Abstract
This paper has considered a task to expand the scope of application of fuzzy mathematics methods, which is important from a theoretical and practical point of view. A case was examined where the parameters of fuzzy numbers’ membership functions are also fuzzy numbers with their membership functions. The resulting bifuzziness does not make it possible to implement the standard procedure of building a membership function. At the same time, there are difficulties in performing arithmetic and other operations on fuzzy numbers of the second order, which practically excludes the possibility of solving many practical problems. A computational procedure for calculating the membership functions of such bifuzzy numbers has been proposed, based on the universal principle of generalization and rules for operating on fuzzy numbers. A particular case was tackled where the original fuzzy number’s membership function contains a single fuzzy parameter. It is this particular case that more often occurs in practice. It has been shown that the correct description of the original fuzzy number, in this case, involves a family of membership functions, rather than one. The simplicity of the proposed and reported analytical method for calculating a family of membership functions of a bifuzzy quantity significantly expands the range of adequate analytical description of the behavior of systems under the conditions of multi-level uncertainty. A procedure of constructing the membership functions of bifuzzy numbers with the finite and infinite carrier has been considered. The method is illustrated by solving the examples of using the developed method for fuzzy numbers with the finite and infinite carrier. It is clear from these examples that the complexity of analytic description of membership functions with hierarchical uncertainty is growing rapidly with the increasing number of parameters for the original fuzzy number’s membership function, which are also set in a fuzzy fashion. Possible approaches to overcoming emerging difficulties have been described.
Highlights
Fuzzy mathematics [1, 2] offers an effective toolkit for building models of systems that operate in uncertain environments
The most important feature of a fuzzy set theory is the structural easing of the requirements for the basic concept of this theory – the membership function of a fuzzy number
The reason for the Mathematics and cybernetics – applied aspects problems here is the hierarchical nature of the description of the membership functions of fuzzy second-order numbers
Summary
Fuzzy mathematics [1, 2] offers an effective toolkit for building models of systems that operate in uncertain environments. Numerous studies in recent years [6, 7] have drawn attention to the fact that the totality of observations of real objects operating in uncertain conditions does not fully meet the requirements of the basic axioms of probability theory This fact naturally calls into question the correctness of the results from the theoretically probabilistic analysis of them. An analytical description of membership functions is formed on the basis of statistical treatment of actual data This procedure is fundamentally impossible for fuzzy second-order numbers. Problems arise when solving numerous problems of system analysis, as well as their structural and parametric synthesis It is clear, that the use of fuzzy second-order numbers improves the adequacy of an analytical description of the actual uncertainty of the original data. That makes the task of developing a technique to build a membership function for bifuzzy numbers undoubtedly relevant
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