# A Novel Pythagorean Group Decision-Making Method Based on Evidence Theory and Interactive Power Averaging Operator

Authors: Zhaohe Wan; Minghua Shi; Feng Yang; Guang Zhu
Journal: Complexity
 (2021)

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## Abstract

Since Pythagorean fuzzy sets can better reflect the cognition of the decision objects for experts, researchers have begun to pay increasingly more attention to them in recent years. The majority of the research on Pythagorean fuzzy environment assumes that the decision maker is completely rational and does not consider the correlation among the attribute variables. In view of the above, this paper proposes a method to solve the multiple attribute group decision-making problem based on D-S theory and interactive power averaging operator. First, the new Pythagorean fuzzy interactive weighted power average operator is designed to aggregate the attribute evaluation information given by experts one by one, and the comprehensive evaluation information of each expert is obtained. Then, the expert comprehensive evaluation information is aggregated by the rule of evidence combination to obtain the comprehensive evidence information and confidence interval of each candidate. Then, the decision-making method for candidate alternatives is performed by the possibility discriminant rule. The design method considers not only the decision makers’ bounded rationality but also the correlation among the attribute variables. Finally, the selection of the energy exploitation plan illustrates the feasibility and effectiveness of the proposed group decision method.


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## Body

## 1. Introduction

Since Bellman and Zadeh formally proposed the concept of fuzzy decision in 1970, it has attracted a large number of scholars to study this field of decision-making due to its wide application [1]. With the deepening of research, traditional fuzzy sets can no longer meet the needs of modern decision science. Some extended fuzzy set concepts have been proposed, such as interval fuzzy sets, triangular fuzzy sets, trapezoidal fuzzy sets, type-2 fuzzy sets, and type-n fuzzy sets. However, these fuzzy sets have not broken through the limitations of the dichotomy and cannot represent the psychological characteristics of negation and hesitation that people show in the decision-making process. For this reason, Atanassov expands fuzzy sets and proposes the concept of intuitionistic fuzzy sets (IFS), which uses membership, nonmembership, and hesitation to represent information more delicately [2]. In view of the fact that intuitionistic fuzzy sets can flexibly represent decision-making information from multiple perspectives of membership, nonmembership, and hesitation, it has attracted the attention of many scholars and has been widely used to solve fuzzy decision problems. In 2014, Yager proposed Pythagorean fuzzy sets on the basis of intuitionistic fuzzy sets, and the constraint conditions of intuitionistic fuzzy sets were reduced such that the sum of the squares of membership and nonmembership was less than 1 [3].After several years of development, Pythagorean fuzzy set theory and its application research have obtained some achievements [4–7]. Zhang et al. defined some similarity measures of the Pythagorean fuzzy sets and discussed their properties [8]. They also proposed a Pythagorean fuzzy decision-making method to solve the intelligent medical diagnostic problem [8]. Wan et al. further proposed a new order relation for elements of Pythagorean fuzzy sets and then used it to solve decision-making problems [9]. Ejegwa defined the triparametric correlation coefficients of Pythagorean fuzzy sets and verified that they can capture the three orthodox parameters of Pythagorean fuzzy sets [10]. In order to measure the closeness between Pythagorean fuzzy information units, Thao analyzed the existing defects of intuitionistic fuzzy similarity calculation methods, proposed a new similarity calculation formula to measure the Pythagorean fuzzy numbers, and used it to solve the multiattribute decision-making problem [11]. Chinnadurai et al. proposed the Pythagorean fuzzy soft sets and defined its basic operations such as addition, intersection, union, and complement. Based on this, a multiple attribute decision-making method using the integration operator was designed to solve the human resource management problem [12]. Zhang et al. introduced Pythagorean fuzzy preference relations and its additive consistency, which expanded the application scope of preference relations [13]. Given that the Muirhead mean operator can capture the correlation between multiple decision variables, Liu et al. discussed the relevant properties of the Muirhead mean operator in the Pythagorean fuzzy linguistic environment and proposed a Pythagorean linguistic multiple attribute decision-making method [14]. Considering the cross relationship between the degree of membership and the degree of nonmembership, Wang et al. proposed the interactive algorithm of Pythagorean fuzzy number and some collections based on interactive operations and used it to solve the multiple attribute decision-making problem of enterprise resource planning [15]. Combining the dual hesitant fuzzy sets and Pythagorean fuzzy sets, Ji et al. proposed probabilistic dual hesitant Pythagorean fuzzy sets and designed a new decision-making method to solve the multiple attribute decision-making problem in portfolio selection [16]. Wan et al. developed a Pythagorean fuzzy group decision method based on the degree of truth and information entropy [17]. Akram et al. proposed a novel approach based on the Hamacher operator for complex Pythagorean fuzzy decision-making [18]. Considering group utility measures and individual regret measures, Ma et al. developed a method to solve Pythagorean fuzzy multiple attribute group decision-making [19]. Wan et al. developed a novel TOPSIS method for Pythagorean fuzzy multiple attribute group decision-making [20]. Akram et al. proposed Pythagorean fuzzy Yager aggregation operators and then used them to solve the multicriteria decision-making problem [21].However, the abovementioned Pythagorean fuzzy decision-making problems are solved under the assumption that the decision maker is completely rational, and this assumption is difficult to satisfy in actual decision-making. Evidence theory, also known as Dempster–Shafer theory, or D-S theory for short, is based on decision makers’ understanding of objective things; based on the evidence and knowledge that decision makers possess, D-S theory provides uncertainty measures for fuzzy decision information. This way of portraying uncertainty is closer to human thinking habits and can reflect the bounded rational behaviour of experts in the decision-making process. This feature makes evidence theory widely used to solve fuzzy decision-making problems. However, to date, no evidence has been used to synthesize Pythagorean fuzzy information. In addition, the power average operator [22] is a classic fuzzy information gathering tool. Since it was proposed, it has received attention from many scholars and has been extended to different fuzzy environments. The common power average operators are the hesitant fuzzy power aggregation operator, intuitionistic fuzzy power average operator, and triangular intuitionistic fuzzy power average operator; they are used to solve different ambiguities in the decision-making information aggregation problem in a fuzzy environment. However, these power average operators can gather fuzzy information only from the perspective of the overall equilibrium and cannot capture the relevant information between attributes. He et al. stated that interactive computing can capture the relevant information between attributes and the information between membership and nonmembership [24]. In view of the above analysis, this paper proposes an improved power averaging operator, the Pythagorean interactive power averaging operator, combined with evidence theory, and a Pythagorean fuzzy group decision method under bounded rationality.


