# Pythagorean Fuzzy Interaction Partitioned Bonferroni Mean Operators and Their Application in Multiple-Attribute Decision-Making

Authors: Wei Yang; Jiarong Shi; Yong Liu; Yongfeng Pang; Ruiyue Lin
Journal: Complexity
 (2018)

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## Abstract

The aim of this paper is to develop partitioned Pythagorean fuzzy interaction Bonferroni mean operators based on the Pythagorean fuzzy set, Bonferroni mean, and interaction between membership and nonmembership. Several new aggregation operators are developed including the Pythagorean fuzzy interaction partitioned Bonferroni mean (PFIPBM) operator, the Pythagorean fuzzy weighted interaction partitioned Bonferroni mean (PFWIPBM) operator, the Pythagorean fuzzy interaction partitioned geometric Bonferroni mean (PFIPGBM) operator, and the Pythagorean fuzzy weighted interaction partitioned geometric Bonferroni mean (PFWIPGBM) operator. Some main properties and some special particular cases of the new operators are studied. Many existing operators are the special cases of new aggregation operators. Moreover, a multiple-attribute decision-making method based on the proposed operator has been developed and the investment company selection problem is presented to illustrate feasibility and practical advantages of the new method.


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## Body

## 1. Introduction

Pythagorean fuzzy set was first developed by Yager [1, 2], which is the extension of intuitionistic fuzzy set [3–5].In Pythagorean fuzzy set, the square sum of membership and nonmembership is no more than 1, which can lead to larger feasible space than that of intuitionistic fuzzy set. Hence, comparing with the existing tools to model fuzzy and uncertain information, the Pythagorean fuzzy set is more powerful and flexible. In the literature, many studies have been conducted for decision-making problems with complex uncertainty in Pythagorean fuzzy environment [6–12].Yager [12] developed the Pythagorean fuzzy weighted averaging (PFWA) operator and Pythagorean fuzzy weighted geometric averaging (PFWGA) operator. Garg proposed some Pythagorean fuzzy Einstein aggregation operations in [13] and Pythagorean fuzzy Einstein geometric aggregation operators using t-norm and t-conorm in [14]. Some Pythagorean fuzzy interaction weighted geometric aggregation operators were proposed in [15]. Yang and Pang [16] developed some Pythagorean fuzzy interaction Maclaurin symmetric mean operators. Peng and Yang [17] defined the Pythagorean fuzzy Choquet integral aggregation operator. Zhang et al. [18] proposed generalized Pythagorean fuzzy Bonferroni mean operator. Liang et al. [19] developed Pythagorean fuzzy geometric Bonferroni mean and weighted Pythagorean fuzzy geometric Bonferroni mean operator. Wei [20] presented some Pythagorean fuzzy interaction aggregation operators. Wei and Lu proposed some Pythagorean fuzzy power aggregation operators in [21] and presented some Pythagorean fuzzy Maclaurin symmetric mean operators in [22]. Zeng [23] developed Pythagorean fuzzy probabilistic ordered weighted averaging operator by considering probabilistic information in aggregating Pythagorean fuzzy values. Garg [24] proposed some probabilistic Pythagorean fuzzy aggregation operators by considering probabilistic information and decision maker’s attitudinal character. Peng and Dai [25] proposed Pythagorean fuzzy stochastic decision-making method based on the prospect theory and regret theory. Some Pythagorean fuzzy multiple-attribute decision-making methods have been developed including the TOPSIS [26], QUALIFLEX [27], clustering analysis [28], TODIM [29], and VIKOR [30]. Pythagorean fuzzy set has been further extended to accommodate interval values [31, 32], linguistic variables [33] and so on.Though several studies have been conducted in Pythagorean fuzzy environments, interaction between membership and nonmembership is considered less in existing studies and partitioned Pythagorean fuzzy values to be aggregated are rarely considered yet.Bonferroni mean (BM) was introduced by Bonferroni [34], which has the capability of capturing interrelationship among arguments to be aggregated by considering conjunction among each pair of aggregated arguments. Yager [35] provided an interpretation of Bonferroni mean as involving a product of each argument with the average of the other arguments. Beliakov et al. [36] developed generalized Bonferroni mean to extend the Bonferroni mean in a more general form. Beliakov and James [37] extended the generalized Bonferroni mean to intuitionistic fuzzy environment. Xu and Yager [38] extended the Bonferroni mean to accommodate intuitionistic fuzzy values. Zhu and Xu [39] developed hesitant fuzzy Bonferroni mean operator and weighted hesitant fuzzy Bonferroni mean operator. Zhu et al. [40] explored the geometric Bonferroni mean under hesitant fuzzy environment. Xia et al. [41] introduced the Bonferroni geometric mean and further developed intuitionistic fuzzy geometric Bonferroni mean operator. Blanco-Mesa et al. [42] developed Bonferroni ordered weighted averaging index of maximum and minimum level operators by using Bonferroni mean, OWA operators, and some distance measures. Liang et al. [43] proposed the Pythagorean fuzzy Bonferroni mean and the weighted Pythagorean fuzzy Bonferroni mean. Dutta and Guha [44] presented the partitioned Bonferroni mean for 2-tuple linguistic information by considering the partitioned attribute class. Z. Liu and P. Liu [45] developed intuitionistic uncertain linguistic partitioned Bonferroni mean.In some cases, the interrelationship does not exist in the whole attributes, but in some of the attributes. For example, consider a candidate selection problem for research sector in a university where the best candidate is selected among several candidates based on the following attributes: management skill (A1), interpersonal relationship A2, research ability A3, and grant A4. The attributes should be partitioned into two classes P1=A1,A2 and P2=A3,A4. Obviously, A1 and A2 are interrelated and they belong to P1. A3 and A4 are interrelated and they belong to P2. But there is no interrelation between P1 and P2. Hence, there is a need to partition attributes into several classes when there is no interrelationship among all the attributes but there is interrelationship among parts of the attributes. Though many useful Bonferroni mean operators have been developed in various environments, the partitioned aggregation operators in Pythagorean fuzzy environment have not been considered yet. Moreover, interaction between the membership and nonmembership of Pythagorean fuzzy values should be considered in the partitioned Pythagorean fuzzy aggregation operator. Hence in this paper, based on the partitioned Bonferroni mean operator, we develop Pythagorean fuzzy interaction partitioned Bonferroni mean operators by considering partitioned values and interaction between membership and nonmembership. Then, we give a new multiple-attribute decision-making method based on the partitioned Bonferroni mean operators. Comparing with the existing methods based on the Bonferroni mean operators, our proposed method is a good complement to the existing work and it can be used to solve more complex multiple-attribute decision-making problems.The structure of the paper is as follows. In Section2, some basic concepts on Pythagorean fuzzy set and Bonferroni mean have been reviewed. In Section 3, some Bonferroni mean operators in Pythagorean fuzzy environments considering interaction have been developed including the Pythagorean fuzzy interaction partitioned Bonferroni mean (PFIPBM) operator, the Pythagorean fuzzy weighted interaction partitioned Bonferroni mean (PFWIPBM) operator, the Pythagorean fuzzy interaction partitioned geometric Bonferroni mean (PFIPGBM) operator, and the Pythagorean fuzzy weighted interaction partitioned geometric Bonferroni mean (PFWIPGBM) operator. Some properties and some special cases of the new aggregation operators have been studied. In Section 4, a new multiple-attribute group decision-making method based on the new proposed operators has been proposed. In Section 5, the problem of investment company selection has been presented to illustrate the new method and some comparisons with other methods have been conducted. Conclusions have been given in the last section.


## 2. Preliminaries

Pythagorean fuzzy set [1, 2] is the extension of fuzzy set and intuitionistic fuzzy set. We review some concepts of Pythagorean fuzzy set and their operations in the following.Definition 1 (see [26]).
LetX be a fixed set. A Pythagorean fuzzy set P on X can be represented as follows:
(1)P=<x,μPx,νPx>x∈X,where μPx:X→0,1 is the membership function and νPx:X→0,1 is the nonmembership function. For each x∈X, it satisfies the following condition 0≤μPx2+νPx2≤1. πPx=1−μPx2−νPx2 is the indeterminacy degree of x to X. For simplicity, μPx,νPx is called a Pythagorean fuzzy number (PFN), denoted by μP,νP, where μP,νP∈0,1, πP=1−μP2−νP2, and 0≤μP2+νP2≤1.Definition 2 (see [26]).
Letα=μα,να,α1=μα1,να1, and α2=μα2,να2 be three PFNs. The operations are as follows:
(1)
α1⊕α2=μα12+μα22−μα12μα22,να1να2.
(2)
α1⊗α2=μα1μα2,να12+να22−να12να22.
(3)
kα=1−1−μα2k,ναk,k≥0.
(4)
αk=μαk,1−1−να2k,k≥0.
Letα=μα,να be a PFN; the score function [26] is defined as
(2)Sα=μα2−να2.
The accuracy function [46] is defined as
(3)Aα=μα2+να2.Definition 3 (see [2]).
Letα1=μα1,να1 and α2=μα2,να2 be two PFNs. Yager and Abbasov defined the following method to compare two PFNs:
(1)
IfSα1<Sα2, then α1<α2.
(2)
IfSα1=Sα2,
(i)
IfAα1<Aα2, then α1<α2.
(ii)
IfAα1=Aα2, then α1=α2.Example 1.
Supposeα1=0.6,0.4, α2=0.5,0.6, and α3=0.7,0.0, the corresponding weight vector is 0.25,0.4,0.35, then α=w1α1⊕w2α2⊕w3α3=0.6045,0. This means that nonmemberships have no effects on the overall results, which is not reasonable. In order to overcome this shortcoming, some new operational laws on Pythagorean fuzzy set were developed.Definition 4 (see [20]).
Letα=μα,να,α1=μα1,να1, and α2=μα2,να2 be three PFNs. The operation laws can be defined as follows:
(1)
α1⊕α2=μα12+μα22−μα12μα22,να12+να22−να12να22−μα12να22−να12μα22.
(2)
α1⊗α2=μα12+μα22−μα12μα22−να12μα22−μα12να22,να12+να22−να12να22.
(3)
λα=1−1−μα2λ,1−μα2λ−1−μα2+να2λ,λ>0.
(4)
αλ=1−να2λ−1−μα2+να2λ,1−1−να2λ,λ>0.
Equations (1) and (2) can be rewritten as follows:
(1)
α1⊕α2=1−1−μα121−μα22,1−μα121−μα22−1−μα12+να121μα22+να221/2=1−∏j=121−μαj2,∏j=121−μαj2−∏j=121−μαj2+ναj2.
(2)
α1⊗α2=1−να121−να22−1−μα12+να121−μα22+να221−1−να121−να22∏j=121−ναj2−∏j=121−μαj2+ναj2,1−∏j=121−ναj2.Definition 5.
Letα1=μα1,να1 and α2=μα2,να2 be two Pythagorean fuzzy numbers. The Hamming distance between α1 and α2 can be defined as follows:
(4)dα1,α2=12μα12−μα22+να12−να22.


