Open AccessArticle

Two Types of Single Valued Neutrosophic Covering Rough Sets and an Application to Decision Making

Jingqian Wang

¹ and

Xiaohong Zhang

^2,*

College of Electrical and Information Engineering, Shaanxi University of Science and Technology, Xi’an 710021, China

School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, China

Author to whom correspondence should be addressed.

Symmetry 2018, 10(12), 710; https://doi.org/10.3390/sym10120710

Submission received: 8 November 2018 / Revised: 25 November 2018 / Accepted: 27 November 2018 / Published: 3 December 2018

(This article belongs to the Special Issue Fuzzy Techniques for Decision Making 2018)

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Abstract

In this paper, to combine single valued neutrosophic sets (SVNSs) with covering-based rough sets, we propose two types of single valued neutrosophic (SVN) covering rough set models. Furthermore, a corresponding application to the problem of decision making is presented. Firstly, the notion of SVN

β

-covering approximation space is proposed, and some concepts and properties in it are investigated. Secondly, based on SVN

β

-covering approximation spaces, two types of SVN covering rough set models are proposed. Then, some properties and the matrix representations of the newly defined SVN covering approximation operators are investigated. Finally, we propose a novel method to decision making (DM) problems based on one of the SVN covering rough set models. Moreover, the proposed DM method is compared with other methods in an example.

Keywords:

covering; single valued neutrosophic; matrix representation; decision making

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 [3,4,5] 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 [10,11,12]. In theory, covering-based rough set theory has been connected with matroid theory [13,14,15,16], lattice theory [17,18] and fuzzy set theory [19,20,21,22].

Zadeh’s fuzzy set theory [23] addresses the problem of how to understand and manipulate imperfect knowledge. It has been used in various applications [24,25,26,27]. 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 [39,40,41,42]. 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

⋃ C = U

, then

C

is called a covering of U.

The pair

(U, 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 = {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 : x \in U},

where

T_{A} (x) : U \to [0, 1]

is a truth-membership function,

I_{A} (x) : U \to [0, 1]

is an indeterminacy-membership function and

F_{A} (x) : U \to [0, 1]

is a falsity-membership function for any

x \in U

. They satisfy

0 \leq T_{A} (x) + I_{A} (x) + F_{A} (x) \leq 3

for all

x \in U

. The family of all single valued neutrosophic sets in U is denoted by

S V N (U)

. For convenience, a SVN number is represented by

α = 〈 a, b, c 〉

, where

a, b, c \in [0, 1]

and

a + b + c \leq 3

Specially, for two SVN numbers

α = 〈 a, b, c 〉

and

β = 〈 d, e, f 〉

α \leq β \Leftrightarrow a \leq d

b \geq e

and

c \geq f

. Some operations on

S V N (U)

are listed as follows [29,32]: for any

A, B \in S V N (U)

(1): $A \subseteq B$ iff $T_{A} (x) \leq T_{B} (x)$ , $I_{B} (x) \leq I_{A} (x)$ and $F_{B} (x) \leq F_{A} (x)$ for all $x \in U$ .
(2): $A = B$ iff $A \subseteq B$ and $B \subseteq A$ .
(3): $A \cap B = {〈 x, T_{A} (x) \land T_{B} (x), I_{A} (x) \lor I_{B} (x), F_{A} (x) \lor F_{B} (x) 〉 : x \in U}$ .
(4): $A \cup B = {〈 x, T_{A} (x) \lor T_{B} (x), I_{A} (x) \land I_{B} (x), F_{A} (x) \land F_{B} (x) 〉 : x \in U}$ .
(5): $A^{'} = {〈 x, F_{A} (x), 1 - I_{A} (x), T_{A} (x) 〉 : x \in U}$ .
(6): $A \oplus B = {〈 x, T_{A} (x) + T_{B} (x) - T_{A} (x) \cdot T_{B} (x), I_{A} (x) \cdot I_{B} (x), F_{A} (x) \cdot F_{B} (x) 〉 : x \in U}$ .

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

S V N (U)

be the SVN power set of U. For a SVN number

β = 〈 a, b, c 〉

, we call

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

, with

C_{i} \in S V N (U) (i = 1, 2, \dots, m)

, a SVN β-covering of U, if for all

x \in U

C_{i} \in \hat{C}

exists such that

C_{i} (x) \geq β

. We also call

(U, \hat{C})

a SVN β-covering approximation space.

Definition 4.

Let

\hat{C}

be a SVN β-covering of U and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. For any

x \in U

, the SVN β-neighborhood

{\tilde{N}}_{x}^{β}

of x induced by

\hat{C}

can be defined as:

{\tilde{N}}_{x}^{β} = \cap {C_{i} \in \hat{C} : C_{i} (x) \geq β} .

(1)

Note that

C_{i} (x)

is a SVN number

〈 T_{C_{i}} (x), I_{C_{i}} (x), F_{C_{i}} (x) 〉

in Definitions 3 and 4. Hence,

C_{i} (x) \geq β

means

T_{C_{i}} (x) \geq a

I_{C_{i}} (x) \leq b

and

F_{C_{i}} (x) \leq c

where SVN number

β = 〈 a, b, c 〉

Remark 1.

Let

\hat{C}

be a SVN β-covering of U,

β = 〈 a, b, c 〉

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. For any

x \in U

{\tilde{N}}_{x}^{β} = \cap {C_{i} \in \hat{C} : T_{C_{i}} (x) \geq a, I_{C_{i}} (x) \leq b, F_{C_{i}} (x) \leq c} .

