Fuzzy Numbers and Fuzzy Arithmetics

This chapter starts with an overview of basic concepts and results with fuzzy numbers and the corresponding fuzzy arithmetic operations, including the fuzzy extension of functions. Fuzzy numbers, a special kind of fuzzy sets on the real space, are characterized by a membership function which is normal, quasi-concave and upper semicontinuos and with nonempty bounded support, so that the level sets are nonempty compact convex sets. We show the general arithmetical and topological properties of the space of fuzzy numbers and we illustrate some parametric representations of the classical LR frame and in the level-cut setting, the LU (lower-upper) form. Corresponding arithmetic operators are discussed and the procedures are presented by adopting an algorithmic approach. In particular, the fuzzy extension of single and multiple variables functions is discussed in detail (including an account of the under- and over-estimation effects) and a new differential evolution algorithm is presented, particularly adapted for high dimensional problems. Finally, the proposed LU-fuzzy representation is applied to the integration and the (generalized) differentiation of fuzzy valued functions. A brief account of the current trends in relevant applications concludes the chapter.