A Decomposition Approach to Type 2 Interval Arithmetic

The classic interval has precise borders A=[a_,a¯]A = \left[ {\underline{a},\bar a} \right]. Therefore, it can be called a type 1 interval. Because of great practical importance of such interval data, several versions of type 1 interval arithmetic have been created. However, sometimes precise borders a_\underline{a} and ā of intervals cannot be determined in practice. If the borders are uncertain, then we have to do with type 2 intervals. A type 2 interval can be denoted as AT2=[[a_L,a_R],[a¯L,a¯R]]{A_{T2}} = \left[ {\left[ {{\underline{a}_L},{\underline{a}_R}} \right],\left[ {{{\bar a}_L},{{\bar a}_R}} \right]} \right]. The paper presents multidimensional decomposition RDM type 2 interval arithmetic (D-RDM-T2-I arithmetic), where RDM means relative-distance measure. The decomposition approach considerably simplifies calculations and is transparent for users. Apart from this arithmetic, examples of its applications are also presented. To the authors’ best knowledge, no papers on this arithmetic exist. D-RDM-T2-I arithmetic is necessary to create type 2 fuzzy arithmetic based on horizontal μ-cuts, which the authors aim to do.