Rough Neutrosophic Multisets: Operations Over Rough Neutrosophic Multisets

Handbook of Research on Advances and Applications of Fuzzy Sets and Logic

Introduction

This chapter deliberates the definition of Rough Neutrosophic Multisets (RNM) based on the equivalence relation of the universe of discourse by raising the lower and upper approximations for a single-valued neutrosophic multisets (SVNM) introduced by Ye & Ye (2014) in Pawlak’s approximation space (Pawlak, 1982). Generally, rough set are deemed as a powerful tool in explaining imprecise information with a set of prediction levels (lower and upper approximation). While the lower approximation in Pawlak’s approximation space encompasses element sets that certainly belongs to the object, the upper approximation on the other hand contains element sets that possibly belongs to the object. Moreover, the RNM allows an object's multiplicity element to occur more than once with the same or different value. A RNM is also a generalization of a hybrid method of a neutrosophic set by Smarandache (1999) and a rough neutrosophic set by Broumi et al. (2014). Nevertheless, these previous theories are unable to deal with multiplicity element in the sets of prediction level. Therefore, gaining the determination from the hybrid method based on the generalization theory, RNM will overcome the complex decision-making situation with the aim to provide the exact value of interpretation of uncertainty data (uncertainty data interpretation) that are not explored. Relatively, a pair of lower and upper approximation of neutrosophic multisets is called rough neutrosophic multisets (RNM). The new notion of RNM theory is useful in solving uncertainty information by considering the same contingent (equivalence relation) of information.

This paper aims to introduce a new notion for uncertainty information set theory namely RNM with a number of operators and properties. The basic operations such as complement, union, intersection, contained, and equality is discussed using examples. Subsequently, the algebraic operations over RNM namely addition, multiplication, scalar multiplication, and power of rough neutrosophic multisets are defined with an illustrative example. Finally, the “AND” and “OR” operation for RNM is presented.