Pythagorean Fuzzy Einstein Hybrid Averaging Aggregation Operator and its Application to Multiple-Attribute Group Decision Making

1 Introduction

Multi-criteria decision making is one of the processes for finding the optimal alternative from all feasible alternatives according to some criteria or attributes. Traditionally, it has been generally assumed that all data that access the alternative in terms of criteria and their corresponding weights are expressed in the form of crisp numbers. However, most of the decisions in real-life situations are taken in the environment where the goals and constraints are generally imprecise or vague in nature. In order to handle the uncertainties, the intuitionistic fuzzy set [1] theory, one of the successful extensions of the fuzzy set theory [36], which is characterized by the degree of membership and degree of non-membership, has been presented. Xu [25] developed some basic arithmetic aggregation operators, including intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid averaging operator. Xu and Yager [29] defined some basic geometric aggregation operators, such as intuitionistic fuzzy weighted geometric operator, intuitionistic fuzzy ordered weighted geometric operator, and intuitionistic fuzzy hybrid geometric operator. Wang and Liu [22], [23] introduced the notion of some Einstein aggregation operators, such as intuitionistic fuzzy Einstein weighted geometric operator, intuitionistic fuzzy Einstein ordered weighted geometric operator, intuitionistic fuzzy Einstein weighted averaging operator, and intuitionistic fuzzy Einstein ordered weighted averaging operator, and applied them to group decision making. In Refs. [5], [6], [8], [9], [20], [21], [24], [26], [27], [30], [31], [35], many scholars worked in the field of intuitionistic fuzzy sets and introduced many aggregation operators and applied them to group decision making.

However, there are many cases where the decision maker may provide the degree of membership and non-membership of a particular attribute in such a way that their sum is greater than 1. For example, suppose a man expresses his preferences toward the alternative in such a way that degree of their satisfaction is 0.6 and the degree of rejection is 0.8. Obviously, its sum is greater than 1. Therefore, Yager [33] introduced the concept of another set called Pythagorean fuzzy set. Pythagorean fuzzy set is a more powerful tool to solve uncertain problems. Like intuitionistic fuzzy aggregation operators, Pythagorean fuzzy aggregation operators have also become an interesting and important area for research, after the advent of the Pythagorean fuzzy sets theory. In 2013, Yager and Abbasov [34] introduced the notion of two new Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy weighted averaging operator and Pythagorean fuzzy ordered weighted averaging operator. In Refs. [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], Rahman et al. introduced the concept of many aggregation operators using Pythagorean fuzzy numbers and also applied them to group decision making. In Refs. [2], [3], [4], Garg introduced the notion of Einstein averaging aggregation operator and Einstein geometric aggregation operator, and applied them to group decision making. In Ref. [37], Zang and Xu introduced the notion of TOPSIS for multiple-criteria decision making with Pythagorean fuzzy sets. Xue et al. [32], Liang et al. [7], and Xu and Da [28] developed some methods and aggregation operators using Pythagorean fuzzy information.

Thus, keeping the advantages and applications of the above-mentioned aggregation operators, in this paper, we introduce the notion of Pythagorean fuzzy Einstein hybrid averaging (PFEHA) aggregation operator along with its desirable properties, namely idempotency, boundedness, and monotonicity. Actually, Pythagorean fuzzy Einstein weighted averaging (PFEWA) aggregation operator weights only the Pythagorean fuzzy arguments and Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA) aggregation operator weights only the ordered positions of the Pythagorean fuzzy arguments instead of weighting the Pythagorean fuzzy arguments themselves. To overcome these limitations, we introduce the concept of PFEHA aggregation operator, which weights both the given Pythagorean fuzzy value and its ordered position. Thus, the method proposed in this paper is more general, more flexible, and provides more accurate and precise results compared to the existing methods.

The remainder of this paper is structured as follows. In Section 2, we give some basic definitions and results, which will be used in later sections. In Section 3, we introduce the notion of PFEHA aggregation operator and method. In Section 4, we apply the proposed aggregation operator to multiple-attribute group decision-making problems with Pythagorean fuzzy information. In Section 5, we construct a numerical example. In Section 6, we compare the proposed method to other methods. In Section 7, we provide our conclusion.

2 Preliminaries

In the following, we developed Pythagorean fuzzy set, score function, and accuracy function.

