Operations on Concavoconvex Type-2 Fuzzy Sets

about to represent. Once the membership function has been established (estimated or defined), the concept is described very precisely as the membership values are exact numerical quantities. This seems to raise a certain dilemma of excessive precision in describing imprecise phenomena. In fact, this concern has already sparked a lot of debates starting from the very inception of fuzzy sets.” Concavoconvex fuzzy set is the result of the combination of the concepts of convex and concave fuzzy sets. This paper investigates concavoconvex type-2 fuzzy sets. Basic operations, union, intersection and complement on concavoconvex type-2 fuzzy sets using min and product t-norm and max t-conorm are studied and some of their algebraic properties are explored. In reality, there are situations in which the grade of membership itself is frequently ill-defined [29] and may not be determined precisely. This can be explained by the fact that on one hand a crisp value, as a result of a measurement, is not a suitable representative for a membership value [18] and, on the other hand, many researchers believe that assigning an exact number to an experts’ opinion is too restrictive [10]. Type-2 fuzzy set enable capturing the uncertainty on membership functions of fuzzy sets through relating one or more crisp numbers as membership values to an entity and that with not necessarily equal strengths. This will introduce the third dimension in type-2 fuzzy sets. Although once the membership function of a type-2 fuzzy set is chosen it is totally precise, the additional dimension of type-2 fuzzy sets provides a further degree of freedom in handling uncertainties in membership degrees. Of course, this in turn may raise debates on the deficiencies of type-2 fuzzy sets and promote the use of higher-order fuzzy sets and eventually lead to the idea of type-∞ fuzzy set, which is not practically possible. We have to stick to a finite-type fuzzy set; the type-2 fuzzy set constitutes a sensible trade-off between computational complexity and ability to handle uncertainty. Mendel [22] argued: “The original F[uzzy ]L[ogic], funded by Lotfi Zadeh ... is unable to handle uncertainties. By handle, I mean to model and minimize the effect of. ... The expanded FL—type-2 FL—is able to handle uncertainties because it can model them and minimize their effect.” John and Coupland [12] further noted that “fuzzy logic, as it is commonly used, is essentially precise in nature and that for many applications it is unable to model knowledge from an expert adequately. We argued that the modeling of imprecision can be enhanced by the use of type-2 fuzzy sets – providing a higher level of imprecision. ... The use of type-2 fuzzy sets allows for a better representation of uncertainty and imprecision in particular applications and domains. The more imprecise or vague the data are, then type-2 fuzzy sets offer a significant