Isomorphic Operators and Ranking Methods for Pythagorean and Intuitionistic Fuzzy Sets

Abstract

With three pairs of fuzzy sets, intuitionistic fuzzy sets and Pythagorean fuzzy sets, interval-valued intuitionistic fuzzy sets and interval-valued Pythagorean fuzzy sets, dual hesitation fuzzy sets, and dual hesitation Pythagorean fuzzy sets as the research objects, the isomorphic relation between each pair of fuzzy sets is studied from three aspects, such as operational laws, aggregation operators, and ranking methods. Firstly, based on an automorphism on the unit interval, the transformation relationship between intuitionistic fuzzy dual triple (T_I, S_I, N_I) and Pythagorean fuzzy dual triple (T_P, S_P, N_P) is investigated, and the mathematical isomorphism for each pair of fuzzy sets is developed. Subsequently, the isomorphic operations for these three pairs of fuzzy sets are provided, which lays the foundation for the operators’ isomorphism. Secondly, the isomorphic Archimedean aggregation operators of each pairs of fuzzy sets are constructed based on the isomorphic operational laws. Finally, the isomorphism ranking methods are developed by exploring the conversion relationship between the score function and the accuracy function of each pair of fuzzy sets.

1 Introduction

After decades of research, the theories and methods of intuitionistic fuzzy sets (IFSs) [1] have formed a relatively perfect decision-making system and been applied to multi-attribute decision-making (MADM) problems in various fields. The aggregation operators and the ranking methods of fuzzy sets are the key components of the decision-making method. In the process of solving MADM problems, fuzzy information aggregation operators and sorting methods play a key role. Operators are mainly used to aggregate multiple attribute evaluation information to obtain the comprehensive evaluation values of the alternatives. Sorting methods are mainly applied to get the ranking of comprehensive evaluation value to obtain the optimal alternative.

Because intuitionistic fuzzy sets have the advantage of considering both membership degree and nonmembership degree, in recent years, scholars have cross-fusing other types of fuzzy sets, such as interval-valued fuzzy sets (IVFSs) [2] and hesitation fuzzy sets (HFSs) [4], with intuitionistic fuzzy sets to derive interval-valued intuitionistic fuzzy sets (IVIFSs) [3] and dual hesitation fuzzy sets (DHFSs) [5]. These new fuzzy sets integrate the advantages of the combiner, which makes the related operation rules, aggregation operators, and ranking methods get close attention of scholars. A series of fuzzy multi-attribute decision-making (FMADM) methods are developed and applied. In 2014, the concept of Pythagorean fuzzy sets (PFSs) [6] is introduced, which expands the value range of membership degree and nonmembership degree from triangle region to quarter circle region. In other words, intuitionistic fuzzy sets belong to the category of PFSs. Following the development of intuitionistic fuzzy sets, interval-valued fuzzy sets and hesitant fuzzy sets are extended to Pythagorean fuzzy environment, and interval-valued Pythagorean fuzzy sets (IVPFSs) [7, 8] and dual hesitant Pythagorean fuzzy sets (DHPFSs) [9] are derived. The aggregation operators and sorting methods of these extended fuzzy sets have been a hot topic in recent years.

The aggregation operators of fuzzy sets are generally based on operational laws, mainly including four types of operations such as addition, multiplication, scalar multiplication, and exponentiation. Dual Archimedean t-norm and s-norm [10, 11] are the core tools to define these laws, especially Algebraic t-norm/s-norm, Einstein t-norm/s-norm, Hamming t-norm/s-norm, and frank t-norm/s-norm are the four most commonly used tools. On this basis, a series of Archimedes fuzzy aggregation operators are generated. For example, Pythagorean fuzzy interactive Hamacher power aggregation operators [32] and generalized Pythagorean fuzzy Einstein geometric aggregation operators [33] are developed to construct multiple attribute group decision-making methods. In terms of sorting methods, the score function and accuracy function of fuzzy numbers are generally used to construct sorting methods to distinguish different fuzzy numbers. The Pythagorean fuzzy sets and its extended sets mainly follow the relevant theories and methods of intuitionistic fuzzy sets in terms of aggregation operators and ranking methods. It is mainly reflected in the operators based on Archimedean t-norm and s-norm, and the ranking methods are based on score function and accuracy function. However, the current researches on Pythagorean fuzzy sets mainly focus on how to extend the theory and method of intuitionistic fuzzy sets to Pythagorean fuzzy decision-making environment, and pay less attention to the internal relationship between the two kinds of fuzzy sets. Therefore, from the perspective of isomorphism, this study will reveal the internal relationship and transformation relationship between the relevant theoretical methods of two types of fuzzy sets, and provides method support for solving the two kinds of fuzzy multi-attribute decision-making problems.

The purpose of this study is to reveal the substantial relationship between three kinds of intuitionistic fuzzy sets and three kinds of Pythagorean fuzzy sets, including IFSs and PFSs, IVIFSs and IVPFSs, and DHFSs and DHPFSs. The isomorphism between three pairs of fuzzy sets is studied from three aspects: operational laws, aggregation operators, and ranking methods.

2 Preliminaries

Some basic concepts of several kinds of fuzzy sets and the related concepts of t-norm and s-norm are reviewed.

2.1 Related Definitions of Intuitionistic Fuzzy Sets and Pythagorean Fuzzy Sets

Atanassov [1] generalizes Zadeh’s fuzzy set theory [12] with the concept of intuitionistic fuzzy sets (IFSs) as defined below:

Definition 1

([1]) Let $X$ be a universe of discourse. An IFS $I$ in $X$ is given by

$$I = \left\{ {\left\langle {x,\mu_{I} \left( x \right),v_{I} \left( x \right)} \right\rangle \left| {x \in X} \right.} \right\},$$

(1)

where the function $\mu_{I} :X \to \left[ {0,1} \right]$ defines the degree of membership and $v_{I} :X \to \left[ {0,1} \right]$ defines the degree of nonmembership of the element $x \in X$ to $I$, respectively, and for every $x \in X$, it holds that $\mu_{I} \left( x \right) + v_{I} \left( x \right) \le 1$.

Pythagorean fuzzy set (PFS) is a generalization of intuitionistic fuzzy sets (IFSs) and its definition is given as follows:

Definition 2

([6]) Let $X$ be a universe of discourse. A PFS $P$ in $X$ is given by

$$P = \left\{ {\left\langle {x,\mu_{P} \left( x \right),v_{P} \left( x \right)} \right\rangle \left| {x \in X} \right.} \right\},$$

(2)

where the function $\mu_{P} :X \to \left[ {0,1} \right]$ defines the degree of membership and $v_{P} :X \to \left[ {0,1} \right]$ defines the degree of nonmembership of the element $x \in X$ to $P$, respectively, and for every $x \in X$, it holds that $\left( {\mu_{P} \left( x \right)} \right)^{2} + \left( {v_{P} \left( x \right)} \right)^{2} \le 1$.The degree of indeterminacy $\pi_{P} \left( x \right) = \sqrt {1 - \left( {\mu_{P} \left( x \right)} \right)^{2} - \left( {v_{P} \left( x \right)} \right)^{2} }$.

For simplicity, we called $\left( {\mu_{P} \left( x \right),v_{P} \left( x \right)} \right)$ a Pythagorean fuzzy number (PFN) denoted by $P = \left( {\mu_{P} ,v_{P} } \right)$, where $\mu_{P} ,v_{P} \in \left[ {0,1} \right]$,$\pi_{P} = \sqrt {1 - \mu_{P}^{2} - v_{P}^{2} },$ and $\mu_{P}^{2} + v_{P}^{2} \le 1$.

Interval-valued intuitionistic fuzzy set (IVIFS) is a combination of interval-valued fuzzy sets and intuitionistic fuzzy sets and its definition is given as follows:

Definition 3

([3]) We let $K$ denote a finite universal set. An IVIFS $B$ on $K$ is provided as

$${B} = \left\{ {\left\langle {q,{\mu }_{B} \left( q \right),{v}_{B} \left( q \right)} \right\rangle \left| {q \in K} \right.} \right\},$$

(3)

where ${\mu }_{B} :K \to \left[ {0,1} \right]$ is the interval membership function, ${v}_{B} :K \to \left[ {0,1} \right]$ is the interval nonmembership function of $B$, satisfying ${\mu }_{B} = \left[ {\mu_{B}^{ - } \left( q \right),\mu_{B}^{ + } \left( q \right)} \right],{v}_{B} = \left[ {v_{B}^{ - } \left( q \right),v_{B}^{ + } \left( q \right)} \right]$, $0 \le \mu_{B}^{ - } \left( q \right) \le \mu_{B}^{ + } \left( q \right) \le 1$, $0 \le v_{B}^{ - } \left( q \right) \le v_{B}^{ + } \left( q \right) \le 1$, $\mu_{B}^{ + } \left( q \right) + v_{B}^{ + } \left( q \right) \le 1$ for any $q \in K$. Especially, ${\pi }_{B} = \left[ {1 - \mu_{B}^{ + } \left( q \right) - v_{B}^{ + } \left( q \right),1 - \mu_{B}^{ - } \left( q \right) - v_{B}^{ - } \left( q \right)} \right]$ is the indeterminacy degree of $q \in K$. The pair ${\beta } = \left( {\left[ {\mu_{B}^{ - } \left( q \right),\mu_{B}^{ + } \left( q \right)} \right],\left[ {v_{B}^{ - } \left( q \right),v_{B}^{ + } \left( q \right)} \right]} \right)$ is called IVIFN and simply expressed as ${\beta } = \left( {\left[ {\mu^{ - } ,\mu^{ + } } \right],\left[ {v^{ - } ,v^{ + } } \right]} \right)$, where $\left[ {\mu^{ - } ,\mu^{ + } } \right],\left[ {v^{ - } ,v^{ + } } \right] \subseteq \left[ {0,1} \right]$ and $\mu^{ + } + v^{ + } \le 1$.

According to Definition 1, if $\mu^{ - } = \mu^{ + }$ and $v^{ - } = v^{ + }$, then IVIFN ${\beta }$ reduces to an intuitionistic fuzzy number (IFN) [1].

Interval-valued Pythagorean fuzzy set (IVPFS) is a combination of interval-valued fuzzy sets and Pythagorean fuzzy sets and its definition is given as follows:

Definition 4

([7, 8]) Given a finite universal set $K$, an interval-valued PFS (IVPFS) ${P}$ on $K$ is given by

$${P} = \left\{ {\left\langle {y,{\mu }_{P} \left( y \right),{v}_{P} \left( y \right)} \right\rangle \left| {y \in K} \right.} \right\},$$

(4)

where ${\mu }_{P} \left( y \right):K \to \varepsilon \left( {\left[ {0,1} \right]} \right)$ is the membership degree, ${v}_{P} \left( y \right):K \to \varepsilon \left( {\left[ {0,1} \right]} \right)$ is the nonmembership degree, and $\sup \left( {{\mu }_{P}^{2} \left( y \right)} \right) + \sup \left( {{v}_{P}^{2} \left( y \right)} \right) \le 1$. And $\varepsilon \left( {\left[ {0,1} \right]} \right)$ is the set of all closed intervals in the unit interval, and we call the two-tuples $\left( {{\mu }_{P} \left( y \right),{v}_{P} \left( y \right)} \right)$ as interval-valued Pythagorean fuzzy number (IVPFN). Let ${\mu }_{P} \left( y \right) = \left[ {a^{ - } ,a^{ + } } \right]$ and ${v}_{P} \left( y \right) = \left[ {b^{ - } ,b^{ + } } \right]$, then the IVPFN can be expressed as ${\beta } = \left( {\left[ {a^{ - } ,a^{ + } } \right],\left[ {b^{ - } ,b^{ + } } \right]} \right)$, and $\left( {b^{ + } } \right)^{2} + \left( {a^{ + } } \right)^{2} \le 1$.

According to Definition 2, if $a^{ - } = a^{ + }$ and $b^{ - } = b^{ + }$, then IVPFN ${\beta }$ reduces to an Pythagorean fuzzy number (PFN) [6].

Considering the advantages of IFSs and hesitant fuzzy sets, the concept of dual hesitant fuzzy sets (DHFSs) is defined, which are composed of membership hesitant fuzzy sets and nonmembership hesitant fuzzy sets.

Definition 5

Let $X$ be a universe of discourse. A DHFS $\delta$ in $X$ is given by

$$\delta = \left\{ {\left\langle {x,p_{I} \left( x \right),q_{I} \left( x \right)} \right\rangle \left| {x \in X} \right.} \right\}$$

(5)

in which $p_{I} \left( x \right)$ and $q_{I} \left( x \right)$ are two sets of some values in $\left[ {0,1} \right]$, denoting the possible membership degrees and nonmembership degrees of the element $x \in X$ to $\delta$, respectively, with the conditions: $\max_{{\gamma \in p_{I} }} \left\{ \gamma \right\} + \max_{{\eta \in q_{I} }} \left\{ \eta \right\} \le 1$, where $\gamma \in p_{I} \left( x \right)$ and $\eta \in q_{I} \left( x \right)$ for all $x \in X$. For convenience, the pair $\delta = \left( {p_{I} \left( x \right),q_{I} \left( x \right)} \right)$ is called a hesitant Pythagorean fuzzy number (HPFN) denoted by $\delta = \left( {p,q} \right)$, with the conditions: $\gamma \in p,\eta \in q$$\gamma ,\eta \in \left[ {0,1} \right]$, and $\max_{\gamma \in p} \left\{ \gamma \right\} + \max_{\eta \in q} \left\{ \eta \right\} \le 1$.

Considering the advantages of PFSs and hesitant fuzzy sets, the concept of dual hesitant fuzzy sets (DHFSs) is defined, which are composed of membership hesitant fuzzy sets and nonmembership hesitant fuzzy sets.

Definition 6

Let $X$ be a universe of discourse. A DHPFS $\psi$ in $X$ is given by

$$\psi = \left\{ {\left\langle {x,a_{P} \left( x \right),b_{P} \left( x \right)} \right\rangle \left| {x \in X} \right.} \right\}$$

(6)

in which $a_{P} \left( x \right)$ and $b_{P} \left( x \right)$ are two sets of some values in $\left[ {0,1} \right]$, denoting the possible membership degrees and nonmembership degrees of the element $x \in X$ to $\psi$, respectively, with the conditions: $\max_{{\tau \in a_{P} }} \left\{ {\tau^{2} } \right\}{ + }\max_{{\varsigma \in b_{P} }} \left\{ {\varsigma^{2} } \right\} \le 1$, where $\tau \in a_{P} \left( x \right)$ and $\varsigma \in b_{P} \left( x \right)$ for all $x \in X$. For convenience, the pair $\psi = \left( {a_{P} \left( x \right),b_{P} \left( x \right)} \right)$ is called a hesitant Pythagorean fuzzy number (HPFN) denoted by $\psi = \left( {a,b} \right)$, with the conditions: $\tau \in a,\varsigma \in b$$\tau ,\varsigma \in \left[ {0,1} \right]$, and $\max_{\tau \in a} \left\{ {\tau^{2} } \right\}{ + }\max_{\varsigma \in q} \left\{ {\varsigma^{2} } \right\} \le 1$.

Figure 1 describes the relationship between various fuzzy sets. From Fig. 1, we have

Fig. 1

Inclusion relations of six kinds of fuzzy sets

• (1) An intuitionistic fuzzy number (IFN) is also a Pythagorean fuzzy number (PFN), but not all PFNs are IFNs;

• (2) A dual hesitant fuzzy number (DHFN) is also a dual hesitant Pythagorean fuzzy number (DHPFN), but not all DHPFNs are DHFNs;

• (3) An interval-valued intuitionistic fuzzy number (IVIFN) is also an interval-valued Pythagorean fuzzy number (IVPFN), but not all IVPFNs are IVIFNs.

