## 5 Illustrative Example
### Comparisons and Discussion  
To show the effectiveness of the proposed PFPNs, a comparison analysis with the IFPNs [40] and the conventional FPNs [34] are made in this part. The expression level of the target gene \(d_{9}\) by using the IFPNs is \((0.725,0.149)\)[40]. For the given gene regulatory network, the knowledge parameters in the FPNs are shown as follows:  
\(U=[0.96\ 0.96\ 0.90\ 0.99\ 0.99\ 0.98]\),  
\(Th=[0.3\ 0.3\ 0.2\ 0.2\ 0.3\ 0.2\ 0.2\ 0.2\ 1]^{T}\),  
\(M_{0}=[0.8\ 0.9\ 0.6\ 0.5\ 0.7\ 0.8\ 0\ 0\ 0]^{T}\). Based on the reasoning algorithm of FPNs, the result is obtained as:  
\(M_{3}=[0.8\ 0.9\ 0.6\ 0.5\ 0.7\ 0.8\ 0.75\ 0.792\ 0.776]^{T}\). According to above three FPN models, the ranking results of the intermediate places \(p_{7}\), \(p_{8}\), and the terminating place \(p_{9}\) are listed in Table 7. First, we can find that the ranking results of PFPNs and IFPNs are the same. This can prove the feasibility of the proposed PFPNs, but IFSs are utilized in the IFPN model to handle uncertainty and vagueness in knowledge representation and reasoning. Although IFSs have been successfully applied in various areas, there are situations that cannot be represented by IFSs [19]. As a generalization of IFSs, the PFSs that consider the degree of positive membership, the degree of neutral membership, and the degree of negative membership are more suitable to describe uncertain information and data. For example, the expression level of the target gene \(d_{9}\) is \((0.757,0,0.125)\) in PFPNs and \((0.725,0.149)\) in IFPNs. The neutral membership degree is ignored in the IFPNs. Thus, the proposed PFPNs have a wider range of applicability than the IFPNs. The ranking results derived by the PFPNs and the FPNs are different. The main reason is that the information concerning neutral membership degree and negative membership degree is ignored when the FPN model is used. Thus, the original information will be lost in the knowledge representation and acquisition processes. Furthermore, the global weights are not taken into consideration in the traditional FPNs. This implies a lack of precision in the final reasoning result of FPNs. In addition, the proposed model considers the conflict and inconsistency among expert evaluations in acquiring knowledge parameters. If conflicts and inconsistencies among experts are ignored, i.e., the five experts are treated equally, then the knowledge parameters are obtained as:  
\(GW^{\prime}=[1\ 0.296\ 0.326\ 0.377\ 0.404\ 0.596]\)  
\(\widetilde{U}^{\prime}=[(0.965,0,0)\ (0.945,0,0.021)\ (0.909,0,0)\ (0.988,0,0)\ (0.973,0,0.021)\ (0.9 88,0,0)]\)  
\(LW^{\prime}=[0.454\ 0.315\ 0.231\ 1\ 1\ 1\ 1\ 0]^{T}\) \begin{table}
\begin{tabular}{c c} \hline \hline
**FPN Models** & **Ranking Results** \\ \hline PFPNs & \(p_{7}>p_{9}>p_{8}\) \\ IFPNs & \(p_{7}>p_{9}>p_{8}\) \\ FPNs & \(p_{8}>p_{9}>p_{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Rankings of \(p_{7}\), \(p_{8}\), and \(p_{9}\) by PFPNs, intuitionistic FPNs (IFPNs), and FPNs. \(\widehat{Th^{\prime}}= \begin{array}{ll}\left[\left(0.379,\ 0.197,\ 0.371\right)\ (0.286,\ 0.087,\ 0.521)\ (0.136,\ 0.204,\ 0.583)\ (0.209,\ 0.042,\ 0.728) \right.\\ \left.\left(0.247,\ 0.109,\ 0.54\right)\ (0.146,\ 0.184,\ 0.562)\ (0.255,\ 0.076,\ 0.619)\ (0.257,\ 0.0,\ 0.715)\ (1,0,0)\right]^{T}\). \\ \end{array} \)  
Then the reasoning result of PFPNs is obtained as:  
\(\widehat{M}_{3}^{\prime}=\begin{array}{ll}\left[\left(0.742,\ 0.108,\ 0.06\right)\ (0.878,\ 0,\ 0.06)\ (0.45,\ 0.305,\ 0.194)\ (0.408,\ 0.204,\ 0.287) \right.\\ \left.\left(0.608,\ 0.19,\ 0.110\right)\ (0.754,\ 0.095,\ 0.110)\ (0.807,\ 0,\ 0.093)\ (0.717,\ 0,\ 0.172) \right.\\ \left.\left.\left(0.739,\ 0.148\right)\right]^{T}\). Comparing \(\widehat{M}_{3}^{\prime}\) with \(\widehat{M}_{3}\), it is evident that the obtained expression levels of the target gene \(d_{9}\) are different. This difference can be explained by the fact that the knowledge parameters acquired without considering the conflict between expert evaluations are inaccurate.