Pythagorean trapezoidal fuzzy geometric aggregation operators based on Einstein operations and their application in group decision making

The theory of aggregation operators has an important role since in the beginning of fuzzy set theory. Einstein operations is a kind of various t-norms and t-conorms families [22, 23] can be used to accomplish the corresponding intersections and unions of IFSs, Pythagorean Einstein operations includes the Pythagorean Einstein product and the Pythagorean Einstein sum, respectively, defined as follows;

a ⊗ ε b = a . b 1 + ( 1 − a 2 ) . ( 1 − b 2 ) , a ⊕ ε b = a 2 + b 2 ( 1 + a 2 . b 2 ) , for all ( a , b ) ∈ [ 0 , 1 ] 2(7)

Definition 11. [20] Let α ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and α ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) , be two PTF numbers, and δ ≥ 0. Then, ( 1 ) α ˜ 1 ⊕ α ˜ 2 = ( [ p 1 + p 2 , q 1 + q 2 , r 1 + r 2 , s 1 + s 2 ] ; ( Ψ α ˜ 1 ) 2 + ( Ψ α ˜ 2 ) 2 - ( Ψ α ˜ 1 Ψ α ˜ 2 ) 2 , ϒ α ˜ 1 ϒ α ˜ 2 ) ,

( 2 ) α ˜ 1 ⊗ α ˜ 2 = ( [ p 1 p 2 , q 1 q 2 , r 1 r 2 , s 1 s 2 ] ; Ψ α ˜ 1 Ψ α ˜ 2 , ( ϒ α ˜ 1 ) 2 + ( ϒ α ˜ 2 ) 2 - ( ϒ α ˜ 1 ϒ α ˜ 2 ) 2 ) ,

( 3 ) δ α ˜ = ( [ δ p , δ q , δ r , δ s ] ; 1 - ( 1 - Ψ α ˜ 2 ) δ ; ( ϒ α ˜ ) δ ) ,

( 4 ) α ˜ δ = ( [ p δ , q δ , r δ , s δ ] ; Ψ α ˜ δ , 1 - ( 1 - ϒ α ˜ 2 ) δ ) .

In the following, we define some operational laws of Pythagorean trapezoidal fuzzy numbers, based on Einstein operations.

Definition 12. Let α ˜ = ( [ p , q , r , s ] ; Ψ α ˜ , ϒ α ˜ ) ,

α ˜ 1 = ( [ p 1 , q 1 , r 1 , s 1 ] ; Ψ α ˜ 1 , ϒ α ˜ 1 ) , and

α ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) , be three Pythagorean trapezoidal fuzzy numbers, and δ ≥ 0. Then, ( 1 ) α ˜ 1 ⊕ ɛ α ˜ 2 = ( [ p 1 + p 2 , q 1 + q 2 , r 1 + r 2 , s 1 + s 2 ] ; ( Ψ α ˜ 1 ) 2 + ( Ψ α ˜ 2 ) 2 1 + ( Ψ α ˜ 1 ) 2 ( Ψ α ˜ 2 ) 2 , ϒ α ˜ 1 ϒ α ˜ 2 1 + ( 1 - ϒ α ˜ 1 2 ) ( 1 - ϒ α ˜ 2 2 ) ) , ( 2 ) α ˜ 1 ⊗ ɛ α ˜ 2 = ( [ p 1 p 2 , q 1 q 2 , r 1 r 2 , s 1 s 2 ] ; Ψ α ˜ 1 Ψ α ˜ 2 1 + ( 1 - Ψ α ˜ 1 2 ) ( 1 - Ψ α ˜ 2 2 ) , ( ϒ α ˜ 1 ) 2 + ( ϒ α ˜ 2 ) 2 1 + ( ϒ α ˜ 1 ) 2 ( ϒ α ˜ 2 ) 2 , ) , ( 4 ) α ˜ ∧ ɛ δ = ( [ p δ , q δ , r δ , s δ ] ; 2 ( Ψ α ˜ 2 ) δ ( 2 - Ψ α ˜ ) δ + ( Ψ α ˜ ) δ , ( 1 + ϒ α ˜ 2 ) δ - ( 1 - ϒ α ˜ 2 ) δ ( 1 + ϒ α ˜ 2 ) δ + ( 1 - ϒ α ˜ 2 ) δ ) .

