Abstract
The need to improve the adequacy of conventional models of the source data uncertainty in order to undertake research using fuzzy mathematics methods has led to the development of natural improvement in the analytical description of the fuzzy numbers' membership functions. Given this, in particular, in order to describe the membership functions of the three-parametric fuzzy numbers of the (L-R)-type, the modification implies the following. It is accepted that these functions' parameters (a modal value, the left and right fuzzy factors) are not set clearly by their membership functions. The numbers obtained in this way are termed the second-order fuzzy numbers (bi-fuzzy). The issue, in this case, is that there are no rules for operating on such fuzzy numbers. This paper has proposed and substantiated a system of operating rules for a widely used and effective class of fuzzy numbers of the (L-R)-type whose membership functions' parameters are not clearly defined. These rules have been built as a result of the generalization of known rules for operating on regular fuzzy numbers. We have derived analytical ratios to compute the numerical values of the membership functions of the fuzzy results from executing arithmetic operations (addition, subtraction, multiplication, division) over the second-order fuzzy numbers. It is noted that the resulting system of rules is generalized for the case when the numbers-operands' fuzziness order exceeds the second order. The examples of operations execution over the second-order fuzzy numbers of the (L-R)-type have been given.
Highlights
The bi-fuzzy numbers were introduced in [1] as a natural generalization of the regular fuzzy numbers
A system is proposed that forms the rules for the execution of arithmetic operations on fuzzy numbers whose order is higher than the first order
We have considered a case where the membership functions’ parameters of the fuzzy numbers of the (L-R) type are themselves set by fuzzy numbers of the (L-R)-type
Summary
The bi-fuzzy numbers (the second-order fuzzy numbers) were introduced in [1] as a natural generalization of the regular fuzzy numbers. The fundamental novelty of the technology that formally defines these numbers is that the parameters of their membership functions are themselves fuzzy with their own membership functions. An important positive feature of the emerging complexity of the structure of mathematical notation of fuzzy numbers is the arising possibility to radically improve the adequacy of the uncertainty models of objects in the external world and within their functioning environment. The resulting formalism significantly expands the space of the mathematical models of real-world objects operating under conditions of hierarchical uncertainty. The two-stage character of the analytical description of the membership functions’ parameters of the second-order fuzzy numbers makes it possible to more accurately and strictly determine the real inaccuracy of the conventional descriptions of objects and their functioning environment. The practical usefulness and effectiveness of using this new field of fuzzy mathematics are largely limited by the lack of the algebra of operations on the corresponding fuzzy numbers, which requires further research
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