641–658 Slobodan R, Dalbelo BB, Pavešić N (1999) A model for fuzzy temporal knowledge representation and reasoning. In: Proceedings of the IEEE international fuzzy systems, pp 216–221 Song S, Lee S (2002) A Strategy of dynamic reasoning in knowledge-based system with fuzzy production rules. J Intell Inf Syst 19(3):303–318 Sun YB, Chen YP, Zhang B et al (2015) Fault diagnosis for power system using time sequence fuzzy Petri net. In: Proceedings of the 3rd international conference on mechanical engineering and intelligent systems, pp 729–735 Sun FF, Cheng YC, Cheng CH et al (2017) Intelligent evaluation of fault status in power transf transformer ormer based on fuzzy fault Petri net. In: Proceedings of the IEEE electrical insulation conference, pp 104–107 Sun FC, Zhang WC, Chen JH et al (2018) Fused fuzzy Petri nets: a shared control method for brain computer interface systems. IEEE Trans Cogn Dev Syst 2(11):188–199 Suraj Z (2012a) Knowledge representation and reasoning based on generalized fuzzy Petri nets. In: Proceedings of the 12th 12t h IEEE international conference on intelligent systems design and applications, pp 101–106 Suraj Z (2012b) Parameterised fuzzy Petri nets for approximate reasoning in decision support systems. In: Proceedings of the international conference on advanced machine learning technologies and applications, pp 33–42 Suraj Z (2013) Matrix representation of parameterised fuzzy Petri nets. In: Proceedings of the international workshop on rough sets fuzzy sets data mining and granular-soft computing, pp 200–207 Suraj Z (2015) Modified generalised fuzzy petri nets for rule-based systems. In: Proceedings of the international workshop on rough sets, fuzzy sets, data mining, and granular computing, pp 196–206 Suraj Z, Grochowalski P, P, Bandyopadhyay S (2016) Flexible generalized fuzzy petri nets for rule-based systems. In: Proceedings of the international conference on theory and practice of natural computing, pp 196–207 Tan MY, Li JM, Chen XQ et al (2019) Power grid fault diagnosis method using intuitionistic fuzzy Petri nets based on time series matching. Complexity 4:1–14 Tao XH, Miao Y, Zhang YC et al (2014) Collaborative medical diagnosis through fuzzy Petri net based agent argumentation. In: Proceedings of the IEEE international conference on fuzzy systems, pp 1197–1204 Thiagarajan Thiagaraja n PS (1986) Elementary net systems. In: Proceedings of the advanced course on petri nets: central models and their properties, advances in Petri nets, pp 26–59 Tian Z, Zhang Z, Dong H et al (2014) Real-time control for traffic signal based on fuzzy hybrid petri net. J Comput Inf Syst 10(5):2079–2090 Ting Y, Lu WB, Chen CH et al (2008) A fuzzy reasoning design for fault detection and diagnosis of a computer-controlled computer-c ontrolled system. Eng Appl Artif Intell 21(2):157–170 Tsang ECC, Yeung DS, John WTL (1999) Learning capability in fuzzy Petri nets. In: Proceedings of the IEEE conference conference on systems man and cybernetics, pp 355–360

 1 3

 

