Picture fuzzy interactional partitioned Heronian mean aggregation operators: an application to MADM process

Abstract

The picture fuzzy sets (PFSs) state or model the voting information accurately without information loss. However, their existing operational laws usually generate unreasonable computing results, especially when the agreement degree (AD) or neutrality degree (ND) or opposition degree (OD) is zero. To tackle this issue, we propose the interactional operational laws (IOLs) to compute picture fuzzy numbers (PFNs), which can capture the interaction between the ADs and NDs in two PFNs, as well as the interaction between the ADs and ODs in two PFNs. Based on the proposed novel IOLs, partitioned Heronian mean (PHM) operator, and partitioned geometric Heronian mean (PGHM) operator, some picture fuzzy interactional PHM (PFIPHM), weighted PFIPHM (PFIWPHM), geometric PFIPHM (PFIPGHM), and weighted PFIPGHM (PFIWPGHM) operators are proposed in this paper. Afterwards, we investigate the properties of these operators. Using the PFIWPHM and PFIWPGHM operators, a novel multiple attribute decision-making (MADM) method with PFNs is elaborated. Finally, a study example that involves the service quality ranking of nursing facilities is provided to show the decision procedure of the proposed MADM method and we also give the comparative analysis between the proposed operators and the existing aggregation operators developed for PFNs.

Appendix A. Proof of Theorems

Theorem 1

Given any two PFNs $p_{1} = \left( {\mu_{1} ,\eta_{1} ,\nu_{1} } \right)$ and $p_{2} = \left( {\mu_{2} ,\eta_{2} ,\nu_{2} } \right)$, then the computing result from $p_{1} \oplus p_{2}$ is also an PFN.

Proof

Since $0 \le \mu_{l} \le 1$, then we can derive $1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} \Rightarrow 0 \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} \le 1$.

Let $g = \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)}$, then the derivative of the mathematical function $g$ with regard to the variables $\mu_{1}$, $\mu_{2}$, $\eta_{1}$, $\eta_{2}$ can be computed and we obtain the following equations:$g^{\prime}\left( {\mu_{1} } \right) = - \eta_{2} \le 0$, $g^{\prime}\left( {\mu_{2} } \right) = - \eta_{1} \le 0$, $g^{\prime}\left( {\eta_{1} } \right) = 1 - \mu_{2} - \eta_{2} \ge 0$, $g^{\prime}\left( {\eta_{2} } \right) = 1 - \mu_{1} - \eta_{1} \ge 0$.

Hence, it can be noted that the function value of $g$ becomes smaller as the value of variables $\mu_{1}$ or $\mu_{2}$ become bigger, and it becomes bigger as the value of variables $\eta_{1}$ or $\eta_{ 2}$ becomes bigger.

Since $0 \le \eta_{l} \le 1$, then we can derive that$\prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} \le g \le \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} \Rightarrow 0 \le g = \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} \le 1$.

Similarly, the OD $\prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)}$ can also be proved to be bounded between 0 and 1.

The sum of AD, ND, and OD is

$$\begin{aligned} 1 - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} + \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} + \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} \hfill \\ = 1 + \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} \hfill \\ \end{aligned}$$

Let $g = \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)}$, then the derivative of the function $g$ with regard to the different variables $\mu_{l}$, $\eta_{l}$, $\nu_{l}$ can be computed and we obtain the following equations:

$$g^{\prime}\left( {\mu_{1} } \right) = 1 - \mu_{2} - \eta_{2} - \nu_{2} \ge 0,g^{\prime}\left( {\mu_{2} } \right) = {\kern 1pt} 1 - \mu_{1} - \eta_{1} - \nu_{1} \ge 0$$

$$g^{\prime}\left( {\eta_{1} } \right) = 1 - \mu_{2} - \eta_{2} \ge 0,g^{\prime}\left( {\eta_{2} } \right) = 1 - \mu_{1} - \eta_{1} \ge 0,g^{\prime}\left( {\nu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0,g^{\prime}\left( {\nu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0$$

Hence, it can be noted that the function value of $g$ becomes bigger as the value of variables $\mu_{l}$, $\eta_{l}$, or $\nu_{l}$ becomes bigger.

When $\mu_{l} = 1$, $\eta_{l} = 0$, $\nu_{l} = 0$, then we can derive that

$$g \le \left( {\prod\limits_{l = 1}^{2} {\left( {1 - 1} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} } \right) = 0 \Rightarrow 1 + \left( {\prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} } \right) \le 1$$

When $\eta_{l} = 1$, $\mu_{l} = 0$, $\nu_{l} = 0$, then we can derive that

$$g \le \left( {\prod\limits_{l = 1}^{2} {\left( {1 - 0} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 0 - 0} \right)} } \right) = 0 \Rightarrow 1 + \left( {\prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} } \right) \le 1$$

When $\nu_{l} = 1$, $\mu_{l} = 0$, $\eta_{l} = 0$, then we can derive that

$$g \le \left( {\prod\limits_{l = 1}^{2} {\left( {1 - 0} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 0 - 0} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} } \right) = 0 \Rightarrow 1 + \left( {\prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} } \right) \le 1$$

Then, it can be observed that the sum of AD, ND, and OD

$$1 - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} + \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} + \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} \le 1$$

The above inference process can prove Theorem 1.

Theorem 2

Given any two PFNs $p_{1} = \left( {\mu_{1} ,\eta_{1} ,\nu_{1} } \right)$ and $p_{2} = \left( {\mu_{2} ,\eta_{2} ,\nu_{2} } \right)$, then the computing result from $p_{1} \otimes p_{2}$ is also an PFN.