(2)

Example 1.

Let

U = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}

\hat{C} = {C_{1}, C_{2}, C_{3}, C_{4}}

and

β = 〈 0.5, 0.3, 0.8 〉

. We can see that

\hat{C}

is a SVN β-covering of U in Table 1.

Then,

\begin{matrix} {\tilde{N}}_{x_{1}}^{β} = C_{1} \cap C_{2}, {\tilde{N}}_{x_{2}}^{β} = C_{1} \cap C_{2} \cap C_{4}, {\tilde{N}}_{x_{3}}^{β} = C_{3} \cap C_{4}, {\tilde{N}}_{x_{4}}^{β} = C_{1} \cap C_{4}, {\tilde{N}}_{x_{5}}^{β} = C_{2} \cap C_{3} \cap C_{4} . \end{matrix}

Hence, all SVN

β

-neighborhoods are shown in Table 2.

In a SVN

β

-covering approximation space

(U, \hat{C})

, we present the following properties of the SVN

β

-neighborhood.

Theorem 1.

Let

\hat{C}

be a SVN β-covering of U and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then, the following statements hold:

(1): ${\tilde{N}}_{x}^{β} (x) \geq β$ for each $x \in U$ .
(2): $\forall x, y, z \in U$ , if ${\tilde{N}}_{x}^{β} (y) \geq β$ , ${\tilde{N}}_{y}^{β} (z) \geq β$ , then ${\tilde{N}}_{x}^{β} (z) \geq β$ .
(3): For two SVN numbers $β_{1}, β_{2}$ , if $β_{1} \leq β_{2} \leq β$ , then ${\tilde{N}}_{x}^{β_{1}} \subseteq {\tilde{N}}_{x}^{β_{2}}$ for all $x \in U$ .

Proof.

(1): For any $x \in U$ , ${\tilde{N}}_{x}^{β} (x) = (⋂_{C_{i} (x) \geq β} C_{i}) (x) = \underset{C_{i} (x) \geq β}{⋀} C_{i} (x) \geq β$ .
(2): Let $I = {1, 2, \dots, m}$ . Since ${\tilde{N}}_{x}^{β} (y) \geq β$ , for any $i \in I$ , if $C_{i} (x) \geq β$ , then $C_{i} (y) \geq β$ . Since ${\tilde{N}}_{y}^{β} (z) \geq β$ , for any $i \in I$ , $C_{i} (z) \geq β$ when $C_{i} (y) \geq β$ . Then, for any $i \in I$ , $C_{i} (x) \geq β$ implies $C_{i} (z) \geq β$ . Therefore, ${\tilde{N}}_{x}^{β} (z) \geq β$ .
(3): For all $x \in U$ , since $β_{1} \leq β_{2} \leq β$ , ${C_{i} \in \hat{C} : C_{i} (x) \geq β_{1}} \supseteq {C_{i} \in \hat{C} : C_{i} (x) \geq β_{2}}$ . Hence, ${\tilde{N}}_{x}^{β_{1}} = \cap {C_{i} \in \hat{C} : C_{i} (x) \geq β_{1}} \subseteq \cap {C_{i} \in \hat{C} : C_{i} (x) \geq β_{2}} = {\tilde{N}}_{x}^{β_{2}}$ for all $x \in U$ .

□

Proposition 1.

Let

\hat{C}

be a SVN β-covering of U. For any

x, y \in U

{\tilde{N}}_{x}^{β} (y) \geq β

if and only if

{\tilde{N}}_{y}^{β} \subseteq {\tilde{N}}_{x}^{β}

Proof.

Suppose the SVN number

β = 〈 a, b, c 〉

(\Rightarrow)

: Since

{\tilde{N}}_{x}^{β} (y) \geq β

T_{{\tilde{N}}_{x}^{β}} (y) = T_{\underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{⋂_{T_{C_{i}} (x) \geq a}}} C_{i}} (y) = \underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{\underset{T_{C_{i}} (x) \geq a}{⋀}}} T_{C_{i}} (y) \geq a, I_{{\tilde{N}}_{x}^{β}} (y) = I_{\underset{F_{C_{i}} (x) \leq b}{\underset{I_{C_{i}} (x) \leq c}{⋂_{T_{C_{i}} (x) \geq a}}} C_{i}} (y) = \underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{\underset{T_{C_{i}} (x) \geq a}{⋁}}} I_{C_{i}} (y) \leq b

, and

F_{{\tilde{N}}_{x}^{β}} (y) = F_{\underset{F_{C_{i}} (x) \leq b}{\underset{I_{C_{i}} (x) \leq c}{⋂_{T_{C_{i}} (x) \geq a}}} C_{i}} (y) = \underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{\underset{T_{C_{i}} (x) \geq a}{⋁}}} F_{C_{i}} (y) \leq c

Then,

{C_{i} \in \hat{C} : T_{C_{i}} (x) \geq a, I_{C_{i}} (x) \leq b, F_{C_{i}} (x) \leq c} \subseteq {C_{i} \in \hat{C} : T_{C_{i}} (y) \geq a, I_{C_{i}} (y) \leq b, F_{C_{i}} (y) \leq c}

