Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 41

α j ( = , ..., )

Theorem 1.12 (Boundedness) Let j 1 2 n be a collection of IVIFVs,

then

α − ≤ ( α 1 , α 2 ,..., α n ) ≤ α +

IVIFPWA (1.90)

α − ≤ ( α 1 , α 2 ,..., α n ) ≤ α + IVIFPWG (1.91)

α − ≤ ( α 1 , α 2 ,..., α n ) ≤ α + IVIFPOWA (1.92)

α − ≤ ( α 1 , α 2 ,..., α n ) ≤ α + IVIFPOWG (1.93)

where

α − =

mi j { μ α j } , mi j { μ α j }] , [ v α j } , v α j }] , [ mi j { μ α j }− v α j }] ,

[

ma j { ma j {

−

ma j {

1

mi j { μ α j }− v α j }] [ 1 − ma j { (1.94)

α + = ma j { μ α j } , ma j { μ α j }] , [ v α j } , v α j }] ,

[ mi j { mi j {

ma j { μ α j }− v α j }] , [ ma j { μ α j }− v α j }] [ 1 − mi j { 1 − mi j { (1.95)

1.3.3 Approaches to Multi-Attribute Group Decision Making with
Interval-Valued Intuitionistic Fuzzy Information

In the following, we investigate the application of the interval-valued intuitionistic
fuzzy power aggregation operators to multi-attribute group decision making with
interval-valued intuitionistic fuzzy information:
For a multi-attribute group decision making problem with interval-valued intu-
itionistic fuzzy information, suppose that Y , G and E are defined as in Sect. 1.2.4 .Let
B ( k ) = ( b ( k )
ij

) m × n be an interval-valued intuitionistic fuzzy decision matrix, where

b ( k )
ij

= ( t ( k )
ij

, f ( k )
ij

, π ( k )
ij

) is an attribute value provided by the expert e k , denoted by

an IVIFV, where t ( k )
ij

t − ( k )
ij

t + ( k )
ij

=[ , ]

indicates the degree range that the alternative

f ( k )
ij

f − ( k )
ij

f + ( k )
ij

indicates the degree range
that the alternative y j does not satisfy the attribute G i , and

y j satisfies the attribute G i , while =[ , ]

π ( k )
ij

=[ π − ( k )
ij

,π + ( k )

ij ]
indicates the degree range of uncertainty of the alternative y j to the attribute G i ,
such that