Pythagorean fuzzy Einstein weighted geometric aggregation operator and their application to multiple attribute group decision making

Abstract

There are many aggregation operators and its applications have been developed up to date, but Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator was not introduced. Therefore in this paper, we introduce Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator. We also discuss some basic properties of the proposed operator, including idempotency, boundedness, and monotonicity. Finally, we apply the PFEWG operator to deal with multiple attribute group decision making problem under Pythagorean fuzzy information. For Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator, we construct an algorithm for multiple attribute group decision making problem also. In the last we construct a numerical example also for multiple attribute group decision making problem

Keywords

Pythagorean fuzzy set Pythagorean fuzzy Einstein weighted geometric aggregation operator multiple attribute group decision making

1 Introduction

The idea of intuitionistic fuzzy set (AIFS) was introduced by Atanassov [8] to generalize the idea of Zadeh’s fuzzy set [14]. In an AIFS each element can be written in the form of ordered pair and each ordered pair is characterized by a membership degree and a non-membership degree. Atanassov also presented some basic and important operations and relations of AIFSs, such as union, intersection, complement, algebraic product and algebraic sum [8, 9, 11].He also introduced the concept of pseudo-fixed points of all operators defined over the AIFSs [10]. De et al. [21] introduced the concentration, dilation and normalization of AIFSs, and showed some propositions also. Deschrijver and Kerre [3] defined and explored the properties of a generalized union and a generalized intersection of AIFSs, using a general t-norm and t-conorm. Since the emergence of AIFS in 1986, many scholars [5 , 30] have done works in the field of AIFS and its applications. Particularly, information aggregation is a very crucial research area in AIFS theory that has been receiving more and more focus. Xu [28] developed some basic arithmetic aggregation operators, including IFWA operator, IFOWA operator, and IFHA operator. Xu and Yager [27] defined some basic geometric aggregation operators, such as intuitionistic fuzzy weighted geometric (IFWG) operator, intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and intuitionistic fuzzy hybrid geometric (IFHG) operator, and applied them to multiple attribute decision making (MADM) based on intuitionistic fuzzy information. In [2 , 31] many aggregation operators have been developed. Xu [33], Tan and Chen [1] used the Choquet integral to developed some intuitionistic fuzzy aggregation operators, which not only consider the importance of the elements or their ordered positions, but also can reflect the correlations among the elements or their ordered positions. Xu and Xia [35] applied Choquet integral and Dempster–Shafer theory of evidence to aggregate intuitionistic fuzzy information and introduced the induced generalized aggregation operators under intuitionistic fuzz information. Xu [34] defined an intuitionistic fuzzy Bonferroni mean (IFBM) and also applied the weighted IFBM to MADM. Xu and Cai [32] also provided a study of the aggregation techniques of intuitionistic fuzzy information and their applications in several fields. Wang and Liu [22] introduced an intuitionistic fuzzy Einstein weighted geometric (IFEWG) operator and an intuitionistic fuzzy Einstein ordered weighted geometric (IFEOWG) operator. In 2013, Yager [17, 20] introduced the basic concept of Pythagorean fuzzy set (PFS) characterized by a membership degree and a nonmembership degree, with condition that the square sum of its membership degree and nonmembership degree is equal to or less than one. Actually PFS is a generalization of IFS. Similarly intuitionistic fuzzy aggregation operators, Pythagorean fuzzy aggregation operators are also become an interesting and important area for research, after the advent of Pythagorean fuzzy set theory. 1n 2012, Wang and Liu [23] introduced the concept of intuitionistic fuzzy Einstein weighted averaging operator and the intuitionistic fuzzy Einstein ordered weighted averaging operator. There are two types of Einstein operations, such as Einstein sum and Einstein product. In [7] H. Garg used the Einstein sum and introduced the notion of the Pythagorean fuzzy arithmetic aggregating operators such as, Pythagorean fuzzy Einstein weighted averaging (PFEWA) operator, Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA) operator, generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA) operator, and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA) operator. But in this paper we use the Einstein product and introduce the notion of Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator. These both are new methods for decision making problem under Pythagorean fuzzy information but Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator is more reliable than arithmetic aggregating operators. Yager and Abbasov [19] introduced the idea of membership grades and the related idea of Pythagorean fuzzy subset. [18] Yager also defined Pythagorean fuzzy subset. Chiclana et al. [36] introduced the notion of induced ordered weighted geometric operators. Pérez et al. [37] worked on a dynamic group decision making problems. Cabrerizo et al. [38] worked on decision support system and developed a quality management in academic digital libraries. Porcel and Herrera [39] studied the incomplete information in a fuzzy linguistic recommender system. Tejeda et al. [40] introduced the recommender system for researchers based on bibliometrics. Dong et al. [41] introduced the concept of integrating experts’ weights generated dynamically into the consensus reaching process. Cabrerizo [42] worked on fuzzy decision making problem. Cabrerizo [43] worked on group decision making.

This paper consists of six section. In Section 1, we give an introduction related to the research background. In Section 2, we give some basic definitions and results which will be used in our later sections. In Section 3, we introduce some Einstein operations on Pythagorean fuzzy sets and analysis some desirable properties of the proposed operations. In Section 4, we introduce a Pythagorean fuzzy Einstein weighted geometric (PFEWG) aggregation operator, we also give a numerical example to develop (PFEWG) operator. In Section 5 we apply the (PFEWG) operator to MADM with Pythagorean fuzzy information. In Section 6, we have conclusion.

2 Preliminaries

Definition 2.1. [8] Let Z be a fixed set, then an IFS, I in Z can be defined as follows: $I = {(z, μ_{I} (z), ν_{I} (z)) | z \in Z},$ (1) where μ_I (z) and ν_I (z) are mappings from Z to [0, 1], such that 0 ≤ μ_I (z) ≤1, 0 ≤ ν_I (z) ≤1 and 0 ≤ μ_I (z) + ν_I (z) ≤1, for all z ∈ Z.

Let π_I (z) =1 - μ_I (z) - ν_I (z), then it is commonly called the intuitionistic fuzzy index of element z ∈ Z to set I, representing the degree of indeterminacy z to I. Also 0 ≤ π_I (z) ≤1 for everyz ∈ Z .

Definition 2.2. [17] Let Z be a fixed set, then a PFS, H in Z can be defined as follows: $H = {(z, μ_{H} (z), ν_{H} (z)) | z \in Z},$ (2) where μ_H (z) and ν_H (z) are mappings from Z to [0, 1], such that 0 ≤ μ_H (z) ≤1, 0 ≤ ν_H (z) ≤1 and $0 \leq μ_{H}^{2} (z) + ν_{H}^{2} (z) \leq 1,$ for all z ∈ Z, and they denote the membership degree and nonmembership degree of element z ∈ Z to set H, respectively. Let $π_{H} (z) = \sqrt{1 - μ_{H}^{2} (z) - ν_{H}^{2} (z)}$ , then it is commonly called the Pythagorean fuzzy index of element z ∈ Z to set H, representing the degree of indeterminacy of z to H. Also 0 ≤ π_H (z) ≤1, for every z ∈ Z .

