Generating fuzzy sets using rearranged nested family and the mathematical formula in decomposition theorem

The purpose of this paper is to propose a mechanical procedure to generate fuzzy data from the observed real number data. The observed data may be imprecise owing to some unexpected situation. In this case, the decision-makers may treat the observed data to be fuzzy data by providing suitable membership functions. However, the determination of those membership functions is frequently subjective for the different decision-makers. To avoid this biased subjectivity raised by the decision-makers, this paper proposes a mechanical procedure to generate fuzzy data. The essence of this mechanical procedure is to propose an extended form to generate a fuzzy set from a pre-determined family. This extended form is based on two functions, such that the conventional form can be covered when these these two functions are taken to be the identity functions. One of these two functions is adopted to rearrange the pre-determined family to be a nested family. A more general form is also proposed by considering a countable subfamily.