On Different Types of Single-Valued Neutrosophic Covering Rough Set with Application in Decision-Making

. This paper aims to propose the notion of Type-1 single-valued neutrosophic complementary β -neighborhood (brieﬂy, Type-1 SVN complementary β -neighborhood) and use it to introduce a novel class of 1-single-valued neutrosophic β -covering rough set (brieﬂy, 1-SVN β -CRS). Then, we will merge the 1-SVN β -neighborhood and 1-SVN complementary β -neighborhood to create new two models of 1-SVN β -CRS. Furthermore, we will discuss the relationships between the present work and Wang and Zhang’s work. For further study on Type-2 Wang and Zhang’s models, we will deﬁne the 2-SVN complementary β -neighborhood and use it to present a novel class of 2-SVN β -CRS. Also, we combine the 2-SVN β -neighborhood and 2-SVN complementary β -neighborhood to investigate the new two models of 2-SVN β -CRS. Lately, we will demonstrate two illustrative examples as real problems to show the diﬀerences between two of our approaches and Wang and Zhang’s approach.

e notion of single-valued neutrosophic sets (briefly, SVNS) was developed by Wang et al. [39]. SVNS is a natural extension of the intuitionistic fuzzy set (briefly, IFS) [40]. Smarandache [41] investigated a new set called neutrosophic set as a generalization of mathematical tools (i.e., fuzzy set [35], interval-valued fuzzy set [42], IFS [40], and interval-valued intuitionistic fuzzy set [43]). In 2015, Mondal and Pramanik [44] demonstrated a new terminology called rough neutrosophic set. By using SVN relation, Yang et al. [45] introduced the SVN rough set model, and based on the notion of Type-1 SVN β-neighborhoods, Wang and Zhange [46] proposed two models of Type-1 SVNβ-covering rough sets (briefly, SVNβ-CRS). Furthermore, they presented a new kind of SVNβ-CRS called Type-2 SVNβ-CRS utilizing Type-2 SVN β-neighborhoods in [47]. e notions of neutrosophic soft rough sets and its generalizations are presented in [48][49][50][51][52][53]. By the above discussion and extend the other work (see [46,47]) in SVNβ-CRS. We will generalize these methods in [46,47] by boosting the lower approximation and minimizing the upper approximation, which is a big challenge to every author. Consequently, the motivation of this paper is to improve this area is obtained by introducing the notion of 1-SVN complementary β-neighborhood (resp., 2-SVN complementary β-neighborhood) to build novel classes of 1-SVNβ-CRS (resp., 2-SVNβ-CRS). And, by joining 1-SVN β-neighborhoods (resp., 2-SVN β-neighborhoods) and 1-SVN complementary β-neighborhood (resp., 2-SVN complementary β-neighborhood), we obtain two new SVN β-neighborhoods which establish two new models of 1-SVNβ-CRS (resp., 2-SVNβ-CRS). Also, we discuss the properties of the two proposed covering methods. Finally, we apply our work (i.e., two proposed methods) to solve decision-making problems. e organization of this article is as follows. In Section 2, we give a basic notion about the presented study. Section 3 establishes the definition of 1-SVN complementary β-neighborhood, and hence, a new model of 1-SVNβ-CRS is proposed. Also, by merging between the 1-SVN β-neighborhoods and its complementary, we set up two other models of 1-SVNβ-CRS.
us, the relevant characteristics are also studied. Section 4 constructs the notion of 2-SVN complementary β-neighborhood, and thus, a new model of 2-SVNβ-CRS is proposed. By merging between the 2-SVN β-neighborhoods and its complementary, we also set up two other models of 2-SVN β-CRS. en, the relevant properties are also discussed. e decision-making approaches to the two methods are mentioned in Sections 3 and 4 are investigated in Section 5. Also, in this section, we compare between our approach and Wang and Zhang's approach. Section 6 shows the overall benefits of our study.

Preliminaries
In this section, we review some basic terminologies about the subject of this study.
Definition 1 (Cf. [26]). Assume that Ω is a universe and Γ is a family of subsets of Ω. If no element in Γ is empty and Ω � ∪ C∈Γ C, then Γ is called a covering of Ω, and (Ω, Γ) is called a covering approximation space (briefly, CAS).
Definition 3 (Cf. [54,55]). Assume that Ω is not an empty set. For each x ∈ Ω, define the SVN set A ⊆ Ω as the following formula: where In 2018, Wang et al. [39] established the notion of SVN β-covering approximation space, and then, Wang and Zhang [46,47] used this notion to create two types of the covering method as in the following definition.
Definition 4 (Cf. [46,47]). Let Ω be a universe and SVN (Ω) be the SVN power set of Ω. For a SVN number β � (a, b, c), Here, ∀A, B ∈ SVN(Ω), and we have the following relation, union, and intersection operations.