These results mean we can use the PFNs (DHPFNs/IVPFNs) with less limitation than IFNs (DHIFNs/IVIFNs) in some special situation.

2.2 T-Norm and Its Dual T-Conorm

A triple $\left( {T,S,N} \right)$, where $T$ is a t-norm, $S$ a t-conorm and $N$ a fuzzy complement is called a dual triple if $T$ and $S$ are dual with respect to $N$. The dual triple $\left( {T,S,N} \right)$ is a utility tool to define the generalized operational laws for a variety of different fuzzy environments.

The concepts of t-norm and t-conorm are given as follows:

Definition 7

([10]). A function $T:\left[ {0,1} \right] \times \left[ {0,1} \right] \to \left[ {0,1} \right]$ is called a t-norm if it satisfies the following four conditions:

• (1) (Neutral element): $T\left( {1,x} \right) = x$, for all $x$.

• (2) (Commutativity): $T\left( {y,x} \right) = T\left( {x,y} \right)$, for all $x$ and $y$.

• (3) (Associativity): $T\left( {T\left( {x,y} \right),z} \right) = T\left( {x,T\left( {y,z} \right)} \right)$, for all $x$, $y,$ and $z$.

• (4) (Monotonicity): If $x \le x^{\prime}$ and $y \le y^{\prime}$, then $T\left( {x,y} \right) \le T\left( {x^{\prime},y^{\prime}} \right)$.

Definition 8

([10]) A function $S:\left[ {0,1} \right] \times \left[ {0,1} \right] \to \left[ {0,1} \right]$ is called a t-conorm if it satisfies the following four conditions:

• (1) (Neutral element): $S\left( {0,x} \right) = x$, for all $x$.

• (2) (Commutativity): $S\left( {y,x} \right) = S\left( {x,y} \right)$, for all $x$ and $y$.

• (3) (Associativity): $S\left( {S\left( {x,y} \right),z} \right) = S\left( {x,S\left( {y,z} \right)} \right)$, for all $x$, $y,$ and $z$.

• (4) (Monotonicity): if $x \le x^{\prime}$ and $y \le y^{\prime}$, then $S\left( {x,y} \right) \le S\left( {x^{\prime},y^{\prime}} \right)$.

Continuous Archimedean t-norms are very useful because they can be represented via additive generator. Now, the definitions of Archimedean t-norm and t-conorm are given as follows:

Definition 9

([10]) A t-norm is called Archimedean t-norm if for each $\left( {a,b} \right) \in \left] {0,1} \right[^{2}$ there is an $n \in \left\{ {1,2, \cdots } \right\}$ with $T\overbrace {{\left( {a, \cdots ,a} \right)}}^{n - times} < b$.

Definition 10

([10]) A t-conorm is called Archimedean t-conorm if for each $\left( {a,b} \right) \in \left] {0,1} \right[^{2}$ there is an $n \in \left\{ {1,2, \cdots } \right\}$ with $S\overbrace {{\left( {a, \cdots ,a} \right)}}^{n - times} > b$.

Any continuous Archimedean t-norm and t-conorm can be represented with the help of a continuous additive generator.

Proposition 1

([10]) A continuous Archimedean t-norm $T$ is expressed via its additive generator $g:\left[ {0,1} \right] \to \left[ {0,\infty } \right]$, which verifies $g\left( 1 \right) = 0$ as $T\left( {x,y} \right) = g^{ - 1} \left( {g\left( x \right) + g\left( y \right)} \right)$, where $g$ is a continuous strictly decreasing function and $g^{ - 1} \left( t \right) = \sup \left\{ {z \in \left[ {0,1} \right]\left| {g\left( z \right) > t,t \in \left[ {0,\infty } \right]} \right.} \right\}$ is the pseudo-inverse of $g$.

Proposition 2

([10]) A continuous Archimedean t-conorm $S$ is expressed via its additive generator $h:\left[ {0,1} \right] \to \left[ {0,\infty } \right]$, which verifies $h\left( 0 \right) = 0$ as $S\left( {x,y} \right) = h^{ - 1} \left( {h\left( x \right) + h\left( y \right)} \right)$, where $h$ is a continuous strictly increasing function and $h^{ - 1} \left( t \right) = \sup \left\{ {z \in \left[ {0,1} \right]\left| {h\left( z \right) < t,t \in \left[ {0,\infty } \right]} \right.} \right\}$ is the pseudo-inverse of $h$.

Definition 11

([10]) A fuzzy complement is a mapping denoted by $N:\left[ {0,1} \right] \to \left[ {0,1} \right]$ that satisfies the following:

• (1) Boundary conditions: $N\left( 0 \right) = 1$ and $N\left( 1 \right) = 0$.

• (2) Monotonicity: for all $a,b \in \left[ {0,1} \right]$, if $a \le b$, then $N\left( a \right) \ge N\left( b \right)$,

• (3) Continuity

• (4) Involution: $N\left( {N\left( a \right)} \right) = a$ for all $a \in \left[ {0,1} \right]$.

Yager class of fuzzy complements [13, 14] is defined by $N\left( a \right) = \left( {1 - a^{p} } \right)^{{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0pt} p}}} \;$, where $p \in \left( {0,\infty } \right)$. When $p = 1$, this function becomes the classical fuzzy complements $N_{I} \left( a \right) = 1 - a$. while $p = 2$ will make this function become the Pythagorean complement [6] $N_{P} \left( a \right) = \sqrt {1 - a^{2} }$.

Klir and Yuan [11] first put forward the concept of dual triple $\left( {T,S,N} \right)$, which denote that $T$ and $S$ are dual with respect to $N$, and let any such triple be called a dual triple.

Definition 12

([11]) A t-norm $T$ and a t-conorm $S$ are dual with respect to a fuzzy complement $N$ iff $T\left( {x,y} \right) = N\left( {S\left( {N\left( x \right),N\left( y \right)} \right)} \right)$ and $S\left( {x,y} \right) = N\left( {T\left( {N\left( x \right),N\left( y \right)} \right)} \right)$.

Theorem 1

([11]). Given a t-norm $T$ and an involutive fuzzy complement $N$, the binary operation $S$ on $\left[ {0,1} \right]$ defined by $S\left( {x,y} \right) = N\left( {T\left( {N\left( x \right),N\left( y \right)} \right)} \right)$ for all $x$, $y \in \left[ {0,1} \right]$ is a t-conorm such that $\left( {T,S,N} \right)$ is a dual triple.

Proposition 3

([11]) Let $T$ be a t-norm, $S$ its dual t-conorm, and $g:\left[ {0,1} \right] \to \left[ {0,\infty } \right]$ an additive generator of $T$. The function $h:\left[ {0,1} \right] \to \left[ {0,\infty } \right]$ defined by $h\left( t \right) = g\left( {N\left( t \right)} \right)$ is an additive generator of $S$.

2.3 Four Types of Dual Archimedean T-Norm and S-Norm

It is found that the traditional triple $\left( {T,S,N_{I} } \right)$ is mainly used to define the operational laws and aggregation operators of intuitionistic fuzzy sets, dual hesitant fuzzy sets, and interval-valued intuitionistic fuzzy sets. For convenience, this study denotes such triples as $\left( {T_{I} ,S_{I} ,N_{I} } \right)$. Triple $\left( {T,S,N_{P} } \right)$ is mainly used in the operation laws and aggregation operators of Pythagorean fuzzy sets, dual hesitant Pythagorean fuzzy sets, and interval-valued Pythagorean fuzzy sets. For convenience, this study denotes such triples as $\left( {T_{P} ,S_{P} ,N_{P} } \right)$.

This subsection introduces four classic types of dual T-norm and S-norm, whose specific forms are different in the intuitionistic fuzzy environment and the Pythagorean fuzzy environment. But there is a certain transformational relationship. See Table 1 for details.

Table 1 Generators of operations for IFSs and PFSs

Theorem 2

If the additive generators of $S_{I}$ and $S_{P}$ satisfy $g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right)$ , then

$$S_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {S_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right),\,\,T_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {T_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right),$$

where $T_{I}$ and $T_{P}$ are the dual t-norm of $S_{I}$ and $S_{P}$, respectively, and $\phi \left( t \right) = t^{2}$ is an automorphism on $\left[ {0,1} \right]$.

Theorem 2 reveals the transformation between intuitionistic fuzzy triple $\left( {T_{I} ,S_{I} ,N_{I} } \right)$ and Pythagorean fuzzy triple $\left( {T_{P} ,S_{P} ,N_{P} } \right)$.

3 Operations Isomorphism

Operations are mainly used to construct aggregation operators. As mentioned in the previous section, the operational laws of the three types of fuzzy sets are all based on t-norm and s-norm. From the isomorphism perspective, this section will reveal the correlation between the operational laws of the three types of intuitionistic fuzzy sets and Pythagorean fuzzy sets.

3.1 Operations Isomorphism Between IFSs and PFSs

Firstly, the relationship between the intuitionistic fuzzy operations and the Pythagorean fuzzy operations is analyzed.

Dual Archimedean t-norm and s-norm are used to define the generalized intuitionistic fuzzy operations, which can be reduced to special operations such as Einstein, Hamacher, and Frank operation.

Definition 13

([15]). For three IFNs $\vartheta = \left( {\rho ,\sigma } \right)$ and $\vartheta_{i} = \left( {\rho_{i} ,\sigma_{i} } \right)\left( {i = 1,2} \right)$, then we have

• (1) $\begin{aligned} \vartheta_{1} \oplus \vartheta_{2} &= \left( {S_{I} \left( {\rho_{1} ,\rho_{2} } \right),T_{I} \left( {\sigma_{1} ,\sigma_{2} } \right)} \right) \\ & = \left( {h_{I}^{ - 1} \left( {h_{I} \left( {\rho_{1} } \right) + h_{I} \left( {\rho_{2} } \right)} \right),g_{I}^{ - 1} \left( {g_{I} \left( {\sigma_{1} } \right) + g_{I} \left( {\sigma_{2} } \right)} \right)} \right) \end{aligned}$;

• (2) $\begin{aligned} \vartheta_{1} \otimes \vartheta_{2} &= \left( {T_{I} \left( {\rho_{1} ,\rho_{2} } \right),S_{I} \left( {\sigma_{1} ,\sigma_{2} } \right)} \right)\\ & = \left( {g_{I}^{ - 1} \left( {g_{I} \left( {\rho_{1} } \right) + g_{I} \left( {\rho_{2} } \right)} \right),h_{I}^{ - 1} \left( {h_{I} \left( {\sigma_{1} } \right) + h_{I} \left( {\sigma_{2} } \right)} \right)} \right) \end{aligned}$;

• (3) $\lambda \vartheta = \left( {h_{I}^{ - 1} \left( {\lambda h_{I} \left( \rho \right)} \right),g_{I}^{ - 1} \left( {\lambda g_{I} \left( \sigma \right)} \right)} \right),\lambda > 0$;

• (4) $\vartheta^{\lambda } = \left( {g_{I}^{ - 1} \left( {\lambda g_{I} \left( \rho \right)} \right),h_{I}^{ - 1} \left( {\lambda h_{I} \left( \sigma \right)} \right)} \right),\lambda > 0$;

where $T_{I}$ and $S_{I}$ are dual with respect to $N_{I}$ and $N_{I} \left( a \right) = 1 - a$. $g_{I}$ and $h_{I}$ are the additive generators of $T_{I}$ and $S_{I}$, respectively.

By referring to the intuitionistic fuzzy generalized operations, some new dual Archimedean t-norm and s-norm suitable for Pythagorean fuzzy environment are proposed and used to construct Pythagorean fuzzy generalized operations.

Definition 14

([16]). For three PFNs $\alpha = \left( {\mu_{\alpha } ,v_{\alpha } } \right)$ and $\alpha_{i} { = }\left( {\mu_{{\alpha_{i} }} ,v_{{\alpha_{i} }} } \right)\left( {i = 1,2} \right)$, then we have

• (1) $\begin{aligned} \alpha_{1} \oplus \alpha_{2} & = \left( {S_{P} \left( {\mu_{1} ,\mu_{2} } \right),T_{P} \left( {v_{1} ,v_{2} } \right)} \right)\\& = \left( {h_{p}^{ - 1} \left( {h_{P} \left( {\mu_{1} } \right) + h_{P} \left( {\mu_{2} } \right)} \right),g_{p}^{ - 1} \left( {g_{P} \left( {v_{1} } \right) + g_{P} \left( {v_{2} } \right)} \right)} \right) \end{aligned}$;

• (2) $\begin{aligned} \alpha_{1} \otimes \alpha_{2} &= \left( {T_{P} \left( {\mu_{1} ,\mu_{2} } \right),S_{P} \left( {v_{1} ,v_{2} } \right)} \right)\\ &= \left( {g_{p}^{ - 1} \left( {g_{P} \left( {\mu_{1} } \right) + g_{P} \left( {\mu_{2} } \right)} \right),h_{p}^{ - 1} \left( {h_{P} \left( {v_{1} } \right) + h_{P} \left( {v_{2} } \right)} \right)} \right) \end{aligned}$;

• (3) $\lambda \alpha = \left( {h_{p}^{ - 1} \left( {\lambda h_{P} \left( \mu \right)} \right),g_{p}^{ - 1} \left( {\lambda g_{P} \left( v \right)} \right)} \right),\lambda > 0$;

• (4) $\alpha^{\lambda } = \left( {g_{p}^{ - 1} \left( {\lambda g_{P} \left( \mu \right)} \right),h_{p}^{ - 1} \left( {\lambda h_{P} \left( v \right)} \right)} \right),\lambda > 0$;

where $T_{P}$ and $S_{P}$ are dual with respect to $N_{P}$ and $N_{P} \left( a \right) = \sqrt {1 - a^{2} }$. $g_{P}$ and $h_{P}$ are the additive generators of $T_{P}$ and $S_{P}$, respectively.

For convenience, in this study, the set of all the intuitionistic fuzzy sets is denoted as $A_{I}$, and the set of all the Pythagorean fuzzy sets is denoted as $A_{P}.$

Definition 15

Let $\alpha_{i} = \left( {\mu_{i} ,\nu_{i} } \right)_{P} \in A_{P}$, a mapping $\wp_{P \to I} :A_{P} \to A_{I}$ is defined as follows:

$$\wp_{P \to I} \left( {\alpha_{i} } \right) = \left( {\phi \left( {\mu_{i} } \right),\phi \left( {\nu_{i} } \right)} \right)_{I} \in A_{I} ,$$

where $\varphi \left( x \right) = x^{2}$ is an automorphism on $\left[ {0,1} \right]$.

Definition 15 reveals that IFSs and PFSs have a mathematical isomorphism $\wp_{P \to I}$.

