-
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 [email protected]
Saleem [email protected]
Asad [email protected]
Fazli [email protected]
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
123
http://crossmark.crossref.org/dialog/?doi=10.1007/s40747-018-0076-x&domain=pdf
-
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:
123
-
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
123
-
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
⎤⎥⎥⎦
⎞⎟⎟⎠
123
-
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
),
123
-
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])
123
-
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])
123
-
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.
References
1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets
Syst20(1):87–96
2. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–3533. Xu ZS,
Yager RR (2006) Some geometric aggregation operators
based on intuitionistic fuzzy sets. Int J Gen Syst
35(4):417–4334. Xu ZS (2007) Intuitionistic fuzzy aggregation
operators. IEEE
Trans Fuzzy Syst 15(6):1179–11875. Wang W, Liu X (2011)
Intuitionistic fuzzy geometric aggrega-
tion operators based on Einstein operations. Int J Intell
Syst26(11):1049–1075
6. Wang WZ, Liu XW (2012) Intuitionistic fuzzy
informationaggregation using Einstein operations. IEEE Trans Fuzzy
Syst20(5):923–938
7. Xu ZS (2007b) Methods for aggregating interval-valued
intuition-istic fuzzy information and their application to
decision-making.Control Decis 22(2):215–219 (in Chinese)
8. Xu ZS, Chen J (2007b) On geometric aggregation over
interval-valued intuitionistic fuzzy information. Fourth Int Conf
Fuzzy SystKnowl Discov FSKD 2:466–471
9. Wang W, Liu X (2013) Interval-valued intuitionistic fuzzy
hybridweighted averaging operator based on Einstein operation and
itsapplication to decision making. J Intell Fuzzy Syst
25(2):279–290
10. WangW,LiuX (2013) Themulti-attribute
decisionmakingmethodbased on interval-valued intuitionistic fuzzy
Einstein hybridweighted geometric operator. Comput Math Appl
66(10):1845–56
11. Rahman K, Abdullah S, Jamil M, Khan MY (2018) Some
general-ized intuitionistic fuzzy Einstein hybrid aggregation
operators andtheir application to multiple attribute group decision
making. Int JFuzzy Syst 20(5):1567–1575
12. Liao H, Xu Z (2014) Some new hybrid weighted
aggregationoperators under hesitant fuzzy multi-criteria decision
making envi-ronment. J Intell Fuzzy Syst 26(4):1601–1617
13. Liao H, Xu Z (2014) Intuitionistic fuzzy hybrid weighted
aggrega-tion operators. Int J Intell Syst 29(11):971–993
14. Liao H, Xu Z (2015) Extended hesitant fuzzy hybrid
weightedaggregation operators and their application in decision
making.Soft Comput 19(9):2551–2564
15. Yu D, Liao H (2016) Visualization and quantitative research
onintuitionistic fuzzy studies. J Intell Fuzzy Syst
30(6):3653–3663
16. Liu W, Liao H (2017) A bibliometric analysis of fuzzy
decisionresearch during 1970–2015. Int J Fuzzy Syst 19(1):1–14
17. Yager R.R (2013) Pythagorean fuzzy subsets. In Proc Joint
IFSAWorldCongress andNAFIPSAnnualMeeting, Edmonton,Canada,pp
57–61
18. Yager RR (2014) Pythagorean membership grades in
multi-criteriadecision making. IEEE Trans Fuzzy Syst
22(4):958–965
19. Yager RR, Abbasov AM (2013) Pythagorean membership
grades,complex numbers and decisionmaking. Int J Intell Syst
28(5):436–452
20. Peng X, Yang Y (2015) Some results for Pythagorean fuzzy
sets.Int J Intell Syst 30(11):1133–1160
123
http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/
-
52 Complex & Intelligent Systems (2019) 5:41–52
21. Garg H (2016) A new generalized Pythagorean fuzzy
informationaggregation using Einstein operations and its
application to deci-sion making. Int J Intell Syst
31(9):886–920
22. Garg H (2017) Generalized Pythagorean fuzzy geometric
aggreg-tion operators using Einstein t-Norm and t-Conorm for
multicrite-ria decision-making process. Int J Intell Syst
32:597–630. https://doi.org/10.1002/int.21860
23. RahmanK,Abdullah S,Husain F,AliKhanMS (2016)Approachesto
Pythagorean fuzzy geometric aggregation operators. Int J Com-put
Sci Inf Secur IJCSIS 4(9):174–200
24. Rahman K, Khan MSA, Ullah M, Fahmi A (2017)
Multipleattribute group decision making for plant location
selection withPythagorean fuzzy weighted geometric aggregation.
OperatorNucleus 54(1):66–74
25. Rahman K, Abdullah S, Husain F, Ali Khan MS, Shakeel M(2017)
Pythagorean fuzzy orderedweighted geometric aggregationoperator and
their application to multiple attribute group decisionmaking. J
Appl Environ Biol Sci 7(4):67–83
26. Rahman K, Abdullah S, Ali Khan MS, Shakeel M
(2016)Pythagorean fuzzy hybrid geometric aggregation operator and
theirapplications tomultiple attribute decisionmaking. Int J Comput
SciInf Secur IJCSIS 837–854
27. Rahman K, Ali A, Shakeel M, Khan MSA, Ullah Murad
(2017)Pythagorean fuzzy weighted averaging aggregation operator
andits application to decision making theory. Nucleus
54(3):190–196
28. Rahman K, Abdullah S, Ahmed R, Ullah Murad (2017)Pythagorean
fuzzy Einstein weighted geometric aggregation oper-ator and their
application to multiple attribute group decisionmaking. J Intell
Fuzzy Syst 33(1):635–647
29. Peng X, Yang Y (2016) Fundamental properties of
interval-valued Pythagorean fuzzy aggregation operators. Int J
Intell Syst31(5):444–487
30. Rahman K, Ali Asad, Khan MSA (2018) Some
interval-valuedPythagorean fuzzy weighted averaging aggregation
operators andtheir application tomultiple attribute decisionmaking,
PunjabUni-versity. J Math 50(2):113–129
31. Rahman K, Abdullah S, Shakeel M, Khan MSA, Ullah Murad(2017)
Interval-valued Pythagorean fuzzy geometric aggregationoperators
and their application to decision making. Cogent Math4(1):1338638.
https://doi.org/10.1515/jisys-2017-0212
32. Rahman K, Abdullah S, Khan MSA (2018) Some
interval-valuedPythagorean fuzzy Einstein weighted averaging
aggregation oper-ator and their application to group decision
making. J Intell Syst.https://doi.org/10.1515/jisys-2017-0212
33. Rahman K, Abdullah S (2018) Generalized
interval-valuedPythagorean fuzzy aggregation operators and their
application togroup decision making. Granul Comput.
https://doi.org/10.1007/s41066-018-0082-9
Publisher’s Note Springer Nature remains neutral with regard to
juris-dictional claims in published maps and institutional
affiliations.
123
https://doi.org/10.1002/int.21860https://doi.org/10.1002/int.21860https://doi.org/10.1515/jisys-2017-0212https://doi.org/10.1515/jisys-2017-0212https://doi.org/10.1007/s41066-018-0082-9https://doi.org/10.1007/s41066-018-0082-9
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