Membership Functions, Set-Theoretic Operations, Distance Measurement Methods Based on Ambiguous Set Theory: A Solution to a Decision-Making Problem in Selecting the Appropriate Colleges

To deal with uncertainty in a very precise way, the ambiguous set theory has been proposed. This theory supports the representation of ambiguities in terms of four membership degrees, namely “true,” “false,” “true-ambiguous,” and “false-ambiguous.” These membership degrees are linearly dependent on each other, because they are defined over the same event. However, these four membership degrees have the difficulty of unclear margins that prevent them to be separated from each other in the ambiguities representation of an event. To address this problem of unclear margins in the ambiguous set, the membership degrees are defined using four different functions, and called ambiguous membership functions (AMFs). These membership degrees are defined in such a way that their sum should be less than or equal to 2, and the individual membership degree should be in the range [0, 1]. To visually represent the uncertain margins for the membership degrees, the concept of an ambiguous region (AR) is introduced. In this study, various set-theoretic operations, their properties, and methods for measuring distances in support of the ambiguous sets are discussed. Finally, we discuss a real-time application of the ambiguous set theory with respect to solving the decision-making problem of choosing the appropriate colleges for students.