## 3. Pythagorean Fuzzy Weighted Interaction Partitioned Bonferroni Mean Operator

The Bonferroni mean (BM) aggregation operator was defined by Bonferroni [34] in 1950. It was generalized by Yager [35] and others. The BM operator has the following forms.Definition 6 (see [35]).
For anyp,q≥0 with p+q>0, the BM aggregation operator of dimension n is a mapping BM: R+n→R+, such that
(5)BMp,qa1,a2,…,an=1nn−1∑i,j=1,i≠jnaipajq1/p+q.Definition 7 (see [44]).
For anyp,q≥0 with p+q>0 and T=a1,a2,…,an with ak≥0k=1,2,…,n, which is partitioned into d distinct sorts P1,P2,…,Pd, where ⋃h=1dPh=T, the partitioned Bonferroni mean aggregation operator of dimension n is a mapping PBM:
(6)PBMp,qa1,a2,…,an=1d∑h=1d1Ph∑i∈Phaip1Ph−1∑j∈Ph,j≠iajq1/p+q,where Ph denotes the cardinality of Ph, d is the number of partitioned sorts, and ∑h=1dPh=n.Definition 8.
LetT=α1,α2,…,αn be a collection of PFNs, which is partitioned into d distinct sorts P1,P2,…,Pd, where αi=μi,νii=1,2,…,n and ⋃h=1dPh=T. The Pythagorean fuzzy interaction partitioned Bonferroni mean (PFIPBM) operator is defined as follows:
(7)PFIPBMp,qα1,α2,…,αn=1d⊕h=1d1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q,
wherep,q≥0, Ph denotes the cardinality of Ph, d is the number of the partitioned sorts, and ∑h=1dPh=n.Theorem 1.
Letαi=μi,νii=1,2,…,n be a collection of PFNs and p,q≥0. The aggregated result of PFIPBM operator is still of a PFN, which has the following form:
(8)PFIPBMp,qα1,α2,…,αn=1d⊕h=1d1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q=1−∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d1/2,∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d−∏h=1d∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d1/2,where ξ=∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1 and η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1.Proof 1.
(9)αjq=1−νj2q−1−μj2+νj2q1/2,1−1−νj2q,αip=1−νi2p−1−μi2+νi2p1/2,1−1−νi2p,⊕j∈Ph,j≠iαjq=1−∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/2,∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q−∏j∈Ph,j≠i1−μj2+νj2q1/2,1Ph−1⊕j∈Ph,j≠iαjq=1−∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−11/2,∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1−∏j∈Ph,j≠i1−μj2+νj2q1/Ph−11/2.
Let(10)ξ=∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1,η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1,αip⊗1∣Ph∣−1⊕j∈Ph,j≠iαjq=1−νi2p1−ξ+η−1−μi2+νi2pη,1−1−νi2p1−ξ+η,⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq=1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/2,∏i∈Ph1−1−νi2p∗1−ξ+η+1−μi2+νi2pη−∏i∈Ph1−μi2+νi2pη1/2,1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q=1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q−∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/2,1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/2,1d⊕h=1d1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q=1−∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d1/2,∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d−∏h=1d∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d1/2.
Moreover,(11)μPFIPBMp,q2+νPFIPBMp,q2=1−∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d+∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d−∏h=1d∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d=1−∏h=1d∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d,since
(12)η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1,then 0≤η≤1 and
(13)0≤1−∏h=1d∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d≤1.
Hence, the aggregated result of thePFIPBMp,q operator is still a PFN.Theorem 2 (idempotency).
Letαi=μi,νii=1,2,…,n be a collection of PFNs and p,q≥0. If all αii=1,2,…,n are equal, that is, αi=α=μ,νi=1,2,…,n, then
(14)PFIPBMp,qα1,α2,…,αn=α.Proof 2.
LetPFIPBMp,qα1,α2,…,αn=σ,τ. Because μi=μ and νi=ν, then we get
(15)ξ=∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1=∏j∈Ph,j≠i1−1−ν2q+1−μ2+ν2q1/Ph−1=1−1−ν2q+1−μ2+ν2q,η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1=∏j∈Ph,j≠i1−μ2+ν2q1/Ph−1=1−μ2+ν2q.
Therefore, we have(16)σ=1−∏h=1d1−1−∏i∈Ph1−1−ν2p1−ν2q+1−μ2+ν2p1−μ2+ν2q1/Ph+1−μ2+ν2p1−μ2+ν2q1/p+q+1−μ2+ν2p∗1−μ2+ν2q1/p+q1/d1/2=1−∏h=1d1−1−1−1−ν2p+q+1−μ2+ν2p+q+1−μ2+ν2p+q1/p+q+1−μ2+ν2p+q1/p+q1/d1/2=μ,τ=∏h=1d1−1−∏i∈Ph1−1−ν2p1−ν2q+1−μ2+ν2p1−μ2+ν2q1/Ph+∏i∈Ph1−μ2+ν2p1−μ2+ν2q1/Ph1/p+q+∏i∈Ph1−μ2+ν2p1−μ2+ν2q1/Ph1/p+q1/d−∏h=1d∏i∈Ph1−μ2+ν2p1−μ2+ν2q1/Ph1/p+q1/d1/2=∏h=1d1−1−1−1−ν2p+q+1−μ2+ν2p+q+1−μ2+ν2p+q1/p+q+1−μ2+ν2p+q1/p+q1/d−∏h=1d1−μ2+ν2p+q1/p+q1/d1/2=1−1−ν2+1−μ2+ν2−1−μ2+ν21/2=ν.
Hence, we getσ,τ=μ,ν and PFIPBMp,qα1,α2,…,αn=α.Theorem 3 (commutativity).
Letαi=μi,νii=1,2,…,n and α′i=μ′i,ν′ii=1,2,…,n be two collections of PFNs. If α′i=μ′i,ν′i is any permutation of αi=μi,νi, then
(17)PFIPBMp,qα1,α2,…,αn=PFIPBMp,qα′1,α′2,…,α′n.Proof 3.
By using (8), we can get
(18)PFIPBMp,qα1,α2,…,αn=1−∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d1/2∏h=1d1−1−∏i∈Ph1−1−νi2p1−ξ+η+1−μi2+νi2pη1/Ph+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q+∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d−∏h=1d∏i∈Ph1−μi2+νi2pη1/Ph1/p+q1/d1/2,where
(19)ξ=∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1,η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1.PFIPBMp,qα′1,α′2,…,α′n=1−∏h=1d1−1−∏i∈Ph1−1−ν′i2p1−ξ′+η′+1−μ′i2+ν′i2pη1/Ph+∏i∈Ph1−μ′i2+ν′i2pη′1/Ph1/p+q+∏i∈Ph1−μ′i2+ν′i2pη′1/Ph1/p+q1/d1/2,∏h=1d1−1−∏i∈Ph1−1−ν′i2p1−ξ′+η′+1−μ′i2+ν′i2pη′1/Ph+∏i∈Ph1−μ′i2+ν′i2pη′1/Ph1/p+q+∏i∈Ph1−μ′i2+ν′i2pη′1/Ph1/p+q1/d−∏h=1d∏i∈Ph1−μ′i2+ν′i2pη′1/Ph1/p+q1/d1/2,where
(20)ξ′=∏j∈Ph,j≠i1−1−ν′j2q+1−μ′j2+ν′j2q1/Ph−1,η′=∏j∈Ph,j≠i1−μ′j2+ν′j2q1/Ph−1.
Sinceα′i=μ′i,ν′i is any permutation of αi=μi,νi, then we can get
(21)PFIPBMp,qα1,α2,…,αn=PFIPBMp,qα1′,α2′,…,αn′.Theorem 4 (boundedness).
Letα~=1,0 and αˇ=0,1, then
(22)αˇ≤PFIPBMp,qα1,α2,…,αn≤α~Proof 4.
The property of boundedness can be proved easily by using Theorem1.
Some of the special cases of the proposedPFIPBMp,q operator regarding parameters p and q are as follows:
(i)
Whenq→0, we can get
(23)ξ=∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1=1,η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1=1.
Thus, we can get(24)PFIPBMp,0α1,α2,…,αn=1−∏h=1d1−1−∏i∈Ph1−1−νi2p+1−μi2+νi2p1/Ph+∏i∈Ph1−μi2+νi2p1/Ph1/p+∏i∈Ph1−μi2+νi2p1/Ph1/p1/d1/2,∏h=1d1−1−∏i∈Ph1−1−νi2p+1−μi2+νi2p1/Ph+∏i∈Ph1−μi2+νi2p1/Ph1/p+∏i∈Ph1−μi2+νi2p1/Ph1/p1/d−∏h=1d∏i∈Ph1−μi2+νi2p1/Ph1/p1/d1/2.
(ii)
Whenq→0 and p=1, we can get
(25)ξ=∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1=1,η=∏j∈Ph,j≠i1−μj2+νj2q1/Ph−1=1.
Thus, we can get(26)PFIPBM1,0α1,α2,…,αn=1−∏h=1d∏i∈Ph1−μi21/Ph1/d1/2,∏h=1d∏i∈Ph1−μi21/Ph1/d−∏h=1d∏i∈Ph1−μi2+νi21/Ph1/d1/2.
(iii)
Whenp→0, we can get
(27)PFIPBMp,qα1,α2,…,αn=1−∏h=1d1−1−∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1+∏j∈Ph,j≠i1−μj2+νj2q1/Ph−11/q+∏j∈Ph,j≠i1−μj2+νj2q1/Ph−11/q1/d1/2,∏h=1d1−1−∏j∈Ph,j≠i1−1−νj2q+1−μj2+νj2q1/Ph−1+∏j∈Ph,j≠i1−μj2+νj2q1/Ph−11/q+∏j∈Ph,j≠i1−μj2+νj2q1/Ph−11/q1d−∏h=1d∏j∈Ph,j≠i1−μj2+νj2q1/Ph−11/q1/d1/2.
(iv)
Whenp→0 and ,q=1, we can get
(28)PFIPBMp,qα1,α2,…,αn=1−∏h=1d∏j∈Ph,j≠i1−μj21/Ph−11/d1/2,∏h=1d∏j∈Ph,j≠i1−μj21/Ph−11/d−∏h=1d∏j∈Ph,j≠i1−μj2+νj21/Ph−11/d1/2.
If all the PFNs are partitioned into one sort, then the PFIPBM operator reduces to the Pythagorean fuzzy interaction Bonferroni mean (PFIBM) operator as follows:(29)PFIBMα1,α2,…,αn=1nn−1⊕i,j=1,i≠jnαip⊗αjq1/p+q=1−∑i,j=1,i≠jn1−1−νi2p1−νj2q+1−μi2+νi2p1−μj2+νj2q1/p+q+∑i,j=1,i≠jn1−μi2+νi2p1−μj2+νj2q1/nn−11/p+q−∑i,j=1,i≠jn1−μi2+νi2p∗1−μj2+νj2q1/nn−11/p+q1/2,1−1−∑i,j=1,i≠jn1−1−νi2p1−νj2q+1−μi2+νi2p1−μj2+νj2q1/nn−1+∑i,j=1,i≠jn1−μi2+νi2p∗1−μj2+νj2q1/nn−11/p+q1/2.Definition 9.
LetT=α1,α2,…,αn be a collection of PFNs, which is partitioned into d distinct sorts P1,P2,…,Pd, where αi=μi,νii=1,2,…,n and ⋃h=1dPh=T. The Pythagorean fuzzy weighted interaction partitioned Bonferroni mean (PFWIPBM) operator is defined as follows:
(30)PFWIPBMp,qα1,α2,…,αn=1d⊕h=1d1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q,where p,q≥0, Ph denotes the cardinality of Ph, d is the number of the partitioned sorts, and ∑h=1dPh=n. W=w1,w2,…,wn is the weight vector of α1,α2,…,αn satisfying wj∈0,1, j=1,2,…,n, and ∑j=1nwj=1.Theorem 5.
Letαi=μi,νii=1,2,…,n be a collection of PFNs and p,q≥0. The aggregated result of PFWIPBM operator is still of a PFN, which has the following form:
(31)PFWIPBMp,qα1,α2,…,αn=1d⊕h=1d1∣Ph∣∣Ph∣−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+qd1/2,where ξ=1−1−μi2wi+1−μi2+νi2wip1−1−μj2wj+1−μj2+νj2wjq and η=1−μi2+νi2wip1−μj2+νj2wjq.Proof 5.
(32)wiαi=1−1−μi2wi,1−μi2wi−1−μi2+νi2wi,wjαj=1−1−μj2wj,1−μj2wj−1−μj2+νj2wj1/2,wiαip⊗wjαjq1−1−μi2wi+1−μi2+νi2wip1−1−μj2wj+1−μj2+νj2wjq−1−μi2+νi2wip1−μj2+νj2wjq1/2,1−1−1−μi2wi+1−μi2+νi2wip∗1−1−μj2wj+1−μj2+νj2wjq1/2.
Let(33)ξ=1−1−μi2wi+1−μi2+νi2wip1−1−μj2wj+1−μj2+νj2wjq,η=1−μi2+νi2wip1−μj2+νj2wjq,⊕i,j∈Ph,j≠iwiαip⊗wjαjq=1−∏i,j∈Ph,j≠i1−ξ+η,∏i,j∈Ph,j≠i1−ξ+η−∏i,j∈Ph,j≠iη,1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q−∏i,j∈Ph,j≠iη1/PhPh−1p+q1/2,1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q1/2,1d⊕h=1d1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+qd1/2.
Moreover,(34)μPFWIPBMp,q2+νPFWIPBMp,q2=1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d+∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+qd=1−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+qd,since η=1−μi2+νi2wip1−μj2+νj2wjq, then 0≤η≤1 and we can get 0≤1−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+qd≤1. Hence, the aggregated result of the PFWIPBMp,q operator is still a PFN.Some special cases of the PFWIPBM operator are discussed as follows:(i)