Proof

Let $g = \prod\limits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)}$, then the derivative of the function $g$ with regard to the variables $\mu_{1}$, $\mu_{2}$, $\nu_{1}$, $\nu_{2}$ can be computed and we can get the following formulas: $g^{\prime}\left( {\mu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0$, $g^{\prime}\left( {\mu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0$, $g^{\prime}\left( {\nu_{1} } \right) = - \mu_{2} \le 0$, $g^{\prime}\left( {\nu_{2} } \right) = - \mu_{1} \le 0$.

Hence, the function value of $g$ becomes bigger as the value of variables $\mu_{1}$ or $\mu_{2}$ becomes bigger, and it becomes smaller as the value of variables $\nu_{1}$ or $\nu_{ 2}$ becomes bigger.

Since $0 \le \mu_{l} \le 1$, $0 \le \nu_{l} \le 1$, then we can derive that

$$\prod\limits_{l = 1}^{2} {\left( {1 - 1} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} \le g \le \prod\limits_{l = 1}^{2} {\left( {1 - 0} \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} \Rightarrow 0 \le g = \prod\limits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} \le 1$$

Since $0 \le \eta_{l} \le 1$, then we can derive that $1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} \Rightarrow 0 \le 1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} \le 1$.

Similarly, the ND $1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)}$ and OD $1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)}$ can also be proved to be between 0 and 1.

The sum of AD, ND, and OD is

$$\prod\limits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} + 1 - \prod\limits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} + 1 - \prod\limits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} = 2 - \prod\limits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} - \prod\limits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)}$$

Let $g = 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)}$, then the derivative of the function $g$ with regard to some variables $\mu_{l}$, $\eta_{l}$, $\nu_{l}$ can be computed and we can get the following formulas:$g^{\prime}\left( {\mu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0$, $g^{\prime}\left( {\mu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0$, $g^{\prime}\left( {\nu_{1} } \right) = 1 - \mu_{2} - \nu_{2} \ge 0$, $g^{\prime}\left( {\nu_{2} } \right) = 1 - \mu_{1} - \nu_{1} \ge 0$, $g^{\prime}\left( {\eta_{1} } \right) = 1 - \eta_{2} \ge 0$, $g^{\prime}\left( {\eta_{2} } \right) = 1 - \eta_{1} \ge 0$.

Hence, it can be noted that the value of $g$ increases as the value of $\mu_{l}$, $\eta_{l}$, or $\nu_{l}$ increases.

When $\mu_{l} = 1$, $\eta_{l} = 0$, $\nu_{l} = 0$, then $g \le 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 - 0} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} = 1$.

When $\eta_{l} = 1$, $\mu_{l} = 0$, $\nu_{l} = 0$, then $g \le 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0 - 0} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1} \right)} = 1$.

When $\nu_{l} = 1$, $\mu_{l} = 0$, $\eta_{l} = 0$, then $g \le 2 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0 - 1} \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 0} \right)} = 1$.

Hence, it can be observed that the sum of AD, ND, and OD is less than or equal to 1. The inference process can prove Theorem 2.

Theorem 3

Given three PFNs $p = \left( {\mu ,\eta ,\nu } \right)$, $p_{1} = \left( {\mu_{1} ,\eta_{1} ,\nu_{1} } \right)$ and $p_{2} = \left( {\mu_{2} ,\eta_{2} ,\nu_{2} } \right)$, then we have

• 1. $p_{1} \oplus p_{2} = p_{2} \oplus p_{1}$;

• 2. $\lambda \left( {p_{1} \oplus p_{2} } \right) = \lambda p_{1} \oplus \lambda p_{2}$;

• 3. $\left( {\lambda_{1} + \lambda_{2} } \right)p = \lambda_{1} p \oplus \lambda_{2} p$;

• 4. $p_{1} \otimes p_{2} = p_{2} \otimes p_{1}$;

• 5. $\left( {p_{1} \otimes p_{2} } \right)^{\lambda } = p_{1}^{\lambda } \otimes p_{2}^{\lambda }$;

• 6. $p^{{\lambda_{1} + \lambda_{2} }} = p^{{\lambda_{1} }} \otimes p^{{\lambda_{2} }}$.

Proof

Let us define $\delta_{1} { = }1 - \mu_{1}$, $\delta_{2} { = }1 - \mu_{2}$, $\varepsilon_{1} { = }1 - \mu_{1} - \eta_{1}$, $\varepsilon_{2} { = }1 - \mu_{2} - \eta_{2}$, $\gamma_{1} = 1 - \mu_{1} - \nu_{1}$, $\gamma_{2} = 1 - \mu_{2} - \nu_{2}$, $\kappa_{1} = 1 - \nu_{1}$, $\kappa_{2} = 1 - \nu_{2}$, $\chi_{1} = 1 - \eta_{1}$, $\chi_{2} = 1 - \eta_{2}$.

