ABSTRACT
In the previous chapters, the fuzzy sets and numbers were presented as individual expressions that classify any phenomenon into its subjective and overlapping sub-domains in the forms of MFs. The relationships either between one linguistic variable sub-domain or between two linguistic variables’ sub-domains provide key information in all modeling and estimation work. In crisp studies, the correlation coefcient and association indices are used objectively to present the global presence or absence of association, interaction, or interconnectedness between the elements of two crisp sets. These classical methods do not consider the relationships between the elements’ sub-domains (fuzzy sets) of the phenomenon, but yield an overall result with some restrictive mathematical assumptions. For instance, the correlation coefcient measures the linear dependence and requires that the data are normally (Gaussian) distributed, stationary, have constant variance (homoscedasticity), etc. (S¸ en, 1978, 1979, 2001). None of these or any other assumptions are necessary in the fuzzy relationship denitions. Fuzzy relations are generalizations of crisp relations and they allow for various degrees of associations between elements or sub-domains, where degree of association can be represented by MDs and hence the fuzzy relations themselves are fuzzy sets.
