Abstract
In this paper, some new operations and basic properties of picture fuzzy relations are intensively studied. First, a new inclusion relation (called type-2 inclusion relation) of picture fuzzy relations is introduced, as well as the corresponding type-2 union, type-2 intersection and type-2 complement operations. Second, the notions of anti-reflexive kernel, symmetric kernel, reflexive closure and symmetric closure of a picture fuzzy relation are introduced and their properties are explored. Moreover, a new method to solve picture fuzzy comprehensive evaluation problems is proposed by defining the new composition operation of picture fuzzy relations, and the picture fuzzy comprehensive evaluation model is built. Finally, an application example (about investment risk) of picture fuzzy comprehensive evaluation is given, and the effective experiment results are obtained.
Highlights
We meet many concepts in our everyday life
Uncertainty is an unintelligible expression without a straightforward description, and many theories were established, such as probability theory, fuzzy set theory [1,2,3], intuitionistic fuzzy set theory [4], hesitant fuzzy set theory [5,6,7,8], soft set theory [9,10,11,12], rough set theory [13,14,15,16,17], granular computing [18,19,20,21,22,23,24,25,26,27], et al an intuitionistic fuzzy set has been successfully applied in different areas, but there are situations that cannot be represented by it in real life, such as voting, we may face human opinions involving more answers of the type: yes, abstain, no and refusal
According to the new composition operation, a new method to solve picture fuzzy comprehensive evaluation problems is proposed, and we prove that this method is doable by an application example
Summary
We meet many concepts in our everyday life. Most of them are vague than precise, and uncertainty is a common research topic of many branches of science (economics, engineering, environment, management science, medical science, and so on). After the fuzzy set was defined, the definition of fuzzy relations was proposed by Zadeh in paper [1] as an extension of classic relationship. Cuong proposed the notion of picture fuzzy relations and studied their operations and properties [28,29]. Their computational formulas and some properties are obtained. According to the new composition operation, a new method to solve picture fuzzy comprehensive evaluation problems is proposed, and we prove that this method is doable by an application example.
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