An algorithmic study of relative cardinalities for interval-valued fuzzy sets

• We define generalised relative cardinality for IVFS using different t-norms.
• We study properties of such relative cardinality especially as a subsethood measure.
• We construct efficient algorithms to compute such relative cardinality for IVFS.
• We discuss practical implementation and applications of proposed algorithms.

The main topic of this paper is the notion of relative cardinality for interval-valued fuzzy sets – its definition, properties and computation. First we define relative cardinality for interval-valued fuzzy sets following the concept of uncertainty modelling given by Mendel's Wavy-Slice Representation Theorem. We expand on previous approaches by considering relative cardinality based on different t-norms and scalar cardinalities and we initiate an investigation of its properties and possible applications. Drawing on the Nguyen–Kreinovich and Karnik–Mendel algorithms, we propose efficient algorithms to compute relative cardinality depending on a chosen t-norm. This seems to be the first such broad and consistent analysis to have been made of relative cardinality for interval-valued fuzzy sets. As a promising application we consider using interval-valued relative cardinality to construct the family of parameterised subsethood measures.