FORTRAN IV programs for fuzzy set analysis

Abstract

This paper describes a series of programs, written in FORTRAN IV for the DEC 1091 computer, that performs various kinds of data analysis using fuzzy set concepts and techniques. These programs are based on standard FORTRAN 77 and call no special subroutines, and thus are highly portable. Since fuzzy set theory and its applications still are rather unfamiliar to most behavioral and social scientists, this paper begins with a brief review of concepts, followed by descriptions of the models and statistics used in these programs. The subsequent section describes the input and output of the programs themselves. Finally, program limitations and system requirements are discussed. Fuzzy Set Concepts and Techniques. A set is fuzzy when an element may belong partially (or in degree) to it, rather than belonging either totally or not at all. Formally, fuzzy sets are those in which the characteristic function is allowed to take values in the [0,1] interval instead of the classic restriction to the pair of values {O,1}. Zadeh (1965) proposed the first complete fuzzy set theory, which since has seen many modifications and refinements. The programs described in this paper, however, are based on foundations in fuzzy set theory that have rather wide acceptance among both theorists and researchers. Given data points Xi, the customary notation is to denote membership values in a fuzzy set by !-ti, which formally distinguishes between elements and their respective degrees of membership in the referent set. In the discussion to follow, measures and operators will be described for the fuzziness of a set, intersection between two fuzzy sets, and fuzzy set inclusion. Since one of the appeals of fuzzy set theory is its conceptual basis for categories with blurred edges, it seems crucial to have an appropriate measure of how fuzzy a category actually is. Measures of fuzziness have been much discussed in the fuzzy set literature (see De Luca & Termini, 1972; Knopfmacher, 1975; Loo, 1977). The approach used in the programs described here follows Smithson's (1982) generalization of relative variation. The specific measure used is