Page 274 - Materials Chemistry, Second Edition

272 13. Multi-criteria decision-making after life cycle sustainability assessment under hybrid information

U
L
(1) P(x y )¼1 if and only if y x ;

L
U
(2) P(x y )¼0 if and only if x y ;

(3) P(x y )¼0.5 if and only if x ¼y ; and

Px y + Py x ¼ 1

Definition 13.4 Distance between two interval numbers (Xu, 2008).
L U L U
Let x ¼ x , x and y ¼ y , y be two interval numbers; the distance between x and y

can be determined by Eq. (13.7).
1 L U
dx , y ¼ x y L U (13.7)
+ x y

2
L
L

Where d(x ,y ) represents the distance between x ¼ x , x U and y ¼ y , y U .

Definition 13.5 Intuitionistic fuzzy set (Atanassov, 1986; Szmidt and Kacprzyk, 2001).
An intuitionistic fuzzy set A in X was defined by Atanassov (1986); the intuitionistic fuzzy set
A on X can be defined as (Atanassov, 1986):
(13.8)
A
A ¼ ð f A, μ xðÞ, υ A xðÞÞj x 2 Xg
where
μ A (x):X![0,1] and υ A (x):X![0,1] should satisfy:
0 μ xðÞ + υ A xðÞ 1 (13.9)
A
for all x2X.
μ A (x):X![0,1] and υ A (x):X![0,1] represent the degree of membership of x to A and that of
non-membership of x to A, respectively.
After determining the degree of membership and that of the non-membership, the indeter-
minacy degree which represents the hesitancy degree of the decision-makers forx to A can
be determined, as presented in Eq. (13.10).
π A xðÞ ¼ 1 μ xðÞ υ A xðÞ,x 2 X (13.10)
A
where π A (x) represents the indeterminacy degree of x to X.
The indeterminacy degree π A (x) is different from the degree of membership μ β (x)and the de-
gree of non-membership υ β (x)of x to X; it can be used as a measure of the degree of indeter-
minacy of x to X. Accordingly, an intuitionistic fuzzy number A can usually be represented
by A¼(μ A ,υ A ,π A ) which consists of the degree of membership, non-membership, and
indeterminacy.
Definition 13.6 Transfer intuitionistic fuzzy set into interval number (Zhou et al., 2005).
Let A¼(μ A ,υ A ,π A ) be an intuitionistic fuzzy set, and it can be transferred into an interval
number by Eq. (13.11).
(13.11)
A
A ¼ μ½ 1 υ A Š