Is Fuzzy Number the Right Result of Arithmetic Operations on Fuzzy Numbers?
Present versions of fuzzy arithmetic (FA) are not ideal. For some computational problems they deliver credible results. However for many other problems the results are less credible or sometimes clearly incredible. Reason of this state of matter is the fact that present FA-versions partially or fully (depending on a method) do not possess mathematical properties that are necessary for achieving correct calculation results as: distributivity law, cancellation law, neutral elements of addition and multiplication, property of restoration, possibility of decomposition of calculation in parts, ability of credible equations’ solving, property of delivering universal algebraic solutions, possibility of formula transformation, and other. Lack of above properties is, in the authors’ opinion, caused by incorrect assumption of all existing FA-versions that result of arithmetic operations on unidimensional fuzzy intervals is also a unidimensional fuzzy interval. In the paper authors show that the correct result is a multidimensional fuzzy set and present a fuzzy arithmetic based on this proposition, which possess all necessary mathematical properties and delivers credible results.
KeywordsFuzzy arithmetic Fuzzy computations Uncertainty theory Granular computing Soft computing Artificial intelligence
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