Behavioral TOPSIS technique based on probabilistic picture hesitant fuzzy probability splitting algorithm and novel interactive operations

Abstract

Hotel recommendation (HR) for tourists is an obvious multi-attribute group decision-making (MAGDM) problem, which is a very important part of tourist travel. To address the uncertainty of information in HR, probabilistic picture hesitant fuzzy set (PPHFS) has been applied as an effective mathematical tool in this study to express decision information. This article proposes a novel MAGDM technique for HR. Firstly, to address the issue of inconsistent lengths of possible positive-membership hesitant degree (PPOMHD), possible neutral-membership hesitant degree (PNEUMHD), and possible negative-membership hesitant degree (PNEGMHD) for two PPHFEs that require normalization, we propose a normalized method named probability splitting algorithm to avoid the drawback of changing the original information by adding new elements based on the risk preferences of decision-makers in the past. Then, the basic operation of the PPHF elements (PPHFEs) does not satisfy the defect of closeness, we propose some novel operations for aggregating PPHFEs. In addition, based on these operations, the PPHF weighted average (PPHFWA) and PPHF weighted geometric (PPHFWG) operators were proposed, and their excellent properties were studied in detail. To better measure the difference between two PPHFEs, we have also proposed several novel PPHF distance measures (PPHFDisMs) for PPHFEs. Considering the irrational behavior of decision-makers (DMs), we extend the behavioral TOPSIS (BTOPSIS) technique to the PPHF environment and propose the PPHFWA and PPHFWG based BTOPSIS (PPHFBTOPSIS) technique, which is applied to HR. Finally, the robustness of the proposed PPHFBTOPSIS technique is demonstrated through parameter analysis.

Introduction

Hotel recommendation (HR) for tourists is an obvious multi-attribute group decision-making (MAGDM) problem, which is a very important part of tourist travel. With the improvement of people’s living standards, a large number of tourists travel to other places, but there is a shortage of hotel accommodation. Therefore, it is crucial to provide a more scientific and reasonable decision-making model for recommending hotels to tourists. Therefore, the strategic role of HR is more important than before. From various studies, HR is a typical type of MAGDM issue, Research on ranking methods is relatively mature, many scholars have proposed many different decision methods from different perspectives, including TOPSIS method¹, VIKOR method², LINMAP method³, EDAS method⁴, CODAS method⁵, BWM method⁶, ARAS method⁷. These methods have been used in many fields. But the most frequently used method is TOPSIS method^{8,9,10,11,12,13,14}. However, corresponding examples in¹⁵ were used to illustrate some unavoidable shortcomings of the traditional TOPSIS method. Therefore, a more practical behavioral TOPSIS (BTOPSIS) method was proposed. This method overcomes the shortcomings of the traditional TOPSIS method. Therefore, Yoon and Kim¹⁵ built the BTOPSIS method, which well reflects the psychological behavior of DMs when facing losses and gains. Liu and Huang¹⁶ proposed the PLq-ROFWA and PLq-ROFOWA operators, and fused the BTOPSIS method to PLq-ROF setting and a fresh MAGDM approach, finally, it was utilized to the selection of optimal green enterprise. Liu and Luo¹⁷ built the Pearson correlation coefficient and extended BTOPSIS to IF setting, and a fresh MAGDM methodology is built and utilized to the selection of suitable raw material supplier. Chen, Chen, Chen, Yang, Jin, Herrera-Viedma and Pedrycz¹⁸ extended the BTOPSIS to linguistic term setting, and utilized it to the MAGDM issue. Liu, Yang and Jiang¹⁹ fused the prospect theory and BTOPSIS, and built the PT- BTOPSIS approach, furthermore, fused the regret theory and BTOPSIS, and built the RT-TOPSIS approach, and utilized them to the investment projects selection. Salamirad, Kheybari, Ishizaka and Farazmand²⁰ combined the BWM and BTOPSIS and utilized it to calculation of the performance of seven WWT technologies.

With the rapid development of the economy and society, the situation where crisp numbers were used to express decision information has become increasingly powerless, while fuzzy information often occurs. With the frequent occurrence of such situations, in order to handle them more scientifically, Zadeh²¹ proposed fuzzy sets (FSs). Fuzzy numbers and crisp numbers are completely different concepts. Fuzzy numbers are no longer black and white like crisp numbers, and provide decision-makers (DMs) with more choices. Fuzzy numbers use values in the unit interval [0,1] to express DM’ fuzzy attitudes and achieve decision-making on problems. However, in real life, we may encounter situations where some people choose to vote in favor while others vote against, which cannot be solved by fuzzy numbers. Therefore, Atanassov²² provided another intuitionistic FS (IFS) that can express such situations. It has certain advantages over fuzzy numbers and is a binary structure composed of membership and non-membership degrees (MD and NMD), and satisfies that the sum of the two does not exceed 1. Although IF numbers (IFNs) have certain advantages over fuzzy numbers, they also have certain drawbacks. For example, when we still encounter voting problems, some people choose to vote in favor, some choose to vote against, and there are still some who abstain or even refuse to vote. In such situations, IFS appears powerless. According to practical needs, picture fuzzy set (PFS) has emerged, which is fully capable of handling the situation we just mentioned. It is composed of three membership degrees, namely, the membership degree (MD), the neutrality degree (ND), and the non-membership degree (NMD), and can also be used to express rejection degree.

To more accurately measure the degree of hesitation of DMs, Jang, Park and Son²³ proposed the mathematical tool named as PPHFS. Currently, there is only two research articles on PPHFS^23,24, but the contents of two articles are also incorrect. The main flaw is that:

• (1). When the lengths of PPOMHD, PNEUMHD, and PNEGMHD for two PPHFEs are different, new elements need to be added to ensure that the three membership lengths of each PPHFEs are the same. This addition method will change the original information of the original decision, resulting in a defect where the decision result does not match or even contradict the actual situation.

• (2). The basic operation rules of PPHFEs proposed in the original literature²³ do not satisfy closure, resulting in several types of aggregation operators also not satisfying closure, leading to the defect of incorrect aggregation PPHF information.

In general, the HR can be viewed as a PPHFMADM process. In order to improve the existing PPHFMADM methods, this paper aims to develop the BTOPSIS method based on the PPHFDisM and PPHFWA or PPHFWG operators. The main contributions and advancements of this paper are summarized as follows:

• (1). We propose a reasonable normalization method for PPHFEs that does not alter the errors in decision results caused by changes in the original information.

• (2). We have proposed some novel operations for PPHFEs based on strict t-norm and t-conorm, which not only ensure the closure of operations but also guarantee the reliability of decision results.

• (3). Since PPHF MADM methods in reference²³ can lead to unreasonable results, we extend the BTOPSIS method to picture fuzzy environment based on the PPHFDisM and PPHFWA or PPHFWG operators, thereby establishing a new MAGDM method.

• (4). Combining PPHF data acquisition and heterogeneous data processing, we use the proposed PPHFBTOPSIS technique to HR and conduct numerical analysis on the results.

• (5). The proposed PPHFBTOPSIS technique can also overcome the drawbacks of the model in²³, which cannot accurately aggregate the picture fuzzy information under each decision attribute.

The remainder of this paper is arranged as follows: Section "Preliminaries" presents PPHFS and some basic concepts, and some existed operations. In Section "Some novel operations and PPHFDisMs for PPHFEs", some shortcomings of existed operations are analyzed and propose some novel operations for PPHFEs based on probability splitting algorithm and strict t-norm and t-conorm, a novel linear order is introduced and some novel distances, which is admissible with the linear order are developed. In Section "Some picture fuzzy average/geometric operators for PPHFEs", PPHFWA and PPHFWG operators are developed based on novel operations of PPHFEs, and some valuable properties are investigated. In Section “The PPHFBTOPSIS technique”, the PPHFBTOPSIS technique is introduced and the monotonicity of PPHFBTOPSIS technique is investigated. In Section “Numerical case”, the proposed MAGDM technique is applied to address a practical problem on the HR, including the calculation process, sensitivity analysis. Finally, Section “Conclusion” concludes this paper.

Preliminaries

Probabilistic picture hesitant fuzzy set

Definition 2.1

A PPHFS $M$ on a non-empty and finite set $X$ is defined by²³

$$M = \left\{ {\left. {\left\langle {x,\left. {w\left( x \right)} \right|p\left( x \right),\left. {e\left( x \right)} \right|q\left( x \right),\left. {r\left( x \right)} \right|l\left( x \right)} \right\rangle } \right|x \in X} \right\}$$

(1)

The components $\left. {w\left( x \right)} \right|p\left( x \right)$, $\left. {e\left( x \right)} \right|q\left( x \right)$ and $\left. {r\left( x \right)} \right|l\left( x \right)$ are three sets of some possible elements, where $w\left( x \right)$, $e\left( x \right)$ and $r\left( x \right)$ represent the possible positive-membership hesitant degree (PPMHD), possible neutral-membership hesitant degree (PNMHD), and possible negative-membership hesitant degree (PNMHD) to the set $X$ of $x$. $p\left( x \right)$, $q\left( x \right)$ and $l\left( x \right)$ are the probability for $w\left( x \right)$, $e\left( x \right)$. And,

$$0 \le \mu ,\nu ,o \le 1,\;0 \le \mu^{ + } + \nu^{ + } + o^{ + } \le 1;\;p_{i} \in \left[ {0,1} \right],\;q_{j} \in \left[ {0,1} \right],\;r_{k} \in \left[ {0,1} \right],\,\sum\nolimits_{i = 1}^{\# w} {p_{i} } \le 1,\,\sum\nolimits_{j = 1}^{\# e} {q_{j} } \le 1,\;\sum\nolimits_{k = 1}^{\# r} {l_{k} } \le 1.$$

where $\mu \in w\left( x \right)$, $\nu \in e\left( x \right)$, $o \in r\left( x \right)$. $\mu^{ + } \in w^{ + } \left( x \right) = \cup_{\mu \in w\left( x \right)} \max \mu$, $\nu^{ + } \in e^{ + } \left( x \right) = \cup_{\nu \in e\left( x \right)} \max \nu$, $o^{ + } \in r^{ + } \left( x \right) = \cup_{o \in r\left( x \right)} \max o$, $p_{i} \in p$, $q_{j} \in q$, $l_{k} \in l$. The symbols $\# w$, $\# e$ and $\# r$ are total numbers of elements in the components $\left. {w\left( x \right)} \right|p\left( x \right)$, $\left. {e\left( x \right)} \right|q\left( x \right)$ and $\left. {r\left( x \right)} \right|l\left( x \right)$.

For convenience, we call $m = \left\langle {\left. {w\left( x \right)} \right|p\left( x \right),\left. {e\left( x \right)} \right|q\left( x \right),\left. {r\left( x \right)} \right|l\left( x \right)} \right\rangle$ a PPHFE and denoted by $m = \left\langle {\left. w \right|p,\left. e \right|q,\left. r \right|l} \right\rangle$.

If $\sum\nolimits_{i = 1}^{\# w} {p_{i} } < 1$, $\sum\nolimits_{j = 1}^{\# e} {q_{j} } < 1$ and $\sum\nolimits_{k = 1}^{\# r} {l_{k} } < 1$, then the normalized form of a generalized PPHFS is denoted by

$$\widetilde{M} = \left\{ {\left. {\left\langle {x,\left. {w\left( x \right)} \right|\widetilde{p}\left( x \right),\left. {e\left( x \right)} \right|\widetilde{q}\left( x \right),\left. {r\left( x \right)} \right|\widetilde{l}\left( x \right)} \right\rangle } \right|x \in X} \right\}$$

where the elements in $\widetilde{p}\left( x \right)$, $\widetilde{q}\left( x \right)$ and $\widetilde{l}\left( x \right)$ can be calculated by $\widetilde{p}_{i} = {{p_{i} } \mathord{\left/ {\vphantom {{p_{i} } {\sum\nolimits_{i = 1}^{\# w} {p_{i} } }}} \right. \kern-0pt} {\sum\nolimits_{i = 1}^{\# w} {p_{i} } }}$, $\widetilde{q}_{j} = {{q_{j} } \mathord{\left/ {\vphantom {{q_{j} } {\sum\nolimits_{j = 1}^{\# e} {q_{j} } }}} \right. \kern-0pt} {\sum\nolimits_{j = 1}^{\# e} {q_{j} } }}$ and $\widetilde{l}_{k} = {{l_{k} } \mathord{\left/ {\vphantom {{l_{k} } {\sum\nolimits_{k = 1}^{\# r} {l_{k} } }}} \right. \kern-0pt} {\sum\nolimits_{k = 1}^{\# r} {l_{k} } }}$, respectively.

In the process of using PPHFE for actual MADM problems, we need to rank PPHFEs. Therefore, we will propose indicators for comparing two PPHFEs, namely score function and accurate function of PPHFE, and a theorem for determining the size relationship of two PPHFEs is given.

Definition 2.2.

Let $m = \left\langle {\left. w \right|p,\left. e \right|q,\left. r \right|l} \right\rangle$ be a PPHFE, then the score and accuracy functions are defined as²³

$$s\left( m \right) = \frac{{1 + \sum\nolimits_{i = 1}^{\# w} {\mu_{i} p_{i} } - \sum\nolimits_{j = 1}^{\# e} {\nu_{j} q_{j} } - \sum\nolimits_{k = 1}^{\# r} {o_{k} l_{k} } }}{2},s\left( m \right) \in \left[ {0,1} \right].$$

(2)

and

$$h\left( m \right) = \sum\nolimits_{i = 1}^{\# w} {\mu_{i} p_{i} } + \sum\nolimits_{j = 1}^{\# e} {\nu_{j} q_{j} } + \sum\nolimits_{k = 1}^{\# r} {o_{k} l_{k} } ,h\left( m \right) \in \left[ {0,1} \right].$$

(3)

Theorem 2.1

Let $m_{1} = \left\langle {\left. {w_{1} } \right|p_{1} ,\left. {e_{1} } \right|q_{1} ,\left. {r_{1} } \right|l_{1} } \right\rangle$ and $m_{2} = \left\langle {\left. {w_{2} } \right|p_{2} ,\left. {e_{2} } \right|q_{2} ,\left. {r_{2} } \right|l_{2} } \right\rangle$ be two PPHFEs, then²³

(1) If $s\left( {m_{1} } \right)> s\left( {m_{1} } \right),$ then $m_{1}> m_{2} ;$

(2) If $s\left( {m_{1} } \right) = s\left( {m_{1} } \right),$ then.

(a) If $h\left( {m_{1} } \right)> h\left( {m_{1} } \right),$ then $m_{1}> m_{2} ;$

(b) If $h\left( {m_{1} } \right) = h\left( {m_{1} } \right),$ then $m_{1} = m_{2} ;$

(c) If $h\left( {m_{1} } \right) < h\left( {m_{1} } \right),$ then $m_{1} < m_{2} .$

Existed operations for PPHFEs

In the process of using PPHFE for actual MADM problems, we need to rank PPHFEs. Therefore, we will propose indicators for comparing two PPHFEs, namely score function and accurate function of PPHFE, and a theorem for determining the size relationship of two PPHFEs is given.

Definition 2.3

Let $m = \left\langle {\left. w \right|p,\left. e \right|q,\left. r \right|l} \right\rangle$, $m_{1} = \left\langle {\left. {w_{1} } \right|p_{1} ,\left. {e_{1} } \right|q_{1} ,\left. {r_{1} } \right|l_{1} } \right\rangle$ and $m_{2} = \left\langle {\left. {w_{2} } \right|p_{2} ,\left. {e_{2} } \right|q_{2} ,\left. {r_{2} } \right|l_{2} } \right\rangle$ be three PPHFEs, $\xi> 0$, and $m^{C}$ represents the complement operation, and some operations of PPHFEs are defined as follows²³.

(1) $m^{C} = \bigcup\limits_{\mu \in w,\nu \in e,o \in r} {\left\langle {\left\{ {\left. o \right|l_{o} } \right\},\left\{ {\left. \nu \right|q_{\nu } } \right\},\left\{ {\left. \mu \right|p_{\mu } } \right\}} \right\rangle } ;$

(2) $\begin{gathered} m_{1} \oplus m_{2} = \left\langle {w_{1} \oplus w_{2} ,e_{1} \otimes e_{2} ,r_{1} \otimes r_{2} } \right\rangle \hfill \\ \, = \bigcup\limits_{\begin{subarray}{l} \mu_{1} \in w_{1} ,\nu_{1} \in e_{1} ,o_{1} \in r_{1} ; \\ \mu_{2} \in w_{2} ,\nu_{2} \in e_{2} ,o_{2} \in r_{2} \end{subarray} } {\left\langle {\left\{ {\left. {\mu_{1} + \mu_{2} - \mu_{1} \mu_{2} } \right|p_{1} p_{2} } \right\},\left\{ {\left. {\nu_{1} \nu_{2} } \right|q_{1} q_{2} } \right\},\left\{ {\left. {o_{1} o_{2} } \right|l_{1} l_{2} } \right\}} \right\rangle } ; \hfill \\ \end{gathered}$

(3) $\begin{gathered} m_{1} \otimes m_{2} = \left\langle {w_{1} \otimes w_{2} ,e_{1} \oplus e_{2} ,r_{1} \oplus r_{2} } \right\rangle \hfill \\ \, = \bigcup\limits_{\begin{subarray}{l} \mu_{1} \in w_{1} ,\nu_{1} \in e_{1} ,o_{1} \in r_{1} ; \\ \mu_{2} \in w_{2} ,\nu_{2} \in e_{2} ,o_{2} \in r_{2} \end{subarray} } {\left\langle {\left\{ {\left. {\mu_{1} \mu_{2} } \right|p_{1} p_{2} } \right\},\left\{ {\left. {\nu_{1} + \nu_{2} - \nu_{1} \nu_{2} } \right|q_{1} q_{2} } \right\},\left\{ {\left. {o_{1} + o_{2} - o_{1} o_{2} } \right|l_{1} l_{2} } \right\}} \right\rangle } ; \hfill \\ \end{gathered}$

(4) $\xi m = \bigcup\limits_{\mu \in w,\nu \in e,o \in r} {\left\langle {\left\{ {\left. {1 - \left( {1 - \mu } \right)^{\xi } } \right|p_{\mu } } \right\},\left\{ {\left. {\nu^{\xi } } \right|q_{\nu } } \right\},\left\{ {\left. {o^{\xi } } \right|l_{o} } \right\}} \right\rangle } ;$

(5) $m^{\xi } = \bigcup\limits_{\mu \in w,\nu \in e,o \in r} {\left\langle {\left\{ {\left. {\mu^{\xi } } \right|p_{\mu } } \right\},\left\{ {\left. {1 - \left( {1 - \nu } \right)^{\xi } } \right|q_{\nu } } \right\},\left\{ {\left. {1 - \left( {1 - o} \right)^{\xi } } \right|l_{o} } \right\}} \right\rangle } .$

Definition 2.4

A mapping $T:\left[ {0,1} \right]^{2} \to \left[ {0,1} \right]$ is said to be a triangular norm (or briefly, t-norm) on $\left[ {0,1} \right]$ if, for any $x,y,z \in \left[ {0,1} \right]$, the following conditions are satisfied^25,26:

(T1) $T\left( {x,y} \right) = T\left( {y,x} \right)$ (commutativity);

(T2) $T\left( {x,T\left( {y,z} \right)} \right) = T\left( {T\left( {x,y} \right),z} \right)$ (associativity);

(T3) $T\left( {x,y} \right) \le T\left( {x,z} \right)$ for $y\le z$ (monotonicity);

(T3) $T\left( {x,1} \right) = x$ (neutrality).

