Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 140

If S α F n < S α F n and S α F n > S α F n , then by using Xu and Yager

(2006)'s ranking method, we have

α F n GIFPWAF w (α 1 ,α 2 ,...,α n )<α F n < (1.358)

S α F n , i.e., If S α F n =

μ F n

v F n

μ α F n − v = max

j

− min

j

(1.359)

κ α j ,λ α j ( α j

)

κ α j ,λ α j ( α j

)

α F n

then by Eqs. ( 1.349 ) and ( 1.355 ), we have

v F n

μ α F n = max

j

μ F n , α F n = min

j

(1.360)

κ α j ,λ α j ( α j

)

κ α j ,λ α j ( α j

)

then

v F n

h

α F n

H α F n = μ α F n +

= μ F n + =

v α F n max

j

min
j

κ α j ,λ α j ( α j

)

κ α j ,λ α j ( α j

)

So we have

GIFPWAF w (α 1 ,α 2 ,...,α m ) = α F n

(1.361)

S α F n , i.e., If S α F n =

v F n

μ α F n − v α F n = min

j

μ F n − max

j

(1.362)

κ α j ,λ α j ( α j

)

κ α j ,λ α j ( α j

)

then by Eqs. ( 1.350 ) and ( 1.354 ), we have

μ F n

v F n

μ α F n = , =

min
j

v α F n max

j

(1.363)

κ α j ,λ α j (α

)

κ α j ,λ α j (α

)

j j

hence

μ F n

v F n

h α F n

(1.364)

H α F n = μ α F n +

v = min

j

+ max

j

=

α

κ α j ,λ α j (α j ) κ α j ,λ α j (α j )

F n

Thus, it follows that

GIFPWAF w (α 1 ,α 2 ,...,α n ) = α F n (1.365)

and then from Eqs. ( 1.358 ), ( 1.361 ) and ( 1.365 ), we know that (2) always holds.