Measurement-theoretic frameworks for fuzzy set theory

Abstract

Abstract. Two di erent but related measurement problems are considered within the fuzzy set theory. The rst problem is the membership measurement and the second is property ranking. These two measurement problems are combined and two axiomatizations of fuzzy set theory are obtained. In the rst one, the indi erence is transitive but in the second one this drawback is removed by utilizing interval orders.

Elicitation of membership functions: how far can theory take us?

“…In such a theory one can discuss the representation of a qualitative structure and the meaningfulness of such a representation (see Figure 2). This approach has been applied to fuzzy membership functions [19,13,14,18,6,2,3,4] which resulted in valuable insights as to what membership functions mean. Specifically, the ordinal nature of the membership function is agreed upon [19,18,6,3] and the min-max calculus is justified in this framework.…”

“…Specifically, the ordinal nature of the membership function is agreed upon [19,18,6,3] and the min-max calculus is justified in this framework. Also, the axioms required to move into "cardinal" (interval and ratio) scales with appropriate triangular norms and conorms are extracted and their suitability to fuzzy set theory (based on different interpretations) is discussed [2,3,4] (see Figure 3).…”

“…A fuzzy set theorist has a very unique problem at hand: consider the interpretation of the membership function before anything else and then choose an appropriate calculus for that interpretation in the light of measurement theory. Figure 4 shows which type of elicitation method is appropriate for each interpretation of fuzzy membership function based on results from measurement theory [2,3,4]. There is a large body of empirical research (mostly in the psychology literature) about elicitation methods of fuzzy membership functions (see [5] for a review).…”

Model-based localization for an autonomous mobile robot equipped with sonar sensors

“…In our earlier studies [1,2], we investigated representations of vague preferences and came up with consistent languages based on intervalvalued fuzzy sets [3] to represent such vagueness. We also constructed interval-valued preference structures and developed techniques under which a crisp choice is possible.…”

“…The search mechanism utilizes the concepts of scaling and interval-valued preference structures developed in [2] to make a choice between possible orientations and/or positions. The search procedures are described in Section 3.…”