A multidimensional approach to rank fuzzy numbers based on the concept of magnitude

Abstract

Ranking fuzzy numbers have become of growing importance in recent years, especially as decision-making is increasingly performed under greater uncertainty. In this paper, we extend the concept of magnitude to rank fuzzy numbers to a more general definition to increase in flexibility and generality. More precisely, we propose a multidimensional approach to rank fuzzy numbers considering alternative magnitude definitions with three novel features: multidimensionality, normalization, and a ranking based on a parametric distance function. A multidimensional magnitude definition allows us to consider multiple attributes to represent and rank fuzzy numbers. Normalization prevents meaningless comparison among attributes due to scaling problems, and the use of the parametric Minkowski distance function becomes a more general and flexible ranking approach. The main contribution of our multidimensional approach is the representation of a fuzzy number as a point in a $n$-dimensional normalized space of attributes in which the distance to the origin is the magnitude value. We illustrate our methodology and provide further insights into different normalization approaches and parameters through several numerical examples. Finally, we describe an application of our ranking approach to a multicriteria decision-making problem within an economic context in which the main goal is to rank a set of credit applicants considering different financial ratios used as evaluation criteria.