• (1) $\begin{aligned} p_{1} \oplus p_{2} = \left( {1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \eta_{l} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \left( {1 - \delta_{1} \delta_{2} ,\delta_{1} \delta_{2} - \varepsilon_{1} \varepsilon_{2} ,\delta_{1} \delta_{2} - \gamma_{1} \gamma_{2} } \right) = p_{2} \oplus p_{1} \hfill \\ \end{aligned}$;

• (2) $\lambda \left( {p_{1} \oplus p_{2} } \right) = \lambda \left( {1 - \delta_{1} \delta_{2} ,\delta_{1} \delta_{2} - \varepsilon_{1} \varepsilon_{2} ,\delta_{1} \delta_{2} - \gamma_{1} \gamma_{2} } \right) = \left( {1 - \left( {\delta_{1} \delta_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\varepsilon_{1} \varepsilon_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } } \right)$.According to Definition 10, we have $\lambda p_{1} = \left( {1 - \delta_{1}^{\lambda } ,\delta_{1}^{\lambda } - \varepsilon_{1}^{\lambda } ,\delta_{1}^{\lambda } - \gamma_{1}^{\lambda } } \right)$ and $\lambda p_{2} = \left( {1 - \delta_{2}^{\lambda } ,\delta_{2}^{\lambda } - \varepsilon_{2}^{\lambda } ,\delta_{2}^{\lambda } - \gamma_{2}^{\lambda } } \right)$, then $\lambda p_{1} \oplus \lambda p_{2} = \left( {1 - \left( {\delta_{1} \delta_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\varepsilon_{1} \varepsilon_{2} } \right)^{\lambda } ,\left( {\delta_{1} \delta_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } } \right) = \lambda \left( {p_{1} \oplus p_{2} } \right)$.

• (3) According to Definition 10, it can be derived that $\left( {\lambda_{1} + \lambda_{2} } \right)p = \left( {1 - \delta^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \varepsilon^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} } \right)$, $\lambda_{1} p = \left( {1 - \delta^{{\lambda_{1} }} ,\delta^{{\lambda_{1} }} - \varepsilon^{{\lambda_{1} }} ,\delta^{{\lambda_{1} }} - \gamma^{{\lambda_{1} }} } \right)$, and $\lambda_{2} p = \left( {1 - \delta^{{\lambda_{2} }} ,\delta^{{\lambda_{2} }} - \varepsilon^{{\lambda_{2} }} ,\delta^{{\lambda_{2} }} - \gamma^{{\lambda_{2} }} } \right)$. Then we can get $\begin{aligned} \lambda_{1} p \oplus \lambda_{2} p \hfill \\ = \left( {1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} - \delta^{{\lambda_{l} }} + \varepsilon^{{\lambda_{l} }} } \right)} ,\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \delta^{{\lambda_{l} }} - \delta^{{\lambda_{l} }} + \gamma^{{\lambda_{l} }} } \right)} } \right) \hfill \\ {\kern 1pt} = \left( {1 - \delta^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \varepsilon^{{\lambda_{1} + \lambda_{2} }} ,\delta^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} } \right) = \left( {\lambda_{1} + \lambda_{2} } \right)p \hfill \\ \end{aligned}$.

• (4) $p_{1} \otimes p_{2} = \left( {\prod\nolimits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \mu_{l} - \nu_{l} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \eta_{l} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - \nu_{l} } \right)} } \right){\kern 1pt} = p_{2} \otimes p_{2}$.

• (5) According to Definition 10, we can have $\left( {p_{1} \otimes p_{2} } \right)^{\lambda } = \left( {\kappa_{1} \kappa_{2} - \gamma_{1} \gamma_{2} ,1 - \chi_{1} \chi_{2} ,1 - \kappa_{1} \kappa_{2} } \right)^{\lambda } {\kern 1pt} {\kern 1pt} = \left( {\left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } ,1 - \left( {\chi_{1} \chi_{2} } \right)^{\lambda } ,1 - \left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } } \right)$. Since $p_{1}^{\lambda } = \left( {\kappa_{1}^{\lambda } - \gamma_{1}^{\lambda } ,1 - \chi_{1}^{\lambda } ,1 - \kappa_{1}^{\lambda } } \right)$ and $p_{2}^{\lambda } = \left( {\kappa_{2}^{\lambda } - \gamma_{2}^{\lambda } ,1 - \chi_{2}^{\lambda } ,1 - \kappa_{2}^{\lambda } } \right)$, then it can be derived that $\begin{aligned} p_{1}^{\lambda } \otimes p_{2}^{\lambda } = \left( {\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa_{l}^{\lambda } } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \kappa_{l}^{\lambda } + \gamma_{l}^{\lambda } - 1 + \kappa_{l}^{\lambda } } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \chi_{l}^{\lambda } } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa_{l}^{\lambda } } \right)} } \right) \hfill \\ {\kern 1pt} = \left( {\left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } - \left( {\gamma_{1} \gamma_{2} } \right)^{\lambda } ,1 - \left( {\chi_{1} \chi_{2} } \right)^{\lambda } ,1 - \left( {\kappa_{1} \kappa_{2} } \right)^{\lambda } } \right) = \left( {p_{1} \otimes p_{2} } \right)^{\lambda } \hfill \\ \end{aligned}$.