Schweizer and Sklar²⁷ introduced triangular conorms as a dual concept of t-norms as follows.

A triangular conorm (or briefly, t-conorm) is a mapping $S:\left[ {0,1} \right]^{2} \to \left[ {0,1} \right]$, which, for any $x,y,z \in \left[ {0,1} \right]$, satisfies (T1)–(T3) and (S4):

(S4) $S\left( {x,0} \right) = x$ (neutrality).

Proposition 2.1

A mapping $T$ is a t-norm if and only if there is a t-conorm $S$ such that, for any $x,y \in \left[ {0,1} \right]^{2}$²⁶,

$$T\left( {x,y} \right) = 1 - S\left( {1 - x,1 - y} \right)$$

(4)

The t-norm $T$ given by formula (4) is called the dual t-norm of $S$. Analogously, we can give the definition of the dual t-conorm of a t-norm $T$. If $\tau$ is additive generator of $T$, then $T\left( {x,y} \right) = \tau^{ - 1} \left( {\tau \left( x \right) + \tau \left( y \right)} \right).$

Some novel operations and PPHFDisMs for PPHFEs

Some defects of operations for PPHFEs

In this section, we will focus on analyzing the shortcomings of the basic operations proposed in reference²³ using several examples, especially the defects that do not satisfy closeness.

Example 3.1.

Let $m = \left\langle {\left\{ {\left. {0.25} \right|0.3,\left. {0.2} \right|0.7} \right\},\left\{ {\left. {0.25} \right|0.3,\left. {0.2} \right|0.7} \right\},\left\{ {\left. {0.5} \right|0.5,\left. {0.3} \right|0.5} \right\}} \right\rangle$ be a PPHFE, where $\lambda = 2.$

(1) Defect analysis of the basic operations of PPHFEs.

By direct calculation, we have.

(a) $m \otimes m = \left\langle \begin{gathered} \left\{ {\left. {0.0625} \right|0.09,\left. {0.05} \right|0.21,\left. {0.05} \right|0.21,\left. {0.04} \right|0.49} \right\}, \hfill \\ \left\{ {\left. {0.4375} \right|0.09,\left. {0.4} \right|0.21,\left. {0.4} \right|0.21,\left. {0.36} \right|0.49} \right\}, \hfill \\ \left\{ {\left. {0.75} \right|0.25,\left. {0.65} \right|0.25,\left. {0.65} \right|0.25,\left. {0.51} \right|0.25} \right\} \hfill \\ \end{gathered} \right\rangle \notin {\mathbb{P}}$, since $0.0625 + 0.4375 + 0.75> 1;$

(b) When $\lambda = 2$, we have $m^{2} = \left\langle \begin{gathered} \left\{ {\left. {0.0625} \right|0.3,\left. {0.04} \right|0.7} \right\},\left\{ {\left. {0.4375} \right|0.3,\left. {0.36} \right|0.7} \right\}, \hfill \\ \left\{ {\left. {0.75} \right|0.5,\left. {0.51} \right|0.5} \right\} \hfill \\ \end{gathered} \right\rangle \notin {\mathbb{P}}$, since $0.0625 + 0.4375 + 0.75> 1$. This implies that there exists $\lambda> 0$ such that, for any $\lambda \in \left( {0,\lambda_{0} } \right)$, $m^{\lambda } \notin {\mathbb{P}}$;

The above show that the operational laws $\otimes$, and $m^{\lambda }$ are not closed in ${\mathbb{P}}$.

(2) Defect analysis of the PPHFWG operator.

The result is very obvious according to (1).

Probability distributions of the threeNovel operations for PPHFEs

In this subsection, we will address the Defects of the operations and ranking method of PPHFEs mentioned in Section "Some defects of operations for PPHFEs", propose some novel operations, and provide practical examples to testify closeness of novel operations. Before presenting these new contents, we will first provide the probability splitting algorithm of PPHFEs.

When the lengths of DPOM, DNEM, and DNOM for two PPHFEs are different, new elements Because the probability distributions of the threeneed to be added to ensure that the three membership lengths of each PPHFEs are the same. This addition method will change the original information of the original decision, resulting in a defect where the decision result does not match or even contradict the actual situation. Taking inspiration from the element normalization method mentioned in reference²⁷, we propose a normalization method for PPHFEs, named “probability splitting algorithm”. Below, we provide the definition of the method.

Definition 3.1 (Probability splitting algorithm).

Let $m_{v} = \left\langle {\left. {w_{v} } \right|p_{v} ,\left. {e_{v} } \right|q_{v} ,\left. {r_{v} } \right|l_{v} } \right\rangle$ $= \left\langle {\left\{ {\left. {\mu_{{w_{v} }} } \right|p_{{w_{v} }} } \right\},\left\{ {\left. {v_{{e_{v} }} } \right|q_{{e_{v} }} } \right\},\left\{ {\left. {o_{{r_{v} }} } \right|l_{{r_{v} }} } \right\}} \right\rangle$, 

$m_{1} = \left\langle {\left. {w_{{m_{1} }} } \right|p_{{m_{1} }} ,\left. {e_{{m_{1} }} } \right|q_{{m_{1} }} ,\left. {r_{{m_{1} }} } \right|l_{{m_{1} }} } \right\rangle$ $= \left\langle {\left\{ {\left. {\mu_{{w_{{m_{1} }} }} } \right|p_{{w_{{m_{1} }} }} } \right\},\left\{ {\left. {v_{{e_{{m_{1} }} }} } \right|q_{{e_{{m_{1} }} }} } \right\},\left\{ {\left. {o_{{r_{{m_{1} }} }} } \right|l_{{r_{{m_{1} }} }} } \right\}} \right\rangle$

and $m_{2} = \left\langle {\left. {w_{{m_{2} }} } \right|p_{{m_{2} }} ,\left. {e_{{m_{2} }} } \right|q_{{m_{2} }} ,\left. {r_{{m_{2} }} } \right|l_{{m_{2} }} } \right\rangle$ $= \left\langle {\left\{ {\left. {\mu_{{w_{{m_{2} }} }} } \right|p_{{w_{{m_{2} }} }} } \right\},\left\{ {\left. {v_{{e_{{m_{2} }} }} } \right|q_{{e_{{m_{2} }} }} } \right\},\left\{ {\left. {o_{{r_{{m_{2} }} }} } \right|l_{{r_{{m_{2} }} }} } \right\}} \right\rangle$ be three PPHFEs. Taking the normalization of PPMHD of PPHFEs as an example, we give the normalization process of PPHFEs. The normalization processed of PNMHD and PNMHD are similar with PPMHD, the steps are as follows:

Step 1. Firstly, we rank all PPMHD values of PPHFEs in ascending order;

Step 2. Determine the first normalized PPHFE, where $\widetilde{m}_{v} = \left\langle {\left. {\widetilde{w}_{v} } \right|\widetilde{p}_{v} ,\left. {\widetilde{e}_{v} } \right|\widetilde{q}_{v} ,\left. {\widetilde{r}_{v} } \right|\widetilde{l}_{v} } \right\rangle = \left\langle {\left\{ {\left. {\mu_{{\widetilde{w}_{v} }} } \right|p_{{\widetilde{w}_{v} }} } \right\},\left\{ {\left. {v_{{\widetilde{e}_{v} }} } \right|q_{{\widetilde{e}_{v} }} } \right\},\left\{ {\left. {o_{{\widetilde{r}_{v} }} } \right|l_{{\widetilde{r}_{v} }} } \right\}} \right\rangle$ denotes the normalized PPHFEs. If $p_{{w_{{m_{1} }} }}^{1} < p_{{w_{{m_{1} }} }}^{1}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{1} } \right|p_{{w_{{m_{1} }} }}^{1} = \left. {\mu_{{w_{{m_{1} }} }}^{1} } \right|p_{{w_{{m_{1} }} }}^{1}$ and $\left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|p_{{w_{{m_{2} }} }}^{1} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|p_{{w_{{m_{2} }} }}^{1}$, otherwise, $\left. {\mu_{{w_{{m_{1} }} }}^{1} } \right|p_{{w_{{m_{1} }} }}^{1} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|p_{{w_{{m_{2} }} }}^{1}$ and $\left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|p_{{w_{{m_{2} }} }}^{1} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|p_{{w_{{m_{2} }} }}^{1}$;

Step 3. Determine the second normalized PDHFE. If $p_{{w_{{m_{1} }} }}^{1} < p_{{w_{{m_{2} }} }}^{1}$ and $p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} \le p_{{w_{{m_{1} }} }}^{2}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|\left( {p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} } \right)$ and $\left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|p_{{w_{{m_{2} }} }}^{2} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|\left( {p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} } \right)$. If $p_{{w_{{m_{1} }} }}^{1} < p_{{w_{{m_{2} }} }}^{1}$ and $p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1}> p_{{w_{{m_{1} }} }}^{2}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2}$ and $\left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|p_{{w_{{m_{2} }} }}^{2} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|p_{{w_{{m_{1} }} }}^{2}$. If $p_{{w_{{m_{1} }} }}^{1}> p_{{w_{{m_{2} }} }}^{1}$ and $p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} \le p_{{w_{{m_{1} }} }}^{2}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2} = \left. {\mu_{{w_{{m_{1} }} }}^{1} } \right|\left( {p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1} } \right)$ and $\left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|p_{{w_{{m_{2} }} }}^{2} = \left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|\left( {p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1} } \right)$. If $p_{{w_{{m_{1} }} }}^{1}> p_{{w_{{m_{2} }} }}^{1}$ and $p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1}> p_{{w_{{m_{2} }} }}^{2}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2} = \left. {\mu_{{w_{{m_{1} }} }}^{1} } \right|p_{{w_{{m_{2} }} }}^{2}$ and $\left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|p_{{w_{{m_{2} }} }}^{2} = \left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|p_{{w_{{m_{2} }} }}^{2}$;

Step 4. Determine the third normalized PHFE. If $p_{{w_{{m_{1} }} }}^{1} \ge p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1} \le p_{{w_{{m_{2} }} }}^{2}$ and $p_{{w_{{m_{2} }} }}^{1} \le p_{{w_{{m_{2} }} }}^{2} - p_{{w_{{m_{1} }} }}^{1} + p_{{w_{{m_{2} }} }}^{1}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2}$ and $\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|p_{{w_{{m_{1} }} }}^{2}$. If $p_{{w_{{m_{1} }} }}^{1} \ge p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1} \le p_{{w_{{m_{2} }} }}^{2}$, and $p_{{w_{{m_{2} }} }}^{1}> p_{{w_{{m_{2} }} }}^{2} - p_{{w_{{m_{1} }} }}^{1} + p_{{w_{{m_{2} }} }}^{1}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|\left( {p_{{w_{{m_{2} }} }}^{2} + p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} } \right)$ and $\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|\left( {p_{{w_{{m_{2} }} }}^{2} + p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} } \right)$. If $p_{{w_{{m_{1} }} }}^{1} \ge p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1}> p_{{w_{{m_{2} }} }}^{2}$ and $p_{{w_{{m_{1} }} }}^{2}> p_{{w_{{m_{2} }} }}^{2} + p_{{w_{{m_{2} }} }}^{3}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|p_{{w_{{m_{2} }} }}^{3}$, and $\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3}$. If $p_{{w_{{m_{1} }} }}^{1} \ge p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{2} }} }}^{1}> p_{{w_{{m_{2} }} }}^{2}$ and $p_{{w_{{m_{1} }} }}^{2} < p_{{w_{{m_{2} }} }}^{2} + p_{{w_{{m_{2} }} }}^{3}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|\left( {p_{{w_{{m_{1} }} }}^{2} - p_{{w_{{m_{2} }} }}^{2} } \right)$, and $\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|\left( {p_{{w_{{m_{1} }} }}^{2} - p_{{w_{{m_{2} }} }}^{2} } \right)$. If $p_{{w_{{m_{1} }} }}^{1} < p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} \le p_{{w_{{m_{1} }} }}^{2}$, and $p_{{w_{{m_{1} }} }}^{2} + p_{{w_{{m_{1} }} }}^{1} \le p_{{w_{{m_{2} }} }}^{2} + p_{{w_{{m_{2} }} }}^{1}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{2} } \right|\left( {p_{{w_{{m_{1} }} }}^{2} - p_{{w_{{m_{2} }} }}^{1} + p_{{w_{{m_{1} }} }}^{1} } \right)$, and $\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{2} } \right|\left( {p_{{w_{{m_{1} }} }}^{2} - p_{{w_{{m_{2} }} }}^{1} + p_{{w_{{m_{1} }} }}^{1} } \right)$. If $p_{{w_{{m_{1} }} }}^{1} < p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1}> p_{{w_{{m_{1} }} }}^{2}$, and $p_{{w_{{m_{2} }} }}^{1} + p_{{w_{{m_{1} }} }}^{2} \le p_{{w_{{m_{1} }} }}^{3} + p_{{w_{{m_{1} }} }}^{1}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|\left( {p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{1} }} }}^{2} } \right), \text{and}\:\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|\left( {p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1} - p_{{w_{{m_{1} }} }}^{2} } \right)$ . If $p_{{w_{{m_{1} }} }}^{1} < p_{{w_{{m_{2} }} }}^{1}$, $p_{{w_{{m_{2} }} }}^{1} - p_{{w_{{m_{1} }} }}^{1}> p_{{w_{{m_{1} }} }}^{2}$, and $p_{{w_{{m_{2} }} }}^{1} + p_{{w_{{m_{1} }} }}^{2}> p_{{w_{{m_{1} }} }}^{3} + p_{{w_{{m_{1} }} }}^{1}$, then $\left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|p_{{w_{{m_{1} }} }}^{3} = \left. {\mu_{{w_{{m_{1} }} }}^{3} } \right|\left( {p_{{w_{{m_{1} }} }}^{3} } \right)$ and $\left. {\mu_{{w_{{m_{2} }} }}^{3} } \right|p_{{w_{{m_{2} }} }}^{3} = \left. {\mu_{{w_{{m_{2} }} }}^{1} } \right|\left( {p_{{w_{{m_{1} }} }}^{3} } \right)$. Where $p_{{w_{{m_{1} }} }}^{1} + p_{{w_{{m_{1} }} }}^{2} + \cdots + p_{{w_{{m_{1} }} }}^{{\# h_{1} }} = 1$, and $p_{{w_{{m_{2} }} }}^{1} + p_{{w_{{m_{2} }} }}^{2} + \cdots + p_{{w_{{m_{2} }} }}^{{\# h_{2} }} = 1$.

Remark 3.1.

To facilitate calculations and propose novel concepts, we assume that the probability distributions of PPMHD, PNMHD and PNMHD of PPHFEs are same involved in the subsequent content. And this manner does not alter the original information or change the decision outcome. And denote $m = \left\langle {\left. w \right|p,\left. e \right|q,\left. r \right|l} \right\rangle$ is simply written as $m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{m} }}^{s} ,\nu_{{e_{m} }}^{s} ,o_{{r_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$, indicating that three possible membership degrees have the same probability distribution, where $M$ represents the number of elements after probability splitting algorithm.

Example 3.2.

$m_{1} = \left\langle {\left\{ {\left. {0.2} \right|0.3,\left. {0.5} \right|0.2,\left. {0.3} \right|0.5} \right\},\left\{ {\left. {0.1} \right|0.5,\left. {0.2} \right|0.5} \right\},\left\{ {\left. {0.3} \right|0.3,\left. {0.2} \right|0.7} \right\}} \right\rangle$ and $m_{2} = \left\langle {\left\{ {\left. {0.1} \right|0.4,\left. {0.3} \right|0.6} \right\},\left\{ {\left. {0.4} \right|1} \right\},\left\{ {\left. {0.2} \right|0.3,\left. {0.1} \right|0.3,\left. {0.3} \right|0.4} \right\}} \right\rangle$ be two PPHFEs. The normalized process according to probability splitting algorithm is displayed as follows.

Step 1. Since $m_{1}$ and $m_{2}$ are normalized PPHFEs, then, we can obtain the normalized PPHFEs $m_{1}^{*}$ and $m_{2}^{*}$ are listed below.

$m_{1}^{*} = \left\langle {\left\{ {\left. {\left( {0.2,0.1,0.3} \right)} \right|0.3,\left. {\left( {0.5,0.1,0.2} \right)} \right|0.1,\left. {\left( {0.5,0.1,0.2} \right)} \right|0.1,\left. {\left( {0.3,0.2,0.2} \right)} \right|0.1,\left. {\left( {0.3,0.2,0.2} \right)} \right|0.4} \right\}} \right\rangle$,

$m_{2}^{*} = \left\langle {\left\{ {\left. {\left( {0.1,0.4,0.2} \right)} \right|0.3,\left. {\left( {0.1,0.4,0.1} \right)} \right|0.1,\left. {\left( {0.3,0.4,0.1} \right)} \right|0.1,\left. {\left( {0.3,0.4,0.1} \right)} \right|0.1,\left. {\left( {0.3,0.4,0.3} \right)} \right|0.4} \right\}} \right\rangle$.

Therefore, the corresponding PPMHD, PNMHD, and PNMHD of normalized $m_{1}^{*}$, and $m_{2}^{*}$ has same probability distribution.