Theorem 3

Let $\alpha_{i} = \left\langle {\mu_{i} ,\nu_{i} } \right\rangle_{P} \in A_{P}$, $\wp_{P \to I} \left( {\alpha_{i} } \right) = \left( {\phi \left( {\mu_{i} } \right),\phi \left( {\nu_{i} } \right)} \right)_{I} \in A_{I}$, and the operations of $\alpha_{i} = \left\langle {\mu_{i} ,\nu_{i} } \right\rangle_{P}$ on $A_{P}$ and the operations of $\wp_{P \to I} \left( {\alpha_{i} } \right) = \left( {\phi \left( {\mu_{i} } \right),\phi \left( {\nu_{i} } \right)} \right)_{I}$ on $A_{I}$ are defined based on the t-norm and s-norm, then we have

• (1) $\wp_{P \to I} \left( {\alpha_{1} \oplus \alpha_{2} } \right) = \wp_{P \to I} \left( {\alpha_{1} } \right) \oplus \wp_{P \to I} \left( {\alpha_{2} } \right)$;

• (2) $\wp_{P \to I} \left( {\alpha_{1} \otimes \alpha_{2} } \right) = \wp_{P \to I} \left( {\alpha_{1} } \right) \otimes \wp_{P \to I} \left( {\alpha_{2} } \right)$;

• (3) $\wp_{P \to I} \left( {\lambda \alpha_{1} } \right) = \lambda \wp_{P \to I} \left( {\alpha_{1} } \right)$;

• (4) $\wp_{P \to I} \left( {\alpha_{1}^{\lambda } } \right) = \wp_{P \to I} \left( {\alpha_{1} } \right)^{\lambda }$.

From Theorem 3, $\wp_{P \to I}$ can be proved to be an isomorphic mapping from $A_{P}$ to $A_{I}$. Theorem 3 reveals that IFSs and PFSs do not only have a mathematical isomorphism, but also have a operations isomorphism. The later is a substantial relationship between IFSs and PFSs not reported up to now in literature.

3.2 Operations Isomorphism Between IVIFSs and IVPFSs

Atanassov [18] proposed the interval-valued intuitionistic fuzzy Algebraic operations, and several classical t-norm and s-norm are applied to interval-valued intuitionistic fuzzy environment, and various operations, such as Hamacher, Frank and Einstein operations, are proposed [19,20,21]. In this study, the uniform generalized interval-valued intuitionistic operations are provided as follow.

Definition 16

For three IVIFNs $\tilde{\vartheta } = \left( {\left[ {\rho^{ - } ,\rho^{ + } } \right],\left[ {\sigma^{ - } ,\sigma^{ + } } \right]} \right)$ and $\tilde{\vartheta }_{i} = \left( {\left[ {\rho_{i}^{ - } ,\rho_{i}^{ + } } \right],\left[ {\sigma_{i}^{ - } ,\sigma_{i}^{ + } } \right]} \right)\left( {i = 1,2} \right)$, then

• (1)
  $$\tilde{\vartheta }_{1} \oplus \tilde{\vartheta }_{2} = \left( {\left[ {S_{I} \left( {\rho_{1}^{ - } ,\rho_{2}^{ - } } \right),S_{I} \left( {\rho_{1}^{ + } ,\rho_{2}^{ + } } \right)} \right],\left[ {T_{I} \left( {\sigma_{1}^{ - } ,\sigma_{2}^{ - } } \right),T_{I} \left( {\sigma_{1}^{ + } ,\sigma_{2}^{ + } } \right)} \right]} \right);$$
• (2)
  $$\tilde{\vartheta }_{1} \otimes \tilde{\vartheta }_{2} = \left( {\left[ {T_{I} \left( {\rho_{1}^{ - } ,\rho_{2}^{ - } } \right),T_{I} \left( {\rho_{1}^{ + } ,\rho_{2}^{ + } } \right)} \right],\left[ {S_{I} \left( {\sigma_{1}^{ - } ,\sigma_{2}^{ - } } \right),S_{I} \left( {\sigma_{1}^{ + } ,\sigma_{2}^{ + } } \right)} \right]} \right);$$
• (3)
  $$\lambda \tilde{\vartheta } = \left( {\left[ {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\rho^{ - } } \right)} \right),h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\rho^{ + } } \right)} \right)} \right],\left[ {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\sigma^{ - } } \right)} \right),g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\sigma^{ + } } \right)} \right)} \right]} \right),\lambda > 0;$$
• (4)
  $$\tilde{\vartheta }^{\lambda } = \left( {\left[ {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\rho^{ - } } \right)} \right),g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\rho^{ + } } \right)} \right)} \right],\left[ {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\sigma^{ - } } \right)} \right),h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\sigma^{ + } } \right)} \right)} \right]} \right),\lambda > 0.$$

where $T_{I} \left( {x,y} \right) = g_{I}^{ - 1} \left( {g_{I} \left( x \right) + g_{I} \left( y \right)} \right)$ and $S_{I} \left( {x,y} \right) = h_{I}^{ - 1} \left( {h_{I} \left( x \right) + h_{I} \left( y \right)} \right)$ are dual with respect to $N_{I}$ and $N_{I} \left( a \right) = 1 - a$. $g_{I}$ and $h_{I}$ are the additive generators of $T_{I}$ and $S_{I}$, respectively.

Similar to interval-valued intuitionistic fuzzy operations, some kinds of interval-valued Pythagorean fuzzy operations are proposed. The uniform generalized interval-valued Pythagorean operations are provided as follow.

Definition 17

([22]). For three IVPFNs $\tilde{\alpha } = \left( {\left[ {\mu^{ - } ,\mu^{ + } } \right],\left[ {v^{ - } ,v^{ + } } \right]} \right)$ and $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)\left( {i = 1,2} \right)$, then

• (1)
  $$\tilde{\alpha }_{1} \oplus \tilde{\alpha }_{2} = \left( {\left[ {S_{P} \left( {\mu_{1}^{ - } ,\mu_{2}^{ - } } \right),S_{P} \left( {\mu_{1}^{ + } ,\mu_{2}^{ + } } \right)} \right],\left[ {T_{P} \left( {v_{1}^{ - } ,v_{2}^{ - } } \right),T_{P} \left( {v_{1}^{ + } ,v_{2}^{ + } } \right)} \right]} \right);$$
• (2)
  $$\tilde{\alpha }_{1} \otimes \tilde{\alpha }_{2} = \left( {\left[ {T_{P} \left( {\mu_{1}^{ - } ,\mu_{2}^{ - } } \right),T_{P} \left( {\mu_{1}^{ + } ,\mu_{2}^{ + } } \right)} \right],\left[ {S_{P} \left( {v_{1}^{ - } ,v_{2}^{ - } } \right),S_{P} \left( {v_{1}^{ + } ,v_{2}^{ + } } \right)} \right]} \right);$$
• (3)
  $$\lambda \tilde{\alpha } = \left( {\left[ {h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\mu^{ - } } \right)} \right),h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\mu^{ + } } \right)} \right)} \right],\left[ {g_{P}^{ - 1} \left( {\lambda g_{P} \left( {v^{ - } } \right)} \right),g_{P}^{ - 1} \left( {\lambda g_{P} \left( {v^{ + } } \right)} \right)} \right]} \right),\lambda > 0;$$
• (4)
  $$\tilde{\alpha }^{\lambda } = \left( {\left[ {g_{P}^{ - 1} \left( {\lambda g_{P} \left( {\mu^{ - } } \right)} \right),g_{P}^{ - 1} \left( {\lambda g_{P} \left( {\mu^{ + } } \right)} \right)} \right],\left[ {h_{P}^{ - 1} \left( {\lambda h_{P} \left( {v^{ - } } \right)} \right),h_{P}^{ - 1} \left( {\lambda h_{P} \left( {v^{ + } } \right)} \right)} \right]} \right),\lambda > 0.$$

where $T_{P} \left( {x,y} \right) = g_{P}^{ - 1} \left( {g_{P} \left( x \right) + g_{P} \left( y \right)} \right)$ and $S_{P} \left( {x,y} \right) = h_{P}^{ - 1} \left( {h_{P} \left( x \right) + h_{P} \left( y \right)} \right)$ are dual with respect to $N_{P}$ and $N_{P} \left( a \right) = \sqrt {1 - a^{2} }$. $g_{P}$ and $h_{P}$ are the additive generators of $T_{P}$ and $S_{P}$, respectively.

For convenience, in this study, the set of all the interval-valued intuitionistic fuzzy sets is denoted as $\tilde{A}_{I}$, and the set of all the interval-valued Pythagorean fuzzy sets is denoted as $\tilde{A}_{P}$.

Definition 18

Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)_{P} \in A_{P}$, a mapping $\tilde{\wp }_{P \to I} :\tilde{A}_{P} \to \tilde{A}_{I}$ is defined as follows:

$$\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right) = \left( {\left[ {\phi \left( {\mu_{i}^{ - } } \right),\phi \left( {\mu_{i}^{ + } } \right)} \right],\left[ {\phi \left( {v_{i}^{ - } } \right),\phi \left( {v_{i}^{ + } } \right)} \right]} \right)_{I} \in \tilde{A}_{I} ,$$

where $\varphi \left( x \right) = x^{2}$ is an automorphism on $\left[ {0,1} \right]$.

Definition 18 reveals that IVIFSs and IVPFSs have a mathematical isomorphism $\tilde{\wp }_{P \to I}$.

Theorem 4

Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)_{P} \in \tilde{A}_{P}$, $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right) = \left( {\left[ {\phi \left( {\mu_{i}^{ - } } \right),\phi \left( {\mu_{i}^{ + } } \right)} \right],\left[ {\phi \left( {v_{i}^{ - } } \right),\phi \left( {v_{i}^{ + } } \right)} \right]} \right)_{I} \in \tilde{A}_{I}$, and the operations of $\tilde{\alpha }_{i} \in \tilde{A}_{P}$ on $\tilde{A}_{P}$ and the operations of $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right) \in \tilde{A}_{I}$ on $\tilde{A}_{I}$ are defined based on the t-norm and s-norm, then we have

• (1) $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} \oplus \tilde{\alpha }_{2} } \right) = \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right) \oplus \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)$;

• (2) $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} \otimes \tilde{\alpha }_{2} } \right) = \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right) \otimes \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)$;

• (3) $\tilde{\wp }_{P \to I} \left( {\lambda \tilde{\alpha }_{1} } \right) = \lambda \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)$;

• (4) $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1}^{\lambda } } \right) = \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)^{\lambda }$.

From Theorem 4, $\tilde{\wp }_{P \to I}$ can be proved to be an isomorphic mapping from $\tilde{A}_{P}$ to $\tilde{A}_{I}$. Theorem 4 reveals that IVIFSs and IVPFSs do not only have a mathematical isomorphism, but also have a operations isomorphism. The later is a substantial relationship between IVIFSs and IVPFSs not reported up to now in literature.

3.3 Operations Isomorphism Between DHFSs and DHPFSs

Inspired by intuitionistic fuzzy Archimedean operations and Pythagorean fuzzy Archimedean operations, some scholars developed dual hesitant fuzzy Archimedean operations and dual hesitant Pythagorean fuzzy Archimedean operations by applying Archimedes t-norm and s-norm [23, 24].

Definition 19

([23]) For three DHFNs $\delta = \left( {p,q} \right)$ and $\delta_{i} = \left( {p_{i} ,q_{i} } \right)\left( {i = 1,2} \right)$, then we have

• (1) $\begin{aligned} \delta_{1} \oplus \delta_{2} & = \cup_{{\gamma_{i} \in p_{i} ,\eta_{i} \in q_{i} }} \left\{ {\left\{ {S_{I} \left( {\gamma_{1} ,\gamma_{2} } \right)} \right\},\left\{ {T_{I} \left( {\eta_{1} ,\eta_{2} } \right)} \right\}} \right\} \\ & = \cup_{{\gamma_{i} \in p_{i} ,\eta_{i} \in q_{i} }} \left\{ {\left\{ {h_{I}^{ - 1} \left( {h_{I} \left( {\gamma_{1} } \right) + h_{I} \left( {\gamma_{2} } \right)} \right)} \right\},\left\{ {g_{I}^{ - 1} \left( {g_{I} \left( {\eta_{1} } \right) + g_{I} \left( {\eta_{2} } \right)} \right)} \right\}} \right\} \\ \end{aligned}$;

• (2) $\begin{aligned} \delta_{1} \otimes \delta_{2} & = \cup_{{\gamma_{i} \in p_{i} ,\eta_{i} \in q_{i} }} \left\{ {\left\{ {T_{I} \left( {\gamma_{1} ,\gamma_{2} } \right)} \right\},\left\{ {S_{I} \left( {\eta_{1} ,\eta_{2} } \right)} \right\}} \right\} \\ & = \cup_{{\gamma_{i} \in p_{i} ,\eta_{i} \in q_{i} }} \left\{ {\left\{ {g_{I}^{ - 1} \left( {g_{I} \left( {\gamma_{1} } \right) + g_{I} \left( {\gamma_{2} } \right)} \right)} \right\},\left\{ {h_{I}^{ - 1} \left( {h_{I} \left( {\eta_{1} } \right) + h_{I} \left( {\eta_{2} } \right)} \right)} \right\}} \right\} \\ \end{aligned}$;

• (3) $\lambda \delta = \cup_{\gamma \in p,\eta \in q} \left\{ {\left\{ {h_{I}^{ - 1} \left( {\lambda h_{I} \left( \gamma \right)} \right)} \right\},\left\{ {g_{I}^{ - 1} \left( {\lambda g_{I} \left( \eta \right)} \right)} \right\}} \right\},\lambda > 0$;

• (4) $\delta^{\lambda } = \cup_{\gamma \in p,\eta \in q} \left\{ {\left\{ {g_{I}^{ - 1} \left( {\lambda g_{I} \left( \gamma \right)} \right)} \right\},\left\{ {h_{I}^{ - 1} \left( {\lambda h_{I} \left( \eta \right)} \right)} \right\}} \right\},\lambda > 0$.

where $T_{I}$ and $S_{I}$ are dual with respect to $N_{I} \left( a \right) = 1 - a$. $g_{I}$ and $h_{I}$ are the additive generators of $T_{I}$ and $S_{I}$, respectively.

Definition 20

([24]) For three DHPFNs $\psi = \left( {a,b} \right)$ and $\psi_{i} = \left( {a_{i} ,b_{i} } \right)\left( {i = 1,2} \right)$, then we have

• (1) $\begin{aligned} \psi_{1} \oplus \psi_{2} & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {S_{P} \left( {\tau_{1} ,\tau_{2} } \right)} \right\},\left\{ {T_{P} \left( {\varsigma_{1} ,\varsigma_{2} } \right)} \right\}} \right\} \\ & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {h_{P}^{ - 1} \left( {h_{P} \left( {\tau_{1} } \right) + h_{P} \left( {\tau_{2} } \right)} \right)} \right\},\left\{ {g_{P}^{ - 1} \left( {g_{P} \left( {\varsigma_{1} } \right) + g_{P} \left( {\varsigma_{2} } \right)} \right)} \right\}} \right\} \\ \end{aligned}$;

• (2) $\begin{aligned} \psi_{1} \otimes \psi_{2} & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {T_{P} \left( {\tau_{1} ,\tau_{2} } \right)} \right\},\left\{ {S_{P} \left( {\varsigma_{1} ,\varsigma_{2} } \right)} \right\}} \right\} \\ & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {g_{P}^{ - 1} \left( {g_{P} \left( {\tau_{1} } \right) + g_{P} \left( {\tau_{2} } \right)} \right)} \right\},\left\{ {h_{P}^{ - 1} \left( {h_{P} \left( {\varsigma_{1} } \right) + h_{P} \left( {\varsigma_{2} } \right)} \right)} \right\}} \right\} \, \\ \end{aligned}$;

• (3) $\lambda \psi = \cup_{\tau \in a,\varsigma \in b} \left\{ {\left\{ {h_{P}^{ - 1} \left( {\lambda h_{P} \left( \tau \right)} \right)} \right\},\left\{ {g_{P}^{ - 1} \left( {\lambda g_{P} \left( \varsigma \right)} \right)} \right\}} \right\},\lambda > 0$;

• (4) $\delta^{\lambda } = \cup_{\tau \in a,\varsigma \in b} \left\{ {\left\{ {g_{P}^{ - 1} \left( {\lambda g_{P} \left( \tau \right)} \right)} \right\},\left\{ {h_{P}^{ - 1} \left( {\lambda h_{P} \left( \varsigma \right)} \right)} \right\}} \right\},\lambda > 0$.

where $T_{P}$ and $S_{P}$ are dual with respect to $N_{P} \left( a \right) = \sqrt {1 - a^{2} }$. $g_{P}$ and $h_{P}$ are the additive generators of $T_{P}$ and $S_{P}$, respectively.

