Fuzzy Petri nets and industrial applications: a review

Abstract

Fuzzy Petri net (FPN) provides an extremely competent basis for the implementation of computing reasoning processes and the modeling of systems with uncertainty. This paper reviews recent developments of the FPN and its industrial applications. Several important aspects of FPN’s background, history and formalisms are discussed, including the reasoning algorithm and relevant industrial applications; after which we present our conclusions and suggestions for future research.

Appendix 1: formal definitions of FPN (2000 to 2013)

Formalism	Highlight
$\textit{FPN}=(P,T,D,I,O,f,\alpha ,\beta )$ (Chen 2000; Sun et al. 2004; Shih et al. 2010)	(P,T,D,I,O) is the basic net structure of FPN; $(\alpha ,\beta )$describes correspondence between KBS and FPN
$\textit{AFPN}=(P,T,D,I,O,\alpha ,\beta ,Th,W)$ (Li et al. 2000)	Weight and threshold considered and added to formalism
$\textit{FPN}=(P,T,F,M_0 ,D,H,\alpha ,\theta ,\lambda )$ (Wang et al. 2001)	Transitions classified into five types: $t^{1},^{and}t,^{or}t,^{and},t^{or};$ to control scale of FPN and simplify analysis
$\textit{WFPN}=(P,T,D,I,O,f,\alpha ,\beta ,W)$ (Chen 2002)	Chen considered the function of weight in his research and enhanced the proposed formalism
$\textit{HLFPN}=(P,T,F,C,V,\alpha ,\beta ,\delta )$ (Shen 2003, 2006; Shih et al. 2010)	This formalism extended the description range for FPN
$\textit{FRPN}=(P,R,I,O,H,\theta ,\gamma ,C)$ (Gao et al. 2003, 2004; Luo and Kezunovic 2008; Zhou et al. 2012; Guan and Kezunovic 2013)	This formalism used an algebraic form to explore parallel operation ability and used I, O, H to define relevance matrices
$\textit{FPN}=(P,R,D,G,R,\Delta ,\Gamma ,\Theta ,M_0 )$ (Gniewek and Kluska 2004)	This formalism focused on one-to-one correspondence between KBS and FPN
$\textit{FAPN}=(P,T,I,O,M,\tau ,\alpha ,\lambda )$ (Tang et al. 2006)	This formalism was based on the disassembly issue; human factors were considered. For example, places divided into two modules, one for operators, another for product subassembly or component
$\textit{IFPN}=(P,T,D,I,O,\mu ,f,w,H,\beta )$ (Heng et al. 2006)	This formalism considered parameters and correspondence between KBS and FPN. Moreover, dynamic certainty given and marked by f
$\textit{APN}=(P,T,S,D,\Lambda ,\Gamma ,I,O,C,\alpha ,\beta ,W,Th)$ (Shih et al. 2007)	This formalism derived from FPN. Compared with above, a special element, ’square’, was added in the APN model
$\textit{FPN}=(P,T,I,O,M,\theta ,\alpha ,\delta ,\tau ,\lambda )$ (Tang 2009)	This formalism also focused on the disassembly issue. Compared with the 8-tuple of FPN by Gao et al. (2004), this formalism used 10-tuple to describe the FPN
$\textit{FPN}=(P,T,I,O,\alpha ,\beta ,M_0 )$ (Cao and Chen 2010)	This formalism focused on computing with words, weight value of one for every situation
$\textit{FIPN}=(P,T,\Omega ,\Psi ,R,\Delta ,K,W,\Gamma ,\Theta ,M_0 ,e)$ (Gniewek 2013)	This formalism proposed a strategy to settle conflict. For instance, places were summarized in two modules, one associated with modeling a processes, another associated with modeling resources
$\hbox {WFSN P system}=(O,N_p ,N_r ,syn,IN,OUT)$ (Wang et al. 2013)	This formalism proposed to model weighted FPRs and implement weighted reasoning based on the SNP system model
$DAFPN=(P;T;I;O;D;\alpha ;\beta ;W;U;Th;M)$ (Liu et al. 2013a)	This formalism proposed to overcome unreasonable points in the defection of FPN
$\textit{DAFPN}=(P;T;I;O;D;\alpha ;\beta ;W;U;Th_I ;Th_O ;M)$ Liu (2013b)	This formalism divided the threshold in two: input and output values, respectively