Intuitionistic Fuzzy Aggregation Techniques - Intuitionistic Fuzzy Aggregation and Clustering - page 44

r ( 1 )
ij

r ( 2 )
ij

r ( s )
ij

r ij = ˜ IVIFPWG ( ˜ , ˜ ,..., ˜ )

s

1

− ξ ( k )
ij

− ξ ( k )
ij

s

1 1

1 (μ − ( k ) 1 (μ + ( k )

= ) , ) ,

s − 1 s −

ij ij

k = k =

1

1

s

− ξ (

k )

s

− ξ (

k )

1 1

ij ij

v − ( k )
ij

v + ( k )
ij

−

1 (

− ) , −

1 (

− ) ,

1 1 1

s − 1 s −

k = k =

s

− ξ (

k )

s

− ξ (

k )

1 1

ij ij

v + ( k )
ij

1 (μ + ( k )

1 (

1 − ) − ) ,

s − 1 s − 1

ij

k = k =

s

− ξ (

k )

s

− ξ (

k )

1 1

ij ij

v − ( k )
ij

1 (μ − ( k ) 1 (

1 − ) − )

s − 1 s − 1

ij

k = k =

⎛
⎝

⎡
⎣

⎤
⎦ ,

1 1

r (

k )

r (

k )

η k ( 1 + T ( ˜ )) η k ( 1 + T ( ˜ ))

ij ij

s

1

s

1

− −

k k

s − 1

r (

k ) s − 1

r (

k )

1 (μ − ( k ) 1 (μ + ( k )

1 η k (

1 + T ( ˜ ))

1 η k (

1 + T ( ˜ ))

= ) , )

= ij = ij

ij ij

k = k =

⎡
⎣

1

r (

k )

η k ( 1 + T ( ˜ ))

ij

s

1

−

k

s − 1

r (

k )

v − ( k )
ij

1 η k (

1 + T ( ˜ ))

1 −

1 (

1 − ) ,

= ij

k =

⎤
⎦ ,

1

r (

k )

η k ( 1 + T ( ˜ ))

s

1 ij

−

s
k = 1 η k (

s − 1

r (

k )

v + ( k )
ij

1 + T ( ˜ ))

1 −

1 (

1 − )

ij

k =

⎡
⎣

r (

k )

r (

k )

η k ( 1 + T ( ˜ )) η k ( 1 + T ( ˜ ))

s

1 ij

s

1 ij

1 − 1 − s

k

s
k

s − 1

r (

k ) s − 1

r (

k )

v + ( k )
ij

1 (μ + ( k )

1 η k (

1 + T ( ˜ ))

1 η k (

1 + T ( ˜ ))

1 (

− ) − ) ,

1

= ij = ij

ij

k = k =

⎤
⎦

⎞
⎠

1 1

r ( k )
ij

r ( k )
ij

η k ( 1 + T ( ˜ )) η k ( 1 + T ( ˜ ))

1 1

s s

− − s

k

s
k

s − 1

r ( k )
ij

s − 1

r ( k )
ij

v − ( k )
ij

1 (μ − ( k )

1 η k (

1 + T ( ˜ ))

1 η k (

1 + T ( ˜ ))

1 (

1 − ) − )

= =

ij

k = k =

(1.104)

to aggregate all the individual interval-valued intuitionistic fuzzy decision matrices
R ( k ) = ( ˜

r ( k )
ij

) m × n ( k = 1 , 2 ,..., s ) into the collective interval-valued intuitionistic

R

fuzzy decision matrix = ( ˜ r ij ) m × n , where

v ij ) = [ μ ij ,μ ij ] , [ v ij ] , [ π ij ,π ij ] ,

v ij , r ij = ( μ ij , ˜ ˜

i = 1 , 2 ,..., m ; j = 1 , 2 ,..., n (1.105)

Step 4 To get the overall preference value r j corresponding to the alternative y j , ˜

in the j th column of R by
using the interval-valued intuitionistic fuzzy weighted average (IVIFWA) operator
(Xu and Cai 2009):

r ij (

˜ = , ,..., )

we aggregate all the preference values j 1 2 n