IT2FSs

Interval Type-2 Fuzzy Systems (IT2FSs) are a class of fuzzy logic systems that use interval type-2 fuzzy sets to represent and manage uncertainty in reasoning. In IT2FSs, the membership function for a given input is not a single value in [0,1] as in type-1 fuzzy sets, but an interval of possible membership grades. The collection of these intervals across the input space forms the footprint of uncertainty (FOU), which accounts for imprecision in measurements, linguistic terms, or rule interpretation.

A typical IT2FS architecture includes fuzzification, a rule base, an inference engine, type-reduction, and defuzzification. Fuzzification maps crisp inputs to intervals defined by the upper membership function (UMF) and the lower membership function (LMF). The rule base contains if-then rules built from IT2FSs or their interval-valued representations. The inference engine combines the rules to produce an IT2 fuzzy output, which remains an interval-valued fuzzy set until it is reduced to a form suitable for defuzzification.

Type-reduction is a key step that collapses an IT2 fuzzy set into a type-1 representation or a crisp value. The Karnik-Mendel (KM) algorithms are the most widely used method for IT2 type-reduction; they iteratively determine the lower and upper bounds of the output in a monotone fashion. Some implementations use approximate or alternative type-reduction schemes to reduce computational load.

IT2FSs offer robustness to uncertainty and noise, making them attractive for control, decision support, and pattern recognition in imperfect environments. They face challenges in design and computation, including selecting appropriate UMFs and LMFs and managing higher computational complexity compared with type-1 systems.

Applications span process control, robotics, financial forecasting, and diagnostic systems, where explicit uncertainty modeling improves reliability and interpretability. IT2FSs generalize IT1FSs and converge to them when the FOUs collapse to single values.