Inspired by the comparison of two PFNs in reference²⁸, we propose a novel partial order to better compare two PPHFEs.

Definition 3.2.

For two PPHFEs $m_{1}$ and $m_{2}$, $m_{1} { \preccurlyeq }_{N} m_{2}$ if and only if $\mu_{{w_{{m_{1} }} }}^{s} \le \mu_{{w_{{m_{2} }} }}^{s}$, $\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} \le \mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s}$, and $o_{{r_{{m_{1} }} }}^{s} \ge o_{{r_{{m_{2} }} }}^{s}$.

Definition 3.3.

Let $m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{m} }}^{s} ,\nu_{{e_{m} }}^{s} ,o_{{r_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$, $m_{1} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\nu_{{e_{{m_{1} }} }}^{s} ,o_{{r_{{m_{1} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$ and $m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{2} }} }}^{s} ,\nu_{{e_{{m_{2} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$ be three PPHFEs, $\xi> 0$, $\tau$ is additive generator of $T$ and $\zeta$ is additive generator of $S$ and $\zeta \left( x \right) = \tau \left( {1 - x} \right)$, and $m^{\rm{\complement }}$ represents the complement operation, and some interactive operations of PPHFEs are defined as follows.

(1) $m^{\rm{\complement }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {o_{{r_{m} }}^{s} ,\pi_{{e_{m} }}^{s} ,\mu_{{w_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } ,$ where $\pi = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {1 - o_{{r_{m} }}^{s} - \mu_{{w_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } ;$

(2) $m_{1} \oplus_{T} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) \hfill \\ - S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

(3) $m_{1} \otimes_{T} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} T\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),T\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) \hfill \\ - T\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),S\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

(4) $\xi_{T} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

(5) $m^{{\xi_{T} }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. \begin{gathered} \left( {\tau^{ - 1} \left( {\xi \cdot \tau \left( {\mu_{{w_{m} }}^{s} } \right)} \right),} \right)\tau^{ - 1} \left( {\xi \cdot \tau \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \tau^{ - 1} \left( {\xi \cdot \tau \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right|k^{s} } \right\}} \right\rangle } .$

The following Theorem 3.1 demonstrates that all operations defined in Definition 3.3 are closed in ${\mathbb{P}}$.

Theorem 3.1.

Let $m_{1} ,m_{2} ,m \in {\mathbb{P}}$ and $T$ be a strict t-norm. Then, for any $\xi> 0$, $m_{1} \oplus_{T} m_{2}$, $m_{1} \otimes_{T} m_{2}$, $\lambda_{T} m$ and $m^{{\lambda_{T} }}$ are PPHFEs.

Proof

(1) Because the probability distributions of the three possible membership parts are the same, we only need to focus on the three possible membership parts. By Definition 3.3, we have

$$m_{1} \oplus_{T} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) \hfill \\ - S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle }$$

Noting that $\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{w_{{m_{1} }} }}^{s} \le 1 - o_{{w_{{m_{1} }} }}^{s}$ and $\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{w_{{m_{2} }} }}^{s} \le 1 - o_{{w_{{m_{2} }} }}^{s}$, since $S$ is increasing, we have,

$$\begin{gathered} S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right) + S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) - S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right) + T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right) \hfill \\ = S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) + T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right) \le S\left( {1 - o_{{r_{{m_{1} }} }}^{s} ,1 - o_{{r_{{m_{2} }} }}^{s} } \right) = 1 \hfill \\ \end{gathered}$$

implying that $m_{1} \oplus_{T} m_{2} \in {\mathbb{P}}$. Similarly, we can prove that $m_{1} \otimes_{T} m_{2} \in {\mathbb{P}}$.

(2) Fix $\lambda> 0$. Since $T$ is a strict t-norm, by Definition 3.3, we have

$$\xi_{T} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle }$$

$$\xi_{T} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle }.$$

This, together with $\zeta \left( x \right) = \tau \left( {1 - x} \right)$, implies that

$$\begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right) + \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) - \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right) + \tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ = \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) + \tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) = 1 - \tau^{ - 1} \left( {\lambda \cdot G_{E} \left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)} \right) + \tau^{ - 1} \left( {\lambda \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) = 1 \hfill \\ \end{gathered}.$$

Since $\tau$ is strictly decreasing, by $o_{{r_{m} }}^{s} \le 1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s}$, we have$1 - \tau^{ - 1} \left( {\lambda \cdot \tau \left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)} \right) + \tau^{ - 1} \left( {\lambda \cdot \tau \left( {v_{\alpha } } \right)} \right)$ 

$\le 1 - \tau^{ - 1} \left( {\lambda \cdot \tau \left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)} \right) + \tau^{ - 1} \left( {\lambda \cdot \tau \left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)} \right) = 1$, implying that $\xi_{T} m \in {\mathbb{P}}$. Similarly, we can prove that $m^{{\xi_{T} }} \in {\mathbb{P}}$.

If we assign specific forms of t-norms to Definition 3.3, we have the following results.

(1) If we take $T = T_{{\text{P}}} = xy$, $S = S_{{\text{P}}} = 1 - \left( {1 - x} \right)\left( {1 - y} \right)$, i.e., $\tau \left( x \right) = - \ln x$ and $\zeta \left( x \right) = - \ln \left( {1 - x} \right)$, then.

$\text{i)}\:\:\:\:\:\:\:m_{1} \oplus_{{T_{{\text{P}}} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \mu_{{w_{{m_{1} }} }}^{s} + \mu_{{w_{{m_{2} }} }}^{s} - \mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} ,\nu_{{e_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} \nu_{{e_{{m_{2} }} }}^{s} - \mu_{{w_{{m_{1} }} }}^{s} \nu_{{e_{{m_{2} }} }}^{s} - \mu_{{w_{{m_{2} }} }}^{s} \nu_{{e_{{m_{1} }} }}^{s} \hfill \\ - \mu_{{w_{{m_{1} }} }}^{s} + \mu_{{w_{{m_{2} }} }}^{s} - \mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} ,o_{{r_{{m_{1} }} }}^{s} o_{{r_{{m_{2} }} }}^{s} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{ii)}\:\:\:\:\:\:\:m_{1} \otimes_{{T_{{\text{P}}} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} ,\nu_{{e_{{m_{1} }} }}^{s} \nu_{{e_{{m_{2} }} }}^{s} + \mu_{{w_{{m_{1} }} }}^{s} \nu_{{e_{{m_{2} }} }}^{s} + \mu_{{w_{{m_{2} }} }}^{s} \nu_{{e_{{m_{1} }} }}^{s} ,o_{{r_{{m_{1} }} }}^{s} + o_{{r_{{m_{2} }} }}^{s} - o_{{r_{{m_{1} }} }}^{s} o_{{r_{{m_{2} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iii)}\:\:\:\:\:\:\:\lambda_{{T_{{\text{P}}} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {1 - \left( {1 - \mu_{{w_{m} }}^{s} } \right)^{\lambda } ,\left( {1 - \mu_{{w_{m} }}^{s} } \right)^{\lambda } - \left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)^{\lambda } ,\left( {o_{{r_{m} }}^{s} } \right)^{\lambda } } \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{ix)}\:\:\:\:\:\:\:m^{{\lambda_{{T_{{\text{P}}} }} }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\left( {\mu_{{w_{m} }}^{s} } \right)^{\lambda } ,\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)^{\lambda } - \left( {\mu_{{w_{m} }}^{s} } \right)^{\lambda } ,1 - \left( {1 - o_{{r_{m} }}^{s} } \right)^{\lambda } } \right)} \right|k^{s} } \right\}} \right\rangle }.$

(2) If we take $T$ as Schweizer-Sklar t-norms, i.e., $T = T_{\gamma }^{SS} = \left( {x^{\gamma } + y^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \left( {\gamma \in \left( { - \infty ,0} \right)} \right)$, $S = S_{\gamma }^{SS} = 1 - \left( {\left( {1 - x} \right)^{\gamma } + \left( {1 - y} \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}}$, i.e., $\tau \left( x \right) = \frac{{1 - x^{\gamma } }}{\gamma }$ and $\zeta \left( x \right) = \frac{{1 - \left( {1 - x} \right)^{\gamma } }}{\gamma }$, then.

$\text{i)}\:\:\:\:\:\:\:m_{1} \oplus_{{T_{\gamma }^{SS} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \left( {\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {1 - \mu_{{w_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ \left( {\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {1 - \mu_{{w_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} - \hfill \\ \left( {\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {1 - \mu_{{w_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ \left( {\left( {o_{{r_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {o_{{r_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

$\text{ii)}\:\:\:\:m_{1} \otimes_{{T_{\gamma }^{SS} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \left( {\left( {\mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {\mu_{{w_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ \left( {\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ - \left( {\left( {\mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {\mu_{{w_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ 1 - \left( {\left( {1 - o_{{r_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( {1 - o_{{r_{{m_{2} }} }}^{s} } \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

$\text{iii)}\:\:\:\:\:\:\:\lambda_{{T_{\gamma }^{SS} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \left( {1 - \lambda \left( {1 - \left( {1 - \gamma \mu_{{w_{m} }}^{s} } \right)^{\gamma } } \right)} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ \left( {1 - \lambda \left( {1 - \left( {1 - \gamma \mu_{{w_{m} }}^{s} } \right)^{\gamma } } \right)} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} - \hfill \\ \left( {1 - \lambda \left( {1 - \left( {1 - \gamma \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right)^{\gamma } } \right)} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ \left( {1 - \lambda \left( {1 - o_{{r_{m} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

$\text{iv)}\:\:\:\:\:\:\:m^{{\lambda_{{T_{\gamma }^{SS} }} }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \left( {1 - \lambda \left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ \left( {1 - \lambda \left( {1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} - \hfill \\ \left( {1 - \lambda \left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} , \hfill \\ 1 - \left( {1 - \lambda \left( {1 - \left( {1 - \gamma o_{{r_{{m_{1} }} }}^{s} } \right)^{\gamma } } \right)} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } .$

(3) If we take $T$ as Hamacher t-norms, i.e., $T = T_{\gamma }^{H} = \frac{xy}{{\gamma + \left( {1 - \gamma } \right)\left( {x + y - xy} \right)}}\left( {\gamma \in \left( {0, + \infty } \right)} \right),$

$S = S_{\gamma }^{H} = 1 - \frac{{\left( {1 - x} \right)\left( {1 - y} \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {1 - xy} \right)}}\left( {\gamma \in \left( {0, + \infty } \right)} \right)$, i.e., $\tau \left( x \right) = \log_{\gamma }^{{\frac{{\gamma + \left( {1 - \gamma } \right)x}}{x}}}$ and $\zeta \left( x \right) = \log_{\gamma }^{{\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - x} \right)}}{1 - x}}}$, then.

$\text{i)}\:\:\:\:\:\:\:m_{1} \oplus_{{T_{\gamma }^{H} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \frac{{\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)\left( {1 - \mu_{{w_{{m_{2} }} }}^{s} } \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} } \right)}}, \hfill \\ \frac{{\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)\left( {1 - \mu_{{w_{{m_{2} }} }}^{s} } \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} } \right)}} - \hfill \\ \frac{{\left( {1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} } \right)\left( {1 - \mu_{{w_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{2} }} }}^{s} } \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right)\left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right)} \right)}}, \hfill \\ \frac{{o_{{r_{{m_{1} }} }}^{s} o_{{r_{{m_{2} }} }}^{s} }}{{\gamma + \left( {1 - \gamma } \right)\left( {o_{{r_{{m_{1} }} }}^{s} + o_{{r_{{m_{2} }} }}^{s} - o_{{r_{{m_{1} }} }}^{s} o_{{r_{{m_{2} }} }}^{s} } \right)}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

 $\text{ii)}\:\:\:\:\:\:\:\begin{gathered} m_{1} \otimes_{{T_{\gamma }^{H} }} m_{2} \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} }}{{\gamma + \left( {1 - \gamma } \right)\left( {\mu_{{w_{{m_{1} }} }}^{s} + \mu_{{w_{{m_{2} }} }}^{s} - \mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} } \right)}}, \hfill \\ \frac{{\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right)\left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} + \mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} - \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right)\left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right)} \right)}} \hfill \\ - \frac{{\mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} }}{{\gamma + \left( {1 - \gamma } \right)\left( {\mu_{{w_{{m_{1} }} }}^{s} + \mu_{{w_{{m_{2} }} }}^{s} - \mu_{{w_{{m_{1} }} }}^{s} \mu_{{w_{{m_{2} }} }}^{s} } \right)}}, \hfill \\ 1 - \frac{{\left( {1 - o_{{r_{{m_{1} }} }}^{s} } \right)\left( {1 - o_{{r_{{m_{2} }} }}^{s} } \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {1 - o_{{r_{{m_{1} }} }}^{s} o_{{r_{{m_{2} }} }}^{s} } \right)}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ;} \hfill \\ \end{gathered}$

$\text{iii)}\:\:\:\:\:\:\:\lambda_{{T_{\gamma }^{H} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \mu_{{w_{m} }}^{s} } \right)^{\lambda } }}{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} } \right)^{\lambda } }}, \hfill \\ \frac{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} - \mu_{{e_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \mu_{{w_{m} }}^{s} - \mu_{{e_{m} }}^{s} } \right)^{\lambda } }}{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} - \mu_{{e_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} - \mu_{{e_{m} }}^{s} } \right)^{\lambda } }} \hfill \\ - \frac{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \mu_{{w_{m} }}^{s} } \right)^{\lambda } }}{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{m} }}^{s} } \right)^{\lambda } }}, \hfill \\ \frac{{\gamma \left( {o_{{r_{m} }}^{s} } \right)^{\lambda } }}{{\left( {\gamma + \left( {1 - \gamma } \right)o_{{r_{m} }}^{s} } \right)^{{^{\lambda } }} - \left( {1 - \gamma } \right)\left( {o_{{r_{m} }}^{s} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ;$

$\text{iv)}\:\:\:\:\:\:\:m^{{\lambda_{{T_{\gamma }^{H} }} }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\gamma \left( {\mu_{{w_{m} }}^{s} } \right)^{\lambda } }}{{\left( {\gamma + \left( {1 - \gamma } \right)\mu_{{w_{m} }}^{s} } \right)^{{^{\lambda } }} - \left( {1 - \gamma } \right)\left( {\mu_{{w_{m} }}^{s} } \right)^{\lambda } }}, \hfill \\ \frac{{\gamma \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)^{\lambda } }}{{\left( {\gamma + \left( {1 - \gamma } \right)\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right)^{{^{\lambda } }} - \left( {1 - \gamma } \right)\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)^{\lambda } }} \hfill \\ - \frac{{\gamma \left( {\mu_{{w_{m} }}^{s} } \right)^{\lambda } }}{{\left( {\gamma + \left( {1 - \gamma } \right)\mu_{{w_{m} }}^{s} } \right)^{{^{\lambda } }} - \left( {1 - \gamma } \right)\left( {\mu_{{w_{m} }}^{s} } \right)^{\lambda } }}, \hfill \\ \frac{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - o_{{r_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - o_{{r_{m} }}^{s} } \right)^{\lambda } }}{{\left( {1 + \left( {1 - \gamma } \right)\left( {1 - o_{{r_{m} }}^{s} } \right)} \right)^{\lambda } - \left( {1 - \gamma } \right)\left( {1 - o_{{r_{m} }}^{s} } \right)^{\lambda } }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } .$

In particular, when $\gamma = 2$, $T_{2}^{H}$ and $S_{2}^{H}$ reduce to the Einstein product and Einstein sum, respectively; when $\gamma = 1$, $T_{1}^{H}$ and $S_{1}^{H}$ reduce to the $T_{{\text{P}}}$ and $S_{{\text{P}}}$, respectively.

(4) If we take $T$ as Frank t-norms, i.e.,

$T = T_{\gamma }^{F} = \log_{\gamma }^{{1 + \frac{{\left( {\gamma^{x} - 1} \right)\left( {\gamma^{y} - 1} \right)}}{\gamma - 1}}} \left( {\gamma \in \left( {0,1} \right) \cup \left( {1, + \infty } \right)} \right),$

$S = S_{\gamma }^{H} = 1 - \log_{\gamma }^{{1 + \frac{{\left( {\gamma^{1 - x} - 1} \right)\left( {\gamma^{1 - y} - 1} \right)}}{\gamma - 1}}} \left( {\gamma \in \left( {0,1} \right) \cup \left( {1, + \infty } \right)} \right)$, i.e., $\tau \left( x \right) = \log_{\gamma }^{{\frac{\gamma - 1}{{\gamma^{x} - 1}}}}$ and $\zeta \left( x \right) = \log_{\gamma }^{{\frac{\gamma - 1}{{\gamma^{1 - x} - 1}}}}$, then.

 $\text{i)}\:\:\:\:\:\:\:m_{1} \oplus_{{T_{\gamma }^{F} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - \mu_{{w_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{1 - \mu_{{w_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right), \hfill \\ \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - \mu_{{w_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{1 - \mu_{{w_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right) \hfill \\ - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{1 - \mu_{{w_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right), \hfill \\ \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{o_{{r_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{o_{{r_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{ii)}\:\:\:\:\:\:\:m_{1} \otimes_{{T_{\gamma }^{F} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{\mu_{{w_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{\mu_{{w_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right), \hfill \\ \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right) \hfill \\ - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{\mu_{{w_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{\mu_{{w_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right), \hfill \\ 1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - o_{{r_{{m_{1} }} }}^{s} }} - 1} \right)\left( {\gamma^{{1 - o_{{r_{{m_{2} }} }}^{s} }} - 1} \right)}}{\gamma - 1}} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iii)}\:\:\:\:\:\:\:\lambda_{{T_{\gamma }^{F} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{1 - \mu_{{w_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right), \hfill \\ \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right) - \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{1 - \mu_{{w_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right), \hfill \\ \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{o_{{r_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iv)}\:\:\:\:\:\:\:m^{{\lambda_{{T_{\gamma }^{F} }} }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{\mu_{{w_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right), \hfill \\ \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right) - \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{\mu_{{w_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right), \hfill \\ 1 - \log_{\gamma } \left( {1 + \frac{\gamma - 1}{{\left( {\frac{\gamma - 1}{{\gamma^{{1 - o_{{r_{m} }}^{s} }} - 1}}} \right)^{\lambda } }}} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle }.$

(5) If we take $T$ as Dombi t-norms, i.e.,

$T = T_{\gamma }^{D} = \frac{1}{{1 + \left( {\left( {\frac{1 - x}{x}} \right)^{\gamma } + \left( {\frac{1 - y}{y}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}\left( {\gamma \in \left( {0, + \infty } \right)} \right),$

$S = S_{\gamma }^{D} = 1 - \frac{1}{{1 + \left( {\left( {\frac{x}{1 - x}} \right)^{\gamma } + \left( {\frac{y}{1 - y}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}$, i.e., $\tau \left( x \right) = \left( {\frac{1 - x}{x}} \right)^{\gamma }$ and $\zeta \left( x \right) = \left( {\frac{x}{1 - x}} \right)^{\gamma }$, then.