Ifq→0, we can get ξ=1−1−μi2wi+1−μi2+νi2wip and η=1−μi2+νi2wip. Thus, we can get
(35)PFWIPBMp,0α1,α2,…,αn=1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+∏i,j∈Ph,j≠iη1/PhPh−1p1/d1/2,∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+∏i,j∈Ph,j≠iη1/PhPh−1p1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1pd1/2.
(ii)
Ifq→0 and p=1, we can get ξ=1−1−μi2wi+1−μi2+νi2wi and η=1−μi2+νi2wi. Thus, we can get
(36)PFWIPBM1,0α1,α2,…,αn=1−∏h=1d∏i,j∈Ph,j≠i1−μi2wi/PhPh−1d1/d1/2,∏h=1d∏i,j∈Ph,j≠i1−μi2wi/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μi2+νi2wi/PhPh−1d1/2.
(iii)
Ifp→0, we can get ξ=1−1−μj2wj+1−μj2+νj2wjq and η=1−μj2+νj2wjq. Thus, we can get
(37)PFWIPBM0,qα1,α2,…,αn=1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/q+∏i,j∈Ph,j≠iη1/PhPh−1q1/d1/2,∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/q+∏i,j∈Ph,j≠iη1/PhPh−1q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1qd1/2.
(iv)
Ifp→0 and q=1, we can get ξ=1−1−μj2wj+1−μj2+νj2wj and η=1−μj2+νj2wj. Thus, we can get
(38)PFWIPBM0,1α1,α2,…,αn=1−∏h=1d∏i,j∈Ph,j≠i1−μj2wj/PhPh−11/d,∏h=1d∏i,j∈Ph,j≠i1−μj2wj/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μj2+νj2wj/PhPh−1d.If all the PFNs are partitioned into one sort, the PFWIPBM operator reduces to the Pythagorean fuzzy weighted interaction Bonferroni mean (PFWIBM) operator as follows:(39)1∣Ph∣∣Ph∣−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q−∏i,j∈Ph,j≠iη1/PhPh−1p+q1/2,1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q1/2,where ξ=1−1−μi2wi+1−μi2+νi2wip1−1−μj2wj+1−μj2+νj2wjq; η=1−μi2+νi2wip1−μj2+νj2wjq; p,q≥0; and w1,w2,…,wn is the weight vector of α1,α2,…,αn satisfying wj∈0,1, j=1,2,…,n, and ∑j=1nwj=1.Definition 10.
LetT=α1,α2,…,αn be a collection of PFNs, which is partitioned into d distinct sorts P1,P2,…,Pd, where αi=μi,νii=1,2,…,n and ⋃h=1dPh=T. The Pythagorean fuzzy interaction partitioned geometric Bonferroni mean (PFIPGBM) operator is defined as follows:
(40)PFIPGBMp,qα1,α2,…,αn=⊗h=1d1p+q⊗i,j∈Ph,j≠ipαi⊕qαj1/PhPh−11/d,where p,q≥0, Ph denotes the cardinality of Ph, d is the number of the partitioned sorts, and ∑h=1dPh=n.Theorem 6.
Letαi=μi,νii=1,2,…,n be a collection of PFNs and p,q≥0. The aggregated result of PFIPGBM operator is still of a PFN, which has the following form:
(41)PFIPGBMp,qα1,α2,…,αn=⊗h=1d1p+q⊗i,j∈Ph,j≠ipαi⊕qαj1/PhPh−11/d=∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,where ξ=1−μi2p1−μj2q and η=1−μi2+νi2p1−μj2+νj2q.Proof 6.
(42)pαi=1−1−μi2p,1−μi2p−1−μi2+νi2pqαj=1−1−μj2q,1−μj2q−1−μj2+νj2q1/2,pαi⊕qαj=1−1−μi2p1−μj2q,1−μi2p1−μj2q−1−μi2+νi2p1−μj2+νj2q1/2.
Let=1−μi2p1−μj2q and η=1−μi2+νi2p1−μj2+νj2q, then
(43)⊗i,j∈Ph,j≠ipαi⊕qαj=∏i,j∈Ph,j≠i1−ξ+η−∏i,j∈Ph,j≠iη,1−∏i,j∈Ph,j≠i1−ξ+η,1p+q⊗i,j∈Ph,j≠ipαi⊕qαj1/PhPh−1=1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q1/21−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q−∏i,j∈Ph,j≠iη1/PhPh−1p+q1/2,⊗h=1d1p+q⊗i,j∈Ph,j≠ipαi⊕qαj1/PhPh−11/d=∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2.
Moreover,(44)μPFIPGBMp,q2+νPFIPGBMp,q2=∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d+1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d=1−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d,since η=1−μi2+νi2p1−μj2+νj2q, then 0≤η≤1 and we can get
(45)0≤1−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d≤1.
Hence, the aggregated result of thePFIPGBMp,q operator is still a PFN.Theorem 7
(idempotency). LetT=α1,α2,…,αn be a collection of PFNs. If all αkk=1,2,…,n are equal, that is, αk=α=μ,νk=1,2,…,n, then
(46)PFIPGBMp,qα1,α2,…,αn=α.Theorem 8
(commutativity). Letαk=μk,νkk=1,2,…,n and α′kk=1,2,…,n be two collections of PFNs. If α′k=μ′k,ν′kk=1,2,…,n is any permutation of αk=μk,νk, then
(47)PFIPGBMp,qα1,α2,…,αn=PFIPGBMp,qα1′,α2′,…,αn′.Theorem 9
(boundedness). Letα~=1,0 and αˇ=0,1, then
(48)αˇ≤PFIPGBMp,qα1,α2,…,αn≤α~.Some special cases of the PFIPGBM operator based on the parametersp and q are discussed as follows:
(i)
Whenq→0, we can get ξ=1−μi2p,η=1−μi2+νi2p,
(49)PFIPGBMp,0α1,α2,…,αn=∏h=1d1−1−∏i,j∈Ph,j≠i1−1−μi2p+1−μi2+νi2p1/PhPh−1+∏i,j∈Ph,j≠i1−μi2+νi2p/PhPh−11/p+∏i,j∈Ph,j≠i1−μi2+νi2p1/PhPh−1p1/d−∏i,j∈Ph,j≠i1−1−μi2p+1−μi2+νi2p1/PhPh−1+∏i,j∈Ph,j≠i1−∏h=1d∏i,j∈Ph,j≠i1−μi2+νi2p1/PhPh−1p1/d1/2,1−∏h=1d1−1−μi2+νi2p1/PhPh−11/p+∏i,j∈Ph,j≠i1−μi2+νi2p1/PhPh−1p1/d1/2.
(ii)
Whenq→0 and p=1, we can get ξ=1−μi2,η=1−μi2+νi2, and
(50)PFIPGBMp,0α1,α2,…,αn=∏h=1d∏i,j∈Ph,j≠i1−νi21/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μi2+νi21/PhPh−11/d1/2,1−∏h=1d∏i,j∈Ph,j≠i1−νi21/PhPh−11/d1/2.
(iii)
Whenp→0, we can get ξ=1−μj2q and η=1−μj2+νj2q,
(51)PFIPGBM0,qα1,α2,…,αn=∏h=1d1−1−∏i,j∈Ph,j≠i1−1−μj2q+1−μj2+νj2q1/PhPh−1+∏i,j∈Ph,j≠i1−μj2+νj2q/PhPh−11/q+∏i,j∈Ph,j≠i1−μj2+νj2q/PhPh−1q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1q1/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−1−μj2q+1−μj2+νj2q1/PhPh−1+∏i,j∈Ph,j≠i1−μj2+νj2q/PhPh−11/q+∏i,j∈Ph,j≠i1−μj2+νj2q/PhPh−11/d1/2.
(iv)
Whenp→0 and q=1, we can get ξ=1−μj2 and η=1−μj2+νj2,
(52)PFIPGBM0,1α1,α2,…,αn=∏h=1d∏i,j∈Ph,j≠i1−νj21/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μj2+νj21/PhPh−11/d1/2,1−∏h=1d∏i,j∈Ph,j≠i1−νj21/PhPh−11/d1/2.If all PFNs are partitioned into one sort, the PFIPGBM operator reduces to the Pythagorean fuzzy interaction geometric Bonferroni mean (PFIGBM) operator as follows:(53)PFIGBMp,qα1,α2,…,αn=1p+q⊗i,j∈Ph,j≠ipαi⊕qαj1/mm−1=1−1−∏i,j∈Ph,j≠i1−1−μi2p1−μj2q+1−μi2+νi2p1−μj2+νj2q1/PhPh−1+∏i,j∈Ph,j≠i1−μi2+νi2p1−μj2+νj2q1/PhPh−11/p+q1/2,1−∏i,j∈Ph,j≠i1−1−μi2p1−μj2q+1−μi2+νi2p1−μj2+νj2q1/PhPh−1+∏i,j∈Ph,j≠i1−μi2+νi2p1−μj2+νj2q1/PhPh−11/p+q−∏i,j∈Ph,j≠i1−μi2+νi2p1−μj2+νj2q1/PhPh−1p+q1/2,where p,q≥0.Definition 11.
LetT=α1,α2,…,αn be a collection of PFNs, which is partitioned into d distinct sorts P1,P2,…,Pd and ⋃h=1dPh=T. The Pythagorean fuzzy weighted interaction partitioned geometric Bonferroni mean (PFWIPGBM) operator is defined as follows:
(54)PFWIPGBMp,qα1,α2,…,αn=⊗h=1d1p+q⊗i,j∈Ph,j≠ipαiwi⊕qαjwj1/PhPh−11/d,where αi=μi,νii=1,2,…,n; p,q≥0; Ph denotes the cardinality of Ph; d is the number of the partitioned sorts; and ∑h=1dPh=n. w=w1,w2,…,wn is the weight vector of α1,α2,…,αn, wj≥0,j=1,2,…,n, and ∑j=1nwj=1.Theorem 10.
Letαi=μi,νii=1,2,…,n be a collection of PFNs. For any p,q≥0, the aggregated result of the PFWIPGBM operator is still a PFN, which has the following forms:
(55)PFWIPGBMp,qα1,α2,…,αn=⊗h=1d1p+q⊗i,j∈Ph,j≠ipαiwi⊕qαjwj1/PhPh−11/d=∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,where ξ=1−1−νi2wi+1−μi2+νi2wip1−1−νj2wj+1−μj2+νj2wjq and η=1−μi2+νi2wip1−μj2+νj2wjq.Proof 7.
(56)αiwi=1−νi2wi−1−μi2+νi2wi1/2,1−1−νi2wi1/2,αjwj=1−νj2wj−1−μj2+νj2wj1/2,1−1−νj2wj1/2,pαiwi⊕qαjwj=1−1−1−νi2wi+1−μi2+νi2wip1−1−νj2wj+1−μj2+νj2wjq1/2,1−1−νi2wi+1−μi2+νi2wip1−1−νj2wj+1−μj2+νj2wjq−1−μi2+νi2wip1−μj2+νj2wjq1/2.
Let(57)⊗i,j∈Ph,j≠ipαiwi⊕qαjwj=ξ=1−1−νi2wi+1−μi2+νi2wip1−1−νj2wj+1−μj2+νj2wjq,η=1−μi2+νi2wip1−μj2+νj2wjq,∏i,j∈Ph,j≠i1−ξ+η−∏i,j∈Ph,j≠iη,1−∏i,j∈Ph,j≠i1−ξ+η,1p+q⊗i,j∈Ph,j≠ipαiwi⊕qαjwj1/PhPh−1=1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q1/2,1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q−∏i,j∈Ph,j≠iη1/PhPh−1p+q1/2.⊗h=1d1p+q⊗i,j∈Ph,j≠ipαiwi⊕qαjwj1/PhPh−11/d=∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d1/2.Moreover,(58)μPFWIPGBMp,q2+νPFWIPGBMp,q2=∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d+1−∏h=1d1−1−∏i,j∈Ph,j≠i1−ξ+η1/PhPh−1+∏i,j∈Ph,j≠iη1/PhPh−11/p+q+∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d=1−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d.Sinceη=1−μi2+νi2wip1−μj2+νj2wjq, then 0≤η≤1 and we can get
(59)0≤1−∏h=1d∏i,j∈Ph,j≠iη1/PhPh−1p+q1/d≤1.Hence, the aggregated result of thePFWIPGBMp,q is still a PFN.Some special cases of the PFWIPGBM operator are discussed as follows:(i)
Ifq=0, we can get ξ=1−1−νi2wi+1−μi2+νi2wip and η=1−μi2+νi2wip.
(60)PFWIPGBMp,0α1,α2,…,αn=∏h=1d1−1−∏i,j∈Ph,j≠i1−1−1−νi2wi+1−μi2+νi2wip+1−μi2+νi2wip1/PhPh−1+∏i,j∈Ph,j≠i1−μi2+νi2wip/PhPh−11/p+∏i,j∈Ph,j≠i1−μi2+νi2wi/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μi2+νi2wi/PhPh−11/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−1−1−νi2wi+1−μi2+νi2wip+1−μi2+νi2wip/PhPh−1+∏i,j∈Ph,j≠i1−μi2+νi2wip/PhPh−11/p+∏i,j∈Ph,j≠i1−μi2+νi2wi/PhPh−11/d1/2.
(ii)
Ifq=0 and p=1, we can get ξ=1−1−νi2wi+1−μi2+νi2wi and η=1−μi2+νi2wi.
(61)PFWIPGBM1,0α1,α2,…,αn∏h=1d∏i,j∈Ph,j≠i1−νi2wi/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μi2+νi2wi/PhPh−11/d1/2,1−∏h=1d∏i,j∈Ph,j≠i1−νi2wi/PhPh−11/d1/2.
(iii)
Ifp=0, we can get ξ=1−1−νj2wj+1−μj2+νj2wjq and η=1−μj2+νj2wjq.
(62)PFWIPGBM0,qα1,α2,…,αn=∏h=1d1−1−∏i,j∈Ph,j≠i1−1−1−νj2wj+1−μj2+νj2wjq+1−μj2+νj2wjq1/PhPh−1+∏i,j∈Ph,j≠i1−μj2+νj2wjq/PhPh−11/q+∏i,j∈Ph,j≠i1−μj2+νj2wj/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μj2+νj2wj/PhPh−11/d1/2,1−∏h=1d1−1−∏i,j∈Ph,j≠i1−1−1−νj2wj+1−μj2+νj2wjq+1−μj2+νj2wjq1/PhPh−1+∏i,j∈Ph,j≠i1−μj2+νj2wiq/PhPh−11/q+∏i,j∈Ph,j≠i1−μj2+νj2wj/PhPh−11/d1/2.
(iv)
Ifp=0,q=1, we can get ξ=1−1−νj2wj+1−μj2+νj2wj and η=1−μj2+νj2wj.
(63)PFWIPGBM0,1α1,α2,…,αn=∏h=1d∏i,j∈Ph,j≠i1−νj2wj/PhPh−11/d−∏h=1d∏i,j∈Ph,j≠i1−μj2+νj2wj/PhPh−11/d1/2,1−∏h=1d∏i,j∈Ph,j≠i1−νj2wj/PhPh−11/d1/2.If all PFNs are partitioned into one sort, the PFWIPGBM operator reduces to the Pythagorean fuzzy weighted interaction geometric Bonferroni mean (PFWIGBM) operator as follows:(64)PFWIGBMp,qα1,α2,…,αn=1p+q⊗i,j∈Ph,j≠ipαiwi⊕qαjwj1/mm−1=1−1−∏i,j∈Ph,j≠i1−ξ+η1/mm−1+∏i,j∈Ph,j≠iη1/mm−11/p+q1/2,1−∏i,j∈Ph,j≠i1−ξ+η1mm−1+∏i,j∈Ph,j≠iη1/mm−11/p+q−∏i,j∈Ph,j≠iη1/mm−1p+q1/2.


