Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 54

⎧
⎨
⎩

⎛
⎝

⎞
⎠

1
p + q

1

q

n

1
n ( n − 1 )

p

1 − 1 − − ( 1 − μ α i ) ( 1 − μ α j )

i , j = 1

i = j

⎛
⎝

⎞
⎠

1
p + q

1

q

n

1
n ( n − 1 )

p

≤ 1 − 1 − − ( 1 − μ β i ) ( 1 − μ β j )

i , j = 1
i = j

⇒

⎛
⎝

⎞
⎠

1
p + q

1 − v p

α j

n

1

α i v q 1 −

n ( n − 1 )

i , j = 1
i = j

⎛
⎝

⎞
⎠

1
p + q

1

β j

n

1

− v p

β i v q

n ( n − 1 )

≥ −

1

i , j = 1

i = j

⎛
⎝

⎞
⎠

1
p + q

n

1

q

1
n ( n − 1 )

p

⇒ 1 − 1 − − ( 1 − μ α i ) ( 1 − μ α j )

i , j = 1
i

= j

⎛
⎝

⎞
⎠

1
p + q

1

α j

n

1
n ( n − 1 )

− v p

α i v q

− −

1

i , j = 1

i = j

⎛
⎝

⎞
⎠

1
p + q

n

1

q

1
n ( n − 1 )

p

≤ 1 − 1 − − ( 1 − μ β i ) ( 1 − μ β j )

i , j = 1
i = j

⎛
⎝

⎞
⎠

1
p + q

1

β j

n

1

v p

β i v q

n ( n − 1 )

− 1 − − (1.135)

i , j = 1

i = j

which completes the proof.

α − and α + be given by Eqs. ( 1.35 ) and ( 1.36 ), then (4) (Boundedness) Let

α − ≤ IFGBM p , q (α 1 ,α 2 ,...,α n ) ≤ α +

(1.136)

which can be obtained easily by the monotonicity.

If the values of the parameters p and q change in the IFBGM, then some special
cases can be obtained as follows (Xia et al. 2012a):
Case 1 If q

→ 0, then by Eq. ( 1.119 ), we have