Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 19

therefore,

− μ α ) λ 1

v λ 1

− μ α ) λ 2

v λ 2

− ( −

α ≥

− ( −

1 1 1 1 (1.20)

α

1 − λ 1 1 − λ 2 , with the

which implies λ 1 α ≥ λ 2 α , similarly, we can prove that α ≥ α

condition 0 <λ 1 ,λ 2 ≤ 1.

(2) If μ α 1 ≥ μ α 2 , v

α 1 ≤

v

α 2 , then

− μ α 1 ) λ − v α 1 ≥ − μ α 2 ) λ − v α 2 1 − ( 1 1 − ( 1 (1.21)

and

μ α 1 − ( ) λ ) ≥ μ α 2 − ( ) λ ) 1 − ( 1 − v 1 − ( 1 − v (1.22)

α α 1 2

α 1 ≥ α 2 .

λα 1 ≥ λα 2 and

thus

(3) If μ α 1 ≥ μ α 3 ,μ α 2 ≥ μ α 4 , v α 1 ≤ v α 3 and v α 2 ≤ v α 4 , then

μ α 1 + μ α 2 − μ α 1 μ α 2 − v

α 1 v α 2 =

1 − ( 1 − μ α 1 )( 1 − μ α 2 ) − v

α 1 v

α 2

≥ 1 − ( 1 − μ α 3 )( 1 − μ α 4 ) − v

α 3 v

α 4

= μ α 3 + μ α 4 − μ α 3 μ α 4 − v

α 3 v

(1.23)

α 4

and

μ α 1 μ α 2 − ( v + v − v

1 v

) = μ α 1 μ α 2 − 1 + ( 1 − v )( 1 − v )

α α α α α α 1 2 2 1 2

≥ μ α 3 μ α 4 − 1 + ( 1 − v )( 1 − v )

α α 3 4

= μ α 1 μ α 2 − ( v + v − v

3 v

) (1.24)

α α α α 3 4 4

thus, α 1 ⊕ α 3 ≥ α 2 ⊕ α 4 and α 1 ⊗ α 3 ≥ α 2 ⊗ α 4 .

Moreover, the relations of the operational laws above are given as below:

Theorem 1.3 (Xu and Yager 2006; Xu 2007)

(1) α 1 ⊕ α 2 = α 2 ⊕ α 1 .

(2) α 1 ⊗ α 2 = α 2 ⊗ α 1 .

(3) λ(α 1 ⊕ α 2 ) = λα 1 ⊕ λα 2 , λ> 0.

(α 1 ⊗ α 2 ) λ = α 1 ⊗ α 2 ,

(4) λ> 0.

(5) λ 1 α ⊕ λ 2 α = (λ 1 + λ 2 )α , λ> 0.

α λ 1 ⊗ α λ 2 = α λ 1 + λ 2 , (6) λ> 0.

Based on the ranking method given by Xu and Yager (2006), and Definition 1.3,
Xu (2011) developed a series of intuitionistic fuzzy power aggregation operators,
which allow the input data values to support each other in the aggregation process.
In what follows, we shall give a detailed introduction to them.