Definition 2.3. [23] Let p = (μ_p, ν_p) , p₁ = (μ_{p
₁}, ν_{p
₁}) , p₂ = (μ_{p
₂}, ν_{p
₂}) , are three PFNs, and δ > 0, then their operations are defined asfollows: $p_{1} \oplus p_{2} = (\sqrt{μ_{p_{1}}^{2} + μ_{p_{2}}^{2} - μ_{p_{1}}^{2} μ_{p_{2}}^{2}}, ν_{p_{1}} ν_{p_{2}})$ (3) $p_{1} \otimes p_{2} = (μ_{p_{1}} μ_{p_{2}}, \sqrt{ν_{p_{1}}^{2} + ν_{p_{2}}^{2} - ν_{p_{1}}^{2} ν_{p_{2}}^{2}})$ (4) $δ p = (\sqrt{1 - {(1 - μ_{p}^{2})}^{δ}}, ν_{p}^{δ}), δ > 0$ (5) $p^{δ} = (μ_{p}^{δ}, \sqrt{1 - {(1 - ν_{p}^{2})}^{δ}}), δ > 0 .$ (6)

Definition 2.4. [24] Let p = (μ_p, ν_p) be a PFN, then the score function and accuracy function of p can be defined as follows: $S (p) = μ_{p}^{2} - ν_{p}^{2},$ (7) $H (p) = μ_{p}^{2} + ν_{p}^{2},$ (8) where S (p) ∈ [-1, 1] and H (p) ∈ [0, 1] .

Definition 2.5. [24] Let p₁ = (μ_{p
₁}, ν_{p
₁}) and p₂ = (μ_{p
₂}, ν_{p
₂}) be the two PFVs. Then $S (p_{1}) = μ_{p_{1}}^{2} - ν_{p_{1}}^{2}$ and $S (p_{2}) = μ_{p_{2}}^{2} - ν_{p_{2}}^{2}$ are the scores of p₁ and p₂ respectively, and $H (p_{1}) = μ_{p_{1}}^{2} + ν_{p_{1}}^{2},$ $H (p_{2}) = μ_{p_{2}}^{2} + ν_{p_{2}}^{2}$ are the accuracy degrees of p₁, p₂ respectively. Then we have

If S (p₁) < S (p₂) , then p₁ is smaller than p₂, denoted by p₁ < p₂,

If S (p₁) = S (p₂) , then

If H (p₁) = H (p₂) , then p₁ and p₂ represent the same information, i.e., $μ_{p_{1}}^{2} = μ_{p_{2}}^{2}$ and $ν_{p_{1}}^{2} = ν_{p_{2}}^{2}$ denoted by p₁ = p₂.

If H (p₁) < H (p₂) then p₁ is smaller than p₂ denoted by p₁ < p₂,

If H (p₁) > H (p₂) then p₁ is greater than p₂ denoted by p₁ > p₂ .

Definition 2.6. [7] The generalized intersection (∧) and generalized union (∨) of two PFNs p₁ and p₂ can be defined as follows:

p₁ ∨ p₂ = {(x, T (μ_p
₁ (x) , μ_p
₂ (x)) , S (ν_p
₁ (x) , ν_p
₂ (x))) |x ∈ X} ,

p₁ ∧ p₂ = {(x, S (μ_p
₁ (x) , μ_p
₂ (x)) , T (ν_p
₁ (x) , ν_p
₂ (x))) |x ∈ X} ,

where T and S denotes a t-norm and a t-conorm respectively. Einstein product ⊗_ɛ and Einstein sum ⊕_ɛ are common examples of t-norm and t-conorm respectively, and it can be defined for PFNs as follows:

S_{ɛ} (p_{1}, p_{2}) = \frac{\sqrt{p_{1}^{2} + p_{2}^{2}}}{\sqrt{1 + p_{1}^{2} \cdot_{ɛ} p_{2}^{2}}},

(9)

T_{ɛ} (p_{1}, p_{2}) = \frac{p_{1} \cdot_{ɛ} p_{2}}{\sqrt{1 + (1 - p_{1}^{2}) \cdot_{ɛ} (1 - p_{2}^{2})}},

(10) where S_ɛ (p₁, p₂) is a well-known t-conorm satisfying the following properties (1) Boundary: S_P (1, 1) = 1, S_P (z, 0) = S_P (0, z) = z,

(2) Monotonicity): If z₁ < z₂ and z₃ < z₄, then S_P (z₁, z₃) < S_P (z₂, z₄) ,

(3) Commutativity: S_P (z₁, z₂) = S_P (z₂, z₁)

(4) Associativity: S_P (z₁, S_P (z₂, z₃)) = S_P (S_P (z₁, z₂) , z₃) , and T_ɛ (p₁, p₂) is a well-known t-norm satisfying the following properties

(1) Boundary: T_P (0, 0) = 0, T_P (z, 1) = z,

(2) Monotonicity: If z₁ < z₂ and z₃ < z₄, then T_P (z₁, z₃) < T_P (z₂, z₄) ,

(3) Commutativity: T_P (z₁, z₂) = T_P (z₂, z₁) ,

(4) Associativity: T_P (z₁, T_P (z₂, z₃)) = T_P (T_P (z₁, z₂) , z₃) .

Definition 2.7. [7] Let P = (μ_P, ν_P) , P₁ = (μ_{P
₁}, ν_{P
₁}) and P₂ = (μ_{P
₂}, ν_{P
₂}) be the three PFVs, and δ > 0 be any real number. Then $P_{1} \oplus_{ɛ} P_{2} = (\frac{\sqrt{μ_{P_{1}}^{2} + μ_{P_{2}}^{2}}}{\sqrt{1 + μ_{P_{1}}^{2} \cdot_{ɛ} μ_{P_{2}}^{2}}}, \frac{ν_{P_{1}} \cdot_{ɛ} ν_{P_{2}}}{\sqrt{1 + (1 - ν_{P_{1}}^{2}) \cdot_{ɛ} (1 - ν_{P_{2}}^{2})}})$ (11) $P_{1} \otimes_{ɛ} P_{2} = (\frac{μ_{P_{1}} \cdot_{ɛ} μ_{P_{2}}}{\sqrt{1 + (1 - μ_{P_{1}}^{2}) \cdot_{ɛ} (1 - μ_{P_{2}}^{2})}}, \frac{\sqrt{ν_{P_{1}}^{2} + ν_{P_{2}}^{2}}}{\sqrt{1 + ν_{P_{1}}^{2} \cdot_{ɛ} ν_{P_{2}}^{2}}})$ (12) $P^{\land_{ɛ} δ} = (\frac{\sqrt{2 {(μ_{P}^{2})}^{δ}}}{\sqrt{{(2 - μ_{P}^{2})}^{δ} + {(μ_{P}^{2})}^{δ}}}, \frac{\sqrt{{(1 + ν_{P}^{2})}^{δ} - {(1 - ν_{P}^{2})}^{δ}}}{\sqrt{{(1 + ν_{P}^{2})}^{δ} + {(1 - ν_{P}^{2})}^{δ}}}), δ > 0$ (13) $δ \cdot_{ɛ} P = (\frac{\sqrt{{(1 + μ_{P}^{2})}^{δ} - {(1 - μ_{P}^{2})}^{δ}}}{\sqrt{{(1 + μ_{P}^{2})}^{δ} + {(1 - μ_{P}^{2})}^{δ}}}, \frac{\sqrt{2 {(ν_{P}^{2})}^{δ}}}{\sqrt{{(2 - ν_{P}^{2})}^{δ} + {(ν_{P}^{2})}^{δ}}}), δ > 0 .$ (14)