• (6) According to Definition 10, then we can have $p^{{\lambda_{1} + \lambda_{2} }} = \left( {\kappa^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} ,1 - \chi^{{\lambda_{1} + \lambda_{2} }} ,1 - \kappa^{{\lambda_{1} + \lambda_{2} }} } \right)$. Since $p^{{\lambda_{1} }} = \left( {\kappa^{{\lambda_{1} }} - \gamma^{{\lambda_{1} }} ,1 - \chi^{{\lambda_{1} }} ,1 - \kappa^{{\lambda_{1} }} } \right)$ and $p^{{\lambda_{2} }} = \left( {\kappa^{{\lambda_{2} }} - \gamma^{{\lambda_{2} }} ,1 - \chi^{{\lambda_{2} }} ,1 - \kappa^{{\lambda_{2} }} } \right)$, then it can be derived that $\begin{aligned} p^{{\lambda_{1} }} \otimes p^{{\lambda_{2} }} = \left( {\prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa^{{\lambda_{l} }} } \right)} - \prod\nolimits_{l = 1}^{2} {\left( {1 - \kappa^{{\lambda_{l} }} + \gamma^{{\lambda_{l} }} - 1 + \kappa^{{\lambda_{l} }} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \chi^{{\lambda_{l} }} } \right)} ,1 - \prod\nolimits_{l = 1}^{2} {\left( {1 - 1 + \kappa^{{\lambda_{l} }} } \right)} } \right) \hfill \\ {\kern 1pt} {\kern 1pt} = \left( {\kappa^{{\lambda_{1} + \lambda_{2} }} - \gamma^{{\lambda_{1} + \lambda_{2} }} ,1 - \chi^{{\lambda_{1} + \lambda_{2} }} ,1 - \kappa^{{\lambda_{1} + \lambda_{2} }} } \right) = p^{{\lambda_{1} + \lambda_{2} }} \hfill \\ \end{aligned}$.

Theorem 4

Given a collection of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, then the fused result from the PFIPHM operator is still an PFN as follows:

$$\begin{aligned} PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) \hfill \\ = \left( \begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned}$$

where $1 - \nu_{\rho } = a_{\rho }$, $1 - \nu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }$, $1 - \eta_{\rho } = c_{\rho }$, $1 - \eta_{\tau } = c_{\tau }$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$.

Proof

According to IOLs defined in Definition 10, it can be derived that

$$p_{\rho }^{\alpha } { = }\left( {\left( {1 - \nu_{\rho } } \right)^{\alpha } - \left( {1 - \mu_{\rho } - \nu_{\rho } } \right)^{\alpha } ,1 - \left( {1 - \eta_{\rho } } \right)^{\alpha } ,1 - \left( {1 - \nu_{\rho } } \right)^{\alpha } } \right)$$

and $p_{\tau }^{\beta } { = }\left( {\left( {1 - \nu_{\tau } } \right)^{\beta } - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{\beta } ,1 - \left( {1 - \eta_{\tau } } \right)^{\beta } ,1 - \left( {1 - \nu_{\tau } } \right)^{\beta } } \right)$.

Let $1 - \nu_{\rho } = a_{\rho }$, $1 - \nu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }$, $1 - \eta_{\rho } = c_{\rho }$, and $1 - \eta_{\tau } = c_{\tau }$, then

$$p_{\rho }^{\alpha } { = }\left( {a_{\rho }^{\alpha } - b_{\rho }^{\alpha } ,1 - c_{\rho }^{\alpha } ,1 - a_{\rho }^{\alpha } } \right),p_{\tau }^{\beta } { = }\left( {a_{\tau }^{\beta } - b_{\tau }^{\beta } ,1 - c_{\tau }^{\beta } ,1 - a_{\tau }^{\beta } } \right)$$

Hence, we can get $p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } = \left( {a_{\rho }^{\alpha } a_{\tau }^{\beta } - b_{\rho }^{\alpha } b_{\tau }^{\beta } ,1 - c_{\rho }^{\alpha } c_{\tau }^{\beta } ,1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } } \right)$.

Then

$$\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right) = \left( \begin{aligned} 1 - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} , \hfill \\ \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} , \hfill \\ \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} \hfill \\ \end{aligned} \right)$$

$$\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right) = \left( \begin{aligned} 1 - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} \hfill \\ \end{aligned} \right)$$

Let us define that $\left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, and $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$, then we have $\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right) = \left( {1 - \chi ,\chi - \varsigma ,\chi - \psi } \right)$ and $\left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} = \left( {\left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} - \psi^{{\frac{1}{\alpha + \beta }}} ,1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} ,1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)$.

Then we can continue to obtain that

$$\mathop \oplus \limits_{\ell = 1}^{s} \left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} = \left( \begin{aligned} 1 - \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} - \prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} - \prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} \hfill \\ \end{aligned} \right)$$

Thus,

$$\frac{1}{s}\left( {\mathop \oplus \limits_{\ell = 1}^{s} \left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {p_{\rho }^{\alpha } \otimes p_{\tau }^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} } \right) = \left( \begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

The sum of the AD, ND, and OD of the aggregation result of the PFIPHM operator is

$$\begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} { + }\left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} { + }\left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ = 1 + \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned}$$

Since $\left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \ge 0$ and $\left( {\prod\nolimits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \le 1$, thus, $1 + \left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\nolimits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \ge 0$.

Since $- \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} \le 0$, $\left( {\prod\nolimits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} \le 1$, then,

$$\begin{aligned} 1 + \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} { + }1} \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \le 1 + 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \le 2 - \left( {\prod\limits_{\ell = 1}^{s} {\left( 1 \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \le 1 \hfill \\ \end{aligned}$$

The above inference process can prove Theorem 4.

Theorem 5

(Idempotency Property). Given a collection of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, if $p_{l} = p = \left( {\mu ,\eta ,\nu } \right)$ for each $l$, then we have $PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = p = \left( {\mu ,\eta ,\nu } \right)$, where $\alpha$ and $\beta$ are two parameters, which satisfy that $\alpha ,\beta \ge 0$.

Proof

Since $\mu_{\rho } { = }\mu_{\tau } { = }\mu$, $\nu_{\rho } { = }\nu_{\tau } { = }\nu$, $\eta_{\rho } { = }\eta_{\tau } { = }\eta$, then we have $1 - \nu_{\rho } = 1 - \nu_{\tau } = a_{\rho } = a_{\tau } = a$, $1 - \mu_{\rho } - \nu_{\rho } = 1 - \mu_{\tau } - \nu_{\tau } = b_{\rho } = b_{\tau } = b$, $1 - \eta_{\rho } = 1 - \eta_{\tau } = c_{\rho } = c_{\tau } = c$.

