**1. Introduction**

Rough set theory, as a a tool to deal with various types of data in data mining, was proposed by Pawlak [1,2] in 1982. Since then, rough set theory has been extended to generalized rough sets based on other notions such as binary relations, neighborhood systems and coverings.

Covering-based rough sets [3–5] were proposed to deal with the type of covering data. In application, they have been applied to knowledge reduction [6,7], decision rule synthesis [8,9], and other fields [10–12]. In theory, covering-based rough set theory has been connected with matroid theory [13–16], lattice theory [17,18] and fuzzy set theory [19–22].

Zadeh's fuzzy set theory [23] addresses the problem of how to understand and manipulate imperfect knowledge. It has been used in various applications [24–27]. Recent investigations have attracted more attention on combining covering-based rough set and fuzzy set theories. There are many fuzzy covering rough set models proposed by researchers, such as Ma [28] and Yang et al. [20].

Wang et al. [29] presented single valued neutrosophic sets (SVNSs) which can be regarded as an extension of IFSs [30]. Neutrosophic sets and rough sets both can deal with partial and uncertain information. Therefore, it is necessary to combine them. Recently, Mondal and Pramanik [31] presented the concept of rough neutrosophic set. Yang et al. [32] presented a SVN rough set model based on SVN relations. However, SVNSs and covering-based rough sets have not been combined up to now. In this paper, we present two types of SVN covering rough set models. This new combination is a bridge, linking SVNSs and covering-based rough sets.

As we know, the multiple criteria decision making (MCDM) is an important tool to deal with more complicated problems in our real world [33,34]. There are many MCDM methods presented based on different problems or theories. For example, Liu et al. [35] dealt with the challenges of many criteria in the MCDM problem and decision makers with heterogeneous risk preferences. Watróbski et al. [36] proposed a framework for selecting suitable MCDA methods for a particular decision situation. Faizi et al. [37,38] presented an extension of the MCDM method based on hesitant fuzzy theory. Recently, many researchers have studied decision making (DM) problems by rough set models [39–42]. For example, Zhan et al. [39] applied a type of soft rough model to DM problems. Yang et al. [32] presented a method for DM problems under a type of SVN rough set model. By investigation, we have observed that no one has applied SVN covering rough set models to DM problems. Therefore, we construct the covering SVN decision information systems according to the characterizations of DM problems. Then, we present a novel method to DM problems under one of the SVN covering rough set models. Moreover, the proposed decision making method is compared with other methods, which were presented by Yang et al. [32], Liu [43] and Ye [44].

The rest of this paper is organized as follows. Section 2 reviews some fundamental definitions about covering-based rough sets and SVNSs. In Section 3, some notions and properties in SVN *β*-covering approximation space are studied. In Section 4, we present two types of SVN covering rough set models, based on the SVN *β*-neighborhoods and the *β*-neighborhoods. In Section 5, some new matrices and matrix operations are presented. Based on this, the matrix representations of the SVN approximation operators are shown. In Section 6, a novel method to decision making (DM) problems under one of the SVN covering rough set models is proposed. Moreover, the proposed DM method is compared with other methods. This paper is concluded and further work is indicated in Section 7.

## **2. Basic Definitions**

Suppose *U* is a nonempty and finite set called universe.

**Definition 1** (Covering [45,46])**.** *Let U be a universe and C a family of subsets of U. If none of subsets in C is empty and* A *C* = *U, then C is called a covering of U.*

The pair (*<sup>U</sup>*, **C**) is called a covering approximation space.

**Definition 2** (Single valued neutrosophic set [29])**.** *Let U be a nonempty fixed set. A single valued neutrosophic set (SVNS) A in U is defined as an object of the following form:*

$$A = \{ \langle \mathbf{x}, T\_A(\mathbf{x}), I\_A(\mathbf{x}), F\_A(\mathbf{x}) \rangle : \mathbf{x} \in \mathcal{U} \},$$

*where TA*(*x*) : *U* → [0, 1] *is a truth-membership function, IA*(*x*) : *U* → [0, 1] *is an indeterminacy-membership function and FA*(*x*) : *U* → [0, 1] *is a falsity-membership function for any x* ∈ *U. They satisfy* 0 ≤ *TA*(*x*) + *IA*(*x*) + *FA*(*x*) ≤ 3 *for all x* ∈ *U. The family of all single valued neutrosophic sets in U is denoted by SVN*(*U*)*. For convenience, a SVN number is represented by α* = *a*, *b*, *c, where a*, *b*, *c* ∈ [0, 1] *and a* + *b* + *c* ≤ 3*.*

Specially, for two SVN numbers *α* = *a*, *b*, *c* and *β* = *d*,*e*, *f*, *α* ≤ *β* ⇔ *a* ≤ *d*, *b* ≥ *e* and *c* ≥ *f* . Some operations on *SVN*(*U*) are listed as follows [29,32]: for any *A*, *B* ∈ *SVN*(*U*),

(1) *A* ⊆ *B* iff *TA*(*x*) ≤ *TB*(*x*), *IB*(*x*) ≤ *IA*(*x*) and *FB*(*x*) ≤ *FA*(*x*) for all *x* ∈ *U*.