Theorem 1. Let α ˜ 1 , α ˜ 2 and α ˜ be three Pythagorean trapezoidal fuzzy numbers n, m be any positive real number. Then, ( 1 ) α ˜ 1 ⊕ α ˜ 2 . ( 2 ) α ˜ 1 ⊗ α ˜ 2 . ( 3 ) δ ∈ α ˜ . ( 4 ) α ˜ ∧ ɛ δ , are also Pythagorean trapezoidal fuzzy numbers.

Proof: (1) and (2) result are obvious.

(3) Let n be any positive integer and α ˜ is Pythagorean trapezoidal fuzzy number. Then, n ∈ α ˜ = α ˜ ⊕ ε α ˜ ⊕ ε ... ⊕ ε α ˜ ︷ n(8)

By using mathematical induction we prove that Equation (8) holds for all positive integers n. First, we show that Equation (8) holds for n = 2. Since α ˜ + α ˜ = ( [ p + p , q + q , r + r , s + s ] ; ( ψ α ˜ ) 2 + ( ψ α ˜ ) 2 1 + ( ψ α ˜ ) 2 + ( ψ α ˜ ) 2 , ϒ α ˜ ϒ α ˜ 1 + ( 1 − ϒ α ˜ 2 ) ( 1 − ϒ α ˜ 2 ) ) ; = ( [ 2 p , 2 q , 2 r , 2 s ] ; ( 1 + ψ α ˜ 2 ) 2 − ( 1 − ψ α ˜ 2 ) 2 ( 1 + ψ α ˜ 2 ) 2 + ( 1 − ψ α ˜ 2 ) 2 , 2 ( ϒ α ˜ 2 ) 2 ( 2 − ϒ α ˜ ) 2 + ( ϒ α ˜ ) 2 ) = 2 ε ˙ α ˜ . Therefore, the Equation (8) holds for n = 2. If Equation (8) holds for n = k. Then, α ˜ 1 ∧ ε m = α ˜ 1 ⊗ ε α ˜ 1 ⊗ ε ... ⊗ ε α ˜ 1 ︷ m When n = k + 1, we have α ˜ ⊕ ε α ˜ ⊕ ε … ⊕ ε α ˜ ︷ n + 1 = α ˜ ⊕ ε α ˜ ⊕ ε … ⊕ ε α ˜ ︷ n ⊕ ε α ˜ = n ε ˙ α ˜ ⊕ ε α ˜ = ( [ n p , n q , n r , n s ] ; ( 1 + ψ α ˜ 2 ) 2 − ( 1 − ψ α ˜ 2 ) n ( 1 + ψ α ˜ 2 ) n + ( 1 − ψ α ˜ 2 ) n , 2 ( ϒ α ˜ 2 ) n ( 2 − ϒ α ˜ 2 ) n + ( ϒ α ˜ 2 ) n ) ⊕ ε ( [ p , q , r , s ] ; ψ α ˜ , ϒ α ˜ ) = ( [ ( n + 1 ) p , ( n + 1 ) q , ( n + 1 ) r , ( n + 1 ) s ] ; ( 1 + ψ α ˜ 2 ) n + 1 − ( 1 − ψ α ˜ 2 ) n + 1 ( 1 + ψ α ˜ 2 ) n + 1 + ( 1 − ψ α ˜ 2 ) n + 1 , 2 ( ϒ α ˜ 2 ) n + 1 ( 2 − ϒ α ˜ 2 ) n + 1 + ( ϒ α ˜ 2 ) n + 1 ) = ( n + 1 ) ε α ˜

Hence Equation (8) holds for n = k + 1, therefore Equation (8) holds for all n.

(4) Let m be any positive integer and α ˜ is Pythagorean trapezoidal fuzzy number, then α ˜ ∧ ɛ m = α ˜ ⊗ ɛ α ˜ ⊗ ɛ . . . ⊗ ɛ α ˜ ︷ m(9) By using mathematical induction we prove that Equation (9) holds for all positive integers m. First, we show that Equation (9) holds for m = 2. Since α ˜ ⊗ ɛ α ˜ = ( [ p . p , q . q , r . r , s . s ] ; Ψ α ˜ Ψ α ˜ 1 + ( 1 - Ψ α ˜ 2 ) ( 1 - Ψ α ˜ 2 ) ( ϒ α ˜ ) 2 + ( ϒ α ˜ ) 2 1 + ( ϒ α ˜ ) 2 ( ϒ α ˜ ) 2 ) = ( [ p 2 , q 2 , r 2 , s 2 ] ; 2 ( Ψ α ˜ 2 ) 2 ( 2 - Ψ α ˜ 2 ) 2 + ( ϒ α ˜ 2 ) 2 ( 1 + ϒ α ˜ 2 ) 2 - ( 1 - ϒ α ˜ 2 ) 2 ( 1 + ϒ α ˜ 2 ) 2 + ( 1 - ϒ α ˜ 2 ) 2 ) = α ˜ ∧ ɛ 2 . Therefore, the Equation (9) holds for m = 2. If Equation (9) holds for m = k, such that α ˜ 1 ∧ ɛ m = α ˜ 1 ⊗ ɛ α ˜ 1 ⊗ ɛ . . . ⊗ ɛ α ˜ 1 ︷ m

