Page 332 - Materials Chemistry, Second Edition

332 16. Life cycle sustainability improvement

μ A (x):X![0,1] and υ A (x):X![0,1] represent the degree of membership and the degree of
nonmembership of the element x in the set A, respectively.

π A (x)¼1 μ A (x) υ A (x), x2X represents the hesitancy or nondeterminacy degree of the users
of the element x to A, the so-called “the indeterminacy degree” or “intuitionistic index”
(Wang et al., 2011). It can be determined by Eq. (16.2).
π A xðÞ ¼ 1 μ xðÞ υ A xðÞ,x 2 X (16.2)
A
where π A (x) represents the indeterminacy degree of x to A.
The ordered triple elements α A (x)¼(μ α (x),υ α (x),π α (x)) is a typical intuitionistic fuzzy number
with the condition that μ α (x), υ α (x)2[0 1] and μ α (x)+υ α (x) 1(Xu, 2007). The α A (x)¼(μ α (x),
υ α (x),π α (x)) can usually be abbreviated as α A (x)¼(μ α (x),υ α (x)).
Definition 16.2 Arithmetic operations (Xu, 2007). Let A¼(μ A ,υ A ) and B¼(μ B ,υ B ) be two
intuitionistic fuzzy numbers:
Addition:
(16.3)
A B A B A B
A B ¼ μ , υ A Þ μ , υ B Þ ¼ μ + μ μ μ , υ A υ B Þ
ð
ð
ð
Multiplication:
(16.4)
A B A B
ð
ð
A
B ¼ μ , υ A Þ
μ , υ B Þ ¼ μ μ , υ A + υ B υ A υ B Þ
ð
Definition 16.3 Intuitionistic fuzzy comparison matrix (Wang et al., 2011). Assume that there
are a total of n items (I 1 I 2 ⋯ I n ), the intuitionistic fuzzy numbers are usually to compare
each pair of items in the intuitionistic fuzzy comparison matrix (as presented in Eq. 16.5).
⋯
I 1 I 2 I n
11
12
1n
I 1 ð μ , υ 11 Þ ð μ , υ 12 Þ ⋯ ð μ , υ 1n Þ
(16.5)
21
2n
22
I 2 ð μ , υ 21 Þ ð μ , υ 22 Þ ⋯ ð μ , υ 2n Þ
⋮ ⋮ ⋮ ⋱ ⋮
n2
nn
n1
ð
ð
I n ð μ , υ n1 Þ μ , υ n2 Þ ⋯ μ , υ nn Þ
where (μ ij ,υ ij ) represents the relative preference of the i-th item over the j-th item.
For instance, (μ ij ,υ ij )¼(0.6,0.2) presents the relative preference of the i-th item over the
j-th item, 0.6 represents the certainty degree of the i-th item be superior to the j-th
item, and 0.2 represents the certainty degree of the j-th item be superior to the i-th item.
Accordingly, the indeterminacy degree 1 0.6 0.2¼0.2 represents the uncertainty degree
of the i-th item be preferred than the j-th item. The nine-scale system can be employed to
determine the elements in the intuitionistic fuzzy comparison matrix (as presented in
Table 16.1).
The elements in the diagonal of the intuitionistic fuzzy comparison matrix representing the
relative preference of one item over itself equals (05, 0.5) based on the nine-scale system
presented in the work of Xu and Liao (2014).
Definition 16.4 Normalized intuitionistic fuzzy vector (Qian and Feng, 2008).
The intuitionistic fuzzy vector:
μ μ (16.6)
1 2 n
X ¼ x 1 x 2 ⋯ x n Þ ¼ ðð μ υ 1 π 1 Þ ð υ 2 π 2 Þ ⋯ ð υ n π n ÞÞ
ð