Shadowed numbers and their standard and multidimensional arithmetic

Abstract

A shadowed set was introduced by W. Pedrycz as a concept of modeling vagueness. There are methods and algorithms for obtaining a shadowed set on the basis of a fuzzy set; a shadowed number can also be obtained from a fuzzy number. This article presents definitions of a shadowed number and two concepts of its arithmetic. The first arithmetic is called standard shadowed arithmetic (SSA) and relies on standard interval arithmetic (SIA), while the second is called multidimensional RDM shadowed arithmetic (RDMSA) and is based on multidimensional relative distance measure interval arithmetic (RDMIA). This paper presents the basic properties of operations on shadowed numbers with SSA and RDMSA. It also provides examples that show the difference between the results obtained with SSA and those obtained with multidimensional RDMSA. RDMSA introduces a multidimensional approach to the concept of uncertainty calculation results. Theories and examples presented in this paper will help to develop three-way decision methods and models, and can be applied in granular computing.