fuzzy modeling using the similarity-based approximate reasoning system

Just as we humans use many different types of inferential procedures to help us understand things or to make decisions,  there are many different fuzzy logic inferential procedures, including similarity-based approaches. similarity measures can be seen not only as a general notion but also as a particular family of fuzzy relations which play  crucial roles for the motivation and the whole design of similarity-based reasoning. in the context of similarity-based reasoning, several issues merit concern. one is the representation of implication relation and two is the composition of a fuzzy implication relation with an observed system fact. the others are  continuity and robustness of these systems which are the soul that must be inherited in the newly setup frameworks. therefore, the purpose of this study is to introduce a new similarity-based approximate reasoning system which is based on introducing a new class of similarity measure on the space of $lr$-fuzzy numbers. therefore,  first, a new class of similarity measures is introduced between fuzzy sets. the similarity measure is needed in order to activate rules which are in terms of linguistic variables. second,  it is proved that the proposed measures satisfy the properties of the axiomatic definition as well as the other properties by a theorem. next,  we validate the effectiveness of the   proposed similarity measure in a bidirectional approximate reasoning system in order to provide a  nonlinear mapping of fuzzy  input data into fuzzy output data. finally,  using existing experimental data from uniaxial compressive strength (ucs) testing,  the fuzzy inference system constitutive model is produced to describe the influence of joint geometry (joint location, trace length and orientation) on the ucs of rock. the numerical results will show that the proposed model based on similarity-based approximate reasoning systems has better performance compared with the mamdani fuzzy inference systems and the  multivariate regression.