## 4. A New Method for Multiple-Attribute Decision-Making Based on the Proposed PFWIPBM and PFWIPGBM Operator

For a MAGDM problem with PFNs, letE1,E2,…,Et be a collection of decision-makers, let A1,A2,…,Am be a collection of alternatives, and let C=C1,C2,…,Cn be a collection of attributes. Dk=dijkm×n is a decision matrix given by decision-maker Ek, where dijk=μijk,νijk is the evaluation value of Ai with respect to attribute Cj given by decision-maker Ek. λ1,λ2,…,λt is the weight vector of decision-makers, where λk≥0 and ∑k=1tλk=1. Let w1,w2,…,wn be the weight vector of attributes with wj≥0 and ∑j=1nwj=1. The proposed method based on the new operators is presented as follows.Step 1.
Decision-makerEk gives the evaluation value of alternative Ai with respect to attribute Cj using Pythagorean fuzzy number dijk. Then, the decision matrix is formed as Dk=dijkm×n,k=1,2,…,t.Step 2.
Aggregate different decision matrices into a collective one by using the PFIPBM operator or the PFIPGBM operator.(65)rij=μij,νij=PFIPBMp,qrij1,rij2,…,rijt=1d⊕h=1d1Ph⊕l∈Phαijlp⊗1Ph−1⊕k∈Ph,k≠lαijkq1/p+q,rij=μij,νij=PFIPGBMp,qrij1,rij2,…,rijt=⊗h=1d1p+q⊗l,k∈Ph,l≠kpαijl⊕qαijk1/PhPh−11/d,where p,q≥0, Ph denotes the cardinality of Ph, d is the number of the partitioned sorts and ∑h=1dPh=t. By using Eq.(8) and Eq. (41), rij=μij,νij can be calculated.Step 3.
Calculate the collective evaluation values of each alternatives by using the proposed PFWIPBM operator or PFWIPGBM operator.(66)ri=μi,νi=PFWIPBMp,qri1,ri2,…,rin=1d⊕h=1d1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q,ri=μi,νi=PFWIPGBMp,qri1,ri2,…,rin⊗h=1d1p+q⊗i,j∈Ph,j≠ipαiwi⊕qαjwj1/PhPh−11/d,where p,q≥0, Ph denotes the cardinality of Ph, d is the number of the partitioned sorts, and ∑h=1dPh=n. w1,w2,…,wn is the weight vector with wi≥0 and ∑j=1nwj=1. By using (31) and (55), the collective evaluation values ri=μi,νii=1,2,…,m of alternatives can be calculated.Step 4.
Calculate the score valueSri and accuracy value Ari of the collective evaluation value ri of alternative Ai by using (2) and (3).Step 5.
Rank alternatives according the method introduced in Definition3 and select the optimal alternative.The new method has some desirable advantages as follows: (1) Pythagorean fuzzy numbers are used as the evaluation values, which are more powerful and flexible comparing with other existing tools to model uncertain and fuzzy information; (2) Bonferroni mean has been used to model interrelationship of attributes; (3) partitioned Bonferroni mean operator is used to depict the interrelationship among different sorts, which can lead to more accurate decision results; and (4) the interaction between membership and nonmembership has been considered to avoid unreasonable results caused by extremely small values of membership or nonmembership.


## 5. An Illustrative Example

An investment company wants to invest a large amount of money to the following five possible areas:A1—real estate, A2—energy industry, A3—gold, A4—stock market, and A5—artificial intellectual company. The company’s board has decided to appoint an expert panel consisting of three decision-makers Eii=1,2,3 to evaluate the investment opinions on the basis of following five interrelated attributes: C1—market potential, C2—growth potential, C3—risk of losing capital sum, C4—the amount of interests received, and C5—inflation. Based on the interrelationship, the attributes have been partitioned into the following two sets P1=C1,C2 and P2=C3,C4,C5. The proposed multiple-attribute group decision-making method is applied for the selection of the best investment option as follows.


### 5.1. Decision-Making Steps

Step 1.
Decision matricesDkk=1,2,3 are presented by decision-makers when evaluating alternatives with respect to attributes. The results are shown in Tables 1–3.Step 2.
The collective decision matrix is obtained by aggregating different decision-makers’ evaluation values. The decision-makers are partitioned into one sort, and (30) is used to calculate the collective decision matrix, and the results are shown in Table 4. Here, p=1 and q=2.Step 3.
The attribute weight vector is given as (0.10, 0.20, 0.25, 0.30, 0.15). The alternative’s collective evaluation values are calculated by using the PFWIPBM operator andp=1,q=2. The results are as follows
(67)r1=0.3563,0.3563,r2=0.3514,0.1674,r3=0.3892,0.1629,r4=0.3222,0.1721,r5=0.3588,0.1374.Step 4.
The scores ofri can be calculated as follows:
(68)Sr1=0.0904,Sr2=0.0955,Sr3=0.1249,Sr4=0.0742,Sr5=0.1099.Step 5.
The alternatives can be ranked according to the ranking of scoresSri i=1,2,⋯,5 to get
(69)A3>A5>A2>A1>A4.
The optimal alternative isA3.Table 1
Pythagorean fuzzy decision matrixD1.
C1
C2
C3
C4
C5
A1
(0.65, 0.30)
(0.60, 0.10)
(0.80, 0.30)
(0.75, 0.40)
(0.80, 0.10)
A2
(0.70, 0.20)
(0.50, 0.30)
(0.60, 0.40)
(0.80, 0.10)
(0.70, 0.30)
A3
(0.50, 0.40)
(0.80, 0.20)
(0.40, 0.30)
(0.90, 0.20)
(0.60, 0.30)
A4
(0.50, 0.60)
(0.40, 0.20)
(0.70, 0.20)
(0.60, 0.40)
(0.85, 0.15)
A5
(0.80, 0.30)
(0.90, 0.00)
(0.50, 0.10)
(0.40, 0.20)
(0.50, 0.30)Table 2
Pythagorean fuzzy decision matrixD2.
C1
C2
C3
C4
C5
A1
(0.50, 0.40)
(0.60, 0.20)
(0.85, 0.35)
(0.70, 0.30)
(0.80, 0.20)
A2
(0.75, 0.25)
(0.40, 0.50)
(0.70, 0.30)
(0.85, 0.20)
(0.60, 0.20)
A3
(0.60, 0.30)
(0.85, 0.10)
(0.50, 0.40)
(0.90, 0.10)
(0.75, 0.25)
A4
(0.50, 0.65)
(0.30, 0.40)
(0.80, 0.15)
(0.65, 0.30)
(0.80, 0.20)
A5
(0.85, 0.20)
(0.95, 0.05)
(0.60, 0.20)
(0.40, 0.30)
(0.50, 0.40)Table 3
Pythagorean fuzzy decision matrixD3.
C1
C2
C3
C4
C5
A1
(0.60, 0.35)
(0.50, 0.30)
(0.80, 0.40)
(0.50, 0.25)
(0.85, 0.25)
A2
(0.80, 0.20)
(0.30, 0.40)
(0.65, 0.20)
(0.90, 0.20)
(0.50, 0.40)
A3
(0.70, 0.30)
(0.80, 0.25)
(0.60, 0.30)
(0.80, 0.30)
(0.50, 0.30)
A4
(0.50, 0.50)
(0.40, 0.20)
(0.85, 0.10)
(0.60, 0.20)
(0.90, 0.10)
A5
(0.75, 0.35)
(0.80, 0.05)
(0.55, 0.25)
(0.30, 0.40)
(0.40, 0.50)Table 4
Pythagorean fuzzy collective decision matrixD.
C1
C2
C3
C4
C5
A1
(0.5889, 0.3510)
(0.5706, 0.2119)
(0.8206, 0.3515)
(0.6765, 0.3335)
(0.8210, 0.1927)
A2
(0.7541, 0.2187)
(0.4115, 0.4089)
(0.6545, 0.3081)
(0.8589, 0.1742)
(0.6122, 0.3090)
A3
(0.6106, 0.3355)
(0.8186, 0.1889)
(0.5109, 0.3386)
(0.8738, 0.2065)
(0.6350, 0.2869)
A4
(0.5043, 0.5882)
(0.3712, 0.2815)
(0.7715, 0.1524)
(0.6197, 0.3102)
(0.8555, 0.1521)
A5
(0.8035, 0.2864)
(0.9010, 0.0448)
(0.5531, 0.1958)
(0.3709, 0.3100)
(0.4711, 0.4070)


### 5.2. Comparison Analysis and Discussions

#### 5.2.1. Influence of the Parametersp and q

In order to illustrate influence of parametersp and q on the ranking results, we consider different p and q in Steps 2 and 3. For simplicity, the same p and q are used in Steps 2 and 3. For example, if p=1 and q=2 are used in Step 2, then p=1 and q=2 are also used in Step 3. The results are shown in Table 5; here, OA means the optimal alternative. From the results, we can see that A5 is the optimal alternative in the cases of p=1 and q=3 and p=2 and q=2 and A3 becomes the optimal alternative in other cases. If p=1 and q=1 and p=1 and q=2, the optimal alternative is A3 and the suboptimal alternative is A5. With the increase of p and q, the suboptimal becomes A2 and A5 is ranked last. A3 has relatively larger memberships and relatively smaller nonmemberships comparing with other alternatives. Though A5 has the largest membership and the smallest nonmembership among all the evaluation values, it is still ranked last with increasing p and q due to the intersection between membership and nonmembership that is considered. The rankings of alternatives change with different p and q.Table 5
Results of differentp and q considering interaction.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
0.0904
0.0950
0.1256
0.0734
0.1091
A3>A5>A2>A1>A4
A3
p=1, q=2
0.0904
0.0955
0.1249
0.0742
0.1099
A3>A5>A2>A1>A4
A3
p=1, q=3
0.0264
0.0958
0.1163
0.0754
0.2574
A5>A3>A2>A4>A1
A5
p=2, q=2
−0.0268
0.0891
0.0630
0.0743
0.2040
A5>A2>A4>A3>A1
A5
p=1, q=4
−0.1183
0.2049
0.5881
0.1483
−0.3674
A3>A2>A4>A1>A5
A3
p=2, q=3
−0.2481
0.1444
0.4812
0.0797
−0.4889
A3>A2>A4>A1>A5
A3
p=1, q=5
−0.0570
0.2625
0.6405
0.2327
−0.3190
A3>A2>A4>A1>A5
A3
p=2, q=4
−0.2037
0.1983
0.5309
0.1647
−0.4507
A3>A2>A4>A1>A5
A3
p=3, q=3
−0.2567
0.1791
0.4917
0.1474
−0.5051
A3>A2>A4>A1>A5
A3The larger thep and q, the more interaction can be emphasized. But in special cases of p=0 or q=0, there is no interaction between input arguments. From the viewpoint of the risk attitudes, decision-makers are more risk-seeking with the increase of p and q. By taking different p and q in the PFIPBM operator, the PFIPGBM operator, the PFWIPBM, or the PFWIPGBM operator, different risk attitudes of decision-makers can be reflected and different aspects of decision problem can also be reflected, since the arithmetic aggregation operator stresses the impact of the overall input arguments while the geometric aggregation operator emphasizes the balance of the input arguments [47]. In real decision-making, decision-makers can select the corresponding aggregation operator and p and q according to their preferences and real needs. For simplicity, the decision-makers can select p=1 and q=1 if the decision-maker is risk averse, which is simple and intuitive.