Then, we have α ˜ ⊗ ɛ α ˜ ⊗ ɛ . . . ⊗ ɛ α ˜ ︷ m + 1 = α ˜ ⊗ ɛ α ˜ ⊗ ɛ . . . ⊗ ɛ α ˜ ︷ ⊗ ɛ α ˜ = m α ˜ ∧ ɛ m ⊗ ɛ α ˜ = ( [ p m , q m , r m , s m ] ; 2 ( Ψ α ˜ 2 ) m ( 2 - Ψ α ˜ 2 ) m + ( Ψ α ˜ 2 ) m ( 1 + ϒ α ˜ 2 ) m - ( 1 - ϒ α ˜ 2 ) m ( 1 + ϒ α ˜ 2 ) m + ( 1 - ϒ α ˜ 2 ) m , ) ⊗ ɛ ( [ p , q , r , s ] ; Ψ α ˜ , ϒ α ˜ ) = ( [ p ( m + 1 ) , q ( m + 1 ) , r ( m + 1 ) , s ( m + 1 ) ] ; 2 ( Ψ α ˜ 2 ) m + 1 ( 2 - Ψ α ˜ 2 ) m + 1 + ( Ψ α ˜ 2 ) m + 1 . ( 1 + ϒ α ˜ 2 ) m + 1 - ( 1 - ϒ α ˜ 2 ) m + 1 ( 1 + ϒ α ˜ 2 ) m + 1 + ( 1 - ϒ α ˜ 2 ) m + 1 ) = α ˜ 1 ∧ ɛ m + 1 Hence Equation (9) holds for m = k + 1. Therefore Equation (9) holds for all m.

Theorem 2. and α ˜ 2 = ( [ p 2 , q 2 , r 2 , s 2 ] ; Ψ α ˜ 2 , ϒ α ˜ 2 ) , be any two Pythagorean trapezoidal fuzzy numbers, then the operational laws between α ˜ 1 and α ˜ 2 are holds as follows:

(1) α ˜ 1 ⊗ α ˜ 2 = α ˜ 2 ⊗ α ˜ 1 ; (2) ( α ˜ 1 ⊗ α ˜ 2 ) δ = α ˜ 1 δ ⊗ α ˜ 2 δ δ ≥ 0 ; (3) α ˜ δ 1 ⊗ α ˜ δ 2 = α ˜ δ 1 + δ 2 δ 1 , δ 2 ≥ 0 .

3.1 Pythagorean Trapezoidal Fuzzy Einstein Geometric Aggregation Operators

Definition 13. Let α ˜ j = ( [ p j , q j , r j , s j ] , Ψ α ˜ j , ϒ α ˜ j ) where j = (1, 2 . . . , n) be a collection of Pythagorean trapezoidal fuzzy numbers. Pythagorean trapezoidal fuzzy Einstein weighted geometric (PTFEWG) operator of dimension n is mapping PTFEWG : Ω ⁿ  - > Ω, such that; PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ n ) = ⊗ ɛ j = 1 n ( α ˜ j ) w j(10) where w = (w₁, w₂,. . . w _n )  ^T is the weighting vector of α ˜ j ( j = 1 , 2 . . . , n ) , with w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1 .

Theorem 3. Let α ˜ j ( j = 1 , 2 , . . . , n ) be a collection of Pythagorean trapezoidal fuzzy numbers, then their aggregated value by using the PTFEWG operator is also and Pythagorean trapezoidal fuzzy number as follows; PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ n ) = ⊗ ɛ j = 1 n ( α ˜ j ) w j ( [ ∏ j = 1 n p j w j , ∏ j = 1 n q j w j , ∏ j = 1 n r j w j , ∏ j = 1 n s j w j ] ; ∏ j = 1 n ( Ψ α ˜ j 2 ) w j ∏ j = 1 n ( 2 - Ψ α ˜ j 2 ) w j + ∏ j = 1 n ( Ψ α ˜ j 2 ) w j , ∏ j = 1 n ( 1 + ϒ α ˜ j 2 ) w j - ∏ j = 1 n ( 1 - ϒ α ˜ j 2 ) w j ∏ j = 1 n ( 1 + ϒ α ˜ j 2 ) w j + ∏ j = 1 n ( 1 - ϒ α ˜ j 2 ) w j )(11)where w = (w₁, w₂,. . . w _n )  ^T is weight vector of α ˜ j ( j = 1 , 2 . . . , n ) , with w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1 ,

Proof. In the following, we prove the second result by using mathematical induction on n.

