A case study to illustrate the use of non-convex membership functions for linguistic terms

Terms used in fuzzy systems are almost invariably normalised, convex and distinct. The shapes of these terms are generated by certain accepted membership functions: piecewise linear functions, Gaussians or Sigmoids are almost exclusively used. This paper extends previous work in which it was suggested that non-convex membership functions might be considered for use in the context of modelling human decision making utilising fuzzy expert systems. In particular, the merits of non-convex fuzzy sets are discussed and a case study is presented to investigate inferencing with non-convex fuzzy sets in a practical implementation. It is shown that it is indeed possible to build a fuzzy expert system featuring usual Mamdani style fuzzy inference in which a time-related non-convex fuzzy set is used together with 'traditional' fuzzy sets. An examination is made of the resultant output surface generated by four different sub-classes of non-convex membership functions.