Study on weighted Nagar-Bardini algorithms for centroid type-reduction of general type-2 fuzzy logic systems

General type-2 fuzzy logic systems (GT2 FLSs) have drawn great attentions since the alpha-planes representation of general type-2 fuzzy sets (GT2 FSs) was proposed. The iterative of type-reduction (TR) algorithms are difficult to apply in practical applications. In the enhanced types of algorithms, the Nagar-Bardini (NB) algorithms decrease the computation complexity greatly. In terms of the Newton-Cotes quadrature formulas of numerical integration techniques, the paper extends the NB algorithms to three different forms of weighted NB (WNB) algorithms according to the comparisons between the sum operation in NB algorithms and the integral operation in continuous version of NB (CNB) algorithms. The NB algorithms just become a special case of the WNB algorithms. Four simulation examples are used to illustrate and analyze the performances of the WNB algorithms while performing the centroid TR of GT2 FLSs. It also shows that, in general, the WNB algorithms have smaller absolute error and faster convergence speed compared with the NB algorithms, which provides the potential value for T2 FLSs designers and users.