$\text{i)}\:\:\:\:\:\:\:m_{1} \oplus_{{T_{\gamma }^{D} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \frac{1}{{1 + \left( {\left( {\frac{{\mu_{{w_{{m_{1} }} }}^{s} }}{{1 - \mu_{{w_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{\mu_{{w_{{m_{2} }} }}^{s} }}{{1 - \mu_{{w_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}, \hfill \\ \frac{1}{{1 + \left( {\left( {\frac{{\mu_{{w_{{m_{1} }} }}^{s} }}{{1 - \mu_{{w_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{\mu_{{w_{{m_{2} }} }}^{s} }}{{1 - \mu_{{w_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ - \frac{1}{{1 + \left( {\left( {\frac{{\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} }}{{1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} }}{{1 - \mu_{{w_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}, \hfill \\ \frac{1}{{1 + \left( {\left( {\frac{{1 - o_{{r_{{m_{1} }} }}^{s} }}{{o_{{r_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{1 - o_{{r_{{m_{2} }} }}^{s} }}{{o_{{r_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{ii)}\:\:\:\:\:\:\:m_{1} \otimes_{{T_{\gamma }^{D} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{1}{{1 + \left( {\left( {\frac{{1 - \mu_{{w_{{m_{1} }} }}^{s} }}{{\mu_{{w_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{1 - \mu_{{w_{{m_{2} }} }}^{s} }}{{\mu_{{w_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}, \hfill \\ \frac{1}{{1 + \left( {\left( {\frac{{1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} }}{{\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{1 - \mu_{{w_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{2} }} }}^{s} }}{{\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ - \frac{1}{{1 + \left( {\left( {\frac{{1 - \mu_{{w_{{m_{1} }} }}^{s} }}{{\mu_{{w_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{1 - \mu_{{w_{{m_{2} }} }}^{s} }}{{\mu_{{w_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}, \hfill \\ 1 - \frac{1}{{1 + \left( {\left( {\frac{{o_{{r_{{m_{1} }} }}^{s} }}{{1 - o_{{r_{{m_{1} }} }}^{s} }}} \right)^{\gamma } + \left( {\frac{{o_{{r_{{m_{2} }} }}^{s} }}{{1 - o_{{r_{{m_{2} }} }}^{s} }}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iii)}\:\:\:\:\:\:\:\lambda_{{T_{\gamma }^{D} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\sqrt[\gamma ]{\lambda }\mu_{{w_{m} }}^{s} }}{{1 - \mu_{{w_{m} }}^{s} - \sqrt[\gamma ]{\lambda }\mu_{{w_{m} }}^{s} }}, \hfill \\ \frac{{\sqrt[\gamma ]{\lambda }\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)}}{{1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} - \sqrt[\gamma ]{\lambda }\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)}} - \frac{{\sqrt[\gamma ]{\lambda }\mu_{{w_{m} }}^{s} }}{{1 - \mu_{{w_{m} }}^{s} - \sqrt[\gamma ]{\lambda }\mu_{{w_{m} }}^{s} }}, \hfill \\ \frac{{\sqrt[\gamma ]{\lambda }o_{{r_{m} }}^{s} }}{{o_{{r_{m} }}^{s} + \sqrt[\gamma ]{\lambda }\left( {1 - o_{{r_{m} }}^{s} } \right)}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iv)}\:\:\:\:\:\:\:\lambda_{{T_{\gamma }^{D} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\sqrt[\gamma ]{\lambda }\mu_{{w_{m} }}^{s} }}{{\mu_{{w_{m} }}^{s} + \sqrt[\gamma ]{\lambda }\left( {1 - \mu_{{w_{m} }}^{s} } \right)}}, \hfill \\ \frac{{\sqrt[\gamma ]{\lambda }\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)}}{{\left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right) + \sqrt[\gamma ]{\lambda }\left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)}} - \frac{{\sqrt[\gamma ]{\lambda }\mu_{{w_{m} }}^{s} }}{{\mu_{{w_{m} }}^{s} + \sqrt[\gamma ]{\lambda }\left( {1 - \mu_{{w_{m} }}^{s} } \right)}}, \hfill \\ \frac{{\sqrt[\gamma ]{\lambda }o_{{r_{m} }}^{s} }}{{1 - o_{{r_{m} }}^{s} - \sqrt[\gamma ]{\lambda }o_{{r_{m} }}^{s} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle }.$

(6) If we take $T$ as Aczél-Alsina t-norms, i.e., $T = T_{\gamma }^{AA} = e^{{ - \left( {\left( { - \ln x} \right)^{\gamma } + \left( { - \ln y} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }},$

$S = S_{\gamma }^{AA} = 1 - e^{{ - \left( {\left( { - \ln \left( {1 - x} \right)} \right)^{\gamma } + \left( { - \ln \left( {1 - y} \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}$, i.e., $\tau \left( x \right) = \left( { - \ln x} \right)^{\gamma }$ and $\zeta \left( x \right) = \left( { - \ln \left( {1 - x} \right)} \right)^{\gamma }$, then.

$\text{i)}\:\:\:\:\:\:\:m_{1} \oplus_{{T_{\gamma }^{AA} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - e^{{ - \left( {\left( { - \ln \left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)} \right)^{\gamma } + \left( { - \ln \left( {1 - \mu_{{w_{{m_{2} }} }}^{s} } \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} , \hfill \\ e^{{ - \left( {\left( { - \ln \left( {1 - \mu_{{w_{{m_{1} }} }}^{s} } \right)} \right)^{\gamma } + \left( { - \ln \left( {1 - \mu_{{w_{{m_{2} }} }}^{s} } \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ - e^{{ - \left( {\left( { - \ln \left( {1 - \mu_{{w_{{m_{1} }} }}^{s} - \nu_{{e_{{m_{1} }} }}^{s} } \right)} \right)^{\gamma } + \left( { - \ln \left( {1 - \mu_{{w_{{m_{2} }} }}^{s} - \nu_{{e_{{m_{2} }} }}^{s} } \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} , \hfill \\ e^{{ - \left( {\left( { - \ln o_{{r_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( { - \ln o_{{r_{{m_{2} }} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{ii)}\:\:\:\:\:\:\:m_{1} \otimes_{{T_{\gamma }^{AA} }} m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} e^{{ - \left( {\left( { - \ln \mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( { - \ln \mu_{{w_{{m_{2} }} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} , \hfill \\ e^{{ - \left( {\left( { - \ln \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right)} \right)^{\gamma } + \left( { - \ln \left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ - e^{{ - \left( {\left( { - \ln \mu_{{w_{{m_{1} }} }}^{s} } \right)^{\gamma } + \left( { - \ln \mu_{{w_{{m_{2} }} }}^{s} } \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} , \hfill \\ 1 - e^{{ - \left( {\left( { - \ln \left( {1 - o_{{r_{{m_{1} }} }}^{s} } \right)} \right)^{\gamma } + \left( { - \ln \left( {1 - o_{{r_{{m_{2} }} }}^{s} } \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iii)}\:\:\:\:\:\:\:\lambda_{{T_{\gamma }^{AA} }} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \left( {1 - \mu_{{w_{m} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} , \hfill \\ \left( {1 - \mu_{{w_{m} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} - \left( {1 - \mu_{{w_{m} }}^{s} - \nu_{{e_{m} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} \hfill \\ \left( {o_{{r_{{m_{1} }} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle };$

$\text{iv)}\:\:\:\:\:\:\:m^{{\lambda_{{T_{\gamma }^{AA} }} }} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \left( {\mu_{{w_{m} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} , \hfill \\ \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} - \left( {\mu_{{w_{m} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} \hfill \\ 1 - \left( {1 - o_{{r_{{m_{1} }} }}^{s} } \right)^{{\sqrt[\gamma ]{\lambda }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle }.$

Theorem 3.2.

Let $m_{1} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\nu_{{e_{{m_{1} }} }}^{s} ,o_{{r_{{m_{1} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$, $m_{2} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{2} }} }}^{s} ,\nu_{{e_{{m_{2} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$ and $m_{3} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{3} }} }}^{s} ,\nu_{{e_{{m_{3} }} }}^{s} ,o_{{r_{{m_{3} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle }$ be three PPHFEs, $\xi ,\lambda> 0$, then.

(i) $m_{1} \oplus_{T} m_{2} = m_{2} \oplus_{T} m_{1}$;

(ii) $m_{1} \otimes_{T} m_{2} = m_{2} \otimes_{T} m_{1}$;

(iii) $\left( {m_{1} \oplus_{T} m_{2} } \right) \oplus_{T} m_{3} = m_{1} \oplus_{T} \left( {m_{2} \oplus_{T} m_{3} } \right)$;

(iv) $\left( {m_{1} \otimes_{T} m_{2} } \right) \otimes_{T} m_{3} = m_{1} \otimes_{T} \left( {m_{2} \otimes_{T} m_{3} } \right)$;

(v) $\xi_{T} m_{1} \oplus_{T} \lambda_{T} m_{1} = \left( {\xi + \lambda } \right)_{T} m_{1}$;

(vi) $m_{1}^{{\xi_{T} }} \otimes_{T} m_{1}^{{\lambda_{T} }} = m_{1}^{{\left( {\xi + \lambda } \right)_{T} }}$

(vii) $\lambda_{T} \left( {m_{1} \oplus_{T} m_{2} } \right) = \lambda_{T} m_{1} \oplus_{T} \lambda_{T} m_{2}$;

(viii) $\left( {m_{1} \otimes_{T} m_{2} } \right)^{{\lambda_{T} }} = m_{1}^{{\lambda_{T} }} \otimes_{T} m_{2}^{{\lambda_{T} }}$;

(ix) $\xi_{T} \left( {\lambda_{T} m_{1} } \right) = \left( {\xi \cdot \lambda } \right)_{T} m_{1}$;

(x) $\left( {m_{1}^{{\lambda_{T} }} } \right)^{{\xi_{T} }} = m_{1}^{{\left( {\lambda \cdot \xi } \right)_{T} }}$;

(xi) $\left( {m_{1} \oplus_{T} m_{2} } \right)^{\rm{\complement }} = m_{1}^{\rm{\complement }} \otimes_{T} m_{2}^{\rm{\complement }}$.

Proof

The statements (i)-(ii) follow directly from the commutativity of $T$ and $S$.

(iii) By direct calculation, we have.

$$\begin{gathered} \left( {m_{1} \oplus_{T} m_{2} } \right) \oplus_{T} m_{3} \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left( {\left. {S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) - S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right)} \right|} \right)k^{s} } \right\}} \right\rangle } \oplus_{T} m_{3} \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} S\left( {S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),\mu_{{w_{{m_{3} }} }}^{s} } \right),S\left( {S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right),\mu_{{w_{{m_{3} }} }}^{s} + \nu_{{e_{{m_{3} }} }}^{s} } \right) \hfill \\ - S\left( {S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right),\mu_{{w_{{m_{3} }} }}^{s} } \right),T\left( {T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right),o_{{r_{{m_{3} }} }}^{s} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,S\left( {\mu_{{w_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{3} }} }}^{s} } \right)} \right),S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,S\left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{3} }} }}^{s} + \nu_{{e_{{m_{3} }} }}^{s} } \right)} \right) \hfill \\ - S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,S\left( {\mu_{{w_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{3} }} }}^{s} } \right)} \right),T\left( {o_{{r_{{m_{1} }} }}^{s} ,T\left( {o_{{r_{{m_{2} }} }}^{s} ,o_{{r_{{m_{3} }} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = m_{1} \oplus_{T} \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {S\left( {\mu_{{w_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{3} }} }}^{s} } \right),S\left( {\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{3} }} }}^{s} + \nu_{{e_{{m_{3} }} }}^{s} } \right) - S\left( {\mu_{{w_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{3} }} }}^{s} } \right),T\left( {o_{{r_{{m_{2} }} }}^{s} ,o_{{r_{{m_{3} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = m_{1} \oplus_{T} \left( {m_{2} \oplus_{T} m_{3} } \right). \hfill \\ \end{gathered}$$

(v) By $T\left( {x,y} \right) = \tau^{ - 1} \left( {\tau \left( x \right) + \tau \left( y \right)} \right)$ and $S\left( {x,y} \right) = \zeta^{ - 1} \left( {\zeta \left( x \right) + \zeta \left( y \right)} \right)$, we have.

$$\begin{gathered} \xi_{T} m_{1} \oplus_{T} \lambda_{T} m_{1} \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \oplus_{T} \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\lambda \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \left\langle {\left\{ {\left. {\left( \begin{gathered} S\left( {\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right)} \right), \hfill \\ S\left( {\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right)} \right) \hfill \\ - S\left( {\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right)} \right), \hfill \\ T\left( {\tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\lambda \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle \hfill \\ = \left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\left( {\xi + \lambda } \right) \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\left( {\xi + \lambda } \right) \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\left( {\xi + \lambda } \right) \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\left( {\xi + \lambda } \right) \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle \hfill \\ = \left( {\xi + \lambda } \right)_{T} m_{1} . \hfill \\ \end{gathered}$$

(vii) By $T\left( {x,y} \right) = \tau^{ - 1} \left( {\tau \left( x \right) + \tau \left( y \right)} \right)$ and $S\left( {x,y} \right) = \zeta^{ - 1} \left( {\zeta \left( x \right) + \zeta \left( y \right)} \right)$, we have

(ix) By direct calculation, we have.

$$\begin{gathered} \xi_{T} \left( {\lambda_{T} m_{1} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\xi_{T} \cdot \left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\xi \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\xi \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\xi \cdot \lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\xi \cdot \lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\xi \cdot \lambda \cdot \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\xi \cdot \lambda \cdot \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \left( {\xi \cdot \lambda } \right)_{T} m_{1} . \hfill \\ \end{gathered}$$

By using similar arguments to the proofs of (iii), (v), (vii), and (ix), it can be verified that the statements (iv), (vi), (viii), and (x) hold.

(xi) By direct calculation, we have.

$$\begin{gathered} \left( {m_{1} \oplus_{T} m_{2} } \right)^{\rm{\complement }} \hfill \\ = \left( {\bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right),1 - S\left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} + \nu_{{e_{{m_{2} }} }}^{s} } \right) - T\left( {o_{{w_{{m_{1} }} }}^{s} ,o_{{w_{{m_{2} }} }}^{s} } \right),S\left( {\mu_{{w_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle } } \right), \hfill \\ \end{gathered}$$

and

$$\begin{gathered} m_{1}^{\rm{\complement }} \otimes_{T} m_{2}^{\rm{\complement }} = \left\langle {o_{{r_{{m_{1} }} }}^{s} ,\pi_{{r_{{m_{1} }} }}^{s} ,\mu_{{w_{{m_{1} }} }}^{s} } \right\rangle \otimes_{T} \left\langle {o_{{r_{{m_{2} }} }}^{s} ,\pi_{{r_{{m_{2} }} }}^{s} ,\mu_{{w_{{m_{2} }} }}^{s} } \right\rangle \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right),T\left( {o_{{r_{{m_{1} }} }}^{s} + \pi_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} + \pi_{{r_{{m_{2} }} }}^{s} } \right) - T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right),S\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right),T\left( {1 - \mu_{{r_{{m_{1} }} }}^{s} + \nu_{{r_{{m_{1} }} }}^{s} ,1 - \mu_{{r_{{m_{2} }} }}^{s} + \nu_{{r_{{m_{2} }} }}^{s} } \right) - T\left( {o_{{r_{{m_{1} }} }}^{s} ,o_{{r_{{m_{2} }} }}^{s} } \right),S\left( {\mu_{{r_{{m_{1} }} }}^{s} ,\mu_{{r_{{m_{2} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \end{gathered}$$

This, together with $S\left( {x,y} \right) = 1 - T\left( {1 - x,1 - y} \right).$ Implies that $\left( {m_{1} \oplus_{T} m_{2} } \right)^{\rm{\complement }} = m_{1}^{\rm{\complement }} \otimes_{T} m_{2}^{\rm{\complement }} .$

Some PPHFDisMs for PPHFEs

Definition 3.4

Let $X$ be a UOD. A mapping $D:PPHFS\left( X \right) \times PPHFS\left( X \right) \to \left[ {0,1} \right]$ is called a picture fuzzy distance measure (PPHFDisM) on $PPHFS\left( X \right)$ if it meets the below conditions: For any $M,N,K \in PFS\left( X \right),$

(Dis.1) $0 \le D\left( {M,N} \right) = D\left( {N,M} \right) \le 1;$

(Dis.2) $D\left( {M,N} \right) = 0$ iff $M = N;$

(Dis.3) $D\left( {M,K} \right) \le D\left( {M,N} \right) + D\left( {N,K} \right);$

(Dis.4) If $M \subseteq N \subseteq K,$ then $D\left( {M,K} \right) \ge D\left( {M,N} \right)$ and $D\left( {M,K_{3} } \right) \ge D\left( {N,K} \right).$

Let $\Delta w\left( x \right) = \sum\nolimits_{s \in M} {\left| {\mu_{{w_{M} }}^{s} \left( x \right) - \mu_{{w_{N} }}^{s} \left( x \right)} \right|k^{s} } ,$ $\Delta e\left( x \right) = \sum\nolimits_{s \in M} {\left| {\mu_{{w_{M} }}^{s} \left( x \right) + \nu_{{e_{M} }}^{s} \left( x \right) - \mu_{{w_{N} }}^{s} \left( x \right) - \nu_{{e_{N} }}^{s} \left( x \right)} \right|k^{s} }$ and $\Delta r\left( x \right) = \sum\nolimits_{s \in M} {\left| {o_{{r_{M} }}^{s} \left( x \right) - o_{{r_{N} }}^{s} \left( x \right)} \right|k^{s} } ,$ where $\lambda> 0.$

(1) The PPHF normalized Hamming distance measure (PPHFNHDM) between $M$ and $N.$

$$D_{PPHFNHDM} \left( {M,M} \right) = \frac{1}{3n}\left( {\Delta w\left( x \right) + \Delta e\left( x \right) + \Delta r\left( x \right)} \right).$$

(2) The PPHF normalized Euclidean distance measure (PPHFNEDM) between $M$ and $N.$

$$D_{PPHFNEDM} \left( {M,M} \right) = \sqrt {\frac{1}{3n}\left( {\Delta w^{2} \left( x \right) + \Delta e^{2} \left( x \right) + \Delta r^{2} \left( x \right)} \right)} .$$

(3) The generalized PPHF normalized distance measure (GPPHFNDM) between $M$ and $N$

$$D_{GPPHFNDM} \left( {M,M} \right) = \sqrt[\lambda ]{{\frac{1}{3n}\left( {\Delta w^{\lambda } \left( x \right) + \Delta e^{\lambda } \left( x \right) + \Delta r^{\lambda } \left( x \right)} \right)}}.$$

(4) The PPHF weighted Hamming distance measure (PPHFWHDM) between $M$ and $N.$

$$D_{PPHFWHDM} \left( {M,M} \right) = \frac{1}{3}w_{i} \left( {\Delta w\left( x \right) + \Delta e\left( x \right) + \Delta r\left( x \right)} \right).$$

(5) The PPHF weighted Euclidean distance measure (PPHFWEDM) between $M$ and $N.$

$$D_{PPHFWEDM} \left( {M,M} \right) = \sqrt {\frac{1}{3}w_{i} \left( {\Delta w^{2} \left( x \right) + \Delta e^{2} \left( x \right) + \Delta r^{2} \left( x \right)} \right)} .$$

(6) The generalized PPHF weighted distance measure (GPPHFWDM) between $M$ and $N.$

$$D_{GPPHFWDM} \left( {M,M} \right) = \sqrt[\lambda ]{{\frac{1}{3}w_{i} \left( {\Delta w^{\lambda } \left( x \right) + \Delta e^{\lambda } \left( x \right) + \Delta r^{\lambda } \left( x \right)} \right)}}.$$

Some picture fuzzy average/geometric operators for PPHFEs

Definition 4.1.

