Keywords: Pythagorean fuzzy set, Pythagorean fuzzy weighted geometric ( PFWG ) operator, Pythagorean fuzzy ordered weigh...
International Journal of Computer Science and Information Security (IJCSIS), Vol. 14, No. 9, September 2016
Approaches to Pythagorean Fuzzy Geometric Aggregation Operators Khaista Rahman1, Saleem Abdullah2, Fawad Husain3, Muhammad Sajjad Ali Khan4 1,4
Department of Mathematics, Hazara University, Mansehra, KPK, Pakistan
[email protected],
[email protected]
2
Department of Mathematics, Abdul Wali Khan University Mardan, KPK, Pakistan
[email protected]
3
Department of Mathematics Abbottabad University of Science and Technology Abbottabad, KPK, Pakistan
[email protected]
Abstract : There are many aggregation operators have been introduced so far, but in this paper, we introduce some new geometric aggregation operators for aggregating Pythagorean fuzzy numbers, such as Pythagorean fuzzy weighted geometric
( PFWG )
operator, Pythagorean fuzzy ordered weighted geometric
hybrid geometric
( PFHG )
( PFOWG )
operator and Pythagorean fuzzy
operator. We also discuss some basic properties of these operators such as monotonicity,
idempotency, and boundedness of the proposed operator. Some numerical examples are given to outline the developed operators. Keywords: Pythagorean fuzzy set, Pythagorean fuzzy weighted geometric weighted geometric
( PFOWG )
( PFWG )
operator and Pythagorean fuzzy hybrid geometric
I.
operator, Pythagorean fuzzy ordered
( PFHG )
operator.
NTRODUCTION
In 1965, L. A. Zadeh presented the notion of fuzzy set [13]. In 1986, Atanassov presented the idea of intuitionistic fuzzy set, which is the generalization of the fuzzy set [5]. The intuitionistic fuzzy set has gotten increasingly consideration since its development [5,6,7,8,9,10,11]. Chen and Tan [19] and Hong and Choi [1] characterized some fundamental standards multi-criteria fuzzy decision making problems based on vague sets. Bustince and Burillo [3] demonstrated that vague sets are intuitionistic fuzzy sets. De et al [20] defined concentration, dilation and normalization of intuitionistic fuzzy sets. He additionally demonstrated some recommendations in this field. Bustince et al. [4] introduced the notion of intuitionistic fuzzy generators and also studied the complementary of an intuitionistic fuzzy set from the intuitionistic fuzzy generators. Yager [17, 18] introduced the notion of Pythagorean fuzzy set categorized by a membership degree and nonmembership degree which holds the condition that the square sum of its membership degree and nonmembership degree is equal to or less than one. Xu [25] developed some basic arithmetic aggregation operators, like as
IFWA operator, IFOWA operator and IFHA operator.
Xu and Yager [24] developed some basic geometric aggregation operators, such as
174
IFWG
operator,
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IFOWG operator and IFHG operator. They also applied them to MADM
IFSs . Wei [2]
base on
introduced the notion of some induced geometric aggregation operators with intuitionistic fuzzy information and also applied them to group decision making. Liu [21] introduced the notion of operator, and applied the IFWGε to in the
MCDM
problems.
MADM
IFWGε operator, IFOWGε
also. Bellman and Zadeh [14] presented the theory of fuzzy sets
IFSs have been mostly applied in real-life MCDM MCDM
studies of both methods and applications of
problems with
IFSs
[12,15,16,20,21,26,27]. In 2015, X. Peng and Y. Yang [23] introduced the notion of
problems, and the
have got great focus
PFWA
operator,
PFWPA operator, and PFWPG operator. This paper consists of five section. In section 1, we give an introduction of the research background. In section 2, we give some basic definitions and results which will be used in later sections. In section 3, we introduce some new operational laws and relations on Pythagorean fuzzy sets and analysis some desirable properties of the proposed operational laws. In section 4, we introduce Pythagorean fuzzy weighted geometric
( PFWG )
operator, Pythagorean fuzzy ordered weighted geometric
( PFOWG )
( PFHG ). In section 5, we have conclusion.
operator and Pythagorean fuzzy hybrid geometric
2. PRELIMINERS
In this section, we define the basic concept of fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, intuitionistic fuzzy weighted geometric ( IFWG ) operator, intuitionistic fuzzy ordered weighted geometric
( IFOWG ) operator and intuitionistic fuzzy hybrid geometric ( IFHG ) operator. We also define score function
( PFNs )
and accuracy function for Pythagorean fuzzy numbers Definition 2.1: [13] Let
X be a fixed set, then a fuzzy set ( FS ), F=
where
.
μ F : X → [0,1]
{ x, μ
F
( x)
, and for each x ∈ X ,
}
(1)
| x∈ X ,
μF ( x )
F in X can be is defined as:
is called the degree of membership of
x in X
. Definition 2.2: [5] Let X
be a fixed set, then an intuitionistic fuzzy set
( IFS ) ,
I in X can be is defined
as:
I = { x, μ I ( x),ν I ( x) | x ∈ X } , where
μ I ( x)
and
ν I ( x)
are mappings from
X
0 ≤ μ I ( x) ≤ 1, 0 ≤ ν I ( x) ≤ 1 and 0 ≤ μ I ( x) + ν I ( x) ≤ 1, ∀ membership and nonmembership of element
x∈ X
to set
175
( 2)
to the closed interval
[0,1] , such that
x ∈ X , and they are denote the degrees of
I , respectively. Let π I ( x) = 1 − μ I ( x) −ν I ( x)
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x∈ X
, then it is usually called the intuitionistic fuzzy index of element
I , representing the degree of
x to A . It is obvious that 0 ≤ π A ( x) ≤ 1 for every x ∈ X .
indeterminacy or hesitation of Definition 2.3: [17] Let
to set
X be a fixed set, then a Pythagorean fuzzy set ( PFS ) , P in X
can be is
defined as:
( 3)
P = { x, μ P ( x),ν P ( x) | x ∈ X }, where
μ P ( x)
ν P ( x)
and
are mappings from
X
0 ≤ μ P ( x) ≤ 1, 0 ≤ ν P ( x) ≤ 1 and 0 ≤ μ P2 ( x) + ν P2 ( x ) ≤ 1, ∀ membership
and
nonmembership
π P ( x) = 1 − μ P2 ( x) −ν P2 ( x)
of
to
set
P
,
respectively.
, then it is generally called the Pythagorean fuzzy index of member
set P representing the degree of indeterminacy or hesitation of all
x ∈ X , and they are denote the degrees of
x∈ X
member
[0,1] , such that
to the closed interval
x∈ X
Let to
x to P . It is clear that 0 ≤ π P ( x) ≤ 1 for
x∈ X.
In this paper, we consider the interval [ rP ( x ) ,1 − sP ( x )] is a Pythagorean fuzzy value, and substitute equation
(3) with P = {x, ⎡⎣ rP ( x ) ,1 − sP ( x ) ⎤⎦ | x ∈ X }, correspondingly. The Pythagorean fuzzy value membership
μ P ( x)
of
( 4)
[rP ( x ) ,1 − sP ( x )]
indicates that the exact degree of
x may be unknown. But it is bounded by rP ( x ) ≤ μ P ( x) ≤ 1 − sP ( x ) . Where
rP2 ( x ) + sP2 ( x ) ≤ 1 . Definition 2.4: [22] Let
( PFNs )
(
a%1 = μa%1 ,ν a%1
)
and
a%2 = (ua%2 , va%2 )
be the two Pythagorean fuzzy numbers
s ( a%1 ) = μa2%1 −ν a2%1 and s ( a%2 ) = μa2%2 −ν a%22 be the score of a%1 , a%2 , h ( a%1 ) = μa2%1 + ν a2%1
, then
and h ( a%2 ) = μ a%2 + ν a%2 2
2
be the accuracy degree of a%1 and a%2 , respectively, then
(1)
If s (a%1 ) < s ( a%2 ), then a%1 < a%2 .
( 2)
If s (a%1 ) = s ( a%2 ), then
(a)
If h( a%1 ) = h( a%2 ), then a%1 and a%2 represent the same information.
(b)
If h(α1 ) < h(α1 )
then a%1 is smaller than a% 2 denoted by a%1 < a%2 .