For convenience, in this study, the set of all the dual hesitant fuzzy sets is denoted as $\hat{A}_{I}$, and the set of all the Pythagorean fuzzy sets is denoted as $\hat{A}_{P}$.

Definition 21

Let $\psi_{i} = \left( {a_{i} ,b_{i} } \right)_{P} \in \hat{A}_{P}$, a mapping $\hat{\wp }_{P \to I} :\hat{A}_{P} \to \hat{A}_{I}$ is defined as follows

$$\hat{\wp }_{P \to I} \left( {\psi_{i} } \right) = \left( {\hat{\phi }\left( {a_{i} } \right),\hat{\phi }\left( {b_{i} } \right)} \right)_{I} \in A_{I} ,$$

where $\varphi \left( x \right) = x^{2}$ is an automorphism on $\left[ {0,1} \right]$, and $\hat{\phi }\left( {a_{i} } \right) = \cup_{{\tau_{i} \in a_{i} }} \left\{ {\phi \left( {\tau_{i} } \right)} \right\}$ and $\hat{\phi }\left( {b_{i} } \right) = \cup_{{\tau_{i} \in a_{i} }} \left\{ {\phi \left( {\varsigma_{i} } \right)} \right\}$.

Definition 20 reveals that DHFSs and DHPFSs have a mathematical isomorphism $\tilde{\wp }_{P \to I}$.

Theorem 5

Let $\psi_{i} = \left( {a_{i} ,b_{i} } \right)_{P} \in \hat{A}_{P}$, $\hat{\wp }_{P \to I} \left( {\psi_{i} } \right) = \left( {\hat{\phi }\left( {a_{i} } \right),\hat{\phi }\left( {b_{i} } \right)} \right)_{I} \in \hat{A}_{I}$, and the operations of $\psi_{i}$ on $\hat{A}_{P}$ and the operations of $\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)$ on $\hat{A}_{I}$ are defined based on the t-norm and s-norm, then we have

• (1) $\hat{\wp }_{P \to I} \left( {\psi_{1} \oplus \psi_{2} } \right) = \hat{\wp }_{P \to I} \left( {\psi_{1} } \right) \oplus \wp_{P \to I} \left( {\psi_{2} } \right)$;

• (2) $\hat{\wp }_{P \to I} \left( {\psi_{1} \otimes \psi_{2} } \right) = \hat{\wp }_{P \to I} \left( {\psi_{1} } \right) \otimes \wp_{P \to I} \left( {\psi_{2} } \right)$;

• (3) $\hat{\wp }_{P \to I} \left( {\lambda \psi_{1} } \right) = \lambda \hat{\wp }_{P \to I} \left( {\psi_{1} } \right)$;

• (4) $\hat{\wp }_{P \to I} \left( {\psi_{1}^{\lambda } } \right) = \hat{\wp }_{P \to I} \left( {\psi_{1} } \right)^{\lambda }$.

From Theorem 5, $\hat{\wp }_{P \to I}$ can be proved to be an isomorphic mapping from $\hat{A}_{P}$ to $\hat{A}_{I}$. Theorem 5 reveals that DHFSs and DHPFSs do not only have a mathematical isomorphism, but also have an operations isomorphism. The later is a substantial relationship between DHFSs and DHPFSs not reported up to now in literature.

4 Aggregation Operators Isomorphism

In the previous section, the isomorphic relationship between three kinds of Archimedean intuitionistic fuzzy operations and three kinds of Archimedean Pythagorean fuzzy operations is studied. On this basis, in this section, the relationship between three kinds of intuitionistic fuzzy aggregation operators and three kinds of Pythagorean fuzzy aggregation operators will be studied.

4.1 Aggregation Operators Isomorphism Between PFSs and IFSs

By extending four classical Archimedean t-norm and s-norm to the intuitionistic fuzzy environment, the scholars proposed Archimedean intuitionistic fuzzy aggregation operators, which can be used to aggregate intuitionistic fuzzy information and constitute a systematic intuitionistic fuzzy information aggregation system to support the construction of intuitionistic fuzzy decision-making method. It provides a reference for the generalized operators of interval-valued intuitionistic fuzzy sets, Pythagorean fuzzy sets, and so on.

Definition 22

([16]) Let $\vartheta_{i} = \left( {\rho_{{\alpha_{i} }} ,\sigma_{{\alpha_{i} }} } \right)\;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of IFNs, $w_{i} \;\left( {i = 1,2, \ldots ,n} \right)$ is the weights of $\alpha_{i} \;\left( {i = 1,2, \ldots ,n} \right)$, and $w_{i} \; \in \left[ {0,1} \right]\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {w_{n} = 1\;}$. Then,

• (1) the Archimedean intuitionistic fuzzy weighted average operator is defined as follows:

$${\text{Arch}} - IFWA\left( {\vartheta_{1} ,\vartheta_{2} , \ldots ,\vartheta_{n} } \right) = \left( {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\rho_{i} } \right)} } \right),g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\sigma_{i} } \right)} } \right)} \right),$$

• (2) the Archimedean intuitionistic fuzzy weighted geometric operator is defined as follows:

$${\text{Arch}} - IFWG\left( {\vartheta_{1} ,\vartheta_{2} , \ldots ,\vartheta_{n} } \right) = \left( {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\rho_{i} } \right)} } \right),h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\sigma_{i} } \right)} } \right)} \right).$$

Inspired by the Archimedean intuitionistic fuzzy aggregation operators, the Archimedean fuzzy aggregation operator is proposed based on the Pythagorean fuzzy operational rules in the previous section.

Definition 23

([15]) Let $\alpha_{i} = \left( {\mu_{{\alpha_{i} }} ,\nu_{{\alpha_{i} }} } \right)\;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of PFNs, $w_{i} \;\left( {i = 1,2, \ldots ,n} \right)$ is the weights of $\alpha_{i} \;\left( {i = 1,2, \ldots ,n} \right)$, and $w_{i} \; \in \left[ {0,1} \right]\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {w_{n} = 1\;}$. Then,

• (1) the Archimedean Pythagorean fuzzy weighted average operator is defined as follows:

$${\text{Arch}} - PFWA\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right) = \left( {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i} } \right)} } \right),g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i} } \right)} } \right)} \right).$$

• (2) the Archimedean Pythagorean fuzzy weighted geometric operator is defined as follows:

$${\text{Arch}} - PFWG\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right) = \left( {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {\mu_{i} } \right)} } \right),h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {v_{i} } \right)} } \right)} \right).$$

Based on the mapping relation $\wp_{P \to I}$ in Sect. 3.1, the relationship between two kinds of fuzzy aggregation operators is studied

Theorem 6

Let $\alpha_{i} = \left\langle {\mu_{i} ,\nu_{i} } \right\rangle_{P} \in A_{P} \left( {i = 1,2, \ldots ,n} \right)$ be a collection of PFNs, $\wp_{P \to I} \left( {\alpha_{i} } \right) = \left( {\phi \left( {\mu_{i} } \right),\phi \left( {\nu_{i} } \right)} \right)_{I} \in A_{I}$ be a collection of IFNs, then

• (1) $\wp_{P \to I} \left( {Arch - PFWA\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right)} \right) = Arch - IFWA\left( \wp_{P \to I} \left( {\alpha_{1} } \right),\right.$$\left. \wp_{P \to I} \left( {\alpha_{2} } \right), \ldots ,\wp_{P \to I} \left( {\alpha_{n} } \right) \right)$.

• (2) $\wp_{P \to I} \left( {Arch - PFWG\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right)} \right) = Arch - IFWG\left( \wp_{P \to I} \left( {\alpha_{1} } \right),\right.$$\left. \wp_{P \to I} \left( {\alpha_{2} } \right), \ldots ,\wp_{P \to I} \left( {\alpha_{n} } \right) \right)$.

From Theorem 6, the Archimedean Pythagorean and intuitionistic fuzzy aggregation operators are isomorphic with respect to the mapping $\wp_{P \to I}$.

4.2 Aggregation Operators Isomorphism Between IVPFSs and IVIFSs

In recent years, based on the classical Algebraic operations of interval-valued intuitionistic fuzzy sets, scholars extended Eistein, Hamacher, and Frank t-norm and s-norm to the environment of interval-valued intuitionistic fuzzy sets [19,20,21]. For convenience, these results are expressed in a general form as follows.

Definition 24

Let $\tilde{\vartheta }_{i} = \left( {\left[ {\rho_{i}^{ - } ,\rho_{i}^{ + } } \right],\left[ {\sigma_{i}^{ - } ,\sigma_{i}^{ + } } \right]} \right)\;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVIFNs, $w_{i} \;\left( {i = 1,2, \ldots n} \right)$ is the weights of $\tilde{\alpha }_{i} \;\left( {i = 1,2, \ldots ,n} \right)$, and $w_{i} \; \in \left[ {0,1} \right]\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {w_{n} = 1\;}$. Then,

• (1) the Archimedean interval-valued intuitionistic fuzzy weighted average operator is defined as follows:

$$\begin{aligned} &{\text{Arch-IV}}IFWA\left( {\tilde{\vartheta }_{1} ,\tilde{\vartheta }_{2} , \ldots ,\tilde{\vartheta }_{n} } \right) \\ & = \left( {\left[ {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\rho_{i}^{ - } } \right)} } \right),h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\rho_{i}^{ + } } \right)} } \right)} \right],} \right. \\ & \,\,\left. { \;\;\;\;\left[ {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\sigma_{i}^{ - } } \right)} } \right),g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\sigma_{i}^{ + } } \right)} } \right)} \right]} \right) \\ \end{aligned}$$

• (2) the Archimedean interval-valued intuitionistic fuzzy weighted geometric operator is defined as follows:

$$\begin{aligned} & {\text{Arch-IV}}IFWG\left( {\tilde{\vartheta }_{1} ,\tilde{\vartheta }_{2} , \ldots ,\tilde{\vartheta }_{n} } \right) \\ & = \left( {\left[ {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\rho_{i}^{ - } } \right)} } \right),g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\rho_{i}^{ + } } \right)} } \right)} \right],} \right. \\ & \left. { \;\;\;\;\left[ {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\sigma_{i}^{ - } } \right)} } \right),h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\sigma_{i}^{ + } } \right)} } \right)} \right]} \right) \\ \end{aligned}.$$

The Archimedean interval-valued Pythagorean fuzzy aggregation operators are proposed to aggregate IVPFNs based on the operations.

Definition 25

([22]) Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)\;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVPFNs, $w_{i} \;\left( {i = 1,2, \ldots ,n} \right)$ is the weights of $\tilde{\alpha }_{i} \;\left( {i = 1,2, \ldots ,n} \right)$, and $w_{i} \; \in \left[ {0,1} \right]\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {w_{n} = 1\;}$. Then,

• (1) the Archimedean interval-valued Pythagorean fuzzy weighted average operator is defined as follows:

$$\begin{aligned} & {\text{Arch-IV}}PFWA\left( {\tilde{\alpha }_{1} ,\tilde{\alpha }_{2} , \ldots ,\tilde{\alpha }_{n} } \right) \\ & = \left( {\left[ {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i}^{ - } } \right)} } \right),h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i}^{ + } } \right)} } \right)} \right],} \right. \\ & \left. { \;\;\;\;\left[ {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i}^{ - } } \right)} } \right),g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i}^{ + } } \right)} } \right)} \right]} \right) \\ \end{aligned}.$$

• (2) the Archimedean interval-valued Pythagorean fuzzy weighted geometric operator is defined as follows:

$$\begin{aligned} & {\text{Arch-IV}}PFWG\left( {\tilde{\alpha }_{1} ,\tilde{\alpha }_{2} , \ldots ,\tilde{\alpha }_{n} } \right) \\ & \; = \left( {\left[ {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {\mu_{i}^{ - } } \right)} } \right),g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {\mu_{i}^{ + } } \right)} } \right)} \right],} \right. \\ & \left. { \;\;\;\;\;\left[ {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {v_{i}^{ - } } \right)} } \right),h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {v_{i}^{ + } } \right)} } \right)} \right]} \right) \\ \end{aligned}.$$

Based on the mapping relation $\tilde{\wp }_{P \to I}$ in Sect. 3.2, the relationship between two kinds of fuzzy aggregation operators is studied.

Theorem 7

Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)_{P} \in \tilde{A}_{P} \left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVPFNs, $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right) = \left( {\left[ {\phi \left( {\mu_{i}^{ - } } \right),\phi \left( {\mu_{i}^{ + } } \right)} \right],\left[ {\phi \left( {v_{i}^{ - } } \right),\phi \left( {v_{i}^{ + } } \right)} \right]} \right)_{I} \in \tilde{A}_{I} \left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVIFNs, then

• (1)
  $$\begin{aligned} & \tilde{\wp }_{P \to I} \left( {Arch - IVPFWA\left( {\tilde{\alpha }_{1} ,\tilde{\alpha }_{2} , \ldots ,\tilde{\alpha }_{n} } \right)} \right) \\ & \;\;\; = Arch - IVIFWA\left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right),\wp_{P \to I} \left( {\tilde{\alpha }_{2} } \right), \ldots ,\wp_{P \to I} \left( {\tilde{\alpha }_{n} } \right)} \right) \\ \end{aligned}$$
• (2)
  $$\begin{aligned} & \tilde{\wp }_{P \to I} \left( {Arch - IVPFWA\left( {\tilde{\alpha }_{1} ,\tilde{\alpha }_{2} , \ldots ,\tilde{\alpha }_{n} } \right)} \right) \\ & \;\;\;\; = Arch - IVIFWA\left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right),\wp_{P \to I} \left( {\tilde{\alpha }_{2} } \right), \ldots ,\wp_{P \to I} \left( {\tilde{\alpha }_{n} } \right)} \right) \\ \end{aligned}.$$

From Theorem 7, the Archimedean interval-valued Pythagorean and intuitionistic fuzzy aggregation operators are isomorphic with respect to the mapping $\tilde{\wp }_{P \to I}$.

4.3 Aggregation Operators Isomorphism Between HPFSs and DHFSs

Based on the Archimedean dual hesitant fuzzy operations in the previous section, some series of Archimedean dual hesitant fuzzy power weighted aggregation operators are constructed. In this study, the degenerate operator Archimedes dual hesitant fuzzy weighted aggregation operators are mainly studied.