#### 5.2.2. Comparison with Other Methods

If interactions between the memberships and nonmemberships are not considered, the PFIPBM operator and the PFWIPBM operator reduce to the Pythagorean fuzzy partitioned Bonferroni mean (PFPBM) operator and the Pythagorean fuzzy weighted partitioned Bonferroni mean (PFWPBM) operator as follows:(70)PFPBMp,qα1,α2,…,αn=1d⊕h=1d1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q=1−∏h=1d1−1−∏i∈Ph1−μipξ21/Ph1/p+q1/d1/2,∏h=1d1−1−∏i∈Ph1−1−νi2p1−η21/Ph1/p+q1/d1/2,where
(71)ξ=1−∏j∈Ph,j≠i1−μjq21/Ph−1,η=∏j∈Ph,j≠i1−1−νj2q1/Ph−1,PFWPBMp,qα1,α2,…,αn=1d⊕h=1d1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2.If thePFPBMp,q operator is used in Step 2 and the PFWPBMp,q operator is used in Step 3; the interactions between memberships and nonmemberships are not considered and the results are shown in Table 6. From the results, we can see that the ranking of alternatives is different from that of considered interaction between memberships and nonmemberships. A5 becomes the optimal alternative and A3 becomes the suboptimal alternative in all cases. In fact, the rankings of alternatives are the same except for the case of p=1 and q=1. If interactions are not considered, the effect of memberships will reduce if one membership is nearly approaching zero in multiply operation no matter what about the other memberships and the effect of nonmemberships will be reduced if one nonmembership is nearly approaching zero in sum operation no matter what about the other nonmemberships. These shortcomings can be overcome by considering interactions between memberships and nonmemberships.Table 6
Results of differentp and q without considering interaction.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
−0.5132
−0.5198
−0.5082
−0.5213
−0.4771
A5>A3>A1>A2>A4
A5
p=1, q=2
−0.5041
−0.5030
−0.4830
−0.5110
−0.4519
A5>A3>A2>A1>A4
A5
p=1, q=3
−0.4892
−0.4790
−0.4437
−0.4946
−0.4128
A5>A3>A2>A1>A4
A5
p=2, q=2
−0.5103
−0.5073
−0.4989
−0.5185
−0.4678
A5>A3>A2>A1>A4
A5
p=1, q=4
−0.4738
−0.4557
−0.4063
−0.4791
−0.3758
A5>A3>A2>A1>A4
A5
p=2, q=3
−0.5046
−0.4962
−0.4833
−0.5121
−0.4522
A5>A3>A2>A1>A4
A5
p=1, q=5
−0.4599
−0.4349
−0.3743
−0.4656
−0.3444
A5>A3>A2>A1>A4
A5
p=2, q=4
−0.4943
−0.4798
−0.4563
−0.5009
−0.4260
A5>A3>A2>A1>A4
A5
p=3, q=3
−0.5073
−0.4965
−0.4914
−0.5155
−0.4585
A5>A3>A2>A1>A4
A5If the TOPSIS method has been used, the Pythagorean fuzzy interaction averaging (PFIA) operator is used to aggregate different evaluation values given by different decision-makers into collective ones. Then, calculate the weighted decision matrixD′ as in Table 7. The Pythagorean fuzzy positive ideal solution (PFPIS) can be determined as r+=r1+,r2+,…,r5+=maxjr1j,maxjr2j,…,maxjr5j=0.3148,0.1513,0.5329,0.0396,0.7021,0.1073,0.5935,0.1907,0.4240,0.1022. The Pythagorean fuzzy negative ideal solution (PFNIS) can be determined as r−=r1−,r2−,…,r5−=minjr1j,minjr2j,…,minjr5j=0.1684,0.2434,0.1908,0.2058,0.2700,0.1954,0.2082,0.1828,0.1918,0.1847. The distances of each alternative evaluation values to the PFPIS and the PFNIS can be calculated by using the following equations, respectively, di+=∑j=15drij,rj+, di−=∑j=15drij,rj−i=1,2,…,5. We can get d1+=0.3196, d2+=0.2887, d3+=0.2150,d4+=0.3023,d5+=0.2978,d1−=0.3098,d2−=0.2454,d3−=0.3047,d4−=0.2278, and d5−=0.2175. The closeness coefficients can be calculated by the equation CCi=di−/di−+di+, and we can get CC1=0.4923,CC2=0.4594,CC3=0.5863,CC4=0.4297 and CC5=0.4221. The alternatives can be ranked according to the ranking of CCi to get A3>A2>A1>A4>A5. The optimal alternative is A3. The optimal alternative is the same as the most case of the proposed method, but the rankings of alternatives are slightly different.
(72)αij=PFIAαij1,αij2,…,αijt=1t⊕k=1tαijk=1−∏k=1t1−μijk21/t,∏k=1t1−μijk21/t−∏k=1t1−μijk2+νijk21/t.Table 7
Pythagorean fuzzy weighted decision matrixD′.
C1
C2
C3
C4
C5
A1
(0.2046,0.1402)
(0.2751,0.1129)
(0.4920,0.2945)
(0.4046,0.2437)
(0.3914,0.1297)
A2
(0.2839,0.1038)
(0.1908,0.2058)
(0.5840,0.1981)
(0.5721,0.1664)
(0.2607,0.1515)
A3
(0.2141,0.1358)
(0.4459,0.1350)
(0.4507,0.1954)
(0.5935,0.1907)
(0.2740,0.1408)
A4
(0.1684,0.2434)
(0.1706,0.1366)
(0.7021,0.1073)
(0.3664,0.2098)
(0.4240,0.1022)
A5
(0.3148,0.1513)
(0.5329,0.0396)
(0.4892,0.1139)
(0.2082,0.1828)
(0.1918,0.1847)If interaction is not considered in TOPSIS as the method in [48], the Pythagorean fuzzy averaging (PFA) operator is first used to aggregate evaluation values given by different decision-makers into collective ones. The PFA operator is defined as
(73)αij=PFAαij1,αij2,…,αijt=1t⊕k=1tαijk=1−∏k=1t1−μijk21/t,∏k=1tνijk1/t.The weighted collective decision matrixD′′ is calculated as in Table 8, where the weight vector is also taken as 0.10,0.20,0.25,0.30,0.15. The PFPIS can be determined as r+=0.2893,0.8577,0.5329,0.0000,0.4501,0.6163,0.5721,0.5757,0.4240,0.7479. The PFNIS can be determined as r−=0.1684,0.9470,0.1908,0.8290,0.2700,0.7580,0.2082,0.6887,0.1918,0.8688. The distances of alternative’s weighted evaluation values to the PFPIS and the PFNIS can be calculated as d1+=0.7299, d2+=0.7030, d3+=0.6430, d4+=0.6901, d5+=0.4851, d1−=0.4651, d2−=0.4159, d3−=0.5009,d4−=0.4361, d5−=0.6523. The closeness coefficients can be calculated as CC1=0.3892,CC2=0.3717,CC3=0.4379,CC4=0.3872, and CC5=0.5735. The alternatives can be ranked as A5>A3>A1>A4>A2 and the optimal alternative is A5.Table 8
Pythagorean fuzzy weighted decision matrixD″.
C1
C2
C3
C4
C5
A1
(0.2046,0.8997)
(0.2751,0.7110)
(0.4920,0.7678)
(0.4046,0.7042)
(0.3914,0.7673)
A2
(0.2839,0.8577)
(0.3245,0.8290)
(0.3603,0.7329)
(0.5721,0.5757)
(0.2607,0.8299)
A3
(0.2141,0.8951)
(0.6970,0.7024)
(0.2700,0.7580)
(0.5935,0.5995)
(0.2740,0.8272)
A4
(0.1684,0.9470)
(0.2912,0.7591)
(0.4501,0.6163)
(0.3664,0.6887)
(0.4240,0.7479)
A5
(0.3148,0.8792)
(0.7955,0.0000)
(0.2952,0.6431)
(0.2082,0.6887)
(0.1918,0.8688)If attributes are all are partitioned into one sort, (37) is used to aggregate alternative evaluation values into collective ones in Step 3 and other steps are the same. Then, the results are shown in Table 9. From the results, we can see that A3 becomes the optimal alternative in the cases of p=1, q=1, p=1, and q=2 and A5 becomes the optimal alternative in the cases of p=1 and q=3 and p=2 and q=2. A2 becomes the optimal alternative in the other cases. The results are different from those of the partitioned one. If attributes can be divided into several classes and there is interaction relationships among the same class and there is no interaction between classes, the PFWIPBM operator can be used to assure accuracy and reasonableness of decision results.Table 9
Results of differentp and q considering interaction with one sort.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1,q=1
0.1709
0.1893
0.2253
0.1583
0.1706
A3>A2>A1>A5>A4
A3
p=1, q=2
0.0979
0.1073
0.1274
0.0903
0.0953
A3>A2>A1>A5>A4
A3
p=1, q=3
0.4854
0.1068
0.5529
0.0920
0.5541
A5>A3>A1>A2>A4
A5
p=2, q=2
0.4507
0.5012
0.5184
0.0904
0.5241
A5>A2>A3>A1>A4
A5
p=1, q=4
0.4996
0.6237
0.5562
0.5878
0.5462
A2>A4>A3>A5>A1
A2
p=2, q=3
0.4183
0.5855
0.4759
0.5463
0.4801
A2>A4>A5>A3>A1
A2
p=1, q=5
0.5807
0.6883
0.6175
0.6697
0.6104
A2>A4>A3>A5>A1
A2
p=2, q=4
0.5076
0.6526
0.5392
0.6334
0.5463
A2>A4>A5>A3>A1
A2
p=3, q=3
0.4883
0.6419
0.5146
0.6245
0.5285
A2>A4>A5>A3>A1
A2If attributes have been partitioned into one sort and interaction between membership and nonmembership is not considered, the Pythagorean fuzzy Bonferroni meanPFBMp,q operator [43] is used in Step 2 and Pythagorean fuzzy weighted Bonferroni mean PFWBMp,q operator is used in Step 3; the results are shown in Table 10. A5 becomes the optimal alternative and A3 becomes the suboptimal alternative. Though the optimal and suboptimal alternatives are the same as those of partitioned cases, the ranking of alternatives is different.
(74)PFBMp,qα1,α2,…,αn=1nn−1⊕i,j∈Ph,j≠iαip⊗αjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2,PFWBMp,qα1,α2,…,αn=1nn−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2.Table 10
Results of differentp and q without considering interaction with one sort.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
−0.4835
−0.4958
−0.4628
−0.4836
−0.4528
A5>A3>A1>A4>A2
A5
p=1, q=2
−0.4628
−0.4627
−0.4270
−0.4496
−0.4147
A5>A3>A4>A2>A1
A5
p=1, q=3
−0.4399
−0.4171
−0.3838
−0.4183
−0.3669
A5>A3>A2>A4>A1
A5
p=2, q=2
−0.4567
−0.4625
−0.4195
−0.4358
−0.4100
A5>A3>A4>A1>A2
A5
p=1, q=4
−0.4199
−0.3720
−0.3441
−0.3920
−0.3225
A5>A3>A2>A4>A1
A5
p=2, q=3
−0.4424
−0.4384
−0.3939
−0.4156
−0.3830
A5>A3>A4>A2>A1
A5
p=1, q=5
−0.4029
−0.3317
−0.3094
−0.3726
−0.2843
A5>A3>A2>A4>A1
A5
p=2, q=4
−0.4272
−0.4057
−0.3649
−0.3968
−0.3507
A5>A3>A2>A4>A1
A5
p=3, q=3
−0.4347
−0.4335
−0.3821
−0.4036
−0.3734
A5>A3>A4>A2>A1
A5If the PFIA operator is used in aggregating different decision-makers’ evaluation values into collective ones in Step2 and the Pythagorean fuzzy weighted interaction averaging (PFIWA) operator is used in aggregating alternatives’ evaluation values into collective ones in Step 3, the collective evaluation values of alternatives are as r1=0.7243,0.3242, r2=0.7188,0.2748, r3=0.7586,0.2535, r4=0.6825,0.2825, and r5=0.6367,0.2416. The scores of alternatives can be calculated as Sr1=0.4195, Sr2=0.4411, Sr3=0.5112, Sr4=0.3860, and Sr5=0.3990, then Sr3>Sr2>Sr1>Sr5>Sr4. The alternatives can be ranked accordingly as A3>A2>A1>A5>A4. The optimal alternative is A3.
(75)αi=PFIWAαi1,αi2,…,αin=⊕j=1nwjαij=1−∏j=1n1−μij2wj,∏j=1n1−μij2wj−∏j=1n1−μij2+νij2wj.If the PFA operator is used in Step2 and the PFWA operator is used in the second phase [38], the results are as r1=0.7243,0.2654, r2=0.7188,0.2489, r3=0.7586,0.2364, r4=0.6825,0.2282, and r5=0.6367,0.0000, where
(76)αi=PFWAαi1,αi2,…,αin=⊕j=1nwjαij=1−∏j=1n1−μij2wj,∏j=1nνijwj.The scores can be calculated asSr1=0.4542, Sr2=0.4547, Sr3=0.5195, Sr4=0.4138, and Sr5=0.4574. The alternatives can be ranked accordingly to the ranking of scores to get A3>A5>A2>A1>A4. The optimal alternative is A3.As discussed above, we summarize differences between our proposed method with the existing methods in Table11. In a word, we can know from Table 11 that our proposed method is based on the Pythagorean fuzzy numbers and partitioned Bonferroni mean operator with parameters p and q. Moreover, interaction between membership and nonmembership has been considered by using the Pythagorean fuzzy interaction operation laws and interaction between attributes have been considered by using the Bonferroni mean. Hence the new method is more general and flexible than the existing methods. Partitioned the input arguments into several sorts can accurately model the interrelationship between attributes.Table 11
The characteristic comparisons of different methods.
Methods
Information by Pythagorean fuzzy number
Whether to consider the interrelationships between aggregating arguments
Liang et al. [43]
Yes
Yes
Xu and Yager [38]
No
Yes
Zhang and Xu [26]
Yes
No
Our proposed method
Yes
Yes
Methods
Whether to consider the partition of the input arguments
Whether to consider the interactions between membership and nonmembership
Liang et al. [43]
No
No
Xu and Yager [38]
No
No
Zhang and Xu [26]
No
No
Our proposed method
Yes
Yes