Definition 2.5: [24] Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) be a collection of all intuitionistic fuzzy values,
⎣
and let
⎦
IFWG : Ω n → Ω, if
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IFWGw ( a%1 , a%2 ,..., a%n ) = a%1w1 ⊗ a%2w2 ⊗ ... ⊕ a%nwn ,
( 5)
IFWG is called intuitionistic fuzzy weighted geometric ( IFWG ) operator of dimension n , where
then
w = ( w1 , w2 ,..., wn )
T
Especially, if
w j ∈ [ 0,1] and
is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
n
∑w
j
= 1.
j =1
w = ( 1n , 1n ,..., 1n ) , then the IFWG operator is reduced to an intuitionistic fuzzy geometric T
( IFG ) operator of dimension n, which is defined as follows: IFG ( a%1 , a%2 ,..., a%n ) = ( a%1 ⊗ a%2 ⊗ ... ⊗ a%n ) n .
(6)
1
Definition 2.6:
[24] Let
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦ ( j = 1, 2,3,..., n ) be a collection of intuitionistic fuzzy values. An
intuitionistic fuzzy ordered weighted geometric
( IFOWG )
à w 1 , w 2 , . . . , w n ÄT ,
IFOWG : Ω n → Ω, that has an associated vector w n
∑w
and
j
operator of dimension
n
is a mapping
such that
w j ∈ [ 0,1]
= 1. Furthermore
j =1
( ) ⊗ ( a% ( ) )
IFOWGw ( a%1 , a%2 ,..., a%n ) = a%σ (1)
(σ (1) , σ ( 2 ) ,..., σ ( n ) )
where
Especially, if
w1
w2
σ 2
is a permutation of
(
⊗ ... ⊗ a%σ ( n )
(1, 2,..., n )
such that
)
wn
,
(7)
a%σ ( j −1) ≥ a%σ ( j ) for all
j,
w = ( 1n , 1n ,..., 1n ) , then the IFOWG operator is reduced to a IFG operator of dimension T
n . Definition 2.7: [24] Intuitionistic fuzzy hybrid Averaging geometric
( IFHG ) of dimension n is a mapping
IFHG : Ω n → Ω, which has an associated vector w = ( w1 , w2 ,..., wn ) , such that w j ∈ [ 0,1] and T
n
∑w
j
= 1. Furthermore
j =1
w1
w2
wn
⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ IFHGw, w ( a%1 , a%2 ,..., a%n ) = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ .... ⊗ ⎜ a% σ ( n ) ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⋅
a% σ ( j )
where
is the
w = ( w1 , w2 ,..., wn )
T
n
and
∑w
j
jth
largest of the weighted Pythagorean fuzzy values
is the weighted vector of a% j ( j = 1, 2,..., n )
(8) ⋅ nw ⎛⋅ a% σ ( j ) ⎜ a% σ ( j ) = a% j j ⎝
⎞ ⎟, ⎠
[ ]
such that w j ∈ 0,1 ( j = 1, 2,3,..., n)
= 1, and n is the balancing coefficient, which plays a role of balance (in such a case, if the vector
j =1
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( w1 , w2 ,..., wn )
T
( a%1 , a%2 ,..., a%n )
T
( 1n , 1n ,..., 1n )
T
approaches
,
then the vector
( a%
nw1 1
, a%2nw2 ,..., a%nnwn
)
T
approaches
. 3. Operational laws and relations
In this study, we present two new operational laws and relations on Pythagorean fuzzy sets and analysis necessary properties of the suggested operational laws. We also define the score and the accuracy function of Pythagorean fuzzy numbers. Definition 3.1: Let a%1 = [ ra%1 ,1 − sa%1 ] and a%2 = [ ra%2 ,1 − sa%2 ] be the two Pythagorean fuzzy value, then we have
(
)(
)
(1) a%1 ⊗ a%2 = ⎡⎣ ra%1 ra%2 , 1 − sa%1 1 − sa%2 ⎤⎦ λ ( 2 ) a% λ = ⎡⎣ ra%λ , (1 − sa% ) ⎤⎦ , λ > 0. a%1 = [ra%1 ,1 − sa%1 ] and a%2 = [ra%2 ,1 − sa%2 ] be the two Pythagorean fuzzy value, and let
Theorem 3.2: Let
( λ > 0) ,
a = a%1 ⊗ a%2 and β = a% λ
then we show that
α
and
β
are also Pythagorean fuzzy valves.
Proof: Since a%1 = [ ra%1 ,1 − sa%1 ] and a%2 = [ ra%2 ,1 − sa%2 ] are the two Pythagorean fuzzy value, then we have
ra%1 ∈ [ 0,1] ,
(1) ,
sa%1 ∈ [ 0,1] , ra%2 ∈ [ 0,1] , sa%2 ∈ [ 0,1]
(
we have 0 ≤ ra%1 ra%2
)
2
((
and
)(
≤ 1, 0 ≤ 1 − sa%1 1 − sa%2
))
2
ra%21 + sa2%1 ≤ 1, ra%22 + sa2%2 ≤ 1, then by operational ≤ 1. Then
( r r ) + 1 − ( (1 − s )(1 − s ) ) ≤ ( (1 − s )(1 − s ) ) + 1 − (1 − s ) (1 − s ) ) = ( (1 − s )(1 − s ) ) + 1 − (1 − s ) (1 − s ) ) 2
2
a%1 a%2
a%1
a%2
2
a%1
a%2
a%1
a%2
a%1
a%2
a%1
a%2
2
2
2
= 1. Thus
(r r )
(1 − sa% )
a%1 a% 2
λ
2
((
)(
+ 1 − 1 − sa%1 1 − sa%2
))
2
≤ 1. Thus α is a Pythagorean fuzzy value. Also ra%λ ≥ 0 and
≥ 0. Since
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(r )
λ 2
a%
(
+ 1 − (1 − sa% )
) ( ) = ( (1 − s ) ) = ( (1 − s ) )
λ 2
≤ (1 − sa% )
λ 2
λ 2
a%
λ 2
λ 2
a%
λ 2
a%
( ) + 1 − ( (1 − s ) ) + 1 − ( (1 − s ) ) + 1 − (1 − sa% )
λ 2
a%
= 1. Thus
(r )
λ 2
a%
(
+ 1 − (1 − sa2% )
Theorem 3.3:
β
is a Pythagorean fuzzy value.
then
a%1 ⊗ a%2 = a%2 ⊗ a%1 ,
( 2 ) ( a%1 ⊗ a%2 ) ( 3)
Thus
a% = [ra% ,1 − sa% ], a%1 = [ra%1 ,1 − sa%1 ] and a%2 = [ra%2 ,1 − sa%2 ] be the two Pythagorean
λ , λ1 , λ2 > 0,
fuzzy values,
(1)
Let
) ≤ 1.
λ 2
λ
= a%1λ ⊗ a%2λ , λ + λ2
a% λ1 ⊗ a% λ2 = ( a% ) 1
( 4 ) ( a% )
λ1λ2
Proof: (1)
,
= ( a% λ1 ) . λ2
Since
( (
)( )(
a%1 ⊗ a%2 = ⎡⎣ ra%1 ra%2 , 1 − sa%1 1 − sa%2 = ⎡⎣ ra%2 ra%1 , 1 − sa%2 1 − sa%1 = a%2 ⊗ a%1.
