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A multicriteria intuitionistic fuzzy group decision-making method 361

α
α ¼ ð f x, μ xðÞ, υ α xðÞÞj x 2 Xg (13.1)

where μ α (x):X![0,1], x2X!μ α (x)2[0,1] and it represents the degree of
membership of the element x2X to the set α, and υ α (x):X![0,1],
x2X!υ α (x)2[0,1] and it represents the degree of nonmembership of
the element x2X to the set α, and they satisfy the condition of 0 μ α (x)
+υ α (x) 1 for all x2X.
Besides the membership and nonmembership, the degree of indetermi-
nacy of x2X to the set α can be determined by Eq. (13.2)
π α x ðÞ ¼ 1 μ x ðÞ υ α x ðÞ,x 2 X (13.2)
α
It is worth pointing out that π α (x) can be recognized as a measure of the
certainty of knowledge about x, and the smaller the value of π α (x), the more
certainty the knowledge about x (Boran et al., 2009).
Definition 2 Arithmetic operations
Assuming that there are two intuitionistic fuzzy numbers α denoted
by α¼(μ α ,υ α ,π α ), where μ α 2[0,1], υ α 2[0,1], μ α (x)+υ α (x) 1, and
π α ¼1 μ α υ α and β denoted by β¼(μ β ,υ β ,π β ), where μ β 2[0,1],
υ β 2[0,1], μ β (x)+υ β (x) 1, and π β ¼1 μ β υ β , the arithmetic operations
including addition, multiplication, and scale multiplication between α and β
are presented in Table 13.1.

Definition 3 Score, accuracy, and indeterminancy
The score, accuracy, and indeterminancy of the intuitionistic fuzzy number
α¼(μ α ,υ α ,π α ) can be determined by Eqs. (13.9)–(13.11), respectively.

S α ðÞ ¼ μ υ α (13.9)
α
A αðÞ ¼ μ + υ α (13.10)
α
I αðÞ ¼ μ υ α (13.11)
α
where S(α), A(α), and I(α) are the score, accuracy, and indeterminancy of
the intuitionistic fuzzy number α¼(μ α ,υ α ,π α ).
Definition 4 Comparisons of intuitionistic fuzzy numbers (Beliakov et al.,
2011)
As for the two intuitionistic fuzzy numbers α¼(μ α ,υ α ,π α ) and β¼(μ β ,
υ β ,π β ), a defuzzied score can be determined for these two intuitionistic fuzzy
numbers to identify the preference relationship between them based on
the work of Szmidt and Kacprzyk (2009) and that of Liao and Xu (2014),
as presented in Eqs. (13.12), (13.13).