A fuzzy directional distance measure

Abstract

The measure of distance between two fuzzy sets is a fundamental tool within fuzzy set theory, however, distance measures currently within the literature use a crisp value to represent the distance between fuzzy sets. A real valued distance measure is developed into a fuzzy distance measure which better reflects the uncertainty inherent in fuzzy sets and a fuzzy directional distance measure is presented, which accounts for the direction of change between fuzzy sets. A multiplicative version is explored as a full maximal assignment is computationally intractable so an intermediate solution is offered.

Measuring the Directional or Non-directional Distance Between Type-1 and Type-2 Fuzzy Sets With Complex Membership Functions

“…The functiond may be (16) for directional distance, and Consider three α-cuts, where A α = [2, 4], B α = ( [3,5], [7,9]) and C α = [5, 7] (A α and C α and convex, B α is non-convex). Using (18),d(A α , B α ) =d(A α , C α ); the distance is the same even though the α-cuts are fundamentally different.…”

“…While this may not be desired, it will always be an issue when simplifying the distance of fuzzy sets to a singleton. If it necessary to show non-convexity in the distance when it is present in the fuzzy sets that are compared, we recommend modelling distance as a fuzzy set [7]. When distance must be represented as a singleton, we recommend the proposed method (18) for measuring non-convex fuzzy sets.…”

“…For example, Fig. 3 shows two fuzzy sets and their distance as a fuzzy set (using [7]) and as a singleton (using (18)). Note that the singleton result is the centroid of the fuzzy result.…”

“…For example, a distance measure on T1 FSs would return a T1 FS distance. Such a distance measure is, for example, discussed in [7]. It is clear that returning a set of the same type as the input sets minimises the potential for information loss.…”

“…For example, it will not be possible to know from the value of distance if the fuzzy sets compared were non-normal or non-convex. In a future publication, we are looking to revisit some of the contributions made in this paper in respect to more complex (distance) measures which return non-numeric distances, such as proposed in [7].…”