## 2. Evidence Theory

The recognition framework refers to a limited and complete set of all possible results of a certain proposition. The construction of the recognition framework can transform abstract propositional operations into set operations. Important concepts such as the trust function, likelihood function, and evidence combination function of D-S theory are all based on the recognition framework.Definition 1 (see [25]).
F is set as the DS identification framework, andF=F1,F2,…,Fnwhich is a set of mutually independent conclusions. Then, the basic probability assignment (BPA) function of the framework, also known as the mass function, is the mapping: m:2F⟶0,1, which satisfies the following conditions: (1) m∅=0; 2) and (2) ∑A∈FmA=1. Among the components, ∅ is the empty set, A is any subset of F, and2F=∅,F1,…,Fn,F1,F2,…,F1,Fn,…,F is the set consisting of all subsets of F.Definition 2 (see [25]).
LetF be a D-S recognition framework, F=F1,F2,…,Fn be a set of mutually independent conclusions, and A and B be subsets of F. Then, the belief measure (Bel) function and the plausibility measure (Pl) function are as follows:(1)BelA=∑B⊆AmB,PlA=∑B∩A≠∅mB,where BelA represents the credibility that the evidence A must occur and PlA represents the credibility that the evidence A may occur. Obviously, there is PlA≥BelA for any subset A of F, so the support interval of A can be expressed as [BelA, PlA], also known as the confidence interval of A. In order to fuse multiple pieces of evidentiary information to implement optimal decision-making, Dempster proposed the following orthogonal sum rule.Theorem 1.
(see [26]). Suppose that m1,m2,…,mn is the mass function under the same recognition framework, and the corresponding evidence is A1,A2,…,An. Then,(2)mA=m1A1⊕m2A2⊕⋯mnAn=0,A=∅,∑A1∩A2∩⋯∩An=Am1A1m2A2⋯mnAn1−∑A1∩A2∩⋯∩An=∅m1A1m2A2⋯mnAn,A≠∅.


## 3. Pythagorean Interactive Power Average Operator

### 3.1. Pythagorean Fuzzy Sets

Definition 3.
(see [7]). Suppose that X is a given universe of discourse. Then, a Pythagorean fuzzy set A on X is A=x,rx,dx|x∈X, where rx is the strength of the commitment given x and dxis the bias value of the commitment, rx∈0,1 and dx∈0,1.
Zhang and Xu further proposed the concept of a Pythagorean fuzzy number. That is, the binary array ofα=Pμα,υα is called a Pythagorean fuzzy number, where μα and υα satisfy the constraint conditions of μα2+υα2≤1. The following algorithms and binary relations are given to open the application space of Pythagorean fuzzy sets.Definition 4.
(see [8]). Let αi=Pμαi,υαii=1,2 and α=Pμα,υα be three Pythagorean fuzzy numbers. Then,(1)
α1⊕α2=Pμα12+μα22−μα12μα22,υα1υα2(2)
α1⊗α2=Pμα1μα2,υα12+υα22−υα12υα22(3)
λα=P1−1−μα2λ,υαλ; λ>0(4)
αλ=Pμαλ,1−1−υα2λ; λ>0
However, the above algorithm does not consider the interaction between the degree of membership and the degree of nonmembership, and abnormal phenomena may occur in the calculation.Example 1.
Supposeα1=P0.6,0.4, α2=P0.5,0.6, and α3=P0.7,0.0 are three Pythagorean fuzzy numbers; the corresponding weight vector is w=w1,w2,w3T=0.25,0.4,0.35T. Then, the arithmetic weighted average of αii=1,2,3 obtained by Definition 4 is(3)α=w1α1⊕w2α2⊕w3α3=P0.6045,0.
This means that when calculating the arithmetic weighted average ofαii=1,2,3, the nonmembership of α1 and α2 has no effect, which is puzzling. For this reason, Wei and others stipulated the following interactive Pythagorean fuzzy number algorithm.Definition 5.
(see [27]). Let αi=Pμαi,υαii=1,2 and α=Pμα,υα be three Pythagorean fuzzy numbers. Then,(1)
α1⊕α2=Pμα12+μα22−μα12μα22,υα12+υα22−υα12υα22−μα12υα22−υα12μα22(2)
α1⊗α2=Pμα12+μα22−μα12μα22−υα12μα22−μα12υα22,υα12+υα22−υα12υα22(3)
λα=P1−1−μα2λ,1−μα2λ−1−μα2+υα2λ; λ>0(4)
αλ=P1−υα2λ−1−μα2+υα2λ,1−1−υα2λ; λ>0
Using the interactive algorithm, recalculate Example1 to obtain(4)α=w1α1⊕w2α2⊕w3α3=P0.6084,0.4226.
minυα1,υα2,υα3≤υα=0.4226≤maxυα1,υα2,υα3 shows that the interactive algorithm is closer to the actual situation.
Huang et al. stipulated the following Pythagorean fuzzy number ranking criteria and distance measure.Definition 6.
(see [30]). Set α=Pμα,υα as a Pythagorean fuzzy number. Then, the score function of α is sα=μα2−υα2. For any two Pythagorean fuzzy numbers α1 and α2, the following sorting criteria exist:(1)
Ifsα1>sα2, then α1>α2(2)
Ifsα1=sα2, then α1=α2(3)
Ifsα1<sα2, then α1<α2Definition 7.
(see [30]). Let αi=Pμαi,υαii=1,2 be two Pythagorean fuzzy numbers. Then, the distance between α1 and α2 is(5)dα1,α2=12μα1−μα2+υα1−υα2+πα1−πα2,where παi=1−μαi2−υαi2i=1,2 is the hesitation of αii=1,2.