Definition 26

([23]) Let $\delta_{i} = \left( {p_{i} ,q_{i} } \right)\;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVIFNs, $w_{i} \;\left( {i = 1,2, \ldots ,n} \right)$ is the weights of $\delta_{i} \;\left( {i = 1,2, \ldots ,n} \right)$, and $w_{i} \; \in \left[ {0,1} \right]\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {w_{n} = 1\;}$. Then,

• (1) the Archimedean dual hesitant fuzzy weighted average operator is defined as follows:

$$\begin{aligned} & {\text{Arch}} - DHFWA\left( {\delta_{1} ,\delta_{2} , \ldots ,\delta_{n} } \right) \\ & \; = \cup_{{\gamma_{i} \in p_{i} ,\eta_{i} \in q_{i} }} \left\{ {\left\{ {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\gamma_{i} } \right)} } \right)} \right\},\left\{ {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\eta_{i} } \right)} } \right)} \right\}} \right\} \\ \end{aligned}.$$

• (2) the Archimedean dual hesitant fuzzy weighted geometric operator is defined as follows:

$$\begin{aligned} & {\text{Arch}} - DHFWG\left( {\delta_{1} ,\delta_{2} , \ldots ,\delta_{n} } \right) \\ & \; = \cup_{{\gamma_{i} \in p_{i} ,\eta_{i} \in q_{i} }} \left\{ {\left\{ {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\gamma_{i} } \right)} } \right)} \right\},\left\{ {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\eta_{i} } \right)} } \right)} \right\}} \right\} \\ \end{aligned}.$$

The Archimedean dual hesitant Pythagorean fuzzy aggregation operators are proposed to aggregate DHPFNs based on the operations.

Definition 27

([24]) Let $\psi_{i} = \left( {a_{i} ,b_{i} } \right)_{P} \in \hat{A}_{P} \;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVPFNs, $w_{i} \;\left( {i = 1,2, \ldots ,n} \right)$ is the weights of $\psi_{i} \left( {i = 1,2, \ldots ,n} \right)$, and $w_{i} \; \in \left[ {0,1} \right]\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {w_{n} = 1\;}$. Then,

• (1) the Archimedean dual hesitant Pythagorean fuzzy weighted average operator is defined as follows:

$$\begin{aligned} & {\text{Arch}} - {\text{DH}}PFWA\left( {\psi_{1} ,\psi_{2} , \ldots ,\psi_{n} } \right) \\ & \; = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\tau_{i} } \right)} } \right)} \right\},\left\{ {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {\varsigma_{i} } \right)} } \right)} \right\}} \right\} \\ \end{aligned}.$$

• (2) the Archimedean dual hesitant Pythagorean fuzzy weighted geometric operator is defined as follows:

$$\begin{aligned} & {\text{Arch}} - {\text{DH}}PFWG\left( {\psi_{1} ,\psi_{2} , \ldots ,\psi_{n} } \right) \\ & \; = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {\tau_{i} } \right)} } \right)} \right\},\left\{ {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\varsigma_{i} } \right)} } \right)} \right\}} \right\} \\ \end{aligned}.$$

Based on the mapping relation $\hat{\wp }_{P \to I}$ in Sect. 3.3, the relationship between two kinds of fuzzy aggregation operators is studied.

Theorem 8

Let $\psi_{i} = \left( {a_{i} ,b_{i} } \right)_{P} \in \hat{A}_{P} \;\left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVPFNs, $\hat{\wp }_{P \to I} \left( {\psi_{i} } \right) = \left( {\hat{\phi }\left( {a_{i} } \right),\hat{\phi }\left( {b_{i} } \right)} \right)_{I} \in \hat{A}_{I} \left( {i = 1,2, \ldots ,n} \right)$ be a collection of IVIFNs, then

• (1)
  $$\begin{aligned} & \hat{\wp }_{P \to I} \left( {Arch - DHPFWA\left( {\psi_{1} ,\psi_{2} , \ldots ,\psi_{n} } \right)} \right) \\ & \; = Arch - DHFWA\left( {\hat{\wp }_{P \to I} \left( {\psi_{1} } \right),\hat{\wp }_{P \to I} \left( {\psi_{2} } \right), \ldots ,\hat{\wp }_{P \to I} \left( {\psi_{n} } \right)} \right) \\ \end{aligned}$$
• (2)
  $$\begin{aligned} & \hat{\wp }_{P \to I} \left( {Arch - DHPFWG\left( {\psi_{1} ,\psi_{2} , \ldots ,\psi_{n} } \right)} \right) \\ & = Arch - DHFWG\left( {\hat{\wp }_{P \to I} \left( {\psi_{1} } \right),\hat{\wp }_{P \to I} \left( {\psi_{2} } \right), \ldots ,\hat{\wp }_{P \to I} \left( {\psi_{n} } \right)} \right) \\ \end{aligned}.$$

From Theorem 8, the Archimedean dual hesitant Pythagorean and dual hesitant fuzzy aggregation operators are isomorphic with respect to the mapping $\hat{\wp }_{P \to I}$.

The effectiveness of isomorphic operators is verified and illustrated by aggregating the evaluation information of Pythagorean fuzzy multi-attribute decision-making problem.

Example 1

Let $X_{i} \left( {i = 1,2} \right)$ be two alternatives, $C_{j} \left( {j = 1,2,3,4} \right)$ be four attributes, and $w_{j} = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}$ is the weight of $C_{j} \left( {j = 1,2,3,4} \right)$. Suppose the Pythagorean fuzzy decision matrix $D = \left( {\alpha_{ij} } \right)_{2 \times 4} = \left( {\left( {\mu_{ij} ,v_{ij} } \right)} \right)_{2 \times 4}$ is provided in Table 2.

Table 2 Pythagorean fuzzy decision matrix $D$

We use the Pythagorean fuzzy weighted average (PFWA) operator [6] to aggregation decision information $\alpha_{ij} \left( {j = 1,2,3,4} \right)$ for alternatives $X_{i} \left( {i = 1,2} \right)$, and obtain the comprehensive evaluation information $\alpha_{i} \left( {i = 1,2} \right)$:

$$\begin{aligned} \alpha_{1} & = PFWA\left( {\alpha_{11} ,\alpha_{12} ,\alpha_{13} ,\alpha_{14} } \right) = \left( {\sqrt {1{ - }\prod\nolimits_{j = 1}^{4} {\left( {1 - \mu_{1j}^{2} } \right)^{{w_{j} }} } } ,\prod\nolimits_{j = 1}^{4} {v_{1j}^{{w_{j} }} } } \right) \\ & = \left( {0.3375,0.3130} \right) \\ \end{aligned}$$

$$\begin{aligned} & \alpha_{2} = PFWA\left( {\alpha_{21} ,\alpha_{22} ,\alpha_{23} ,\alpha_{24} } \right) = \left( {\sqrt {1{ - }\prod\nolimits_{j = 1}^{4} {\left( {1 - \mu_{2j}^{2} } \right)^{{w_{j} }} } } ,\prod\nolimits_{j = 1}^{4} {v_{2j}^{{w_{j} }} } } \right) \\ & = \left( {0.5715,0.5384} \right) \\ \end{aligned}.$$

Based on the mapping relation $\wp_{P \to I}$, we obtain the intuitionistic fuzzy decision matrix $D_{I} = \left( {\alpha_{ij}^{'} } \right)_{2 \times 4}$, which is shown in Table 3. We use the intuitionistic fuzzy weighted average (IFWA) operator [31] to aggregation decision information $\alpha_{ij}^{'} \left( {j = 1,2,3,4} \right)$ for alternatives $X_{i} \left( {i = 1,2} \right)$, and obtain the comprehensive evaluation information $\alpha_{i}^{'} \left( {i = 1,2} \right).$

Table 3 Intuitionistic fuzzy decision matrix $D_{I}$ with mapping relation $\wp_{P \to I}$

$$\begin{aligned} \alpha_{1}^{'} & = IFWA\left( {\alpha_{1}^{'} ,\alpha_{12}^{'} ,\alpha_{13}^{'} ,\alpha_{14}^{'} } \right) \\ & = IFWA\left( {\hat{\wp }_{P \to I} \left( {\alpha_{11} } \right),\hat{\wp }_{P \to I} \left( {\alpha_{12} } \right),\hat{\wp }_{P \to I} \left( {\alpha_{13} } \right),\hat{\wp }_{P \to I} \left( {\alpha_{14} } \right)} \right) \\ & = \left( {1{ - }\prod\nolimits_{j = 1}^{4} {\left( {1 - \rho_{1j} } \right)^{{w_{j} }} } ,\prod\nolimits_{j = 1}^{4} {\sigma_{1j}^{{w_{j} }} } } \right) = \hat{\wp }_{P \to I} \left( {\alpha_{1} } \right) = \left( {0.1139,0.0980} \right) \\ \end{aligned}$$

$$\begin{aligned} \alpha_{2}^{'} & = IFWA\left( {\alpha_{21}^{'} ,\alpha_{22}^{'} ,\alpha_{23}^{'} ,\alpha_{24}^{'} } \right)\\ & = IFWA\left( {\hat{\wp }_{P \to I} \left( {\alpha_{21} } \right),\hat{\wp }_{P \to I} \left( {\alpha_{22} } \right),\hat{\wp }_{P \to I} \left( {\alpha_{23} } \right),\hat{\wp }_{P \to I} \left( {\alpha_{24} } \right)} \right) \\ & = \left( {1{ - }\prod\nolimits_{j = 1}^{4} {\left( {1 - \rho_{1j} } \right)^{{w_{j} }} } ,\prod\nolimits_{j = 1}^{4} {\sigma_{1j}^{{w_{j} }} } } \right) = \hat{\wp }_{P \to I} \left( {\alpha_{2} } \right){ = }\left( {0.3266,0.2898} \right) \\ \end{aligned}.$$

Example 1 shows that isomorphism operators can transform Pythagorean fuzzy multi-attribute decision-making (FMADM) problem into intuitionistic FMADM problem, and the aggregation result obtained is consistent with that before transformation. Similarly, the hesitant Pythagorean FMADM problem and interval-valued Pythagorean FMADM problem can be transformed into dual hesitant FMADM problem and interval-valued intuitionistic FMADM problem, respectively.

5 Ranking Methods Isomorphism

For fuzzy multi-attribute decision-making approach, the ranking method of fuzzy numbers is used to sort the comprehensive attribute values to determine the evaluation results. The ranking method of three groups of research objects in this study is mainly based on score function and accuracy function. Based on the isomorphic perspective, this section will study the relationship between three sets of fuzzy sets.

5.1 Ranking Methods Isomorphism Between IFNs and PFNs

The score function and accuracy function of intuitionistic fuzzy sets constitute the sorting method of intuitionistic fuzzy numbers and also lay a foundation for the sorting methods of other fuzzy sets.

Definition 28

([25]) For any two IFNs, $\vartheta_{i} = \left( {\rho_{i} ,\sigma_{i} } \right)\left( {i = 1,2} \right)$:

• (i) if $sco_{I} \left( {\vartheta_{1} } \right) < sco_{I} \left( {\vartheta_{2} } \right)$, then $\vartheta_{1} <_{I} \vartheta_{2}$;

• (ii) if $sco_{I} \left( {\vartheta_{1} } \right) = sco_{I} \left( {\vartheta_{2} } \right)$, then

  • (a) if $acc_{I} \left( {\vartheta_{1} } \right) < acc_{I} \left( {\vartheta_{2} } \right),$ then $\vartheta_{1} <_{I} \vartheta_{2}$;

  • (b) if $acc_{I} \left( {\vartheta_{1} } \right) = acc_{I} \left( {\vartheta_{2} } \right),$ then $\vartheta_{1} = \vartheta_{2}$,

where $sco_{I} \left( {\vartheta_{i} } \right) = \rho_{i} - \sigma_{i}$ and $acc_{I} \left( {\vartheta_{i} } \right) = \rho_{i} + \sigma_{i} \left( {i = 1,2} \right)$ are the score functions and accuracy functions, respectively.

Based on the score function and accuracy function of intuitionistic fuzzy sets, scholars put forward the score function and accuracy function of the Pythagorean fuzzy sets, and then constructed the sorting method for ranking Pythagorean fuzzy numbers.

Definition 29

([26, 27]) For any two PFNs $\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2} \right)$:

• (i) if $sco_{P} \left( {\alpha_{1} } \right) < sco_{P} \left( {\alpha_{2} } \right)$, then $\alpha_{1} <_{p} \alpha_{2}$;

• (ii) if $sco_{P} \left( {\alpha_{1} } \right) = sco_{P} \left( {\alpha_{2} } \right)$, then

  • (a) if $acc_{P} \left( {\alpha_{1} } \right) < acc_{P} \left( {\alpha_{2} } \right),$ then $\alpha_{1} <_{p} \alpha_{2}$;

  • (b) if $acc_{P} \left( {\alpha_{1} } \right) = acc_{P} \left( {\alpha_{2} } \right),$ then $\alpha_{1} \sim_{p} \alpha_{2}$,

where $sco_{P} \left( {\alpha_{i} } \right) = \mu_{i}^{2} - v_{i}^{2} \left( {i = 1,2} \right)$ are the score functions of $\alpha_{i} \left( {i = 1,2} \right)$, and $acc_{P} \left( {\alpha_{i} } \right) = \mu_{i}^{2} + v_{i}^{2} \left( {i = 1,2} \right)$ are the accuracy functions.

Based on the mapping relation $\wp_{P \to I}$ in Sect. 3.1, the relationship between two kinds of fuzzy ranking methods is studied.

Theorem 9

Let $\alpha_{i} = \left\langle {\mu_{i} ,\nu_{i} } \right\rangle_{P} \in A_{P}$, $\wp_{P \to I} \left( {\alpha_{i} } \right) = \left( {\phi \left( {\mu_{i} } \right),\phi \left( {\nu_{i} } \right)} \right)_{I} \in A_{I}$, then we have

• (1) $sco_{P} \left( {\alpha_{i} } \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right)$;

• (2) $acc_{P} \left( {\alpha_{i} } \right) = acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right)$.

Theorem 10

Let $\alpha_{i} = \left\langle {\mu_{i} ,\nu_{i} } \right\rangle_{P} \in A_{P}$, $\wp_{P \to I} \left( {\alpha_{i} } \right) = \left( {\phi \left( {\mu_{i} } \right),\phi \left( {\nu_{i} } \right)} \right)_{I} \in A_{I}$. Then,

$\alpha_{1} <_{p} \alpha_{2}$ if and only if $\wp_{P \to I} \left( {\alpha_{1} } \right) <_{I} \wp_{P \to I} \left( {\alpha_{2} } \right)$

and

$\alpha_{1} \sim_{p} \alpha_{2}$ if and only if $\wp_{P \to I} \left( {\alpha_{1} } \right)\sim_{I} \wp_{P \to I} \left( {\alpha_{2} } \right).$

Theorem 10 reveals IFSs and PFSs have a ranking method isomorphism.

5.2 Ranking Methods Isomorphism Between IVIFNs and IVPFNs

The ranking methods of interval intuitionistic fuzzy numbers and interval Pythagorean fuzzy numbers are mainly defined by using score function and accuracy function. In recent years, scholars have proposed a series of scoring functions and accuracy functions of interval-valued Pythagorean fuzzy numbers (IVPFNs) from different perspectives, and used to construct corresponding decision analysis methods. For example, fuzzy linear programming model [34] and TOPSIS method [35] based on improved score function, decision analysis methods based on improved accuracy function [36, 37]. Considering the generality, this paper mainly analyzes the traditional score function and accuracy function of IVPFNs.