Thus,

$$\chi = \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = 1 - a^{\alpha + \beta } + b^{\alpha + \beta }$$

$$\varsigma = \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = - a^{\alpha + \beta } + b^{\alpha + \beta } + c^{\alpha + \beta }$$

$$\psi = \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = b^{\alpha + \beta }$$

Bring the above equation into (2), then $PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right){\kern 1pt} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {a - b,1 - c,1 - a} \right) = \left( {\mu ,\eta ,\nu } \right) = p$. Hence, the proof of Theorem 5 is finished.

Theorem 6

(Commutativity Property). Given a collection of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, if there is a permutation of $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$, denoted by $P^{\prime}_{\ell } = \left\{ {p^{\prime}_{\ell 1} ,p^{\prime}_{\ell 2} , \cdots ,p^{\prime}_{{\ell \left| {P_{\ell } } \right|}} } \right\}$, then we have $PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right)$, where $\alpha$ and $\beta$ are two parameters, which satisfy that $\alpha ,\beta \ge 0$, $\left| {P_{\ell } } \right|$ means the quantity of PFNs belonging to the cluster $P_{\ell }$.

Proof

According to Eq. (2), we have

$$PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = \left( \begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

$$PFIPHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right) = \left( \begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

For each $\ell$, we have

$$\left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} = \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}}$$

$$\left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} = \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}}$$

$$\left( {\prod\limits_{\ell = 1}^{s} {\left( {\left( {\psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} = \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}}$$

Thus, $PFIPHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right)$. Hence, the proof Theorem 6 is finished.

Theorem 7

Given a collection of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, then the fused result of the PFIWPHM operator is still an PFN and

$$\begin{aligned} PFIWPHM^{\alpha ,\beta } \hfill \\ = \left( \begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned}$$

where $1 - \mu_{\rho } = a_{\rho }$, $1 - \mu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }$, $\left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right) = d_{\rho }$, $\left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right) = d_{\tau }$, $c_{\rho }^{{\omega_{\rho } }} = e_{\rho }$, $c_{\tau }^{{\omega_{\tau } }} = e_{\tau }$, $\left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right) = f_{\rho }$, $\left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right) = f_{\tau }$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, and $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$.

Proof

According to IOLs defined in Definition 10, it can be derived that

$$\omega_{\rho } p_{\rho } = \left( {1 - \left( {1 - \mu_{\rho } } \right)^{{\omega_{\rho } }} ,\left( {1 - \mu_{\rho } } \right)^{{\omega_{\rho } }} - \left( {1 - \mu_{\rho } - \eta_{\rho } } \right)^{{\omega_{\rho } }} ,\left( {1 - \mu_{\rho } } \right)^{{\omega_{\rho } }} - \left( {1 - \mu_{\rho } - \nu_{\rho } } \right)^{{\omega_{\rho } }} } \right)$$

and $\omega_{\tau } p_{\tau } = \left( {1 - \left( {1 - \mu_{\tau } } \right)^{{\omega_{\tau } }} ,\left( {1 - \mu_{\tau } } \right)^{{\omega_{\tau } }} - \left( {1 - \mu_{\tau } - \eta_{\tau } } \right)^{{\omega_{\tau } }} ,\left( {1 - \mu_{\tau } } \right)^{{\omega_{\tau } }} - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{{\omega_{\tau } }} } \right)$.

Let $1 - \mu_{\rho } = a_{\rho }$, $1 - \mu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }$, and $1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }$, then we have $\omega_{\rho } p_{\rho } = \left( {1 - a_{\rho }^{{\omega_{\rho } }} ,a_{\rho }^{{\omega_{\rho } }} - b_{\rho }^{{\omega_{\rho } }} ,a_{\rho }^{{\omega_{\rho } }} - c_{\rho }^{{\omega_{\rho } }} } \right)$ and $\omega_{\tau } p_{\tau } = \left( {1 - a_{\tau }^{{\omega_{\tau } }} ,a_{\tau }^{{\omega_{\tau } }} - b_{\tau }^{{\omega_{\tau } }} ,a_{\tau }^{{\omega_{\tau } }} - c_{\tau }^{{\omega_{\tau } }} } \right)$.

Hence, the exponential operation is

$$\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } = \left( {\left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } - \left( {c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } ,1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } ,1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } } \right)$$

$$\left( {\omega_{\tau } p_{\tau } } \right)^{\beta } = \left( {\left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( {c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } ,1 - \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } ,1 - \left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } } \right)$$

$$\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } = \left( \begin{aligned} \left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( {c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } , \hfill \\ 1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } , \hfill \\ 1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } \hfill \\ \end{aligned} \right)$$

Let $\left( {1 - a_{\rho }^{{\omega_{\rho } }} + c_{\rho }^{{\omega_{\rho } }} } \right) = d_{\rho }$, $\left( {1 - a_{\tau }^{{\omega_{\tau } }} + c_{\tau }^{{\omega_{\tau } }} } \right) = d_{\tau }$, $c_{\rho }^{{\omega_{\rho } }} = e_{\rho }$, $c_{\tau }^{{\omega_{\tau } }} = e_{\tau }$, $\left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right) = f_{\rho }$, $\left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right) = f_{\tau }$, then it can be derived that $\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } = \left( {d_{\rho }^{\alpha } d_{\tau }^{\beta } - e_{\rho }^{\alpha } e_{\tau }^{\beta } ,1 - f_{\rho }^{\alpha } f_{\tau }^{\beta } ,1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } } \right)$.