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $T$ be t-norm. Then.

$$\begin{gathered} PPHFWA_{T,\Omega } :{\mathbb{P}}^{n} \to {\mathbb{P}} \hfill \\ \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \to \left( {\omega_{1} } \right)_{T} m_{1} \oplus_{T} \left( {\omega_{2} } \right)_{T} m_{2} \oplus_{T} \cdots \oplus_{T} \left( {\omega_{n} } \right)_{T} m_{n} . \hfill \\ \end{gathered}$$

(5)

and

$$\begin{gathered} PPHFWG_{T,\Omega } :{\mathbb{P}}^{n} \to {\mathbb{P}} \hfill \\ \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \to m_{1}^{{\left( {\omega_{1} } \right)_{T} }} \otimes_{T} m_{2}^{{\left( {\omega_{2} } \right)_{T} }} \otimes_{T} \cdots \otimes_{T} m_{n}^{{\left( {\omega_{n} } \right)_{T} }} . \hfill \\ \end{gathered}$$

(6)

respectively.

Based on the interactional operations defined in Definition 3.3, this section is devoted to aggregating the PPHF information.

Theorem 4.1.

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ be a group of PPHFEs, and $\Omega = \left( {\omega_{1} ,\omega_{2} , \cdots ,\omega_{n} } \right)^{T}$ be the weight vector such that $\omega_{j} \in \left[ {0,1} \right]$ and $\sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1$, $T$ and $S$ be t-norm and t-conorm. Define the PPHF interactional weighted average operator $PPHFWA_{T,\Omega }$ and PPHF interactional weighted geometric operator $PPHFWG_{T,\Omega }$ induced by $T$ as.

$$\begin{gathered} PPHFWA_{{T,\Omega }} \left( {m_{1} ,\;m_{2} , \cdots ,m_{n} } \right) = \oplus _{{Tj = 1}}^{n} \left( {\omega _{j} } \right)_{T} m_{j} \hfill \\ = \bigcup\limits_{{s \in M}} {\left\langle {\left\{ {\left( {\begin{array}{*{20}c} {\xi ^{{ - 1}} \left( {\sum\limits_{{j = 1}}^{n} {\omega _{j} \xi \left( {\mu _{{w_{{m_{j} }} }}^{s} } \right)} } \right),\xi ^{{ - 1}} \left( {\sum\limits_{{j = 1}}^{n} {\omega _{j} \xi \left( {\mu _{{w_{{m_{j} }} }}^{s} } \right) + \nu _{{e_{{m_{j} }} }}^{s} } } \right)} \\ { - \zeta ^{{ - 1}} \left( {\sum\limits_{{j = 1}}^{n} {\omega _{j} \xi \left( {\mu _{{w_{{m_{j} }} }}^{s} } \right)} } \right),\tau ^{{ - 1}} \left( {\sum\limits_{{j = 1}}^{n} {\omega _{j} \tau \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} } \right)} \\ \end{array} } \right)K^{s} } \right\}} \right\rangle } , \hfill \\ \end{gathered}$$

(7)

and

$$\begin{gathered} PPHFWG_{{T,\Omega }} \left( {m_{1} ,\;m_{2} , \cdots ,m_{n} } \right) = \oplus _{{Tj = 1}}^{n} m_{j}^{{\left( {\omega _{j} } \right)_{T} }} \hfill \\ = \bigcup\limits_{{s \in M}} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \tau ^{{ - 1}} \left( {\sum\nolimits_{{j = 1}}^{n} {\omega _{j} \tau \left( {\mu _{{w_{{m_{j} }} }}^{s} } \right)} } \right),\tau ^{{ - 1}} \left( {\sum\nolimits_{{j = 1}}^{n} {\omega _{j} \tau \left( {\mu _{{w_{{m_{j} }} }}^{s} + \nu _{{e_{{m_{j} }} }}^{s} } \right)} } \right) \hfill \\ - \tau ^{{ - 1}} \left( {\sum\nolimits_{{j = 1}}^{n} {\omega _{j} \tau \left( {\mu _{{w_{{m_{j} }} }}^{s} } \right)} } \right),\zeta ^{{ - 1}} \left( {\sum\nolimits_{{j = 1}}^{n} {\omega _{j} \zeta \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \end{gathered}$$

(8)

where $\zeta \left( x \right) = \tau \left( {1 - x} \right)$.

Proof

From formulas (5) and (6), Theorem 4.1, and Definition 3.3, it follows that.

$$\begin{gathered} PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right)} \right) - \zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\omega_{1} \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right)} \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \, \oplus_{T} \cdots \oplus_{T} \hfill \\ \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\zeta^{ - 1} \left( {\omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right) - \zeta^{ - 1} \left( {\omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\omega_{n} \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} S^{\left( n \right)} \left( {\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right), \cdots ,\omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right)} \right), \hfill \\ S^{\left( n \right)} \left( {\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right), \cdots ,\omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right)} \right) \hfill \\ - S^{\left( n \right)} \left( {\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right), \cdots ,\omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right)} \right), \hfill \\ T^{\left( n \right)} \left( {\tau^{ - 1} \left( {\omega_{1} \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right)} \right), \cdots ,\tau^{ - 1} \left( {\omega_{n} \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\omega_{1} \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \tau^{ - 1} \left( {\omega_{1} \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right)} \right) + \cdots + \tau^{ - 1} \left( {\omega_{n} \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ,} \hfill \\ \end{gathered}$$

and

$$\begin{gathered} PPHFWG_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{w_{{m_{1} }} }}^{s} } \right)} \right) - \tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {o_{{r_{{m_{1} }} }}^{s} } \right)} \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \, \oplus_{T} \cdots \oplus_{T} \hfill \\ \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\tau^{ - 1} \left( {\omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right) - \tau^{ - 1} \left( {\omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\omega_{n} \zeta \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} T^{\left( n \right)} \left( {\tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right), \cdots ,\omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right)} \right), \hfill \\ T^{\left( n \right)} \left( {\tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right), \cdots ,\omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right)} \right) \hfill \\ - T^{\left( n \right)} \left( {\tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right), \cdots ,\omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right)} \right), \hfill \\ S^{\left( n \right)} \left( {\zeta^{ - 1} \left( {\omega_{1} \zeta \left( {o_{{r_{{m_{1} }} }}^{s} } \right)} \right), \cdots ,\zeta^{ - 1} \left( {\omega_{n} \zeta \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ - \tau^{ - 1} \left( {\omega_{1} \tau \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \tau \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\omega_{1} \zeta \left( {o_{{r_{{m_{1} }} }}^{s} } \right)} \right) + \cdots + \zeta^{ - 1} \left( {\omega_{n} \zeta \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \end{gathered}$$

If we assign specific forms of t-norms to Theorem 4.1, we have the following results.

If we take $T = T_{{\text{P}}} = xy$,

$S = S_{{\text{P}}} = 1 - \left( {1 - x} \right)\left( {1 - y} \right)$, i.e., $\tau \left( x \right) = - \ln x$ and $\zeta \left( x \right) = - \ln \left( {1 - x} \right)$, then.

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWA_{{T_{{\text{P}}} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } ,\prod\limits_{j = 1}^{n} {\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } - \prod\limits_{j = 1}^{n} {\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } ,\prod\limits_{j = 1}^{n} {\left( {o_{{r_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } } \right)} \right|k^{s} } \right\}} \right\rangle ;} \hfill \\ \end{gathered}$

$\text{ii)}\:\:\:\:\:\:\:\begin{gathered} PPHFWG_{{T_{{\text{P}}} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\prod\limits_{j = 1}^{n} {\left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } ,\prod\limits_{j = 1}^{n} {\left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } - \prod\limits_{j = 1}^{n} {\left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } ,1 - \prod\limits_{j = 1}^{n} {\left( {1 - o_{{r_{{m_{j} }} }}^{s} } \right)^{{\omega_{j} }} } } \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ \end{gathered}$

If we take $T$ as Schweizer-Sklar t-norms, i.e.,

$T = T_{\gamma }^{SS} = \left( {x^{\gamma } + y^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \left( {\gamma \in \left( { - \infty ,0} \right)} \right)$,

$S = S_{\gamma }^{SS} = 1 - \left( {\left( {1 - x} \right)^{\gamma } + \left( {1 - y} \right)^{\gamma } - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}}$, i.e., $\tau \left( x \right) = \frac{{1 - x^{\gamma } }}{\gamma }$ and $\zeta \left( x \right) = \frac{{1 - \left( {1 - x} \right)^{\gamma } }}{\gamma }$, then.

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWA_{{T_{\gamma }^{SS} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} ,\left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} - \hfill \\ \left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} ,\left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {o_{{r_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } ; \hfill \\ \end{gathered}$

$\text{ii)}\:\:\:\:\:\:\:\begin{gathered} PPHFWG_{{T_{\gamma }^{SS} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} ,\left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} - \hfill \\ \left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} ,1 - \left( {\sum\limits_{j = 1}^{n} {\omega_{j} \left( {1 - o_{{r_{{m_{j} }} }}^{s} } \right)^{\gamma } } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ \end{gathered}$

If we take $T$ as Hamacher t-norms, i.e., $T = T_{\gamma }^{H} = \frac{xy}{{\gamma + \left( {1 - \gamma } \right)\left( {x + y - xy} \right)}}\left( {\gamma \in \left( {0, + \infty } \right)} \right)$,

$S = S_{\gamma }^{H} = 1 - \frac{{\left( {1 - x} \right)\left( {1 - y} \right)}}{{\gamma + \left( {1 - \gamma } \right)\left( {1 - xy} \right)}}\left( {\gamma \in \left( {0, + \infty } \right)} \right)$, i.e., $\tau \left( x \right) = \log_{\gamma }^{{\frac{{\gamma + \left( {1 - \gamma } \right)x}}{x}}}$ and $\zeta \left( x \right) = \log_{\gamma }^{{\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - x} \right)}}{1 - x}}}$, then.

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWA_{{T_{\gamma }^{H} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)}}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1}}{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)}}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }}, \hfill \\ \frac{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)}}{{1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1}}{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)}}{{1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }} \hfill \\ - \frac{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)}}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1}}{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)}}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }}, \hfill \\ \frac{\gamma }{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)o_{{r_{{m_{j} }} }}^{s} }}{{o_{{r_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ;} \hfill \\ \end{gathered}$

$\text{ii)}\:\:\:\:\:\:\:\begin{gathered} PPHFWG_{{T_{\gamma }^{H} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{\gamma }{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\mu_{{w_{{m_{j} }} }}^{s} }}{{\mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }},\frac{\gamma }{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {\mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)}}{{\mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }} \hfill \\ - \frac{\gamma }{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\mu_{{w_{{m_{j} }} }}^{s} }}{{\mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }},\frac{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - o_{{r_{{m_{j} }} }}^{s} } \right)}}{{1 - o_{{r_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1}}{{\prod\limits_{j = 1}^{n} {\left( {\frac{{\gamma + \left( {1 - \gamma } \right)\left( {1 - o_{{r_{{m_{j} }} }}^{s} } \right)}}{{1 - o_{{r_{{m_{j} }} }}^{s} }}} \right)^{{\omega_{j} }} } - 1 + \gamma }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } . \hfill \\ \end{gathered}$

If we take $T$ as Frank t-norms, i.e., $T = T_{\gamma }^{F} = \log_{\gamma }^{{1 + \frac{{\left( {\gamma^{x} - 1} \right)\left( {\gamma^{y} - 1} \right)}}{\gamma - 1}}} \left( {\gamma \in \left( {0,1} \right) \cup \left( {1, + \infty } \right)} \right)$,

$S = S_{\gamma }^{H} = 1 - \log_{\gamma }^{{1 + \frac{{\left( {\gamma^{1 - x} - 1} \right)\left( {\gamma^{1 - y} - 1} \right)}}{\gamma - 1}}} \left( {\gamma \in \left( {0,1} \right) \cup \left( {1, + \infty } \right)} \right)$, i.e., $\tau \left( x \right) = \log_{\gamma }^{{\frac{\gamma - 1}{{\gamma^{x} - 1}}}}$ and $\zeta \left( x \right) = \log_{\gamma }^{{\frac{\gamma - 1}{{\gamma^{1 - x} - 1}}}}$, then.

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWA_{{T_{\gamma }^{F} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{1 - \mu_{{w_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} , \hfill \\ \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{1 - \mu_{{w_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} - \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} ,\log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{o_{{r_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ;} \hfill \\ \end{gathered}$

$\text{ii)}\:\:\:\:\:\:\:\begin{gathered} PPHFWG_{{T_{\gamma }^{F} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{\mu_{{w_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} , \hfill \\ \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} - \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{\mu_{{w_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} ,1 - \log_{\gamma }^{{\prod\limits_{j = 1}^{n} {\left( {\left( {\gamma^{{1 - o_{{r_{{m_{j} }} }}^{s} }} - 1} \right)^{{\omega_{j} }} + 1} \right)} }} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ \end{gathered}$

(5) If we take $T$ as Dombi t-norms, i.e., $T = T_{\gamma }^{D} = \frac{1}{{1 + \left( {\left( {\frac{1 - x}{x}} \right)^{\gamma } + \left( {\frac{1 - y}{y}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}$,

$S = S_{\gamma }^{D} = 1 - \frac{1}{{1 + \left( {\left( {\frac{x}{1 - x}} \right)^{\gamma } + \left( {\frac{y}{1 - y}} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}\left( {\gamma \in \left( {0, + \infty } \right)} \right),$ i.e., $\tau \left( x \right) = \left( {\frac{1 - x}{x}} \right)^{\gamma }$ and $\zeta \left( x \right) = \left( {\frac{x}{1 - x}} \right)^{\gamma } ,$ then.

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWA_{{T_{\gamma }^{D} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{{\sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{\mu_{{w_{{m_{j} }} }}^{s} }}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{\mu_{{w_{{m_{j} }} }}^{s} }}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}},\frac{{\sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} }}{{1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} }}{{1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}} \hfill \\ - \frac{{\sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{\mu_{{w_{{m_{j} }} }}^{s} }}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{\mu_{{w_{{m_{j} }} }}^{s} }}{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}},\frac{1}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{1 - o_{{r_{{m_{j} }} }}^{s} }}{{o_{{r_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ;} \hfill \\ \end{gathered}$

$\text{ii)}\:\:\:\:\:\:\:\begin{gathered} PPHFWG_{{T_{\gamma }^{D} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \frac{1}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}{{\mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}},\frac{1}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} }}{{\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}} \hfill \\ - \frac{1}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{1 - \mu_{{w_{{m_{j} }} }}^{s} }}{{\mu_{{w_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}},\frac{{\sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{o_{{r_{{m_{j} }} }}^{s} }}{{1 - o_{{r_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}}{{1 + \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( {\frac{{o_{{r_{{m_{j} }} }}^{s} }}{{1 - o_{{r_{{m_{j} }} }}^{s} }}} \right)^{\gamma } } }}}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ \end{gathered}$

If we take $T$ as Aczél-Alsina t-norms, i.e., $T = T_{\gamma }^{AA} = e^{{ - \left( {\left( { - \ln x} \right)^{\gamma } + \left( { - \ln y} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}$,

$S = S_{\gamma }^{AA} = e^{{ - \left( {\left( { - \ln \left( {1 - x} \right)} \right)^{\gamma } + \left( { - \ln \left( {1 - y} \right)} \right)^{\gamma } } \right)^{{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-0pt} \gamma }}} }}$, i.e., $\tau \left( x \right) = \left( { - \ln x} \right)^{\gamma }$ and $\zeta \left( x \right) = \left( { - \ln \left( {1 - x} \right)} \right)^{\gamma }$, then.

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWA_{{T_{\gamma }^{AA} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} 1 - e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} , \hfill \\ e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {1 - \mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} - e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {1 - \mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} ,e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {o_{{r_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ;} \hfill \\ \end{gathered}$

$\text{i)}\:\:\:\:\:\:\:\begin{gathered} PPHFWG_{{T_{\gamma }^{AA} ,\Omega }} \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} , \hfill \\ e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {\mu_{{w_{{m_{j} }} }}^{s} - \nu_{{e_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} - e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} ,1 - e^{{ - \sqrt[\gamma ]{{\sum\limits_{j = 1}^{n} {\omega_{j} \left( { - \ln \left( {1 - o_{{r_{{m_{j} }} }}^{s} } \right)^{\gamma } } \right)} }}}} \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ \end{gathered}$

The operators $PPHFWA_{T,\Omega }$ and $PPHFWG_{T,\Omega }$ have the following desirable properties.