Definition 2.8. [22] Let a t-norm T be the Einstein product $\dot{ɛ}$ and a t-conorm S be Einstein sum $\overset{+}{ɛ}$ , then the generalized intersection and union on two IFSs A₁ and A₂ become the Einstein product (denoted by A₁ ⊗ _ɛA₂) and Einstein sum (denoted by A₁ ⊕ _ɛA₂) of two IFSs A₁ and A₂, respectively, as follows: $\begin{matrix} A_{1} \otimes_{ɛ} A_{2} & = & {(x, μ_{A_{1}} (x) \overset{\cdot}{ɛ} μ_{A_{2}} (x), \\ ν_{A_{1}} (x) \overset{+}{ɛ} ν_{A_{2}} (x)) | x \in X}, \end{matrix}$ (15) $\begin{matrix} A_{1} \oplus_{ɛ} A_{2} & = & {(x, μ_{A_{1}} (x) \overset{+}{ɛ} μ_{A_{2}} (x), \\ ν_{A_{1}} (x) \overset{\cdot}{ɛ} ν_{A_{2}} (x)) | x \in X} . \end{matrix}$ (16)

Definition 2.9. [22] The Einstein product (A₁ ⊗ _ɛA₂) and the Einstein sum (A₁ ⊕ _ɛA₂) on two IFSs A₁ and A₂, based on the Einstein t-norm and Einstein t-conorm can be defined as follows: $\begin{matrix} A_{1} \otimes_{ɛ} A_{2} \\ = {(\begin{matrix} x, \frac{μ_{A_{1}} (x) . μ_{A_{2}} (x)}{1 + (1 - μ_{A_{1}} (x)) ((1 - μ_{A_{2}} (x)))}, \\ \frac{ν_{A_{1}} (x) + ν_{A_{2}} (x)}{1 + ν_{A_{1}} (x) . ν_{A_{2}} (x)} \end{matrix}) | x \in X}, \end{matrix}$ (17) $\begin{matrix} A_{1} \oplus_{ɛ} A_{2} \\ = {(\begin{matrix} x, \frac{μ_{A_{1}} (x) + μ_{A_{2}} (x)}{1 + μ_{A_{1}} (x) . μ_{A_{2}} (x)}, \\ \frac{ν_{A_{1}} (x) . ν_{A_{2}} (x)}{1 + (1 - ν_{A_{1}} (x)) ((1 - ν_{A_{2}} (x)))} \end{matrix}) | x \in X} . \end{matrix}$ (18)

Definition 2.10. [13] Let P_j = (μ_{P
_j}, ν_{P
_j}) j = 1, …, n be a collection of fuzzy Pythagorean values, and let PFWG : Φⁿ → Φ, if

$\begin{matrix} {PFWG}_{w} (P_{1}, P_{2}, \dots, P_{n}) \\ = P_{1}^{w_{1}} \otimes P_{2}^{w_{2}} \otimes \dots \oplus P_{n}^{w_{n}} . \end{matrix}$ (19)

Then PFWG, is called Pythagorean fuzzy weighted geometric (PFWG) operator of dimension n, where w = (w₁, …, w_n) ^T is the weighted vector of P_j (j = 1, …, n) with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$

Definition 2.11. [13] Let P_j = (μ_{P
_j}, ν_{P
_j}) (j = 1, …, n) be a collection of PFNs A Pythagorean fuzzy ordered weighted geometric (PFOWG) operator of dimension n is a mapping PFOWG : Φⁿ → Φ, that has an associated vector w = (w₁, …, w_n) ^T, such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Furthermore

$\begin{matrix} {PFOWG}_{w} (P_{1}, P_{2}, \dots, P_{n}) \\ = {(P_{η (1)})}^{w_{1}} \otimes {(P_{η (2)})}^{w_{2}} \otimes \dots \otimes {(P_{η (n)})}^{w_{n}} \end{matrix}$ (20)

Where (η (1) , η (2) , …, η (n)) is a permutation of (1, 2, …, n) such that P_η(j-1) ≥ P_η(j) for all j . Especially, if $w = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ . Then PFOWG operator is reduced to PFG operator ofdimension n.

3 Einstein operations of Pythagorean fuzzy sets

The Einstein product (A ⊗ _ɛA₁) and the Einstein sum (A ⊕ _ɛA₁) on two PFSs A and A₁ can be defined in the following forms: $\begin{matrix} A \otimes_{ɛ} A_{1} \\ = {(\begin{matrix} x, \frac{μ_{A} (x) . μ_{A_{1}} (x)}{\sqrt{1 + (1 - μ_{A}^{2} (x)) ((1 - μ_{A_{1}}^{2} (x)))}}, \\ \frac{\sqrt{ν_{A}^{2} (x) + ν_{A_{1}}^{2} (x)}}{\sqrt{1 + ν_{A}^{2} (x) . ν_{A_{1}}^{2} (x)}} \end{matrix}) | x \in X}, \end{matrix}$ (21) $\begin{matrix} A \oplus_{ɛ} A_{1} \\ = {(\begin{matrix} x, \frac{\sqrt{μ_{A}^{2} (x) + μ_{A_{1}}^{2} (x)}}{\sqrt{1 + μ_{A}^{2} (x) . μ_{A_{1}}^{2} (x)}}, \\ \frac{ν_{A} (x) . ν_{A_{1}} (x)}{\sqrt{1 + (1 - ν_{A}^{2} (x)) ((1 - ν_{A_{1}}^{2} (x)))}} \end{matrix}) | x \in X} . \end{matrix}$ (22)

Theorem 3.1.Letnbe any positive integer andAis a PFS, then the exponentiation operationA^∧_ɛnis a mapping fromZ⁺ × ΩtoΩ : $A^{\land_{ɛ} n} = {(\begin{matrix} x, \frac{\sqrt{2 {(μ_{A}^{2} (x))}^{n}}}{\sqrt{{(2 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}}}, \\ \frac{\sqrt{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}}{\sqrt{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}}} \end{matrix}) | x \in X}$ (23)

Where $A^{\land_{ɛ} n} = \overset{n}{\overset{︷}{A \otimes_{ɛ} A \otimes_{ɛ} A \otimes_{ɛ} \dots A \otimes_{ɛ} A}} .$ Moreover A^∧_ɛn, is a Pythagorean fuzzy set (PFS), even if n is any positive real number.