Type-1 SVN Complementary β-Neighborhood and Three New Kinds of Type-1 SVNβ-CRS
We will propose the concept of a type-1 SVN complementary β-neighborhood and three new kinds of Type-1 SVNβ-CRS and introduce several definitions, propositions, and examples as indicated below.
In Table 1 Table 2 contains the results of type-1 SVN β-neighborhood.
us, we can obtain the values of type-1 SVN complementary β-neighborhood as in Table 3.
Hence, we can merge 1 N  Table 4.
x as set in Table 5.

Proof
(1) It follows directly from Definitions 5 and 7.

Mathematical Problems in Engineering
□ Now, we present the three new types of 1-1-SVNβCRSs based on Definitions 5 and 7 as indicated below.
en, the following statements hold: Proof. We shall only prove (SVNL1), (SVNL2), (SVNL3), and (SVNL4). (SVNL1): (SVNL2): let A, B ∈ SVN(Ω) such that A⊆B (i.e., T A ≤ T B , I B ≤ I A and F B ≤ F A ) and x ∈ Ω. en, we get the following result: Mathematical Problems in Engineering □ Now, we proceed to explain some relationships among these models.
us, we can merge 2 M  Table 8. Furthermore, we can calculate 2 N β x ⊔ 2 M β x , as set in Table 9. Proposition 6. Let (Ω, Γ) be a 2-SVNβCAS, for some β � 〈a, b, c〉 and for each x, y, z ∈ Ω. en, the following statements hold:

□
Here, we construct three new types of 2-1-SVNβCRSs based on Definitions 5 and 9 as seen below.

Decision-Making Approach to DM Based on SVNβCRSs
indicates the symptom value for each patient which is known by a doctor D, for some β � 〈a, b, c〉, and (Ω, Γ) is a Type-1 SVN β-CRS, where T C i (x r ) ∈ [0, 1] (i.e., the degree that doctor D confirms the patient x r has symptom y i ), I C i (x r ) ∈ [0, 1] (i.e., the degree that doctor D is not sure if the patient x r has symptom y i ), F C i (x r ) ∈ [0, 1] (i.e., the degree that doctor D confirms the patient x r does not have any symptom y i ), and According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps: Step 1: consider, for each x r ∈ Ω, there is at least one y i ∈ V such that the symptom value C i for patient x r is not less than β, where β is a critical value.
Step 2: consider A(x r ) � 〈d, e, f〉 is the evaluation by a decision maker D, where d is a possible degree, e is an indeterminacy degree, and f is an impossible degree of A disease.
Step 3: based on this information, use Definition 8 and 3-1-SVNβCRSs model to calculate the lower and upper approximation of A.
Step 4: calculate R A by the following equation: where Step 5: calculate the decision method by the following formula: hence, ranking the alternatives.
Based on these steps, we give an algorithm to solve the decision-making problems based on Definition 8. e steps corresponding to it are summarized in Algorithm 1.

Method II.
Suppose that Ω � x r : r � 1, . . . , k is the set of alternatives (papers), m is main attributes (symptoms) (e.g., spot and steak) V � y i : i � 1, 2, . . . , m of A paper trouble, indicates the symptom value for each paper which known by an investigator I, for some β � 〈a, b, c〉, and (Ω, Γ) is a Type-2 SVN β-CRS, where T C i (x r ) ∈ [0, 1] (i.e., the degree that the investigator I asserts the paper x r has symptom y i ), I C i (x r ) ∈ [0, 1] (i.e., the degree that the investigator I is not sure whether the paper x r has symptom y i ), F C i (x r ) ∈ [0, 1] (i.e., the degree that investigator I affirms paper x r does not have any symptom y i ), and 0 According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps: Step 1: consider, for each x r ∈ Ω, there is at least one y i ∈ V such that the symptom value C i for paper x r is not less than β (i.e., C i (x)≽β), where β is a critical value.
Step 2: consider A(x r ) � 〈d, e, f〉 is the evaluation by a decision maker I, where d is a possible degree, e is an indeterminacy degree, and f is an impossible degree of A disease.
Step 3: based on this information, use Definition 10 and 3-2-SVNβCRSs model to calculate the lower and upper approximation of A.
Step 4: calculate R A by the following equation: where Step 5: calculate the decision method by the following formula.
hence, ranking the alternatives.
Based on these steps, we give an algorithm to solve the decision-making problems based on Definition 10. e steps corresponding to it are summarized in Algorithm 2.

