Complex & Intelligent Systems (2019) 5:41–52https://doi.org/10.1007/s40747-018-0076-x

ORIG INAL ART ICLE

Interval-valued Pythagorean fuzzy Einstein hybrid weighted averagingaggregation operator and their application to group decision making

Khaista Rahman1 · Saleem Abdullah2 · Asad Ali1 · Fazli Amin1

Received: 15 May 2017 / Accepted: 26 May 2018 / Published online: 5 June 2018© The Author(s) 2018

AbstractThe objective of the present work is divided into two folds. Firstly, interval-valued Pythagorean fuzzy Einstein hybridweightedaveraging aggregation operator has been introduced along with their several properties, namely idempotency, boundednessand monotonicity. Secondly, we apply the proposed operator to deal with multi-attribute group decision-making problemunder Pythagorean fuzzy information. For this, we construct an algorithm for multi-attribute group decision making. At thelast, we construct a numerical example for multi-attribute group decision making. The main advantage of using the proposedoperator is that this operator provides more accurate and precise results is compared to the existing methods.

Keywords IVPFS · IVPFEHWA averaging operator · MAGDM problems

Introduction

Multi-criteria group decision making is one of the success-ful processes for finding the optimal alternative from all thefeasible alternatives according to some criteria or attributes.Traditionally, it has been generally assumed that all the infor-mation that access the alternative in terms of criteria and theircorresponding weights are expressed in the form of crispnumbers. But most of the decisions in the real-life situationsare taken in the environment where the goals and constraintsare generally imprecise or vague in nature. In order to han-dle the uncertainties and fuzziness intuitionistic fuzzy set [1]theory is one of the successful extensions of the fuzzy set the-ory [2], which is characterized by the degree of membershipand degree of non-membership has been presented. After the

B Khaista Rahmankhaista355@yahoo.com

Saleem Abdullahsaleemabdullah81@yahoo.com

Asad Aliasad_maths@hu.edu.pk

Fazli Aminfazliamin@hu.edu.pk

1 Department of Mathematics, Hazara University, Mansehra,KPK, Pakistan

2 Department of Mathematics, Abdul Wali Khan UniversityMardan, Mardan, KPK, Pakistan

successful and positive applications of intuitionistic fuzzyset, aggregation operators become more interesting topic forresearch. Thus, many scholars in [3–16] developed severalaggregation operators for group decision making using intu-itionistic fuzzy information.

However, there are many cases where the decision makermay provide the degree of membership and nonmembershipof a particular attribute in such a way that their sum is greaterthan one. To solve these types of problems, Yager [17,18]introduced the concept of another set called Pythagoreanfuzzy set. Pythagorean fuzzy set is more powerful tool tosolve uncertain problems. Like intuitionistic fuzzy aggrega-tion operators, Pythagorean fuzzy aggregation operators arealso become an interesting and important area for research,after the advent of Pythagorean fuzzy set theory. Severalresearchers in [19–28] introduced many aggregation oper-ators for decision using Pythagorean fuzzy information.

But, in some real decision-making problems, due to insuf-ficiency in available information, it may be difficult fordecision makers to exactly quantify their opinions with acrisp number, but they can be represented by an interval num-ber within [0, 1]. Therefore, it is so important to present theidea of interval-valued Pythagorean fuzzy sets, which permitthe membership degrees and non- membership degrees to agiven set to have an interval value. Thus in [29] Peng andYang introduced the concept of interval-valued Pythagoreanfuzzy set. Rahman et al. [30–33] introduced many aggre-gation operators using interval-valued Pythagorean fuzzy

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42 Complex & Intelligent Systems (2019) 5:41–52

numbers and applied them to multi-attribute group decisionmaking.

Thus, keeping the advantages of these operators, in thispaper,we introduce thenotionof interval-valuedPythagoreanfuzzy Einstein hybrid weighted averaging operator. More-over, we introduce some of their basic properties such asidempotency, boundedness and monotonicity. This motiva-tion comes from [32], in which the authors introduced thenotion of IVPFEWA operator and IVPFEOWA operator andapplied them to group decision making. But in this paper weintroduce the notion of IVPFEHWA operator, which is thegeneralization of the above mention operators.

The remainder of this paper is structured as follows.In Sect. “Preliminaries”, we give some basic definitionsand results which will be used in our later sections. InSect. “Interval-valued Pythagorean fuzzy Einstein hybridweighted averaging aggregation operator”, we introduce thenotion of interval-valued Pythagorean fuzzy Einstein hybridweighted averaging operator. In Sect. “An approach to mul-tiple attribute group decision-making problems based onintervalvalued Pythagorean fuzzy information”, we applythe proposed operator to multi-attribute group decision-making problem with Pythagorean fuzzy information. InSect. “Illustrative example”, we develop a numerical exam-ple. In Sect. “Conclusion”, we have conclusion.

Preliminaries

Definition 1 [17,18] Let K be a fixed set, then a Pythagoreanfuzzy set can be defined as:

P = {〈k, uP (k), vP (k)〉|k ∈ K }, (1)

where uP (k) : P → [0, 1], vP (k) : K → [0, 1] are calledmembership function and non-membership function, respec-tively, with condition 0 ≤ (uP (k))2 + (vP (k))2 ≤ 1, for allk ∈ K .

Let

πP (k) =√1 − u2P (k) − v2P (k). (2)

Then, it is called the Pythagorean fuzzy index of k ∈ K ,with condition 0 ≤ πP (k) ≤ 1, for every k ∈ K .Definition 2 [29] Let K be a fixed set, then an interval-valuedPythagorean fuzzy set can be defined as:

I = {〈k, uI (k), vI (k)〉|k ∈ K }, (3)

where

uI (k) = [uaI (k), ubI (k)] ⊂ [0, 1], (4)

and

vI (k) = [vaI (k), vbI (k)] ⊂ [0, 1]. (5)

Also

uaI (k) = inf(uI (k)), (6)ubI (k) = sup(uI (k)), (7)vaI (k) = inf(vI (k)), (8)vbI (k) = sup(vI (k)), (9)

and

0 ≤(ubI (k)

)2 +(vbI (k)

)2 ≤ 1. (10)

If

πI (k) =[πaI (k), π

bI (k)], for all k ∈ K . (11)

Then, it is called the interval-valued Pythagorean fuzzyindex of k to I , where

πaI (k) =√1 − (ubI (k)

)2 − (vbI (k))2

, (12)

and

πbI (k) =√1 − (uaI (k)

)2 − (vaI (k))2

. (13)

Definition 3 [29] Let λ = ([uλ, vλ], [xλ, yλ]) be an IVPFN,then the score function and accuracy function of λ can bedefined as follows, respectively:

s(λ) = 12

[(uλ)

2 + (vλ)2 − (xλ)2 − (yλ)2], (14)

and

h(λ) = 12

[(uλ)

2 + (vλ)2 + (xλ)2 + (yλ)2]. (15)

If λ1 and λ2 are two IVPFNs, then

1. If s(λ1) ≺ s(λ2), then λ1 ≺ λ2.2. If s(λ1) = s(λ2), then we have the following three con-

ditions.

