Intuitionistic fuzzy set

An intuitionistic fuzzy set . intuitionistic fuzzy set is a generalization of the concept of fuzzy set and was introduced in 1986 by K. Atanassov. While a fuzzy set is characterized solely by its membership function , an intuitionistic fuzzy set also has the non-membership function .

Definitions

Let the so-called universe, i.e. H. the base set on which the examinations take place is frequent . An intuitionistic fuzzy set is characterized by a membership function and a non-membership function . The following applies to these functions: ${\ displaystyle U}$ ${\ displaystyle U = \ mathbb {R}}$ ${\ displaystyle A}$ ${\ displaystyle m_ {A}}$ ${\ displaystyle n_ {A}}$

{\ displaystyle \ forall x \ in U: \ quad 0 \ leq m_ {A} (x) \ leq 1, \ quad 0 \ leq n_ {A} (x) \ leq 1, \ quad 1-m_ {A} (x) -n_ {A} (x) \ geq 0}

Here is interpreted as a degree of acceptance that to be heard and that the degree of acceptance not to be heard. If , then we have the special case of a classic fuzzy set: In this case, the degree of non-membership results as . If, however , this value stands for the degree of uncertainty, i.e. H. with this degree one cannot decide whether to belong or not to belong. For fixed numbers , the pair is also called an intuitionistic fuzzy number . ${\ displaystyle m_ {A} (x)}$ ${\ displaystyle x}$ ${\ displaystyle A}$ ${\ displaystyle n_ {A} (x)}$ ${\ displaystyle x}$ ${\ displaystyle A}$ ${\ displaystyle 1-m_ {A} (x) -n_ {A} (x) = 0}$ ${\ displaystyle n_ {A} (x) = 1-m_ {A} (x)}$ ${\ displaystyle 1-m_ {A} (x) -n_ {A} (x)> 0}$ ${\ displaystyle x}$ ${\ displaystyle A}$ ${\ displaystyle A}$ ${\ displaystyle x}$ ${\ displaystyle (m_ {A} (x); n_ {A} (x)) = (m; n)}$

example

A resolution is voted on. There are 5 votes in favor, 1 against and 4 abstentions. This “fuzzy” approval result can be well expressed by the intuitionistic fuzzy number , where the degree of approval, the degree of disapproval and the degree of uncertainty. ${\ displaystyle (0 {,} 5; 0 {,} 1)}$ ${\ displaystyle 0 {,} 5}$ ${\ displaystyle 0 {,} 1}$ ${\ displaystyle 1-0 {,} 5-0 {,} 1 = 0 {,} 4}$

additional

Algebraic operations such as addition, subtraction, multiplication, division can be defined for intuitionistic fuzzy numbers. In addition, intuitionistic fuzzy functions can be defined and a differential and integral calculus can be established for them.

Individual evidence

↑ Atanassov, K. (1986). Intuitionistic fuzzy sets . Fuzzy Sets and Systems 20, 87-96
^ Lei, Q. and Xu, Z. (2017). Intuitionistic Fuzzy Calculus. Studies in Fuzziness and Soft Computing 353, Springer International Publishing 2017. e-Book ISBN 978-3-319-54148-8

[1] Atanassov, K. (1986). Intuitionistic fuzzy sets . Fuzzy Sets and Systems 20, 87-96

[2] Lei, Q. and Xu, Z. (2017). Intuitionistic Fuzzy Calculus. Studies in Fuzziness and Soft Computing 353, Springer International Publishing 2017. e-Book ISBN 978-3-319-54148-8