New closed-form solutions for Karnik-Mendel algorithm+defuzzification of an interval type-2 fuzzy set

Karnik-Mendel (KM) algorithms used for type reduction (TR) followed by defuzzification are commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Up until now no closed-form solution has existed for KM+defuzzification. Our paper gives, for the first time, a Taylor-series approximation to KM+defuzzification. The first term in the approximation is the Nie-Tan (NT) formula, and the second-term (which is actually the third-order term in the Taylor series) is a correction to it. One very interesting fact is that the Taylor series approximation is itself iterative (as is the KM Algorithm) in that the second term depends on the first term. Simulations show that the first term is a very good approximation to KM+defuzzification.