1) If h(λ1) = h(λ2), then λ1 = λ2.2) If h(λ1) ≺ h(λ2), then λ1 ≺ λ2.3) If h(λ1) h(λ2), then λ1 λ2.

Definition 4 [32] Let λ = ([u, v], [x, y]), λ1 = ([u1, v1],[x1, y1]), λ2 = ([u2, v2], [x2, y2]) are three IVPFNs, andδ 0, then some Einstein operations for λ, λ1, λ2 can bedefined as follows:

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Complex & Intelligent Systems (2019) 5:41–52 43

1.

λ1 ⊕ε λ2 =⎛⎝⎡⎣√u21 + u22√1 + u21u22

,

√v21 + v22√1 + v21v22

⎤⎦ ,

⎡⎣ x1x2√

1 + (1 − x21) (1 − x22

) ,

y1y2√1 + (1 − y21

) (1 − y22

)

⎤⎦⎞⎠

2.

λ1 ⊗ε λ2 =⎛⎝⎡⎣ u1u2√

1 + (1 − u21) (1 − u22

) ,

v1v2√1 + (1 − v21

) (1 − v22

)

⎤⎦ ,

⎡⎣√x21 + x22√1 + x21 x22

,

√y21 + y22√1 + y21 y22

⎤⎦⎞⎠

3.

δλ =([√

(1 + u2)δ − (1 − u2)δ√(1 + u2)δ + (1 − u2)δ ,

√(1 + v2)δ − (1 − v2)δ√(1 + v2)δ + (1 − v2)δ

],

[ √2(x2)δ√

(2 − x2)δ + (x2)δ ,√2(y2)δ√

(2 − y2)δ + (y2)δ])

4.

λδ =([ √

2(u2)δ√(2 − u2)δ + (u2)δ ,

√2(v2)δ√

(2 − v2)δ + (v2)δ

],

[√(1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ ,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

])

Definition 5 [32] Let λ j = ([u j , v j ], [x j , y j ])( j = 1, 2, 3,..., n) be the collection of IVPFVs, then IVPFEWA operatorcan be defined as:

IVPFEWAw(λ1, λ2, λ3, ..., λn)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√n∏j=1

(1+u2λ j

)w j −n∏j=1

(1−u2λ j

)w j√

n∏j=1

(1+u2λ j

)w j +n∏j=1

(1−u2λ j

)w j ,

√n∏j=1

(1+v2λ j

)w j −n∏j=1

(1−v2λ j

)w j√

n∏j=1

(1+v2λ j

)w j +n∏j=1

(1−v2λ j

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

n∏j=1

(x2λ j

)w j√

n∏j=1

(2−x2λ j

)w j +n∏j=1

(x2λ j

)w j ,

√2

n∏j=1

(y2λ j

)w j√

n∏j=1

(2−y2λ j

)w j +n∏j=1

(y2λ j

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(16)

where w = (w1, w2, w3, ..., wn)T is the weighted vector ofλ j ( j = 1, 2, 3, ..., n), such thatw j ∈ [0, 1] and∑nj=1 w j =1.

Definition 6 [32] Let λ j ( j = 1, 2, 3, ..., n) be a collection ofIVPFVs, then IVPFEOWA operator can be defined as:

IVPFEOWAw(λ1, λ2, λ3, ..., λn)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√n∏j=1

(1+u2λσ( j)

)w j −n∏j=1

(1−u2λσ( j)

)w j√

n∏j=1

(1+u2λσ( j)

)w j +n∏j=1

(1−u2λσ( j)

)w j ,

√n∏j=1

(1+v2λσ( j)

)w j −n∏j=1

(1−v2λσ( j)

)w j√

n∏j=1

(1+v2λσ( j)

)w j +n∏j=1

(1−v2λσ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

n∏j=1

(x2λσ( j)

)w j√

n∏j=1

(2−x2λσ( j)

)w j +n∏j=1

(x2λσ( j)

)w j ,

√2

n∏j=1

(y2λσ( j)

)w j√

n∏j=1

(2−y2λσ( j)

)w j +n∏j=1

(y2λσ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(17)

where (σ (1), σ (2), σ (3), ..., σ (n)) is a permutation of (1, 2,3, ..., n) such that σ( j) ≤ σ( j − 1) for all jand w =(w1, w2, w3, ..., wn)

T is the weighted vector of λσ( j)( j =1, 2, 3, ..., n) such that w j ∈ [0, 1] and∑nj=1 w j = 1.

Interval-valued Pythagorean fuzzy Einsteinhybrid weighted averaging aggregationoperator

In this section, we introduce the notion of interval-valuedPythagorean fuzzyEinstein hybridweighted averaging aggre-gation operator. We also discuss some desirable propertiessuch as idempotency, boundedness and monotonicity.

Definition 7 An interval-valued Pythagorean fuzzy Einsteinhybrid weighted averaging operator of dimension n is amapping IVPFEHWA : �n → �,which has associatedvectorw = (w1, w2, w3, ..., wn)T , such that w j ∈ [0, 1] and∑n

j=1 w j = 1. Furthermore

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44 Complex & Intelligent Systems (2019) 5:41–52

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√n∏j=1

(1+u2

λ̇σ ( j)

)w j −n∏j=1

(1−u2

λ̇σ ( j)

)w j

√n∏j=1

(1+u2

λ̇σ ( j)

)w j +n∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√n∏j=1

(1+v2

λ̇σ ( j)

)w j −n∏j=1

(1−v2

λ̇σ ( j)

)w j

√n∏j=1

(1+v2

λ̇σ ( j)

)w j +n∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

n∏j=1

(x2λ̇σ ( j)

)w j

√n∏j=1

(2−x2

λ̇σ ( j)

)w j +n∏j=1

(x2λ̇σ ( j)

)w j ,

√2

n∏j=1

(y2λ̇σ ( j)

)w j

√n∏j=1

(2−y2

λ̇σ ( j)

)w j +n∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(18)

where λ̇σ ( j) is the j th largest of the weighted interval-valuedPythagorean fuzzy values, λ̇σ ( j)(λ̇σ ( j) = nω jλ j ). ω =(ω1, ω2, ω3, ..., ωn)

T is the weighted vector of λ j ( j = 1, 2,3, ..., n) such that ω j ∈ [0, 1], ∑nj=1 ω j = 1, and nis the balancing coefficient, which plays a role of bal-ance. If the vector w = (w1, w2, w3, ..., wn)T approachesto( 1n ,

1n ,

1n , ...,

1n

)T, then the vector (nω1λ1, nω2λ2, ...,

nωnλn)T approaches to (λ1, λ2, λ3, ..., λn)T .