### 3.2. Pythagorean Interactive Power Average Operator

Based on the integrity of the data, Yager proposed a power average operator that can reduce the impact of abnormal information.Definition 8.
(see [22]). Let aii=1,2,…,n be a sequence of real numbers. If(6)PAa1,a2,…,an=∑i=1n1+Tai∑t=1n1+Tatai,then PA is called the power average operator. Among them, Tai=∑j=1,j≠inSupai,aj; Supai,aj represents the degree of support of ai and aj, and it meets the following conditions:(1)
Supai,aj∈0,1(2)
Supai,aj=Supaj,ai(3)
Ifdai,aj≤dal,ak, then Supai,aj≥Supal,ak
However, the power average operator can only capture the balance of the attribute information, not the associated information. For this reason, this paper proposes an improved power average operator based on the Pythagorean interactive operation: the Pythagorean interactive power average operator.Definition 9.
Letαi=Pμαi,υαii=1,2,…,n be a list of Pythagorean fuzzy numbers with weights of wii=1,2,…,n, where wi∈0,1 and ∑i=1nwi=1. Then, the Pythagorean fuzzy interactive weighted power average operator is PFIWPA:(7)PFIWPAα1,α2,…,αn=⊕i=1nwi1+Tαi∑t=1nwt1+Tαtαi,where Tαi=∑j=1,j≠inSupαi,αj and Supαi,αj is the degree of support of αi and αj, which satisfies the constraints:(1)
Supαi,αj∈0,1(2)
Supαi,αj=Supαj,αi(3)
Ifdαi,αj≤dαl,αk, then Supαi,αj≥Supαl,αkTheorem 2.
Letαi=Pμαi,υαii=1,2,…,n be a list of Pythagorean fuzzy numbers with weights of wii=1,2,…,n, where wi∈0,1 and ∑i=1nwi=1. Then, the aggregation value obtained by the PFIWPA operator is also a Pythagorean fuzzy number, and(8)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1n1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.Proof.
(1)
Whenα1,α2=⊕i=12wi1+Tαi/∑t=12wt1+Tαtαi,(9)PFIWPAα1,α2=⊕i=12wi1+Tαi∑t=12wt1+Tαtαi.It can be seen from Definition5 that(10)w11+Tα1∑t=12wt1+Tαtα1=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt,1−μα12w11+Tα1/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt,w21+Tα2∑t=12wt1+Tαtα2=P1−1−μα22w21+Tα2/∑t=12wt1+Tαt,1−μα22w21+Tα2/∑t=12wt1+Tαt−1−μα22+υα22w21+Tα2/∑t=12wt1+Tαt.There is(11)PFIWPAα1,α2=⊕i=12wi1+Tαi∑t=12wt1+Tαtαi=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt,1−μα12w11+Tα1/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt,1−μα12w11+Tα1/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt⊕P1−1−μα22w21+Tα2/∑t=12wt1+Tαt,1−μα22w21+Tα2/∑t=12wt1+Tαt−1−μα22+υα22w21+Tα2/∑t=12wt1+Tαt=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt1−μα22w21+Tα2/∑t=12wt1+Tαt,1−1−μα12w11+Tα1/∑t=12wt1+Tαt1−μα22w21+Tα2/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt1−μα22+υα22w21+Tα2/∑t=12wt1+Tαt=P1−∏j=121−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=121−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=121−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.Therefore, whenα1,α2,…,αk⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi, formula (8) is established.(2)
Assumingα1,α2,…,αk=⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi (α1,α2,…,αk=⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi),formula (8) is established; considering α1,α2,…,αk=⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi, formula (8) is still established:(12)PFIWPAα1,α2,…,αk=⊕i=1kwi1+Tαi∑t=1nwt1+Tαtαi=P1−∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1k1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.
Then,(13)PFIWPAα1,α2,…,αk+1=⊕i=1k+1wi1+Tαi∑t=1nwt1+Tαtαi=⊕i=1kwi1+Tαi∑t=1nwt1+Tαtαi⊕wi1+Tαi∑t=1nwt1+Tαtαk+1=P1−∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1k1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt⊕P1−1−μαk+12wk+11+Tαk+1/∑t=1k+1wt1+Tαt,1−μαk+12wk+11+Tαk+1/∑t=1k+1wt1+Tαt−1−μαk+12+υαk+12wk+11+Tαk+1/∑t=1k+1wt1+Tαt=P1−∏j=1k+11−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1k+11−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1k+11−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.
Therefore, whenn≥2, formula (8) is also established.
In summary, formula (8) is established when n≥2 from the mathematical induction; obviously, the aggregation value of the PFIWPA operator is still a Pythagorean fuzzy number.
Note that ifTαi=ci=1,2,…,n, then(14)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1n1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt=P1−∏j=1n1−μαj2wj1+c/∑t=1nwt1+c,∏j=1n1−μαj2wj1+c/∑t=1nwt1+c−∏j=1n1−μαj2+υαj2wj1+c/∑t=1nwt1+c=P1−∏j=1n1−μαj2wj,∏j=1n1−μαj2wj−∏j=1n1−μαj2+υαj2wj=PFIWAα1,α2,…,αn.
That is, the PFIWPA operator degenerates into the Pythagorean fuzzy interactive weighted average (PFIWA) operator.Theorem 3.
Letαi=Pμαi,υαi be a list of Pythagorean fuzzy numbers. If αi=α=Pμα,υαi=1,2,…,n, then PFIWPA α1,α2,…,αn=αProof.
(15)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μα2wj1+Tα/∑t=1nwt1+Tα,∏j=1n1−μα2wj1+Tα/∑t=1nwt1+Tα−∏j=1n1−μα2+υα2wj1+Tα/∑t=1nwt1+Tα=P1−∏j=1n1−μα2wj,∏j=1n1−μα2wj−∏j=1n1−μα2+υα2wj=P1−1−μα2∑j=1nwj,1−μα2∑j=1nwj−1−μα2+υα2∑j=1nwj=P1−1−μα2,1−μα2−1−μα2+υα2.Theorem 4.
Letαi,αi′i=1,2,…,n can be two columns of Pythagorean fuzzy numbers.
Ifα1′,α2′,…,αn′ is any permutation of α1,α2,…,αn, then PFIWPAα1,α2,…,αn=PFIWPAα1′,α2′,…,αn′.Proof.
(16)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1n1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt=P1−∏j=1n1−μαj′2wj′1+Tαj′/∑t=1nwt′1+Tαt′,∏j=1n1−μαj′2wj′1+Tαj′/∑t=1nwt′1+Tαt′−∏j=1n1−μαj′2+υαj′2wj′1+Tαj′/∑t=1nwt′1+Tαt′=PFIWPAα1′,α2′,…,αn′.