Theorem 4.2. (Monotonicity)

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $m_{j}^{*} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j}^{*} }} }}^{s} ,\nu_{{e_{{m_{j}^{*} }} }}^{s} ,o_{{r_{{m_{j}^{*} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ be two collections of PPHFEs such that $\mu_{{w_{{m_{j} }} }}^{s} \le \mu_{{w_{{m_{j}^{*} }} }}^{s}$, $\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} \le \mu_{{w_{{m_{j}^{*} }} }}^{s} + \nu_{{e_{{m_{j}^{*} }} }}^{s}$ and $o_{{r_{{m_{j} }} }}^{s} \ge o_{{r_{{m_{j}^{*} }} }}^{s}$, i.e., $m_{j} { \preccurlyeq }_{N} m_{j}^{*}$. Then

$$PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right){ \preccurlyeq }_{N} PPHFWA_{T,\Omega } \left( {m_{1}^{*} ,m_{2}^{*} , \cdots ,m_{n}^{*} } \right).$$

(9)

Proof

By Theorem 4.1 we have.

$$\begin{gathered} PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)} } \right),\zeta^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right)} } \right) \hfill \\ - \zeta^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)} } \right),\tau^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \tau \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ,} \hfill \\ \triangleq \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {w_{1} ,e_{1} ,r_{1} } \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \end{gathered}$$

and.

$$\begin{gathered} PPHFWA_{T,\Omega } \left( {m_{1}^{*} ,m_{2}^{*} , \cdots ,m_{n}^{*} } \right) = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \zeta \left( {\mu_{{w_{{m_{j}^{*} }} }}^{s} } \right)} } \right),\zeta^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \zeta \left( {\mu_{{w_{{m_{j}^{*} }} }}^{s} + \nu_{{e_{{m_{j}^{*} }} }}^{s} } \right)} } \right) \hfill \\ - \zeta^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \zeta \left( {\mu_{{w_{{m_{j}^{*} }} }}^{s} } \right)} } \right),\tau^{ - 1} \left( {\sum\nolimits_{j = 1}^{n} {\omega_{j} \tau \left( {o_{{r_{{m_{j}^{*} }} }}^{s} } \right)} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ \triangleq \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {w_{2} ,e_{2} ,r_{2} } \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \end{gathered}$$

Noting that $\tau$ is strictly decreasing and $\zeta$ is strictly increasing, by $w_{1} \le w_{2}$, $w_{1} + e_{1} \le w_{2} + e_{2}$ and $r_{1} \ge r_{2}$, according to the partial order in Definition 3.2, we have $PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right){ \preccurlyeq }_{N} PPHFWA_{T,\Omega } \left( {m_{1}^{*} ,m_{2}^{*} , \cdots ,m_{n}^{*} } \right)$.

Theorem 4.3. (Idempotency)

If $m_{j} = m \in {\mathbb{P}}$ for all $j = 1,2, \cdots ,n$, then.

$$PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) = m.$$

(10)

Proof

By Theorem 4.1, we have.

$$\begin{gathered} PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) = \bigcup\limits_{s \in M} {\left\langle {\left. {\left\{ \begin{gathered} \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{m} }}^{s} } \right) \cdot \sum\nolimits_{j = 1}^{n} {\omega_{j} } } \right),\zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right) \cdot \sum\nolimits_{j = 1}^{n} {\omega_{j} } } \right) \hfill \\ - \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{m} }}^{s} } \right) \cdot \sum\nolimits_{j = 1}^{n} {\omega_{j} } } \right),\tau^{ - 1} \left( {\tau \left( {o_{{r_{m} }}^{s} } \right) \cdot \sum\nolimits_{j = 1}^{n} {\omega_{j} } } \right) \hfill \\ \end{gathered} \right\}} \right|k^{s} } \right\rangle } \hfill \\ \bigcup\limits_{s \in M} {\left\langle {\left. {\left\{ \begin{gathered} \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right\}} \right|k^{s} } \right\rangle } = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{m} }}^{s} ,\nu_{{e_{m} }}^{s} ,o_{{r_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } . \hfill \\ \end{gathered}$$

Theorem 4.4. (Boundedness)

If $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$, then.

$$m^{ - } { \preccurlyeq }_{N} PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right){ \preccurlyeq }_{N} m^{ + } .$$

(11)

where 

$m^{ - } = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mathop {\min }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\min }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right) - \mathop {\min }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\max }\limits_{1 \le j \le n} \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle }$

and

$m^{ + } = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mathop {\max }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\max }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right) - \mathop {\max }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\min }\limits_{1 \le j \le n} \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle .}$

Proof

Clearly, $m^{ - } { \preccurlyeq }_{N} m_{j} { \preccurlyeq }_{N} m^{ + } \left( {j = 1,2, \cdots ,n} \right)$. This, together with Theorems 4.2 and 4.3, implies that.

$$\begin{gathered} m^{ - } = PPHFWA_{T,\Omega } \left( {m^{ - } ,m^{ - } , \cdots ,m^{ - } } \right){ \preccurlyeq }_{N} PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ \, { \preccurlyeq }_{N} PPHFWA_{T,\Omega } \left( {m^{ + } ,m^{ + } , \cdots ,m^{ + } } \right) = m^{ + } . \hfill \\ \end{gathered}$$

Theorem 4.5. (Shift Invariance)

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{m} }}^{s} ,\nu_{{e_{m} }}^{s} ,o_{{r_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}$. Then.

$$PPHFWA_{T,\Omega } \left( {m_{1} \oplus_{T} m,m_{2} \oplus_{T} m, \cdots ,m_{n} \oplus_{T} m} \right) = PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \oplus_{T} m.$$

(12)

Proof

By $T\left( {x,y} \right) = \tau^{ - 1} \left( {\tau \left( x \right) + \tau \left( y \right)} \right)$ and $S\left( {x,y} \right) = \zeta^{ - 1} \left( {\zeta \left( x \right) + \zeta \left( y \right)} \right)$, it follows from Definition 3.3 that.

$$\begin{gathered} m_{j} \oplus_{T} m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {S\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\mu_{{w_{m} }}^{s} } \right),S\left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} ,\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right) - S\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\mu_{{w_{m} }}^{s} } \right),T\left( {o_{{r_{{m_{j} }} }}^{s} ,o_{{r_{m} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \, = \left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) - \zeta^{ - 1} \left( {\zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right), \hfill \\ \tau^{ - 1} \left( {\tau \left( {o_{{r_{{m_{j} }} }}^{s} } \right) + \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle , \hfill \\ \end{gathered}$$

This, together with Theorem 4.1, implies that.

$$\begin{gathered} PPHFWA_{T,\Omega } \left( {m_{1} \oplus_{T} m,m_{2} \oplus_{T} m, \cdots ,m_{n} \oplus_{T} m} \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right), \hfill \\ \tau^{ - 1} \left( {\omega_{1} \cdot \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right) + \tau \left( {o_{{r_{m} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\zeta \left( {\zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right)} \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\zeta \left( {\zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right)} \right)} \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\zeta \left( {\zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right) + \zeta \left( {\mu_{{w_{m} }}^{s} } \right)} \right)} \right)} \right), \hfill \\ \tau^{ - 1} \left( {\tau \left( {\tau^{ - 1} \left( {\omega_{1} \cdot \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right) + \tau \left( {o_{{r_{m} }}^{s} } \right)} \right)} \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} S\left( {\zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right),\mu_{{w_{m} }}^{s} } \right), \hfill \\ S\left( {\zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right),\mu_{{w_{m} }}^{s} + \nu_{{e_{m} }}^{s} } \right) \hfill \\ - S\left( {\zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right),\mu_{{w_{m} }}^{s} } \right), \hfill \\ T\left( {\tau^{ - 1} \left( {\omega_{1} \cdot \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right),o_{{r_{m} }}^{s} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle .} \hfill \\ = PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \oplus_{T} m \hfill \\ \end{gathered}$$

Theorem 4.6. (Homogeneity)

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $\lambda> 0$. Then.

$$PPHFWA_{T,\Omega } \left( {\lambda_{T} \cdot m_{1} ,\lambda_{T} \cdot m_{2} , \cdots ,\lambda_{T} \cdot m_{n} } \right) = \lambda_{T} \cdot PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right).$$

(13)

Proof

Noting that.

$$\lambda_{T} \cdot m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)} \right),\zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\lambda \cdot \zeta \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)} \right),\tau^{ - 1} \left( {\lambda \cdot \tau \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \left( {j = 1,2, \cdots ,n} \right),$$

from Definition 3.3 and Theorem 4.1, it follows that.

$$\begin{gathered} PPHFWA_{T,\Omega } \left( {\lambda_{T} \cdot m_{1} ,\lambda_{T} \cdot m_{2} , \cdots ,\lambda_{T} \cdot m_{n} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\omega_{1} \cdot \lambda \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \lambda \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\omega_{1} \cdot \lambda \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \lambda \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\omega_{1} \cdot \lambda \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \lambda \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \tau^{ - 1} \left( {\omega_{1} \cdot \lambda \cdot \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \lambda \cdot \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ,} \hfill \\ \end{gathered}$$

and.

$$\begin{gathered} \lambda_{T} \cdot PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \hfill \\ = \lambda_{T} \cdot \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right), \hfill \\ \tau^{ - 1} \left( {\omega_{1} \cdot \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\lambda \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right)} \right), \hfill \\ \zeta^{ - 1} \left( {\lambda \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} + \nu_{{e_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} + \nu_{{e_{{m_{n} }} }}^{s} } \right)} \right)} \right) \hfill \\ - \zeta^{ - 1} \left( {\lambda \left( {\omega_{1} \cdot \zeta \left( {\mu_{{w_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \zeta \left( {\mu_{{w_{{m_{n} }} }}^{s} } \right)} \right)} \right), \hfill \\ \tau^{ - 1} \left( {\lambda \left( {\omega_{1} \cdot \tau \left( {o_{{r_{{m_{1} }} }}^{s} } \right) + \cdots + \omega_{n} \cdot \tau \left( {o_{{r_{{m_{n} }} }}^{s} } \right)} \right)} \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle ,} \hfill \\ \end{gathered}$$

implying that $PPHFWA_{T,\Omega } \left( {\lambda_{T} \cdot m_{1} ,\lambda_{T} \cdot m_{2} , \cdots ,\lambda_{T} \cdot m_{n} } \right) = \lambda_{T} \cdot PPHFWA_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right).$

Similarly, we have the following results.

Theorem 4.7. (Monotonicity)

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $m_{j}^{*} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j}^{*} }} }}^{s} ,\nu_{{e_{{m_{j}^{*} }} }}^{s} ,o_{{r_{{m_{j}^{*} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ be two collections of PPHFEs such that $\mu_{{w_{{m_{j} }} }}^{s} \le \mu_{{w_{{m_{j}^{*} }} }}^{s}$, $\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} \le \mu_{{w_{{m_{j}^{*} }} }}^{s} + \nu_{{e_{{m_{j}^{*} }} }}^{s}$ and $o_{{r_{{m_{j} }} }}^{s} \ge o_{{r_{{m_{j}^{*} }} }}^{s}$, i.e., $m_{j} { \preccurlyeq }_{N} m_{j}^{*}$. Then.

$$PPHFWG_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right){ \preccurlyeq }_{N} PPHFWG_{T,\Omega } \left( {m_{1}^{*} ,m_{2}^{*} , \cdots ,m_{n}^{*} } \right).$$

(14)

Theorem 4.8. (Idempotency)

If $m_{j} = m \in {\mathbb{P}}$ for all $j = 1,2, \cdots ,n$, then.

$$PPHFWG_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) = m.$$

(15)

Theorem 4.9. (Idempotency)

If $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$, then.

$$m^{ - } { \preccurlyeq }_{N} PPHFWG_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right){ \preccurlyeq }_{N} m^{ + } .$$

(16)

where

$$m^{ - } = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mathop {\min }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\min }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right) - \mathop {\min }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\max }\limits_{1 \le j \le n} \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle }$$

and

$$m^{ + } = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mathop {\max }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\max }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right) - \mathop {\max }\limits_{1 \le j \le n} \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right),\mathop {\min }\limits_{1 \le j \le n} \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} \right)} \right|k^{s} } \right\}} \right\rangle } .$$

Theorem 4.10. (Shift Invariance)

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $m = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{m} }}^{s} ,\nu_{{e_{m} }}^{s} ,o_{{r_{m} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}$. Then.

$$PPHFWG_{T,\Omega } \left( {m_{1} \oplus_{T} m,m_{2} \oplus_{T} m, \cdots ,m_{n} \oplus_{T} m} \right) = PPHFWG_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right) \oplus_{T} m.$$

(17)

Theorem 4.11. (Homogeneity)

Let $m_{j} = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( {\mu_{{w_{{m_{j} }} }}^{s} ,\nu_{{e_{{m_{j} }} }}^{s} ,o_{{r_{{m_{j} }} }}^{s} } \right)} \right|k^{s} } \right\}} \right\rangle } \in {\mathbb{P}}\left( {j = 1,2, \cdots ,n} \right)$ and $\lambda> 0$. Then.

$$PPHFWG_{T,\Omega } \left( {\lambda_{T} \cdot m_{1} ,\lambda_{T} \cdot m_{2} , \cdots ,\lambda_{T} \cdot m_{n} } \right) = \lambda_{T} \cdot PPHFWG_{T,\Omega } \left( {m_{1} ,m_{2} , \cdots ,m_{n} } \right).$$

(18)

The PPHFBTOPSIS technique

Consider a multi-attribute group decision-making (MAGDM) problem in the DHF environment. Let $\Re = \left\{ {\Re_{1} ,\Re_{2} , \cdots ,\Re_{m} } \right\}$ represent the set of alternatives, and $G = \left\{ {G_{1} ,G_{2} , \cdots ,G_{n} } \right\}$ denote the set of attribute with the weight vector $\Omega = \left( {\omega_{1} ,\omega_{2} , \cdots ,\omega_{n} } \right)^{T}$, where $\omega_{j} \in \left( {0,1} \right]$ and $\sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1$. The set of decision-makers/experts is $D = \left\{ {D_{1} ,D_{2} , \cdots ,D_{p} } \right\}$, with the weight vector $\Theta = \left( {\theta_{1} ,\theta_{2} , \cdots ,\theta_{p} } \right)^{T}$, where $\theta_{k} \in \left( {0,1} \right]$ and $\sum\nolimits_{k = 1}^{p} {\theta_{k} } = 1$. The rating of an alternative $\Re_{i} \left( {i = 1,2, \cdots ,m} \right)$ under criterion $G_{j} \left( {j = 1,2, \cdots ,n} \right)$ is provided by decision-maker $D_{s}$, resulting in DHFE $y_{ij}^{k} = \left\langle {w_{ij}^{k} ,e_{ij}^{k} ,r_{ij}^{k} } \right\rangle = \left\langle {\left\{ {\left. {\mu_{{w_{{m_{ij} }} }}^{k} } \right|p_{{w_{{m_{ij} }} }}^{k} } \right\},\left\{ {\left. {\nu_{{e_{{m_{ij} }} }}^{k} } \right|q_{{e_{{m_{ij} }} }}^{k} } \right\},\left\{ {\left. {o_{{r_{{m_{ij} }} }}^{k} } \right|l_{{r_{{m_{ij} }} }}^{k} } \right\}} \right\rangle$. Here, $\mu_{{w_{{m_{ij} }} }}^{k}$, $\nu_{{e_{{m_{ij} }} }}^{k}$ and $o_{{r_{{m_{ij} }} }}^{k}$ are the PPMHD, PNMHD, and PNMHD given by DM with corresponding $p_{{w_{{m_{ij} }} }}^{k}$, $q_{{e_{{m_{ij} }} }}^{k}$ and $l_{{r_{{m_{ij} }} }}^{k}$$l_{{r_{{m_{ij} }} }}$ for evaluating alternative $\Re_{i}$ against attribute $G_{j}$. To rank the alternatives, we introduce a method called PPHF-BTOPSIS, which adapts the BTOPSIS method to the PPHF framework.

Alternatives	$G_{1}$	$G_{2}$	$\cdots$	$G_{n}$
$\Re_{1}$	$\left\langle {w_{11}^{k} ,e_{11}^{k} ,r_{11}^{k} } \right\rangle$	$\left\langle {w_{12}^{k} ,e_{12}^{k} ,r_{12}^{k} } \right\rangle$	$\cdots$	$\left\langle {w_{1n}^{k} ,e_{1n}^{k} ,r_{1n}^{k} } \right\rangle$
$\Re_{2}$	$\left\langle {w_{21}^{k} ,e_{21}^{k} ,r_{21}^{k} } \right\rangle$	$\left\langle {w_{22}^{k} ,e_{22}^{k} ,r_{22}^{k} } \right\rangle$	$\cdots$	$\left\langle {w_{2n}^{k} ,e_{2n}^{k} ,r_{2n}^{k} } \right\rangle$
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$
$\Re_{m}$	$\left\langle {w_{m1}^{k} ,e_{m1}^{k} ,r_{m1}^{k} } \right\rangle$	$\left\langle {w_{m2}^{k} ,e_{m2}^{k} ,r_{m2}^{k} } \right\rangle$	$\cdots$	$\left\langle {w_{mn}^{k} ,e_{mn}^{k} ,r_{mn}^{k} } \right\rangle$

Step 1: Collect the evaluation information provided by the $k^{{{\text{th}}}}$ decision-maker $D_{k}$ and construct the PPHF decision matrix $Y_{k}$ in the following form:

Step 2: Use probability splitting algorithm to transform normalized $Y_{k} = \left[ {y_{ij}^{k} } \right]_{m \times n}$ into a novel normalization matrix $\overline{Y}_{k}^{s} = \left[ {\overline{y}_{ij}^{sk} } \right]_{m \times n}$.