Theorem 1 Let λ, λ1, λ2 be the three interval-valuedPythagorean fuzzy numbers and δ, δ1, δ2 0, then the fol-lowing conditions always hold:

1. λ1 ⊕ε λ2 = λ2 ⊕ε λ1,2. λ1 ⊗ε λ2 = λ2 ⊗ε λ1,3. δ(λ1 ⊕ε λ2) = δλ1 ⊕ε δλ2,4. (λ1 ⊗ε λ2)δ = (λ1)δ ⊗ε (λ2)δ ,5. δ1(λ) ⊕ε δ2(λ) = (δ1 ⊕ε δ2)λ,6. (λ)δ1 ⊗ε (λ)δ2 = λ(δ1⊗εδ2).

Proof The proof is trivial, so it is omitted here.

Theorem 2 Let λ j = ([u j , v j ], [x j , y j ])( j = 1, 2, 3, ..., n)be a collection of IVPFVs, then their aggregated value usingthe IVPFEHWA operator is also an IVPFV, and

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√n∏j=1

(1+u2

λ̇σ ( j)

)w j −n∏j=1

(1−u2

λ̇σ ( j)

)w j

√n∏j=1

(1+u2

λ̇σ ( j)

)w j +n∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√n∏j=1

(1+v2

λ̇σ ( j)

)w j −n∏j=1

(1−v2

λ̇σ ( j)

)w j

√n∏j=1

(1+v2

λ̇σ ( j)

)w j +n∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

n∏j=1

(x2λ̇σ ( j)

)w j

√n∏j=1

(2−x2

λ̇σ ( j)

)w j +n∏j=1

(x2λ̇σ ( j)

)w j ,

√2

n∏j=1

(y2λ̇σ ( j)

)w j

√n∏j=1

(2−y2

λ̇σ ( j)

)w j +n∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(19)

where λ̇σ ( j) is the j th largest of the weighted interval-valuedPythagorean fuzzy values, λ̇σ ( j)(λ̇σ ( j) = nω jλ j ), w =(w1, w2, w2, . . . , wn)

T is the weighted vector of IVPFE-HWA, such that w j ∈ [0, 1], ∑nj=1 w j = 1. ω =(ω1, ω2, ω2, . . . , ωn)

T is theweighted vector of λ j ( j = 1, 2,3, . . . , n) such that ω j ∈ [0, 1], ∑nj=1 ω j = 1, and n

is the balancing coefficient, which plays a role of bal-ance. If the vector w = (w1, w2, w2, . . . , wn)T approaches( 1n ,

1n ,

1n , . . . ,

1n

)T, then the vector (nwωλ1,

nω2λ2, . . . , nωnλn)T approaches(λ1, λ2, λ3, . . . , λn)T .

Proof We can prove this theorem by mathematical inductionon n.

For n = 2

w1λ̇1 =

⎛⎜⎜⎝

⎡⎢⎢⎣

√(1 + u2

λ̇1

)w1 −(1 − u2

λ̇1

)w1√(

1 + u2λ̇1

)w1 +(1 − u2

λ̇1

)w1 ,

√(1 + v2

λ̇1

)w1 −(1 − v2

λ̇1

)w1√(

1 + v2λ̇1

)w1 +(1 − v2

λ̇1

)w1

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2(x2λ̇1

)w1√(

2 − x2λ̇1

)w1 +(x2λ̇1

)w1 ,

√2(y2λ̇1

)w1√(

2 − y2λ̇1

)w1 +(y2λ̇1

)w1

⎤⎥⎥⎦

⎞⎟⎟⎠

and

w2λ̇2 =

⎛⎜⎜⎝

⎡⎢⎢⎣

√(1 + u2

λ̇2

)w2 −(1 − u2

λ̇2

)w2√(

1 + u2λ̇2

)w2 +(1 − u2

λ̇2

)w2 ,

√(1 + v2

λ̇2

)w2 −(1 − v2

λ̇2

)w2√(

1 + v2λ̇2

)w2 +(1 − v2

λ̇2

)w2

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2(x2λ̇2

)w2√(

2 − x2λ̇2

)w2 +(x2λ̇2

)w2 ,

√2(y2λ̇2

)w2√(

2 − y2λ̇2

)w2 +(y2λ̇2

)w2

⎤⎥⎥⎦

⎞⎟⎟⎠

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Complex & Intelligent Systems (2019) 5:41–52 45

Then

IVPFEHWAω,w(λ1, λ2)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√2∏j=1

(1+u2

λ̇σ ( j)

)w j −2∏j=1

(1−u2

λ̇σ ( j)

)w j

√2∏j=1

(1+u2

λ̇σ ( j)

)w j +2∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√2∏j=1

(1+v2

λ̇σ ( j)

)w j −2∏j=1

(1−v2

λ̇σ ( j)

)w j

√2∏j=1

(1+v2

λ̇σ ( j)

)w j +2∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

2∏j=1

(x2λ̇σ ( j)

)w j

√2∏j=1

(2−x2

λ̇σ ( j)

)w j +2∏j=1

(x2λ̇σ ( j)

)w j ,

√2

2∏j=1

(y2λ̇σ ( j)

)w j

√2∏j=1

(2−y2

λ̇σ ( j)

)w j +2∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Thus, the result is true for n = 2, now we assume that Eq.(19) holds for n = k. ThusIVPFEHWAω,w(λ1, λ2, λ3, ..., λk )

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√k∏j=1

(1+u2

λ̇σ ( j)

)w j −k∏j=1

(1−u2

λ̇σ ( j)

)w j

√k∏j=1

(1+u2

λ̇σ ( j)

)w j +n∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√k∏j=1

(1+v2

λ̇σ ( j)

)w j −k∏j=1

(1−v2

λ̇σ ( j)

)w j

√k∏j=1

(1+v2

λ̇σ ( j)

)w j +k∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

k∏j=1

(x2λ̇σ ( j)

)w j

√k∏j=1

(2−x2

λ̇σ ( j)

)w j +k∏j=1

(x2λ̇σ ( j)

)w j ,

√2

k∏j=1

(y2λ̇σ ( j)

)w j

√k∏j=1

(2−y2

λ̇σ ( j)

)w j +k∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

If Eq. (19) holds for n = k, then we show that Eq. (19)holds for n = k + 1. ThusIVPFEHWAω,w(λ1, λ2, λ3, ..., λk+1)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√k∏j=1

