An extended type-reduction method for spiked concave type-2 fuzzy sets

In this paper, an extended type-reduction method for spiked concave type-2 fuzzy sets has been studied. First, the concave type-2 fuzzy set is decomposed with α-level, and the α-level sets are obtained. Second, to the certain α-level set, there exists more than one combination (vertical interval sets defined on U) for the concave type-2 fuzzy set that parallels to the x-u plane; then, enhanced Karnik-Mendel (EKM) algorithm can be utilized to resolve the problem of type-reduction of the α-level set. Third, based on the type-reduction result that is a union of the horizontal interval set, and the type-reduction result is parallel to the x-axis. Spikes are considered for centroid calculation for the general concave type-2 fuzzy set, and the centroid for each discretized point on the x-axis is modified to have impact on the EKM results. Finally, examples on Gaussian concave type-2 fuzzy set and triangular concave type-2 fuzzy set are presented to show that spikes have impact on the concave type-2 fuzzy set's type-reduction, and the spikes should be considered if its happened whether in fuzzy modeling or fuzzy inference process.