William Voxman

Elementary fuzzy calculus

Abstract In [2] Dubois and Prade introduced the notion of fuzzy numbers and defined the basic operations of addition, subtraction, multiplication, and division. A slightly modified definition of fuzzy numbers was presented in [4], and in that paper a metric was defined for this family of fuzzy sets. Another less restrictive definition of fuzzy numbers can be found in [1]. In the present paper we consider fuzzy numbers from a somewhat different perspective. Basically, we shall view fuzzy numbers in a topological vector space setting. Using the customary vector space operations together with the metric given in [4] we will define differentiation and integration of fuzzy-valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration.

On a canonical representation of fuzzy numbers

Abstract Fuzzy numbers, and more generally linguistic values, are approximate assessments, given by experts and accepted by decision-makers when obtaining more accurate values is impossible or unnecessary. To simplify the task of representing and handling fuzzy numbers, several authors have introduced real indices in order to capture the information contained in a fuzzy number. In this paper we propose two parameters, value and ambiguity, for this purpose. We use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision-making problems. Several examples illustrate these ideas.

Topological properties of fuzzy numbers

Fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets. Topological properties of the resulting metric space are studied. The notion of @q-crisp fuzzy numbers is introduced, and, in particular, it is shown that this class of fuzzy numbers is homeomorphic to the Hilbert cube.

Some remarks on distances between fuzzy numbers

In this paper we consider different approaches to assigning distances between fuzzy numbers. A pseudo-metric on the set of fuzzy numbers arising from the idea of the value of a fuzzy number is described, and some of its topological properties are noted. Reducing functions are used to define a family of metrics on the space of fuzzy numbers; some convergent properties for these metrics are illustrated. Finally, a fuzzy distance between fuzzy numbers is introduced and its basic properties are studied.

Canonical representations of discrete fuzzy numbers

In this paper we obtain two canonical representations of discrete fuzzy numbers. In particular, it is shown that any discrete fuzzy number can be represented by a triangular or a block discrete fuzzy number having the same value, ambiguity and fuzziness as the original number.

A fuzziness measure for fuzzy numbers: applications

Abstract Extending the ideas of Delgado et al. (1995) to include the notion of fuzziness, we obtain canonical representations of fuzzy numbers. In particular, we show that any fuzzy number can be represented by either a trapezoidal or a “quasitrapezoidal” number having the same basic attributes as the original fuzzy number.

Bases of fuzzy matroids

In this paper we continue our study of fuzzy matroids. We define the notion of a basis for a fuzzy matroid and investigate which properties of crisp matroid bases carry over to the fuzzy case.

Eigen fuzzy number sets

In this paper we extend certain results of Sanchez (Resolution of eigen fuzzy sets equations, Fuzzy Sets and Systems 1 (1978) 69-75) to the context of fuzzy numbers. A slight modification of the Dubois-Prade definition of fuzzy numbers is used.

Fuzzy rank functions

Abstract In this paper we obtain a characterization of fuzzy matroids in terms of the fuzzy rank function defined in [1]. This characterization parallels to some extent a well known characterization theorem for crisp matroids.

Fuzzy matroid sums and a greedy algorithm

Abstract In this paper we establish a necessary and sufficient condition for a greedy algorithm to find a maximal basis for a fuzzy matroid.