Type–reduction of Interval Type–2 fuzzy numbers via the Chebyshev inequality

The centroid is one of the most important methods for type–reduction of Type–2 fuzzy sets/numbers and it is also a popular expectation measure, but available methods to compute it are iterative/algorithmic which is an important issue for real–world implementations. This paper presents some theoretical results about the Chebyshev integral inequality for a class of Interval Type–2 fuzzy numbers which leads to obtain non–iterative closed forms of the centroid and its bounds for Type–1 and interval Type–2 fuzzy numbers (by extension). An analysis of the obtained results and a comparison to four well known type–reduction methods: the Karnik–Mendel, Yager Index, Mitchell and Nie–Tan methods are provided where the experimental evidence shows that the proposed method is equivalent to the K–M algorithms.