Therefore, for each

z \in U

\begin{matrix} T_{{\tilde{N}}_{x}^{β}} (z) = \underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{\underset{T_{C_{i}} (x) \geq a}{⋀}}} T_{C_{i}} (z) \geq \underset{F_{C_{i}} (y) \leq c}{\underset{I_{C_{i}} (y) \leq b}{\underset{T_{C_{i}} (y) \geq a}{⋀}}} T_{C_{i}} (z) = T_{{\tilde{N}}_{y}^{β}} (z), \\ I_{{\tilde{N}}_{x}^{β}} (z) = \underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{\underset{T_{C_{i}} (x) \geq a}{⋁}}} I_{C_{i}} (z) \leq \underset{F_{C_{i}} (y) \leq c}{\underset{I_{C_{i}} (y) \leq b}{\underset{T_{C_{i}} (y) \geq a}{⋁}}} I_{C_{i}} (z) = I_{{\tilde{N}}_{y}^{β}} (z), \\ F_{{\tilde{N}}_{x}^{β}} (z) = \underset{F_{C_{i}} (x) \leq c}{\underset{I_{C_{i}} (x) \leq b}{\underset{T_{C_{i}} (x) \geq a}{⋁}}} F_{C_{i}} (z) \leq \underset{F_{C_{i}} (y) \leq c}{\underset{I_{C_{i}} (y) \leq b}{\underset{T_{C_{i}} (y) \geq a}{⋁}}} F_{C_{i}} (z) = F_{{\tilde{N}}_{y}^{β}} (z) . \end{matrix}

Hence,

{\tilde{N}}_{y}^{β} \subseteq {\tilde{N}}_{x}^{β}

(\Leftarrow)

: For any

x, y \in U

, since

{\tilde{N}}_{y}^{β} \subseteq {\tilde{N}}_{x}^{β}

T_{{\tilde{N}}_{x}^{β}} (y) \geq T_{{\tilde{N}}_{y}^{β}} (y) \geq a, I_{{\tilde{N}}_{x}^{β}} (y) \leq I_{{\tilde{N}}_{y}^{β}} (y) \leq b and F_{{\tilde{N}}_{x}^{β}} (y) \leq F_{{\tilde{N}}_{y}^{β}} (y) \leq c

Therefore,

{\tilde{N}}_{x}^{β} (y) \geq β

. □

The notion of SVN

β

-neighborhood in the SVN

β

-covering approximation space in the following definition.

Definition 5.

Let

(U, \hat{C})

be a SVN β-covering approximation space and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. For each

x \in U

, we define the β-neighborhood

{\bar{N}}_{x}^{β}

of x as:

{\bar{N}}_{x}^{β} = {y \in U : {\tilde{N}}_{x}^{β} (y) \geq β} .

(3)

Note that

{\tilde{N}}_{x}^{β} (y)

is a SVN number

〈 T_{{\tilde{N}}_{x}^{β}} (y), I_{{\tilde{N}}_{x}^{β}} (y), F_{{\tilde{N}}_{x}^{β}} (y) 〉

in Definition 5.

Remark 2.

Let

\hat{C}

be a SVN β-covering of U,

β = 〈 a, b, c 〉

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. For each

x \in U

{\bar{N}}_{x}^{β} = {y \in U : T_{{\tilde{N}}_{x}^{β}} (y) \geq a, I_{{\tilde{N}}_{x}^{β}} (y) \leq b, F_{{\tilde{N}}_{x}^{β}} (y) \leq c} .

(4)

Example 2

(Continued from Example 1). Let

β = 〈 0.5, 0.3, 0.8 〉

, then we have

\begin{matrix} {\bar{N}}_{x_{1}}^{β} = {x_{1}, x_{2}}, {\bar{N}}_{x_{2}}^{β} = {x_{2}}, {\bar{N}}_{x_{3}}^{β} = {x_{3}, x_{5}}, {\bar{N}}_{x_{4}}^{β} = {x_{2}, x_{4}}, {\bar{N}}_{x_{5}}^{β} = {x_{5}} . \end{matrix}

Some properties of the

β

-neighborhood in a SVN

β

-covering of U are presented in Theorem 2 and Proposition 2.

Theorem 2.

Let

\hat{C}

be a SVN β-covering of U and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then, the following statements hold:

(1): $x \in {\bar{N}}_{x}^{β}$ for each $x \in U$ .
(2): $\forall x, y, z \in U$ , if $x \in {\bar{N}}_{y}^{β}$ , $y \in {\bar{N}}_{z}^{β}$ , then $x \in {\bar{N}}_{z}^{β}$ .

Proof.

(1): According to Theorem 1 and Definition 5, it is straightforward.
(2): For any $x, y, z \in U$ , $x \in {\bar{N}}_{y}^{β} \Leftrightarrow {\tilde{N}}_{y}^{β} (x) \geq β \Leftrightarrow {\tilde{N}}_{x}^{β} \subseteq {\tilde{N}}_{y}^{β}$ , and $y \in {\bar{N}}_{z}^{β} \Leftrightarrow {\tilde{N}}_{z}^{β} (y) \geq β \Leftrightarrow {\tilde{N}}_{y}^{β} \subseteq {\tilde{N}}_{z}^{β}$ . Hence, ${\tilde{N}}_{x}^{β} \subseteq {\tilde{N}}_{z}^{β}$ . By Proposition 1, we have ${\tilde{N}}_{z}^{β} (x) \geq β$ , i.e., $x \in {\bar{N}}_{z}^{β}$ .

□

Proposition 2.

Let

\hat{C}

be a SVN β-covering of U and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then, for all

x \in U

x \in {\bar{N}}_{y}^{β}

if and only if

{\bar{N}}_{x}^{β} \subseteq {\bar{N}}_{y}^{β}

Proof.

