Multicriteria Decision-Making Under Conditions of Uncertainty

3Design and Processing Aspects of Fuzzy Sets

In this chapter, we focus on the fundamentals of fuzzy sets by concentrating on the three main processing issues; (i) design of fuzzy sets (membership functions), (ii) logic operations and aggregation of fuzzy sets, and (iii) transformations (mappings) of fuzzy sets including fuzzy arithmetic. Those are essentials, which make the framework of fuzzy sets fully operational when supporting a wide range of applications.

3.1 The Development of Fuzzy Sets: Elicitation of Membership Functions

The problem of elicitation and interpretation of fuzzy sets (their membership functions) is of significant relevance from the conceptual, algorithmic, and application‐oriented standpoints (Klir and Yuan 1995; Nguyen and Walker 1999). In the existing literature, we encounter a great number of methods that support the construction of membership functions. In general, we distinguish here between user‐driven and data‐driven approaches with a number of techniques that share some features specific to both data‐ and user‐driven techniques, and hence are located somewhere in between. The determination of membership functions has been a debatable issue for a long time, almost since the very inception of fuzzy sets. In contrast to interval analysis and set theory where the estimation of bounds of the interval constructs did not attract a great deal of attention and seemed to be somewhat taken for granted, an estimation of membership degrees (and membership ...