Proof. By mathematical induction we can prove that, Equation (23) holds for all positive integer n. First we show that Equation (23) holds for n = 1. $A^{\land_{ɛ} 1} = {(\begin{matrix} x, \frac{\sqrt{2 (μ_{A}^{2} (x))}}{\sqrt{(2 - μ_{A}^{2} (x)) + (μ_{A}^{2} (x))}}, \\ \frac{\sqrt{(1 + ν_{A}^{2} (x)) - (1 - ν_{A}^{2} (x))}}{\sqrt{(1 + ν_{A}^{2} (x)) + (1 - ν_{A}^{2} (x))}} \end{matrix}) | x \in X} .$

Taking left hand side of the above equation, $A^{\land_{ɛ} 1} = A = {(x, μ_{A} (x), ν_{A} (x) | x \in X)} .$ (24)

Taking right hand side of the above equation

$\begin{matrix} {(\begin{matrix} x, \frac{\sqrt{2 (μ_{A}^{2} (x))}}{\sqrt{(2 - μ_{A}^{2} (x)) + (μ_{A}^{2} (x))}}, \\ \frac{\sqrt{(1 + ν_{A}^{2} (x)) - (1 - ν_{A}^{2} (x))}}{\sqrt{(1 + ν_{A}^{2} (x)) + 1 - ν_{A}^{2} x}} \end{matrix}) | x \in X} \\ = {(x, μ_{A} (x), ν_{A} (x) | x \in X)} . \end{matrix}$ (25)

From Equations (24 and 25) we have Equation (23) holds for n = 1. Next we show that Equation (23) holds for n = k. If Equation (23) holds for n = k, then Equation (23) also holds for n = k + 1. $\begin{matrix} A^{\land_{ɛ} k + 1} \\ = {(\begin{matrix} x, \frac{\sqrt{2 {(μ_{A}^{2} (x))}^{k}}}{\sqrt{{(2 - μ_{A}^{2} (x))}^{k} + {(μ_{A}^{2} (x))}^{k}}}, \\ \frac{\sqrt{{(1 + ν_{A}^{2} (x))}^{k} - {(1 - ν_{A}^{2} (x))}^{k}}}{\sqrt{{(1 + ν_{A}^{2} (x))}^{k} + {(1 - ν_{A}^{2} (x))}^{k}}} \end{matrix}) | x \in X} \\ \otimes_{ɛ} {(x, μ_{A} (x), ν_{A} (x)) | x \in X} \\ = {(\begin{matrix} x, \frac{\sqrt{2 {(μ_{A}^{2} (x))}^{k + 1}}}{\sqrt{{(2 - μ_{A}^{2} (x))}^{k + 1} + {(μ_{A}^{2} (x))}^{k + 1}}}, \\ \frac{\sqrt{{(1 + ν_{A}^{2} (x))}^{k + 1} - {(1 - ν_{A}^{2} (x))}^{k + 1}}}{\sqrt{{(1 + ν_{A}^{2} (x))}^{k + 1} + {(1 - ν_{A}^{2} (x))}^{k + 1}}} \end{matrix}) | x \in X} . \end{matrix}$

Now we are going to prove that Equation (23) holds for any positive integer n. For this, we have $\begin{matrix} 0 & \leq & {(\frac{\sqrt{2 {(μ_{A}^{2} (x))}^{n}}}{\sqrt{{(2 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}}})}^{2} \\ + {(\frac{\sqrt{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}}{\sqrt{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}}})}^{2} \leq 1 \end{matrix}$ even if n is any positive real number. Since 0 ≤ μ_A (x) ≤ 1, 0 ≤ ν_A (x) ≤ 1, $0 \leq μ_{A}^{2} (x) + ν_{A}^{2} (x) \leq 1 .$ Then $1 - μ_{A}^{2} (x) \geq ν_{A}^{2} (x) \geq 0 .$ So $\begin{matrix} 0 & \leq & {(\sqrt{\frac{2 {(μ_{A}^{2} (x))}^{n}}{{(2 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}}})}^{2} \\ + {(\sqrt{\frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}}})}^{2} \leq 1 \\ = & 0 \leq \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(2 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ + \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}} \leq 1 . \end{matrix}$

Since $\begin{matrix} \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(2 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ = \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(1 + (1 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ \leq \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} . \end{matrix}$

Thus

$\begin{matrix} \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(2 - μ_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ \leq \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} . \end{matrix}$ (26)

Again $1 - ν_{A}^{2} (x) \geq μ_{A}^{2} (x) \geq 0 .$

Since $\begin{matrix} \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}} \\ = \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}} \\ \leq \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} . \end{matrix}$

Thus

$\begin{matrix} \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(1 - ν_{A}^{2} (x))}^{n}} \\ \leq \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} . \end{matrix}$ (27)

From Equations (26) and (27) we have $\begin{matrix} \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ + \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ = \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ + \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ \leq \frac{2 {(μ_{A}^{2} (x))}^{n} + {(1 + ν_{A}^{2} (x))}^{n} - {(μ_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ = 1 . \end{matrix}$

Thus $\begin{matrix} 0 & \leq & \frac{2 {(μ_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \\ + \frac{{(1 + ν_{A}^{2} (x))}^{n} - {(1 - ν_{A}^{2} (x))}^{n}}{{(1 + ν_{A}^{2} (x))}^{n} + {(μ_{A}^{2} (x))}^{n}} \leq 1 . \end{matrix}$

Thus a PFS A^∧_ɛn define above is a PFS for any positive real number n.

4 Pythagorean fuzzy Einstein weighted geometric aggregation operator

Definition 4.1. Let P_j = (μ_{P
_j}, ν_{P
_j}) (j = 1, …, n) be the collection of PFVs with ≤_L, then a PFWG^ɛ operator of dimension n is a mapping ${PFWG}_{w}^{ɛ} : Θ^{n} \to Θ,$ and

$\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \\ = P_{1}^{\land_{ɛ} w_{1}} \otimes_{ɛ} P_{2}^{\land_{ɛ} w_{2}} \otimes_{ɛ} \dots \otimes_{ɛ} P_{n}^{\land_{ɛ} w_{n}}, \end{matrix}$ (28) where w = (w₁, …, w_n) ^T is the weighted vector of P_j (j = 1, …, n) such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$

Theorem 4.2.LetP_j = (μ_{P
_j}, ν_{P
_j}) (j = 1, …, n) be a collection of PFVs with ≤_L, then their aggregated value by using the ${PFWG}_{w}^{ɛ}$ operator is also a PFV, and

$\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \\ = (\begin{matrix} \frac{\sqrt{2 \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}, \\ \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \end{matrix}), \end{matrix}$ (29)

where w = (w₁, w₂, w₃, …, w_n) ^T is the weighted vector of P_j (j = 1, …, n) such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Especially, if $ν_{P_{j}}^{2} = 1 - μ_{P_{j}}^{2} (j = 1, \dots n)$ then Equation (29) is reduced to the following forms:

$\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \\ = \frac{\sqrt{2 \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}, \end{matrix}$ (30) which becomes the fuzzy Einstein weighted geometric $({FWG}_{w}^{ɛ})$ operator of dimension n for aggregating fuzzy information.

Proof. We can prove this Theorem by mathematical induction. First we show that Equation (29) holds for n = 1.

Taking left hand side, ${PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) = {PFWG}_{w}^{ɛ} (P) = P^{\land_{ɛ} w} .$ (31)

Now taking right hand side, $(\begin{matrix} \frac{\sqrt{2 \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}, \\ \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \end{matrix}) = P^{\land_{ɛ} w} .$ (32)

From Equations (30) and (31) we have Equation (29) holds for n = 1 . Now we show that Equation (29) holds for n = k . $\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{k}) \\ = (\begin{matrix} \frac{\sqrt{2 \prod_{j = 1}^{k} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k} {(μ_{P_{j}}^{2})}^{w_{j}}}}, \\ \frac{\sqrt{\prod_{j = 1}^{k} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{k} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \end{matrix}) . \end{matrix}$

Next we are going to show that Equation (29) holds for n = k + 1 .