(1+u2

λ̇σ ( j)

)w j −k∏j=1

(1−u2

λ̇σ ( j)

)w j

√k∏j=1

(1+u2

λ̇σ ( j)

)w j +n∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√k∏j=1

(1+v2

λ̇σ ( j)

)w j −k∏j=1

(1−v2

λ̇σ ( j)

)w j

√k∏j=1

(1+v2

λ̇σ ( j)

)w j +k∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

k∏j=1

(x2λ̇σ ( j)

)w j

√k∏j=1

(2−x2

λ̇σ ( j)

)w j +k∏j=1

(x2λ̇σ ( j)

)w j ,

√2

k∏j=1

(y2λ̇σ ( j)

)w j

√k∏j=1

(2−y2

λ̇σ ( j)

)w j +k∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⊕ε

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√(1+u2

λ̇k+1

)wk+1 −(1−u2

λ̇k+1

)wk+1

√(1+u2

λ̇k+1

)wk+1 +(1−u2

λ̇k+1

)wk+1 ,

√(1+v2

λ̇k+1

)wk+1 −(1−v2

λ̇k+1

)wk+1

√(1+v2

λ̇k+1

)wk+1 +(1−v2

λ̇k+1

)wk+1

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

(x2λ̇k+1

)wk+1

√(2−x2

λ̇k+1

)wk+1 +(x2λ̇k+1

)wk+1 ,

√2

(y2λ̇k+1

)wk+1

√(2−y2

λ̇k+1

)wk+1 +(y2λ̇k+1

)wk+1

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(20)

Let

t1 =√√√√

k∏j=1

(1 + u2

λ̇σ ( j)

)w j −k∏j=1

(1 − u2

λ̇σ ( j)

)w j

t2 =√√√√

k∏j=1

(1 + u2

λ̇σ ( j)

)w j +n∏j=1

(1 − u2

λ̇σ ( j)

)w j

p1 =√√√√

k∏j=1

(1 + v2

λ̇σ ( j)

)w j −k∏j=1

(1 − v2

λ̇σ ( j)

)w j

p2 =√√√√

k∏j=1

(1 + v2

λ̇σ ( j)

)w j +k∏j=1

(1 − v2

λ̇σ ( j)

)w j

m1 =√(

1 + u2λ̇k+1

)wk+1 −(1 − u2

λ̇k+1

)wk+1

m2 =√(

1 + u2λ̇k+1

)wk+1 +(1 − u2

λ̇k+1

)wk+1

a1 =√(

1 + v2λ̇k+1

)wk+1 −(1 − v2

λ̇k+1

)wk+1

a2 =√(

1 + v2λ̇k+1

)wk+1 +(1 − v2

λ̇k+1

)wk+1

r2 =√√√√

k∏j=1

(2 − x2

λ̇σ ( j)

)w j +k∏j=1

(x2λ̇σ ( j)

)w j

r1 =√√√√2

k∏j=1

(x2λ̇σ ( j)

)w j, s1 =

√√√√2k∏j=1

(y2λ̇σ ( j)

)w j

s2 =√√√√

k∏j=1

(2 − y2

λ̇σ ( j)

)w j +k∏j=1

(y2λ̇σ ( j)

)w j

b2 =√(

2 − x2λ̇k+1

)wk+1 +(x2λ̇k+1

)wk+1

b1 =√2(x2λ̇k+1

)wk+1, c1 =

√2(y2λ̇k+1

)wk+1

c2 =√(

2 − y2λ̇k+1

)wk+1 +(y2λ̇k+1

)wk+1

Now putting these values in Eq. (20), we have

IVPFEHWAω,w(λ1, λ2, λ3, ..., λk+1)

=([

t1t2

,p1p2

],

[r1r2

,s1s2

])⊕ε([

m1m2

,a1a2

],

[b1b2

,c1c2

])

=

⎛⎜⎜⎝

⎡⎢⎢⎣

√(t1t2

)2 +(m1m2

)2√1 +(t1t2

)2 (m1m2

)2 ,

√(p1p2

)2 +(a1a2

)2√1 +(p1p2

)2 (a1a2

)2

⎤⎥⎥⎦ ,

⎡⎢⎢⎢⎢⎣

(r1r2

) (b1b2

)√1 +(1 −(r1r2

)2)(1 −(b1b2

)2),

(s1s2

) (c1c2

)√1 +(1 −(s1s2

))2 (1 −(c1c2

))2

⎤⎥⎥⎦

⎞⎟⎟⎠

=⎛⎝⎡⎣√

(t1m2)2 + (t2m1)2√(t2m2)2 + (t1m1)2

,

√(p1a2)2 + (a1 p2)2√(p2a2)2 + (p1a1)2

⎤⎦ ,

123

46 Complex & Intelligent Systems (2019) 5:41–52

⎡⎣ r1b1√

2r22b22 + r21b21 − r22b21 − r21b22

,

s1c1√2s22c

22 + s21c21 − s22c21 − s21c22

⎤⎦⎞⎠ . (21)

Again putting the values of (t1m2)2 + (t2m1)2, (t2m2)2 +(t1m1)2, (p1a2)2+(a1 p2)2, (p2a2)2+(p1a1)2, r1b1, 2r22b22+r21b

21 − r22b21 − r21b22, s1c1, 2s22c22 + s21c21 − s22c21 − s21c22, in

Eq. (21), then

IVPFEHWAω,w(λ1, λ2, λ3, ..., λk+1)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√k+1∏j=1

(1+u2

λ̇σ ( j)

)w j −k+1∏j=1

(1−u2

λ̇σ ( j)

)w j

√k+1∏j=1

(1+u2

λ̇σ ( j)

)w j +k+1∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√k+1∏j=1

(1+v2

λ̇σ ( j)

)w j −k+1∏j=1

(1−v2

λ̇σ ( j)

)w j

√k+1∏j=1

(1+v2

λ̇σ ( j)

)w j +k+1∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2k+1∏j=1

(x2λ̇σ ( j)

)w j

√k+1∏j=1

(2−x2

λ̇σ ( j)

)w j +k+1∏j=1

(x2λ̇σ ( j)

)w j ,

√2k+1∏j=1

(y2λ̇σ ( j)

)w j

√k+1∏j=1

(2−y2

λ̇σ ( j)

)w j +k+1∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Hence, Eq. (19) holds for n = k + 1. Thus, Eq. (19) holdsfor all n.

Remark 1 In the following, let us look δλ andλδ some specialcases of δ and λ.

