Cardinality approach to fuzzy number arithmetic
To solve problems from the area of computing with words, arithmetic operations often have to be carried out on fuzzy numbers. The author proposes to call fuzzy numbers fuzzy sets of numbers (FSofN) to emphasize the fact that they are sets of many numbers (in the continuous case of infinitely many numbers) and not one, single number as the name fuzzy number suggests, because it occasionally leads to incorrect interpretations of their concept. To realize arithmetic operations on FSofN the standard extension principle is used. This principle was intended for possibilistic FSofN. However, in practical tasks many FSofN are of probabilistic character. If, for such FSofN, the standard extension principle is used, the results will be incorrect. In this paper a cardinality extension principle is proposed. It realizes arithmetic operations on probabilistic FSofN. This principle is proposed in 2 versions: as a normal and as a generalized version that enables context-dependent constraints to be taken into account. In addition, a new form of the possibilistic, constraint-extension principle of Klir is presented. In this paper, many examples are given that illustrate computations with both extension principles to make them less abstract and more user-friendly.
fuzzy numbers computing with words extension principle fuzzy arithmetic