(\Rightarrow)

: For any

z \in {\bar{N}}_{x}^{β}

, we know

{\tilde{N}}_{x}^{β} (z) \geq β

. Since

x \in {\bar{N}}_{y}^{β}

{\tilde{N}}_{y}^{β} (x) \geq β

. According to

(2)

in Theorem 1, we have

{\tilde{N}}_{y}^{β} (z) \geq β

. Hence,

z \in {\bar{N}}_{y}^{β}

. Therefore,

{\bar{N}}_{x}^{β} \subseteq {\bar{N}}_{y}^{β}

(\Leftarrow)

: According to

(1)

in Theorem 2,

x \in {\bar{N}}_{x}^{β}

for all

x \in U

. Since

{\bar{N}}_{x}^{β} \subseteq {\bar{N}}_{y}^{β}

x \in {\bar{N}}_{y}^{β}

. □

The relationship between SVN

β

-neighborhoods and

β

-neighborhoods is presented in the following proposition.

Proposition 3.

Let

\hat{C}

be a SVN β-covering of U. For any

x, y \in U

{\tilde{N}}_{x}^{β} \subseteq {\tilde{N}}_{y}^{β}

if and only if

{\bar{N}}_{x}^{β} \subseteq {\bar{N}}_{y}^{β}

Proof.

According to Propositions 1 and 2, it is straightforward. □

4. Two Types of Single Valued Neutrosophic Covering Rough Set Models

In this section, we propose two types of SVN covering rough set models on basis of the SVN

β

-neighborhoods and the

β

-neighborhoods, respectively. Then, we investigate the properties of the defined lower and upper approximation operators.

Definition 6.

Let

(U, \hat{C})

be a SVN β-covering approximation space. For each

A \in S V N (U)

where

A = {〈 x, T_{A} (x),

I_{A} (x), F_{A} (x) 〉 : x \in U}

, we define the single valued neutrosophic (SVN) covering upper approximation

\tilde{C} (A)

and lower approximation

\underset{\sim}{C} (A)

of A as:

\begin{matrix} \tilde{C} (A) = {〈 x, \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land T_{A} (y)], \lor_{y \in U} [I_{{\tilde{N}}_{x}^{β}} (y) \land I_{A} (y)], \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor F_{A} (y)] 〉 : x \in U}, \\ \underset{\sim}{C} (A) = {〈 x, \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{A} (y)], \land_{y \in U} [(1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor I_{A} (y)], \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land F_{A} (y)] 〉 : x \in U} . \end{matrix}

(5)

\tilde{C} (A) \neq \underset{\sim}{C} (A)

, then A is called the first type of SVN covering rough set.

Example 3

(Continued from Example 1). Let

β = 〈 0.5, 0.3, 0.8 〉

A = \frac{(0.6, 0.3, 0.5)}{x_{1}} + \frac{(0.4, 0.5, 0.1)}{x_{2}} + \frac{(0.3, 0.2, 0.6)}{x_{3}} + \frac{(0.5, 0.3, 0.4)}{x_{4}} + \frac{(0.7, 0.2, 0.3)}{x_{5}}

. Then,

\begin{matrix} \tilde{C} (A) = {〈 x_{1}, 0.6, 0.3, 0.5 〉, 〈 x_{2}, 0.4, 0.3, 0.6 〉, 〈 x_{3}, 0.6, 0.5, 0.5 〉, 〈 x_{4}, 0.5, 0.3, 0.6 〉, 〈 x_{5}, 0.6, 0.5, 0.5 〉}, \\ \underset{\sim}{C} (A) = {〈 x_{1}, 0.6, 0.5, 0.5 〉, 〈 x_{2}, 0.6, 0.5, 0.4 〉, 〈 x_{3}, 0.4, 0.4, 0.5 〉, 〈 x_{4}, 0.4, 0.5, 0.4 〉, 〈 x_{5}, 0.6, 0.4, 0.3 〉} . \end{matrix}

Some basic properties of the SVN covering upper and lower approximation operators are proposed in the following proposition.

Proposition 4.

Let

\hat{C}

be a SVN β-covering of U. Then, the SVN covering upper and lower approximation operators in Definition 6 satisfy the following properties: for all

A, B \in S V N (U)

(1): $\tilde{C} (A^{'}) = {(\underset{\sim}{C} (A))}^{'}$ , $\underset{\sim}{C} (A^{'}) = {(\tilde{C} (A))}^{'}$ .
(2): If $A \subseteq B$ , then $\underset{\sim}{C} (A) \subseteq \underset{\sim}{C} (B)$ , $\tilde{C} (A) \subseteq \tilde{C} (B)$ .
(3): $\underset{\sim}{C} (A ⋂ B) = \underset{\sim}{C} (A) ⋂ \underset{\sim}{C} (B)$ , $\tilde{C} (A ⋃ B) = \tilde{C} (A) ⋃ \tilde{C} (B)$ .
(4): $\underset{\sim}{C} (A ⋃ B) \supseteq \underset{\sim}{C} (A) ⋃ \underset{\sim}{C} (B)$ , $\tilde{C} (A ⋂ B) \subseteq \tilde{C} (A) ⋂ \tilde{C} (B)$ .

Proof.