1. If λ = ([u, v], [x, y]) = ([1, 1], [0, 0]) i. e,. u = v = 1and u = v = 1, then

λδ =⎛⎝⎡⎣

√2(u2)δ

√(2 − u2)δ + (u2)δ

,

√2(v2)δ

√(2 − v2)δ + (v2)δ

⎤⎦ ,

⎡⎣√(

1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ

,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

⎤⎦⎞⎠

=([ √

2(1)δ√(2 − 1)δ + (1)δ ,

√2(1)δ√

(2 − 1)δ + (1)δ

],

[√(1 + 0)δ − (1 − 0)δ√(1 + 0)δ + (1 − 0)δ

,

√(1 + 0)δ − (1 − 0)δ√(1 + 0)δ + (1 − 0)δ

])

= ([1, 1], [0, 0]).

Thus λδ = ([1, 1], [0, 0]) and δλ = ([0, 0], [1, 1]).2. If λ = ([u, v], [x, y]) = ([0, 0], [1, 1]) i. e,. u = v = 0

and x = y = 1, then

λδ =⎛⎝⎡⎣

√2(u2)δ

√(2 − u2)δ + (u2)δ

,

√2(v2)δ

√(2 − v2)δ + (v2)δ

⎤⎦ ,

⎡⎣√(

1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ

,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

⎤⎦⎞⎠

=([ √

2(0)δ√(2 − 0)δ + (0)δ ,

√2(0)δ√

(2 − 0)δ + (0)δ

],

[√(1 + 1)δ − (1 − 1)δ√(1 + 1)δ + (1 − 1)δ ,

√(1 + 1)δ − (1 − 1)δ√(1 + 1)δ + (1 − 1)δ

])

= ([0, 0], [1, 1]).

Thus λδ = ([0, 0], [1, 1]) and δλ = ([1, 1], [0, 0]).3. If λ = ([u, v], [x, y]) = ([0, 0], [0, 0]) i. e,. u = v = 0

and x = y = 0, then

λδ =⎛⎝⎡⎣

√2(u2)δ

√(2 − u2)δ + (u2)δ

,

√2(v2)δ

√(2 − v2)δ + (v2)δ

⎤⎦ ,

⎡⎣√(

1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ

,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

⎤⎦⎞⎠

=([ √

2(0)δ√(2 − 0)δ + (0)δ ,

√2(0)δ√

(2 − 0)δ + (0)δ

],

[√(1 + 0)δ − (1 − 0)δ√(1 + 0)δ + (1 − 0)δ ,

√(1 + 0)δ − (1 − 0)δ√(1 + 0)δ + (1 − 0)δ

])

= ([0, 0], [0, 0]).

Thus λδ = ([0, 0], [0, 0]) and δλ = ([0, 0], [0, 0]).4. If δ → 0 and 0 ≤ u, v, x, y ≤ 1, then

λδ =⎛⎝⎡⎣

√2(u2)δ

√(2 − u2)δ + (u2)δ

,

√2(v2)δ

√(2 − v2)δ + (v2)δ

⎤⎦ ,

⎡⎣√(

1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ

,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

⎤⎦⎞⎠

=([ √

2(u2)0√(2 − u2)0 + (u2)0 ,

√2(v2)0√

(2 − v2)0 + (v2)0

],

123

Complex & Intelligent Systems (2019) 5:41–52 47

⎡⎣√

(1 + x2)0 − (1 − x21 )0√(1 + x2)0 + (1 − x2)0 ,

√(1 + y2)0 − (1 − y2)0√(1 + y2)0 + (1 − y2)0

])

= ([1, 1], [0, 0]).

Thus λδ = ([1, 1], [0, 0]) and δλ = ([0, 0], [1, 1]).5. If δ → +∞ and 0 ≤ u, v, x, y ≤ 1, then

λδ =⎛⎝⎡⎣

√2(u2)δ

√(2 − u2)δ + (u2)δ

,

√2(v2)δ

√(2 − v2)δ + (v2)δ

⎤⎦ ,

⎡⎣√(

1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ

,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

⎤⎦⎞⎠

=([ √

2(u2)∞√(2 − u2)∞ + (u2)∞ ,

√2(v2)∞√

(2 − v2)∞ + (u2)∞

],

[√(1 + x2)∞ − (1 − x2)∞√(1 + x2)∞ + (1 − x2)∞ ,

√(1 + y2)∞ − (1 − y2)∞√(1 + y2)∞ + (1 − y2)∞

])= ([[0, 0], 1, 1]).

Thus, λδ = ([0, 0], [1, 1]) and δλ = ([1, 1], [0, 0]).6. If δ = 1 and 0 ≤ u, v, x, y ≤ 1, then

λδ =⎛⎝⎡⎣

√2(u2)δ

√(2 − u2)δ + (u2)δ

,

√2(v2)δ

√(2 − v2)δ + (v2)δ

⎤⎦ ,

⎡⎣√(

1 + x2)δ − (1 − x2)δ√(1 + x2)δ + (1 − x2)δ

,

√(1 + y2)δ − (1 − y2)δ√(1 + y2)δ + (1 − y2)δ

⎤⎦⎞⎠

=⎛⎝⎡⎣

√2(u2)1

√(2 − u2)1 + (u2)1

,

√2(v2)1

√(2 − v2)1 + (u2)1

⎤⎦ ,

⎡⎣√(

1 + x2)1 − (1 − x2)1√(1 + x2)1 + (1 − x2)1

,

√(1 + y2)1 − (1 − y2)1√(1 + y2)1 + (1 − y2)1

⎤⎦⎞⎠ = λ.

Thus, λδ = λ and δλ = λ.

Lemma 1 [6] Let λ j 0, w j 0( j = 1, 2, 3, ..., n) and∑nj=1 w j = 1, then

n∏j=1

(λ j )w j ≤

n∑j=1

w jλ j , (22)

where the equality holds if and only if λ1 = λ2 = · · · = λn .

Theorem 3 Let λ j = ([u j , v j ], [x j , y j ])( j = 1, 2, 3, ..., n)be a collection of IVPFVs, where the w = (w1, w2, w3, ...,wn)

T is the weighted vector of IVPFEHWA and IVPFHWA,such thatw j ∈ [0, 1]and∑nj=1 w j = 1. ω = (ω1, ω2, ω2, ...,ωn)

T is the weighted vector of λ j ( j = 1, 2, 3, ..., n) suchthatω j ∈ [0, 1],∑nj=1 ω j = 1, then

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

≤ IVPFHWAω,w(λ1, λ2, λ3, ..., λn). (23)

Proof Straight forward.

