## 4 Picture Fuzzy Petri Nets (PFNs)
### Knowledge Acquisition  
In this part, an approach to determine the knowledge parameters of PFPNs, i.e., global weight (\(GW\)), local weight (\(LW\)), threshold value (\(\widetilde{Th}\)), and certainty factor (\(\widetilde{U}\)), is introduced. Particularly, the conflict and inconsistency among expert evaluations and the vagueness in the knowledge parameters can be considered in the knowledge acquisition process. Figure 1: Picture fuzzy Petri net (PFPN) representation of Type 1 rule. Figure 3: PFPN representation of Type 3 rule. Figure 2: PFPN representation of Type 2 rule. Suppose that there are \(N\) WPFPRs \(\{R_{1},R_{2},\ldots,R_{N}\}\) to be evaluated by \(K\) experts \(\{TM_{1},TM_{2},\ldots,TM_{K}\}\) with respect to \(L\) antecedent propositions \(\{P_{1},P_{2},\ldots,P_{L}\}\). For a WPFPR, local weight and threshold value are related to the antecedent propositions; global weight and certainty factor are related to the WPFPRs. To simplify the discussion, only the process of determining the local weight (\(LW\)) is explained here. For the local weight, suppose that an expert \(TM_{k}\) provides the assessment of \(R_{n}\) with respect to \(P_{l}\) by a PFN denoted as \(\widehat{\gamma}^{k}(P_{l})=\left(\mu_{l}^{k},\eta_{l}^{k},\
u_{l}^{k}\right)\). Then, a picture fuzzy evaluation vector for the \(N\) WPFPRs with respect to \(L\) antecedent propositions can be formed as \(\widetilde{X}^{k}=\left(\widehat{\gamma}^{k}(P_{1}),\widehat{\gamma}^{k}(P_ {2}),\ldots,\widehat{\gamma}^{k}(P_{L})\right)^{T}\). The steps to determine the \(LW\) are described as follows:  
**Step 1:** Determine the weights of experts. Let \(\widetilde{X}^{k}(k=1,2,\ldots K)\) be the picture fuzzy evaluation vectors of local weights provided by all the experts. The similarity degree of \(\widetilde{X}^{k}\) to all the others can be calculated by:  
\[SD\Big{(}\widetilde{X}^{k}\Big{)}=\frac{\sum\limits_{i=1,i\
eq k}^{K}SD\Big{(} \widetilde{X}^{k},\widetilde{X}^{i}\Big{)}}{K-1}. \tag{17}\]  
Then, the weight of each expert can be obtained as follows [20]:  
\[\omega_{k}=\left\{ \begin{array}{c}F\Bigg{(}\frac{SD\Big{(}\widetilde{X}_{ \omega(1)}\Big{)}}{T}\Bigg{)},k=1,\\ F\Bigg{(}\frac{\sum\limits_{i=1}^{K}SD\Big{(}\widetilde{X}_{\omega(i)}\Big{)}}{ T}\Bigg{)}-F\Bigg{(}\frac{\sum\limits_{i=1}^{K-1}SD\Big{(}\widetilde{X}_{ \omega(i)}\Big{)}}{T}\Bigg{)},k=2,3,\ldots,K,\end{array} \right., \tag{18}\]  
where \(\omega_{k}\) satisfies \(\sum\limits_{k=1}^{K}\omega_{k}=1\), \(T=\sum\limits_{k=1}^{K}SD\Big{(}\widetilde{X}^{k}\Big{)}\) is the sum of all the similarity degrees, and \(\Big{\{}SD\Big{(}\widetilde{X}_{\omega(1)}\Big{)},SD\Big{(}\widetilde{X}_{ \omega(2)}\Big{)},\ldots,SD\Big{(}\widetilde{X}_{\omega(K)}\Big{)}\Big{\}}\) is a decreasing ranking value of \(\Big{\{}SD\Big{(}\widetilde{X}^{1}\Big{)},SD\Big{(}\widetilde{X}^{2}\Big{)}, \ldots,SD\Big{(}\widetilde{X}^{K}\Big{)}\Big{\}}\). \(F(a)=a^{\lambda}\) is a function to avoid the situation where the generated weights differ greatly from each other. **Step 2:** Aggregate all the picture fuzzy evaluation vectors. The picture fuzzy evaluation vectors \(\widetilde{X}^{k}(k=1,2,\ldots K)\) can be aggregated to obtain a collective picture fuzzy evaluation vector \(\widetilde{X}=(\widehat{\gamma}(P_{1}),\widehat{\gamma}(P_{2}),\ldots, \widehat{\gamma}(P_{L}))^{T}\) with the PFWG operator. For each element of \(\widetilde{X}\), the \(\widehat{\gamma}(P_{l})\) will be computed by:  
\[ \begin{array}{rl}\widetilde{\gamma}(P_{l})=&PFWG\big{(}\widehat{\gamma}^{1} (P_{l}),\widehat{\gamma}^{2}(P_{l}),\ldots,\widehat{\gamma}^{K}(P_{l})\big{)} \\ &=\Bigg{\langle}\prod\limits_{j=1}^{K}\Big{(}\mu_{l}^{j}+\eta_{l}^{j}\Big{)}^ {\omega_{j}}-\prod\limits_{j=1}^{K}\Big{(}\eta_{l}^{j}\Big{)}^{\omega_{j}}, \prod\limits_{j=1}^{K}\Big{(}\eta_{l}^{j}\Big{)}^{\omega_{j}},1-\prod\limits_ {j=1}^{K}\Big{(}1-\
u_{l}^{j}\Big{)}^{\omega_{j}}\Bigg{\rangle}.\end{array}  \tag{19}\]  
**Step 3:** Consensus checking and improving.