Page 273 - Materials Chemistry, Second Edition

13.2 Decision-making under multi-type data condition 271
13.2 Decision-making under multi-type data condition

The basics of interval number and intuitionistic fuzzy numbers were firstly introduced;
then, the multi-criteria decision-making method for life cycle sustainability ranking of energy
and industrial systems under hybrid information was developed.

13.2.1 Preliminary of interval numbers and intuitionistic fuzzy numbers
Definition 13.1 Interval numbers (Xu, 2008; Yue, 2011).
L U L U L U L U
Let x ¼ x , x ¼ xx x x ,x x ,x , x 2 R was defined as an interval number,

L
L
U
U
which varies from x to x , and is a positive interval number if 0 x x . x turns into a

U
L
real number when x ¼x .
Definition 13.2 Arithmetic operations (Xu, 2008; Yue, 2011).
L U L U L U L U
Let x ¼ x , x ¼ x 0 < x x x ,x x ,x , x 2 R and

L U L U L U L U
y ¼ y , y ¼ y 0 < y y y ,y y ,y , y 2 R , and k>0, then,

h i
L U L U
k x ¼ kx , x ¼ kx , kx (13.1)

L
L
L
U
x + y ¼ x , x U L U ¼ x + y , x + y U (13.2)

+ x , x

L U L U L L U U
x y ¼ x , x x , x ¼ x y , x y (13.3)

L
k L U k h k U k i
x ¼ x , x ¼ x , x (13.4)

Definition 13.3 (Xu and Da, 2002)
L U L U
Let x ¼ x , x and y ¼ y , y be two interval numbers; the possibility that x y :

( ! )
U
y x L

Px y ¼ max 1 max ,0 ,0 (13.5)

L x + L y
U
L
where P(x y ) represents the possibility that x y , L x ¼ x x represents the length of

L U U L L U
x ¼ x x , and L y ¼ y y represents the length of y ¼ y y .

In a similar way, the possibility that y x can be determined by Eq. (13.6).

( U L ! )
y x

Px y ¼ max 1 max ,0 ,0 (13.6)

L x + L y

where P(y x ) represents the possibility that y x .

P(x y ) satisfies the following:

0 Px y 1