(2) *A* = *B* iff *A* ⊆ *B* and *B* ⊆ *A*.

(3) *A* ∩ *B* = {*<sup>x</sup>*, *TA*(*x*) ∧ *TB*(*x*), *IA*(*x*) ∨ *IB*(*x*), *FA*(*x*) ∨ *FB*(*x*) : *x* ∈ *<sup>U</sup>*}.


(6) *A* ⊕ *B* = {*<sup>x</sup>*, *TA*(*x*) + *TB*(*x*) − *TA*(*x*) · *TB*(*x*), *IA*(*x*) · *IB*(*x*), *FA*(*x*) · *FB*(*x*) : *x* ∈ *<sup>U</sup>*}.

#### **3. Single Valued Neutrosophic** *β***-Covering Approximation Space**

In this section, we present the notion of SVN *β*-covering approximation space. There are two basic concepts in this new approximation space: SVN *β*-covering and SVN *β*-neighborhood. Then, some of their properties are studied.

**Definition 3.** *Let U be a universe and SVN*(*U*) *be the SVN power set of U. For a SVN number β* = *a*, *b*, *c, we call C* F = {*<sup>C</sup>*1, *C*2, ··· , *Cm*}*, with Ci* ∈ *SVN*(*U*)(*i* = 1, 2, ..., *<sup>m</sup>*)*, a SVN β-covering of U, if for all x* ∈ *U, Ci* ∈ *C* F *exists such that Ci*(*x*) ≥ *β. We also call* (*<sup>U</sup>*,*<sup>C</sup>* F ) *a SVN β-covering approximation space.*

**Definition 4.** *Let C* F *be a SVN β-covering of U and C* F = {*<sup>C</sup>*1, *C*2, ... , *Cm*}*. For any x* ∈ *U, the SVN β-neighborhood* N *β x of x induced by C* F *can be defined as:*

$$\mathbb{N}\_x^\mathbb{R} = \cap \{ \mathbb{C}\_i \in \hat{\mathcal{C}} : \mathbb{C}\_i(x) \ge \beta \}. \tag{1}$$

F

.

Note that *Ci*(*x*) is a SVN number *TCi*(*x*), *ICi*(*x*), *FCi*(*x*) in Definitions 3 and 4. Hence, *Ci*(*x*) ≥ *β* means *TCi*(*x*) ≥ *a*, *ICi*(*x*) ≤ *b* and *FCi*(*x*) ≤ *c* where SVN number *β* = *a*, *b*, *c*.

**Remark 1.** *Let C* F *be a SVN β-covering of U, β* = *a*, *b*, *c and C* F = {*<sup>C</sup>*1, *C*2,..., *Cm*}*. For any x* ∈ *U,*

$$\widetilde{\mathbb{N}}\_{\mathbf{x}}^{\widetilde{\mathbb{B}}} = \cap \{ \mathbb{C}\_{i} \in \widehat{\mathcal{C}} : T\_{\mathbb{C}\_{i}}(\mathbf{x}) \ge a\_{\star} \ I\_{\mathbb{C}\_{i}}(\mathbf{x}) \le b\_{\star} \ F\_{\mathbb{C}\_{i}}(\mathbf{x}) \le c \}. \tag{2}$$

**Example 1.** *Let U* = {*<sup>x</sup>*1, *x*2, *x*3, *x*4, *<sup>x</sup>*5}*, C* F = {*<sup>C</sup>*1, *C*2, *C*3, *<sup>C</sup>*4} *and β* = 0.5, 0.3, 0.8*. We can see that C* F *is a SVN β-covering of U in Table 1.*

**Table 1.** The tabular representation of single valued neutrosophic (SVN) *β*-covering **C** 


*Then,*

$$
\widetilde{\mathbb{N}}\_{\mathbf{x}\_1}^{\beta} = \mathbb{C}\_1 \cap \mathbb{C}\_2,\\
\widetilde{\mathbb{N}}\_{\mathbf{x}\_2}^{\beta} = \mathbb{C}\_1 \cap \mathbb{C}\_2 \cap \mathbb{C}\_4,\\
\widetilde{\mathbb{N}}\_{\mathbf{x}\_3}^{\beta} = \mathbb{C}\_3 \cap \mathbb{C}\_4,\\
\widetilde{\mathbb{N}}\_{\mathbf{x}\_4}^{\beta} = \mathbb{C}\_1 \cap \mathbb{C}\_4,\\
\widetilde{\mathbb{N}}\_{\mathbf{x}\_5}^{\beta} = \mathbb{C}\_2 \cap \mathbb{C}\_3 \cap \mathbb{C}\_4.
$$

Hence, all SVN *β*-neighborhoods are shown in Table 2.

**Table 2.** The tabular representation of N *β xk* (*k* = 1, 2, 3, 4, 5).


In a SVN *β*-covering approximation space (*<sup>U</sup>*, **C** F ), we present the following properties of the SVN *β*-neighborhood.

**Theorem 1.** *Let C* F *be a SVN β-covering of U and C* F = {*<sup>C</sup>*1, *C*2, ... , *Cm*}*. Then, the following statements hold:*