Theorem 4 Idempotency: If λ̇σ ( j) = λ̇ for all j( j = 1, 2,3, ..., n), where λ = ([u, v], [x, y]), then

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn) = λ̇. (24)

Proof Since λ̇σ ( j) = λ̇ for all j , then we haveIVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎣

√n∏j=1

(1+u2

λ̇σ ( j)

)w j −n∏j=1

(1−u2

λ̇σ ( j)

)w j

√n∏j=1

(1+u2

λ̇σ ( j)

)w j +n∏j=1

(1−u2

λ̇σ ( j)

)w j ,

√n∏j=1

(1+v2

λ̇σ ( j)

)w j −n∏j=1

(1−v2

λ̇σ ( j)

)w j

√n∏j=1

(1+v2

λ̇σ ( j)

)w j +n∏j=1

(1−v2

λ̇σ ( j)

)w j

⎤⎥⎥⎦ ,

⎡⎢⎢⎣

√2

n∏j=1

(x2λ̇σ ( j)

)w j

√n∏j=1

(2−x2

λ̇σ ( j)

)w j +n∏j=1

(x2λ̇σ ( j)

)w j ,

√2

n∏j=1

(y2λ̇σ ( j)

)w j

√n∏j=1

(2−y2

λ̇σ ( j)

)w j +n∏j=1

(y2λ̇σ ( j)

)w j

⎤⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎣

√√√√(1+u2

λ̇

) n∑j=1

w j−(1−u2

λ̇

) n∑j=1

w j

√√√√(1+u2

λ̇

) n∑j=1

w j+(1−u2

λ̇

) n∑j=1

w j

,

√√√√(1+v2

λ̇

) n∑j=1

w j−(1−v2

λ̇

) n∑j=1

w j

√√√√(1+v2

λ̇

) n∑j=1

w j+(1−v2

λ̇

) n∑j=1

w j

⎤⎥⎥⎥⎦ ,

⎡⎢⎢⎢⎣

√√√√2(x2λ̇

) n∑j=1

w j

√√√√(2−x2

λ̇

) n∑j=1

w j+(x2λ̇

) n∑j=1

w j

,

√√√√2(y2λ̇

) n∑j=1

w j

√√√√(2−y2

λ̇

) n∑j=1

w j+(y2λ̇

) n∑j=1

w j

⎤⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎣√(

1+u2λ̇

)−(1−u2

λ̇

)

√(1+u2

λ̇

)+(1−u2

λ̇

) ,

√(1+v2

λ̇

)−(1−v2

λ̇

)

√(1+v2

λ̇

)+(1−v2

λ̇

)

⎤⎦ ,

⎡⎣

√2(x2λ̇

)

√(2−x2

λ̇

)+(x2λ̇

) ,

√2(y2λ̇

)

√(2−y2

λ̇

)+(y2λ̇

)

⎤⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

= λ̇.

The proof is completed.

123

48 Complex & Intelligent Systems (2019) 5:41–52

Theorem 5 Boundedness: Let λ j = ([uλ j , vλ j ], [xλ j , yλ j ])( j = 1, 2, 3, ..., n) be a collection of IVPFNs, then

λ̇min ≤ IVPFEHWAω,w(λ1, λ2, λ3, ..., λn) ≤ λ̇max, (25)λ̇max = max

j(λ̇σ ( j)), (26)

λ̇min = minj

(λ̇σ ( j)). (27)

Proof Proof is easy so it is omitted here.

Theorem 6 Monotonicity: If λ j ≤ λ∗j for all j( j =1, 2, 3, ...,n), then

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

≤ IVPFEHWAω,w(λ∗1, λ∗2, λ∗3, ..., λ∗n). (28)

Proof As we know that.

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

= w1λ̇σ (1) ⊕ε w2λ̇σ (2) ⊕ε w3λ̇σ (3) ⊕ε · · · ⊕ε wn λ̇σ (n),(29)

and

IVPFEHWAω,w(λ∗1, λ

∗2, λ

∗3, ..., λ

∗n)

= w1λ̇∗σ(1) ⊕ε w2λ̇∗σ(2) ⊕ε w3λ̇∗σ(3) ⊕ε · · · ⊕ε wn λ̇∗σ(n).(30)

Since λ j ≤ λ∗j for all j , thus Eq. (28) always holds.Theorem 7 Interval-valued Pythagorean fuzzy Einsteinweighted averaging operator is a special case of the interval-valued Pythagorean fuzzy Einstein hybrid weighted averag-ing operator.

Proof Let ω = ( 1n , 1n , 1n , ..., 1n ,)T

, then we have

IVPFEHWAω,w(λ1, λ2, λ3, ..., λn)

= w1λ̇σ (1) ⊕ε w2λ̇σ (2) ⊕ε · · · ⊕ε wn λ̇σ (n)= 1

n(λ̇σ (1) ⊕ε λ̇σ (2) ⊕ε · · · ⊕ε λ̇σ (n))

= 1n(nω1λ1 ⊕ε nω2λ2 ⊕ε · · · ⊕ε nωnλn)

= ω1λ1 ⊕ε ω2λ2 ⊕ε · · · ⊕ε ωnλn= IVPFEWAw(λ1, λ2, λ3, ..., λn).

The proof is completed.

Theorem 8 Interval-valued Pythagorean fuzzy Einsteinordered weighted averaging operator is a special case of theinterval-valued Pythagorean fuzzy Einstein hybrid weightedaveraging operator.

Proof Let w = ( 1n , 1n , 1n , ..., 1n ,)T

, and λ̇σ ( j) = λσ( j), thenwe have

IVPFEHWAω,w(λ1, λ2, λ3, . . . , λn)

= w1λ̇σ (1) ⊕ε w2λ̇σ (2) ⊕ε · · · ⊕ε wn λ̇σ (n)= w1λσ(1) ⊕ε w2λσ(2) ⊕ε · · · ⊕ε wnλσ(n)= IVPFEOWAw(λ1, λ2, λ3, . . . , λn).

The proof completed.

An approach tomultiple attribute groupdecision-making problems based oninterval-valued Pythagorean fuzzyinformation

Algorithm Let X = {X1, X2, X3, ..., Xm} be a finite set ofmalternatives and C = {C1,C2,C3, ...,Cn} be a finite set ofn attributes. Suppose the grade of the alternativesXi (i =1, 2, 3, ...,m)on attributeC j ( j = 1, 2, 3, ..., n) given bydecision makers is interval-valued Pythagorean fuzzy num-bers. Let D = {D1, D2, D3, ..., Dk} be the set of kdecision makers, and let w = (w1, w2, w3, ..., wn)T bethe weighted vector of the attributes C j ( j = 1, 2, 3, ..., n),such that w j ∈ [0, 1],∑nj=1 w j = 1, and let ω =(ω1, ω2, ω3, ..., ωk)

T be the weighted vector of the deci-sion makers Ds(s = 1, 2, 3, ..., k), such that ωs ∈ [0, 1] and∑k

s=1 ωs = 1. Let D = (a ji ) = 〈[u ji , v j i ], [x ji , y ji ]〉(i =1, 2, 3, ...,m, j = 1, 2, 3, ..., n) where [u ji , v j i ] indicatesthe interval degree that the alternative Xi (i = 1, 2, 3, ...,m)satisfies the attribute C j ( j = 1, 2, 3, ..., n) and [x ji , y ji ]indicates the interval degree that the alternative Xi (i =1, 2, 3, ...,m) does not satisfy the attribute C j ( j = 1, 2, 3,..., n), And also [u ji , v j i ] ∈ [0, 1], [x ji , y ji ] ∈ [0, 1] withcondition 0 ≤ (v j i )2 + (y ji )2 ≤ 1, (i = 1, 2, 3, ...,m, j =1, 2, 3, ..., n). This method has the following steps.