$\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{k + 1}) \\ = (\begin{matrix} \frac{\sqrt{2 \prod_{j = 1}^{k} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k} {(μ_{P_{j}}^{2})}^{w_{j}}}}, \\ \frac{\sqrt{\prod_{j = 1}^{k} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{k} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \end{matrix}) \\ \otimes_{ɛ} (\begin{matrix} \frac{\sqrt{2 {(μ_{P_{k + 1}}^{2})}^{w_{k + 1}}}}{\sqrt{{(2 - μ_{P_{k + 1}}^{2})}^{w_{k + 1}} + {(μ_{P_{k + 1}}^{2})}^{w_{k + 1}}}}, \\ \frac{\sqrt{{(1 + ν_{P_{k + 1}}^{2})}^{w_{k + 1}} - {(1 - ν_{P_{k + 1}}^{2})}^{w_{k + 1}}}}{\sqrt{{(1 + ν_{P_{k + 1}}^{2})}^{w_{k + 1}} + {(1 - ν_{P_{k + 1}}^{2})}^{w_{k + 1}}}} \end{matrix}) . \end{matrix}$ (33)

Let $\begin{matrix} t_{1} & = & \sqrt{2 \prod_{j = 1}^{k} {(μ_{P_{j}}^{2})}^{w_{j}}}, \\ q_{1} & = & \sqrt{\prod_{j = 1}^{k} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k} {(μ_{P_{j}}^{2})}^{w_{j}}} \\ r_{1} & = & \sqrt{\prod_{j = 1}^{k} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{k} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}, \\ s_{1} & = & \sqrt{\prod_{j = 1}^{k} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k} {(1 - ν_{P_{j}}^{2})}^{w_{j}}} \\ t_{2} & = & \sqrt{2 {(μ_{P_{k + 1}}^{2})}^{w_{k + 1}}}, \\ q_{2} & = & \sqrt{{(2 - μ_{P_{k + 1}}^{2})}^{w_{k + 1}} + {(μ_{P_{k + 1}}^{2})}^{w_{k + 1}}} \\ r_{2} & = & \sqrt{{(1 + ν_{P_{k + 1}}^{2})}^{w_{k + 1}} - {(1 - ν_{P_{k + 1}}^{2})}^{w_{k + 1}}}, \\ s_{2} & = & \sqrt{{(1 + ν_{P_{k + 1}}^{2})}^{w_{k + 1}} + {(1 - ν_{P_{k + 1}}^{2})}^{w_{k + 1}}} \end{matrix}$

Now putting these values in Equation (32), we have

$\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{k + 1}) \\ = (\frac{t_{1}}{q_{1}}, \frac{r_{1}}{s_{1}}) \otimes_{ɛ} (\frac{t_{2}}{q_{2}}, \frac{r_{2}}{s_{2}}) \\ = (\begin{matrix} \frac{t_{1} t_{2}}{\sqrt{2 q_{1}^{2} q_{2}^{2} + t_{1}^{2} t_{2}^{2} - q_{1}^{2} t_{2}^{2} - t_{1}^{2} q_{2}^{2}}}, \\ \frac{\sqrt{s_{2}^{2} r_{1}^{2} + s_{1}^{2} r_{2}^{2}}}{\sqrt{s_{1}^{2} s_{2}^{2} + r_{1}^{2} r_{2}^{2}}} \end{matrix}) . \end{matrix}$ (34)

Now putting the values of t₁t₂, $s_{2}^{2} r_{1}^{2} + s_{1}^{2} r_{2}^{2}$ $2 q_{1}^{2} q_{2}^{2} + t_{1}^{2} t_{2}^{2} - q_{1}^{2} t_{2}^{2} - t_{1}^{2} q_{2}^{2},$ $s_{1}^{2} s_{2}^{2} + r_{1}^{2} r_{2}^{2}$ in Equation (33), we have $\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{k + 1}) \\ = (\begin{matrix} \frac{\sqrt{2 \prod_{j = 1}^{k + 1} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k + 1} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k + 1} {(μ_{P_{j}}^{2})}^{w_{j}}}}, \\ \frac{\sqrt{\prod_{j = 1}^{k + 1} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{k + 1} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{k + 1} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{k + 1} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \end{matrix}) . \end{matrix}$

Equation (29) holds for n = k + 1. Thus Equation (29) holds for all n .

Lemma 4.3.LetP_j > 0, w_j > 0 (j = 1, …, n) and $\sum_{j = 1}^{n} w_{j} = 1,$ then $\prod_{j = 1}^{n} {(P_{j})}^{w_{j}} \leq \sum_{j = 1}^{n} w_{j} P_{j},$ (35) where the equality holds if and only if P₁ = P₂ = P₃ = … = P_n .

Theorem 4.4.LetP_j = (μ_{P
_j}, ν_{P
_j}) (j = 1, …, n) be a collection of PFVs with ≤_L, then ${PFWG}_{w} (P_{1}, \dots, P_{n}) \leq {PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{n}),$ (36) where w = (w₁, … w_n) ^T is the weighted vector of P_j (j = 1, …, n) such that w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$

Proof. As we know that $\begin{matrix} \sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}} \\ \leq \sqrt{\sum_{j = 1}^{n} w_{j} (2 - μ_{P_{j}}^{2}) + \sum_{j = 1}^{n} w_{j} (μ_{P_{j}}^{2})} . \end{matrix}$

As we also know that $\sqrt{\sum_{j = 1}^{n} w_{j} (2 - μ_{P_{j}}^{2}) + \sum_{j = 1}^{n} w_{j} (μ_{P_{j}}^{2})} = \sqrt{2} .$

So $\sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}} \leq \sqrt{2} .$

Thus $\frac{\sqrt{2 \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}} \geq \prod_{j = 1}^{n} {(μ_{P_{j}})}^{w_{j}},$ (37) where the quality holds if and only if $μ_{P_{j}}^{2} (j = 1, 2, 3, \dots n)$ are equal. Again $\begin{matrix} \sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}} \\ \leq \sqrt{\sum_{j = 1}^{n} w_{j} (1 + ν_{P_{j}}^{2}) + \sum_{j = 1}^{n} w_{j} (1 - ν_{P_{j}}^{2})} . \end{matrix}$

We also know that $\sqrt{\sum_{j = 1}^{n} w_{j} (1 + ν_{P_{j}}^{2}) + \sum_{j = 1}^{n} w_{j} (1 - ν_{P_{j}}^{2})} = \sqrt{2} .$

Thus $\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}} \leq \sqrt{2} .$

Also $\begin{matrix} \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \\ \leq \sqrt{1 - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}} . \end{matrix}$

Thus

$\begin{matrix} \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}} \\ \leq \sqrt{1 - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}, \end{matrix}$ (38) where the quality holds if and only if $ν_{P_{j}}^{2} (j = 1, 2, 3, \dots n)$ are equal. Let ${PFWG}_{w} (P_{1}, P_{2}, P_{3}, \dots, P_{n}) = P = (μ_{P}, ν_{P}) .$

And ${PFWG}_{w}^{ɛ} (P_{1}, P_{2}, P_{3}, \dots, P_{n}) = P^{ɛ} = (μ_{P^{ɛ}}, ν_{P^{ɛ}}) .$

