## 2 Preliminaries
### Defuzzification of Picture Fuzzy Numbers (PFNs)  
Suppose that \(\widetilde{a}=(\mu,\eta,\
u)\) is a PFN, where \(\pi=1-\mu-\eta-\
u\). A defuzzification method to obtain a crisp value of the PFN is introduced as follows [21; 32]:  
**Step 1:** Distribute the neutral degree to the positive degree and negative degree as follows:  
\[\mu^{\prime}=\mu+\frac{\eta}{2}, \tag{11}\]  
\[\
u^{\prime}=\
u+\frac{\eta}{2}. \tag{12}\]  
**Step 2:** Calculate the defuzzification value \(y\) by  
\[y=\mu^{\prime}+\frac{1+\mu^{\prime}-v^{\prime}}{2}\pi. \tag{13}\]  
**3. Fuzzy Petri Nets (FPNs) and Fuzzy Production Rules (FPRs)**  
_3.1. Fuzzy Petri Nets_  
To present the FPRs of a rule-based decision-making system, the concept of FPNs, which is a potential modeling technique for knowledge representation and reasoning, was first proposed by [PERSON] [33]. Later, [PERSON] et al. [34] defined an FPN model to perform knowledge reasoning automatically. **Definition 11** [34]. : _An FPN is denoted as an 8-tuple:_  
\[FPN=(\mathrm{P},\mathrm{T},\mathrm{D},\mathrm{I},\mathrm{O},f,\alpha,\beta), \tag{14}\]  
where:  
1. \(P=\{p_{1},p_{2},\ldots,p_{m}\}\) is a finite set of places;
2. \(T=\{t_{1},t_{2},\ldots,t_{n}\}\) is a finite set of transitions;
3. \(D=\{d_{1},d_{2},\ldots,d_{m}\}\) denotes a finite set of propositions with \(P\cap T\cap D=\varnothing\) and \(|P|{=}|D|\);
4. \(I:T\to P^{\infty}\) denotes the input function, a mapping from transitions to bags of places;
5. \(O:T\to P^{\infty}\) denotes the output function, a mapping from transitions to bags of places;
6. \(f:T\to[0,1]\) denotes an association function, a mapping from transitions to certainty factors between 0 and 1;
7. \(\alpha:P\to[0,1]\) denotes an association function, a mapping from places to truth degrees between 0 and 1;
8. \(\beta:P\to D\) denotes an association function, a bijective mapping from places to propositions. _3.2. Fuzzy Production Rules_  
In many decision support systems, vague and unprecise expert knowledge is hard to capture and store. For representing knowledge properly, FPRs, which are represented in the form of fuzzy if-then rules, have been comprehensively used to represent rule-based knowledge. In FPRs, the antecedent and the consequent of each rule contain the fuzzy concepts expressed by fuzzy sets. The FPR is called a composite FPR if it contains either AND or OR connectors [35; 36]. Later, the weight parameter is been assigned into FPRs to define the weighted FPRs (WFPRs) [35], enhancing the power of FPRs to present fuzzy data. Let \(R=\{R_{1},R_{2},\ldots,R_{n}\}\) be a set of WFPRs. The \(i\)th rule can be defined as follows:  
\[R_{i}:\ \mathrm{IF}\ a\ \mathrm{THEN}\ c\ (\mathrm{CF}\ =\ u),\ Th,\ w, \tag{15}\]  
where:  
1. \(a=\{a_{1},a_{2},\ldots,a_{m}\}\) is the antecedent portion of the rule \(R_{i}\), comprising one or more propositions with some fuzzy variables connected by either \"AND\" or \"OR\";
2. \(c=\{c_{1},c_{2},\ldots,c_{m}\}\) is the consequent portion of the rule \(R_{i}\), comprising one or more propositions with some fuzzy variables connected by either \"AND\" or \"OR\";
3. \(u(u\in[0,1])\) is the certainty factor of the rule \(R_{i}\), representing the belief strength of the rule;
4. \(Th=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{m}\}\) is a set of threshold values specified for each of the propositions in the antecedent;(5) \(w=\{w_{1},w_{2},\ldots,w_{m}\}\) is a set of weights assigned to the propositions in the antecedent portion. In general, the WFPRs can be divided into five types as follows [37; 38; 39]:  
1. A simple WFPR: \(R\): IF \(a\) THEN \(c(u;\lambda;w)\);
2. A composite weighted fuzzy conjunctive rule in the antecedent: \(R\): IF \(a_{1}\) AND \(a_{2}\) AND \(\ldots\) AND \(a_{m}\) THEN \(c(u;\lambda_{1},\lambda_{2},\ldots,\lambda_{m};w_{1},w_{2},\ldots,w_{m})\);
3. A composite weighted fuzzy conjunctive rule in the consequent: \(R\): IF \(a\) THEN \(c_{1}\) AND \(c_{2}\) AND \(\ldots\) AND \(c_{m}(u;\lambda;w)\);
4.