6604

W. Jiang et al.

Vahidipour SM, Meybodi MR, Esnaashari M (2015) Learning automata-based adaptive Petri net and its application to priority assignment in queuing systems with unknown parameters. IEEE Trans Syst Man Cybern Syst 45(10):1373–1384 Valavanis KP, Kokkinaki AI, Tzafestas SG (1994) Knowledge-based systems in engineering applications: a survey.. J Intell Robot Syst 10:113–145 survey Valmari A (1998) The state explosion problem. In: Reisig W, Rozenberg G (eds) Lectures on petri Nets I: basic models. Springer, Berlin, pp 429–528 Wai RJ, Lin YW (2013) Adaptive moving-target tracking control of a vision-based mobile robot via a dynamic Petri recurrent fuzzy neural network. IEEE Trans Fuzzy Syst 21(4):688–701 Wai RJ, Liu CM (2009) Design of dynamic petri recurrent fuzzy neural network and its application to pathtracking control of nonholonomic mobile robot. IEEE Trans Ind Electron 56(7):2667–2683 Wang WM, Peng X, Zhu GN et al (2014a) Dynamic representation of fuzzy knowle knowledge dge based on fuzzy petri net and genetic-particle swarm optimization. Expert Syst Appl 41(4):1369–1376 Wang YN, Ye JF, Xu GJ et al (2014b) Novel hierarchical fault diagnosis approach for smart power grid with information fusion of multi-data resources based on fuzzy petri net. In: Proceedings of the IEEE international conference on fuzzy systems, pp 1183–1189 Wang L, Chen Q, Gao ZJ et al (2015) A new fuzzy Petri net model for power grid fault diagnosis. Appl Mech Mater 700:61–66 Wang YH, Lu YJ, Li M et al (2016) Fault identification method based on fuzzy fault Petri net. In: Proceedings of the international conference on electrical and information technologies for rail transportation, pp 125–133 Wang JF, Fei ZC, Chang Q et al (2019) Energy saving operation of manufacturing system based on dynamic adaptive fuzzy reasoning Petri net. Energies 12(11):2216 Wang C, Li J, Zhu X et al (2020) Adaptive neural fuzzy Petri net algorithm for motor fault diagnosis. https://doi. doi.org/ org/10. 10.1088/ 1088/17551755-1315/ 1315/446/4/ 446/4/042063 042063 IOP Conf Ser: Earth Environ Sci. https:// Weng YS, Huang YS, Pan YL et al (2014) Design of traffic safety control systems for railroads and roadwayss using timed Petri nets. Asian J Control 17(2):626–635 way Wu F (1999) Fuzzy time semirings and fuzzy-timing colored Petri nets. Int J Intell Syst 14(8):747–774 Wu W, Wen F, Xue Y et al (2013) Power system fault diagnosis with a weighted fuzzy time Petri net. In: International conference on sustainable power generation and supply, pp 1–7 Xia JJ, Zhang JH, Wang RB (2016) Modeling of adaptive human–machine systems based on fuzzy infer infer-ence Petri nets. In: Proceedings of the 5th international conference on cognitive neuro dynamics, pp 493–499 Xu B, Yin X, Yin X et al (2019a) Fault diagnosis of power systems based on temporal constrained fuzzy Petri nets. IEEE Access 7:101895–101904 Xu XG, Shi HX, Xu DH et al (2019b) Picture fuzzy Petri nets for knowledge representation and acquisition in considering conflicting opinions. Appl Sci 9(5):12–21 Yang R, Leung WS, Heng PA (2002) Improved algorithm on rule-based reasoning systems modeled by fuzzy Petri nets. IEEE International Conference on Fuzzy Systems, pp 1204–1209 Yang B, Li H (2018) A novel dynamic timed fuzzy Petri nets modeling method with applications to industrial processes. Expert Syst Appl 97(12):276–289 Yang JL, Pu HT, Lian J et al (2018) Modeling and analysis of protein synthesis and DNA mutation using colored Petri nets. IEEE Access 6(2):22386–22400 Yang SH, Lin YN, Chiou GJ et al (2019) novel shot boundary detection in news streams based on fuzzy Petri nets. Appl Artif Intell 33(12):1035–1057 Yao DK, Wang QH, Gan XS (2018) Safety risk assessment on the air traffic control via the improved fuzzy Petri net. J Saf Environ 18:413–417 Yeung DS, Tsang EC (1994a) Fuzzy knowledge representation and reasoning using Petri nets. Expert Syst Appl 7(2):81–289 Yeung DS, Tsang ECC (1994b) Improved fuzzy knowledge representation and rule evaluation using fuzzy petri nets and degree of subset hood. Int J Intell Syst 9(12):1083–1100 Yeung DS, Tsang ECC (1998) A multilevel weighted fuzzy reasoning algorithm for expert systems. IEEE Trans Syst Man Cybern A 28(2):149–158 Yu ZH, Guo F, Ouyang J et al (2016) Object-oriented Petri nets and -calculus-based modeling and analysis of reconfigurable manufacturing systems. Adv Mech Eng 8(11):456–480 Yue WC, Gui WH, Chen XF et al (2019) Knowledge representation and reasoning using self-learning interval type-2 fuzzy Petri nets and extended TOPSIS. Int J Mach Learn Cyber 10:3499–3520 Zaitsev DA (2004) Decomposition of Petri nets. Cybern Syst Anal 40(5):739–746