$$\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } } \right) = \left( \begin{aligned} 1 - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} , \hfill \\ \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} , \hfill \\ \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} \hfill \\ \end{aligned} \right)$$

$$\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } } \right) = \left( \begin{aligned} 1 - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} \hfill \\ \end{aligned} \right)$$

Let $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( { - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$, then it can be derived that

$$\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } } \right) = \left( {1 - \chi ,\chi - \varsigma ,\chi - \psi } \right)$$

$$\left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} = \left( {\left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} - \psi^{{\frac{1}{\alpha + \beta }}} ,1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} ,1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)$$

Thus, we have

$$\begin{aligned} \mathop \oplus \limits_{\ell = 1}^{s} \left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} \hfill \\ = \left( \begin{aligned} 1 - \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} - \prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} - \prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned}$$

Finally, we obtain that

$$\begin{aligned} \frac{1}{s}\left( {\mathop \oplus \limits_{\ell = 1}^{s} \left( {\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}\mathop \oplus \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\left( {\omega_{\rho } p_{\rho } } \right)^{\alpha } \otimes \left( {\omega_{\tau } p_{\tau } } \right)^{\beta } } \right)} \right)^{{\frac{1}{\alpha + \beta }}} } \right) \hfill \\ = \left( \begin{aligned} 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( { - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} + \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \psi } \right)^{{\frac{1}{\alpha + \beta }}} + \psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\psi^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right) \hfill \\ \end{aligned}$$

Theorem 8

Given a collection of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, then the aggregated result of the PFIPGHM operator is still an PFN, and

$$PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

where $1 - \mu_{\rho } = a_{\rho }$, $1 - \mu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} { = }\chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$.

Proof

According to the IOLs presented in Definition 10, it can be derived that $\alpha p_{\rho } = \left( {1 - \left( {1 - \mu_{\rho } } \right)^{\alpha } ,\left( {1 - \mu_{\rho } } \right)^{\alpha } - \left( {1 - \mu_{\rho } - \eta_{\rho } } \right)^{\alpha } ,\left( {1 - \mu_{\rho } } \right)^{\alpha } - \left( {1 - \mu_{\rho } - \nu_{\rho } } \right)^{\alpha } } \right)$, and $\beta p_{\tau } = \left( {1 - \left( {1 - \mu_{\tau } } \right)^{\beta } ,\left( {1 - \mu_{\tau } } \right)^{\beta } - \left( {1 - \mu_{\tau } - \eta_{\tau } } \right)^{\beta } ,\left( {1 - \mu_{\tau } } \right)^{\beta } - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{\beta } } \right)$.

Let $1 - \mu_{\rho } = a_{\rho }$, $1 - \mu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \eta_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \eta_{\tau } = b_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = c_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = c_{\tau }$, then$\alpha p_{\rho } = \left( {1 - a_{\rho }^{\alpha } ,a_{\rho }^{\alpha } - b_{\rho }^{\alpha } ,a_{\rho }^{\alpha } - c_{\rho }^{\alpha } } \right)$, $\beta p_{\tau } = \left( {1 - a_{\tau }^{\beta } ,a_{\tau }^{\beta } - b_{\tau }^{\beta } ,a_{\tau }^{\beta } - c_{\tau }^{\beta } } \right)$, $\alpha p_{\rho } \oplus \beta p_{\tau } { = }\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } ,a_{\rho }^{\alpha } a_{\tau }^{\beta } - b_{\rho }^{\alpha } b_{\tau }^{\beta } ,a_{\rho }^{\alpha } a_{\tau }^{\beta } - c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)$.

Thus,

$$\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right) = \left( \begin{aligned} \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} , \hfill \\ 1 - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} , \hfill \\ 1 - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} \hfill \\ \end{aligned} \right)$$

$$\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( \begin{aligned} \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ 1 - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ 1 - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} \hfill \\ \end{aligned} \right)$$

Let $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} { = }\chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$, then $\left( {\mathop \otimes \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( {\chi - \varsigma ,1 - \psi ,1 - \chi } \right)$.

Thus, $\frac{1}{\alpha + \beta }\left( {\mathop \otimes \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} ,\left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} - \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} ,\left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} - \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)$and

$$\mathop \otimes \limits_{\ell = 1}^{s} \left( {\frac{1}{\alpha + \beta }\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} } \right) = \left( \begin{aligned} \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} - \prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ 1 - \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ 1 - \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} \hfill \\ \end{aligned} \right)$$

Finally,

$$\left( {\mathop \otimes \limits_{\ell = 1}^{s} \left( {\frac{1}{\alpha + \beta }\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha p_{\rho } \oplus \beta p_{\tau } } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} } \right)} \right)^{{\frac{1}{s}}} = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

The sum of the AD, ND, and OD of the aggregation result of the PFIPGHM operator is

$$\begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} { + }1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} { + } \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} { = }2 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \ge 0 \hfill \\ \end{aligned}$$

and

$$2 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \le 2 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( 1 \right)} } \right)^{{\frac{1}{c}}} = 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \le 1$$

Hence, the proof of Theorem 8 is finished.

Theorem 9

(Idempotency Property). Given a collection of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, if $p_{l} = p = \left( {\mu ,\eta ,\nu } \right)$ for each $l$, then we have $PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = p = \left( {\mu ,\eta ,\nu } \right)$, where $\alpha$ and $\beta$ are two parameters, which satisfy that $\alpha ,\beta \ge 0$.

