Environmental Engineering Reference
In-Depth Information
The importance of decision makers in the group decision making process are
shown in Table 10 . These linguistic variables used can be converted into IFNs
(Zhang and Liu 2011 ) and weight of decision makers are calculated using
Eq. ( 16 ).
( k + µ k (
µ k + ν k ))
µ k
k =
(16)
t
µ k
k = 1 k + π k (
µ k + ν k ))
where
K = 1 K = 1 ( K = NUMBER OF DECISION MAKERS ) .
In group decision making process, all the individual decision opinions need to
be fused into a group opinion to construct an aggregated intuitionistic fuzzy deci-
sion matrix. In order to do that, the IFWA operator given by Xu ( 2007 ) is utilized:
T
r ( 1 )
ij
, r ( 2 )
ij
, ... , r ( t )
ij
= 1 r ( 1 )
2 r ( 2 )
⊕···⊕ t r ( t )
r ij = IFWA
ij
ij
ij
t
t
t
t
(17)
k
k
− µ ( k )
ij
ν ( k )
ij
− µ ( k )
ij
) k
(( ν ( k )
ij
) k )
=
1
1
,
,
( 1
k =
1
k =
1
k =
1
k =
1
Therefore according to Eq. 17 all the intuitionistic fuzzy decision matrices
( R a 1 , ... , R a 5 ) aggregated into a complex intuitionistic fuzzy decision matrix R a .
( 0.15, 0.11, 0.04 )( 0.21, 0.15, 05 ) 0.22, 0.16, 0.06 )( 0.33, 0.24, 0.09 )
R A =
( 0.58, 0.48, 0.1 ) 0.65, 0.55, 0.1 ) 0.72, 0.65, 0.07 )( 0.53, 0.43, 0.1 )
( 0.39, 0.31, 0.07 )( 0.34, 0.28, 0.05 )( 0.23, 0.19, 0.04 )( 0.38, 0.31, 0.07 )
(18)
Next point is forming of reference sequence which is the optimal sequence of the
criteria values. In the intuitionistic fuzzy decision matrix, the maximum value
α
+ = ( 1, 0, 0 ) can be used as the reference value. That, in this case the reference
sequence r 0 is:
r 0
= ( r 0 j ) 1 × n =
α
+
α
+
...α
+
( n = number of criteria )
(19)
Equation ( 19 ) is used in order to calculate the distance δ ij between r ij and r 0 j
(Table 11 ). Distance measure between two IFNs is defined as follows:
Let α 1 = α 1 , ν α 1 , π α 1 ) and α 2 = α 2 , ν α 2 , π α 2 ) be two IFNs, then
d (∝ 1 , 2 ) = 1
2 α 1 − µ α 2 + ν α 1 − ν α 2 + π α 1 − π α 2 )
(20)
Table 10 The importance of decision makers
Acronym
Decision maker type
Linguistic variables
ʻ
Environmental expert
Important
0.22
d 1
d 2
Purchase manager
Very important
0.26
Research and development staff
Important
0.22
d 3
Purchase staff
Medium
0.15
d 4
d 5
Purchase staff
Medium
0.15
 
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