( 2)
)⎤⎦ )⎤⎦
Since
(
)(
)
a%1 ⊗ a%2 = ⎡⎣ ra%1 ra%2 , 1 − sa%1 1 − sa%2 ⎤⎦ λ λ ( a%1 ⊗ a%2 ) = ⎡⎢ ra%1 ra%2 , 1 − sa%1 1 − sa%2 ⎣
(
) ((
(
= ⎡ ra%λ1 ra%λ2 , 1 − sa%1 ⎣⎢
)(
) (1 − s ) λ
a%2
λ
))
λ
⎤ ⎦⎥
⎤. ⎦⎥
Also
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( ( ( (
) ) ) )
a%1 = ⎡⎣ ra%1 , 1 − sa%1 ⎤⎦ λ a%1λ = ⎡ ra%λ1 , 1 − sa%1 ⎤ ⎢⎣ ⎥⎦ a%2 = ⎡⎣ ra%2 , 1 − sa%2 ⎤⎦ λ a%2λ = ⎡ ra%λ2 , 1 − sa%2 ⎤ . ⎢⎣ ⎥⎦ Then
(
)
(
λ a%1λ ⊗ a%2λ = ⎡ ra%λ1 , 1 − sa%1 ⎤ ⊗ ⎡ ra%λ2 , 1 − sa%2 ⎢⎣ ⎥⎦ ⎢⎣ λ λ = ⎡ ra%λ1 ra%λ2 , 1 − sa%1 1 − sa%2 ⎤ . ⎣⎢ ⎦⎥
(
( 3)
)(
)
λ
⎤ ⎥⎦
)
Since λ a% λ1 = ⎡ ra%λ1 , (1 − sa% ) 1 ⎤ ⎣ ⎦ λ λ2 λ 2 a% = ⎡ ra% 2 , (1 − sa% ) ⎤ . ⎣ ⎦
Then λ λ a% λ1 ⊗ a% λ2 = ⎡ ra%λ1 , (1 − sa% ) 1 ⎤ ⊗ ⎡ ra%λ2 , (1 − sa% ) 2 ⎤ ⎣ ⎦ ⎣ ⎦ λ λ = ⎡ ra%λ1 ra%λ2 , (1 − sa% ) 1 (1 − sa% ) 2 ⎤ ⎣ ⎦ + λ λ = ⎡ ra%λ1 + λ2 , (1 − sa% ) 1 2 ⎤ ⎣ ⎦ λ + λ2
= ( a% ) 1
( 4)
.
Since
a% λ1 = ⎡ ra%λ1 , (1 − sa% ) 1 ⎤ ⎣ ⎦ λ λ ( a% λ1 ) 2 = ⎡⎢⎣( ra%λ1 ) 2 , (1 − sa% )λ1 λλ = ⎡ ra%λ1λ2 , (1 − sa% ) 1 2 ⎤ ⎣ ⎦ λ1λ2 = a% . λ
(
)
λ2
⎤ ⎥⎦
This completes the proof. Now we going to discuss some special cases of
(1)
If a%
= [ ra% ,1 − sa% ]
λ
and a%.
= [1,1] , i.e., ra% = 1,1 − sa% = 1, then
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λ a% λ = ⎡ ra%λ , (1 − sa% ) ⎤ ⎣ ⎦ λ λ = ⎡(1) , (1 − 0 ) ⎤ ⎣ ⎦ = ⎡⎣1, 1 ⎤⎦ = [1,1] .
( 2)
[
] [
]
If a% = ra% ,1 − sa% = 0, 0 , i.e., ra% = 0,1 − sa% = 0, then λ a% λ = ⎡ ra%λ , (1 − sa% ) ⎤ ⎣ ⎦ λ λ = ⎡( 0 ) , (1 − 1) ⎤ ⎣ ⎦ λ = ⎡0, (1 − 1) ⎤ ⎣ ⎦ = [ 0, 0].
( 3)
[
] [ ]
If a% = ra% ,1 − sa% = 0,1 , i.e., ra% = 0,1 − sa% = 1, then λ a% λ = ⎡ ra%λ , (1 − sa% ) ⎤ ⎣ ⎦ λ λ = ⎡( 0 ) , (1 − 0 ) ⎤ ⎣ ⎦ = [ 0,1]
= [ 0,1] .
( 4)
If
λ →0
[ ]
[ ]( λ → 0)
λ λ λ λ and 0 ≤ ra% , sa% ≤ 1, then a% = ⎡ ra% , (1 − sa% ) ⎤ → 1,1 , i.e., a% → 1,1
⎣
⎦
Since λ a% λ = ⎡ ra%λ , (1 − sa% ) ⎤ ⎣ ⎦ 0 0 = ⎡( ra% ) , (1 − sa% ) ⎤ ⎣ ⎦ = [1,1]
= [1,1].
( 5)
If
λ → +∞
[
]
λ λ λ and 0 ≤ ra% , sa% ≤ 1, then a% = ⎡ ra% , (1 − sa% ) ⎤ → 0, 0 , i. e.,
⎣
⎦
a% λ → [ 0, 0] ( λ → +∞ ) . Since
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λ a% λ = ⎡ ra%λ , (1 − sa% ) ⎤ ⎣ ⎦ ∞ ∞ = ⎡( ra% ) , (1 − sa% ) ⎤ ⎣ ⎦ = [ 0, 0] .
( 6)
If
λ =1
λ λ λ , then a% = ⎡ ra% , (1 − sa% ) ⎤ → a% ,
⎣
i.e., a% → a% ( λ = 1) . λ
⎦
Since λ a% λ = ⎡ ra%λ , (1 − sa% ) ⎤ ⎣ ⎦ 1 = ⎡ ra%1 , (1 − sa% ) ⎤ ⎣ ⎦
= ⎡⎣ ra% , (1 − sa% ) ⎤⎦ = a%. Definition 3.4: Let
( PFVs )
a%1 = ⎡⎣ ra%1 ,1 − sa%1 ⎤⎦ and a%2
, then S ( a%1 ) = ra%1 − sa%1 2
2
= ⎡⎣ ra%2 ,1 − sa%2 ⎤⎦ be the two Pythagorean fuzzy values
and S ( a%2 ) = ra%2 − sa%2 2
be the score of a%1 and a% 2 correspondingly.
2
H ( a%1 ) = ra%21 + sa2%1 and H ( a%2 ) = ra%22 + sa2%2 be the accuracy degree of a%1 and a%2 respectively, then
(1)
If S ( a%1 ) < S ( a%2 ) ,
( 2)
If S ( a%1 ) = S ( a%2 ) , then we have the following two cases
(a)
If
H ( a%1 ) = H ( a%2 ) , then we have a%1 = a%2 .
(b)
If
H ( a%1 ) < H ( a%2 ) then we have a%2 > a%1.
Theorem 3.5: Let
then we have a%2 > a%1.
a%1 = ⎡⎣ ra%1 ,1 − sa%1 ⎤⎦
and
a%2 = ⎡⎣ ra%2 ,1 − sa%2 ⎤⎦
be two Pythagorean fuzzy values, then
a%1 ≤ a%2 ⇐ ra%1 ≤ ra%2 and sa%1 ≥ sa%2 . Proof: As we know that
S ( a%1 ) = ra%21 − sa2%1 S ( a%2 ) = ra%22 − sa2%2 Since
( = (r
) ( ) + (s
S ( a%1 ) − S ( a%2 ) = ra%21 − sa2%1 − ra%22 − sa2%2 2
a%1
If ra%1 = ra%2 , this implies 2
2
− ra%22
2
a%2
− sa2%1
) ).
ra%1 = ra%2 and sa2%2 = sa2%1 , this implies that sa%2 = sa%1 .Then a%1 = a%2 . Otherwise, we
have S ( a%1 ) − S ( a%2 ) < 0,
this implies that S ( a%1 ) < S ( a%2 ) . Thus a%1 < a%2 , which completes the proof.
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4. Pythagorean Fuzzy Aggregation Operators
In this work, we present the idea of Pythagorean fuzzy weighted geometric (PFWG) operator, Pythagorean fuzzy ordered weighted geometric (PFOWG) operator and Pythagorean fuzzy hybrid geometric (PFHG) operator.
4.1 Pythagorean Fuzzy Weighted Geometric Operator Definition 4.1.1: Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) be a collection of all fuzzy Pythagorean values, and
⎣
let
⎦
PFWG : Ω n → Ω, if PFWGw ( a%1 , a%2 ,..., a%n ) = a%1w1 ⊗ a%2w2 ⊗ ... ⊕ a%nwn ,
(9)
PFWG is called Pythagorean fuzzy weighted geometric ( PFWG ) operator of dimension n , where
then
w = ( w1 , w2 ,..., wn )
T
Especially, if
is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
w j ∈ [ 0,1] and
n
∑w
j
= 1.
j =1
w = ( 1n , 1n ,..., 1n ) , then the PFWG operator is reduced to a Pythagorean fuzzy geometric T
( PFG ) operator of dimension n, which is defined as follows: PFG ( a%1 , a%2 ,..., a%n ) = ( a%1 ⊗ a%2 ⊗ ... ⊗ a%n ) n .