(1): $\begin{matrix} \tilde{C} (A^{'}) & = {〈 x, \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land T_{A^{'}} (y)], \lor_{y \in U} [I_{{\tilde{N}}_{x}^{β}} (y) \land I_{A^{'}} (y)], \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor F_{A^{'}} (y)] 〉 : x \in U} \\ = {〈 x, \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land F_{A} (y)], \lor_{y \in U} [I_{{\tilde{N}}_{x}^{β}} (y) \land (1 - I_{A} (y))], \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{A} (y)] 〉 : x \in U} \\ = {(\underset{\sim}{C} (A))}^{'} . \end{matrix}$

If we replace A by $A^{'}$ in this proof, we can also prove $\underset{\sim}{C} (A^{'}) = {(\tilde{C} (A))}^{'}$ .
(2): Since $A \subseteq B$ , so $T_{A} (x) \leq T_{B} (x)$ , $I_{B} (x) \leq I_{A} (x)$ and $F_{B} (x) \leq F_{A} (x)$ for all $x \in U$ . Therefore,

$\begin{matrix} T_{\underset{\sim}{C} (A)} (x) = \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{A} (y)] \leq \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{B} (y)] = T_{\underset{\sim}{C} (B)} (x), \\ I_{\underset{\sim}{C} (A)} (x) = \land_{y \in U} [(1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor I_{A} (y)] \geq \land_{y \in U} [(1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor I_{B} (y)] = I_{\underset{\sim}{C} (B)} (x), \\ F_{\underset{\sim}{C} (A)} (x) = \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land F_{A} (y)] \geq \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land F_{B} (y)] = F_{\underset{\sim}{C} (B)} (x) . \end{matrix}$

Hence, $\underset{\sim}{C} (A) \subseteq \underset{\sim}{C} (B)$ . In the same way, there is $\tilde{C} (A) \subseteq \tilde{C} (B)$ .
(3): $\begin{matrix} \underset{\sim}{C} (A ⋂ B) \\ = & {〈 x, \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{A ⋂ B} (y)], \land_{y \in U} [(1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor I_{A ⋂ B} (y)], \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land F_{A ⋂ B} (y)] 〉 : x \in U} \\ = & {〈 x, \land_{y \in U} [F_{{\tilde{N}}_{x}^{β}} (y) \lor (T_{A} (y) \land T_{B} (y))], \land_{y \in U} [(1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor (I_{A} (y) \lor I_{B} (y))], \lor_{y \in U} [T_{{\tilde{N}}_{x}^{β}} (y) \land (F_{A} (y) \\ \lor F_{B} (y))] 〉 : x \in U} \\ = & {〈 x, \land_{y \in U} [(F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{A} (y)) \land (F_{{\tilde{N}}_{x}^{β}} (y) \lor T_{B} (y))], \land_{y \in U} [((1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor I_{A} (y)) \lor (1 - I_{{\tilde{N}}_{x}^{β}} (y)) \lor \\ I_{B} (y))], \lor_{y \in U} [(T_{{\tilde{N}}_{x}^{β}} (y) \land F_{A} (y)) \lor (T_{{\tilde{N}}_{x}^{β}} (y) \land F_{B} (y))] 〉 : x \in U} \\ = & \underset{\sim}{C} (A) ⋂ \underset{\sim}{C} (B) . \end{matrix}$

Similarly, we can obtain $\tilde{C} (A ⋃ B) = \tilde{C} (A) ⋃ \tilde{C} (B)$ .
(4): Since $A \subseteq A \cup B$ , $B \subseteq A \cup B$ , $A \cap B \subseteq A$ and $A \cap B \subseteq B$ ,

$\underset{\sim}{C} (A) \subseteq \underset{\sim}{C} (A \cup B), \underset{\sim}{C} (B) \subseteq \underset{\sim}{C} (A \cup B), \tilde{C} (A \cap B) \subseteq \tilde{C} (A) and \tilde{C} (A \cap B) \subseteq \tilde{C} (B) .$

Hence,

\underset{\sim}{C} (A ⋃ B) \supseteq \underset{\sim}{C} (A) ⋃ \underset{\sim}{C} (B)

\tilde{C} (A ⋂ B) \subseteq \tilde{C} (A) ⋂ \tilde{C} (B)

. □

We propose the other SVN covering rough set model, which concerns the crisp lower and upper approximations of each crisp set in the SVN environment.

Definition 7.

Let

(U, \hat{C})

be a SVN β-covering approximation space. For each crisp subset

X \in P (U)

(

P (U)

is the power set of U), we define the SVN covering upper approximation

\bar{C} (X)

and lower approximation

\underset{̲}{C} (X)

of X as:

\begin{matrix} \bar{C} (X) = {x \in U : {\bar{N}}_{x}^{β} \cap X \neq \emptyset}, \\ \underset{̲}{C} (X) = {x \in U : {\bar{N}}_{x}^{β} \subseteq X} . \end{matrix}

(6)

\bar{C} (X) \neq \underset{̲}{C} (X)

, then X is called the second type of SVN covering rough set.

Example 4

(Continued from Example 2). Let

β = 〈 0.5, 0.3, 0.8 〉

X = {x_{1}, x_{2}}

Y = {x_{2}, x_{4}, x_{5}}

. Then,

\begin{matrix} \bar{C} (X) = {x_{1}, x_{2}, x_{4}}, \underset{̲}{C} (X) = {x_{1}, x_{2}}, \\ \bar{C} (Y) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}, \underset{̲}{C} (Y) = {x_{2}, x_{4}, x_{5}}, \\ \bar{C} (U) = U, \underset{̲}{C} (U) = U, \bar{C} (\emptyset) = \emptyset, \underset{̲}{C} (\emptyset) = \emptyset . \end{matrix}

Proposition 5.

