Decoupling of Attributes and Aggregation for Fuzzy Number Ranking

Abstract

Intuition, expressed as verbal arguments or axiom formulations, has often been used as a guiding principle for fuzzy number ranking (FNR). This chapter adopts the multi-attribute decision making (MADM) framework to analyze such intuition with three results. First, intuition in FNR should have involved multiple attributes, which are often implicated in the existing ranking methods. Then, we suggest three attributes (i.e., representative x-value, x-value range, overall membership ratio), which can be used to characterize the FNR intuition. Second, we decouple two issues in FNR: selection of attributes and aggregation of values, where aggregation is concerned with the trade-off among attributes to determine a single index for FNR. Then, the discount factors are proposed for the attributes of range and membership ratio to model the trade-off and formulate a ranking index. Third, the decoupling of attributes and aggregation reveals a fundamental tension between information content and the satisfaction of the FNR axioms. That is, if we can consider more information (in terms of attributes) as relevant to FNR, the ranking method will likely violate some FNR axioms. However, if we consider less information, the ranking method will be less sensitive to distinguish some fuzzy numbers for ranking. In the end, the proposed multi-attribute approach can provide a practical aspect to analyze and address the FNR problems