Abstract
The issue of constructing a system of rules to perform binary operations over fuzzy numbers has been formulated and considered. The set problem has been solved regarding the ( L – R )-type fuzzy numbers with a compact carrier. Such a problem statement is predetermined by the simplicity of the analytical notation of these numbers, thereby making it possible to unambiguously set a fuzzy number by a set of values of its parameters. This makes it possible, as regards the ( L – R )-type numbers, to reduce the desired execution rules for fuzzy numbers to the rules for simple arithmetic operations over their parameters. It has been established that many cited works provide ratios that describe the rules for performing operations over the ( L – R )-type fuzzy numbers that contain errors. In addition, there is no justification for these rules in all cases. In order to build a correct system of fuzzy arithmetic rules, a set of metarules has been proposed, which determine the principles of construction and the structure of rules for operation execution. Using this set of metarules has enabled the development and description of the system of rules for performing basic arithmetic operations (addition, subtraction, multiplication, division). In this case, different rules are given for the multiplication and division rules, depending on the position of the number carriers involved in the operation, relative to zero. The proposed rule system makes it possible to correctly solve many practical problems whose raw data are not clearly defined. This system of rules for fuzzy numbers with a compact carrier has been expanded to the case involving a non-finite carrier. The relevant approach has been implemented by a two-step procedure. The advantages and drawbacks of this approach have been identified.
Highlights
Practical tasks related to the analysis and synthesis of systems, decision theory, management theory, are resolved under conditions of uncertainty
The downside of these rules of arithmetic operations is that the fuzzy numbers resulting from their implementations are not normalized
The aim of this study is to develop a sound system of fuzzy arithmetic rules
Summary
Practical tasks related to the analysis and synthesis of systems, decision theory, management theory, are resolved under conditions of uncertainty. Problems in terms of fuzzy mathematics is determined by the correctness of the rules for performing operations over the objects of this theory These rules are introduced as follows [3, 4]. Introduce the * symbol of an arbitrary binary arithmetic queue (addition, subtraction, multiplication, division) Any such operation assigns the numbers x1 and x2 a certain result z. For the operations of subtraction, multiplication, and division, the membership functions operations corresponding to the result of the operation execution are determined by the following ratios: μ(z) = ∫ μ1 (t)μ2 (t − z)dt, The downside of these rules of arithmetic operations is that the fuzzy numbers resulting from their implementations are not normalized. The membership function of such a fuzzy number x takes the form: μ
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