Let

\hat{C}

be a SVN β-covering of U. Then, the SVN covering upper and lower approximation operators in Definition 7 satisfy the following properties: for all

X, Y \in P (U)

(1): $\underset{̲}{C} (\emptyset) = \emptyset$ , $\bar{C} (U) = U$ .
(2): $\underset{̲}{C} (U) = U$ , $\bar{C} (\emptyset) = \emptyset$ .
(3): $\underset{̲}{C} (X^{'}) = {(\bar{C} (X))}^{'}$ , $\bar{C} (X^{'}) = {(\underset{̲}{C} (X))}^{'}$ .
(4): If $X \subseteq Y$ , then $\underset{̲}{C} (X) \subseteq \underset{̲}{C} (Y)$ , $\bar{C} (X) \subseteq \bar{C} (Y)$ .
(5): $\underset{̲}{C} (X ⋂ Y) = \underset{̲}{C} (X) ⋂ \underset{̲}{C} (Y)$ , $\bar{C} (X ⋃ Y) = \bar{C} (X) ⋃ \bar{C} (Y)$ .
(6): $\underset{̲}{C} (X ⋃ Y) \supseteq \underset{̲}{C} (X) ⋃ \underset{̲}{C} (Y)$ , $\bar{C} (X ⋂ Y) \subseteq \bar{C} (X) ⋂ \bar{C} (Y)$ .
(7): $\underset{̲}{C} (\underset{̲}{C} (X)) \subseteq \underset{̲}{C} (X), \bar{C} (\bar{C} (X)) \supseteq \bar{C} (X)$ .
(8): $\underset{̲}{C} (X) \subseteq X \subseteq \bar{C} (X)$ .
(9): $X \subseteq Y$ or $Y \subseteq X \Leftrightarrow \underset{̲}{C} (X \cap Y) = \underset{̲}{C} (X) \cap \underset{̲}{C} (Y), \bar{C} (X \cup Y) = \bar{C} (X) \cup \bar{C} (Y)$ .

Proof.

It can be directly followed from Definitions 5 and 7. □

5. Matrix Representations of These Single Valued Neutrosophic Covering Rough Set Models

In this section, matrix representations of the proposed SVN covering rough set models are investigated. Firstly, some new matrices and matrix operations are presented. Then, we show the matrix representations of these SVN approximation operators defined in Definitions 6 and 7. The order of elements in U is given.

Definition 8.

Let

\hat{C}

be a SVN β-covering of U with

U = {x_{1}, x_{2}, \dots, x_{n}}

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then,

M_{\hat{C}} = {(C_{j} (x_{i}))}_{n \times m}

is named a matrix representation of

\hat{C}

, and

M_{\hat{C}}^{β} = {(s_{i j})}_{n \times m}

is called a β-matrix representation of

\hat{C}

, where

s_{i j} = \{\begin{matrix} 1, & C_{j} (x_{i}) \geq β; \\ 0, & otherwise . \end{matrix}

Example 5

(Continued from Example 1). Let

β = 〈 0.5, 0.3, 0.8 〉

M_{\hat{C}} = (\begin{matrix} 〈 0.7, 0.2, 0.5 〉 & 〈 0.6, 0.2, 0.4 〉 & 〈 0.4, 0.1, 0.5 〉 & 〈 0.1, 0.5, 0.6 〉 \\ 〈 0.5, 0.3, 0.2 〉 & 〈 0.5, 0.2, 0.8 〉 & 〈 0.4, 0.5, 0.4 〉 & 〈 0.6, 0.1, 0.7 〉 \\ 〈 0.4, 0.5, 0.2 〉 & 〈 0.2, 0.3, 0.6 〉 & 〈 0.5, 0.2, 0.4 〉 & 〈 0.6, 0.3, 0.4 〉 \\ 〈 0.6, 0.1, 0.7 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.6, 0.5 〉 & 〈 0.5, 0.3, 0.2 〉 \\ 〈 0.3, 0.2, 0.6 〉 & 〈 0.7, 0.3, 0.5 〉 & 〈 0.6, 0.3, 0.5 〉 & 〈 0.8, 0.1, 0.2 〉 \end{matrix}), M_{\hat{C}}^{β} = (\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \end{matrix}) .

Definition 9.

Let

A = {(a_{i k})}_{n \times m}

and

B = {(〈 b_{k j}^{+}, b_{k j}, b_{k j}^{-} 〉)}_{1 \leq k \leq m, 1 \leq j \leq l}

be two matrices. We define

D = A * B = {(〈 d_{i j}^{+}, d_{i j}, d_{i j}^{-} 〉)}_{1 \leq i \leq n, 1 \leq j \leq l}

, where

〈 d_{i j}^{+}, d_{i j}, d_{i j}^{-} 〉 = 〈 \land_{k = 1}^{m} [(1 - a_{i k}) \lor b_{k j}^{+}], 1 - \land_{k = 1}^{m} [(1 - a_{i k}) \lor (1 - b_{k j})], 1 - \land_{k = 1}^{m} [(1 - a_{i k}) \lor (1 - b_{k j}^{-})] 〉 .

(7)

Based on Definitions 8 and 9, all

{\tilde{N}}_{x}^{β}

for any

x \in U

can be obtained by matrix operations.

Proposition 6.

Let

\hat{C}

be a SVN β-covering of U with

U = {x_{1}, x_{2}, \dots, x_{n}}

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then

M_{\hat{C}}^{β} * M_{\hat{C}}^{T} = {({\tilde{N}}_{x_{i}}^{β} (x_{j}))}_{1 \leq i \leq n, 1 \leq j \leq n},

(8)

where

M_{\hat{C}}^{T}

is the transpose of

M_{\hat{C}}

Proof.