Step 1 Utilize the given information in the form ofmatrices,

Ds =[a(s)j i

]n×m (s = 1, 2, 3, ..., k).

Step 2 If the criteria have two types, such as benefitcriteria and cost criteria, then the interval-valuedPythagorean fuzzydecisionmatrices, Ds =

[a(s)j i

]n×m

(s = 1, 2, 3, ..., k) can be converted into the nor-malized interval-valued Pythagorean fuzzy decision

matrices, Rs =[r (s)j i

]n×m

(s = 1, 2, 3, ..., n), where

r (s)j i ={a(s)j i , for benefit criteria C jā(s)j i , for cost criteria C j ,

(j = 1, 2, 3, ..., ni = 1, 2, 3, ...,m

),

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Complex & Intelligent Systems (2019) 5:41–52 49

and ā(s)j i is the complement of αsji . If all the criteria

have the same type, then there is no need of normal-ization.

Step 3 Utilize the IVPFEWA operator to aggregate all theindividual normalized interval-valued Pythagorean

fuzzy decision matrices, Rs =[r (s)j i

]n×m (s = 1, 2,

3, ..., k) into a single interval-valued Pythagoreanfuzzy decision-matrix, R = [r ji ]n×m, where r ji =〈[u ji , v j i ], [x ji , y ji ]〉.

Step 4 In this step, we calculate ṙ j i = nw j r ji .Step 5 Calculate the scores function of ṙ j i (i = 1, 2, 3, ...,

m, j = 1, 2, 3, ..., n). If there is no differencebetween two or more than two scores, then we mustfind out the accuracy degrees of the collective overallpreference values.

Step 6 Utilize the IVPFEHWA operator to aggregate allpreference values.

Step 7 Arrange the scores of the all alternatives in the formof descending order and select that alternative whichhas the highest score function.

Illustrative example

Suppose in Hazara University, the IT department wants toselect a new information system for the purpose of the bestproductivity. After the first selection, there are only threeXi (i = 1, 2, 3) alternatives have been short listed. There arethree experts Ds(s = 1, 2, 3) from a group to act as decision

makers, whose weight vector is ω = (0.2, 0.3, 0.5)T . Thereare many factors that must be considered while selectingthe most suitable system, but here, we have consider onlythe following four criteria, whose weighted vector is w =(0.1, 0.2, 0.3, 0.4)T

1. C1 : Costs of hardware.2. C2 : Support of the organization.3. C3 : Effort to transform from current systems.4. C4 : Outsourcing software developer reliability,

where C1, C3, are cost type criteria and C2, C4 are benefittype criteria, i.e., the attributes have two types of criteria;thus, we must change the cost type criteria into benefit typecriteria.

Step 1 Construct the decision-making matrices (Tables 1, 2and 3).

Step 2 Construct the normalized decision making matrices(Tables 4, 5 and 6).

Step 3 Utilize the IVPFEWA operator to aggregate all theindividual normalized interval-valued Pythagorean

fuzzy decision matrices, Rs =[r (s)j i

]n×m into a

single interval-valued Pythagorean fuzzy decisionmatrix, R = [r ji ]n×m (Table 7).

Table 1 Interval-valuedPythagorean fuzzy decisionmatrix of D1

X1 X2 X3

C1 ([0.5, 0.8], [0.3, 0.4]) ([0.6, 0.7], [0.3, 0.6]) ([0.3, 0.7], [0.3, 0.5])C2 ([0.3, 0.5], [0.6, 0.7]) ([0.3, 0.7], [0.2, 0.6]) ([0.3, 0.6], [0.4, 0.7])C3 ([0.5, 0.7], [0.3, 0.7]) ([0.5, 0.6], [0.3, 0.7]) ([0.2, 0.6], [0.3, 0.7])C4 ([0.3, 0.6], [0.6, 0.7]) ([0.6, 0.5], [0.2, 0.7]) ([0.3, 0.4], [0.5, 0.6])

Table 2 Interval-valuedPythagorean fuzzy decisionmatrix of D2

X1 X2 X3

C1 ([0.5, 0.6], [0.3, 0.5]) ([0.5, 0.7], [0.3, 0.6]) ([0.2, 0.8], [0.3, 0.4])C2 ([0.3, 0.4], [0.6, 0.8]) ([0.3, 0.8], [0.2, 0.6]) ([0.3, 0.6], [0.3, 0.7])C3 ([0.4, 0.5], [0.3, 0.8]) ([0.5, 0.7], [0.3, 0.6]) ([0.2, 0.6], [0.3, 0.8])C4 ([0.3, 0.6], [0.5, 0.7]) ([0.3, 0.4], [0.2, 0.8]) ([0.3, 0.5], [0.5, 0.7])

Table 3 Interval-valuedPythagorean fuzzy decisionmatrix of D3

X1 X2 X3

C1 ([0.3, 0.8], [0.5, 0.6]) ([0.3, 0.5], [0.5, 0.7]) ([0.2, 0.4], [0.5, 0.7])C2 ([0.5, 0.7], [0.3, 0.4]) ([0.4, 0.6], [0.5, 0.8]) ([0.5, 0.7], [0.2, 0.5])C3 ([0.3, 0.6], [0.4, 0.6]) ([0.3, 0.5], [0.5, 0.6]) ([0.2, 0.8], [0.4, 0.6])C4 ([0.5, 0.7], [0.3, 0.4]) ([0.5, 0.7], [0.2, 0.4]) ([0.5, 0.6], [0.3, 0.5])

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50 Complex & Intelligent Systems (2019) 5:41–52

