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We propose a novel architecture for a higher order fuzzy inference system (FIS) and develop a learning algorithm to build the FIS. The consequent part of the proposed FIS is expressed as a nonlinear combination of the input variables, which can be obtained by introducing an implicit mapping from the input space to a high dimensional feature space. The proposed learning algorithm consists of two phases. In the first phase, the antecedent fuzzy sets are estimated by the kernel-based fuzzy c-means clustering. In the second phase, the consequent parameters are identified by support vector machine whose kernel function is constructed by fuzzy membership functions and the Gaussian kernel. The performance of the proposed model is verified through several numerical examples generally used in fuzzy modeling. Comparative analysis shows that, compared with the zero-order fuzzy model, first-order fuzzy model, and polynomial fuzzy model, the proposed model exhibits higher accuracy, better generalization performance, and satisfactory robustness.

In order to overcome the dimension problem of the traditional fuzzy clustering, we use kernel-based fuzzy c-means clustering (KFCM) to construct first-order TSK fuzzy models. The proposed algorithm is composed of two phases. In the first phase, the antecedent fuzzy sets are obtained by KFCM. We present the expression of the cluster prototypes of KFCM with different kernel functions in original input space. The use of cluster validity indices is a standard approach to determine an appropriate number of clusters in a data set. However, cluster validity index demands running the clustering algorithm for different number of clusters repeatedly. Therefore, a novel method specifying the number of clusters automatically is given for the purpose of reducing the computational complexity and eliminating the outliers. In the second phase, the consequent parameters can be identified by the least squares method. Experiment results show that the proposed method improves the generalization ability and robustness of fuzzy models compared with the traditional techniques.

This paper considers a class of fuzzy linear programming (FLP) problems where the coefficients of the objective function are characterized by LR-fuzzy intervals or LR-fuzzy numbers with the same shapes. The existing methods mainly focus on the problems whose fuzzy coefficients are linear shape. In this paper, we are interested in the solution of FLP problems whose fuzzy coefficients are nonlinear shapes. Specifically, we show that the FLP problem can be transformed into a multi-objective linear programming problem with four objectives if LR-fuzzy numbers have the same shape, and therefore the optimal solutions can be found by the relationships between FLP problems and the resulting parameter linear programming problems. An example is also presented to demonstrate that the method is invalid if fuzzy coefficients have different shapes. Then, we discuss the relationships among FLP problems, possibility and necessity maximization problems, and obtain a conclusion that all Pareto optimal solutions of the possibility-necessity approaches are the subset of the weak dominated solutions of the FLP problem. Finally, one numerical example is given to illustrate the solution procedure.