Proof

Since $\mu_{\rho } { = }\mu_{\tau } { = }\mu$, $\nu_{\rho } { = }\nu_{\tau } { = }\nu$, $\eta_{\rho } { = }\eta_{\tau } { = }\eta$, then we have $1 - \mu_{\rho } = 1 - \mu_{\tau } = a_{\rho } = a_{\tau } = a$, $1 - \mu_{\rho } - \eta_{\rho } = 1 - \mu_{\tau } - \eta_{\tau } = b_{\rho } = b_{\tau } = b$, $1 - \mu_{\rho } - \nu_{\rho } = 1 - \mu_{\tau } - \nu_{\tau } = c_{\rho } = c_{\tau } = c$.

Thus,

$$\chi = \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = 1 - a^{\alpha + \beta } + c^{\alpha + \beta }$$

$$\varsigma = \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {c_{\rho }^{\alpha } c_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = c^{\alpha + \beta }$$

$$\psi = \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - a_{\rho }^{\alpha } a_{\tau }^{\beta } + b_{\rho }^{\alpha } b_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = 1 - a^{\alpha + \beta } + b^{\alpha + \beta }$$

Bring the above equation into (6), then we have

$$PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right) = \left( {1 - a,a - b,a - c} \right) = \left( {\mu ,\eta ,\nu } \right)$$

Hence, the proof of Theorem 9 is finished.

Theorem 10

(Commutativity Property). Given a series of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ groups $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, if there is a permutation of $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$, denoted by $P^{\prime}_{\ell } = \left\{ {p^{\prime}_{\ell 1} ,p^{\prime}_{\ell 2} , \cdots ,p^{\prime}_{{\ell \left| {P_{\ell } } \right|}} } \right\}$, then we have $PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPGHM^{\alpha ,\beta } \left( {p^{\prime}_{1} ,p^{\prime}_{2} , \cdots ,p^{\prime}_{n} } \right)$, where $\alpha$ and $\beta$ are two parameters, which satisfy that $\alpha ,\beta \ge 0$, $\left| {P_{\ell } } \right|$ means the quantity of PFNs belonging to the group $P_{\ell }$.

Proof

According to Eq. (6), we have

$$PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

$$PFIPGHM^{\alpha ,\beta } \left( {p_{ 1}^{\prime } ,p_{ 2}^{\prime } , \cdots ,p_{n}^{\prime } } \right) = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi^{\prime} + \varsigma^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

For each $\ell$, $\prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} = \prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)}$, $\prod\nolimits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} = \prod\nolimits_{\ell = 1}^{s} {\left( {\left( {\varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)}$, and $\prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} = \prod\nolimits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi^{\prime} + \varsigma^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi^{\prime} + \varsigma^{\prime} + \psi^{\prime}} \right)^{{\frac{1}{\alpha + \beta }}} } \right)}$.

Thus, $PFIPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = PFIPGHM^{\alpha ,\beta } \left( {p_{ 1}^{\prime } ,p_{ 2}^{\prime } , \cdots ,p_{n}^{\prime } } \right)$, which completes the proof.

Theorem 11

Given a series of PFNs $P = \left\{ {p_{1} ,p_{2} , \cdots ,p_{n} } \right\}$ with $p_{l} = \left( {\mu_{l} ,\eta_{l} ,\nu_{l} } \right)$, which are grouped into $s$ clusters $P_{1} ,P_{2} , \cdots ,P_{s}$ in which $P_{\ell } = \left\{ {p_{\ell 1} ,p_{\ell 2} , \cdots ,p_{{\ell \left| {P_{\ell } } \right|}} } \right\}\left( {\ell = 1,2, \cdots ,s} \right)$ and $\bigcup\nolimits_{\ell = 1}^{s} {P_{\ell } } = P$, then the aggregated result of the PFIWPGHM operator is still an PFN and

$$PFIWPGHM^{\alpha ,\beta } \left( {p_{1} ,p_{2} , \cdots ,p_{n} } \right) = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$

where $1 - \nu_{\rho } = a_{\rho }$, $1 - \nu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }$, $1 - \eta_{\rho } = c_{\rho }$, $1 - \eta_{\tau } = c_{\tau }$, $1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} = d_{\rho }$, $1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} = d_{\tau }$, $- a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} = e_{\rho }$, $- a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} = e_{\tau }$, $b_{\rho }^{{\omega_{\rho } }} = f_{\rho }$, $b_{\tau }^{{\omega_{\tau } }} = f_{\tau }$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$.

Proof

According to the IOLs presented in Definition 10, it can be derived that $\left( {p_{\rho } } \right)^{{\omega_{\rho } }} = \left( {\left( {1 - \nu_{\rho } } \right)^{{\omega_{\rho } }} - \left( {1 - \mu_{\rho } - \nu_{\rho } } \right)^{{\omega_{\rho } }} ,1 - \left( {1 - \eta_{\rho } } \right)^{{\omega_{\rho } }} ,1 - \left( {1 - \nu_{\rho } } \right)^{{\omega_{\rho } }} } \right)$, and $\left( {p_{\tau } } \right)^{{\omega_{\tau } }} = \left( {\left( {1 - \nu_{\tau } } \right)^{{\omega_{\tau } }} - \left( {1 - \mu_{\tau } - \nu_{\tau } } \right)^{{\omega_{\tau } }} ,1 - \left( {1 - \eta_{\tau } } \right)^{{\omega_{\tau } }} ,1 - \left( {1 - \nu_{\tau } } \right)^{{\omega_{\tau } }} } \right)$.