Suppose

M_{\hat{C}}^{T} = {(C_{k} (x_{j}))}_{m \times n}

M_{\hat{C}}^{β} = {(s_{i k})}_{n \times m}

and

M_{\hat{C}}^{β} * M_{\hat{C}}^{T} = {(〈 d_{i j}^{+}, d_{i j}, d_{i j}^{-} 〉)}_{1 \leq i \leq n, 1 \leq j \leq n}

. Since

\hat{C}

is a SVN β-covering of U, for each i (

1 \leq i \leq n

), there exists k (

1 \leq k \leq m

) such that

s_{i k} = 1

. Then,

\begin{matrix} 〈 d_{i j}^{+}, d_{i j}, d_{i j}^{-} 〉 \\ = & 〈 \land_{k = 1}^{m} [(1 - s_{i k}) \lor T_{C_{k}} (x_{j})], 1 - \land_{k = 1}^{m} [(1 - s_{i k}) \lor (1 - I_{C_{k}} (x_{j}))], 1 - \land_{k = 1}^{m} [(1 - s_{i k}) \lor (1 - F_{C_{k}} (x_{j}))] 〉 \\ = & 〈 \land_{s_{i k} = 1} [(1 - s_{i k}) \lor T_{C_{k}} (x_{j})], 1 - \land_{s_{i k} = 1} [(1 - s_{i k}) \lor (1 - I_{C_{k}} (x_{j}))], 1 - \land_{s_{i k} = 1} [(1 - s_{i k}) \lor (1 - F_{C_{k}} (x_{j}))] 〉 \\ = & 〈 \land_{s_{i k} = 1} T_{C_{k}} (x_{j}), 1 - \land_{s_{i k} = 1} (1 - I_{C_{k}} (x_{j})), 1 - \land_{s_{i k} = 1} (1 - F_{C_{k}} (x_{j})) 〉 \\ = & 〈 \land_{C_{k} (x_{i}) \geq β} T_{C_{k}} (x_{j}), 1 - \land_{C_{k} (x_{i}) \geq β} (1 - I_{C_{k}} (x_{j})), 1 - \land_{C_{k} (x_{i}) \geq β} (1 - F_{C_{k}} (x_{j})) 〉 \\ = & (⋂_{C_{k} (x_{i}) \geq β} C_{k}) (x_{j}) \\ = & {\tilde{N}}_{x_{i}}^{β} (x_{j}), 1 \leq i, j \leq n . \end{matrix}

Hence,

M_{\hat{C}}^{β} * M_{\hat{C}}^{T} = {({\tilde{N}}_{x_{i}}^{β} (x_{j}))}_{1 \leq i \leq n, 1 \leq j \leq n}

. □

Example 6

(Continued from Example 1).

\begin{matrix} M_{\hat{C}}^{β} * M_{\hat{C}}^{T} \\ = & (\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \end{matrix}) * {(\begin{matrix} 〈 0.7, 0.2, 0.5 〉 & 〈 0.6, 0.2, 0.4 〉 & 〈 0.4, 0.1, 0.5 〉 & 〈 0.1, 0.5, 0.6 〉 \\ 〈 0.5, 0.3, 0.2 〉 & 〈 0.5, 0.2, 0.8 〉 & 〈 0.4, 0.5, 0.4 〉 & 〈 0.6, 0.1, 0.7 〉 \\ 〈 0.4, 0.5, 0.2 〉 & 〈 0.2, 0.3, 0.6 〉 & 〈 0.5, 0.2, 0.4 〉 & 〈 0.6, 0.3, 0.4 〉 \\ 〈 0.6, 0.1, 0.7 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.6, 0.5 〉 & 〈 0.5, 0.3, 0.2 〉 \\ 〈 0.3, 0.2, 0.6 〉 & 〈 0.7, 0.3, 0.5 〉 & 〈 0.6, 0.3, 0.5 〉 & 〈 0.8, 0.1, 0.2 〉 \end{matrix})}^{T} \\ = & (\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \end{matrix}) * (\begin{matrix} 〈 0.7, 0.2, 0.5 〉 & 〈 0.5, 0.3, 0.2 〉 & 〈 0.4, 0.5, 0.2 〉 & 〈 0.6, 0.1, 0.7 〉 & 〈 0.3, 0.2, 0.6 〉 \\ 〈 0.6, 0.2, 0.4 〉 & 〈 0.5, 0.2, 0.8 〉 & 〈 0.2, 0.3, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.7, 0.3, 0.5 〉 \\ 〈 0.4, 0.1, 0.5 〉 & 〈 0.4, 0.5, 0.4 〉 & 〈 0.5, 0.2, 0.4 〉 & 〈 0.3, 0.6, 0.5 〉 & 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.6, 0.1, 0.7 〉 & 〈 0.6, 0.3, 0.4 〉 & 〈 0.5, 0.3, 0.2 〉 & 〈 0.8, 0.1, 0.2 〉 \end{matrix}) \\ = & (\begin{matrix} 〈 0.6, 0.2, 0.5 〉 & 〈 0.5, 0.3, 0.8 〉 & 〈 0.2, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.3, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.5, 0.3, 0.8 〉 & 〈 0.2, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.3, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.5, 0.3, 0.4 〉 & 〈 0.3, 0.6, 0.5 〉 & 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.5, 0.3, 0.7 〉 & 〈 0.4, 0.5, 0.4 〉 & 〈 0.5, 0.3, 0.7 〉 & 〈 0.3, 0.2, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.8 〉 & 〈 0.2, 0.3, 0.6 〉 & 〈 0.3, 0.6, 0.7 〉 & 〈 0.6, 0.3, 0.5 〉 \end{matrix}) \\ = & {(N_{x_{i}}^{β} (x_{j}))}_{1 \leq i \leq 5, 1 \leq j \leq 5} . \end{matrix}

Definition 10.