Step 3: Convert each PPHF decision matrix $\overline{Y}_{k}^{s} = \left[ {\overline{y}_{ij}^{sk} } \right]_{m \times n}$ into the normalized PPHF decision matrix as $M_{k}^{s} = \left[ {m_{ij}^{sk} } \right]_{m \times n}$:

$$m_{ij}^{sk} = \left\{ \begin{gathered} \overline{y}_{ij}^{sk} ,\quad \quad for\,benefit\,attribute\,G_{j} , \hfill \\ \left( {\overline{y}_{ij}^{sk} } \right)^{{\complement }} ,\;\;\,for\,\cos t\,attribute\,G_{j} . \hfill \\ \end{gathered} \right.$$

(19)

where $\left( {\overline{y}_{ij}^{sk} } \right)^{{\complement }}$ is the complement of $\overline{y}_{ij}^{sk} .$

Step 4: Use the $PPHFWA_{T,\Omega }$ or $PPHFWG_{T,\Omega }$ operator defined by Eqs. (20) or (21) to compute the group PPHF decision matrix $M^{s} = \left[ {F_{ij}^{s} } \right]_{m \times n}$ as follows:

$$\begin{gathered} F_{ij}^{s} = \left( {m_{ij}^{s} ,m_{ij}^{s} , \cdots ,m_{ij}^{s} } \right) = PPHFWA_{T,\Theta } \left( {m_{ij}^{sk} ,m_{ij}^{sk} , \cdots ,m_{ij}^{sk} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \zeta^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \zeta \left( {\mu_{{w_{{m_{ij}^{k} }} }}^{s} } \right)} } \right),\zeta^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \zeta \left( {\mu_{{w_{{m_{ij}^{k} }} }}^{s} + \nu_{{e_{{m_{ij}^{k} }} }}^{s} } \right)} } \right) \hfill \\ - \zeta^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \zeta \left( {\mu_{{w_{{m_{ij}^{k} }} }}^{s} } \right)} } \right),\tau^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \tau \left( {o_{{r_{{m_{ij}^{k} }} }}^{s} } \right)} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } , \hfill \\ \end{gathered}$$

(20)

and

$$\begin{gathered} F_{ij}^{s} = \left( {m_{ij}^{s} ,m_{ij}^{s} , \cdots ,m_{ij}^{s} } \right) = PPHFWG_{T,\Theta } \left( {m_{ij}^{sk} ,m_{ij}^{sk} , \cdots ,m_{ij}^{sk} } \right) \hfill \\ = \bigcup\limits_{s \in M} {\left\langle {\left\{ {\left. {\left( \begin{gathered} \tau^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \tau \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)} } \right),\tau^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \tau \left( {\mu_{{w_{{m_{j} }} }}^{s} + \nu_{{e_{{m_{j} }} }}^{s} } \right)} } \right) \hfill \\ - \tau^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \tau \left( {\mu_{{w_{{m_{j} }} }}^{s} } \right)} } \right),\zeta^{ - 1} \left( {\sum\nolimits_{k = 1}^{p} {\theta_{k} \zeta \left( {o_{{r_{{m_{j} }} }}^{s} } \right)} } \right) \hfill \\ \end{gathered} \right)} \right|k^{s} } \right\}} \right\rangle } \hfill \\ \end{gathered}$$

(21)

Step 5. Calculate the combined weights of decision attributes.

Step 5.1. Calculate the objective weights of decision attributes using the CRITIC method.

Step 5.1.1. Calculate the scores of PPFE using the score function formula in Definition 2.4 and convert the comprehensive evaluation decision matrix $M$ obtained from the formula in Eq. (20) or (21) into the score decision matrix $D = \left( {d_{ij} } \right)_{m \times n}$.

Step 5.1.2. Convert the score decision matrix $S$ into a standardized score decision matrix $D = \left( {d_{ij} } \right)_{m \times n}$ using Eq. (22).

$$d_{ij} = \frac{{S\left( {m_{ij}^{s} } \right) - \mathop {\min }\limits_{i} S\left( {m_{ij}^{s} } \right)}}{{\mathop {\max }\limits_{i} S\left( {m_{ij}^{s} } \right) - \mathop {\min }\limits_{i} S\left( {m_{ij}^{s} } \right)}}.$$

(22)

where $S\left( {m_{ij}^{s} } \right) = \frac{1}{m}\sum\limits_{i = 1}^{m} {s\left( {m_{ij}^{s} } \right)} .$

Step 5.1.3. Use Eq. (23) to calculate the coefficient of variation for the $j{\text{th}}$ evaluation indicator.

$$v_{j} = \frac{{s_{j} }}{{\overline{d}_{j} }}.$$

(23)

where $\overline{d}_{j} = \frac{1}{m}\sum\nolimits_{i = 1}^{m} {d_{ij} }$ represents the average value of the $j{\text{th}}$ decision-making attribute $G_{j}$, $s_{j} = \sqrt {\frac{1}{m}\sum\nolimits_{i = 1}^{m} {\left( {d_{ij} - \overline{d}_{j} } \right)^{2} } }$ represents the standard deviation of the $j{\text{th}}$ decision-making attribute $G_{j}$.

Step 5.1.4. According to the standardized score decision matrix $D$ in step 3.1.2, calculate the correlation coefficient using Eq. (24) to obtain the corresponding correlation coefficient matrix $C = \left( {\rho_{jk} } \right)_{n \times n}$.

$$\rho_{jk} = \frac{{\sum\nolimits_{i = 1}^{m} {\left( {d_{ij} - \overline{d}_{j} } \right)\left( {d_{ik} - \overline{d}_{k} } \right)} }}{{\sqrt {\sum\nolimits_{i = 1}^{m} {\left( {d_{ij} - \overline{d}_{j} } \right)^{2} } \sum\nolimits_{i = 1}^{m} {\left( {d_{ik} - \overline{d}_{k} } \right)^{2} } } }}.$$

(24)

Step 5.1.5. Calculate the information content of the $j{\text{th}}$ decision-making attribute $G_{j}$ using Eq. (25).

$$c_{j} = v_{j} \sum\nolimits_{k = 1}^{n} {\left( {1 - \left| {\rho_{jk} } \right|} \right)} .$$

(25)

Step 5.1.6. Calculate the objective weight of the $j{\text{th}}$ decision-making attribute $G_{j}$ using Eq. (26).

$$\omega_{j} = \frac{{c_{j} }}{{\sum\nolimits_{j = 1}^{n} {c_{j} } }}.$$

(26)

Step 5.2. Calculate combination weights.

Step 5.2.1. Set the subjective weight vector of the decision attribute as $\xi = \left( {\xi_{1} ,\xi_{2} , \cdots ,\xi_{n} } \right)^{T}$. To ensure that both subjective and objective weights play an important role, the combination weight $\varpi$ should be as close as possible to $\xi$ and $\omega$. According to the principle of minimum discriminative information mentioned in references^29,30, the following objective function is obtained by Eq. (27):

$$\left\{ \begin{gathered} \min F = \sum\nolimits_{j = 1}^{n} {\varpi_{j} \ln \frac{{\varpi_{j} }}{{\xi_{j} }}} + \sum\nolimits_{j = 1}^{n} {\varpi_{j} \ln \frac{{\varpi_{j} }}{{\omega_{j} }}} \hfill \\ s.t.\quad \sum\nolimits_{j = 1}^{n} {\varpi_{j} } = 1,\varpi_{j} \ge 0 \hfill \\ \end{gathered} \right..$$

(27)

Step 5.2.2. According to the Langrange multiplier method, calculate the combined weights of attributes using Eq. (28).

$$\varpi_{j} = \frac{{\sqrt {\xi_{j} \omega_{j} } }}{{\sum\nolimits_{j = 1}^{n} {\sqrt {\xi_{j} \omega_{j} } } }}.$$

(28)

Step 6: Identify the positive ideal solution (PIS) $\Re^{ + }$ and negative ideal solution (NIS) $\Re^{ - }$ by Eqs. (29) and (30):

$$\Re^{ + } = \left\{ {\left. {\left\langle {G_{j} ,\Re_{{w_{j} }}^{s + } ,\Re_{{e_{j} }}^{s + } ,\Re_{{r_{j} }}^{s + } } \right\rangle } \right|j = 1,2, \cdots ,n} \right\} = \left( {\begin{array}{*{20}c} {\left\langle {1,0,0} \right\rangle } & \ldots & {\left\langle {1,0,0} \right\rangle } \\ \vdots & \ddots & \vdots \\ {\left\langle {1,0,0} \right\rangle } & \cdots & {\left\langle {1,0,0} \right\rangle } \\ \end{array} } \right)_{s \times n} ,$$

(29)

and

$$\Re^{ - } = \left\{ {\left. {\left\langle {G_{j} ,\Re_{{w_{j} }}^{s - } ,\Re_{{e_{j} }}^{s - } ,\Re_{{r_{j} }}^{s - } } \right\rangle } \right|j = 1,2, \cdots ,n} \right\} = \left( {\begin{array}{*{20}c} {\left\langle {0,0,1} \right\rangle } & \ldots & {\left\langle {0,0,1} \right\rangle } \\ \vdots & \ddots & \vdots \\ {\left\langle {0,0,1} \right\rangle } & \cdots & {\left\langle {0,0,1} \right\rangle } \\ \end{array} } \right)_{s \times n} .$$

(30)

Step 7: Compute the DisMs of each alternative $\Re_{i}$ from PIS and NIS by Eqs. (31) and (32):

$$D_{PPHFNHDM} \left( {\Re_{i} ,\Re^{ + } } \right) = \sum\limits_{j = 1}^{n} {\frac{{\varpi_{j} }}{3}\left( {\Delta w\left( {\Re_{{w_{ij} }}^{s} ,\Re_{{w_{j} }}^{s + } } \right) + \Delta e\left( {\Re_{{e_{ij} }}^{s} ,\Re_{{e_{j} }}^{s + } } \right) + \Delta r\left( {\Re_{{r_{ij} }}^{s} ,\Re_{{r_{j} }}^{s + } } \right)} \right)} .$$

(31)

and

$$D_{PPHFNHDM} \left( {\Re_{i} ,\Re^{ - } } \right) = \sum\limits_{j = 1}^{n} {\frac{{\varpi_{j} }}{3}\left( {\Delta w\left( {\Re_{{w_{ij} }}^{s} ,\Re_{{w_{j} }}^{s - } } \right) + \Delta e\left( {\Re_{{e_{ij} }}^{s} ,\Re_{{e_{j} }}^{s - } } \right) + \Delta r\left( {\Re_{{r_{ij} }}^{s} ,\Re_{{r_{j} }}^{s - } } \right)} \right)} .$$

(32)

Step 8: Considering the psychological behavior of DMs, use Eq. (33) to calculate the relative closeness ${\mathbb{C}}\left( {\Re_{i} } \right)\left( {i = 1,2, \cdots ,m} \right)$ of the $i{\text{th}}$ alternative.

$${\mathbb{C}}\left( {\Re_{i} } \right) = \left( {D_{PPHFNHDM} \left( {\Re_{i} ,\Re^{ - } } \right)} \right)^{\zeta } - \kappa \left( {D_{PPHFNHDM} \left( {\Re_{i} ,\Re^{ + } } \right)} \right)^{\delta } .$$

(33)

where parameters $\zeta \left( {0 \le \zeta \le 1} \right)$ and $\delta \left( {0 \le \delta \le 1} \right)$ represent the risk attitude of DMs in avoiding risks in returns and seeking risks in losses. $\kappa$ represents the parameter of the DM’s loss aversion rate. $\kappa> 1$ means that the behavior of DM is more sensitive to loss than gain, $\kappa = 1$ means that the DM is neutral towards loss, and $\kappa < 1$ means that the DM’s behavior is more sensitive to gain than loss.

Step 9: Rank the alternatives $\Re_{1} ,\Re_{2} , \cdots ,\Re_{m}$ in descending order according to their relative closeness and choose the one with the highest value as the best alternative.

Similar to Definition 2.3, if there exist two alternatives $\Re_{{i_{1} }}$ and $\Re_{{i_{2} }}$ ($1 \le i_{1} ,i_{2} \le m$), satisfying $m_{{i_{1} ,j}} { \preccurlyeq }_{N} m_{{i_{2} ,j}}$ for all $j \in \left\{ {1,2, \cdots ,n} \right\}$, then $\Re_{{i_{1} }}$ is considered smaller than or equal to $\Re_{{i_{2} }}$ under the linear order ${ \preccurlyeq }_{N}$, denoted as $\Re_{{i_{1} }} { \preccurlyeq }_{N} \Re_{{i_{2} }}$. The proposed PPHFBTOPSIS technique is shown in Figs. 1, 2, 3, 4.

Fig. 1

Methodological flowchart of the proposed PPHFBTOPSIS technique.

Fig. 2

Analysis of the impact of a single parameter on decision results.

Fig. 3

Analysis of the impact of two parameters on decision results.

Fig. 4

Analysis of the impact of three parameters on decision results.

Theorem 5.1.

The method is non-decreasing under the linear order ${ \preccurlyeq }_{N}$. That is, for the MADM problem mentioned above, if $\Re_{{i_{1} }} { \preccurlyeq }_{N} \Re_{{i_{2} }}$ for some $1 \le i_{1} ,i_{2} \le m$, then ${\mathbb{C}}\left( {\Re_{{i_{1} }} } \right) \le {\mathbb{C}}\left( {\Re_{{i_{2} }} } \right)$, i.e., $\Re_{{i_{2} }}$ is not worse than $\Re_{{i_{1} }}$ ranking by Algorithm 1.

Proof.

Let $\Re^{ + } = \left\{ {\left. {\left\langle {G_{j} ,\Re_{{w_{j} }}^{s + } ,\Re_{{e_{j} }}^{s + } ,\Re_{{r_{j} }}^{s + } } \right\rangle } \right|j = 1,2, \cdots ,n} \right\}$ and $\Re^{ - } = \left\{ {\left. {\left\langle {G_{j} ,\Re_{{w_{j} }}^{s - } ,\Re_{{e_{j} }}^{s - } ,\Re_{{r_{j} }}^{s - } } \right\rangle } \right|j = 1,2, \cdots ,n} \right\}$ be the PPHFPIS and PPHFNIS obtained from Eqs. (5.14) and (5.15), respectively. It is clear that, for all $j \in \left\{ {1,2, \cdots ,n} \right\},$

$$\Re_{j}^{ - } { \preccurlyeq }_{N} \Re_{{i_{1} ,j}} { \preccurlyeq }_{N} \Re_{{i_{2} ,j}} { \preccurlyeq }_{N} \Re_{j}^{ + } ,$$

This, together with Proposition 4.4, implies that $Div_{PF} \left( {\Re_{{i_{1} ,j}} ,\Re_{j}^{ - } } \right) \le Div_{PF} \left( {\Re_{{i_{2} ,j}} ,\Re_{j}^{ - } } \right)$ and $Div_{PF} \left( {\Re_{{i_{1} ,j}} ,\Re_{j}^{ + } } \right) \le Div_{PF} \left( {\Re_{{i_{2} ,j}} ,\Re_{j}^{ + } } \right)$. Thus, we have

$$Div_{wPF} \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right) = \sum\nolimits_{j = 1}^{n} {\omega_{j} Div_{PF} \left( {\Re_{{i_{1} ,j}} ,\Re_{j}^{ - } } \right)} \le \sum\nolimits_{j = 1}^{n} {\omega_{j} Div_{PF} \left( {\Re_{{i_{2} ,j}} ,\Re_{j}^{ - } } \right)} = Div_{wPF} \left( {\Re_{{i_{2} }} ,\Re^{ - } } \right).$$

(33)

and

$$Div_{wPF} \left( {\Re_{{i_{1} }} ,\Re^{ + } } \right) = \sum\nolimits_{j = 1}^{n} {\omega_{j} Div_{PF} \left( {\Re_{{i_{1} ,j}} ,\Re_{j}^{ + } } \right)} \le \sum\nolimits_{j = 1}^{n} {\omega_{j} Div_{PF} \left( {\Re_{{i_{2} ,j}} ,\Re_{j}^{ + } } \right)} = Div_{wPF} \left( {\Re_{{i_{2} }} ,\Re^{ + } } \right)$$

(34)

According to Eqs. (33)–(34), we have:

(1) If $Div_{wPF} \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right) = 0$, then.

$${\mathbb{C}}\left( {\Re_{{i_{1} }} } \right) = Div_{wPF}^{\alpha } \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right) - \kappa Div_{wPF}^{\beta } \left( {\Re_{{i_{1} }} ,\Re^{ + } } \right) = 0 \le {\mathbb{C}}\left( {\Re_{{i_{2} }} } \right).$$

(2) If $Div_{wPF} \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right)> 0$, then $Div_{wPF} \left( {\Re_{{i_{2} }} ,\Re^{ - } } \right) \ge Div_{wPF} \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right)> 0$, and thus.

$$\begin{gathered} {\mathbb{C}}\left( {\Re_{{i_{1} }} } \right) = Div_{wPF}^{\alpha } \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right) - \kappa Div_{wPF}^{\beta } \left( {\Re_{{i_{1} }} ,\Re^{ + } } \right) = Div_{wPF}^{\alpha } \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right)\left( {1 - \frac{{\kappa Div_{wPF}^{\beta } \left( {\Re_{{i_{1} }} ,\Re^{ + } } \right)}}{{Div_{wPF}^{\alpha } \left( {\Re_{{i_{1} }} ,\Re^{ - } } \right)}}} \right) \hfill \\ \le Div_{wPF}^{\alpha } \left( {\Re_{{i_{2} }} ,\Re^{ - } } \right)\left( {1 - \frac{{\kappa Div_{wPF}^{\beta } \left( {\Re_{{i_{2} }} ,\Re^{ + } } \right)}}{{Div_{wPF}^{\alpha } \left( {\Re_{{i_{2} }} ,\Re^{ - } } \right)}}} \right) = Div_{wPF}^{\alpha } \left( {\Re_{{i_{2} }} ,\Re^{ - } } \right) - \kappa Div_{wPF}^{\beta } \left( {\Re_{{i_{2} }} ,\Re^{ + } } \right) = {\mathbb{C}}\left( {\Re_{{i_{2} }} } \right). \hfill \\ \end{gathered}$$

Numerical case

Actual case description

Liupanshui is undoubtedly a charming city. Liupanshui is known as the "Cool Capital of China", mainly due to its unique cool climate. Liupanshui is located in the the Yunnan-Guizhou Plateau, and the average temperature in summer is between 19 ℃ and 22 ℃. Such climatic conditions make Liupanshui an excellent summer resort. Compared to other cities with unbearable summer heat, the cool climate of Liupanshui is undoubtedly a huge advantage, attracting a large number of tourists to come for vacation and summer vacation. Here, tourists can escape the troubles of high temperatures and fully enjoy the cool and pleasant natural environment. In addition, Liupanshui, the capital of Cool city, also has abundant tourism resources. Here, there are magnificent mountain landscapes, unique ethnic cultures, rich historical relics, and delicious specialty foods. Whether tourists who want to experience natural scenery or travelers interested in cultural history, they can find their own fun in Liupanshui. For example, Wumeng Mountain National Geopark has attracted countless photography enthusiasts and outdoor explorers with its unique landforms and magnificent scenery; The Meihua Mountain tourist attraction has become a great destination for tourists to relax and vacation due to its beautiful natural scenery and rich cultural heritage. Liupanshui also focuses on building the “Cool city” tourism brand, further enhancing the city’s visibility and influence by hosting various cultural and sports events and activities, such as marathons, hotpot festivals, etc. These activities not only enrich the tourist experience, but also promote the development of the local economy and cultural exchange. Due to the rapid expansion of the “Cool city” Liupanshui area, a large number of tourists have come to Liupanshui for tourism. However, there is a shortage of hotel accommodations. Therefore, it is crucial to provide a more scientific and reasonable decision-making model for hotels recommendation to tourists.

