The Directed Compatibility Between Ordered Fuzzy Numbers - A Base Tool for a Direction Sensitive Fuzzy Information Processing

Revision of the Kosiński’s Theory of Ordered Fuzzy Numbers

“…Therefore, we agree with other scientists [5,6] that the OFN should be called the Kosiński's number.…”

“…For this reason, ordered fuzzy numbers are increasingly called Kosinski's numbers [5,6]. The monograph [7] is a competent source of information about the contemporary state of knowledge on OFN.…”

Ordered Fuzzy Numbers: Sources and Intuitions

“…We outlined disadvantages of standard fuzzy arithmetics and discussed how to look at fuzzy numbers in an alternative way, by representing them as ordered pairs of functions that encode the shapes of fuzzy memberships. In this way, we obtained a mathematical framework that extends the standard approach including a new type of information referred to as a direction of a fuzzy number [18]. We showed a kind of evolution of our way of thinking about fuzzy arithmetics, starting from the classical approach, via quasi-invertible representation of convex fuzzy numbers, and finishing with formal definition of OFNs.…”

“…The general concept that allowed us to control imprecision during calculations can be interpreted as a kind of direction of fuzziness [18]. Chapter 4 contains a complete description of that idea from a mathematical perspective.…”

“…Actually, interpretation related to direction inspired some researchers to propose renaming the presented model as directed fuzzy numbers or fuzzy numbers with direction. Moreover, in papers [16][17][18], the name Kosiński's Fuzzy Numbers is used to honor the contribution of Witold Kosiński in a development of the considered model. Although in this book we keep the original OFN terminology, we believe that the discussion about the most appropriate name is not over.…”