(1) We prove that Equation (11) holds for n = 2. Since α ˜ 1 w 1 = ( [ p 1 w 1 , q 1 w 1 , r 1 w 1 , s 1 w 1 ] ; 2 ( Ψ α ˜ 1 2 ) w 1 ( 2 - Ψ α ˜ 1 2 ) w 1 + ( Ψ α ˜ 1 2 ) w 1 , ( 1 + ϒ α ˜ 1 2 ) w 1 - ( 1 - ϒ α ˜ 1 2 ) w 1 ( 1 + ϒ α ˜ 1 2 ) w 1 + ( 1 - ϒ α ˜ 1 2 ) w 1 ) , and α ˜ 2 w 2 = ( [ p 2 w 2 , q 2 w 2 , r 2 w 2 , s 2 w 2 ] ; 2 ( Ψ α ˜ 1 2 ) w 2 ( 2 - Ψ α ˜ 1 2 ) w 2 + ( Ψ α ˜ 1 2 ) w 2 , ( 1 + ϒ α ˜ 1 2 ) w 2 - ( 1 - ϒ α ˜ 1 2 ) w 2 ( 1 + ϒ α ˜ 1 2 ) w 2 + ( 1 - ϒ α ˜ 1 2 ) w 2 ) . Let x 1 = ( 1 + ϒ α ˜ 1 2 ) w 1 , y 1 = ( 1 - ϒ α ˜ 1 2 ) w 1 , z 1 = ( Ψ α ˜ 1 2 ) w 1 , g 1 = ( 2 - Ψ α ˜ 1 2 ) w 1 , x 2 = ( 1 + ϒ α ˜ 2 2 ) w 2 , y 2 = ( 1 - ϒ α ˜ 2 2 ) w 2 , z 2 = ( Ψ α ˜ 2 2 ) w 2 , g 2 = ( 2 - Ψ α ˜ 2 2 ) w 2 . Then, w 1 α ˜ 1 = ( [ w 1 p 1 , w 1 q 1 , w 1 r 1 , w 1 s 1 ] ; 2 z 1 g 1 + z 1 , x 1 - y 1 x 1 + y 1 ) , and w 2 α ˜ 2 = ( [ w 2 p 2 , w 2 q 2 , w 2 r 2 , w 2 s 2 ] ; 2 z 2 g 2 + z 2 , x 2 - y 2 x 2 + y 2 ) . Thus, Einstein operational law 3 and Definition 12, we have PTFEWG ( α ˜ 1 , α ˜ 2 ) = α ˜ 1 w 1 ⊗ ɛ α ˜ 2 w 2 , = ( [ p 1 w 1 ⊗ p 2 w 2 , q 1 w 1 ⊗ q 2 w 2 , r 1 w 1 ⊗ r 2 w 2 , s 1 w 1 ⊗ s 2 w 2 ] ; 2 z 1 z 2 g 1 g 2 + z 1 z 2 , x 1 x 2 - y 1 y 2 x 1 x 2 + y 1 y 2 ) , = ( [ ∏ j = 1 2 p j w j , ∏ j = 1 2 q j w j , ∏ j = 1 2 r j w j , ∏ j = 1 2 s j w j ] ; 2 ∏ j = 1 2 ( Ψ α ˜ j 2 ) w j ∏ j = 1 2 ( 2 - Ψ α ˜ j 2 ) w j + ∏ j = 1 2 ( Ψ α ˜ j 2 ) w j , ∏ j = 1 2 ( 1 + ϒ α ˜ j 2 ) w j - ∏ j = 1 2 ( 1 - ϒ α ˜ j 2 ) w j ∏ j = 1 2 ( 1 + ϒ α ˜ j 2 ) w j + ∏ j = 1 2 ( 1 - ϒ α ˜ j 2 ) w j ) .