(10 )
1
Theorem 4.1.2:
Let a% j = ⎡ ra% j ,1 − sa% j ⎤
⎣
⎦
then their aggregated value by using the
( j = 1, 2,3,..., n )
be a collection of Pythagorean fuzzy values,
PFWG operator is also a Pythagorean fuzzy value, and
⎡ n w n PFWGw ( a%1 , a%2 ,..., a%n ) = ⎢∏ra% j j , ∏ 1 − sa% j j =1 ⎣ j =1
(
where w = ( w1 , w2 ,..., wn )
T
n
∑w
j
)
wj
⎤ ⎥, ⎦
(11)
is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
w j ∈ [ 0,1] and
= 1.
j =1
(11) holds for all n . First we show that
Proof: By mathematical induction we can prove that equation equation (11)
holds for n = 2. Since
(
)
(
)
a%1w1 = ⎡ ra%w11 , 1 − sa%1 ⎢⎣ a%2w2 = ⎡ ra%w22 , 1 − sa%2 ⎣⎢
⎤ ⎥⎦ w2 ⎤. ⎦⎥
w1
Then
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PFWGw ( a%1 , a%2 ) = a%1w1 ⊕ a%2w2
(
= ⎡ ra%w11 , 1 − sa%1 ⎣⎢
)
w1
(
(
⎤ ⊗ ⎡ r w2 , 1 − s a% 2 ⎦⎥ ⎣⎢ a%2
) (1 − s )
= ⎡ ra%w11 ra%w22 , 1 − sa%1 ⎢⎣
w1
⎡ 2 w 2 = ⎢∏ra% j j , ∏ 1 − sa% j j =1 ⎣ j =1
(
(11)
Thus equation
holds for n = 2. Now we show that
w2
a%2
)
wj
(11)
holds for
n = k , then we show that
(11)
⎤ ⎦⎥
⎤ ⎥⎦
holds for n = k .i.e.,
(
(11)
w2
⎤ ⎥. ⎦
⎡ k wj k % % % PFWGw ( a1 , a2 ,..., ak ) = ⎢∏ra% j , ∏ 1 − sa% j j =1 ⎣ j =1 If equation
)
)
wj
⎤ ⎥ ⎦
holds for n = k + 1
PFWGw ( a%1 , a%2 ,..., a%k +1 ) = a%1w1 ⊗ a%2w2 ⊗ ... ⊗ a%kw+k1+1
a%1w1 ⊗ a%2w2 ⊗ ... ⊗ a%kw+k1+1 wj ⎤ wk +1 ⎡ k w k ⎤ = ⎢∏ra% j j , ∏ 1 − sa% j ⎥ ⊗ ⎡ ra%wkk++11 , 1 − sa%k +1 ⎢ ⎥⎦ ⎣ j =1 ⎣ j =1 ⎦ k wj wk +1 ⎤ ⎡ k w = ⎢∏ra% j j .ra%wkk++11 , ∏ 1 − sa% j . 1 − sa%k +1 ⎥ j =1 ⎣ j =1 ⎦ wj ⎤ ⎡ k +1 w k +1 = ⎢∏ra% j j , ∏ 1 − sa% j ⎥ . j =1 ⎣ j =1 ⎦
(
)
(
(
(
Since equation
) (
)
)
(11) holds for n = k + 1. Therefore equation (11) holds for all n .
Example 4.1.3:
Let
a%1 = [ 0.4, 0.6] ,
a%2 = [ 0.5, 0.7 ] ,
Pythagorean fuzzy values, and w = ( 0.1, 0.2, 0.3, 0.4 )
T
rα1 = 0.4,
)
rα 2 = 0.5,
rα3 = 0.6,
a%3 = [ 0.6, 0.8] ,
a%4 = [ 0.7, 0.9] be four
be the weighted vector of a% j ( j = 1, 2,3, 4 ) ,
then
rα 4 = 0.7 , and 1 − sa%1 = 0.6 ⇒ sa%1 = 0.4 1 − sa%2 = 0.7 ⇒ sa%2 = 0.3 1 − sa%3 = 0.8 ⇒ sa%3 = 0.2 1 − sa%4 = 0.9 ⇒ sa%4 = 0.1.
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⎡ 4 w 4 PFWGw ( a%1 , a%2 , a%3 , a%4 ) = ⎢∏ra% j j , ∏ 1 − sa% j j =1 ⎣ j =1
(
)
wj
⎤ ⎥ ⎦
⎡( 0.4 )0.1 ( 0.5 )0.2 ( 0.6 )0.3 ( 0.7 )0.4 , ⎤ ⎥ =⎢ 0.1 0.2 0.3 0.4 ⎢⎣ ( 0.6 ) ( 0.7 ) ( 0.8 ) ( 0.9 ) ⎥⎦ ⎡( 0.9124 )( 0.8705 )( 0.8579 )( 0.8670 ) , ⎤ =⎢ ⎥ ⎣ ( 0.9505 )( 0.9311)( 0.9352 )( 0.9587 ) ⎦ = [ 0.5907, 0.7934] . Theorem 4.1.4:
T
n
j =1
j
( j = 1, 2,3,..., n )
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦
w = ( w1 , w2 ,..., wn )
and
∑w
Let
be a collection of Pythagorean fuzzy values
a% j ( j = 1, 2,3,..., n )
is the weighted vector of
with
w j ∈ [ 0,1]
and
= 1. If all a% j ( j = 1, 2,3,..., n ) are equal, i.e., a% j = a% , for all j , then PFWGw ( a%1 , a%2 ,..., a%n ) = a%.
(12 )
Proof: As we know that
PFWGw ( a%1 , a%2 ,..., a%n ) = a%1w1 ⊗ a%2w2 ⊗ ... ⊕ a%nwn . Let
a% j ( j = 1, 2,3,..., n ) = a%. Then
PFWGw ( a%1 , a%2 ,..., a%n ) = a% w1 ⊗ a% w2 ⊗ ... ⊕ a% wn n
∑
wj
= a% = a%.
j =1
Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) be a collection of Pythagorean fuzzy values and
Theorem 4.1.5:
⎣
w = ( w1 , w2 ,..., wn )
T
⎦
is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
w j ∈ [ 0,1] and
n
∑w
j
= 1.
j =1
If
( ) ( ) = ⎡ max ( r ) ,1 − min ( s ) ⎤ . ⎣⎢ ⎦⎥
a% − = ⎡ min ra% j ,1 − max sa% j ⎤ j ⎣⎢ j ⎦⎥ a% +
j
a% j
j
a% j
Then
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a% − ≤ PFWGw ( a%1 , a%2 ,..., a%n ) ≤ a% + , for all w.
(13)
Proof: As we know that
( ) (14) min ( s ) ≤ s ≤ max ( s ) . (15) (14 ) , we have min ( r ) ≤ r ≤ max ( r ) ⇔ min ( r ) ≤ r ≤ max ( r ) ⇔ min ( r ) ≤ r ≤ max ( r ) ⇔ ∏ min ( r ) ≤ ∏r ≤ ∏ max ( r ) j
( )
j
a% j
min ra% j ≤ ra% ≤ max ra% .
From equation
j
a% j
j
j
a% j
j
a% j
a% j
a% j
a% j
j
wj
wj
wj
a% j
j
j
a% j
a% j
j
j
a% j
n
a% j
j
wj
n
n
wj
wj
a% j
j
j =1
a% j
j =1
a% j
j
j =1
n
n
∑
( ) ⇔ min ( r ) ≤ ∏r ⇔ min ra% j
wj
j =1
j
≤ ∏ra% j ≤ max ra% w
a% j
Now from equation
(15) ,
wj
j =1
j
j
( )
(16 )
( ) ≤ max ( s )
j
( )
min j s a% ≤ s a% ≤ max j s a%
we have
∑
≤ max ra% .
wj
a% j
j =1
j
j
j =1
n
j
( )
n
j
j
j
( )≤s ⇔ 1 − max ( s ) ≤ 1 − s ≤ 1 − min ( s ) ⇔ (1 − max ( s ) ) ≤ (1 − s ) ≤ (1 − min ( s ) ) ⇔ ∏ (1 − max ( s ) ) ≤ ∏ (1 − s ) ≤ ∏ (1 − min ( s ) ) ⇔ min s a% j
a% j
j
a% j
j
a% j
j
a% j
wj
a% j
n
wj
a% j
j
j =1
wj
wj
a% j
j
a% j
j
a% j
j
n
n
wj
a% j
j =1
j
j =1
n
a% j n
( ( ))∑ ≤ ∏ (1 − s ) ≤ (1 − min ( s ))∑ ⇔ (1 − max ( s ) ) ≤ ∏ (1 − s ) ≤ (1 − min ( s ) ) ⇔ 1 − max ( s ) ≤ ∏ (1 − s ) ≤ 1 − min ( s ) . ⇔ 1 − max s a% j
wj
n
j =1
j =1
j
a% j
j =1
a% j
a% j
a% j
j =1
j =1
j
a% j
wj
n
j
wj
wj
n
j
wj
a% j
j
wj
a% j
j
a% j
(17 )
Let
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PFWGw ( a%1 , a%2 ,..., a%n ) = [ ra% ,1 − sa% ] . Then
S ( a% ) = ra%2 − sa2% 2
≤ ⎡ max ( ra% ) ⎤ − ⎡ min ( sa% ) ⎤ ⎥⎦ ⎢⎣ j ⎥⎦ ⎢⎣ j = S ( a% + ) .
