Fuzzy extension of a classical function: An alternative to the knowledge base for modelling imprecise relations

The aim of this paper is to introduce both the concept of fuzzy extension of a classical function for modelling imprecise relations between variables and the basic arithmetic operations which this entails. The concept of fuzzy extension can be considered as a generalisation of the concept of classical function extended to the field of the fuzzy sets defined in R. The fuzzy extension (f) over bar of a classical continuous function f(x) is a particular kind of fuzzy relation, which describes the correspondence between two variables x and y. The univocal image of each value of x through (f) over bar is a closed interval of y values [f(l)(x),f(u)(x)]. Functions f(l)(x) and f(u)(x) set the limits of the y interval whose extent of correspondence to x is nonzero, being zero for all y : y <= f(l)(x) and for all y : y >= f(u)(x) and they fulfill the condition that for all x is an element of f(l)(x) <= f (x) <= f(u)(x). A fuzzy extension f is defined by its membership function mu((f) over bar)(x,y), which quantifies the extent to which each value of the y variable corresponds to each value of x. The image (y) over bar of each fuzzy set (x) over bar is achieved by means of the composition rule of (f) over bar. In order to perform arithmetic operations with fuzzy extensions, the following basic operations are defined: the addition (a) over bar, subtraction (s) over bar, multiplication (m) over bar and division (d) over bar of two fuzzy extensions. The concept of fuzzy extension of a classical function, the procedure to specify and model mu((f) over bar)(x,y) the procedure to determine the image (y) over bar and the arithmetic operations are graphically illustrated and properly exemplified. A fuzzy extension (f) over bar defined in this way is a useful alternative to the set of fuzzy rules or knowledge base for modelling inherently imprecise relations, which are frequently described in a simplified manner by means of classical functions. For this reason, fuzzy extensions can be applied to a number of different fields, being particularly suitable for environmental assessments, such as the design and evaluation of environmental quality indexes and the environmental impact assessments, among others. (C) 2014 Elsevier Inc. All rights reserved.