(2) If Equation (11) holds for n = k, that is PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ k ) = ⊗ ɛ j = 1 k ( α ˜ j ) w j , = ( [ ∏ j = 1 2 p j w j , ∏ j = 1 2 q j w j , ∏ j = 1 2 r j w j , ∏ j = 1 2 s j w j ] ; 2 ∏ j = 1 k ( Ψ α ˜ j 2 ) w j ∏ j = 1 k ( 2 - Ψ α ˜ j 2 ) w j + ∏ j = 1 k ( Ψ α ˜ j 2 ) w j , ∏ j = 1 k ( 1 + ϒ α ˜ j 2 ) w j - ∏ j = 1 k ( 1 - ϒ α ˜ j 2 ) w j ∏ j = 1 k ( 1 + ϒ α ˜ j 2 ) w j + ∏ j = 1 k ( 1 - ϒ α ˜ j 2 ) w j ) .

When n = k + 1, such that, PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ k + 1 ) = PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ k ) ⊗ ɛ ( α ˜ k + 1 w k + 1 ) , = PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ k ) ⊗ ɛ = ( [ p k + 1 w k + 1 , q k + 1 w k + 1 , r k + 1 w k + 1 , s k + 1 w k + 1 , ] ; 2 ( Ψ α ˜ k + 1 2 ) w k + 1 ( 2 - Ψ α ˜ k + 1 2 ) w k + 1 + ( Ψ α ˜ k + 1 2 ) w k + 1 , ( 1 + ϒ α ˜ k + 1 2 ) w k + 1 - ( 1 - ϒ α ˜ k + 1 2 ) w k + 1 ( 1 + ϒ α ˜ k + 1 2 ) w k + 1 + ( 1 - ϒ α ˜ k + 1 2 ) w k + 1 ) . Let x 1 = ∏ j = 1 k ( 1 + ϒ α ˜ j 2 ) w j , y 1 = ∏ j = 1 k ( 1 - ϒ α ˜ j ) w j , z 1 = ∏ j = 1 k ( Ψ α ˜ j 2 ) w j , g 1 = ∏ j = 1 k ( 2 - Ψ α ˜ j 2 ) w j , x 2 = ( 1 + ϒ α ˜ k + 1 2 ) w j , y 2 = ( 1 - ϒ α ˜ k + 1 2 ) w j , z 2 = ( Ψ α ˜ k + 1 2 ) w j , g 2 = ( 2 - Ψ α ˜ k + 1 2 ) w j . Thus, by the Einstein operational law 3 and Definition 12, we have PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ k + 1 ) = PTFEWG ( α ˜ 1 , α ˜ 2 , . . . , α ˜ k ) ⊗ ɛ ( α ˜ k + 1 ) w k + 1 , = ( [ ∏ j = 1 2 p j w j , ∏ j = 1 2 q j w j , ∏ j = 1 2 r j w j , ∏ j = 1 2 s j w j ] 2 z 1 g 1 + z 1 , x 1 - y 1 x 1 + y 1 , ) ⊗ ɛ ( [ p k + 1 w k + 1 , q k + 1 w k + 1 , r k + 1 w k + 1 , s k + 1 w k + 1 , ] ; 2 z 2 g 2 + z 2 , x 2 - y 2 x 2 + y 2 ) , = ( [ ∏ j = 1 k + 1 p j w j , ∏ j = 1 k + 1 q j w j , ∏ j = 1 k + 1 r j w j , ∏ j = 1 k + 1 s j w j ] ; 2 z 1 z 2 g 1 g 2 + z 1 z 2 , x 1 x 2 - y 1 y 2 x 1 x 2 + y 1 y 2 , ) , = ( [ ∏ j = 1 k + 1 p j w j , ∏ j = 1 k + 1 q j w j , ∏ j = 1 k + 1 r j w j , ∏ j = 1 k + 1 s j w j ] ; 2 ∏ j = 1 k + 1 ( Ψ α ˜ j 2 ) w j ∏ j = 1 k + 1 ( 2 - Ψ α ˜ j 2 ) w j + ∏ j = 1 k + 1 ( Ψ α ˜ j 2 ) w j , ∏ j = 1 k + 1 ( 1 + ϒ α ˜ j 2 ) w j - ∏ j = 1 k + 1 ( 1 - ϒ α ˜ j 2 ) w j ∏ j = 1 k + 1 ( 1 + ϒ α ˜ j 2 ) w j + ∏ j = 1 k + 1 ( 1 - ϒ α ˜ j 2 ) w j , ) . such that, Equation (11) holds for n = k + 1. Therefore, Equation (11) holds for all n.