## 3.1. Pythagorean Fuzzy Sets

Definition 3.
(see [7]). Suppose that X is a given universe of discourse. Then, a Pythagorean fuzzy set A on X is A=x,rx,dx|x∈X, where rx is the strength of the commitment given x and dxis the bias value of the commitment, rx∈0,1 and dx∈0,1.
Zhang and Xu further proposed the concept of a Pythagorean fuzzy number. That is, the binary array ofα=Pμα,υα is called a Pythagorean fuzzy number, where μα and υα satisfy the constraint conditions of μα2+υα2≤1. The following algorithms and binary relations are given to open the application space of Pythagorean fuzzy sets.Definition 4.
(see [8]). Let αi=Pμαi,υαii=1,2 and α=Pμα,υα be three Pythagorean fuzzy numbers. Then,(1)
α1⊕α2=Pμα12+μα22−μα12μα22,υα1υα2(2)
α1⊗α2=Pμα1μα2,υα12+υα22−υα12υα22(3)
λα=P1−1−μα2λ,υαλ; λ>0(4)
αλ=Pμαλ,1−1−υα2λ; λ>0
However, the above algorithm does not consider the interaction between the degree of membership and the degree of nonmembership, and abnormal phenomena may occur in the calculation.Example 1.
Supposeα1=P0.6,0.4, α2=P0.5,0.6, and α3=P0.7,0.0 are three Pythagorean fuzzy numbers; the corresponding weight vector is w=w1,w2,w3T=0.25,0.4,0.35T. Then, the arithmetic weighted average of αii=1,2,3 obtained by Definition 4 is(3)α=w1α1⊕w2α2⊕w3α3=P0.6045,0.
This means that when calculating the arithmetic weighted average ofαii=1,2,3, the nonmembership of α1 and α2 has no effect, which is puzzling. For this reason, Wei and others stipulated the following interactive Pythagorean fuzzy number algorithm.Definition 5.
(see [27]). Let αi=Pμαi,υαii=1,2 and α=Pμα,υα be three Pythagorean fuzzy numbers. Then,(1)
α1⊕α2=Pμα12+μα22−μα12μα22,υα12+υα22−υα12υα22−μα12υα22−υα12μα22(2)
α1⊗α2=Pμα12+μα22−μα12μα22−υα12μα22−μα12υα22,υα12+υα22−υα12υα22(3)
λα=P1−1−μα2λ,1−μα2λ−1−μα2+υα2λ; λ>0(4)
αλ=P1−υα2λ−1−μα2+υα2λ,1−1−υα2λ; λ>0
Using the interactive algorithm, recalculate Example1 to obtain(4)α=w1α1⊕w2α2⊕w3α3=P0.6084,0.4226.
minυα1,υα2,υα3≤υα=0.4226≤maxυα1,υα2,υα3 shows that the interactive algorithm is closer to the actual situation.
Huang et al. stipulated the following Pythagorean fuzzy number ranking criteria and distance measure.Definition 6.
(see [30]). Set α=Pμα,υα as a Pythagorean fuzzy number. Then, the score function of α is sα=μα2−υα2. For any two Pythagorean fuzzy numbers α1 and α2, the following sorting criteria exist:(1)
Ifsα1>sα2, then α1>α2(2)
Ifsα1=sα2, then α1=α2(3)
Ifsα1<sα2, then α1<α2Definition 7.
(see [30]). Let αi=Pμαi,υαii=1,2 be two Pythagorean fuzzy numbers. Then, the distance between α1 and α2 is(5)dα1,α2=12μα1−μα2+υα1−υα2+πα1−πα2,where παi=1−μαi2−υαi2i=1,2 is the hesitation of αii=1,2.


