Fuzzimetric Sets: An Integrated Platform for Both Types of Interval Fuzzy Sets

Type-2 sets are the generalized “fuzzified” sets that can be used in the fuzzy system. Unlike type-1 fuzzy sets, Type-2 allow the fuzzy sets to be “fuzzy” rather than the crisp definition of the set. Although this would improve the flexibility of inferring a decision, the implementation of Type-2 is rather more complex than type-1. Based on this principle, this paper proposes the mechanism of “Fuzzimetric Sets” that is capable of defining a rigid fuzzy set as well as “fuzzy” Fuzzy sets (Type-2). This is based on the concept of Fuzzimetric Arcs which will also be reviewed in this paper. Most of the implementations of this type of fuzzification would be suitable for decision support systems where an example of how to implement Fuzzimetric sets is also presented in this article. The platform of Fuzzimetric sets is composed of the initial definition of the fuzzy sets within the context of Fuzzimetric Arcs and then, the use of mutation and crossover operations on the sets allow the “Fuzziness” property of the set. To control the level of fuzziness in such sets, the introduction of “Degree of Fuzziness” factor was also proposed. DOF is composed of two dimensions: Vertical-DOF: Allowing changes in the level of the fuzzy membership of the set (0-1) and Horizontal-DOF: allowing the fuzziness level between maximum and minimum tolerances of the fuzzy sets, causing the centroid to move between the maximum and minimum allowed tolerances. If V-DOF was equated to zero, and H-DOF range defined between [90-90] then the set becomes type-1, otherwise, fuzziness level of the fuzzy sets can be controlled via this factor.