Decision process

We will apply the developed MAGDM technique to HR for tourists in Liupanshui City. Assuming there are three industry experts $D = \left\{ {D_{1} ,D_{2} ,D_{3} } \right\}$ (where $\Theta = \left( {0.3,0.3,0.4} \right)^{T}$) invited to participate in this recommend work, the these experts will use five decision attributes: $G_{1}$—price, $G_{2}$—comfortability, $G_{3}$—service, $G_{4}$—location and $G_{5}$—convenience to recommend the HR for tourists, assuming there are four hotels $\Re_{1}$, $\Re_{2}$, $\Re_{3}$, and $\Re_{4}$ participating in the recommendation. The evaluation information of three experts will be represented by PPHFEs. The MAGDM technique is applied in HR for tourists visiting Liupanshui City is described as follows.

Step 1: According to their own professional knowledge and experience, three expert decision matrixes are presented in Tables 1,2,3.

Table 1 The first decision matrix $Y_{1} = \left[ {y_{ij}^{1} } \right]_{4 \times 5}$ by $D_{1}$.

Table 2 The second decision matrix $Y_{2} = \left[ {y_{ij}^{2} } \right]_{4 \times 5}$ by $D_{2}$.

Table 3 The third decision matrix $Y_{3} = \left[ {y_{ij}^{3} } \right]_{4 \times 5}$ by $D_{3}$.

Step 2: Use probability splitting algorithm to transform normalized $Y_{k} = \left[ {y_{ij}^{k} } \right]_{4 \times 5}$ into a novel normalization matrix $\overline{Y}_{k} = \left[ {\overline{y}_{ij}^{k} } \right]_{4 \times 5}$. By the calculation process of probability splitting algorithm, the original elements are first sorted, and then the three membership degree parts are normalized together to obtain the common probability distribution of the three membership degree parts as 0.2, 0.1, 0.1, 0.1, 0.1, 0.2, 0.2, i.e., $s = 7$. The probability splitting matrixes are displayed Tables 4,5,6.

Table 4 The first probability splitting decision matrix $\overline{Y}_{1}^{s} = \left[ {\overline{y}_{ij}^{s1} } \right]_{4 \times 5}$ by $D_{1}$.

Table 5 The second probability splitting decision matrix $\overline{Y}_{2}^{s} = \left[ {\overline{y}_{ij}^{s2} } \right]_{4 \times 5}$ by $D_{2}$.

Table 6 The third probability splitting decision matrix $\overline{Y}_{3}^{s} = \left[ {\overline{y}_{ij}^{s3} } \right]_{4 \times 5}$ by $D_{3}$.

Step 3: Convert each PPHF decision matrix $\overline{Y}_{k} = \left[ {\overline{y}_{ij}^{k} } \right]_{4 \times 5}$ into the normalized PPHF decision matrix as $M_{k} = \left[ {m_{ij}^{k} } \right]_{4 \times 5}$, we normalize $G_{1}$ since $G_{1}$ is cost, and displayed in Tables 7,8,9.

Table 7 The first normalized decision matrix $M_{1}^{s} = \left[ {m_{ij}^{s1} } \right]_{4 \times 5}$ by $D_{1}$.

Table 8 The second normalized decision matrix $M_{2}^{s} = \left[ {m_{ij}^{s2} } \right]_{4 \times 5}$ by $D_{2}$.

Table 9 The third normalized decision matrix $M_{3}^{s} = \left[ {m_{ij}^{s3} } \right]_{4 \times 5}$ by $D_{3}$.

Step 4: In this paper, we use the $PPHFWA_{{T_{{\text{P}}} ,\Omega }}$ operator defined by Eq. (20) to compute the group PPHF decision matrix $M^{s} = \left[ {F_{ij}^{s} } \right]_{4 \times 5}$ and displayed in Table 10.

Table 10 The aggregated decision matrix $M^{s} = \left[ {F_{ij}^{s} } \right]_{4 \times 5}$ by PPHFWA operator.

Step 5. Calculate the combined weights of decision attributes.

Step 5.1. Calculate the objective weights of decision attributes using the CRITIC method.

Step 5.1.1. Calculate the scores of PPFE using the score function formula in Definition 2.4 and convert the comprehensive evaluation decision matrix $M$ obtained from the formula in Eq. (20) or (21) into the score decision matrix $D = \left( {d_{ij} } \right)_{4 \times 5}$.

Step 5.1.2. Convert the score decision matrix $S$ into a standardized score decision matrix $D = \left( {d_{ij} } \right)_{4 \times 5}$ using Eq. (22) and displayed in Table 11.

Table 11 The score matrix $D = \left( {d_{ij} } \right)_{4 \times 5}$.

Step 5.1.3. Use Eq. (23) to calculate the coefficient of variation for the $j{\text{th}}$ evaluation indicator.

$$v_{1} = 0.5774,\;v_{2} = 1.3234,\;v_{3} = 1.0865,\;v_{4} = 1.3867,\;v_{5} = 1.1379.$$

Step 5.1.4. According to the standardized score decision matrix $D$ in step 3.1.2, calculate the correlation coefficient using Eq. (24) to obtain the corresponding correlation coefficient matrix $C = \left( {\rho_{jk} } \right)_{n \times n}$ and displayed in Tables 12, 13, 14.

Table 12 The correlation coefficient matrix $C = \left( {\rho_{jk} } \right)_{5 \times 5}$.

Table 13 The comparative results by using GPPHFWA operator.

Table 14 The comparative results by using GPPHFWG operator.

Step 5.1.5. Calculate the information content of the $j{\text{th}}$ decision-making attribute $G_{j}$ using Eq. (25).

$$c_{1} = 1.7841,\;c_{2} = 1.259,\;c_{3} = 0.9092,\;c_{4} = 1.0227,\;c_{5} = 1.2822.$$

Step 5.1.6. Calculate the objective weight of the $j{\text{th}}$ decision-making attribute $G_{j}$ using Eq. (26).

$$\omega_{1} = 0.2851,\;\omega_{2} = 0.2012,\;\omega_{3} = 0.1453,\;\omega_{4} = 0.1634,\;\omega_{5} = 0.2049.$$

Step 5.2. Calculate combination weights and displayed as follows:

$$\varpi_{1} = 0.1757,\;\varpi_{2} = 0.2087,\;\varpi_{3} = 0.2172,\;\varpi_{4} = 0.1629,\;\varpi_{5} = 0.2355.$$

where $\xi = \left( {0.1,0.2,0.3,0.15,0.25} \right)^{T}$.

Step 6: Identify the positive ideal solution (PIS) $\Re^{ + }$ and negative ideal solution (NIS) $\Re^{ - }$ as:

$$\Re^{ + } = \left\{ {\left. {\left\langle {G_{j} ,\Re_{{w_{j} }}^{s + } ,\Re_{{e_{j} }}^{s + } ,\Re_{{r_{j} }}^{s + } } \right\rangle } \right|j = 1,2, \cdots ,n} \right\} = \left( {\begin{array}{*{20}c} {\left\langle {1,0,0} \right\rangle } & \ldots & {\left\langle {1,0,0} \right\rangle } \\ \vdots & \ddots & \vdots \\ {\left\langle {1,0,0} \right\rangle } & \cdots & {\left\langle {1,0,0} \right\rangle } \\ \end{array} } \right)_{s \times n} ,$$

and.

$$\Re^{ - } = \left\{ {\left. {\left\langle {G_{j} ,\Re_{{w_{j} }}^{s - } ,\Re_{{e_{j} }}^{s - } ,\Re_{{r_{j} }}^{s - } } \right\rangle } \right|j = 1,2, \cdots ,n} \right\} = \left( {\begin{array}{*{20}c} {\left\langle {0,0,1} \right\rangle } & \ldots & {\left\langle {0,0,1} \right\rangle } \\ \vdots & \ddots & \vdots \\ {\left\langle {0,0,1} \right\rangle } & \cdots & {\left\langle {0,0,1} \right\rangle } \\ \end{array} } \right)_{s \times n} .$$

Step 7: Compute the DisMs of each alternative $\Re_{i}$ from PIS and NIS as:

$$D_{PPHFNHDM} \left( {\Re_{1} ,\Re^{ + } } \right) = 0.0784,\;D_{PPHFNHDM} \left( {\Re_{2} ,\Re^{ + } } \right) = 0.0527,\;D_{PPHFNHDM} \left( {\Re_{3} ,\Re^{ + } } \right) = 0.0794,\,D_{PPHFNHDM} \left( {\Re_{4} ,\Re^{ + } } \right) = 0.077.$$

and.

$$D_{PPHFNHDM} \left( {\Re_{1} ,\Re^{ - } } \right) = 0.0747,\;D_{PPHFNHDM} \left( {\Re_{2} ,\Re^{ - } } \right) = 0.0868,\;D_{PPHFNHDM} \left( {\Re_{3} ,\Re^{ - } } \right) = 0.0676,\;D_{PPHFNHDM} \left( {\Re_{4} ,\Re^{ - } } \right) = 0.0674.$$

Step 8: Considering the psychological behavior of DMs, use Eq. (33) to calculate the relative closeness ${\mathbb{C}}\left( {\Re_{i} } \right)\left( {i = 1,2, \cdots ,m} \right)$ of the $i{\text{th}}$ alternative.

$${\mathbb{C}}\left( {\Re_{1} } \right) = - 0.0066,\;{\mathbb{C}}\left( {\Re_{i} } \right) = 0.0651,\;{\mathbb{C}}\left( {\Re_{i} } \right) = - 0.0217,\;{\mathbb{C}}\left( {\Re_{i} } \right) = - 0.0178.$$

where $\zeta = 0.5,$ $\delta = 0.5,$ and $\kappa = 1.$.

Step 9: Rank the alternatives $\Re_{1} ,\Re_{2} ,\Re_{3} ,\Re_{4}$ in descending order according to their relative closeness and choose the one with the highest value as the best hotel, then the ranking is $\Re_{3} \prec_{N} \Re_{4} \prec_{N} \Re_{1} \prec_{N} \Re_{2}$, the best hotel is $\Re_{2}$.

Sensitivity analysis

We are well aware that when using the DHF-BTOPSIS method to make decisions on practical problems, the decision results will also be influenced by parameters $\zeta \in \left[ {0,1} \right]$, $\delta \in \left[ {0,1} \right]$, and $\kappa \in \left[ {0,10} \right]$, Therefore, in the following section, we will conduct sensitivity analysis on three parameters to examine the impact of parameter changes on decision results, mainly divided into single parameter analysis, two parameter combination analysis, and three parameter combination analysis, where the step size used in parameter ($\zeta \in \left[ {0,1} \right]$ and $\delta \in \left[ {0,1} \right]$) simulation analysis is 0.01, the step size used in parameter ($\kappa \in \left[ {0,10} \right]$) simulation analysis is 0.1.

Analysis of the impact of a single parameter on decision results

In this section, we analyzed the situation where five alternatives vary with the change of the third parameter when two of the three parameters remain fixed. Through simulation results and Fig. 2, we found that the ranking results of the five alternatives remain unchanged. We also observed that the scores of the five alternatives tend to decrease with the increase of parameter $\kappa$, increase with the increase of parameter $\zeta$, and increase with the increase of parameter $\delta$. This also demonstrates the robustness of the decision-making method proposed in this paper.

Analysis of the impact of two parameters on the decision outcome

In this section, we analyzed the situation where five alternatives change with the variation of the other two parameters when one of the three parameters remains fixed. Through simulation results and Fig. 3, we found that the ranking results of the five alternatives remain unchanged, and at the same time, we can see that the decision results, that is, the changes of the five alternatives, are consistent with the first situation. This also demonstrates the robustness of the decision-making method proposed in this paper.

Analysis of the impact of three parameters on decision results

In this section, we analyzed the changes in five alternatives with three parameters. Through simulation results and Fig. 4, we found that the ranking results of the five alternatives remained unchanged, and further confirmed that the changes in the five alternatives were the same as those in the first and second cases. This also demonstrates the robustness of the decision-making method proposed in this paper.

To further validate the robustness of the proposed PPHFBTOPSIS method, a comprehensive sensitivity analysis was conducted in Section “Sensitivity analysis”. By systematically varying the behavioral parameters $\zeta$, $\delta$, and $\kappa$, three types of analyses—single-parameter, two-parameter joint, and three-parameter joint—were performed. The results consistently showed that although the absolute scores of the alternatives fluctuated with parameter changes, the relative ranking of the alternatives remained stable. Moreover, Figs. 2,3,4 provide a clear graphical illustration of these findings, enabling readers to intuitively observe the stability of outcomes under different parameter settings. These results confirm that the proposed method is not only theoretically sound but also computationally robust, ensuring reliable decision-making performance under input uncertainty.

Comparative analysis

We compared and analyzed the MAGDM method proposed in this article with the two operators GPPHFWA and GPPHFWG proposed in the existing literature²³. We substitute the data in Table 10 into two aggregation operators, and the analysis results are as follows:

A comparison of Table 13 (using the GPPHFWA operator) and Table 14 (using the GPPHFWG operator) reveals both consistencies and differences in the ranking outcomes under varying parameter values.

Stability of rankings

GPPHFWA (Table 13): The rankings remain relatively stable across different parameter settings (1, 2, 5, 10, 20, 50). The best alternative remains unchanged under all scenarios, indicating that the weighted averaging operator exhibits low sensitivity to parameter variation and produces robust decision results. GPPHFWG (Table 14): Although the optimal alternative is consistent with Table 13, the rankings of the second-best and middle-ranked options exhibit minor adjustments across parameter values. This demonstrates that the weighted geometric operator is more sensitive to parameter variation.

Operator characteristics

Weighted Averaging (WA): As a compensatory operator, WA emphasizes balance among attributes. Poor performance on one attribute can be compensated by strong performance on another, resulting in greater ranking stability and robustness when attribute performance is uneven. Weighted Geometric (WG): As a non-compensatory operator, WG places more emphasis on attribute balance. A poor score in any attribute significantly lowers the overall evaluation, which explains the ranking shifts observed in Table 14.

Sensitivity to parameters

Table 13 (GPPHFWA): Results show minimal change in relative rankings as parameters increase, highlighting low sensitivity and strong robustness of this operator. Table 14 (GPPHFWG): Rankings are more affected by parameter variation, suggesting stronger discriminatory power but also higher sensitivity, which may magnify the differences among alternatives.

Decision-making implications

When decision makers require a method that ensures stability and robustness, the GPPHFWA operator is more appropriate. When the decision-making context requires strict penalization of poor performance in any criterion and highlights the need for balance across attributes, the GPPHFWG operator is more suitable. In practice, these two approaches are complementary: the WA operator is preferable for comprehensive and stability-oriented evaluations, while the WG operator is more suitable in risk-sensitive and balance-oriented contexts.

The comparison indicates that although both operators consistently identify the same best hotel, the WA operator (Table 13) produces more stable rankings, while the WG operator (Table 14) better highlights attribute imbalances and shows greater sensitivity to parameter changes. Thus, the choice of operator should depend on the decision makers’ preferences for stability versus strict balance in multi-attribute group decision-making.

Conclusions

This study combines expert experience with picture fuzzy tools to establish a two-phase model for HR. Especially, the MAGDM method is used to select the optimal HR during the refinement stage, while considering factors such as price, comfort, service, location, and convenience. In response to the problem that the basic operations of PPHFEs do not satisfy closure, we first propose a new normalization method for PPHFEs named Probability Splitting Algorithm. Based on this normalization, we use strict t-norm to propose a new interactive operation to ensure the closure of PPHFEs operations, and propose six special operations to ensure the scientific and rational decision results. On the basis of new operation rules, PPHFWA and PPHFWG operators are proposed, and monotonicity, idempotency, boundedness, Shift Invariance, and Homogeneity are proved. Six new operators are also presented. This article also proposes a linear order, and based on this order, we propose some new distance measures. The advantage of these distance measures is that they have good properties. At the same time, in response to the shortcomings of the decision-making method in reference²³, we extend the behavioral TOPSIS method to the PPHF environment, and thus propose a new MAGDM method. Finally, the proposed PPHFBTOPSIS technique was applied to HR, and sensitivity analysis of parameters was conducted to demonstrate the robustness of the method proposed in this paper. However, since there are no new methods for computation, no comparative analysis was conducted in this paper.

Limitations of this paper: First, in the current study, the weights of decision attributes were derived through a score-function-based transformation, which may lead to partial information loss compared to working entirely within the PPHF framework. Second, while we have demonstrated robustness through sensitivity analysis, no direct comparative experiments with classical TOPSIS, fuzzy TOPSIS, or other MAGDM methods were conducted, which may limit the empirical validation of the proposed method.

Future research directions: In future work, we plan to develop weight determination mechanisms that remain fully within the PPHF environment, thereby avoiding potential loss of decision information. We also intend to design systematic benchmarking experiments, applying the proposed method alongside classical TOPSIS, fuzzy TOPSIS, IFS-based BTOPSIS, and other MAGDM approaches, to empirically demonstrate its advantages in accuracy, robustness, and interpretability, to extend our study to include more complex fuzzy sets, such as spherical fuzzy sets³¹, complex picture fuzzy sets³², and complex spherical fuzzy sets³³. It is worth mentioning that the proposed theory and methods can be applied to various research areas, including preference relationships³⁴, estimation issue³⁵, and site selection issue³⁶, energy supplier selection³⁷, among others.