Then Equations (36, 37) can be transformed into the following forms: μ_P ≤ μ_{P
^ɛ} and ν_P ≥ ν_{P
^ɛ} respectively. Hence $S (P) = μ_{P}^{2} - ν_{P}^{2} \leq μ_{P^{ɛ}}^{2} - ν_{P^{ɛ}}^{2} = S (P^{ɛ}) .$ So S (P) ≤ S (P^ɛ) . If S (P) < S (P^ɛ). Then we have

$\begin{matrix} {PFWG}_{w} (P_{1}, \dots, P_{n}) \\ < {PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{n}) . \end{matrix}$ (39)

If S (P) = S (P^ɛ) . Then we have $H (P) = μ_{P}^{2} + ν_{P}^{2} = μ_{P^{ɛ}}^{2} + ν_{P^{ɛ}}^{2} = H (P^{ɛ})$ . Thus we have

$\begin{matrix} {PFWG}_{w} (P_{1}, \dots, P_{n}) \\ = {PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{n}) . \end{matrix}$ (40)

From Equations (38 and 39) we have Equation (35) always holds, where the equality holds if and only if P_j (j = 1, 2, 3, …, n) are equal. Thus $\begin{matrix} {PFWG}_{w} (P_{1}, P_{2}, P_{3}, \dots, P_{n}) \\ \leq {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, P_{3}, \dots, P_{n}) \end{matrix}$

Theorem 4.5.LetP_j = (μ_{P
_j}, ν_{P
_j}) (j = 1, …, n) be a collection of PFVs with ≤_L, andw = (w₁, …, w_n) ^Tis the weight vector ofP_j (j = 1, …, n) such thatw_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Then

(1) (Idempotency): If all P_j (j = 1, …, n) are equal, i.e., P_j (j = 1, …, n) = P, then ${PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) = P .$ (41)

(2) (Boundary):

$\begin{matrix} P_{min} & \leq & {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \\ \leq & P_{max} for every w \end{matrix}$ (42)

Where P_min = min(P_j) , P_max = max(P_j) .

(3) (Monotonicity): Let $P_{j}^{*} = (μ_{P_{j}}^{*}, ν_{P_{j}}^{*})$ (j = 1, …, n) be a collection of PFVs with ≤_L, and $μ_{P_{j}} \leq μ_{P_{j}}^{*}$ , $ν_{P_{j}} \geq ν_{P_{j}}^{*},$ for all, then

$\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{n}) \\ \leq {PFWG}_{w}^{ɛ} (P_{1}^{*}, \dots, P_{n}^{*},) every w . \end{matrix}$ (43)

Proof. (1) Idempotency: As we know that

Let P_j (j = 1, …, n) = P . Then Equation (43) can be written as: $\begin{matrix} {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \\ = (\frac{\sqrt{2 (μ_{P}^{2})}}{\sqrt{(2 - μ_{P}^{2}) + (μ_{P}^{2})}}, \frac{\sqrt{(1 + ν_{P}^{2}) - (1 - ν_{P}^{2})}}{\sqrt{(1 + ν_{P}^{2}) + (1 - ν_{P}^{2})}}) \end{matrix}$

(2) Boundary: $P_{min} \leq {PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{n}) \leq P_{max} for every w,$ (45) where P_min = min(P_j) and P_max = max(P_j) Let $f (x) = \sqrt{\frac{2 - x^{2}}{x^{2}}},$ x ∈ (0, 1] , then $f (x) = \frac{- 2}{x^{3}} \sqrt{\frac{x^{2}}{2 - x^{2}}} < 0 .$ So f (x) is decreasing function on (0, 1] . Since μ_{P
_min} ≤ μ_{P
_j} ≤ μ_{P
_max}, for all j . Then f (μ_{P
_max}) ≤ f (μ_{P
_j}) ≤ f (μ_{P
_min}) (j = 1, …, n) i.e., $\sqrt{\frac{2 - μ_{P_{max}}^{2}}{μ_{P_{max}}^{2}}} \leq \sqrt{\frac{2 - μ_{P_{j}}^{2}}{μ_{P_{j}}^{2}}} \leq \sqrt{\frac{2 - μ_{P_{min}}^{2}}{μ_{P_{min}}^{2}}}$ and let w = (w₁, …, w_n) ^T is the weight vector of P_j (j = 1, …, n) such that w_j ∈ [0, 1] (j = 1, …, n) and $\sum_{j = 1}^{n} w_{j} = 1,$ we have

$\begin{matrix} \Leftrightarrow \sqrt{\prod_{j = 1}^{n} {(\frac{2 - μ_{P_{max}}^{2}}{μ_{P_{max}}^{2}})}^{w_{j}}} \leq \sqrt{\prod_{j = 1}^{n} {(\frac{2 - μ_{P_{j}}^{2}}{μ_{P_{j}}^{2}})}^{w_{j}}} \\ \leq \sqrt{\prod_{j = 1}^{n} {(\frac{2 - μ_{P_{min}}^{2}}{μ_{P_{min}}^{2}})}^{w_{j}}} \\ \Leftrightarrow \sqrt{{(\frac{2 - μ_{P_{max}}^{2}}{μ_{P_{max}}^{2}})}^{\sum_{j = 1}^{n} w_{j}}} \leq \sqrt{\prod_{j = 1}^{n} {(\frac{2 - μ_{P_{j}}^{2}}{μ_{P_{j}}^{2}})}^{w_{j}}} \\ \leq \sqrt{{(\frac{2 - μ_{P_{min}}^{2}}{μ_{P_{min}}^{2}})}^{\sum_{j = 1}^{n} w_{j}}} \\ \Leftrightarrow μ_{P_{min}} \leq \frac{\sqrt{2 \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(2 - μ_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(μ_{P_{j}}^{2})}^{w_{j}}}} \\ \leq μ_{P_{max}} . \end{matrix}$ (46)

Again let $g (y) = \sqrt{\frac{1 - y^{2}}{1 + y^{2}}}, y \in [0, 1] .$ Then $g (y) = \frac{- 2 y}{{(1 + y^{2})}^{2}} \sqrt{\frac{1 + y^{2}}{1 - y^{2}}} < 0 .$ Since g (y) is a decreasing function on [0, 1] . Thus ν_{P
_max} ≤ ν_{P
_j} ≤ ν_{P
_min} for all j . Then g (ν_{P
_min}) ≤ g (ν_{P
_j}) ≤ g (ν_{P
_max}) for all j . i.e., $\sqrt{\frac{1 - ν_{P_{min}}^{2}}{1 + ν_{P_{min}}^{2}}} \leq \sqrt{\frac{1 - ν_{P_{j}}^{2}}{1 + ν_{P_{j}}^{2}}} \leq \sqrt{\frac{1 - ν_{P_{max}}^{2}}{1 + ν_{P_{max}}^{2}}}, (j = 1, \dots, n),$ and let w = (w₁, …, w_n) ^T is the weight vector of P_j (j = 1, 2, 3, …, n) such that w_j ∈ [0, 1] (j = 1, …, n) $\sum_{j = 1}^{n} w_{j} = 1,$ we have