2
(18)
Again
S ( a% ) = ra%2 − sa2% 2
≥ ⎡ min ( ra% ) ⎤ − ⎡ max ( sa% ) ⎤ ⎣⎢ j ⎦⎥ ⎢⎣ j ⎦⎥ = S ( a% − ) . From
(18)
and
(19 ) ,
2
(19 )
we have
a% − < PFWGw ( a%1 , a%2 ,..., a%n ) < a% + .
( 20 )
If
S ( a% ) = S ( a% + ) .
( 21)
Then
( )
( )
2
⇔ ra%2 − sa2% = ⎡ max ra% j ⎤ − ⎡ min sa% j ⎤ ⎥⎦ ⎢⎣ j ⎦⎥ ⎣⎢ j
( ) ⇔ r = max ( r ) , s
( )
2
⇔ ra%2 = ⎡ max ra% j ⎤ , sa2% = ⎡ min sa% j ⎤ ⎢⎣ j ⎥⎦ ⎢⎣ j ⎥⎦ a%
j
a% j
a%
2
2
( )
= min sa% j . j
Since
H ( a% ) = ra%2 + sa2%
( )
2
( )
= ⎡ max ra% j ⎤ + ⎡ min sa% j ⎤ ⎥⎦ ⎢⎣ j ⎥⎦ ⎢⎣ j = H ( a% + ) .
2
Thus
PFWGw ( a%1 , a%2 ,..., a%n ) = a% + .
( 22 )
If
S ( a% ) = S ( a% − ) . Then
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( )
( )
2
⇔ r − s = ⎡ min sa% j ⎤ − ⎡ max ra% j ⎤ ⎢⎣ j ⎥⎦ ⎢⎣ j ⎥⎦ 2 a%
2 a%
( ) ⇔ r = min ( s ) , s
( )
2
⇔ ra%2 = ⎡ min sa% j ⎤ , sa2% = ⎡ max ra% j ⎤ ⎥⎦ ⎢⎣ j ⎣⎢ j ⎦⎥ a%
a% j
j
a%
2
2
( )
= max ra% j . j
Since
H ( a% ) = ra%2 + sa2%
( )
( )
2
= ⎡ min sa% j ⎤ + ⎡ max ra% j ⎤ ⎣⎢ j ⎦⎥ ⎢⎣ j ⎦⎥ − = H ( a% ) .
2
Thus
PFWGw ( a%1 , a%2 ,..., a%n ) = a% − . Thus from equation
( 20 )
to
( 23)
( 23)
, we have
a% − ≤ PFWGw ( a%1 , a%2 ,..., a%n ) ≤ a% + , for all w.
Theorem 4.1.6:
Let
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦ ( j = 1, 2,3,..., n ) and a% ∗j = ⎡⎢ ra%∗ ,1 − sa%∗ ⎤⎥ j ⎦ ⎣ j
be a collection of Pythagorean fuzzy values. If ra% j ≤ ra%∗
sa% j ≥ sa%∗ , then
and
j
j
PFWGw ( a%1 , a%2 ,..., a%n ) ≤ PFWGw ( a%1∗ , a%2∗ ,..., a%n∗ ) . Proof: As we know that, ra% j ≤ ra%∗ j
( j = 1, 2,3,..., n )
( 24 )
and sa% j ≥ sa% ∗ . Then j
⇔ ra% j ≤ ra%∗ j
⇔ ra% j ≤ ra%∗ wj
wj j
n
n
( 25)
⇔ ∏ra% j j ≤ ∏ra%∗j . w
j =1
w
j =1
j
And
⇔ sa%∗ ≤ sa% j j
⇔ 1 − sa% j ≤ 1 − sa%∗ j
) ( ) ⇔ ∏ (1 − s ) ≤ ∏ (1 − s ) (
⇔ 1 − sa% j
wj
wj
j
n
j =1
≤ 1 − sa%∗ n
wj
a% j
j =1
188
a% ∗j
wj
.
( 26 )
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From
( 25 )
and
( 26 ) ,
we have
n
n
(
∏ra% jj + ∏ 1 − sa% j w
j =1
j =1
) ≤ ∏r + ∏ (1 − s ) wj
n
n
wj
j =1
a% ∗j
j =1
a% ∗j
wj
.
( 27 )
Let
a% = PFWGw ( a%1 , a%2 ,..., a%n ) .
( 28 )
a% ∗ = PFWGw ( a%1∗ , a%2∗ ,..., a%n∗ ) .
( 29 )
And
Then by equation
( 27 ) ,
we have
S ( a% ) ≤ S ( a% ∗ ) . If
S ( a% ) < S ( a% ∗ ) . Then
PFWGw ( a%1 , a%2 ,..., a%n ) < PFWGw ( a%1∗ , a%2∗ ,..., a%n∗ ) .
( 30 )
If
S ( a% ) = S ( a% ∗ ) Then
⇔ ra%2 − sa2% = ra%2∗ − sa2%∗ ⇔ ra%2 = ra%2∗ , sa2% = sa2%∗ ⇔ ra% = ra%∗ , sa% = sa%∗ . Since
H ( a% ) = ra%2 + sa2% = ra%2∗ + sa2%∗
= H ( a% ∗ ) . Thus
H ( a% ) = H ( a% ∗ ) . PFWGw ( a%1 , a%2 ,..., a%n ) = PFWGw ( a%1∗ , a%2∗ ,..., a%n∗ ) . Thus from equation
( 30 )
and
( 31) ,
( 31)
we have
PFWGw ( a%1 , a%2 ,..., a%n ) ≤ PFWGw ( a%1∗ , a%2∗ ,..., a%n∗ ) .
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4.2 Pythagorean Fuzzy Ordered Weighted Geometric Operator Definition 4.2.1: Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) be a collection of Pythagorean fuzzy values. Then a
⎣
⎦
Pythagorean fuzzy ordered weighted geometric
( PFOWG )
n
operator of dimension
is a mapping
PFOWG : Ω n → Ω, that has an associated vector w = ( w1 , w2 ,..., wn ) , such that w j ∈ [ 0,1] and T
n
∑w
j
= 1. Furthermore
j =1
( ) ⊗ ( a% ( ) )
PFOWGw ( a%1 , a%2 ,..., a%n ) = a%σ (1) where
(σ (1) , σ ( 2 ) ,..., σ ( n ) )
Especially, if
w = ( 1n , 1n ,..., 1n )
T
w1
w2
σ 2
(
(1, 2,..., n )
is a permutation of
)
⊗ ... ⊗ a%σ ( n )
wn
( 32 )
,
a%σ ( j −1) ≥ a%σ ( j ) for all
such that
, then the Pythagorean fuzzy ordered weighted geometric
operator is reduced to the Pythagorean fuzzy geometric
( PFG )
j,
( PFOWG )
operator of dimension n .
Theorem 4.2.2: Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) be a collection of Pythagorean fuzzy values; then
⎣
⎦
their aggregated value by using the Pythagorean fuzzy ordered weighted geometric
( PFOWG )
operator is
also Pythagorean fuzzy value, and n ⎡ n w PFOWGw ( a%1 , a%2 ,..., a%n ) = ⎢∏ra%σj( j ) , ∏ 1 − sa%σ ( j ) j =1 ⎣ j =1
(
where w = ( w1 , w2 ,..., wn )
T
n
∑w
j
is the weighted vector of
)
wj
⎤ ⎥, ⎦
( 33)
a% j ( j = 1, 2,3,..., n )
with
w j ∈ [ 0,1]
and
= 1.
j =1
Proof: The proof is similar to the Theorem 4.1.2.