Definition 30

([28]) For any two IFNs, $\tilde{\vartheta }_{i} = \left( {\left[ {\rho_{i}^{ - } ,\rho_{i}^{ + } } \right],\left[ {\sigma_{i}^{ - } ,\sigma_{i}^{ + } } \right]} \right)\left( {i = 1,2} \right)$:

• (i) if $Sco_{I} \left( {\tilde{\vartheta }_{1} } \right) < Sco_{I} \left( {\tilde{\vartheta }_{2} } \right)$, then $\tilde{\vartheta }_{1} <_{{\tilde{I}}} \tilde{\vartheta }_{2}$;

• (ii) if $Sco_{I} \left( {\tilde{\vartheta }_{1} } \right) = Sco_{I} \left( {\tilde{\vartheta }_{2} } \right)$, then

  • (a) if $Acc_{I} \left( {\tilde{\vartheta }_{1} } \right) < Acc_{I} \left( {\tilde{\vartheta }_{2} } \right),$ then $\tilde{\vartheta }_{1} <_{{\tilde{I}}} \tilde{\vartheta }_{2}$;

  • (b) if $Acc_{I} \left( {\tilde{\vartheta }_{1} } \right) = Acc_{I} \left( {\tilde{\vartheta }_{2} } \right),$ then $\tilde{\vartheta }_{1} \sim_{{\tilde{I}}} \tilde{\vartheta }_{2}$,

where $Sco_{I} \left( {\tilde{\vartheta }_{i} } \right) = {{\left( {\rho_{i}^{ - } { + }\rho_{i}^{ + } - \sigma_{i}^{ - } - \sigma_{i}^{ + } } \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{i}^{ - } { + }\rho_{i}^{ + } - \sigma_{i}^{ - } - \sigma_{i}^{ + } } \right)} 2}} \right. \kern-0pt} 2}$ and $Acc_{I} \left( {\tilde{\vartheta }_{i} } \right) = {{\left( {\rho_{i}^{ - } { + }\rho_{i}^{ + } + \sigma_{i}^{ - } + \sigma_{i}^{ + } } \right)} \mathord{\left/ {\vphantom {{\left( {\rho_{i}^{ - } { + }\rho_{i}^{ + } + \sigma_{i}^{ - } + \sigma_{i}^{ + } } \right)} 2}} \right. \kern-0pt} 2}\left( {i = 1,2} \right)$ are the score functions and accuracy functions, respectively.

Definition 31

([7, 29]). For any two IVPFNs $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)\left( {i = 1,2} \right)$:

• (i) if $Sco_{P} \left( {\tilde{\alpha }_{1} } \right) < Sco_{P} \left( {\tilde{\alpha }_{2} } \right)$, then $\tilde{\alpha }_{1} <_{{\tilde{P}}} \tilde{\alpha }_{2}$;

• (ii) if $Sco_{P} \left( {\tilde{\alpha }_{1} } \right) = Sco_{P} \left( {\tilde{\alpha }_{2} } \right)$, then

  • (a) if $Acc_{P} \left( {\tilde{\alpha }_{1} } \right) < Acc_{P} \left( {\tilde{\alpha }_{2} } \right),$ then $\tilde{\alpha }_{1} <_{{\tilde{P}}} \tilde{\alpha }_{2}$;

  • (b) if $Acc_{P} \left( {\tilde{\alpha }_{1} } \right) = Acc_{P} \left( {\tilde{\alpha }_{2} } \right),$ then $\tilde{\alpha }_{1} \sim_{{\tilde{P}}} \tilde{\alpha }_{2}$,

where $Sco_{P} \left( {\tilde{\alpha }_{i} } \right) = {{\left( {\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} - \left( {v_{i}^{ - } } \right)^{2} - \left( {v_{i}^{ + } } \right)^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} - \left( {v_{i}^{ - } } \right)^{2} - \left( {v_{i}^{ + } } \right)^{2} } \right)} 2}} \right. \kern-0pt} 2}$ and $Acc_{P} \left( {\tilde{\alpha }_{i} } \right) = {{\left( {\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} { + }\left( {v_{i}^{ - } } \right)^{2} { + }\left( {v_{i}^{ + } } \right)^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} { + }\left( {v_{i}^{ - } } \right)^{2} { + }\left( {v_{i}^{ + } } \right)^{2} } \right)} 2}} \right. \kern-0pt} 2}$ are the score functions and accuracy functions, respectively.

Based on the mapping relation $\hat{\wp }_{P \to I}$ in Sect. 3.2, the relationship between two kinds of fuzzy ranking methods is studied.

Theorem 11

Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)_{P} \in \tilde{A}_{P}$, $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right) = \left( {\left[ {\phi \left( {\mu_{i}^{ - } } \right),\phi \left( {\mu_{i}^{ + } } \right)} \right],\left[ {\phi \left( {v_{i}^{ - } } \right),\phi \left( {v_{i}^{ + } } \right)} \right]} \right)_{I} \in \tilde{A}_{I}$, then

• (1) $Sco_{P} \left( {\tilde{\alpha }_{i} } \right) = Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right)} \right)$;

• (2)
  $$Acc_{P} \left( {\tilde{\alpha }_{i} } \right) = Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right)} \right).$$

Theorem 12

Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)_{P} \in \tilde{A}_{P}$, $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right) = \left( {\left[ {\phi \left( {\mu_{i}^{ - } } \right),\phi \left( {\mu_{i}^{ + } } \right)} \right],\left[ {\phi \left( {v_{i}^{ - } } \right),\phi \left( {v_{i}^{ + } } \right)} \right]} \right)_{I} \in \tilde{A}_{I}$. Then,

• (1) $\tilde{\alpha }_{1} <_{{\tilde{P}}} \tilde{\alpha }_{2}$ if and only if $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right) <_{{\tilde{I}}} \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)$;

• (2) $\tilde{\alpha }_{1} \sim_{{\tilde{P}}} \tilde{\alpha }_{2}$ if and only if $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)\sim_{{\tilde{I}}} \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)$;

Theorem 12 reveals IVIFSs and IVPFSs have a ranking method isomorphism.

5.3 Ranking Methods Isomorphism Between DHFSs and DHPFSs

The score function and accuracy function of DHFSs and DHPFSs are constructed mainly from the perspective of the expected mean of membership degree values and nonmembership degree values.

Definition 32

([30]) For any two DHFNs, $\delta_{i} = \left( {p_{i} ,q_{i} } \right)\;\left( {i = 1,2} \right)$:

• (i) if $\hat{S}co_{I} \left( {\delta_{1} } \right) < \hat{S}co_{I} \left( {\delta_{2} } \right)$, then $\delta_{1} <_{{\hat{I}}} \delta_{2}$;

• (ii) if $\hat{S}co_{I} \left( {\delta_{1} } \right) = \hat{S}co_{I} \left( {\delta_{2} } \right)$, then

  • (a) if $\hat{A}cc_{I} \left( {\delta_{1} } \right) < \hat{A}cc_{I} \left( {\delta_{2} } \right),$ then $\delta_{1} <_{{\hat{I}}} \delta_{2}$;

  • (b) if $\hat{A}cc_{I} \left( {\delta_{1} } \right) = \hat{A}cc_{I} \left( {\delta_{2} } \right),$ then $\delta_{1} \sim_{{\hat{I}}} \delta_{2}$,

where $\hat{S}co_{I} \left( {\delta_{i} } \right) = \frac{1}{{\# p_{i} }}\sum\nolimits_{{\gamma_{i} \in p_{i} }} {\gamma_{i} } - \;\frac{1}{{\# q_{i} }}\sum\nolimits_{{\eta_{i} \in q_{i} }} {\eta_{i} }$ and $\hat{A}cc_{I} \left( {\delta_{i} } \right) = \frac{1}{{\# p_{i} }}\sum\nolimits_{{\gamma_{i} \in p_{i} }} {\gamma_{i} } { + }\;\frac{1}{{\# q_{i} }}\sum\nolimits_{{\eta_{i} \in q_{i} }} {\eta_{i} } \left( {i = 1,2} \right)$ are the score functions and accuracy functions, respectively.

Definition 33

([9]) For any two HPFNs, $\psi_{i} = \left( {a_{i} ,b_{i} } \right)\;\left( {i = 1,2} \right)$:

• (i) if $\hat{S}co_{P} \left( {\psi_{1} } \right) < \hat{S}co_{P} \left( {\psi_{2} } \right)$, then $\psi_{1} <_{{\hat{P}}} \psi_{2}$;

• (ii) if $\hat{S}co_{P} \left( {\psi_{1} } \right) = \hat{S}co_{P} \left( {\psi_{2} } \right)$, then

  • (a) if $\hat{A}cc_{P} \left( {\psi_{1} } \right) < \hat{A}co_{P} \left( {\psi_{2} } \right),$ then $\psi_{1} <_{{\hat{P}}} \psi_{2}$;

  • (b) if $\hat{A}cc_{P} \left( {\psi_{1} } \right) = \hat{A}co_{P} \left( {\psi_{2} } \right),$ then $\psi_{1} \sim_{{\hat{P}}} \psi_{2}$,

where $\hat{S}co_{P} \left( {\psi_{i} } \right) = \frac{1}{{\# a_{i} }}\sum\nolimits_{{\tau \in a_{i} }} {\tau_{i} } - \;\frac{1}{{\# b_{i} }}\sum\nolimits_{{\varsigma_{i} \in b_{i} }} {\varsigma_{i} }$ and $\hat{A}cc_{P} \left( {\psi_{i} } \right) = \frac{1}{{\# a_{i} }}\sum\nolimits_{{\tau \in a_{i} }} {\tau_{i} } { + }\;\frac{1}{{\# b_{i} }}\sum\nolimits_{{\varsigma_{i} \in b_{i} }} {\varsigma_{i} } \left( {i = 1,2} \right)$ are the score functions and accuracy functions, respectively.

Based on the mapping relation $\hat{\wp }_{P \to I}$ in Sect. 3.3, the relationship between two kinds of fuzzy ranking methods is studied.

Theorem 13

Let $\psi_{i} = \left( {a_{i} ,b_{i} } \right)_{P} \in \hat{A}_{P}$, $\hat{\wp }_{P \to I} \left( {\psi_{i} } \right) = \left( {\hat{\phi }\left( {a_{i} } \right),\hat{\phi }\left( {b_{i} } \right)} \right)_{I} \in \hat{A}_{I}$, then we have

• (1) $\hat{S}co_{P} \left( {\psi_{i} } \right) = \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)} \right)$;

• (2)
  $$\hat{A}cc_{P} \left( {\psi_{i} } \right) = \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)} \right)$$

Theorem 14

Let $\tilde{\alpha }_{i} = \left( {\left[ {\mu_{i}^{ - } ,\mu_{i}^{ + } } \right],\left[ {v_{i}^{ - } ,v_{i}^{ + } } \right]} \right)_{P} \in \tilde{A}_{P}$, $\hat{\wp }_{P \to I} \left( {\psi_{i} } \right) = \left( {\hat{\phi }\left( {a_{i} } \right),\hat{\phi }\left( {b_{i} } \right)} \right)_{I} \in \hat{A}_{I}$. Then,

• (1) $\psi_{1} <_{{\hat{P}}} \psi_{2}$ if and only if $\hat{\wp }_{P \to I} \left( {\psi_{1} } \right) <_{{\tilde{I}}} \hat{\wp }_{P \to I} \left( {\psi_{2} } \right)$

• (2) $\psi_{1} \sim_{{\hat{P}}} \psi_{2}$ if and only if $\hat{\wp }_{P \to I} \left( {\psi_{1} } \right)\sim_{{\tilde{I}}} \hat{\wp }_{P \to I} \left( {\psi_{2} } \right).$

Theorem 12 reveals DHFSs and DHPFSs have a ranking method isomorphism.

Example 2

(Continuation Example 1) In this example, the evaluation matrix of Tables 1 and 2 in Example 1 is used. Based on the score functions for Pythagorean and intuitionistic fuzzy numbers, we obtain the score matrix for Tables 1 and 2, which is shown in Table 3. Since $sco_{P} \left( {\alpha_{1j} } \right) < sco_{P} \left( {\alpha_{2j} } \right),j = 1,2,3,4$, then $\alpha_{1j} <_{P} \alpha_{2j} ,j = 1,2,3,4$. Since $sco_{I} \left( {\alpha_{1j}^{'} } \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{1j} } \right)} \right) < sco_{I} \left( {\alpha_{2j}^{'} } \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{2j} } \right)} \right),j = 1,2,3,4$, then $\alpha_{1j}^{'} <_{P} \alpha_{2j}^{'} ,j = 1,2,3,4$. Table 4 shows that the scoring matrix corresponding to Tables 2 and 3 is the same, and the effectiveness of isomorphic ranking method is verified.

Table 4 Score matrix

6 Proofs

Proof of Theorem 2

Proof

Since

$$g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right)$$

and

$$h_{P} \left( t \right) = h_{I} \left( {\phi \left( t \right)} \right) \Rightarrow h_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {h_{I}^{ - 1} \left( t \right)} \right) \Rightarrow h_{I}^{ - 1} \left( t \right) = \phi \left( {h_{P}^{ - 1} \left( t \right)} \right)$$

Then

$$\begin{aligned} S_{P} \left( {x,y} \right) &= h_{P}^{ - 1} \left( {h_{P} \left( x \right) + h_{P} \left( y \right)} \right) = \phi^{ - 1} \left( {h_{I}^{ - 1} \left( {h_{I} \left( {\phi \left( x \right)} \right) + h_{P} \left( {\phi \left( y \right)} \right)} \right)} \right)\\ & \quad= \phi^{ - 1} \left( {S_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right) \end{aligned}$$

and

$$\begin{aligned}T_{P} \left( {x,y} \right) & = g_{P}^{ - 1} \left( {g_{P} \left( x \right) + g_{P} \left( y \right)} \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( {g_{I} \left( {\phi \left( x \right)} \right) + g_{P} \left( {\phi \left( y \right)} \right)} \right)} \right)\\ & \quad = \phi^{ - 1} \left( {T_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right). \end{aligned}$$

Theorem 2 is proven.

Proof of Theorem 3

Proof

Because $\alpha_{i} = \left\langle {\mu_{i} ,\nu_{i} } \right\rangle_{P} \in A_{P}$, we have $\mu_{i}^{2} + \nu_{i}^{2} \le 1$. It means that if $\alpha_{i}$ is a PFN on $A_{P}$, and then $\wp_{P \to I} \left( {\alpha_{i} } \right)$ is an IFN on $A_{I}$.

• (1) Following Definition 14 and Theorem 2, we have

$$\alpha_{1} \oplus \alpha_{2} = \left( {S_{P} \left( {\mu_{1} ,\mu_{2} } \right),T_{P} \left( {v_{1} ,v_{2} } \right)} \right)$$

and

$$S_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {S_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right),\,\,T_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {T_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right)$$

then

$$\begin{aligned} \wp_{P \to I} \left( {\alpha_{1} \oplus \alpha_{2} } \right) = \left( {\phi \left( {S_{P} \left( {\mu_{1} ,\mu_{2} } \right)} \right),\phi \left( {T_{P} \left( {v_{1} ,v_{2} } \right)} \right)} \right) \hfill \\ \, = \left( {S_{I} \left( {\phi \left( {\mu_{1} } \right),\phi \left( {\mu_{2} } \right)} \right),T_{I} \left( {\phi \left( {v_{1} } \right),\phi \left( {v_{2} } \right)} \right)} \right) \hfill \\ \, = \wp_{P \to I} \left( {\alpha_{1} } \right) \oplus \wp_{P \to I} \left( {\alpha_{2} } \right) \hfill \\ \end{aligned}$$

which completes the proof of (1).

• (2) The proof of (2) is similar to (1) and it is hence omitted here.

