Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 87

k
i = 1 w i α i ATS − IFWA (α 1 ,α 2 ,...,α k ) = = w 1 α 1 ⊕ w 2 α 2 ⊕ ... ⊕ w k α k

h − 1 k

, g − 1 k = w i h (μ α i ) w i g( v α i )

i = 1 i = 1

(1.205)

then

ATS − IFWA (α 1 ,α 2 ,...,α k ,α k + 1 )

k
⊕ =

1 w i α i ⊕

w k + 1 α k + 1

i =

h − 1 k

, g − 1 k

= w i h (μ α i ) w i g( v α i )

i = 1 i = 1

⊕ h − 1 w k + 1 h (μ α k + 1 ) , g − 1 w k + 1 g( v α k + 1 )

h − 1 h h − 1 k

h h − 1 w k + 1 h (μ α k + 1 ) = w i h (μ α i ) + ,

i = 1

g − 1 g − 1 k + g g − 1 w k + 1 g( v α k + 1 ) g w i g( v α i )

i = 1

h − 1 k

, g − 1 k

= w i h (μ α i ) + w k + 1 h (μ α k + 1 ) w i g( v α i ) + w k + 1 g( v α k + 1 )

i = 1 i = 1

h − 1 k + 1

, g − 1 k + 1

= w i h (μ α i ) w i g( v α i ) (1.206)

i = 1 i = 1

i.e., Eq. ( 1.203 ) holds for n 1. Thus Eq. ( 1.203 ) holds for all n .

In addition, we have known that h

= k +

( t ) = g( 1 − t ) , and g : [ 0 , 1 ]→ 0 , 1 ] is a strictly

decreasing function, then h ( t ) is a strictly increasing function which indicates that

h − 1 n , g − 1 n

0 ≤ w i h (μ α i ) w i g( v α i ) ≤ 1 (1.207)

i = 1 i = 1

and

h − 1 n + g − 1 n w i h (μ α i ) w i g( v )

α

i

i = 1 i = 1

h − 1 n + g − 1 n ≤ w i h (μ α i ) w i g( 1 − μ α i )

i = 1 i = 1