Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 162

T

λ ≥ , = (

w 1 , w 2 ,..., w n )

, ∈[ , ] , ∈ , = , ,...,

where 1 w w i 0 1 x i X i 1 2 n , and

i = 1 w i =

1.
Eq. ( 2.1 ) can weight not only the deviation of each IFV, but also the deviations of
the correspondingmembership degree, the non-membership degree and the hesitancy
(indeterminancy) degree. It is more general than the similarity measure:

ϑ (

A 1 , A 2 )

n

1
2 n

2 2 2

= 1 −

1 ((μ A 1 (

x j ) − μ A 2 ( x j )) + ( v A 1 ( x j ) − v A 2 ( x j )) + (π A 1 ( x j ) − π A 2 ( x j ))

j =

(2.2)

and thus, Eq. ( 2.1 ) is of high flexibility. If we take ϑ( A 1 , A 2 ) as the function of w ,

then it is a bounded function. Let

n

x i ) | λ + β 2 | x i ) | λ + β 3 | π A 1 ( d ( w ) = w i | μ A 1 ( x i ) − μ A 2 ( v A 1 ( x i ) − v A 2 ( x i )

i = 1

x i ) | λ ), − π A 2 ( λ ≥ 1 (2.3)

then we need to solve the maximum and minimum problem of Eq. ( 2.1 ), which can
be transformed to solve the maximum and minimum problem of d ( w ) . Since

n

x i ) | λ + β 2 | x i ) | λ d ( w ) = w i (β 1 | μ A 1 ( x i ) − μ A 2 ( v A 1 ( x i ) − v A 2 (

i = 1

x i ) | λ ) + β 3 | π A 1 ( x i ) − π A 2 (

x i ) | λ + β 2 | x i ) | λ ≤ ma i { β 1 | μ A 1 ( x i ) − μ A 2 ( v A 1 ( x i ) − v A 2 (

x i ) | λ } ,λ ≥

+ β 3 | π A 1 ( x i ) − π A 2 ( 1 (2.4)

There must exist a positive integer k such that

x i ) | λ + β 2 | x i ) | λ + β 3 | π A 1 ( x i ) | λ } ma i { β 1 | μ A 1 ( x i ) − μ A 2 ( v A 1 ( x i ) − v A 2 ( x i ) − π A 2 (

x k ) | λ + β 2 | x k ) | λ + β 3 | π A 1 ( = β 1 | μ A 1 ( x k ) − μ A 2 ( v A 1 ( x k ) − v A 2 ( x k )

x k ) | λ } ,λ ≥ − π A 2 ( 1 (2.5)

Hence, when w k = 1 and w i = 0, i = k , the equality holds. Also since

n

x i ) | λ + β 2 | x i ) | λ d ( w ) = w i (β 1 | μ A 1 ( x i ) − μ A 2 ( v A 1 ( x i ) − v A 2 (

i = 1

x i ) | λ ) + β 3 | π A 1 ( x i ) − π A 2 (