## 5 Illustrative Example
### Implementation
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**WFFPRs**} & \multirow{2}{*}{**Threshold Values**} & \multicolumn{4}{c}{**Initial Picture Fuzzy Evaluation Vectors**} & \multicolumn{2}{c}{**Collective Picture**} \\ \cline{3-7}  & & **TM\({}_{1}\)** & **TM\({}_{2}\)** & **TM\({}_{3}\)** & **TM\({}_{4}\)** & **TM\({}_{5}\)** & **Fuzzy Evaluations** \\ \hline  & \(\tilde{\lambda}_{11}\) & (0.7,0.1,0.1) & (0.3,0.25,0.35) & (0.4,0.2,0.3) & (0.4,0.2,0.4) & (0.1,0.3,0.6) & (0.196,0.2,0.604) \\
1 & \(\tilde{\lambda}_{21}\) & (0.5,0.1,0.3) & (0.1,0.1,0.7) & (0.3,0.1,0.5) & (0.4,0.1,0.4) & (0.25,0.05,0.6) & (0.252,0.086,0.558) \\  & \(\tilde{\lambda}_{31}\) & (0.1,0.2,0.7) & (0.2,0.1,0.6) & (0.2,0.3,0.4) & (0.1,0.3,0.5) & (0.05,0.2,0.65) & (0.12,0.202,0.599) \\
2 & \(\tilde{\lambda}_{42}\) & (0.1,0.05,0.85) & (0.4,0.05,0.55) & (0.1,0.1,0.7) & (0.3,0.05,0.65) & (0.20,0.1,0.79) & (0.186,0.025,0.789) \\
3 & \(\tilde{\lambda}_{33}\) & (0.2,0.1,0.6) & (0.2,0.1,0.6) & (0.25,0.05,0.6) & (0.4,0.3,0.2) & (0.2,0.1,0.6) & (0.243,0.106,0.547) \\
4 & \(\tilde{\lambda}_{44}\) & (0.1,0.15,0.65) & (0.2,0.15,0.55) & (0.05,0.15,0.7) & (0.5,0.25,0.15) & (0.05,0.25,0.6) & (0.106,0.184,0.608) \\
5 & \(\tilde{\lambda}_{75}\) & (0.3,0.05,0.65) & (0.2,0.25,0.45) & (0.2,0.1,0.6) & (0.2,0.2,0.5) & (0.2,0.01,0.79) & (0.196,0.019,0.785) \\
6 & \(\tilde{\lambda}_{86}\) & (0.15,0.05,0.8) & (0.3,0.1,0.69) & (0.1,0.05,0.85) & (0.4,0.0,0.5) & (0.2,0.1,0.6) & (0.188,0.019,0.793) \\ Similarity degree & 0.582 & 0.647 & 0.633 & 0.56 & 0.653 & & \\ Expert weight & 0.188 & 0.211 & 0.205 & 0.18 & 0.216 & \\ Consensus degree & 0.355 & 0.409 & 0.375 & 0.422 & 0.214 & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Threshold values on six WFFPRs by five experts. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**WFFPRs**} & \multirow{2}{*}{**Certainty Factors**} & \multicolumn{4}{c}{**Initial Picture Fuzzy Evaluation Vectors**} & \multicolumn{2}{c}{**Collective Picture**} \\ \cline{3-7}  & & **TM\({}_{1}\)** & **TM\({}_{2}\)** & **TM\({}_{3}\)** & **TM\({}_{4}\)** & **TM\({}_{5}\)** & **Fuzzy Evaluations** \\ \hline
1 & \(\tilde{\mu}_{1}\) & (0.7,0.3,0) & (0.96,0.0) & (0.99,0.0) & (0.98,0.0) & (0.9,0.0) & (0.966,0.0) \\
2 & \(\tilde{\mu}_{2}\) & (0.96,0.0) & (0.8,0.1,0.1) & (0.99,0.0) & (0.98,0.0) & (0.9,0.0) & (0.946,0.0.02) \\
3 & \(\tilde{\mu}_{3}\) & (0.9,0.0)  
For the gene regulatory network, the truth degrees of the starting places are set as follows:  
\(\widetilde{\alpha}(p_{1})=(0.742,0.108,0.06),\widetilde{\alpha}(p_{2})=(0.878,0,0.06),\widetilde{\alpha}(p_{3})=(0.45,0.305,0.194),\)  
\(\widetilde{\alpha}(p_{4})=(0.408,0.204,0.287),\widetilde{\alpha}(p_{5})=(0.608, 0.19,0.110),\widetilde{\alpha}(p_{6})=(0.754,0.095,0.110).\)  
With the PFPNs established in Figure 4, we can obtain:  
\(GW=[1\ 0.297\ 0.303\ 0.4\ 0.602\ 0.398]\)  
\(\widetilde{U}=[(0.966,0,0)\ (0.946,0.0,0.02)\ (0.907,0)\ (0.988,0,0)\ (0.974,0,0.02)\ (0.988,0,0)]\)  
\(LW=[0.468\ 0.292\ 0.24\ 1\ 1\ 1\ 1\ 0]^{T}\)  
\(\widetilde{Th}=[(0.196,0.2,0.604)\ (0.252,0.086,0.558)\ (0.12,0.202,0.599)\ (0.186,0.025,0.789)\)  
\((0.243,0.106,0.547)\ (0.106,0.184,0.608)\ (0.196,0.019,0.785)\ (0.188,0.019,0.793)\ (1,0,0)]^{T}\)  
\[\widetilde{M}_{2}= [(0.742,0.108,0.06)\ (0.878,0,0.06)\ (0.45,0.305,0.194)\ (0.408,0.204,0.287)\] \[(0.608,0.19,0.110)\ (0.754,0.095,0.110)(0.801,0,0.094)\ (0.72,0,0.168)\ (0.757,0,0.125)]^{T}.\]
7. Since \(\widetilde{M}_{2}\
eq\widetilde{M}_{1}\), we will continue to next iteration, and \(k=3\). \[\widetilde{M}_{3}= [(0.742,0.108,0.06)\ (0.878,0,0.06)\ (0.45,0.305,0.194)\ (0.408,0.204,0.287)\] \[(0.608,0.19,0.110)\ (0.754,0.095,0.110)(0.801,0,0.094)\ (0.72,0,0.168)\ (0.757,0,0.125)]^{T}.\]  
Since \(\widetilde{M}_{3}=\widetilde{M}_{2}\), the reasoning process is over. The final PFNs of all the places are obtained as \(\widetilde{M}_{3}\).