$\begin{matrix} \Leftrightarrow \sqrt{\prod_{j = 1}^{n} {(\frac{1 - ν_{P_{min}}^{2}}{1 + ν_{P_{min}}^{2}})}^{w_{j}}} \leq \sqrt{\prod_{j = 1}^{n} {(\frac{1 - ν_{P_{j}}^{2}}{1 + ν_{P_{j}}^{2}})}^{w_{j}}} \\ \leq \sqrt{\prod_{j = 1}^{n} {(\frac{1 - ν_{P_{max}}^{2}}{1 + ν_{P_{max}}^{2}})}^{w_{j}}} \\ \Leftrightarrow \sqrt{{(\frac{1 - ν_{P_{min}}^{2}}{1 + ν_{P_{min}}^{2}})}^{\sum_{j = i}^{n} w_{j}}} \leq \sqrt{\prod_{j = 1}^{n} {(\frac{1 - ν_{P_{j}}^{2}}{1 + ν_{P_{j}}^{2}})}^{w_{j}}} \\ \leq \sqrt{{(\frac{1 - ν_{P_{max}}^{2}}{1 + ν_{P_{max}}^{2}})}^{\sum_{j = i}^{n} w_{j}}} \\ \Leftrightarrow ν_{P_{max}} \leq \frac{\sqrt{\prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}}}}{\sqrt{\prod_{j = 1}^{n} {(1 - ν_{P_{j}}^{2})}^{w_{j}} + \prod_{j = 1}^{n} {(1 + ν_{P_{j}}^{2})}^{w_{j}}}} \\ \leq ν_{P_{min}} . \end{matrix}$ (47)

Let ${PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) = P .$ Then Equations (45 and 46) can be written as: μ_{P
_min} ≤ μ_P ≤ μ_{P
_max} and ν_{P
_max} ≤ ν_P ≤ ν_{P
_min} respectively. Thus $S (P) = μ_{P}^{2} - ν_{P}^{2} \leq μ_{P_{max}}^{2} - ν_{P_{max}}^{2} = S (P_{max}),$ and $S (P) = μ_{P}^{2} - ν_{P}^{2} \geq μ_{P_{min}}^{2} - ν_{P_{min}}^{2} = S (P_{min}) .$ If S (P) < S (P_max) and S (P) > S (P_min) . Then we have

$\begin{matrix} P_{min} & < & {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \\ < & P_{max} for every w . \end{matrix}$ (48)

If S (P) = S (P_max). Then we have $μ_{P}^{2} = μ_{P_{max}}^{2}$ and $ν_{P}^{2} = ν_{P_{max}}^{2} .$ Thus $H (P) = μ_{P}^{2} + ν_{P}^{2} = μ_{P_{max}}^{2} + ν_{P_{max}}^{2} = H (P_{max}) .$ Therefore ${PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) = P_{max} for every w .$ (49)

If S (P) = S (P_min) i.e., $μ_{P}^{2} - ν_{P}^{2} = μ_{P_{min}}^{2} - ν_{P_{min}}^{2} .$ Then we have $μ_{P}^{2} = μ_{P_{min}}^{2}$ and $ν_{P}^{2} = ν_{P_{min}}^{2} .$ Thus $H (P) = μ_{P}^{2} + ν_{P}^{2} = μ_{P_{min}}^{2} + ν_{P_{min}}^{2} = H (P_{min}) .$ Therefore ${PFWG}_{w}^{ɛ} (P_{1}, \dots, P_{n}) = P_{min} for every w .$ (50)

Thus from Equations (47 to 49) we have $P_{min} \leq {PFWG}_{w}^{ɛ} (P_{1}, P_{2}, \dots, P_{n}) \leq P_{max} for every w .$

(3) (Monotonicity): The proof follows from (2).

5 An application of the Pythagorean fuzzy Einstein weighted geometric (PFEWG) aggregation operator to group decision making problems

In this section, we develop an application of Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator to multiple criteria decision making problem.

Algorithm: Let Y ={ Y₁, Y₂, …, Y_n } be the set of n alternatives, B ={ B₁, B₂, …, B_m } be the set of m attributes, and D ={ D₁, …, D_k } be the set of k decision makers. Let w = (w₁, w₂, …, w_m) ^T be the weighted vector of the attributes B_i (i = 1, 2, …, m) , such that w_i ∈ [0, 1] and $\sum_{i = 1}^{m} w_{i} = 1,$ let η = (η₁, η₂, …, η_k) ^T be the weighted vector of the decision makers D^s (s = 1, 2, …, k) , such that η_s ∈ [0, 1] and $\sum_{s = 1}^{k} η_{s} = 1 .$ This method have the followingsteps.

Step 1. In this step we construct the Pythagorean fuzzy decision making matrices, $D^{s} = {[α_{ji}^{(s)}]}_{n \times m} (s = 1, 2, \dots, k)$ for decision. If the criteria have two types, such as benefit criteria and cost criteria, then the Pythagorean fuzzy decision matrices, $D^{s} = {[α_{ji}^{s}]}_{n \times m}$ can be converted into the normalized Pythagorean fuzzy decision matrices, $R^{s} = {[r_{ji}^{(s)}]}_{n \times m},$ where $r_{ji}^{s} = {\begin{matrix} α_{ji}^{s}, for benefit criteria Y_{i} \\ {\bar{a}}_{ji}^{s}, for cost criteria Y_{i}, \end{matrix} i = 1, \dots, m$ and ${\bar{a}}_{ji}^{s}$ is the complement of $α_{ji}^{s} .$ If all the criteria have the same type, then there is no need of normalization.

Step 2. Using the Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator to aggregate all the individual normalized Pythagorean fuzzy decision matrices, $R^{s} = {[r_{ji}^{(s)}]}_{n \times m} (s = 1, 2, \dots, k)$ into the single Pythagorean fuzzy decision matrix, R = [r_ji] _n×m, where r_ji = (μ_ji, ν_ji) (j = 1, 2, …, n, i = 1, 2, …, m) .

Step 3. Aggregate all the preference values r_ji (j = 1, 2, 3, … n, i = 1, 2, …, m) by using the PFEWG operator and get the overall preference values r_j (j = 1, 2, …, n) corresponding to the alternatives Y_j (j = 1, …, n) .

Step 4. Calculate the scores of r_j (j = 1, 2, 3, … n) . If there is no difference between two or more than two scores then we have must to find out the accuracy degrees of the collective overall preference values.

Step 5. Arrange the scores of the all alternatives in the form of descending order and select that alternative which has the highest score function.

Example 5.1. Suppose in a university, the computer center wants to select a new information system for the purpose of the best productivity. After the first selection, there are only five alternatives Y_j (j = 1, 2, 3, 4, 5) have been short listed. There are three experts D^s (s = 1, 2, 3) , from a group to act as decision makers, whose weight vector η = (0.2, 0.3, 0.5) ^T . There are many factors that must be considered while selecting the most suitable system, but here, we have consider only the following four criteria, whose weighted vector is w = (0.1, 0.2, 0.3, 0.4) ^T .

B₁ : Costs of software/hardware investment,

B₂ : Support of the organization,

B₃ : Effort to transform from current systems,

B₄ : Outsourcing software developer reliability,

where B₁, B₃ are cost type criteria and B₂, B₄ are benefit type criteria i.e., the attributes have two types criteria, thus we have must to change the cost type criteria into benefit type criteria.