## 3.2. Pythagorean Interactive Power Average Operator

Based on the integrity of the data, Yager proposed a power average operator that can reduce the impact of abnormal information.Definition 8.
(see [22]). Let aii=1,2,…,n be a sequence of real numbers. If(6)PAa1,a2,…,an=∑i=1n1+Tai∑t=1n1+Tatai,then PA is called the power average operator. Among them, Tai=∑j=1,j≠inSupai,aj; Supai,aj represents the degree of support of ai and aj, and it meets the following conditions:(1)
Supai,aj∈0,1(2)
Supai,aj=Supaj,ai(3)
Ifdai,aj≤dal,ak, then Supai,aj≥Supal,ak
However, the power average operator can only capture the balance of the attribute information, not the associated information. For this reason, this paper proposes an improved power average operator based on the Pythagorean interactive operation: the Pythagorean interactive power average operator.Definition 9.
Letαi=Pμαi,υαii=1,2,…,n be a list of Pythagorean fuzzy numbers with weights of wii=1,2,…,n, where wi∈0,1 and ∑i=1nwi=1. Then, the Pythagorean fuzzy interactive weighted power average operator is PFIWPA:(7)PFIWPAα1,α2,…,αn=⊕i=1nwi1+Tαi∑t=1nwt1+Tαtαi,where Tαi=∑j=1,j≠inSupαi,αj and Supαi,αj is the degree of support of αi and αj, which satisfies the constraints:(1)
Supαi,αj∈0,1(2)
Supαi,αj=Supαj,αi(3)
Ifdαi,αj≤dαl,αk, then Supαi,αj≥Supαl,αkTheorem 2.
Letαi=Pμαi,υαii=1,2,…,n be a list of Pythagorean fuzzy numbers with weights of wii=1,2,…,n, where wi∈0,1 and ∑i=1nwi=1. Then, the aggregation value obtained by the PFIWPA operator is also a Pythagorean fuzzy number, and(8)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1n1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.Proof.
(1)
Whenα1,α2=⊕i=12wi1+Tαi/∑t=12wt1+Tαtαi,(9)PFIWPAα1,α2=⊕i=12wi1+Tαi∑t=12wt1+Tαtαi.It can be seen from Definition5 that(10)w11+Tα1∑t=12wt1+Tαtα1=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt,1−μα12w11+Tα1/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt,w21+Tα2∑t=12wt1+Tαtα2=P1−1−μα22w21+Tα2/∑t=12wt1+Tαt,1−μα22w21+Tα2/∑t=12wt1+Tαt−1−μα22+υα22w21+Tα2/∑t=12wt1+Tαt.There is(11)PFIWPAα1,α2=⊕i=12wi1+Tαi∑t=12wt1+Tαtαi=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt,1−μα12w11+Tα1/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt,1−μα12w11+Tα1/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt⊕P1−1−μα22w21+Tα2/∑t=12wt1+Tαt,1−μα22w21+Tα2/∑t=12wt1+Tαt−1−μα22+υα22w21+Tα2/∑t=12wt1+Tαt=P1−1−μα12w11+Tα1/∑t=12wt1+Tαt1−μα22w21+Tα2/∑t=12wt1+Tαt,1−1−μα12w11+Tα1/∑t=12wt1+Tαt1−μα22w21+Tα2/∑t=12wt1+Tαt−1−μα12+υα12w11+Tα1/∑t=12wt1+Tαt1−μα22+υα22w21+Tα2/∑t=12wt1+Tαt=P1−∏j=121−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=121−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=121−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.Therefore, whenα1,α2,…,αk⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi, formula (8) is established.(2)
Assumingα1,α2,…,αk=⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi (α1,α2,…,αk=⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi),formula (8) is established; considering α1,α2,…,αk=⊕i=1kwi1+Tαi/∑t=1nwt1+Tαtαi, formula (8) is still established:(12)PFIWPAα1,α2,…,αk=⊕i=1kwi1+Tαi∑t=1nwt1+Tαtαi=P1−∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1k1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.
Then,(13)PFIWPAα1,α2,…,αk+1=⊕i=1k+1wi1+Tαi∑t=1nwt1+Tαtαi=⊕i=1kwi1+Tαi∑t=1nwt1+Tαtαi⊕wi1+Tαi∑t=1nwt1+Tαtαk+1=P1−∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1k1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1k1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt⊕P1−1−μαk+12wk+11+Tαk+1/∑t=1k+1wt1+Tαt,1−μαk+12wk+11+Tαk+1/∑t=1k+1wt1+Tαt−1−μαk+12+υαk+12wk+11+Tαk+1/∑t=1k+1wt1+Tαt=P1−∏j=1k+11−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1k+11−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1k+11−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt.
Therefore, whenn≥2, formula (8) is also established.
In summary, formula (8) is established when n≥2 from the mathematical induction; obviously, the aggregation value of the PFIWPA operator is still a Pythagorean fuzzy number.
Note that ifTαi=ci=1,2,…,n, then(14)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1n1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt=P1−∏j=1n1−μαj2wj1+c/∑t=1nwt1+c,∏j=1n1−μαj2wj1+c/∑t=1nwt1+c−∏j=1n1−μαj2+υαj2wj1+c/∑t=1nwt1+c=P1−∏j=1n1−μαj2wj,∏j=1n1−μαj2wj−∏j=1n1−μαj2+υαj2wj=PFIWAα1,α2,…,αn.
That is, the PFIWPA operator degenerates into the Pythagorean fuzzy interactive weighted average (PFIWA) operator.Theorem 3.
Letαi=Pμαi,υαi be a list of Pythagorean fuzzy numbers. If αi=α=Pμα,υαi=1,2,…,n, then PFIWPA α1,α2,…,αn=αProof.
(15)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μα2wj1+Tα/∑t=1nwt1+Tα,∏j=1n1−μα2wj1+Tα/∑t=1nwt1+Tα−∏j=1n1−μα2+υα2wj1+Tα/∑t=1nwt1+Tα=P1−∏j=1n1−μα2wj,∏j=1n1−μα2wj−∏j=1n1−μα2+υα2wj=P1−1−μα2∑j=1nwj,1−μα2∑j=1nwj−1−μα2+υα2∑j=1nwj=P1−1−μα2,1−μα2−1−μα2+υα2.Theorem 4.
Letαi,αi′i=1,2,…,n can be two columns of Pythagorean fuzzy numbers.
Ifα1′,α2′,…,αn′ is any permutation of α1,α2,…,αn, then PFIWPAα1,α2,…,αn=PFIWPAα1′,α2′,…,αn′.Proof.
(16)PFIWPAα1,α2,…,αn=P1−∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt,∏j=1n1−μαj2wj1+Tαj/∑t=1nwt1+Tαt−∏j=1n1−μαj2+υαj2wj1+Tαj/∑t=1nwt1+Tαt=P1−∏j=1n1−μαj′2wj′1+Tαj′/∑t=1nwt′1+Tαt′,∏j=1n1−μαj′2wj′1+Tαj′/∑t=1nwt′1+Tαt′−∏j=1n1−μαj′2+υαj′2wj′1+Tαj′/∑t=1nwt′1+Tαt′=PFIWPAα1′,α2′,…,αn′.