[
]
[
]
[
]
[
Example 4.2.3: Let a%1 = 0.2, 0.9 , a%2 = 0.3, 0.8 , a%3 = 0.4, 0.6 , a%4 = 0.5, 0.7 fuzzy
values,
and
w = ( 0.1, 0.2, 0.3, 0.4 )
T
be
the
weighted
vector
of
]
be four Pythagorean
a% j ( j = 1, 2,3, 4 ) , then
rα1 = 0.2, rα 2 = 0.3, rα3 = 0.4, rα 4 = 0.5 , and
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1 − sa%1 = 0.9 ⇒ sa%1 = 0.1 1 − sa%2 = 0.8 ⇒ sa%2 = 0.2 1 − sa%3 = 0.6 ⇒ sa%3 = 0.4 1 − sa%4 = 0.7 ⇒ sa%4 = 0.3 Now we calculate the scores of a% j ( j = 1, 2,3, 4 ) ,
( ) = ( 0.2 ) − ( 0.1) S ( ) = ( 0.3) − ( 0.2 ) S ( ) = ( 0.4 ) − ( 0.4 ) S ( ) = ( 0.5 ) − ( 0.3) S ( ) > S ( ) > S ( ) > S ( ) . Thus S
2
2
= 0.04 − 0.01 = 0.03
2
2
= 0.09 − 0.04 = 0.05
2
2
= 0.16 − 0.16 = 0
2
2
= 0.25 − 0.09 = 0.16
a%1
a%2
a%3
a%4
Since
a% 4
a%2
a%1
a%3
a%σ (1) = [ 0.5, 0.7 ] , a%σ ( 2) = [ 0.3, 0.8] , a%σ ( 3) = [ 0.2, 0.9] , a%σ ( 4) = [ 0.4, 0.6] 4 ⎡ 4 w PFOWGw ( a%1 , a%2 , a%3 , a%4 ) = ⎢∏ra%σj( j ) , ∏ 1 − sa%σ ( j ) j =1 ⎣ j =1
(
)
wj
⎤ ⎥ ⎦
⎡( 0.5 )0.1 ( 0.3)0.2 ( 0.2 )0.3 ( 0.4 )0.4 , ⎤ ⎥ =⎢ 0.1 0.2 0.3 0.4 ⎢⎣ ( 0.7 ) ( 0.8 ) ( 0.9 ) ( 0.6 ) ⎥⎦ ⎡( 0.9330 )( 0.7860 )( 0.6170 )( 0.6931) , ⎤ =⎢ ⎥ ⎣ ( 0.9649 )( 0.9563)( 0.9688 )( 0.8151) ⎦ = [ 0.3136, 0.7286] . Theorem 4.2.4:
T
n
j =1
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦
w = ( w1 , w2 ,..., wn )
and
∑w
Let
j
( j = 1, 2,3,..., n )
is the weighted vector of
be a collection of Pythagorean fuzzy values
a% j ( j = 1, 2,3,..., n )
= 1. If all a% j ( j = 1, 2,3,..., n ) are equal, i.e., a% j = a% , for all PFOWGw ( a%1 , a%2 ,..., a%n ) = a%.
with
w j ∈ [ 0,1]
and
j . Then we have the following.
( 34 )
Proof: The proof is similar to the Theorem 4.1.4. Theorem 4.2.5:
Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) be a collection of Pythagorean fuzzy values and
⎣
w = ( w1 , w2 ,..., wn )
T
⎦
is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
w j ∈ [ 0,1] and
n
∑w
j
= 1.
j =1
If
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( ) ( ) = ⎡ max ( r ) ,1 − min ( s ) ⎤ ⎣⎢ ⎦⎥
a% − = ⎡ min ra% j ,1 − max sa% j ⎤ j ⎣⎢ j ⎦⎥ a% +
a% j
j
a% j
j
Then
a% − ≤ PFOWGw ( a%1 , a%2 ,..., a%n ) ≤ a% + , for all w.
( 35)
Proof: The proof is similar to the Theorem 4.1.5. Theorem 4.2.6:
∗ Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,3,..., n ) and a% j = ⎡ ra%∗ ,1 − sa%∗ ⎤ ( j = 1, 2,3,..., n )
be
w = ( w1 , w2 ,..., wn )
a% j
⎣
⎢⎣
⎦
T
the collection of Pythagorean fuzzy values and n
∑w
, a% ∗j ( j = 1, 2,3,..., n ) with w j ∈ [ 0,1] and
j =1
j
j
j
⎥⎦
is the weighted vector of
= 1, if ra% ≤ ra%∗ and s a% ≥ sa%∗ for all j
j
j
j , then
j
PFOWGw ( a%1 , a%2 ,..., a%n ) ≤ PFOWGw ( a%1∗ , a%2∗ ,..., a%n∗ ) for every w.
( 36 )
Proof: The proof is similar to the Theorem 4.1.6. Theorem 4.2.7:
Let a% j = ⎡ ra% j ,1 − sa% j ⎤ ( j = 1, 2,..., n )
⎣
⎡ ⎢⎣
and a% j = ra%´ ,1 − s ´ ´
⎦
a% j
j
⎤ ⎥⎦
( j = 1, 2,..., n )
be
two collection of Pythagorean fuzzy values, then
PFOWGw ( a%1 , a%2 ,..., a%n ) = PFOWGw ( a%1´ , a%2´ ,..., a%n´ ) , where
( a% , a% ,..., a% ) ´ 1
´ 2
´ n
is any permutation of
( 37 )
( a%1 , a%2 ,..., a%n ) .
Proof: As we know that
( ) ⊗ ( a% ( ) )
PFOWGw ( a%1 , a%2 ,..., a%n ) = a%σ (1)
w1
w2
σ 2
(
)
(
)
⊗ ... ⊗ a%σ ( n )
wn
.
( 38)
.
( 39 )
And
( ) ⊗ ( a% ( ) )
PFOWGw ( a%1´ , a%2´ ,..., a%n´ ) = a%σ´ (1) Since
(α ,α ,...α ) ´ 1
´ 2
´ n
w1
´
w2
σ 2
is a permutation of ( a%1 , a%2 ,..., a%n )
⊗ ... ⊗ a%σ´ ( n )
wn
, then we have
a%σ ( j ) = a%σ´ ( j ) ( j = 1, 2,3,..., n ) . Thus equation
(37), always holds.
Theorem 4.2.8:
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦ ( j = 1, 2,3,..., n )
Let
w = ( w1 , w2 , w3 ,..., wn )
T
is
the
weighted
vector
192
of
be a collection of the
PFOWG
PFVs
operator
such
and that
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n
w j ∈ [ 0,1] ( j = 1, 2,3,..., n)
and
∑w
= 1, then
j
j =1
, then
PFOWG ( a%1 , a%2 ,..., a%n ) = max j {a% j }
, then
PFOWG ( a%1 , a%2 ,..., a%n ) = min j {a% j }
(1)
If w = (1, 0, 0,..., 0 )
( 2)
If w = ( 0, 0, 0,...,1)
( 3)
If w j = 1 and wi = 0 ( i ≠ j ) , then
T
T
largest of a%
σ ( j)
PFOWG ( a%1 , a%2 ,..., a%n ) = a% σ ( j ) where a% σ ( j ) is the jth
(i = 1, 2,3,..., n).
Proof: Straightforward.