Numerical Example
Example 5. Diseased people form a set Ω � x 1 , x 2 , x 3 , x 4 , x 5 and their relevant symptoms are collected by the attribute set V � cough(y 1 ), fever(y 2 ), sore(y 3 ), headache(y 4 )} for A disease. Here, the following steps of the algorithm described are implemented.
Step 1: under the attribute set, doctor D estimates each patient and presents its decisions with suitable values which are summarized in Table 1.
Step 4: compute R A as follows: Step 5: according to the above information, we get S(x) as follows: and hence, we get the ranking order as So, by the above computation, the verdict of the decision maker D is x 5 .

Example 6.
Let Ω � x 1 , x 2 , x 3 , x 4 , x 5 be the set of papers, and their relevant symptoms are collected by the attribute set V � spot(y 1 ), steak(y 2 ), crater(y 3 ), fracture(y 4 ) for A paper error. Here, the following steps of the algorithm described are implemented.
Step 1: under the attribute set, investigator I estimates each paper and presents its decisions with suitable values which are summarized in Table 1.
Step 2: consider β � 〈0.5, 0.1, 0.8〉 is a critical and Γ � C 1 , C 2 , C 3 , C 4 is a 2-SVNβCRS. en, we compute the Type-2 SVN β-neighborhood 2 N Step 4: compute R A as follows: Step 5: according to above information, we get S(x) as follows: and hence, we get the ranking order as So, by the above calculations, the verdict of the decision maker I is x 4 .

Comparative Analysis.
e major purpose of our presented work is eligible to raise the lower approximation and reduce the upper approximation of the previous study by Wang and Zhang's methods [46,47], as visible in Examples 2 and 4. To clarify the comparisons between Wang and Zhang's methods [46,47] and our methods, the sorting outcomes of these decision-making models are listed in Table 10 for 1-SVNβCAS and Table 11 for 2-SVNβCAS.
An easy way to explain these outcomes, see Figures 1 and 2 which simplify the comparisons between our presented method and the previous one. Figure 1 explained the differences between the outcomes using our model (3-1-SVNβCAS) and the last one (1-1-SVNβCRSs). Furthermore, Figure 2 illustrated the comparisons between the values through our model (3-2-SVNβCAS) and the previous one (1-2-SVNβCRSs). us, there are slight differences among these distinct methods, and these variations made our model better than others.

Conclusion
is work is extended to Wang and Zhang's studies in [46,47]. We presented the definitions of 1-SVN complementary β-neighborhoods and 2-SVN complementary β-neighborhoods. We use them to set up new models of 1-SVN β-CRS and 2-SVN β-CRS, respectively. Moreover, by merging the Type-1 neighborhoods (resp., Type-2 neighborhoods) and Type-1 complementary neighborhoods (resp., Type-2 complementary neighborhoods), we obtain two new types of Type-1 neighborhoods and Type-2 neighborhoods, respectively. us, two new classes of 1-SVN β-CRS and 2-SVN β-CRS are investigated. To explain the differences between these new and older types of covering methods, see Examples 2 and 4. For more clarification about them, see Figures 1 and 2. ere are some issues in these two covering methods: (1) If β � (0.5, 0.1, 0.8) in Example 2, then Γ is not 1-SVNβCRSs, but it is applicable in 2-SVNβCRSs (2) If β � (0.5, 0.3, 0.8) in Example 4, then Γ is not 2-SVNβCRSs, but it is applicable in 1-SVNβCRSs In short, the two methods are considered complementary to each other, which means if there are some failures in 1-SVNβCRSs, the 2-SVNβCRSs is working instead and vice versa.
In the future, we can extend the results of this study as a combination between 1-SVN (or 2-SVN) complementary β-neighborhoods and published papers (see [50][51][52][53][54][55]). In addition, one may investigate further based on 1-SVN (or 2-SVN) complementary β-neighborhoods with some links to topology as in [26,48]. Finally, there are many areas (for example, several comparative of this proposed method) which can be presented by researchers in the next paper.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.
Different methods Obtain a decision Wang and Zhang's model [47] x 5 ≻x 1 ≻x 2 ≻x 4 ≻x 3 Our model  Our method Wang method Figure 2: e representations of the results by using our model and Wang and Zhang's model [47].