## 4. Pythagorean Fuzzy Multiple Attribute Group Decision-Making Method

The problem of the multiple attribute group method in Pythagorean fuzzy environment can be described as follows: the existing decision plan setY=Y1,Y2,…,Yn, C=c1,c2,…,cm is the m attributes of the plan, the attribute weight is w=w1,w2,…,wm, D1,D2,…,Dl is a group of decision makers, the weight vector is λ=λ1,λ2,…,λm, and hijkrepresents the Pythagorean fuzzy evaluation value given by the kth decision expert Dk to the attribute cj of the scheme Yi. The method matrix provided by the decision expert Dk can be expressed as Hk=αijkn×m. The steps of the group decision-making method of the interactive power average operator are as follows:Step 1: convert the Pythagorean fuzzy decision matrixHk into a canonical matrix H¯k=α¯ijkn×m, where α¯ijk=αijkgj:benefit attributeNαijk=Pυαijk,μαijkgj:cost attribute.Step 2: calculate the degree of supportSupα¯lik,α¯ljk=1−dα¯lik,α¯ljk, i,j=1,2,…,m; l=1,2,…,n,dα¯lik,α¯ljk=1/2μα¯lik−μα¯ljk+υα¯lik−υα¯ljk+πα¯lik−πα¯ljk between variables.Step 3: calculate the supportTα¯lik=Tα¯lik=∑j=1,j≠imwjSupα¯lik,α¯ljk of the variable and the remaining variables and the support indices ηlik=wi1+Tα¯lik/wt1+Tα¯ltk, i=1,2,…,m and l=1,2,…,n of the variable.Step 4: use the PFIWPA operator to gather the attribute valuesα¯ijk, j=1,2,…,m, of the candidate solutions to obtain the comprehensive attribute values of the experts Dk about the candidate solutions xi:(17)αik=PFIWPAα¯i1k,α¯i2k,…,α¯imk,i=1,2,…,n=P1−∏j=1n1−μα¯ijk2wj1+Tαijk/∑t=12wt1+Tαitk,∏j=1n1−μα¯ijk2wj1+Tαijk/∑t=12wt1+Tαitk−∏j=1n1−μα¯ijk2+υα¯ijk2wj1+Tαijk/∑t=12wt1+Tαitk.Step 5: for each expert’s standardized comprehensive evaluation information on the plan, use the evidence synthesis method to gather the results.First, standardize the comprehensive evaluation informationαiksolved in step 3 to obtain standardized comprehensive evaluation information βik:(18)βik=Pμβik,υβik=λkPμαik,υαik,i=1,2,…,n,k=1,2,…,l.Regarding the comprehensive evaluation informationβik of Yi about the expert Dk as a special piece of evidence, βik=Pμβik,υβik contains three conclusions, namely, the degree of satisfaction, the degree of dissatisfaction, and the degree of uncertainty of Yi. If A indicates the degree of satisfaction of Dk regarding Yi, then mkiA=μβik/r; If R represents the degree of dissatisfaction of Dk regarding Yi, then mkiR=υβik/r; If A,R represents the degree of uncertainty of Dk regarding Yi, then mkiA,R=1−μβik2−υβik2/r, where r=1−μβik2−υβik2+υβik+μβik. Evidence synthesis rules are used to calculate the comprehensive evidence Mi of the plan Yi:(19)MiA=m1iA1⊕m2iA2⊕⋯⊕mliAl=∑A1∩A2∩⋯∩Al=Am1iA1m2iA2…mliAl1−∑A1∩A2∩⋯∩Al=∅m1iA1m2iA2…mliAl,where A,A1,A2,…,Al∈A,R,A,R.Step 6: give the confidence intervalIi of the candidate plan Yi regarding the comprehensive evidence Mi=PμMi,υMi, i.e., Ii=Belxi=μMixi=μMi+1−μMi2−υMi2. Then, give the probability matrix P=pijn×n between the candidate plans according to the confidence interval, where pij is the probability that the candidate plan Yi is superior to Yj. The calculation formula is pij=minPlxi−Belxi+Plxj−Belxj,maxPlxi−Belxj,0/Plxi−Belxi+Plxj−Belxj.Step 7: calculate the ranking vectorP1,P2,…,Pn of the possibility matrix P=pijn×n, where Pi=1/nn−1∑j=1npij−1+n/2,i=1,2,…,n.According to the size ofPi, the candidate scheme is selected for the optimal decision; the larger the value Pi, the better the scheme.