## 5.1. Decision-Making Steps

Step 1.
Decision matricesDkk=1,2,3 are presented by decision-makers when evaluating alternatives with respect to attributes. The results are shown in Tables 1–3.Step 2.
The collective decision matrix is obtained by aggregating different decision-makers’ evaluation values. The decision-makers are partitioned into one sort, and (30) is used to calculate the collective decision matrix, and the results are shown in Table 4. Here, p=1 and q=2.Step 3.
The attribute weight vector is given as (0.10, 0.20, 0.25, 0.30, 0.15). The alternative’s collective evaluation values are calculated by using the PFWIPBM operator andp=1,q=2. The results are as follows
(67)r1=0.3563,0.3563,r2=0.3514,0.1674,r3=0.3892,0.1629,r4=0.3222,0.1721,r5=0.3588,0.1374.Step 4.
The scores ofri can be calculated as follows:
(68)Sr1=0.0904,Sr2=0.0955,Sr3=0.1249,Sr4=0.0742,Sr5=0.1099.Step 5.
The alternatives can be ranked according to the ranking of scoresSri i=1,2,⋯,5 to get
(69)A3>A5>A2>A1>A4.
The optimal alternative isA3.Table 1
Pythagorean fuzzy decision matrixD1.
C1
C2
C3
C4
C5
A1
(0.65, 0.30)
(0.60, 0.10)
(0.80, 0.30)
(0.75, 0.40)
(0.80, 0.10)
A2
(0.70, 0.20)
(0.50, 0.30)
(0.60, 0.40)
(0.80, 0.10)
(0.70, 0.30)
A3
(0.50, 0.40)
(0.80, 0.20)
(0.40, 0.30)
(0.90, 0.20)
(0.60, 0.30)
A4
(0.50, 0.60)
(0.40, 0.20)
(0.70, 0.20)
(0.60, 0.40)
(0.85, 0.15)
A5
(0.80, 0.30)
(0.90, 0.00)
(0.50, 0.10)
(0.40, 0.20)
(0.50, 0.30)Table 2
Pythagorean fuzzy decision matrixD2.
C1
C2
C3
C4
C5
A1
(0.50, 0.40)
(0.60, 0.20)
(0.85, 0.35)
(0.70, 0.30)
(0.80, 0.20)
A2
(0.75, 0.25)
(0.40, 0.50)
(0.70, 0.30)
(0.85, 0.20)
(0.60, 0.20)
A3
(0.60, 0.30)
(0.85, 0.10)
(0.50, 0.40)
(0.90, 0.10)
(0.75, 0.25)
A4
(0.50, 0.65)
(0.30, 0.40)
(0.80, 0.15)
(0.65, 0.30)
(0.80, 0.20)
A5
(0.85, 0.20)
(0.95, 0.05)
(0.60, 0.20)
(0.40, 0.30)
(0.50, 0.40)Table 3
Pythagorean fuzzy decision matrixD3.
C1
C2
C3
C4
C5
A1
(0.60, 0.35)
(0.50, 0.30)
(0.80, 0.40)
(0.50, 0.25)
(0.85, 0.25)
A2
(0.80, 0.20)
(0.30, 0.40)
(0.65, 0.20)
(0.90, 0.20)
(0.50, 0.40)
A3
(0.70, 0.30)
(0.80, 0.25)
(0.60, 0.30)
(0.80, 0.30)
(0.50, 0.30)
A4
(0.50, 0.50)
(0.40, 0.20)
(0.85, 0.10)
(0.60, 0.20)
(0.90, 0.10)
A5
(0.75, 0.35)
(0.80, 0.05)
(0.55, 0.25)
(0.30, 0.40)
(0.40, 0.50)Table 4
Pythagorean fuzzy collective decision matrixD.
C1
C2
C3
C4
C5
A1
(0.5889, 0.3510)
(0.5706, 0.2119)
(0.8206, 0.3515)
(0.6765, 0.3335)
(0.8210, 0.1927)
A2
(0.7541, 0.2187)
(0.4115, 0.4089)
(0.6545, 0.3081)
(0.8589, 0.1742)
(0.6122, 0.3090)
A3
(0.6106, 0.3355)
(0.8186, 0.1889)
(0.5109, 0.3386)
(0.8738, 0.2065)
(0.6350, 0.2869)
A4
(0.5043, 0.5882)
(0.3712, 0.2815)
(0.7715, 0.1524)
(0.6197, 0.3102)
(0.8555, 0.1521)
A5
(0.8035, 0.2864)
(0.9010, 0.0448)
(0.5531, 0.1958)
(0.3709, 0.3100)
(0.4711, 0.4070)


## 5.2. Comparison Analysis and Discussions

### 5.2.1. Influence of the Parametersp and q

In order to illustrate influence of parametersp and q on the ranking results, we consider different p and q in Steps 2 and 3. For simplicity, the same p and q are used in Steps 2 and 3. For example, if p=1 and q=2 are used in Step 2, then p=1 and q=2 are also used in Step 3. The results are shown in Table 5; here, OA means the optimal alternative. From the results, we can see that A5 is the optimal alternative in the cases of p=1 and q=3 and p=2 and q=2 and A3 becomes the optimal alternative in other cases. If p=1 and q=1 and p=1 and q=2, the optimal alternative is A3 and the suboptimal alternative is A5. With the increase of p and q, the suboptimal becomes A2 and A5 is ranked last. A3 has relatively larger memberships and relatively smaller nonmemberships comparing with other alternatives. Though A5 has the largest membership and the smallest nonmembership among all the evaluation values, it is still ranked last with increasing p and q due to the intersection between membership and nonmembership that is considered. The rankings of alternatives change with different p and q.Table 5
Results of differentp and q considering interaction.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
0.0904
0.0950
0.1256
0.0734
0.1091
A3>A5>A2>A1>A4
A3
p=1, q=2
0.0904
0.0955
0.1249
0.0742
0.1099
A3>A5>A2>A1>A4
A3
p=1, q=3
0.0264
0.0958
0.1163
0.0754
0.2574
A5>A3>A2>A4>A1
A5
p=2, q=2
−0.0268
0.0891
0.0630
0.0743
0.2040
A5>A2>A4>A3>A1
A5
p=1, q=4
−0.1183
0.2049
0.5881
0.1483
−0.3674
A3>A2>A4>A1>A5
A3
p=2, q=3
−0.2481
0.1444
0.4812
0.0797
−0.4889
A3>A2>A4>A1>A5
A3
p=1, q=5
−0.0570
0.2625
0.6405
0.2327
−0.3190
A3>A2>A4>A1>A5
A3
p=2, q=4
−0.2037
0.1983
0.5309
0.1647
−0.4507
A3>A2>A4>A1>A5
A3
p=3, q=3
−0.2567
0.1791
0.4917
0.1474
−0.5051
A3>A2>A4>A1>A5
A3The larger thep and q, the more interaction can be emphasized. But in special cases of p=0 or q=0, there is no interaction between input arguments. From the viewpoint of the risk attitudes, decision-makers are more risk-seeking with the increase of p and q. By taking different p and q in the PFIPBM operator, the PFIPGBM operator, the PFWIPBM, or the PFWIPGBM operator, different risk attitudes of decision-makers can be reflected and different aspects of decision problem can also be reflected, since the arithmetic aggregation operator stresses the impact of the overall input arguments while the geometric aggregation operator emphasizes the balance of the input arguments [47]. In real decision-making, decision-makers can select the corresponding aggregation operator and p and q according to their preferences and real needs. For simplicity, the decision-makers can select p=1 and q=1 if the decision-maker is risk averse, which is simple and intuitive.


### 5.2.2. Comparison with Other Methods

If interactions between the memberships and nonmemberships are not considered, the PFIPBM operator and the PFWIPBM operator reduce to the Pythagorean fuzzy partitioned Bonferroni mean (PFPBM) operator and the Pythagorean fuzzy weighted partitioned Bonferroni mean (PFWPBM) operator as follows:(70)PFPBMp,qα1,α2,…,αn=1d⊕h=1d1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q=1−∏h=1d1−1−∏i∈Ph1−μipξ21/Ph1/p+q1/d1/2,∏h=1d1−1−∏i∈Ph1−1−νi2p1−η21/Ph1/p+q1/d1/2,where
(71)ξ=1−∏j∈Ph,j≠i1−μjq21/Ph−1,η=∏j∈Ph,j≠i1−1−νj2q1/Ph−1,PFWPBMp,qα1,α2,…,αn=1d⊕h=1d1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2.If thePFPBMp,q operator is used in Step 2 and the PFWPBMp,q operator is used in Step 3; the interactions between memberships and nonmemberships are not considered and the results are shown in Table 6. From the results, we can see that the ranking of alternatives is different from that of considered interaction between memberships and nonmemberships. A5 becomes the optimal alternative and A3 becomes the suboptimal alternative in all cases. In fact, the rankings of alternatives are the same except for the case of p=1 and q=1. If interactions are not considered, the effect of memberships will reduce if one membership is nearly approaching zero in multiply operation no matter what about the other memberships and the effect of nonmemberships will be reduced if one nonmembership is nearly approaching zero in sum operation no matter what about the other nonmemberships. These shortcomings can be overcome by considering interactions between memberships and nonmemberships.Table 6
Results of differentp and q without considering interaction.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
−0.5132
−0.5198
−0.5082
−0.5213
−0.4771
A5>A3>A1>A2>A4
A5
p=1, q=2
−0.5041
−0.5030
−0.4830
−0.5110
−0.4519
A5>A3>A2>A1>A4
A5
p=1, q=3
−0.4892
−0.4790
−0.4437
−0.4946
−0.4128
A5>A3>A2>A1>A4
A5
p=2, q=2
−0.5103
−0.5073
−0.4989
−0.5185
−0.4678
A5>A3>A2>A1>A4
A5
p=1, q=4
−0.4738
−0.4557
−0.4063
−0.4791
−0.3758
A5>A3>A2>A1>A4
A5
p=2, q=3
−0.5046
−0.4962
−0.4833
−0.5121
−0.4522
A5>A3>A2>A1>A4
A5
p=1, q=5
−0.4599
−0.4349
−0.3743
−0.4656
−0.3444
A5>A3>A2>A1>A4
A5
p=2, q=4
−0.4943
−0.4798
−0.4563
−0.5009
−0.4260
A5>A3>A2>A1>A4
A5
p=3, q=3
−0.5073
−0.4965
−0.4914
−0.5155
−0.4585
A5>A3>A2>A1>A4
A5If the TOPSIS method has been used, the Pythagorean fuzzy interaction averaging (PFIA) operator is used to aggregate different evaluation values given by different decision-makers into collective ones. Then, calculate the weighted decision matrixD′ as in Table 7. The Pythagorean fuzzy positive ideal solution (PFPIS) can be determined as r+=r1+,r2+,…,r5+=maxjr1j,maxjr2j,…,maxjr5j=0.3148,0.1513,0.5329,0.0396,0.7021,0.1073,0.5935,0.1907,0.4240,0.1022. The Pythagorean fuzzy negative ideal solution (PFNIS) can be determined as r−=r1−,r2−,…,r5−=minjr1j,minjr2j,…,minjr5j=0.1684,0.2434,0.1908,0.2058,0.2700,0.1954,0.2082,0.1828,0.1918,0.1847. The distances of each alternative evaluation values to the PFPIS and the PFNIS can be calculated by using the following equations, respectively, di+=∑j=15drij,rj+, di−=∑j=15drij,rj−i=1,2,…,5. We can get d1+=0.3196, d2+=0.2887, d3+=0.2150,d4+=0.3023,d5+=0.2978,d1−=0.3098,d2−=0.2454,d3−=0.3047,d4−=0.2278, and d5−=0.2175. The closeness coefficients can be calculated by the equation CCi=di−/di−+di+, and we can get CC1=0.4923,CC2=0.4594,CC3=0.5863,CC4=0.4297 and CC5=0.4221. The alternatives can be ranked according to the ranking of CCi to get A3>A2>A1>A4>A5. The optimal alternative is A3. The optimal alternative is the same as the most case of the proposed method, but the rankings of alternatives are slightly different.
(72)αij=PFIAαij1,αij2,…,αijt=1t⊕k=1tαijk=1−∏k=1t1−μijk21/t,∏k=1t1−μijk21/t−∏k=1t1−μijk2+νijk21/t.Table 7
Pythagorean fuzzy weighted decision matrixD′.
C1
C2
C3
C4
C5
A1
(0.2046,0.1402)
(0.2751,0.1129)
(0.4920,0.2945)
(0.4046,0.2437)
(0.3914,0.1297)
A2
(0.2839,0.1038)
(0.1908,0.2058)
(0.5840,0.1981)
(0.5721,0.1664)
(0.2607,0.1515)
A3
(0.2141,0.1358)
(0.4459,0.1350)
(0.4507,0.1954)
(0.5935,0.1907)
(0.2740,0.1408)
A4
(0.1684,0.2434)
(0.1706,0.1366)
(0.7021,0.1073)
(0.3664,0.2098)
(0.4240,0.1022)
A5
(0.3148,0.1513)
(0.5329,0.0396)
(0.4892,0.1139)
(0.2082,0.1828)
(0.1918,0.1847)If interaction is not considered in TOPSIS as the method in [48], the Pythagorean fuzzy averaging (PFA) operator is first used to aggregate evaluation values given by different decision-makers into collective ones. The PFA operator is defined as
(73)αij=PFAαij1,αij2,…,αijt=1t⊕k=1tαijk=1−∏k=1t1−μijk21/t,∏k=1tνijk1/t.The weighted collective decision matrixD′′ is calculated as in Table 8, where the weight vector is also taken as 0.10,0.20,0.25,0.30,0.15. The PFPIS can be determined as r+=0.2893,0.8577,0.5329,0.0000,0.4501,0.6163,0.5721,0.5757,0.4240,0.7479. The PFNIS can be determined as r−=0.1684,0.9470,0.1908,0.8290,0.2700,0.7580,0.2082,0.6887,0.1918,0.8688. The distances of alternative’s weighted evaluation values to the PFPIS and the PFNIS can be calculated as d1+=0.7299, d2+=0.7030, d3+=0.6430, d4+=0.6901, d5+=0.4851, d1−=0.4651, d2−=0.4159, d3−=0.5009,d4−=0.4361, d5−=0.6523. The closeness coefficients can be calculated as CC1=0.3892,CC2=0.3717,CC3=0.4379,CC4=0.3872, and CC5=0.5735. The alternatives can be ranked as A5>A3>A1>A4>A2 and the optimal alternative is A5.Table 8
Pythagorean fuzzy weighted decision matrixD″.
C1
C2
C3
C4
C5
A1
(0.2046,0.8997)
(0.2751,0.7110)
(0.4920,0.7678)
(0.4046,0.7042)
(0.3914,0.7673)
A2
(0.2839,0.8577)
(0.3245,0.8290)
(0.3603,0.7329)
(0.5721,0.5757)
(0.2607,0.8299)
A3
(0.2141,0.8951)
(0.6970,0.7024)
(0.2700,0.7580)
(0.5935,0.5995)
(0.2740,0.8272)
A4
(0.1684,0.9470)
(0.2912,0.7591)
(0.4501,0.6163)
(0.3664,0.6887)
(0.4240,0.7479)
A5
(0.3148,0.8792)
(0.7955,0.0000)
(0.2952,0.6431)
(0.2082,0.6887)
(0.1918,0.8688)If attributes are all are partitioned into one sort, (37) is used to aggregate alternative evaluation values into collective ones in Step 3 and other steps are the same. Then, the results are shown in Table 9. From the results, we can see that A3 becomes the optimal alternative in the cases of p=1, q=1, p=1, and q=2 and A5 becomes the optimal alternative in the cases of p=1 and q=3 and p=2 and q=2. A2 becomes the optimal alternative in the other cases. The results are different from those of the partitioned one. If attributes can be divided into several classes and there is interaction relationships among the same class and there is no interaction between classes, the PFWIPBM operator can be used to assure accuracy and reasonableness of decision results.Table 9
Results of differentp and q considering interaction with one sort.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1,q=1
0.1709
0.1893
0.2253
0.1583
0.1706
A3>A2>A1>A5>A4
A3
p=1, q=2
0.0979
0.1073
0.1274
0.0903
0.0953
A3>A2>A1>A5>A4
A3
p=1, q=3
0.4854
0.1068
0.5529
0.0920
0.5541
A5>A3>A1>A2>A4
A5
p=2, q=2
0.4507
0.5012
0.5184
0.0904
0.5241
A5>A2>A3>A1>A4
A5
p=1, q=4
0.4996
0.6237
0.5562
0.5878
0.5462
A2>A4>A3>A5>A1
A2
p=2, q=3
0.4183
0.5855
0.4759
0.5463
0.4801
A2>A4>A5>A3>A1
A2
p=1, q=5
0.5807
0.6883
0.6175
0.6697
0.6104
A2>A4>A3>A5>A1
A2
p=2, q=4
0.5076
0.6526
0.5392
0.6334
0.5463
A2>A4>A5>A3>A1
A2
p=3, q=3
0.4883
0.6419
0.5146
0.6245
0.5285
A2>A4>A5>A3>A1
A2If attributes have been partitioned into one sort and interaction between membership and nonmembership is not considered, the Pythagorean fuzzy Bonferroni meanPFBMp,q operator [43] is used in Step 2 and Pythagorean fuzzy weighted Bonferroni mean PFWBMp,q operator is used in Step 3; the results are shown in Table 10. A5 becomes the optimal alternative and A3 becomes the suboptimal alternative. Though the optimal and suboptimal alternatives are the same as those of partitioned cases, the ranking of alternatives is different.
(74)PFBMp,qα1,α2,…,αn=1nn−1⊕i,j∈Ph,j≠iαip⊗αjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2,PFWBMp,qα1,α2,…,αn=1nn−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2.Table 10
Results of differentp and q without considering interaction with one sort.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
−0.4835
−0.4958
−0.4628
−0.4836
−0.4528
A5>A3>A1>A4>A2
A5
p=1, q=2
−0.4628
−0.4627
−0.4270
−0.4496
−0.4147
A5>A3>A4>A2>A1
A5
p=1, q=3
−0.4399
−0.4171
−0.3838
−0.4183
−0.3669
A5>A3>A2>A4>A1
A5
p=2, q=2
−0.4567
−0.4625
−0.4195
−0.4358
−0.4100
A5>A3>A4>A1>A2
A5
p=1, q=4
−0.4199
−0.3720
−0.3441
−0.3920
−0.3225
A5>A3>A2>A4>A1
A5
p=2, q=3
−0.4424
−0.4384
−0.3939
−0.4156
−0.3830
A5>A3>A4>A2>A1
A5
p=1, q=5
−0.4029
−0.3317
−0.3094
−0.3726
−0.2843
A5>A3>A2>A4>A1
A5
p=2, q=4
−0.4272
−0.4057
−0.3649
−0.3968
−0.3507
A5>A3>A2>A4>A1
A5
p=3, q=3
−0.4347
−0.4335
−0.3821
−0.4036
−0.3734
A5>A3>A4>A2>A1
A5If the PFIA operator is used in aggregating different decision-makers’ evaluation values into collective ones in Step2 and the Pythagorean fuzzy weighted interaction averaging (PFIWA) operator is used in aggregating alternatives’ evaluation values into collective ones in Step 3, the collective evaluation values of alternatives are as r1=0.7243,0.3242, r2=0.7188,0.2748, r3=0.7586,0.2535, r4=0.6825,0.2825, and r5=0.6367,0.2416. The scores of alternatives can be calculated as Sr1=0.4195, Sr2=0.4411, Sr3=0.5112, Sr4=0.3860, and Sr5=0.3990, then Sr3>Sr2>Sr1>Sr5>Sr4. The alternatives can be ranked accordingly as A3>A2>A1>A5>A4. The optimal alternative is A3.
(75)αi=PFIWAαi1,αi2,…,αin=⊕j=1nwjαij=1−∏j=1n1−μij2wj,∏j=1n1−μij2wj−∏j=1n1−μij2+νij2wj.If the PFA operator is used in Step2 and the PFWA operator is used in the second phase [38], the results are as r1=0.7243,0.2654, r2=0.7188,0.2489, r3=0.7586,0.2364, r4=0.6825,0.2282, and r5=0.6367,0.0000, where
(76)αi=PFWAαi1,αi2,…,αin=⊕j=1nwjαij=1−∏j=1n1−μij2wj,∏j=1nνijwj.The scores can be calculated asSr1=0.4542, Sr2=0.4547, Sr3=0.5195, Sr4=0.4138, and Sr5=0.4574. The alternatives can be ranked accordingly to the ranking of scores to get A3>A5>A2>A1>A4. The optimal alternative is A3.As discussed above, we summarize differences between our proposed method with the existing methods in Table11. In a word, we can know from Table 11 that our proposed method is based on the Pythagorean fuzzy numbers and partitioned Bonferroni mean operator with parameters p and q. Moreover, interaction between membership and nonmembership has been considered by using the Pythagorean fuzzy interaction operation laws and interaction between attributes have been considered by using the Bonferroni mean. Hence the new method is more general and flexible than the existing methods. Partitioned the input arguments into several sorts can accurately model the interrelationship between attributes.Table 11
The characteristic comparisons of different methods.
Methods
Information by Pythagorean fuzzy number
Whether to consider the interrelationships between aggregating arguments
Liang et al. [43]
Yes
Yes
Xu and Yager [38]
No
Yes
Zhang and Xu [26]
Yes
No
Our proposed method
Yes
Yes
Methods
Whether to consider the partition of the input arguments
Whether to consider the interactions between membership and nonmembership
Liang et al. [43]
No
No
Xu and Yager [38]
No
No
Zhang and Xu [26]
No
No
Our proposed method
Yes
Yes