Step 1. The decision makers give his decision in the following tables. $\begin{matrix} Pythagorean Fuzzy Decision Matrix of Expert 1 \\ D_{5 \times 4}^{1} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.8, 0.3) & (0.7, 0.4) & (0.7, 0.4) & (0.7, 0.5) \\ Y_{2} & (0.8, 0.3) & (0.7, 0.4) & (0.8, 0.5) & (0.8, 0.3) \\ Y_{3} & (0.5, 0.6) & (0.6, 0.5) & (0.7, 0.5) & (0.8, 0.3) \\ Y_{4} & (0.7, 0.5) & (0.7, 0.4) & (0.6, 0.4) & (0.8, 0.4) \\ Y_{5} & (0.6, 0.7) & (0.8, 0.2) & (0.7, 0.3) & (0.6, 0.5) \end{matrix} \\ Pythagorean Fuzzy Decision Matrix of Expert 2 \\ D_{5 \times 4}^{2} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.8, 0.2) & (0.8, 0.4) & (0.6, 0.4) & (0.6, 0.5) \\ Y_{2} & (0.7, 0.3) & (0.8, 0.4) & (0.7, 0.5) & (0.7, 0.4) \\ Y_{3} & (0.6, 0.6) & (0.6, 0.5) & (0.6, 0.6) & (0.7, 0.4) \\ Y_{4} & (0.7, 0.5) & (0.6, 0.6) & (0.7, 0.4) & (0.8, 0.5) \\ Y_{5} & (0.6, 0.4) & (0.7, 0.2) & (0.8, 0.4) & (0.8, 0.4) \end{matrix} \end{matrix}$

$\begin{matrix} Pythagorean Fuzzy Decision Matrix of Expert 3 \\ D_{5 \times 4}^{3} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.7, 0.4) & (0.7, 0.4) & (0.6, 0.5) & (0.6, 0.5) \\ Y_{2} & (0.8, 0.3) & (0.7, 0.3) & (0.8, 0.3) & (0.9, 0.2) \\ Y_{3} & (0.6, 0.5) & (0.6, 0.6) & (0.7, 0.4) & (0.8, 0.3) \\ Y_{4} & (0.7, 0.4) & (0.8, 0.5) & (0.9, 0.1) & (0.6, 0.5) \\ Y_{5} & (0.6, 0.5) & (0.8, 0.2) & (0.8, 0.2) & (0.8, 0.3) \end{matrix} \\ Normalized Pythagorean Fuzzy Decision Matrix \\ R_{5 \times 4}^{1} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.3, 0.8) & (0.7, 0.4) & (0.4, 0.7) & (0.7, 0.5) \\ Y_{2} & (0.3, 0.8) & (0.7, 0.4) & (0.5, 0.8) & (0.8, 0.3) \\ Y_{3} & (0.6, 0.5) & (0.6, 0.5) & (0.5, 0.7) & (0.8, 0.3) \\ Y_{4} & (0.5, 0.7) & (0.7, 0.4) & (0.4, 0.6) & (0.8, 0.4) \\ Y_{5} & (0.7, 0.6) & (0.8, 0.2) & (0.3, 0.7) & (0.6, 0.5) \end{matrix} \\ Normalized Pythagorean Fuzzy Decision Matrix \\ R_{5 \times 4}^{2} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.2, 0.8) & (0.8, 0.4) & (0.4, 0.6) & (0.6, 0.5) \\ Y_{2} & (0.3, 0.7) & (0.8, 0.4) & (0.5, 0.7) & (0.7, 0.4) \\ Y_{3} & (0.6, 0.6) & (0.6, 0.5) & (0.6, 0.6) & (0.7, 0.4) \\ Y_{4} & (0.5, 0.7) & (0.6, 0.6) & (0.4, 0.7) & (0.8, 0.5) \\ Y_{5} & (0.4, 0.6) & (0.7, 0.2) & (0.4, 0.8) & (0.8, 0.4) \end{matrix} \\ Normalized Pythagorean Fuzzy Decision Matrix \\ R_{5 \times 4}^{3} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.4, 0.7) & (0.7, 0.4) & (0.5, 0.6) & (0.6, 0.5) \\ Y_{2} & (0.3, 0.8) & (0.7, 0.3) & (0.3, 0.8) & (0.9, 0.2) \\ Y_{3} & (0.5, 0.6) & (0.6, 0.6) & (0.4, 0.7) & (0.8, 0.3) \\ Y_{4} & (0.4, 0.7) & (0.8, 0.5) & (0.1, 0.9) & (0.6, 0.5) \\ Y_{5} & (0.5, 0.6) & (0.8, 0.2) & (0.2, 0.8) & (0.8, 0.3) \end{matrix} \end{matrix}$

Step 2. Apply the Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator to aggregate all the individual normalized Pythagorean fuzzy decision matrices, $R^{s} = {[r_{ji}^{(s)}]}_{n \times m}$ into a Single Pythagorean fuzzy decision matrix, R = [r_ji] _n×m . $\begin{matrix} Collective Pythagorean Fuzzy Decision Matrix \\ R_{5 \times 4} = \\ \begin{matrix} B_{1} & B_{2} & B_{3} & B_{4} \\ Y_{1} & (0.3, 0.8) & (0.8, 0.4) & (0.4, 0.7) & (0.7, 0.5) \\ Y_{2} & (0.3, 0.8) & (0.8, 0.3) & (0.3, 0.8) & (0.8, 0.2) \\ Y_{3} & (0.6, 0.6) & (0.7, 0.6) & (0.4, 0.7) & (0.8, 0.3) \\ Y_{4} & (0.4, 0.7) & (0.8, 0.6) & (0.2, 0.8) & (0.7, 0.5) \\ Y_{5} & (0.6, 0.6) & (0.8, 0.2) & (0.2, 0.8) & (0.8, 0.3) \end{matrix} \end{matrix}$

Step 3. Aggregate all the preference values r_ji (j = 1, 2, 3, 4, 5, i = 1, 2, 3, 4) $\begin{matrix} r_{1} & = & (0.5702, 0.5957), r_{2} = (0.5618, 0.5702), \\ r_{3} & = & (0.6262, 0.5481), r_{4} = (0.4855, 0.6549), \\ r_{5} & = & (0.5392, 0.5475) . \end{matrix}$

Step 4. Calculate the scores of r_j (j = 1, 2, 3, 4, 5) . $\begin{matrix} S (r_{1}) & = & - 0.0297, S (r_{2}) = - 0.0095, \\ S (r_{3}) & = & 0.0917, S (r_{4}) = - 0.1931, \\ S (r_{5}) & = & - 0.009 \end{matrix}$

Step 5. Thus the most wanted alternative is Y₃ .

6 Conclusion

In this paper, we have defined Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator. We also introduced some basic properties of this operator, such as idempotency, boundary, monotonicity, and also constructed a numerical example. Finally, we applied the Pythagorean fuzzy Einstein weighted geometric (PFEWG) operator to deal with multiple criteria decision making problems under Pythagorean fuzzy information. For Pythagorean fuzzy Einstein weighted geometric (PFEWG) , we have constructed an algorithm for multiple criteria decision making problems. Lastly we developed a numerical example also for decision making problems.

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