## 5. Case Analysis

Example 2.
Energy issues have always plagued mankind. Many countries and regions are committed to the development and utilization of new energy technologies. Nea-Kessani is a rural community in northern Greece. The region’s economy is in a recession. The local government has decided to use clean energy and advanced manufacturing to provide new jobs to society and realize the transformation from traditional agriculture to modern industry. The locals decided to develop low-enthalpy geothermal resources, evaluated five mining methods asxii=1,2,…,5, and formulated the following decision attributes based on its own characteristics: c1 is the net present value of investment, c2 is the number of jobs created, c3 is the resource utilization, c4 is safety, and c5 is social impact force. Its weight vector is w=0.2,0.4,0.1,0.1,0.2. The expert group evaluated the above five attributes with real numbers between 0 and 1 and formed a consensus evaluation opinion, giving the following evaluation matrix Hii=1,2,3. In order to achieve scientific and reasonable decision-making effects, the local government decided to hire three industry experts Dii=1,2,3 to evaluate the mining plan. In addition, weights were assigned according to their professional influence, and the weight vector is λ=0.35,0.3,0.35. After investigation, Pythagorean fuzzy decision matrix provided by the experts is represented in Tables 1–3.
Next, the decision-making method of this paper is used to select the best mining plan for the local area.Step 1: since the five evaluation indicators specified by the company are all benefit-based, there is no need to standardize, that is,H¯k=Hk=αijkn×m, k=1,2,3.Step 2: calculate the degree of support between variablesSupα¯lik,α¯ljk=1−dα¯lik,α¯ljk, i,j=1,2,…,m, l=1,2,…,n, and k=1,2,3. Taking the evaluation matrix H1 given by Expert 1 as an example, the following support matrix can be obtained:(20)S11=10.670.800.930.720.6710.870.600.760.800.8710.730.760.930.600.7310.720.720.760.760.721,S12=10.680.850.600.870.6810.790.280.550.850.7910.450.720.600.280.4510.730.870.550.720.731,S13=10.670.600.730.840.6710.600.670.760.600.6010.870.670.730.670.8710.800.840.760.670.801,S14=10.870.680.800.730.8710.790.670.600.680.7910.670.550.800.670.6710.870.730.600.550.871,S15=10.670.440.890.760.6710.550.670.650.440.5510.550.200.890.670.5510.650.760.650.200.651.Step 3: calculate the supportTα¯lik of the variable and the remaining variables and the support index of the variables ηlik, i=1,2,…,m, l=1,2,…,n, and k=1,2,3. Then, the following support matrix Tk and support index matrix ηk can be obtained:(21)T1=3.122.903.162.982.963.002.302.812.062.872.842.702.743.073.073.082.932.693.012.752.762.541.742.762.26,η1=0.210.390.110.100.200.230.360.110.080.220.200.380.100.110.220.210.400.090.100.190.220.410.070.110.18,T2=2.482.472.672.712.712.732.042.552.332.332.601.952.342.542.332.712.612.163.073.072.162.642.723.083.08,η2=0.190.390.100.110.210.240.350.110.100.200.230.350.100.110.210.200.380.080.110.230.160.390.100.110.23,T3=2.892.762.662.672.282.641.912.422.372.282.632.612.111.762.831.982.862.112.652.822.072.441.932.722.52,η3=0.220410.100.100.170.240.340.110.110.200.210.410.080.070.220.150.440.080.100.220.180.410.080.120.21.Step 4: use the PFIWPA operator to gather the attribute valuesα¯ijk, j=1,2,…,m, of the candidate solutions to obtain expertDk’s comprehensive evaluation value of the candidate solutions xi, as shown in Table 4.Step 5: standardize the comprehensive evaluation value of the expert's candidate program, and then, use the evidence synthesis method to fuse the data. The specific results are obtained, as shown in Table5.Step 6: the comprehensive evidence information of the alternatives shows that the confidence intervals of the alternatives areI1=0.30,0.44, I2=0.57, 0.75, I3=0.31,0.50, I4=0.32,0.51, and I5=0.46,0.65. Based on this, the possibility matrix P=pij5×5 between candidate schemes can be obtained:(22)P=∼00001∼11110∼0.810.22100.19∼0100.781∼.Step 7: from the ranking vector of the degree of possibility matrix0.08,0.28,0.18,0.13,0.21, the ranking of candidate solutions can be obtained as x2≻x5≻x3≻x4≻x1.
Therefore, the best green supplier should isx2.Table 1
Pythagorean decision matrixH1.
c1c2c3c4c5x1P0.4,0.8P0.7,0.6P0.6,0.7P0.3,0.8P0.5,0.6x2P0.7,0.5P0.9,0.2P0.8,0.5P0.3,0.6P0.6,0.5x3P0.3,0.4P0.3,0.7P0.7,0.4P0.6,0.4P0.4,0.5x4P0.6,0.6P0.7,0.5P0.7,0.2P0.4,0.6P0.3,0.7x5P0.5,0.7P0.6,0.4P0.9,0.3P0.6,0.7P0.1,0.7Table 2
Pythagorean decision matrixH2.
c1c2c3c4c5x1P0.3,0.9P0.7,0.6P0.5,0.8P0.3,0.6P0.3,0.6x2P0.7,0.4P0.9,0.2P0.8,0.1P0.3,0.5P0.3,0.5x3P0.3,0.6P0.7,0.7P0.7,0.6P0.4,0.4P0.4,0.3x4P0.4,0.8P0.7,0.5P0.6,0.2P0.4,0.7P0.4,0.7x5P0.2,0.7P0.8,0.2P0.8,0.4P0.6,0.6P0.6,0.6Table 3
Pythagorean decision matrixH3.
c1c2c3c4c5x1P0.6,0.7P0.7,0.6P0.5,0.8P0.5,0.5P0.1,0.6x2P0.6,0.5P0.9,0.2P0.8,0.1P0.3,0.5P0.3,0.4x3P0.4,0.7P0.7,0.5P0.6,0.1P0.2,0.9P0.6,0.5x4P0.2,0.9P0.5,0.6P0.6,0.2P0.1,0.6P0.4,0.7x5P0.1,0.6P0.8,0.2P0.9,0.2P0.6,0.5P0.4,0.6Table 4
Comprehensive evaluation information.
D1D2D3x1P0.58,0.68P0.54,0.71P0.59,0.66x2P0.79,0.39P0.77,0.31P0.75,0.33x3P0.43,0.56P0.56,0.72P0.60,0.57x4P0.61,0.55P0.57,0.62P0.43,0.67x5P0.61,0.56P0.70,0.45P0.69,0.38Table 5
Results of evidence synthesis.
Standardized comprehensive evaluation informationComprehensive evidence informationD1D2D3x1P0.22,0.33P0.19,0.33P0.23,0.320.30,0.56,0.14x2P0.32,0.21P0.31,0.16P0.31,0.170.57,0.25,0.17x3P0.17,0.25P0.20,0.34P0.24,0.260.31,0.50,0.18x4P0.24,0.26P0.21,0.28P0.17,0.300.32,0.49,0.19x5P0.24,0.26P0.27,0.21P0.28,0.180.46,0.35,0.19


