New Tools in Fuzzy Arithmetic with Fuzzy Numbers

We present new tools for fuzzy arithmetic with fuzzy numbers, based on the parametric representation of fuzzy numbers and new fuzzy operations, the generalized difference and the generalized division of fuzzy numbers. The new operations are described in terms of the parametric LR and LU representations of fuzzy numbers and the corresponding algorithms are described. An application to the solution of simple fuzzy equations is illustrated. 1 Parametric Fuzzy Numbers and Fundamental Fuzzy Calculus In some recent papers (see [4], [5]), it is suggested the use of monotonic splines to model LU-fuzzy numbers and derived a procedure to control the error of the approximations. By this approach, it is possible to define a parametric representation of the fuzzy numbers that allows a large variety of possible types of membership functions and is very simple to implement. Following the LUfuzzy parametrization, we illustrate the computational procedures to calculate the generalized difference and division of fuzzy numbers introduced in [6] and [7]; the representation is closed with respect to the operations, within an error tolerance that can be controlled by refining the parametrization. A general fuzzy set over R is usually defined by its membership function μ : R−→ [0, 1] and a fuzzy set u of R is uniquely characterized by the pairs (x, μu(x)) for each x ∈ R; the value μu(x) ∈ [0, 1] is the membership grade of x to the fuzzy set u. Denote by F(R) the collection of all the fuzzy sets over R. Elements of F(R) will be denoted by letters u, v, w and the corresponding membership functions by μu, μv, μw. The support of u is the (crisp) subset of points of R at which the membership grade μu(x) is positive: supp(u) = {x|x ∈ X, μu(x) > 0}. We always assume that supp(u) = ∅. For α ∈]0, 1], the α−level cut of u (or simply the α−cut) is defined by [u]α = {x|x ∈ R, μu(x) ≥ α} and for α = 0 by the closure of the support [u]0 = cl{x|x ∈ R, μu(x) > 0}. The core of u is the set core(u) = {x|x ∈ R, μu(x) = 1} and we say that u is normal if core(u) = ∅. Well-known properties of the level− cuts are: [u]α ⊆ [u]β for α > β and [u]α = ⋂ β<α [u]β for α ∈]0, 1] E. Hüllermeier, R. Kruse, and F. Hoffmann (Eds.): IPMU 2010, Part II, CCIS 81, pp. 471–480, 2010. c © Springer-Verlag Berlin Heidelberg 2010