Table 4 NormalizedPythagorean fuzzy decisionmatrix R1

X1 X2 X3

C1 ([0.3, 0.4], [0.5, 0.8]) ([0.3, 0.6], [0.6, 0.7]) ([0.3, 0.5], [0.3, 0.7])C2 ([0.3, 0.5], [0.6, 0.7]) ([0.3, 0.7], [0.2, 0.6]) ([0.3, 0.6], [0.4, 0.7])C3 ([0.3, 0.7], [0.5, 0.7]) ([0.3, 0.7], [0.5, 0.6]) ([0.3, 0.7], [0.2, 0.6])C4 ([0.3, 0.6], [0.6, 0.7]) ([0.6, 0.5], [0.2, 0.7]) ([0.3, 0.4], [0.5, 0.6])

Table 5 NormalizedPythagorean fuzzy decisionmatrix R2

X1 X2 X3

C1 ([0.3, 0.5], [0.5, 0.6]) ([0.3, 0.6], [0.5, 0.7]) ([0.3, 0.4], [0.2, 0.8])C2 ([0.3, 0.4], [0.6, 0.8]) ([0.3, 0.8], [0.2, 0.6]) ([0.3, 0.6], [0.3, 0.7])C3 ([0.3, 0.8], [0.4, 0.5]) ([0.3, 0.6], [0.5, 0.7]) ([0.3, 0.8], [0.2, 0.6])C4 ([0.3, 0.6], [0.5, 0.7]) ([0.3, 0.4], [0.2, 0.8]) ([0.3, 0.5], [0.5, 0.7])

Table 6 NormalizedPythagorean fuzzy decisionmatrix R3

X1 X2 X3

C1 ([0.5, 0.6], [0.3, 0.8]) ([0.5, 0.7], [0.3, 0.5]) ([0.5, 0.7], [0.2, 0.4])C2 ([0.5, 0.7], [0.3, 0.4]) ([0.4, 0.6], [0.5, 0.8]) ([0.5, 0.7], [0.2, 0.5])C3 ([0.4, 0.6], [0.3, 0.6]) ([0.5, 0.6], [0.3, 0.5]) ([0.4, 0.6], [0.2, 0.8])C4 ([0.5, 0.7], [0.3, 0.4]) ([0.5, 0.7], [0.2, 0.4]) ([0.5, 0.6], [0.3, 0.5])

Table 7 Collective interval-valued Pythagorean fuzzy decision matrix R

X1 X2 X3

C1 ([0.413, 0.537], [0.389, 0.738]) ([0.413, 0.653], [0.405, 0.595]) ([0.413, 0.593], [0.216, 0.562])C2 ([0.413, 0.593], [0.429, 0.563]) ([0.352, 0.692], [0.320, 0.697]) ([0.413, 0.653], [0.260, 0.595])C3 ([0.352, 0.692], [0.363, 0.587]) ([0.413, 0.622], [0.389, 0.576]) ([0.352, 0.692], [0.200, 0.697])C4 ([0.413, 0.653], [0.405, 0.536]) ([0.475, 0.593], [0.200, 0.563]) ([0.413, 0.537], [0.389, 0.576])

Step 4 Calculate λ̇ j i = nwλ j i .

λ̇11 = ([0.262, 0.343], [0.733, 0.897]),λ̇21 = ([0.370, 0.534], [0.523, 0.645])λ̇31 = ([0.385, 0.745], [0.281, 0.513]),λ̇41 = ([0.518, 0.788], [0.201, 0.329])λ̇12 = ([0.262, 0.424], [0.742, 0.837]),λ̇22 = ([0.315, 0.628], [0.420, 0.757])λ̇32 = ([0.452, 0.665], [0.307, 0.501]),λ̇42 = ([0.593, 0.726], [0.061, 0.359])λ̇13 = ([0.262, 0.382], [0.605, 0.823]),λ̇23 = ([0.370, 0.590], [0.357, 0.672])λ̇33 = ([0.385, 0.745], [0.136, 0.638]),λ̇43 = ([0.518, 0.664], [0.188, 0.374]).

Step 5 Calculate the score functions (Table 8).

s(λ̇11) = −0.57, s(λ̇21) = −0.13, s(λ̇31)= 0.18, s(λ̇41) = 0.37

s(λ̇12) = −0.50, s(λ̇22) = −0.12, s(λ̇32)= 0.15, s(λ̇42) = 0.37

s(λ̇13) = −0.41, s(λ̇23) = −0.04, s(λ̇33)= 0.13, s(λ̇43) = 0.26.

Step 6 Utilize the IVPFEHWA aggregation operator toaggregate all preference values.

r1 = ([0.354, 0.567], [0.550, 0.674])

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Complex & Intelligent Systems (2019) 5:41–52 51

Table 8 Pythagorean fuzzy hybrid decision matrix R

X1 X2 X3

C1 ([0.518, 0.788], [0.201, 0.329]) ([0.593, 0.726], [0.061, 0.359]) ([0.518, 0.664], [0.188, 0.374]),C2 ([0.385, 0.745], [0.281, 0.513]) ([0.525, 0.665], [0.307, 0.501]) ([0.585, 0.745], [0.136, 0.638]),C3 ([0.370, 0.534], [0.523, 0.645]) ([0.315, 0.628], [0.420, 0.757]) ([0.370, 0.590], [0.357, 0.672]),C4 ([0.262, 0.343], [0.733, 0.897]) ([0.262, 0.424], [0.742, 0.837]) ([0.262, 0.382], [0.605, 0.823]).

r2 = ([0.367, 0.581], [0.422, 0.686])r3 = ([0.354, 0.571], [0.347, 0.695]).

Step 7 Calculate the score functions.

s(r1) = −0.154, s(r1) = −0.088, s(r1) = −0.076.

Step 8 Arrange the scores of the all alternatives in the formof descending order and select that alternative whichhas the highest score function.

s(r3) s(r2) s(r1)

Thus, the best alternative is X3.

Conclusion

In this paper, we have developed the notion of interval-valued Pythagorean fuzzy Einstein hybrid weighted aver-aging aggregation operator along with their some desir-able properties such as idempotency, boundedness, andmonotonicity. Actually interval-valued Pythagorean fuzzyEinstein weighted averaging aggregation operator weightsonly the Pythagorean fuzzy arguments and interval-valuedPythagorean fuzzy Einstein ordered weighted averagingaggregation operator weights only the ordered positionsof the Pythagorean fuzzy arguments instead of weightingthe Pythagorean fuzzy arguments themselves. To over-come these limitations, we have introduced an interval-valued Pythagorean fuzzy Einstein hybrid weighted aver-aging aggregation operator, which weights both the givenPythagorean fuzzy value and its ordered position. Finally,the proposed operator has been applied to decision-makingproblems to show the validity, practicality and effectivenessof the new approach.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

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Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision makingAbstractIntroductionPreliminariesInterval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operatorAn approach to multiple attribute group decision-making problems based on interval-valued Pythagorean fuzzy informationIllustrative exampleConclusionReferences