Approaches to Pythagorean Fuzzy Geometric Aggregation Operators

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Keywords: Pythagorean fuzzy set, Pythagorean fuzzy weighted geometric ( PFWG ) operator, Pythagorean fuzzy ordered weigh...

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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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International Journal of Computer Science and Information Security (IJCSIS), Vol. 14, No. 9, September 2016

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

197

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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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[2] G. W. Wei. Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Appl Soft Comput (2010) 10, 423-431. [3] H. Bustince and P. Burillo. Vague sets are intuitionistic fuzzy sets. Fuzzy Sets Syst (1996) 79, 403-405. [4] H. Bustince, J. Kacprzyk and V. Mohedano. Intuitionistic fuzzy generators: application to intuitionistic fuzzy complementation. Fuzzy Sets Syst (2000) 114, 485-504. [5] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets Syst (1986) 20, 87-96. [6] K. Atanassov. More on intuitionistic fuzzy sets. Fuzzy Sets Syst (1989) 33, 37-46. [7] K. Atanassov and G. Gargov. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst (1989) 31, 343-349. [8] K. Atanassov. New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets Syst (1994a) 61,137-142. [9] K. Atanassov. Operators over interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst (1994b) 64, 159-174. [10] K. Atanassov. Intuitionistic Fuzzy Sets: Theory and Applications,Heidelberg: Physica-Verlag (1999). [11] K. Atanassov. Two theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst. (2000) 110, 267-269. [12] K. Atanassov, Pasi G, Yager R. R. Intuitionistic fuzzy interpretations of multi-criteria multiperson and multi-measurement tool decision making. Int J Syst Sci (2005) 36, 859-868. [14] L. A. Zadeh. Fuzzy sets. Information and Control. (1965) 8, 338-353. [15] R. E. Bellmanand and L, A. Zadeh. Decision-making in a fuzzy environment. Manage Sci (1970) 17: B-141-B-164 [16] R. R. Yager. OWA aggregation of intuitionistic fuzzy sets. Int J Gen Syst (2009) 38, 617-- 641. [17] R. R. Yager. Level sets and the representation theorem for intuitionistic fuzzy sets. Soft Comput (2010) 14, 1-7. [18] R. R. Yager. Pythagorean fuzzy subsets,In Proc. Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada (2013) 57-61. [19] R. R. Yager. Pythagorean membership grades in multi-criteria decision making. IEEE Trans Fuzzy Syst (2014) 22, 958-965. [20] S. M. Chen and J. M. Tan. Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. (1994) 67,163-172. [21] S. K. De, R. Biswas and A. R. Roy. Some operations on intuitionistic fuzzy sets. Fuzzy Sets Syst. (2000) 114, 477-484. [22] X. W. Liu. Intuitionistic Fuzzy Geometric Aggregation Operators Based on Einstein Operations. (2011) 26,1049-1075. [23] Xu Zhang and Z. S. Xu. Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets (2014) 29, 1061-1078. [24]X. Peng and Y. Yang. Some Results for Pythagorean Fuzzy Sets. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS (2015) 30, 1133-1160. [25] Z. S. Xu and R. R. Yager. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst (2006) 35, 417-433. [26] Z. S. Xu. Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst (2007)15, 1179-1187. [27] Z. S. Xu and R. R. Yager. Dynamic intuitionistic fuzzy multi-attribute decision making. Int J Approx Reason

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(2008) 48, 246-262. [28] Z. S. Xu and R. R. Yager. Intuitionistic fuzzy Bonferroni means. IEEE Trans Syst Man Cybern (2011) 41, 568--578.

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