4.3 Pythagorean Fuzzy Hybrid Geometric (PFHG) Operator
( PFHG ) of dimension n is a mapping
Definition 4.3.1: Pythagorean fuzzy hybrid geometric operator
PFHG : Ω n → Ω, which has an associated vector w = ( w1 , w2 ,..., wn ) , such that w j ∈ [ 0,1] and T
n
∑w
j
= 1. Furthermore
j =1
w1
w2
wn
⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ PFHGw, w ( a%1 , a%2 ,..., a%n ) = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ ... ⊗ ⎜ a% σ ( n ) ⎟ , ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⋅
where
a% σ ( j )
is the
jth
w = ( w1 , w2 ,..., wn )
T
is
the
weighted n
∑w
and
, a%2nw2 ,..., a%nnwn
)
T
goes to ⋅
⎡ ⎢⎣
j
vector
of
a% j ( j = 1, 2,..., n )
such
that
= 1, and n is the balancing coefficient, which plays a role of
j =1
balance (in such a case, if the vector nw1 1
⋅ nw ⎞ ⎛⋅ a% σ ( j ) ⎜ a% σ ( j ) = a% j j ⎟ ⎝ ⎠
largest of the weighted Pythagorean fuzzy values
w j ∈ [ 0,1] , ( j = 1, 2,3,..., n)
( a%
( 40 )
T
( a%1 , a%2 ,..., a%n )
T
Theorem 4.3.2: Let a% j = r⋅ ,1 − s ⋅ a% j
( w1 , w2 ,..., wn )
a% j
goes to
( 1n , 1n ,..., 1n )
T
,
then the vector
.
⎤ j = 1, 2,3,..., n be a collection of Pythagorean fuzzy values; then ) ⎥⎦ (
their aggregated value by using the Pythagorean fuzzy hybrid geometric operator
( PHWG )
operator is also a
Pythagorean fuzzy value, and wj wj n ⎡ n ⎤ PFHGw, w ( a%1 , a%2 ,..., a%n ) = ⎢∏ ⎛⎜ r⋅ ⎞⎟ , ∏ ⎛⎜1 − s ⋅ ⎞⎟ ⎥ . a% j ⎠ j =1 ⎝ ⎣ j =1 ⎝ a% j ⎠ ⎦
193
( 41)
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where n
∑w
j
w = ( w1 , w2 ,..., wn )
T
is the weighted vector of
a% j ( j = 1, 2,3,..., n )
with
w j ∈ [ 0,1]
and
= 1.
j =1
Proof: By mathematical induction we can prove that equation equation (41)
(41) holds for all n . First we show that
holds for n = 2. Since w1 w1 ⎡ w1 ⎛ ⎞ ⎤ ⎛ %⋅ ⎞ ⎜ aσ (1) ⎟ = ⎢ ra%⋅ , ⎜1 − sa%⋅ ⎟ ⎥ . σ (1) ⎠ ⎝ ⎠ ⎢⎣ σ (1) ⎝ ⎥⎦
⎛ %⋅ ⎞ ⎜ aσ ( 2 ) ⎟ ⎝ ⎠
w2
w2 ⎡ w2 ⎛ ⎞ ⎤ = ⎢ r⋅ , ⎜ 1 − s ⋅ ⎟ ⎥ . a%σ ( 2 ) ⎠ ⎣⎢ a%σ ( 2) ⎝ ⎦⎥
Then w1
w2
⎛⋅ ⎞ ⎛⋅ ⎞ PFHGw, w ( a%1 , a%2 ) = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⎝ ⎠ ⎝ ⎠ w1 ⎡ ⎛ ⎞ ⎤ = ⎢ r⋅w1 , ⎜1 − s ⋅ ⎟ ⎥ ⊗ a%σ (1) ⎠ ⎣⎢ a%σ (1) ⎝ ⎦⎥ w2 ⎡ ⎛ ⎞ ⎤ = ⎢ r⋅w2 , ⎜1 − s ⋅ ⎟ ⎥ a%σ ( 2) ⎠ ⎢⎣ a%σ ( 2) ⎝ ⎥⎦ r⋅w1 r⋅w2 , ⎡ ⎤ a%σ (1) a%σ ( 2 ) ⎢ ⎥ w1 w2 ⎥ =⎢ ⎢⎛⎜1 − s ⋅ ⎞⎟ ⎛⎜1 − s ⋅ ⎞⎟ ⎥ a%σ (1) ⎠ ⎝ a%σ ( 2) ⎠ ⎥ ⎢⎣⎝ ⎦ wj ⎡ 2 wj 2 ⎛ ⎞ ⎤ = ⎢∏r⋅ , ∏ ⎜1 − s ⋅ ⎟ ⎥ a%σ ( j ) ⎠ ⎣⎢ j =1 a%σ ( j ) j =1 ⎝ ⎦⎥
Thus equation
( 41)
holds for n = 2. Now we show that
( 41)
holds for
n = k .i.e.,
wj wj k ⎡ k ⎛ ⎤ ⎞ ⎛ ⎞ PFHGw, w ( a%1 , a%2 ,..., a%k ) = ⎢∏ ⎜ r⋅ ⎟ , ∏ ⎜1 − s ⋅ ⎟ ⎥ a%σ ( j ) ⎠ j =1 ⎝ ⎣⎢ j =1 ⎝ a%σ ( j ) ⎠ ⎦⎥
If equation
( 41)
holds for n = k , then we show that
194
( 41)
holds for n = k + 1
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w1
w2
⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ PFHGw, w ( a%1 , a%2 ,..., a%k +1 ) = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ ... ⊗ ⎜ a% σ ( k +1) ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ wj wj k ⎡ k ⎛ ⎞ ⎛ ⎞ ⎤ = ⎢ ∏ ⎜ r⋅ ⎟ , ∏ ⎜ 1 − s ⋅ ⎟ ⎥ ⊗ a%σ ( j ) ⎠ j =1 ⎝ ⎢⎣ j =1 ⎝ a%σ ( j ) ⎠ ⎥⎦ wk +1 wk +1 ⎡⎛ ⎞ ⎛ ⎞ ⎤ = ⎢ ⎜ r⋅ ⎟ , ⎜ 1 − sa%⋅ σ k +1 ⎟ ⎥ ( ) ⎠ ⎝ ⎣⎢⎝ a%σ ( k +1) ⎠ ⎦⎥
wk +1
wk +1 wj ⎡ k ⎛ ⎤ ⎞ ⎛ ⎞ , ⎥ ⎢ ∏ ⎜ r⋅ ⎟ . ⎜ r⋅ ⎟ a% ⎝ a%σ ( k +1) ⎠ j =1 ⎝ σ ( j ) ⎠ ⎥ = ⎢⎢ w k +1 ⎥ k w ⎞ ⎥ ⎢∏ 1 − sa% j . ⎛⎜1 − s ⋅ ⎟ ⎥ j a%σ ( k +1) ⎠ ⎢⎣ j =1 ⎝ ⎦ w w j j k +1 ⎡ k +1 ⎛ ⎞ ⎛ ⎞ ⎤ = ⎢ ∏ ⎜ r⋅ ⎟ , ∏ ⎜ 1 − s ⋅ ⎟ ⎥. a%σ ( k +1) ⎠ j =1 ⎝ ⎢⎣ j =1 ⎝ a%σ ( j ) ⎠ ⎥⎦
(
)
Thus k 1
PFHG w,w à ã 1 , ã 2 , . . . , ã k1 Ä
wj
r
ã HÃjÄ
j 1
Since equation (41)
holds for
k 1
wj
, 1 "s
ã HÃjÄ
.
j 1
n = k + 1. Therefore equation (41) holds for all n , which completes
the proof. Example 4.3.3: Let
a%1 = [ 0.3, 0.7 ] ,
w = ( 0.1, 0.2, 0.3, 0.4 )
T
ra%3 = 0.7,
a%2 = [ 0.5, 0.6] ,
be the weighted vector of
a%3 = [ 0.7, 0.8] ,
a% j ( j = 1, 2,3, 4 ) .