Definition 1 ([37]). Let Z be a universal set, then a Pythagorean fuzzy set can be defined as

where μ_P(z) and η_P(z) are mappings from Z to [0, 1], such that 0≤μ_P(z)≤1, 0≤η_P(z)≤1, and also 0≤μP2(z)+ηP2(z)≤1, for all z∈Z, and they denote the membership degree and non-membership degree of element z∈Z to set P, respectively. Let πP(z)=1−μP2(z)−ηP2(z), then it is called the Pythagorean fuzzy index of element z∈Z to set P, representing the degree of indeterminacy of z to P. Also, 0≤π_P(z)≤1, for every z∈Z.

Definition 2 ([37]). Let α=⟨μ_α, η_α⟩ be a Pythagorean fuzzy number, then the score function of α can be defined as

where s(α)∈[−1, 1].

Definition 3 ([37]). Let α=⟨μ_α, ν_α⟩ be a Pythagorean fuzzy number, then the accuracy function of α can be defined as

where h(α)∈[0, 1].

Definition 4 ([37]). Let α1=〈μα1, ηα1〉 and α2=〈μα2, ηα2〉 be the two Pythagorean fuzzy number, then the following conditions hold:

• 1. If s(α₁)≺s(α₂), then α₁≺α₂.

• 2. If s(α₁)=s(α₂), then

• 1. If h(α₁)=h(α₂), then α₁=α₂.

• 2. If h(α₁)≺h(α₂), then α₁≺α₂.

• 3. If h(α₁)≻h(α₂), then α₁≻α₂.

In the following, we developed some Einstein operational laws for sum and product.

Definition 5 ([2]). Let αj=〈μαj, ηαj〉(j=1, 2) be the three Pythagorean fuzzy values and δ≻0 be any real number, then

(1) α1⊕αε2=(μα12+μα221+μα12⋅μεα22, ηα1⋅ηεα21+(1−ηα12)⋅(1−ηα22)ε).(2) α1⊗αε2=(μα1⋅μεα21+(1−μα12)⋅(1−μα22)ε, ηα12+ηα221+ηα12⋅ηεα22).(3) α∧εδ=(2(μα2)δ(2−μα2)δ+(μα2)δ, (1+ηα2)δ−(1−ηα2)δ(1+ηα2)δ+(1−ηα2)δ).(4) δ⋅αε=((1+μα2)δ−(1−μα2)δ(1+μα2)δ+(1−μα2)δ, 2(ηα2)δ(2−ηα2)δ+(ηα2)δ).

In the following, we developed some aggregation operators, such as Pythagorean fuzzy hybrid averaging (PFHA) operator, PFEWA aggregation operator, and PFEOWA aggregation operator.

Definition 6 ([17]). Let αj=〈μαj, ηαj〉(j=1, 2, 3, …, n) be a collection of fuzzy Pythagorean values, then PFHA aggregation operator can be defined as

where α˙σ(j) is the j^th largest of the weighted Pythagorean fuzzy values α˙j(α˙j=nωjαj), w=(w₁, w₂, w₃, …, w_n)^T is the weighted vector of the PFHA operator, such that w_j∈[0, 1] and ∑j=1nwj=1. ω=(ω₁, ω₂, ω₃, …, ω_n)^T is the weighted vector of α_j(j=1, 2, 3, ..., n), such that ω_j∈[0, 1], ∑j=1nωj=1, and n is the balancing coefficient, which plays a role of balance. If the vector w=(w₁, w₂, w₃, …, w_n)^T approaches to (1n, 1n, 1n, …, 1n)T, then the vector (nω₁α₁, nω₂α₂, nω₃α₃, …, nω_nα_n)^T approaches to (α₁, α₂, α₃, …, α_n)^T.

Definition 7 ([2]). Let αj=〈μαj, ηαj〉(j=1, 2, 3, …, n) be a collection of fuzzy Pythagorean values, then the PFEWA aggregation operator can be defined as

where w=(w₁, w₂, w₃, …, w_n)^T is the weighted vector of α_j(j=1, 2, 3, …, n) such that w_j∈[0, 1] and ∑j=1nwj=1.

Definition 8 ([2]). Let αj=〈μαj, ηαj〉(j=1, 2, 3, …, n) be a collection of fuzzy Pythagorean values, then the PFEOWA aggregation operator can be defined as

where w=(w₁, w₂, w₃, …, w_n)^T is the weighted vector of α_σ(j)(j=1, 2, 3, …, n) such that w_j∈[0, 1] and ∑j=1nwj=1.

Also, (σ(1), σ(2), σ(3), …, σ(n)) is a permutation of (1, 2, 3, …, n) such that α_σ(j)≤α_σ(j−1) for all j.

3 PFEHA Aggregation Operator

In this section, we introduce the concept of PFEHA aggregation operator along with some of its basic properties, such as idempotency, boundedness, and monotonicity.

