Abstract
The effect of uncertainty in information raises human logical thinking due to many variabilities from expert decision making. The generalization of rough neutrosophic multisets theory tries to overcome this problem. The rough neutrosophic multisets allow the multiplicity of information represented by truth membership sequences, indeterminacy membership sequences, and falsity membership sequences. The lower and upper approximations belong to this triple membership sequence, making the information result more valuable since the relationship among this membership sequence makes sense. This chapter introduces the generalization of rough neutrosophic multisets with some operators and an illustrative example. The rough neutrosophic multisets basic operators, algebraic operators, and “AND” and “OR” operators are defined with some property propositions.
Handbook of Research on Advances and Applications of Fuzzy Sets and Logic
TopIntroduction
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.
Key Terms in this Chapter
Uncertainty Information: The unbalance information according to decision-maker when involving human thinking.
Falsity: The rejected value when the decision making is involved.
Multisets: Redundant elements occur more than once and group together.
Indeterminate: The unsure about something, whether to give a chance or not.
Approximation: The nearest value for representing the result in some situations.
Neutrosophic: Term used to replace the philosophy of neutral condition in uncertainty information with a relationship of truth and falsity.
Truth: The acceptable value when the decision making has happened.