## 5.2.1. Influence of the Parametersp and q

In order to illustrate influence of parametersp and q on the ranking results, we consider different p and q in Steps 2 and 3. For simplicity, the same p and q are used in Steps 2 and 3. For example, if p=1 and q=2 are used in Step 2, then p=1 and q=2 are also used in Step 3. The results are shown in Table 5; here, OA means the optimal alternative. From the results, we can see that A5 is the optimal alternative in the cases of p=1 and q=3 and p=2 and q=2 and A3 becomes the optimal alternative in other cases. If p=1 and q=1 and p=1 and q=2, the optimal alternative is A3 and the suboptimal alternative is A5. With the increase of p and q, the suboptimal becomes A2 and A5 is ranked last. A3 has relatively larger memberships and relatively smaller nonmemberships comparing with other alternatives. Though A5 has the largest membership and the smallest nonmembership among all the evaluation values, it is still ranked last with increasing p and q due to the intersection between membership and nonmembership that is considered. The rankings of alternatives change with different p and q.Table 5
Results of differentp and q considering interaction.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
0.0904
0.0950
0.1256
0.0734
0.1091
A3>A5>A2>A1>A4
A3
p=1, q=2
0.0904
0.0955
0.1249
0.0742
0.1099
A3>A5>A2>A1>A4
A3
p=1, q=3
0.0264
0.0958
0.1163
0.0754
0.2574
A5>A3>A2>A4>A1
A5
p=2, q=2
−0.0268
0.0891
0.0630
0.0743
0.2040
A5>A2>A4>A3>A1
A5
p=1, q=4
−0.1183
0.2049
0.5881
0.1483
−0.3674
A3>A2>A4>A1>A5
A3
p=2, q=3
−0.2481
0.1444
0.4812
0.0797
−0.4889
A3>A2>A4>A1>A5
A3
p=1, q=5
−0.0570
0.2625
0.6405
0.2327
−0.3190
A3>A2>A4>A1>A5
A3
p=2, q=4
−0.2037
0.1983
0.5309
0.1647
−0.4507
A3>A2>A4>A1>A5
A3
p=3, q=3
−0.2567
0.1791
0.4917
0.1474
−0.5051
A3>A2>A4>A1>A5
A3The larger thep and q, the more interaction can be emphasized. But in special cases of p=0 or q=0, there is no interaction between input arguments. From the viewpoint of the risk attitudes, decision-makers are more risk-seeking with the increase of p and q. By taking different p and q in the PFIPBM operator, the PFIPGBM operator, the PFWIPBM, or the PFWIPGBM operator, different risk attitudes of decision-makers can be reflected and different aspects of decision problem can also be reflected, since the arithmetic aggregation operator stresses the impact of the overall input arguments while the geometric aggregation operator emphasizes the balance of the input arguments [47]. In real decision-making, decision-makers can select the corresponding aggregation operator and p and q according to their preferences and real needs. For simplicity, the decision-makers can select p=1 and q=1 if the decision-maker is risk averse, which is simple and intuitive.