Definition 9. The PFEHA aggregation operator can be defined as follows:

where α˙σ(j) is the j^th largest of the weighted Pythagorean fuzzy values α˙j(α˙j=nωjαj), w=(w₁, w₂, w₃, …, w_n)^T is the weighted vector of the PFEHA operator such that w_j∈[0, 1], and ∑j=1nwj=1. ω=(ω₁, ω₂, ω₃, …, ω_n)^T is the weighted vector of α_j(j=1, 2, 3, …, n) such that ω_j∈[0, 1], ∑j=1nωj=1, and n is the balancing coefficient, which plays a role of balance. If the vector w=(w₁, w₂, w₃, …, w_n)^T approaches to (1n, 1n, 1n, …, 1n)T, then the vector (nω₁α₁, nω₂α₂, nω₃α₃, …, nω_nα_n)^T approaches to (α₁, α₂, α₃, …, α_n)^T.

Theorem 1. Let αj=〈μαj, ηαj〉(j=1, 2, …, n) be a collection of Pythagorean fuzzy values, then their aggregated value by using the PFEHA aggregation operator is also a Pythagorean fuzzy value, and

Proof. We can prove this theorem by mathematical induction on n.

For n=2

w1α˙1=((1+μα˙12)w1−(1−μα˙12)w1(1+μα˙12)w1+(1−μα˙12)w1, 2(ηα˙12)w1(2−ηα˙12)w1+(ηα˙12)w1)

and

w2α˙2=((1+μα˙22)w2−(1−μα˙22)w2(1+μα˙22)w2+(1−μα˙22)w2, 2(ηα˙22)w2(2−ηα˙22)w2+(ηα˙22)w2).

Then

PFEHAω,w(α1, α2)=(∏j=12(1+μα˙σ(j)2)wj−∏j=12(1−μα˙σ(j)2)wj∏j=12(1+μα˙σ(j)2)wj+∏j=12(1−μα˙σ(j)2)wj, 2∏j=12(ηα˙σ(j)2)wj∏j=12(2−ηα˙σ(j)2)wj+∏j=12(ηα˙σ(j)2)wj).

Thus, the result is true for n=2. Now, we assume that Eq. (8) holds for n=k. Thus

PFEHAω,w(α1, α2, α3, …, αk)=(∏j=1k(1+μα˙σ(j)2)wj−∏j=1k(1−μα˙σ(j)2)wj∏j=1k(1+μα˙σ(j)2)wj+∏j=1k(1−μα˙σ(j)2)wj, 2∏j=1k(ηα˙σ(j)2)wj∏j=1k(2−ηα˙σ(j)2)wj+∏j=1k(ηα˙σ(j)2)wj).

If Eq. (8) is true for n=k, then we show that Eq. (8) is true for n=k+1. Thus

Let

p1=∏j=1k(1+μα˙σ(j)2)wj−∏j=1k(1−μα˙σ(j)2)wj, q1=∏j=1k(1+μα˙σ(j)2)wj+∏j=1k(1−μα˙σ(j)2)wjp2=(1+μα˙k+12)wk+1−(1−μα˙k+12)wk+1, q2=(1+μα˙k+12)wk+1+(1−μα˙k+12)wk+1, r1=2∏j=1k(ηα˙σ(j)2)wjs1=∏j=1k(2−ηα˙σ(j)2)wj+∏j=1k(ηα˙σ(j)2)wj, s2=(2−ηα˙k+12)wk+1+(ηα˙k+12)wk+1, r2=2(ηα˙k+12)wk+1.

Now, putting these values in Eq. (9), we have

PFEHAω,w(α1, α2, α3, …, αk+1)=(p1q1, r1s1)⊕ε(p2q2, r2s2).

By using the Einstein operation law, we have

Now, putting the values of p12q22+p22q12, q12q22+p12p22, r1r2, 2s12s22−s12r22−r12s22+r12r22 in Eq. (10), then

PFEHAω,w(α1, α2, α3, …, αk+1)=(∏j=1k+1(1+μα˙σ(j)2)wj−∏j=1k+1(1−μα˙σ(j)2)wj∏j=1k+1(1+μα˙δ(j)2)wj+∏j=1k+1(1−μα˙σ(j)2)wj, 2∏j=1k+1(ηα˙σ(j)2)wj∏j=1k+1(2−ηα˙σ(j)2)wj+∏j=1k+1(ηα˙σ(j)2)wj) .