## 6. Method Comparison and Discussion

Research on Pythagorean fuzzy group decision-making has just started, and there are few related documents. In this section, we compare the proposed method with the Pythagorean fuzzy TOPSIS group decision-making method [29] and the group decision-making method based on the WA operator [27]. Tables 6 and 7 show that the optimal choices given by the three group decision-making methods in Example 2 are bothx2, which shows that the group decision-making method in this paper is feasible. In-depth analysis from [27, 29] clearly shows that the differences between the candidate solutions are magnified. This is mainly based on the following reasons:(1)
The group decision-making methods proposed by [27, 29] assume that experts involved in decision-making are considered to be fully rational individuals. However, this assumption is not always true because of the complicated decision environment and finiteness of humans’ cognitive abilities.(2)
The PFIWPA operator used in this paper can take into account the balance and relevance of the information when gathering decision information, while the group decision-making methods used in literatures [27, 29] use independent attributes by default. Moreover, the model calculation of [27, 29] is based on Definition 4. As mentioned above, this type of calculation rule has defects, which can easily distort decision information.(3)
When constructing the decision-making model, this paper uses evidence theory to fuse evaluation information among experts, which can be closer to human cognition than a collective settlement in a bounded rational environment [28].(4)
The decision method given in [27, 29] uses the PFWA operator and TOPSIS when aggregating the evaluation information of different attributes and the evaluation information between experts, respectively. This paper uses the PFIWPA operator and the evidence theory to deal with the characteristics of information aggregation at different stages. The proposed method integrates the advantages of multiple methods to comprehensively solve the problem of aggregating of group decision information.Table 6
Calculation results of [27].
Candidate planComprehensive evaluation valueScore valueSort resultx1P0.60,0.71−0.115x2P0.78,0.220.561x3P0.55,0.450.104x4P0.55,0.390.163x5P0.67,0.320.342Table 7
Calculation results of [29].
Candidate planDxi,x+Dxi,x−The relative closeness of the alternativeSort resultx10.2590.1540.3745x20.10.3420.7741x30.2240.2530.533x40.2470.2160.4664x50.1720.2650.6072


## 7. Conclusion

Aiming at the problem of multiple attribute group decision-making in Pythagorean fuzzy environment, this paper uses evidence theory and the Pythagorean fuzzy interactive weighted power average operator to propose a group decision-making method. The new method uses the Pythagorean fuzzy interactive weighted power average operator to aggregate experts’ attribute evaluation information and gives the comprehensive evaluation information of each expert. The operator used is better than the commonly used power average operator and not only satisfies idempotence and exchangeability but also mines attribute associations. Then, the rules of evidence synthesis are used to gather the comprehensive evaluation information of experts and give the comprehensive evidence information of the candidate program. Finally, the evidence information is used to give the best ranking of the candidate solutions. However, this method also has its shortcomings. The situation in which the attribute evaluation value is the interval Pythagorean fuzzy number and the attribute weight is unknown is not discussed. The next step will focus on the Pythagorean fuzzy group decision-making problem in these situations.



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