 1 3

 

Modeling, reasoning, and application of fuzzy Petri net model:…

6605

Zakharov S, Taganov A, Gusev S et al (2018) The analysis and monitoring of ecological risks on the basis of fuzzy Petri nets. In: Proceedings of the 3rd Russian-pacific conference on computer technology and applications, pp 1–5 Zanni C, Goc ML, Frydman CS (2006) A conceptual framework for the analysis, classification and choice of knowledge-based diagnosis systems. Int J Know Knowl-Based l-Based Intell Eng Syst 10(2):113–138 Zhang XJ, Yao SZ (2015) Fuzzy stochastic Petri nets and analysis of the reliability of multi-state systems. Software IET 9(3):83–93 Zhang Y, Liang FC, Su F et al (1991) A fuzzy production rule based expert system. Fuzzy Sets Syst 44(3):391–403 Zhang WH, Yuan JS, Zhang K et al (2014) A reasoning method of transformer fault causes based on fuzzy Petri Net. Appl Mech Mater 631–632:537–542 Zhang Y, Zhang Y, Wen FS et al (2016) A fuzzy Petri net-based approach for fault diagnosis in power systems considering temporal constraints. Int J Electr Power Energy Energy Syst 78(6):215–224 Zhang JF, Reniers G, Zhang L (2017a) A weighted fuzzy Petri-net based approach for security risk assessment in the chemical industry. Chem Eng Sci 174:136–145 17 4:136–145 Zhang JH, Xia JJ, Garibaldi JM et al (2017b) Modeling and control of operator functional state in a unified framework framework of fuzzy inference petri nets. Comput Methods Programs Biomed 144:147–163 Zhang X, Yue S, Zha XB (2018) Method of power grid fault diagnosis using intuitionistic fuzzy Petri nets. IET Gener Transm Distrib 12(2):295–302 Zhang Z, Ge L, Wang P et al (2019) Behavior reconstruction models for large-scale network service systems. Peer-to-Peer Peer-to-Peer Netw Appl 12:502–513 Zhang C, Tian G, Fathollahi-Fard AM et al (2020) Interval-valued Intuitionistic uncertain linguistic cloud petri net and its application to risk assessment for subway fire accident. IEEE Trans Autom Scie Eng 44(9):1–15 Zhou J, Reniers G (2020) Modeling and application of risk assessment considering veto fact factors ors using fuzzy Petri nets. J Loss Prev Process Ind 67:104216 Zhou KQ, Zain AM (2016) Fuzzy Petri nets and industrial applications: a review. Artif Intell Rev 45(4):405–446 Zhou KQ, Zain AM, Mo LP (2015a) Dynamic properties of fuzzy Petri net model and related analysis. J Central South Univ 22(12):4717–4723 Zhou KQ, Zain AM, Mo LP (2015b) A decomposition algorithm of fuzzy Petri net using an index function and incidence matrix. Expert Syst Appl 42(8):3980–3990 Zhou KQ, Gui WH, Mo LP et al (2015c) A bidirectional diagnosis algorithm of fuzzy Petri net using innerreasoning-path. Expert Syst Appl 42(8):3980–3990 Zhou J, Reniers G, Zhang L (2017) A weighted fuzzy Petri-net based approach for security risk assessment in the chemical industry. Chem Eng Sci 174:136–145 174:136–14 5 Zhou KQ, Mo LP, Ding L et al (2018) An automatic algorithm to generate a reachability tree for large-sca large-scale le fuzzy Petri net by and/or graph. Symmetry 10(10):454 Zhou KQ, Mo LP, Jin J et al (2019) An equivalent generating algorithm to model fuzzy Petri net for know knowlledge-based system. J Intell Manuf 30(4):1831–1842 Zurawski R, Zhou MC (1994) Petri nets and industrial applications: a tutorial. IEEE Trans Ind Electron 41(6):567–583

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