Thus, Eq. (8) is true for n=k+1. Thus, Eq. (8) is true for all n.□

Lemma 1 ([24], [27]). Let α_j≻0, w_j≻0(j=0, 2, …n) and ∑j=1nwj=1, then

where the equality holds if and only if α₁=α₂=…=α_n.

Theorem 2. Let αj=〈μαj, ηαj〉(j=1, 2, 3, …, n) be a collection of Pythagorean fuzzy values, then

Proof. As

∏j=1n(1+μα˙σ(j)2)wj+∏j=1n(1−μα˙σ(j)2)wj≤∑j=1nwj(1+μα˙σ(j)2)+∑j=1nwj(1−μα˙σ(j)2).

Also

∑j=1nwj(1+μα˙σ(j)2)+∑j=1nwj(1−μα˙σ(j)2)=2,

then

∏j=1n(1+μα˙σ(j)2)wj+∏j=1n(1−μα˙σ(j)2)wj≤2,

thus

where the quality holds if and only if μα˙σ(j)(j=1, 2, 3, …, n) are equal. Again

∏j=1n(2−ηα˙σ(j)2)wj+∏j=1n(ηα˙σ(j)2)wj≤∑j=1nwj(2−ηα˙σ(j)2)+∑j=1nwj(ηα˙σ(j)2).

Also

∑j=1nwj(2−ηα˙σ(j)2)+∑j=1nwj(ηα˙σ(j)2)=2

then

∏j=1n(2−ηα˙σ(j)2)wj+∏j=1n(ηα˙σ(j)2)wj≤2,

thus,

where the quality holds if and only if ηα˙σ(j)2(j=1, 2, 3, …, n) are equal.

Let

and

Then, Eqs. (13) and (14) can be transformed into the following forms:

thus

If

then

If

then

thus

From Eqs. (20) to (23), Eq. (12) always holds.□

Example 1: Let

α1=(0.4, 0.7), α2=(0.5, 0.8),α3=(0.6, 0.7), α4=(0.7, 0.6),

and w=(0.1, 0.2, 0.3, 0.4)^T, then

α˙1=(0.259, 0.867), α˙2=(0.456, 0.836),α˙3=(0.643, 0.651), α˙4=(0.812, 0.441).

By calculating the scores function, we have

s(α˙1)=−0.684, s(α˙2)=−0.491,s(α˙3)=−0.010, s(α˙4)=0.465.

Hence,

s(α˙4)≻s(α˙3)≻s(α˙2)≻s(α˙1).

Thus

PFHAω,w(α1, α2, α3, α4)=(1−∏j=14(1−μα˙σ(j)2)wj, ∏j=14(ηα˙σ(j))wj)=(0.517, 0.717).

Now applying the PFEHA operator, we have

α˙1=(0.253, 0.882), α˙2=(0.448, 0.841),α˙3=(0.650, 0.641), α˙4=(0.833, 0.402).

By calculating the scores function, we have

s(α˙1)=−0.711, s(α˙2)=−0.505,s(α˙3)=0.012, s(α˙4)=0.532.

As

s(α˙4)≻s(α˙3)≻s(α˙2)≻s(α˙1),

thus

PFEHAω,w(α1, α2, α3, α4)=(∏j=14(1+μα˙σ(j)2)wj−∏j=14(1−μα˙σ(j)2)wj∏j=14(1+μα˙σ(j)2)wj+∏j=14(1−μα˙σ(j)2)wj, 2∏j=14(ηα˙σ(j)2)wj∏j=14(2−ηα˙σ(j)2)wj+∏j=14(ηα˙σ(j)2)wj).=(0.507, 0.742)

Theorem 3. Let αj=〈μαj, ηαj〉(j=1, 2, 3, …, n) be a collection of Pythagorean fuzzy values, then the following properties hold:

1. Idempotency: If α˙σ(j)=α˙, then

   (24) PFEHAω,w(α1, α2, α3, …, αn)=α˙.

2. Boundedness:

   (25) α˙min≤PFEHAω,w(α1, α2, α3, …, αn)≤α˙max,

   where

   (26) α˙min=(minjμα˙σ(j), maxjηα˙σ(j)),

   (27) α˙max=(maxjμα˙σ(j), minjηα˙σ(j)).

3. Monotonicity: Let ασ(j)∗=〈μασ(j)∗, ηασ(j)∗〉(j=1, 2, …, n) be a collection of Pythagorean fuzzy values, and μασ(j)≤μασ(j)∗, ηασ(j)≥ηασ(j)∗, for all j, then

   (28) PFEHAω,w(α1, α2, α3, …, αn)≤PFEHAω,w(α1∗, α2∗, α3∗, …, αn∗).