## 5.2.2. Comparison with Other Methods

If interactions between the memberships and nonmemberships are not considered, the PFIPBM operator and the PFWIPBM operator reduce to the Pythagorean fuzzy partitioned Bonferroni mean (PFPBM) operator and the Pythagorean fuzzy weighted partitioned Bonferroni mean (PFWPBM) operator as follows:(70)PFPBMp,qα1,α2,…,αn=1d⊕h=1d1Ph⊕i∈Phαip⊗1Ph−1⊕j∈Ph,j≠iαjq1/p+q=1−∏h=1d1−1−∏i∈Ph1−μipξ21/Ph1/p+q1/d1/2,∏h=1d1−1−∏i∈Ph1−1−νi2p1−η21/Ph1/p+q1/d1/2,where
(71)ξ=1−∏j∈Ph,j≠i1−μjq21/Ph−1,η=∏j∈Ph,j≠i1−1−νj2q1/Ph−1,PFWPBMp,qα1,α2,…,αn=1d⊕h=1d1PhPh−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2.If thePFPBMp,q operator is used in Step 2 and the PFWPBMp,q operator is used in Step 3; the interactions between memberships and nonmemberships are not considered and the results are shown in Table 6. From the results, we can see that the ranking of alternatives is different from that of considered interaction between memberships and nonmemberships. A5 becomes the optimal alternative and A3 becomes the suboptimal alternative in all cases. In fact, the rankings of alternatives are the same except for the case of p=1 and q=1. If interactions are not considered, the effect of memberships will reduce if one membership is nearly approaching zero in multiply operation no matter what about the other memberships and the effect of nonmemberships will be reduced if one nonmembership is nearly approaching zero in sum operation no matter what about the other nonmemberships. These shortcomings can be overcome by considering interactions between memberships and nonmemberships.Table 6
Results of differentp and q without considering interaction.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
−0.5132
−0.5198
−0.5082
−0.5213
−0.4771
A5>A3>A1>A2>A4
A5
p=1, q=2
−0.5041
−0.5030
−0.4830
−0.5110
−0.4519
A5>A3>A2>A1>A4
A5
p=1, q=3
−0.4892
−0.4790
−0.4437
−0.4946
−0.4128
A5>A3>A2>A1>A4
A5
p=2, q=2
−0.5103
−0.5073
−0.4989
−0.5185
−0.4678
A5>A3>A2>A1>A4
A5
p=1, q=4
−0.4738
−0.4557
−0.4063
−0.4791
−0.3758
A5>A3>A2>A1>A4
A5
p=2, q=3
−0.5046
−0.4962
−0.4833
−0.5121
−0.4522
A5>A3>A2>A1>A4
A5
p=1, q=5
−0.4599
−0.4349
−0.3743
−0.4656
−0.3444
A5>A3>A2>A1>A4
A5
p=2, q=4
−0.4943
−0.4798
−0.4563
−0.5009
−0.4260
A5>A3>A2>A1>A4
A5
p=3, q=3
−0.5073
−0.4965
−0.4914
−0.5155
−0.4585
A5>A3>A2>A1>A4
A5If the TOPSIS method has been used, the Pythagorean fuzzy interaction averaging (PFIA) operator is used to aggregate different evaluation values given by different decision-makers into collective ones. Then, calculate the weighted decision matrixD′ as in Table 7. The Pythagorean fuzzy positive ideal solution (PFPIS) can be determined as r+=r1+,r2+,…,r5+=maxjr1j,maxjr2j,…,maxjr5j=0.3148,0.1513,0.5329,0.0396,0.7021,0.1073,0.5935,0.1907,0.4240,0.1022. The Pythagorean fuzzy negative ideal solution (PFNIS) can be determined as r−=r1−,r2−,…,r5−=minjr1j,minjr2j,…,minjr5j=0.1684,0.2434,0.1908,0.2058,0.2700,0.1954,0.2082,0.1828,0.1918,0.1847. The distances of each alternative evaluation values to the PFPIS and the PFNIS can be calculated by using the following equations, respectively, di+=∑j=15drij,rj+, di−=∑j=15drij,rj−i=1,2,…,5. We can get d1+=0.3196, d2+=0.2887, d3+=0.2150,d4+=0.3023,d5+=0.2978,d1−=0.3098,d2−=0.2454,d3−=0.3047,d4−=0.2278, and d5−=0.2175. The closeness coefficients can be calculated by the equation CCi=di−/di−+di+, and we can get CC1=0.4923,CC2=0.4594,CC3=0.5863,CC4=0.4297 and CC5=0.4221. The alternatives can be ranked according to the ranking of CCi to get A3>A2>A1>A4>A5. The optimal alternative is A3. The optimal alternative is the same as the most case of the proposed method, but the rankings of alternatives are slightly different.
(72)αij=PFIAαij1,αij2,…,αijt=1t⊕k=1tαijk=1−∏k=1t1−μijk21/t,∏k=1t1−μijk21/t−∏k=1t1−μijk2+νijk21/t.Table 7
Pythagorean fuzzy weighted decision matrixD′.
C1
C2
C3
C4
C5
A1
(0.2046,0.1402)
(0.2751,0.1129)
(0.4920,0.2945)
(0.4046,0.2437)
(0.3914,0.1297)
A2
(0.2839,0.1038)
(0.1908,0.2058)
(0.5840,0.1981)
(0.5721,0.1664)
(0.2607,0.1515)
A3
(0.2141,0.1358)
(0.4459,0.1350)
(0.4507,0.1954)
(0.5935,0.1907)
(0.2740,0.1408)
A4
(0.1684,0.2434)
(0.1706,0.1366)
(0.7021,0.1073)
(0.3664,0.2098)
(0.4240,0.1022)
A5
(0.3148,0.1513)
(0.5329,0.0396)
(0.4892,0.1139)
(0.2082,0.1828)
(0.1918,0.1847)If interaction is not considered in TOPSIS as the method in [48], the Pythagorean fuzzy averaging (PFA) operator is first used to aggregate evaluation values given by different decision-makers into collective ones. The PFA operator is defined as
(73)αij=PFAαij1,αij2,…,αijt=1t⊕k=1tαijk=1−∏k=1t1−μijk21/t,∏k=1tνijk1/t.The weighted collective decision matrixD′′ is calculated as in Table 8, where the weight vector is also taken as 0.10,0.20,0.25,0.30,0.15. The PFPIS can be determined as r+=0.2893,0.8577,0.5329,0.0000,0.4501,0.6163,0.5721,0.5757,0.4240,0.7479. The PFNIS can be determined as r−=0.1684,0.9470,0.1908,0.8290,0.2700,0.7580,0.2082,0.6887,0.1918,0.8688. The distances of alternative’s weighted evaluation values to the PFPIS and the PFNIS can be calculated as d1+=0.7299, d2+=0.7030, d3+=0.6430, d4+=0.6901, d5+=0.4851, d1−=0.4651, d2−=0.4159, d3−=0.5009,d4−=0.4361, d5−=0.6523. The closeness coefficients can be calculated as CC1=0.3892,CC2=0.3717,CC3=0.4379,CC4=0.3872, and CC5=0.5735. The alternatives can be ranked as A5>A3>A1>A4>A2 and the optimal alternative is A5.Table 8
Pythagorean fuzzy weighted decision matrixD″.
C1
C2
C3
C4
C5
A1
(0.2046,0.8997)
(0.2751,0.7110)
(0.4920,0.7678)
(0.4046,0.7042)
(0.3914,0.7673)
A2
(0.2839,0.8577)
(0.3245,0.8290)
(0.3603,0.7329)
(0.5721,0.5757)
(0.2607,0.8299)
A3
(0.2141,0.8951)
(0.6970,0.7024)
(0.2700,0.7580)
(0.5935,0.5995)
(0.2740,0.8272)
A4
(0.1684,0.9470)
(0.2912,0.7591)
(0.4501,0.6163)
(0.3664,0.6887)
(0.4240,0.7479)
A5
(0.3148,0.8792)
(0.7955,0.0000)
(0.2952,0.6431)
(0.2082,0.6887)
(0.1918,0.8688)If attributes are all are partitioned into one sort, (37) is used to aggregate alternative evaluation values into collective ones in Step 3 and other steps are the same. Then, the results are shown in Table 9. From the results, we can see that A3 becomes the optimal alternative in the cases of p=1, q=1, p=1, and q=2 and A5 becomes the optimal alternative in the cases of p=1 and q=3 and p=2 and q=2. A2 becomes the optimal alternative in the other cases. The results are different from those of the partitioned one. If attributes can be divided into several classes and there is interaction relationships among the same class and there is no interaction between classes, the PFWIPBM operator can be used to assure accuracy and reasonableness of decision results.Table 9
Results of differentp and q considering interaction with one sort.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1,q=1
0.1709
0.1893
0.2253
0.1583
0.1706
A3>A2>A1>A5>A4
A3
p=1, q=2
0.0979
0.1073
0.1274
0.0903
0.0953
A3>A2>A1>A5>A4
A3
p=1, q=3
0.4854
0.1068
0.5529
0.0920
0.5541
A5>A3>A1>A2>A4
A5
p=2, q=2
0.4507
0.5012
0.5184
0.0904
0.5241
A5>A2>A3>A1>A4
A5
p=1, q=4
0.4996
0.6237
0.5562
0.5878
0.5462
A2>A4>A3>A5>A1
A2
p=2, q=3
0.4183
0.5855
0.4759
0.5463
0.4801
A2>A4>A5>A3>A1
A2
p=1, q=5
0.5807
0.6883
0.6175
0.6697
0.6104
A2>A4>A3>A5>A1
A2
p=2, q=4
0.5076
0.6526
0.5392
0.6334
0.5463
A2>A4>A5>A3>A1
A2
p=3, q=3
0.4883
0.6419
0.5146
0.6245
0.5285
A2>A4>A5>A3>A1
A2If attributes have been partitioned into one sort and interaction between membership and nonmembership is not considered, the Pythagorean fuzzy Bonferroni meanPFBMp,q operator [43] is used in Step 2 and Pythagorean fuzzy weighted Bonferroni mean PFWBMp,q operator is used in Step 3; the results are shown in Table 10. A5 becomes the optimal alternative and A3 becomes the suboptimal alternative. Though the optimal and suboptimal alternatives are the same as those of partitioned cases, the ranking of alternatives is different.
(74)PFBMp,qα1,α2,…,αn=1nn−1⊕i,j∈Ph,j≠iαip⊗αjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2,PFWBMp,qα1,α2,…,αn=1nn−1⊕i,j∈Ph,j≠iwiαip⊗wjαjq1/p+q=1−∏h=1d∏i,j∈Ph,j≠i1−1−1−μip2wi1−1−μjq2wj1/dPhPh−11/2,∏h=1d∏i,j∈Ph,j≠i1−1−1−1−νi2pwi1−1−1−νj2qwj1/dPhPh−11/2.Table 10
Results of differentp and q without considering interaction with one sort.
Sα1
Sα2
Sα3
Sα4
Sα5
Ranking of alternatives
OA
p=1, q=1
−0.4835
−0.4958
−0.4628
−0.4836
−0.4528
A5>A3>A1>A4>A2
A5
p=1, q=2
−0.4628
−0.4627
−0.4270
−0.4496
−0.4147
A5>A3>A4>A2>A1
A5
p=1, q=3
−0.4399
−0.4171
−0.3838
−0.4183
−0.3669
A5>A3>A2>A4>A1
A5
p=2, q=2
−0.4567
−0.4625
−0.4195
−0.4358
−0.4100
A5>A3>A4>A1>A2
A5
p=1, q=4
−0.4199
−0.3720
−0.3441
−0.3920
−0.3225
A5>A3>A2>A4>A1
A5
p=2, q=3
−0.4424
−0.4384
−0.3939
−0.4156
−0.3830
A5>A3>A4>A2>A1
A5
p=1, q=5
−0.4029
−0.3317
−0.3094
−0.3726
−0.2843
A5>A3>A2>A4>A1
A5
p=2, q=4
−0.4272
−0.4057
−0.3649
−0.3968
−0.3507
A5>A3>A2>A4>A1
A5
p=3, q=3
−0.4347
−0.4335
−0.3821
−0.4036
−0.3734
A5>A3>A4>A2>A1
A5If the PFIA operator is used in aggregating different decision-makers’ evaluation values into collective ones in Step2 and the Pythagorean fuzzy weighted interaction averaging (PFIWA) operator is used in aggregating alternatives’ evaluation values into collective ones in Step 3, the collective evaluation values of alternatives are as r1=0.7243,0.3242, r2=0.7188,0.2748, r3=0.7586,0.2535, r4=0.6825,0.2825, and r5=0.6367,0.2416. The scores of alternatives can be calculated as Sr1=0.4195, Sr2=0.4411, Sr3=0.5112, Sr4=0.3860, and Sr5=0.3990, then Sr3>Sr2>Sr1>Sr5>Sr4. The alternatives can be ranked accordingly as A3>A2>A1>A5>A4. The optimal alternative is A3.
(75)αi=PFIWAαi1,αi2,…,αin=⊕j=1nwjαij=1−∏j=1n1−μij2wj,∏j=1n1−μij2wj−∏j=1n1−μij2+νij2wj.If the PFA operator is used in Step2 and the PFWA operator is used in the second phase [38], the results are as r1=0.7243,0.2654, r2=0.7188,0.2489, r3=0.7586,0.2364, r4=0.6825,0.2282, and r5=0.6367,0.0000, where
(76)αi=PFWAαi1,αi2,…,αin=⊕j=1nwjαij=1−∏j=1n1−μij2wj,∏j=1nνijwj.The scores can be calculated asSr1=0.4542, Sr2=0.4547, Sr3=0.5195, Sr4=0.4138, and Sr5=0.4574. The alternatives can be ranked accordingly to the ranking of scores to get A3>A5>A2>A1>A4. The optimal alternative is A3.As discussed above, we summarize differences between our proposed method with the existing methods in Table11. In a word, we can know from Table 11 that our proposed method is based on the Pythagorean fuzzy numbers and partitioned Bonferroni mean operator with parameters p and q. Moreover, interaction between membership and nonmembership has been considered by using the Pythagorean fuzzy interaction operation laws and interaction between attributes have been considered by using the Bonferroni mean. Hence the new method is more general and flexible than the existing methods. Partitioned the input arguments into several sorts can accurately model the interrelationship between attributes.Table 11
The characteristic comparisons of different methods.
Methods
Information by Pythagorean fuzzy number
Whether to consider the interrelationships between aggregating arguments
Liang et al. [43]
Yes
Yes
Xu and Yager [38]
No
Yes
Zhang and Xu [26]
Yes
No
Our proposed method
Yes
Yes
Methods
Whether to consider the partition of the input arguments
Whether to consider the interactions between membership and nonmembership
Liang et al. [43]
No
No
Xu and Yager [38]
No
No
Zhang and Xu [26]
No
No
Our proposed method
Yes
Yes


## 6. Conclusions

Some Pythagorean fuzzy interaction partitioned Bonferroni mean operators have been developed in this paper including the Pythagorean fuzzy interaction partitioned Bonferroni mean operator, the Pythagorean fuzzy weighted interaction partitioned Bonferroni mean operator, the Pythagorean fuzzy interaction partitioned geometric Bonferroni mean operator, and the Pythagorean fuzzy weighted interaction partitioned geometric Bonferroni mean operator. The Bonferroni mean has been used to model interaction between attributes. The attributes have been partitioned into several classes and the attributes in the same class are interrelated, which have been modeled by using Bonferroni mean, while there is no interrelationship between attributes between different classes. Some properties and some special cases of the new aggregation operators have been studied. We have developed new multiple-attribute group decision-making method based on the new aggregation operators. We applied the new method to solve the problem of selecting an investment company. Some comparisons with other existing methods have been made to show its effectiveness and practical advantages.The proposed method has some desirable advantages: (1) the evaluation values are given as Pythagorean fuzzy numbers, which are more flexible than other tools to model fuzzy and uncertain information; (2) interaction operations between Pythagorean fuzzy numbers can overcome the drawback of the existing methods; (3) interrelationship of attributes have been modeled by using the Bonferroni mean. By using the partitioned structure of attributes considering relationship among attributes, the proposed method can model interrelationship among attributes more meaningfully and accurately; and (4) since attributes in different sorts are not related, the new aggregation operators can avoid the conjunction effect of unrelated attributes during aggregation. The disadvantage of the new method is that the computation amount has increased comparing with the existing methods. But it is still a polynomial time algorithm and can be calculated easily by using software such as MATLAB and Excel.The proposed method can be used to handle real-life problems involving fuzziness and uncertainty in the decision-making process. In the future, we will apply it in a wide range of practical problems such as supplier selection problems and site selection problems. Although the proposed operators have been developed in the context of decision-making, they can also be applied in the fields of fuzzy clustering, pattern recognition, and so on. It is also meaningful to investigate other characteristics of the proposed operators, such as combing with Choquet integral and Dempster-Shafer belief structure. We will also extend the partitioned Bonferroni mean operators to other uncertain environments [49–54], such as interval neutrosophic sets, linguistic hesitant intuitionistic fuzzy sets, hesitant Pythagorean fuzzy sets, and q-rung fuzzy sets.



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