• (3) Following Definition 14, we have $\lambda \alpha = \left( {h_{p}^{ - 1} \left( {\lambda h_{P} \left( \mu \right)} \right),g_{p}^{ - 1} \left( {\lambda g_{P} \left( v \right)} \right)} \right),\lambda > 0$,

and

$$g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right),$$

then

$$\begin{aligned} \wp_{P \to I} \left( {\lambda \alpha } \right) & = \left( {\phi \left( {h_{P}^{ - 1} \left( {\lambda h_{P} \left( \mu \right)} \right)} \right),\phi \left( {g_{P}^{ - 1} \left( {\lambda g_{P} \left( v \right)} \right)} \right)} \right) \\ & = \left( {h_{I}^{ - 1} \left( {\lambda h_{P} \left( \mu \right)} \right),g_{I}^{ - 1} \left( {\lambda g_{P} \left( v \right)} \right)} \right) \\ & { = }\left( {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( \mu \right)} \right)} \right),g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( v \right)} \right)} \right)} \right) \\ & = \lambda \wp_{P \to I} \left( \alpha \right) \\ \end{aligned},$$

which completes the proof of (3).

• (4) The proof of (4) is similar to (3) and it is hence omitted here.

Theorem 3 is proven.

Proof of Theorem 4

Proof

(1) Following Definition 17 and Theorem 2, we have

$$\tilde{\alpha }_{1} \oplus \tilde{\alpha }_{2} = \left( {\left[ {S_{P} \left( {\mu_{1}^{ - } ,\mu_{2}^{ - } } \right),S_{P} \left( {\mu_{1}^{ + } ,\mu_{2}^{ + } } \right)} \right],\left[ {T_{P} \left( {v_{1}^{ - } ,v_{2}^{ - } } \right),T_{P} \left( {v_{1}^{ + } ,v_{2}^{ + } } \right)} \right]} \right)$$

and

$$S_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {S_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right),\,\,T_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {T_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right),$$

then

$$\begin{aligned} \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} \oplus \tilde{\alpha }_{2} } \right) & = \left( {\left[ {\phi \left( {S_{P} \left( {\mu_{1}^{ - } ,\mu_{2}^{ - } } \right)} \right),\phi \left( {S_{P} \left( {\mu_{1}^{ + } ,\mu_{2}^{ + } } \right)} \right)} \right],\left[ {\phi \left( {T_{P} \left( {v_{1}^{ - } ,v_{2}^{ - } } \right)} \right),\phi \left( {T_{P} \left( {v_{1}^{ + } ,v_{2}^{ + } } \right)} \right)} \right]} \right) \\ & = \left( \begin{aligned} \left[ {\phi \left( {\phi^{ - 1} \left( {S_{I} \left( {\phi \left( {\mu_{1}^{ - } } \right),\phi \left( {\mu_{2}^{ - } } \right)} \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {S_{I} \left( {\phi \left( {\mu_{1}^{ + } } \right),\phi \left( {\mu_{2}^{ + } } \right)} \right)} \right)} \right)} \right], \hfill \\ \left[ {\phi \left( {\phi^{ - 1} \left( {T_{I} \left( {\phi \left( {v_{1}^{ - } } \right),\phi \left( {v_{2}^{ - } } \right)} \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {T_{I} \left( {\phi \left( {v_{1}^{ + } } \right),\phi \left( {v_{2}^{ + } } \right)} \right)} \right)} \right)} \right] \hfill \\ \end{aligned} \right) \\ & = \left( \left[ {S_{I} \left( {\phi \left( {\mu_{1}^{ - } } \right),\phi \left( {\mu_{2}^{ - } } \right)} \right),S_{I} \left( {\phi \left( {\mu_{1}^{ + } } \right),\phi \left( {\mu_{2}^{ + } } \right)} \right)} \right],\right. \\& \qquad \left. \left[ {T_{I} \left( {\phi \left( {v_{1}^{ - } } \right),\phi \left( {v_{2}^{ - } } \right)} \right),T_{I} \left( {\phi \left( {v_{1}^{ + } } \right),\phi \left( {v_{2}^{ + } } \right)} \right)} \right] \right) \\ & = \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right) \oplus \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right) \\ \end{aligned},$$

which completes the proof of (1).

(2) The proof of (2) is similar to (1) and it is hence omitted here.

(3) Following Definitions 16 and 17, we have $\lambda \tilde{\alpha } = \left( {\left[ {h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\mu^{ - } } \right)} \right),h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\mu^{ + } } \right)} \right)} \right],\left[ {g_{P}^{ - 1} \left( {\lambda g_{P} \left( {v^{ - } } \right)} \right),g_{P}^{ - 1} \left( {\lambda g_{P} \left( {v^{ + } } \right)} \right)} \right]} \right),\lambda > 0$,

and

$$g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right)$$

then

$$\begin{aligned} \tilde{\wp }_{P \to I} \left( {\lambda \tilde{\alpha }} \right) & = \left( \left[ {\phi \left( {h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\mu^{ - } } \right)} \right)} \right),\phi \left( {h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\mu^{ + } } \right)} \right)} \right)} \right], \right.\\&\qquad \left.\left[ {\phi \left( {g_{P}^{ - 1} \left( {\lambda g_{P} \left( {v^{ - } } \right)} \right)} \right),\phi \left( {g_{P}^{ - 1} \left( {\lambda g_{P} \left( {v^{ + } } \right)} \right)} \right)} \right] \right) \\ & = \left( \begin{aligned} \left[ {\phi \left( {\phi^{ - 1} \left( {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( {\mu^{ - } } \right)} \right)} \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( {\mu^{ + } } \right)} \right)} \right)} \right)} \right)} \right], \hfill \\ \left[ {\phi \left( {\phi^{ - 1} \left( {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( {v^{ - } } \right)} \right)} \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( {v^{ + } } \right)} \right)} \right)} \right)} \right)} \right] \hfill \\ \end{aligned} \right) \\ & = \left( \left[ {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( {\mu^{ - } } \right)} \right)} \right),h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( {\mu^{ + } } \right)} \right)} \right)} \right], \right. \\& \qquad \left. \left[ {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( {v^{ - } } \right)} \right)} \right),g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( {v^{ + } } \right)} \right)} \right)} \right] \right) \\ & = \lambda \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }} \right) \\ \end{aligned}$$

which completes the proof of (3).

(4) The proof of (4) is similar to (3) and it is hence omitted here.

Theorem 4 is proven.

Proof of Theorem 5

Proof

• (1) Following Definition 20 and Theorem 2, we have

$$\psi_{1} \oplus \psi_{2} = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {S_{P} \left( {\tau_{1} ,\tau_{2} } \right)} \right\},\left\{ {T_{P} \left( {\varsigma_{1} ,\varsigma_{2} } \right)} \right\}} \right\}$$

and

$$S_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {S_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right),\,\,T_{P} \left( {x,y} \right) = \phi^{ - 1} \left( {T_{I} \left( {\phi \left( x \right),\phi \left( y \right)} \right)} \right)$$

then

$$\begin{aligned} \hat{\wp }_{P \to I} \left( {\psi_{1} \oplus \psi_{2} } \right) & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {\phi \left( {S_{P} \left( {\tau_{1} ,\tau_{2} } \right)} \right)} \right\},\left\{ {\phi \left( {T_{P} \left( {\varsigma_{1} ,\varsigma_{2} } \right)} \right)} \right\}} \right\} \\ & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ \left\{ {\phi \left( {\phi^{ - 1} \left( {S_{I} \left( {\phi \left( {\tau_{1} } \right),\phi \left( {\tau_{2} } \right)} \right)} \right)} \right)} \right\},\right.\\&\qquad\qquad\qquad \left. \left\{ {\phi \left( {\phi^{ - 1} \left( {T_{I} \left( {\phi \left( {\varsigma_{1} } \right),\phi \left( {\varsigma_{2} } \right)} \right)} \right)} \right)} \right\} \right\} \\ & = \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {S_{I} \left( {\phi \left( {\tau_{1} } \right),\phi \left( {\tau_{2} } \right)} \right)} \right\},\left\{ {T_{I} \left( {\phi \left( {\varsigma_{1} } \right),\phi \left( {\varsigma_{2} } \right)} \right)} \right\}} \right\} \\ & = \wp_{P \to I} \left( {\alpha_{1} } \right) \oplus \wp_{P \to I} \left( {\alpha_{2} } \right) \\ \end{aligned}$$

which completes the proof of (1).

• (2) The proof of (2) is similar to (1) and it is hence omitted here.

• (3) Following Definitions 19 and 20, we have $\, \lambda \psi = \cup_{\tau \in a,\varsigma \in b} \left\{ {\left\{ {h_{P}^{ - 1} \left( {\lambda h_{P} \left( \tau \right)} \right)} \right\},\left\{ {g_{P}^{ - 1} \left( {\lambda g_{P} \left( \varsigma \right)} \right)} \right\}} \right\}$,

and

$$g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right)$$

then

$$\begin{aligned} \hat{\wp }_{P \to I} \left( {\lambda \psi_{1} } \right) & = \cup_{{\tau_{1} \in a_{1} ,\varsigma_{1} \in b_{1} }} \left\{ {\left\{ {\phi \left( {h_{P}^{ - 1} \left( {\lambda h_{P} \left( {\tau_{1} } \right)} \right)} \right)} \right\},\left\{ {\phi \left( {g_{P}^{ - 1} \left( {\lambda g_{P} \left( {\varsigma_{1} } \right)} \right)} \right)} \right\}} \right\} \\ & = \cup_{{\tau_{1} \in a_{1} ,\varsigma_{1} \in b_{1} }} \left\{ {\left\{ {\phi \left( {\phi^{ - 1} \left( {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( {\tau_{1} } \right)} \right)} \right)} \right)} \right)} \right\},\left\{ {\phi \left( {\phi^{ - 1} \left( {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( {\varsigma_{1} } \right)} \right)} \right)} \right)} \right)} \right\}} \right\} \\ & { = } \cup_{{\tau_{1} \in a_{1} ,\varsigma_{1} \in b_{1} }} \left\{ {\left\{ {h_{I}^{ - 1} \left( {\lambda h_{I} \left( {\phi \left( {\tau_{1} } \right)} \right)} \right)} \right\},\left\{ {g_{I}^{ - 1} \left( {\lambda g_{I} \left( {\phi \left( {\varsigma_{1} } \right)} \right)} \right)} \right\}} \right\} \\ & = \lambda \hat{\wp }_{P \to I} \left( {\psi_{1} } \right) \\ \end{aligned}$$

which completes the proof of (3).

• (4) The proof of (4) is similar to (3) and it is hence omitted here.

Theorem 5 is proven.

Proof of Theorem 6

Proof

• (1) Since

$$g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right)$$

and

$$h_{P} \left( t \right) = h_{I} \left( {\phi \left( t \right)} \right) \Rightarrow h_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {h_{I}^{ - 1} \left( t \right)} \right) \Rightarrow h_{I}^{ - 1} \left( t \right) = \phi \left( {h_{P}^{ - 1} \left( t \right)} \right).$$

Then, according to Definition 21, we have

$$\begin{aligned} & \wp_{P \to I} \left( {Arch - PFWA\left( {\alpha_{1} ,\alpha_{2} ,\ldots,\alpha_{n} } \right)} \right) \\ & = \wp_{P \to I} \left( {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i} } \right)} } \right),g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i} } \right)} } \right)} \right) \\ & { = }\left( {\phi \left( {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i} } \right)} } \right)} \right),\phi \left( {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i} } \right)} } \right)} \right)} \right) \\ & = \left( {\phi \left( {\phi^{ - 1} \left( {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\mu_{i} } \right)} \right)} } \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {v_{i} } \right)} \right)} } \right)} \right)} \right)} \right) \\ & = \left( {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\mu_{i} } \right)} \right)} } \right),g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {v_{i} } \right)} \right)} } \right)} \right) \\ & = Arch - IFWA\left( {\wp_{P \to I} \left( {\alpha_{1} } \right),\wp_{P \to I} \left( {\alpha_{2} } \right),\ldots,\wp_{P \to I} \left( {\alpha_{n} } \right)} \right) \\ \end{aligned}.$$

• (2) The proof of (2) is similar to (1) and it is hence omitted here.

Theorem 6 is proven.

Proof of Theorem 7

Proof

• (1) Since

$$g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right)$$

and

$$h_{P} \left( t \right) = h_{I} \left( {\phi \left( t \right)} \right) \Rightarrow h_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {h_{I}^{ - 1} \left( t \right)} \right) \Rightarrow h_{I}^{ - 1} \left( t \right) = \phi \left( {h_{P}^{ - 1} \left( t \right)} \right).$$

Then, according to Definitions 24 and 25, we have

$$\begin{aligned} & \wp_{P \to I} \left( {Arch - IVPFWA\left( {\tilde{\alpha }_{1} ,\tilde{\alpha }_{2} ,\ldots,\tilde{\alpha }_{n} } \right)} \right) \\ & = \left( \left[ {\phi \left( {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i}^{ - } } \right)} } \right)} \right),\phi \left( {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\mu_{i}^{ + } } \right)} } \right)} \right)} \right], \right. \\ & \quad \left. \left[ {\phi \left( {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i}^{ - } } \right)} } \right)} \right),\phi \left( {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {v_{i}^{ + } } \right)} } \right)} \right)} \right] \right) \\ & = \left( \begin{aligned} \left[ {\phi \left( {\phi^{ - 1} \left( {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\mu_{i}^{ - } } \right)} \right)} } \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\mu_{i}^{ + } } \right)} \right)} } \right)} \right)} \right)} \right], \hfill \\ \left[ {\phi \left( {\phi^{ - 1} \left( {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {v_{i}^{ - } } \right)} \right)} } \right)} \right)} \right),\phi \left( {\phi^{ - 1} \left( {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {v_{i}^{ + } } \right)} \right)} } \right)} \right)} \right)} \right] \hfill \\ \end{aligned} \right) \\ & = \left( \left[ {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\mu_{i}^{ - } } \right)} \right)} } \right),h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\mu_{i}^{ + } } \right)} \right)} } \right)} \right], \right. \\ & \quad \left. \left[ {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {v_{i}^{ - } } \right)} \right)} } \right),g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {v_{i}^{ + } } \right)} \right)} } \right)} \right] \right) \\ & = Arch - IVIFWA\left( {\wp_{P \to I} \left( {\tilde{\alpha }_{1} } \right),\wp_{P \to I} \left( {\tilde{\alpha }_{2} } \right),\ldots,\wp_{P \to I} \left( {\tilde{\alpha }_{n} } \right)} \right) \\ \end{aligned}.$$

• (2) The proof of (2) is similar to (1) and it is hence omitted here.

Theorem 7 is proven.

Proof of Theorem 8

Proof

• (1) Since $g_{P} \left( t \right) = g_{I} \left( {\phi \left( t \right)} \right) \Rightarrow g_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {g_{I}^{ - 1} \left( t \right)} \right) \Rightarrow g_{I}^{ - 1} \left( t \right) = \phi \left( {g_{P}^{ - 1} \left( t \right)} \right)$ and

$$h_{P} \left( t \right) = h_{I} \left( {\phi \left( t \right)} \right) \Rightarrow h_{P}^{ - 1} \left( t \right) = \phi^{ - 1} \left( {h_{I}^{ - 1} \left( t \right)} \right) \Rightarrow h_{I}^{ - 1} \left( t \right) = \phi \left( {h_{P}^{ - 1} \left( t \right)} \right).$$

Then, according to Definitions 26 and 27, we have

$$\begin{aligned} &\hat{\wp }_{P \to I} \left( {Arch - HPFWA\left( {\psi_{1} ,\psi_{2} ,\ldots,\psi_{n} } \right)} \right) \hfill \\ &= \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {\phi \left( {h_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{p} \left( {\tau_{i} } \right)} } \right)} \right)} \right\},\left\{ {\phi \left( {g_{P}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{p} \left( {\varsigma_{i} } \right)} } \right)} \right)} \right\}} \right\} \hfill \\ &= \cup_{{\tau_{i} \in a_{i} ,\varsigma_{i} \in b_{i} }} \left\{ {\left\{ {h_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} h_{I} \left( {\phi \left( {\tau_{i} } \right)} \right)} } \right)} \right\},\left\{ {g_{I}^{ - 1} \left( {\sum\limits_{i = 1}^{n} {w_{i} g_{I} \left( {\phi \left( {\varsigma_{i} } \right)} \right)} } \right)} \right\}} \right\} \hfill \\ &= Arch - IFWG\left( {\hat{\wp }_{P \to I} \left( \psi \right),\hat{\wp }_{P \to I} \left( {\psi_{2} } \right),\ldots,\hat{\wp }_{P \to I} \left( {\psi_{n} } \right)} \right) \hfill \\ \end{aligned}.$$

• (2) The proof of (2) is similar to (1) and it is hence omitted here.

