Abstract
We first show the interval-valued intuitionistic fuzzy entropy which reflects intuitionism and fuzziness of interval-valued intuitionistic fuzzy set (IvIFS) based on interval-valued intuitionistic fuzzy cross-entropy. As for intuitionism and fuzziness of IvIFS, we propose interval-valued intuitionistic entropy and interval-valued fuzzy entropy, respectively. Furthermore, we establish the interval-valued span entropy describing the uncertainty of membership degree and nonmembership degree and show some concrete measure formulas. Combining intuitionistic factor, fuzzy factor, and span factor, we ultimately put forward the axiomatic definition of the compositive entropy and give a measure formula of compositive entropy. In addition, the effectiveness of the compositive entropy measure is illuminated by comparison with other entropy measures. Furthermore, the compositive entropy is applied to multiple attributes’ decision-making by using the weighted correlation coefficient between IvIFSs and pattern recognition by a similarity measure transformed from the compositive entropy.
Highlights
Since Zadeh [1] first introduced fuzzy set (FS) in 1965, many theories of higher order fuzzy set have been proposed
There are three types of uncertainty factors for interval-valued intuitionistic fuzzy set (IvIFS), including intuitionistic factor, fuzzy factor, and newly proposed span factor which can depict the extent of variation for the interval values of membership degree and nonmembership degree
Let a nonempty set X be the universe of discourse; an intuitionistic fuzzy set on X is defined by Atanassov as A = {⟨x, μA(x), ]A(x)⟩ | x ∈ X}, where μA, ]A : X → [0, 1], with the condition 0 ≤ μA(x)+]A(x) ≤ 1 for all x ∈ X
Summary
Since Zadeh [1] first introduced fuzzy set (FS) in 1965, many theories of higher order fuzzy set have been proposed. There are three types of uncertainty factors for IvIFS, including intuitionistic factor, fuzzy factor, and newly proposed span factor which can depict the extent of variation for the interval values of membership degree and nonmembership degree. Based on these three kinds of uncertainty factors, the main purpose of this paper is to construct a new compositive entropy which can measure uncertain information of IvIFS accurately.
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