Let $1 - \nu_{\rho } = a_{\rho }$, $1 - \nu_{\tau } = a_{\tau }$, $1 - \mu_{\rho } - \nu_{\rho } = b_{\rho }$, $1 - \mu_{\tau } - \nu_{\tau } = b_{\tau }$, $1 - \eta_{\rho } = c_{\rho }$, and $1 - \eta_{\tau } = c_{\tau }$, then

$$\left( {p_{\rho } } \right)^{{\omega_{\rho } }} = \left( {a_{\rho }^{{\omega_{\rho } }} - b_{\rho }^{{\omega_{\rho } }} ,1 - c_{\rho }^{{\omega_{\rho } }} ,1 - a_{\rho }^{{\omega_{\rho } }} } \right),\left( {p_{\tau } } \right)^{{\omega_{\tau } }} = \left( {a_{\tau }^{{\omega_{\tau } }} - b_{\tau }^{{\omega_{\tau } }} ,1 - c_{\tau }^{{\omega_{\tau } }} ,1 - a_{\tau }^{{\omega_{\tau } }} } \right)$$

Then, we can have $\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} = \left( {1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } ,\left( {1 - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } - \left( { - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } ,\left( {1 - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } - \left( {b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } } \right)$ and $\beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} { = }\left( {1 - \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } ,\left( {1 - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( { - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } ,\left( {1 - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( {b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } } \right)$.

Thus,

$$\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} { = }\left( \begin{aligned} 1 - \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } , \hfill \\ \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( { - a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( { - a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } , \hfill \\ \left( {1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } - \left( {b_{\rho }^{{\omega_{\rho } }} } \right)^{\alpha } \left( {b_{\tau }^{{\omega_{\tau } }} } \right)^{\beta } \hfill \\ \end{aligned} \right)$$

Let $1 - a_{\rho }^{{\omega_{\rho } }} + b_{\rho }^{{\omega_{\rho } }} = d_{\rho }$, $1 - a_{\tau }^{{\omega_{\tau } }} + b_{\tau }^{{\omega_{\tau } }} = d_{\tau }$, $- a_{\rho }^{{\omega_{\rho } }} { + }b_{\rho }^{{\omega_{\rho } }} { + }c_{\rho }^{{\omega_{\rho } }} = e_{\rho }$, $- a_{\tau }^{{\omega_{\tau } }} { + }b_{\tau }^{{\omega_{\tau } }} { + }c_{\tau }^{{\omega_{\tau } }} = e_{\tau }$, $b_{\rho }^{{\omega_{\rho } }} = f_{\rho }$, $b_{\tau }^{{\omega_{\tau } }} = f_{\tau }$, then $\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} { = }\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } ,d_{\rho }^{\alpha } d_{\tau }^{\beta } - e_{\rho }^{\alpha } e_{\tau }^{\beta } ,d_{\sigma i}^{\alpha } d_{\sigma j}^{\beta } - f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)$.

Thus,

$$\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right) = \left( \begin{aligned} \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} , \hfill \\ 1 - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} , \hfill \\ 1 - \prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} \hfill \\ \end{aligned} \right)$$

$$\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( \begin{aligned} \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ 1 - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} , \hfill \\ 1 - \left( {\prod\limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} \hfill \\ \end{aligned} \right)$$

Let $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \chi$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {f_{\rho }^{\alpha } f_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \varsigma$, $\left( {\prod\nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} {\left( {1 - d_{\rho }^{\alpha } d_{\tau }^{\beta } + e_{\rho }^{\alpha } e_{\tau }^{\beta } } \right)} } \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \psi$, then we have $\left( {\mathop \otimes \nolimits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} = \left( {\chi - \varsigma ,1 - \psi ,1 - \chi } \right)$ and

$$\begin{aligned} \frac{1}{\alpha + \beta }\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} \hfill \\ = \left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} ,\left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} - \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} ,\left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} - \varsigma^{{\frac{1}{\alpha + \beta }}} } \right) \hfill \\ \end{aligned}$$

Thus,

$$\mathop \otimes \limits_{\ell = 1}^{s} \left( {\frac{1}{\alpha + \beta }\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} } \right) = \left( \begin{aligned} \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} - \prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ 1 - \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} , \hfill \\ 1 - \prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} \hfill \\ \end{aligned} \right)$$

Finally,

$$\left( {\mathop \otimes \limits_{\ell = 1}^{s} \left( {\frac{1}{\alpha + \beta }\left( {\mathop \otimes \limits_{{p_{\rho } ,p_{\tau } \in P_{\ell } }} \left( {\alpha \left( {p_{\rho } } \right)^{{\omega_{\rho } }} \oplus \beta \left( {p_{\tau } } \right)^{{\omega_{\tau } }} } \right)} \right)^{{\frac{2}{{\left| {P_{\ell } } \right|\left( {\left| {P_{\ell } } \right| + 1} \right)}}}} } \right)} \right)^{{\frac{1}{s}}} = \left( \begin{aligned} \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} - \left( {\prod\limits_{\ell = 1}^{s} {\left( {\varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \left( { - \chi + \varsigma + \psi } \right)^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} , \hfill \\ 1 - \left( {\prod\limits_{\ell = 1}^{s} {\left( {1 - \left( {1 - \chi + \varsigma } \right)^{{\frac{1}{\alpha + \beta }}} + \varsigma^{{\frac{1}{\alpha + \beta }}} } \right)} } \right)^{{\frac{1}{s}}} \hfill \\ \end{aligned} \right)$$