Theorem 8 is proven.

Proof of Theorem 9

Proof

By Definitions 28 and 29, we have

$$sco_{P} \left( {\alpha_{i} } \right) = \mu_{i}^{2} - v_{i}^{2} { = }\phi \left( {\mu_{i} } \right) - \phi \left( {v_{i} } \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right)$$

$$acc_{P} \left( {\alpha_{i} } \right) = \mu_{i}^{2} + v_{i}^{2} { = }\phi \left( {\mu_{i} } \right) + \phi \left( {v_{i} } \right) = acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right).$$

Theorem 9 is proven.

Proof of Theorem 10

Proof

According to Theorem 9, we have

$$sco_{P} \left( {\alpha_{i} } \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right),\,\,acc_{P} \left( {\alpha_{i} } \right) = acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right).$$

Therefore,

$$\begin{aligned} sco_{P} \left( {\alpha_{1} } \right) < sco_{P} \left( {\alpha_{2} } \right) \Leftrightarrow sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) < sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right) \hfill \\ sco_{P} \left( {\alpha_{1} } \right) = sco_{P} \left( {\alpha_{2} } \right) \Leftrightarrow sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right) \hfill \\ acc_{P} \left( {\alpha_{1} } \right) < acc_{P} \left( {\alpha_{2} } \right) \Leftrightarrow acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) < acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right) \hfill \\ acc_{P} \left( {\alpha_{1} } \right) = acc_{P} \left( {\alpha_{2} } \right) \Leftrightarrow acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) = acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right) \hfill \\ \end{aligned}.$$

Thus, we have

Case 1

• (i) if $sco_{P} \left( {\alpha_{1} } \right) < sco_{P} \left( {\alpha_{2} } \right)$, then $\alpha_{1} <_{p} \alpha_{2}$, and $sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) < sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right)$, then

$$\wp_{P \to I} \left( {\alpha_{1} } \right) <_{I} \wp_{P \to I} \left( {\alpha_{2} } \right)$$

• (ii) if $sco_{P} \left( {\alpha_{1} } \right) = sco_{P} \left( {\alpha_{2} } \right)$, then $sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right)$, and

  • (a) if $acc_{P} \left( {\alpha_{1} } \right) < acc_{P} \left( {\alpha_{2} } \right)$ then $\alpha_{1} <_{p} \alpha_{2}$, and $acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) < acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right)$, then

    $$\wp_{P \to I} \left( {\alpha_{1} } \right) <_{I} \wp_{P \to I} \left( {\alpha_{2} } \right)$$

• (b) if $acc_{P} \left( {\alpha_{1} } \right) = acc_{P} \left( {\alpha_{2} } \right)$ then $\alpha_{1} \sim_{p} \alpha_{2}$, and $acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{1} } \right)} \right) = acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{2} } \right)} \right)$, then

  $$\wp_{P \to I} \left( {\alpha_{1} } \right)\sim_{I} \wp_{P \to I} \left( {\alpha_{2} } \right).$$

Case 2. If $\wp_{P \to I} \left( {\alpha_{1} } \right) <_{I} \wp_{P \to I} \left( {\alpha_{2} } \right)$ then $\alpha_{1} <_{p} \alpha_{2}$.

The proof of Case 2 is similar to Case 1 and it is hence omitted here.

Theorem 10 is proven.

Proof of Theorem 11.

Proof

According to Definitions 30 and 31, we have

$$\begin{aligned} Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right)} \right) & { = }\frac{{\phi \left( {\mu_{i}^{ - } } \right){ + }\phi \left( {\mu_{i}^{ + } } \right){ - }\phi \left( {v_{i}^{ - } } \right){ - }\phi \left( {v_{i}^{ + } } \right)}}{2} \\ & = \frac{{\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} { - }\left( {v_{i}^{ - } } \right)^{2} { - }\left( {v_{i}^{ + } } \right)^{2} }}{2} = Sco_{P} \left( {\tilde{\alpha }_{i} } \right), \\ \end{aligned}$$

$$\begin{aligned} Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right)} \right) & { = }\frac{{\phi \left( {\mu_{i}^{ - } } \right){ + }\phi \left( {\mu_{i}^{ + } } \right){ + }\phi \left( {v_{i}^{ - } } \right){ + }\phi \left( {v_{i}^{ + } } \right)}}{2} \\ & = \frac{{\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} { + }\left( {v_{i}^{ - } } \right)^{2} { + }\left( {v_{i}^{ + } } \right)^{2} }}{2} = Acc_{P} \left( {\tilde{\alpha }_{i} } \right). \\ \end{aligned}.$$

Theorem 11 is proven.

Proof of Theorem 11.

Proof

According to Definitions 30 and 31, we have

$$\begin{aligned} Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right)} \right) & { = }\frac{{\phi \left( {\mu_{i}^{ - } } \right){ + }\phi \left( {\mu_{i}^{ + } } \right){ - }\phi \left( {v_{i}^{ - } } \right){ - }\phi \left( {v_{i}^{ + } } \right)}}{2} \\ & = \frac{{\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} { - }\left( {v_{i}^{ - } } \right)^{2} { - }\left( {v_{i}^{ + } } \right)^{2} }}{2} = Sco_{P} \left( {\tilde{\alpha }_{i} } \right), \\ \end{aligned}$$

$$\begin{aligned} Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{i} } \right)} \right) & { = }\frac{{\phi \left( {\mu_{i}^{ - } } \right){ + }\phi \left( {\mu_{i}^{ + } } \right){ + }\phi \left( {v_{i}^{ - } } \right){ + }\phi \left( {v_{i}^{ + } } \right)}}{2} \\ & = \frac{{\left( {\mu_{i}^{ - } } \right)^{2} { + }\left( {\mu_{i}^{ + } } \right)^{2} { + }\left( {v_{i}^{ - } } \right)^{2} { + }\left( {v_{i}^{ + } } \right)^{2} }}{2} = Acc_{P} \left( {\tilde{\alpha }_{i} } \right). \\ \end{aligned}$$

Theorem 11 is proven.

Proof of Theorem 12.

Proof

According to Theorem 10, we have

$$sco_{P} \left( {\alpha_{i} } \right) = sco_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right),\,\,acc_{P} \left( {\alpha_{i} } \right) = acc_{I} \left( {\wp_{P \to I} \left( {\alpha_{i} } \right)} \right).$$

Therefore,

$$\begin{aligned} Sco_{P} \left( {\tilde{\alpha }_{1} } \right) < Sco_{P} \left( {\tilde{\alpha }_{2} } \right) \Leftrightarrow Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)} \right) < Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)} \right) \hfill \\ Sco_{P} \left( {\tilde{\alpha }_{1} } \right) = Sco_{P} \left( {\tilde{\alpha }_{2} } \right) \Leftrightarrow Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)} \right) = Sco_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)} \right) \hfill \\ Acc_{P} \left( {\tilde{\alpha }_{1} } \right) < Acc_{P} \left( {\tilde{\alpha }_{2} } \right) \Leftrightarrow Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)} \right) < Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)} \right) \hfill \\ Acc_{P} \left( {\tilde{\alpha }_{1} } \right) = Acc_{P} \left( {\tilde{\alpha }_{2} } \right) \Leftrightarrow Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right)} \right) = Acc_{I} \left( {\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right)} \right) \hfill \\ \end{aligned}.$$

Thus, we have

$\tilde{\alpha }_{1} <_{{\tilde{P}}} \tilde{\alpha }_{2}$ if and only if $\tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{1} } \right) <_{{\tilde{I}}} \tilde{\wp }_{P \to I} \left( {\tilde{\alpha }_{2} } \right).$

Theorem 12 is proven.

Proof of Theorem 13.

Proof

According to Definitions 32 and 33,

$$\begin{aligned} \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)} \right) & = \frac{1}{{\# a_{i} }}\sum\nolimits_{{\tau \in a_{i} }} {\phi \left( {\tau_{i} } \right)} - \;\frac{1}{{\# b_{i} }}\sum\nolimits_{{\varsigma_{i} \in b_{i} }} {\phi \left( {\varsigma_{i} } \right)} \\ & \;\; = \frac{1}{{\# a_{i} }}\sum\nolimits_{{\tau \in a_{i} }} {\tau_{i}^{2} } - \;\frac{1}{{\# b_{i} }}\sum\nolimits_{{\varsigma_{i} \in b_{i} }} {\varsigma_{i}^{2} } { = }\hat{S}co_{P} \left( {\psi_{i} } \right) \\ \end{aligned}$$

$$\begin{aligned} \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)} \right) & = \frac{1}{{\# a_{i} }}\sum\nolimits_{{\tau \in a_{i} }} {\phi \left( {\tau_{i} } \right)} { + }\;\frac{1}{{\# b_{i} }}\sum\nolimits_{{\varsigma_{i} \in b_{i} }} {\phi \left( {\varsigma_{i} } \right)} \\ & = \frac{1}{{\# a_{i} }}\sum\nolimits_{{\tau \in a_{i} }} {\tau_{i}^{2} } { + }\;\frac{1}{{\# b_{i} }}\sum\nolimits_{{\varsigma_{i} \in b_{i} }} {\varsigma_{i}^{2} } = \hat{A}cc_{P} \left( {\psi_{i} } \right) \\ \end{aligned}.$$

Theorem 13 is proven.

Proof of Theorem 14.

Proof

According to Theorem 13, we have

$$\hat{S}co_{P} \left( {\psi_{i} } \right) = \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)} \right),\,\,\hat{A}cc_{P} \left( {\psi_{i} } \right) = \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{i} } \right)} \right).$$

Therefore,

$$\begin{aligned} \hat{S}co_{P} \left( {\psi_{1} } \right) < \hat{S}co_{P} \left( {\psi_{2} } \right) \Leftrightarrow \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{1} } \right)} \right) < \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{2} } \right)} \right) \hfill \\ \hat{S}co_{P} \left( {\psi_{1} } \right) = \hat{S}co_{P} \left( {\psi_{2} } \right) \Leftrightarrow \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{1} } \right)} \right) = \hat{S}co_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{2} } \right)} \right) \hfill \\ \hat{A}cc_{P} \left( {\psi_{1} } \right) < \hat{A}cc_{P} \left( {\psi_{2} } \right) \Leftrightarrow \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{1} } \right)} \right) < \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{2} } \right)} \right) \hfill \\ \hat{A}cc_{P} \left( {\psi_{1} } \right) = \hat{A}cc_{P} \left( {\psi_{2} } \right) \Leftrightarrow \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{1} } \right)} \right) = \hat{A}cc_{I} \left( {\hat{\wp }_{P \to I} \left( {\psi_{2} } \right)} \right) \hfill \\ \end{aligned}.$$

Thus, we have

$\psi_{1} <_{{\hat{P}}} \psi_{2}$ if and only if $\hat{\wp }_{P \to I} \left( {\psi_{1} } \right) <_{{\tilde{I}}} \hat{\wp }_{P \to I} \left( {\psi_{2} } \right).$

Theorem 14 is proven.

7 Conclusion

The main results of this paper are shown in Table 5.

Table 5 Results

The isomorphism operators in Table 5 reveal that the results obtained by the following two patterns are equivalent:

• (1)
  $$A \, collection \, of \, PFNs\mathop{\longrightarrow}\limits{operator}comprehensive \, PFN\mathop{\longrightarrow}\limits{isomorphism}comprehensive \, IFN$$
• (2)
  $$A \, collection \, of \, PFNs\mathop{\longrightarrow}\limits{isomorphism}A \, collection \, of \, IFNs\mathop{\longrightarrow}\limits{operator}comprehensive \, IFN.$$

The isomorphic ranking methods in Table 5 reveal that the ranking relationship of two different PFNs remains unchanged after being converted into IFNs. Similarly, the same conclusion can be obtained for IVPFNs (vs. IVIFNs) and DPHFNs (vs. DHFNs).

When dealing with multiple attribute decision-making problems, isomorphism operators and ranking methods can reveal some interesting facts. Let $X_{i} \left( {i = 1,2} \right)$ be two alternatives, and their related collections of PFNs are $A_{i} \left( {i = 1,2} \right)$. From Table 2, we know that the following two sort results are consistent:

$$\begin{aligned}&Two\,collection\,of\,PFNs\left({{A\mathord{\left/{\vphantom{AB}}\right.\kern-0pt}B}}\right)\\&\Rightarrow\left\{\begin{aligned}&\mathop{\longrightarrow}\limits{operator}\,Two\,comprehensive\,PFN\left({{{\alpha_{1}}\mathord{\left/{\vphantom{{\alpha_{1}}{\alpha_{2}}}}\right.\kern-0pt}{\alpha_{2}}}}\right)\hfill\\&\mathop{\longrightarrow}\limits{isomorphism}Two\,collection\,of\,IFNs\left({{{A^{'}}\mathord{\left/{\vphantom{{A^{'}}{B^{'}}}}\right.\kern-0pt}{B^{'}}}}\right)\\&\mathop{\longrightarrow}\limits{operator}two\,comprehensive\,IFNs\left({{{\alpha_{1}^{'}}\mathord{\left/{\vphantom{{\alpha_{1}^{'}}{\alpha_{2}^{'}}}}\right.\kern-0pt}{\alpha_{2}^{'}}}}\right)\hfill\\\end{aligned}\right.\\ & \mathop{\longrightarrow}\limits{ranking\,}X_{1}<X_{2}\\\end{aligned}.$$

The above analysis shows that the Pythagorean fuzzy multi-attribute decision-making problem can be transformed into intuitionistic fuzzy multi-attribute decision-making problem by isomorphism, and the decision-making results are consistent.

In this study, the isomorphic relation between each pair of fuzzy sets is studied from three aspects, such as the operations, aggregation operators, and sorting methods. Aggregation operator and ranking method play an important role in the construction of fuzzy multi-attribute decision-making methods, which are mainly used for information aggregation and ranking. The study of isomorphic operators and isomorphic ranking methods reveals the essential relationship between intuitionistic FMADM method and Pythagorean FMADM method.

The advantage of this research is that it can transform the Pythagorean FMADM problem into intuitionistic FMADM problem. Especially in the face of large-scale group decision-making problems, some small groups provide Pythagorean fuzzy decision-making matrix, and some small groups provide intuitionistic fuzzy decision-making matrix. The isomorphic aggregation operators and ranking methods proposed in this paper can effectively deal with such decision situations. The limitations of this paper are mainly reflected in two aspects: first, the limitation of isomorphic operators is that the related operation rules are constructed based on Archimedean t-norm and s-norm; second, the limitation of isomorphic ranking method is that the related score functions and accuracy functions are constructed based on the difference, sum value, and expectation between membership and nonmembership degree. In the future research, we will focus on other types of isomorphism operators of various fuzzy sets, such as Bonferroni operator, Heronian operator, etc.