Neutrosophic Sets and Systems, Vol. 21, 2018
36
University of New Mexico
Rough Neutrosophic Multisets Relation with Application in Marketing Strategy Suriana Alias1, Daud Mohamad2, and Adibah Shuib3 1
Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara (UiTM) Kelantan, Campus Machang, Kelantan, 18500, Malaysia. E-mail: suria588@kelantan.uitm.edu.my 2 Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara (UiTM) Shah Alam, Shah Alam, 40450, Malaysia. E-mail: daud@tmsk.uitm.edu.my 3 Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara (UiTM) Shah Alam, Shah Alam, 40450, Malaysia. E-mail: adibah@tmsk.uitm.edu.my
Abstract. The concepts of rough neutrosophic multisets can be easily extended to a relation, mainly since a relation is also a set, i.e. a subset of a Cartesian product. Therefore, the objective of this paper is to define the definition of rough neutrosophic multisets relation of Cartesian product over a universal set. Some of the relation properties of rough neutrosophic multisets such as max, min, the composition of two rough neutrosophic multisets relation, inverse rough neutrosophic multisets relation, and reflexive, symmetric and transitive rough neutrosophic multisets relation over the universe are defined. Subsequently, their properties are successfully proven. Finally, the application of rough neutrosophic multisets relation for decision making in marketing strategy is presented. Keywords: Neutrosophic Multisets, relation, rough set, rough neutrosophic multisets
1 Introduction Imperfect information resulted in an incomplete, imprecision, inconsistency and uncertainty information whereby all the condition must be overcome to represent the perfect information. A relation between each information from the same universe or object is also an important criterion of the information to explain the strong relationship element between them. Fuzzy sets as defined by Zadeh [1] has been used to model the imperfect information especially for uncertainty types of information by representing the membership value between [0, 1]. This indicates the human thinking opinion by replacing the information of linguistic value. Many theories were later introduced with the aim of establishing a fuzzy relation structure [2]. Attanassov introduced an intuitionistic fuzzy set by generalizing the theory of fuzzy sets and introducing two grades of the membership function, namely the degree of membership function and degree of non-membership function [3]. This theory has made the uncertainty decision more interesting. Meanwhile, Burillo et al. studied the intuitionistic fuzzy relation with properties [4]. There are also another theory introduced for solving uncertainty condition such as rough set [5] and soft set [6]. All these studies have extended to rough relation [7] and soft set relation [8]. Smarandache introduced a neutrosophic set as a generalization of the intuitionistic fuzzy set theory [9]. He believed that somehow in a life situation, especially for uncertainty condition, there also exist in-between (indeterminacy) opinion or unexpected condition that cannot be controlled. Instead of two grades of the membership function, neutrosophic set introduced in-between (indeterminacy) function where there exists an element which consists of a set of truth membership function (T), indeterminacy function (I) and falsity membership function (F). Compared to other uncertainty theories, the neutrosophic set can deal with indeterminacy situation. The study in neutrosophic relation with properties are also discussed [10], [11]. Later, Smarandache et al. refined T, I, F to T1, T2, …, Tm and I1, I2, …, In and F1, F2, …, Fr was also known as a neutrosophic refined set or neutrosophic multisets [12], [13]. Instead of one-time occurring for each element of T, I, F, the neutrosophic refined set allowed an element of T, I, F to occur more than once with Suriana Alias, Daud Mohamad and Adibah Shuib. Rough Neutrosophic Multisets Relation with Application in Marketing Strategy