Decision-Making Approach Based on Neutrosophic Rough Information

: Rough set theory and neutrosophic set theory are mathematical models to deal with incomplete and vague information. These two theories can be combined into a framework for modeling and processing incomplete information in information systems. Thus, the neutrosophic rough set hybrid model gives more precision, ﬂexibility and compatibility to the system as compared to the classic and fuzzy models. In this research study, we develop neutrosophic rough digraphs based on the neutrosophic rough hybrid model. Moreover, we discuss regular neutrosophic rough digraphs, and we solve decision-making problems by using our proposed hybrid model. Finally, we give a comparison analysis of two hybrid models, namely, neutrosophic rough digraphs and rough neutrosophic digraphs.


Introduction
The concept of a neutrosophic set, which generalizes fuzzy sets [1] and intuitionistic fuzzy sets [2], was proposed by Smarandache [3] in 1998, and it is defined as a set about the degree of truth, indeterminacy, and falsity.A neutrosophic set A in a universal set X is characterized independently by a truth-membership function (T A (x)), an indeterminacy-membership function (I A (x)), and a falsity-membership function (F A (x)).To apply neutrosophic sets in real-life problems more conveniently, Smarandache [3] and Wang et al., [4] defined single-valued neutrosophic sets which take the value from the subset of [0, 1].
Rough set theory was proposed by Pawlak [5] in 1982.Rough set theory is useful to study the intelligence systems containing incomplete, uncertain or inexact information.The lower and upper approximation operators of rough sets are used for managing hidden information in a system.Therefore, many hybrid models have been built such as soft rough sets, rough fuzzy sets, fuzzy rough sets, soft fuzzy rough sets, neutrosophic rough sets and rough neutrosophic sets for handling uncertainty and incomplete information effectively.Dubois and Prade [6] introduced the notions of rough fuzzy sets and fuzzy rough sets.Liu and Chen [7] have studied different decision-making methods.Mordeson and Peng [8] presented operations on fuzzy graphs.Akram et al., [9][10][11][12] considered several new concepts of neutrosophic graphs with applications.Rough fuzzy digraphs with applications are presented in [13,14].To get the extended notion of neutrosophic sets and rough sets, many attempts have been made.Broumi et al., [15] introduced the concept of rough neutrosophic sets.Yang et al., [16] proposed single valued neutrosophic rough sets by combining single valued neutrosophic sets and rough sets, and established an algorithm for the decision-making problem based on single valued neutrosophic rough sets on two universes.Nabeela et al., [17] and Sayed et al., [18] introduced rough neutrosophic digraphs, in which they have approximated neutrosophic set under the influence of a crisp equivalence relation.In this research article, we apply another hybrid set model, neutrosophic rough, to graph theory.We deal with regular neutrosophic rough digraphs and then solve the decision-making problem by using our proposed hybrid model.
Our paper is organized as follows: Firstly, we develop the notion of neutrosophic rough digraphs and present some numerical examples.Secondly, we explore basic properties of neutrosophic rough digraphs.In particular, we investigate the regularity of neutrosophic rough digraphs.We describe novel applications of our proposed hybrid decision-making method.To compare the two notions, rough neutrosophic digraphs and neutrosophic rough digraphs, we give a comparison analysis.Finally, we end the paper by concluding remarks.

Neutrosophic Rough Information
Definition 1. [4] Let Z be a nonempty universe.A neutrosophic set N on Z is defined as follows: where the functions T, I, F:Z→ [0, 1] represent the degree of membership, the degree of indeterminacy and the degree of falsity.Definition 2. [5] Let Z be a nonempty universe and R an equivalence relation on Z.A pair (Z, R) is called an approximation space.Let N * be a subset of Z and the lower and upper approximations of N * in the approximation space (Z, R) denoted by RN * and RN * are defined as follows: where [x] R denotes the equivalence class of R containing x.A pair (RN * , RN * ) is called a rough set.

Example 5. Consider the two regular neutrosophic rough digraphs G
as shown in Figures 3 and 6, respectively, then the direct sum of G 1 and G 2 as shown in Figure 7 is not a regular neutrosophic rough digraph.
Then, for any two vertices (x 1 , x 2 ) and (y ) and G 2 = (G 2 , G 2 ) be two neutrosophic rough digraphs on the two crisp sets X * 1 = {p, q} and X * 2 = {u, v, w, x} as shown in Figures 8 and 9. Then the residue product of G 1 and G and the respective figures are shown in Figure 10.

Applications to Decision-Making
In this section, we present some real life applications of neutrosophic rough digraphs in decision making.In decision-making, the selection is facilitated by evaluating each choice on the set of criteria.The criteria must be measurable and their outcomes must be measured for every decision alternative.

Online Reviews and Ratings
Customer reviews are increasingly available online for a wide range of products and services.As customers search online for product information and to evaluate product alternatives, they often have access to dozens or hundreds of product reviews from other customers.These reviews are very helpful in product selection.However, only considering the good reviews about a product is not very helpful in decision-making.The customer should keep in mind bad and neutral reviews as well.We use percentages of good reviews, bad reviews and neutral reviews of a product as truth membership, false membership and indeterminacy respectively.

Table 1 .
Companies and their ratings.