a%4 = [ 0.4, 0.9] , and let
Then
ra%1 = 0.3,
ra%2 = 0.5,
ra%4 = 0.4 and 1 − sa%1 = 0.7 ⇒ sa%1 = 0.3 1 − sa%2 = 0.6 ⇒ sa%2 = 0.4 1 − sa%3 = 0.8 ⇒ sa%3 = 0.2 1 − sa%4 = 0.9 ⇒ sa%4 = 0.1
By the operational law (2), we get the weighted Pythagorean fuzzy values as follows:
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⋅
a%1 = ⎡( 0.3) ⎣
, ( 0.7 )
4×0.2
, ( 0.6 )
4×0.2
⎤ = [ 0.5743, 0.6645] ⎦
4×0.3
, ( 0.8 )
4×0.3
⎤ = [ 0.6518, 0.7650] ⎦
4×0.4
, ( 0.9 )
4×0.4
⎤ = [ 0.2308, 0.8448] . ⎦
⋅
a% 2 = ⎡( 0.5 ) ⎣ ⋅
a% 3 = ⎡( 0.7 ) ⎣ ⋅
⎤ = [ 0.6178, 0.8670] ⎦
4×0.1
a% 4 = ⎡( 0.4 ) ⎣
4×0.1
⋅
Now we calculate the scores of a% j ( j = 1, 2,3, 4 ) 2 2 ⎛⋅ ⎞ S ⎜ a%1 ⎟ = ( 0.6178 ) − (1 − 0.8670 ) = 0.364 ⎝ ⎠ 2 2 ⎛⋅ ⎞ S ⎜ a% 2 ⎟ = ( 0.5743) − (1 − 0.6645 ) = 0.2173 ⎝ ⎠ 2 2 ⎛⋅ ⎞ S ⎜ a% 3 ⎟ = ( 0.6518 ) − (1 − 0.7650 ) = 0.3696 ⎝ ⎠ 2 2 ⎛⋅ ⎞ S ⎜ a% 4 ⎟ = ( 0.2308 ) − (1 − 0.8448 ) = 0.1312 ⎝ ⎠
Since
⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ S ⎜ a% 3 ⎟ > S ⎜ a%1 ⎟ > S ⎜ a% 2 ⎟ > S ⎜ a% 4 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Then ⋅
a% σ (1) = [ 0.6518, 0.7650] ⋅
a% σ ( 2) = [ 0.6178, 0.8670] ⋅
a% σ ( 3) = [ 0.5743, 0.6645] ⋅
a% σ ( 4) = [ 0.2308, 0.8448] . Thus wj wj 4 ⎡ 4 ⎛ ⎤ ⎞ ⎛ ⎞ PFHGw, w ( a%1 , a%2 , a%3 , a%4 ) = ⎢∏ ⎜ r⋅ ⎟ , ∏ ⎜1 − s ⋅ ⎟ ⎥ a%σ ( j ) ⎠ j =1 ⎝ ⎣⎢ j =1 ⎝ a%σ ( j ) ⎠ ⎦⎥
⎡( 0.6518 )0.1 ( 0.6178 )0.2 ( 0.5743)0.3 ( 0.2308 )0.4 , ⎤ ⎥ =⎢ 0.1 0.2 0.3 0.4 ⎢⎣ ( 0.7650 ) ( 0.8670 ) ( 0.6645 ) ( 0.8448 ) ⎥⎦ ⎡( 0.9581)( 0.9081)( 0.8467 )( 0.5562 ) , ⎤ =⎢ ⎥ ⎣ ( 0.9735 )( 0.9718 )( 0.8846 ) ( 0.9347 ) ⎦ = [ 0.4097, 0.7822].
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Theorem 4.3.4: Let
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦ ( j = 1, 2,3,..., n ) be a collection of Pythagorean fuzzy values and
w = ( w1 , w2 ,..., wn )
is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
T
⋅
⋅
w j ∈ [ 0,1] and
n
∑w
j
= 1.
j =1
⋅
If all a% σ ( j ) ( j = 1, 2,3,..., n ) are equal, i.e., a% σ ( j ) = a% ,
j , then
for all ⋅
PFHGw, w ( a%1 , a%2 ,..., a%n ) = a% .
( 42 )
Proof: The proof is similar to the Theorem 4.1.4. Theorem 4.3.5:
Let
w = ( w1 , w2 ,..., wn )
T
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦ ( j = 1, 2,3,..., n ) be a collection of Pythagorean fuzzy values and is the weighted vector of a% j ( j = 1, 2,3,..., n ) with
w j ∈ [ 0,1] and
n
∑w
j
= 1.
j =1
If
( ) ( ) = ⎡ max ( r ) ,1 − min ( s ) ⎤ . ⎢⎣ ⎥⎦
a% − = ⎡ min ra%σ ( j ) ,1 − max sa%σ ( j ) ⎤ ⎢⎣ j ⎥⎦ j a% +
a%σ ( j )
j
j
a%σ ( j )
Then
a% − ≤ PFHGw, w ( a%1 , a%2 ,..., a%n ) ≤ a% + , for all w.
( 43)
Proof: The proof is similar to the Theorem 4.1.5. Theorem 4.3.6:
Let
a% j = ⎡⎣ ra% j ,1 − sa% j ⎤⎦ ( j = 1, 2,3,..., n ) and a% ∗j = ⎡⎢ ra%∗ ,1 − sa%∗ ⎤⎥ j ⎦ ⎣ j
be a collection of Pythagorean fuzzy values. If ra%
σ ( j)
≤ ra%∗
σ ( j)
and sa%
σ ( j)
( j = 1, 2,3,..., n )
≥ sa%∗ , then σ ( j)
PFHGw, w ( a%1 , a%2 ,..., a%n ) ≤ PFHGw, w ( a%1∗ , a%2∗ ,..., a%n∗ ) .
( 44 )
Proof: The proof is similar to the Theorem 4.1.6. Theorem 4.3.7:
The Pythagorean fuzzy weighted geometric
Pythagorean fuzzy hybrid geometric Proof: Let
( PFHG )
( PFWG )
operator is a special case of the
operator.
w = ( 1n , 1n , 1n ,... 1n ) , then T
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w1
w2
⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ PFHGw, w ( a%1 , a%2 ,..., a%n ) = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ ... ⊗ ⎜ a% σ ( n ) ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1
1
wn
1
⎛ ⋅ ⎞n ⎛ ⋅ ⎞n ⎛⋅ ⎞n = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ ... ⊗ ⎜ a% σ ( n ) ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1
⋅ ⋅ ⎡⋅ ⎤n = ⎢ a% σ (1) ⊗ a% σ ( 2) ⊗ ... ⊗ a% σ ( n ) ⎥ ⎣ ⎦
= ⎡⎣ a%
nw1 1
⊗ a%
nw2 2
⊗ ... ⊗ a%
nwn σn
⎤⎦
1 n
= ⎡⎣ a%1w1 ⊗ a%2w2 ⊗ ... ⊗ a%3wn ⎤⎦ = PFWGw ( a%1 , a%2 ,..., a%n ) . Theorem 4.3.8:
of the Pythagorean fuzzy hybrid geometric Proof: Let
( PFOWG )
The Pythagorean fuzzy ordered weighted geometric
( PFHG )
operator is a special case
operator.
⋅
w = ( 1n , 1n , 1n ,... 1n ) , then a% j = a% j ( j = 1, 2,3,..., n ) . Thus T
w1
w2
⎛⋅ ⎞ ⎛⋅ ⎞ ⎛⋅ ⎞ PFHGw, w ( a%1 , a%2 ,..., a%n ) = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ ... ⊗ ⎜ a% σ ( n ) ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1
1
wn
1
⎛ ⋅ ⎞n ⎛ ⋅ ⎞n ⎛⋅ ⎞n = ⎜ a% σ (1) ⎟ ⊗ ⎜ a% σ ( 2) ⎟ ⊗ ... ⊗ ⎜ a% σ ( n ) ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1
⋅ ⋅ ⎡⋅ ⎤n = ⎢ a% σ (1) ⊗ a% σ ( 2) ⊗ ... ⊗ a% σ ( n ) ⎥ ⎣ ⎦
= ⎡⎣ a%σnw(11) ⊗ a%σnw( 22 ) ⊗ ... ⊗ a%σnw( nn ) ⎤⎦ = ⎡⎣ a%σw1(1) ⊗ a%σw(22) ⊗ ... ⊗ a%σw(n3) ⎤⎦ = PFOWGw ( a%1 , a%2 ,..., a%n ) .
1 n
5. CONCLUSIONS In this study, we have presented two operational laws of Pythagorean fuzzy values, and developed some new geometric aggregation operators, including the Pythagorean fuzzy weighted geometric (PFWG) operator, the Pythagorean fuzzy ordered weighted geometric (PFOWG) operator, and Pythagorean fuzzy hybrid geometric (PFHG) operator, which extend the notion of IFWG operator and the IFOWG operator to accommodate the situations where the given arguments are Pythagorean fuzzy sets. We have also discussed some basic properties of these operators. References: [1] D. H. Hong and C. H. Choi. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. (2000)114,103-113.
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