Mathematical Foundations of Fuzzy Sets

5Generating Fuzzy Sets

In economics and engineering problems, when the fuzziness is taken into account, the observed data sometimes are fuzzified to be fuzzy numbers. This kind of fuzzification may result in many different types of fuzzy numbers depending on the methodology adopted by the decision makers, where the subjectivity via the viewpoint of decision makers may be biased and not an accurate representation of reality. In this chapter, we propose a general methodology that can get rid of the subjectivity by directly generating the related fuzzy sets based on the observed data without involving the possible biased viewpoint of decision makers. The main idea is based on the solid and nested families of sets.

Suppose that we want to measure the water level in the summer season. Owing to the fluctuation of the water level, we cannot simply say that the water level is now 10 meters. We should say that the water level is around 10 meters. Therefore, the reasonable way to model the water level is to treat it as a fuzzy interval or fuzzy number. Under this consideration, the water level should be taken as a fuzzy number . The problem is that the determination of the membership function of is subjective depending on the viewpoint of decision makers. In other words, there are infinite ...