Let

A = {(〈 c_{i j}^{+}, c_{i j}, c_{i j}^{-} 〉)}_{m \times n}

and

B = {(〈 d_{j}^{+}, d_{j}, d_{j}^{-} 〉)}_{n \times 1}

be two matrices. We define

C = A \circ B = {(〈 e_{i}^{+}, e_{i}, e_{i}^{-} 〉)}_{m \times 1}

and

D = A ⋄ B = {(〈 f_{i}^{+}, f_{i}, f_{i}^{-} 〉)}_{m \times 1}

, where

\begin{matrix} 〈 e_{i}^{+}, e_{i}, e_{i}^{-} 〉 = 〈 \lor_{j = 1}^{n} (c_{i j}^{+} \land d_{j}^{+}), \lor_{j = 1}^{n} (c_{i j} \land d_{j}), \land_{j = 1}^{n} (c_{i j}^{-} \lor d_{j}^{-}) 〉, \\ 〈 f_{i}^{+}, f_{i}, f_{i}^{-} 〉 = 〈 \land_{j = 1}^{n} (c_{i j}^{-} \lor d_{j}^{+}), \land_{j = 1}^{n} [(1 - c_{i j}) \lor d_{j}], \lor_{j = 1}^{n} (c_{i j}^{+} \land d_{j}^{-}) 〉 . \end{matrix}

(9)

According to Proposition 6 and Definition 10, the set representations of

\tilde{C} (A)

and

\underset{\sim}{C} (A)

(for any

A \in S V N (U)

) can be converted to matrix representations.

Theorem 3.

Let

\hat{C}

be a SVN β-covering of U with

U = {x_{1}, x_{2}, \dots, x_{n}}

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then, for any

A \in S V N (U)

\begin{matrix} \tilde{C} (A) = (M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) \circ A, \\ \underset{\sim}{C} (A) = (M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) ⋄ A, \end{matrix}

(10)

where

A = {(a_{i})}_{n \times 1}

with

a_{i} = 〈 T_{A} (x_{i}), I_{A} (x_{i}), F_{A} (x_{i}) 〉

is the vector representation of the SVNS A.

\tilde{C} (A)

and

\underset{\sim}{C} (A)

are also vector representations.

Proof.

According to Proposition 6 and Definitions 6 and 10, for any

x_{i}

(

i = 1, 2, \dots, n

\begin{matrix} ((M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) \circ A) (x_{i}) & = 〈 \lor_{j = 1}^{n} (T_{{\tilde{N}}_{x_{i}}^{β}} (x_{j}) \land T_{A} (x_{j})), \lor_{j = 1}^{n} (I_{{\tilde{N}}_{x_{i}}^{β}} (x_{j}) \land I_{A} (x_{j})), \land_{j = 1}^{n} (F_{{\tilde{N}}_{x_{i}}^{β}} (x_{j}) \lor F_{A} (x_{j})) 〉 \\ = (\tilde{C} (A)) (x_{i}), \end{matrix}

and

\begin{matrix} ((M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) ⋄ A) (x_{i}) & = 〈 \land_{j = 1}^{n} (F_{{\tilde{N}}_{x_{i}}^{β}} (x_{j}) \lor T_{A} (x_{j})), \land_{j = 1}^{n} [(1 - I_{{\tilde{N}}_{x_{i}}^{β}} (x_{j})) \lor I_{A} (x_{j})], \lor_{j = 1}^{n} (T_{{\tilde{N}}_{x_{i}}^{β}} (x_{j}) \land F_{A} (x_{j})) 〉 \\ = (\underset{\sim}{C} (A)) (x_{i}) . \end{matrix}

Hence,

\tilde{C} (A) = (M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) \circ A, \underset{\sim}{C} (A) = (M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) ⋄ A

. □

Example 7

(Continued from Example 3). Let

β = 〈 0.5, 0.3, 0.8 〉

A = \frac{(0.6, 0.3, 0.5)}{x_{1}} + \frac{(0.4, 0.5, 0.1)}{x_{2}} + \frac{(0.3, 0.2, 0.6)}{x_{3}} + \frac{(0.5, 0.3, 0.4)}{x_{4}} + \frac{(0.7, 0.2, 0.3)}{x_{5}}

. Then,

\begin{matrix} \tilde{C} (A) \\ = & (M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) \circ A \\ = & (\begin{matrix} 〈 0.6, 0.2, 0.5 〉 & 〈 0.5, 0.3, 0.8 〉 & 〈 0.2, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.3, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.5, 0.3, 0.8 〉 & 〈 0.2, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.3, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.5, 0.3, 0.4 〉 & 〈 0.3, 0.6, 0.5 〉 & 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.5, 0.3, 0.7 〉 & 〈 0.4, 0.5, 0.4 〉 & 〈 0.5, 0.3, 0.7 〉 & 〈 0.3, 0.2, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.8 〉 & 〈 0.2, 0.3, 0.6 〉 & 〈 0.3, 0.6, 0.7 〉 & 〈 0.6, 0.3, 0.5 〉 \end{matrix}) \circ (\begin{matrix} 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.4, 0.5, 0.1 〉 \\ 〈 0.3, 0.2, 0.6 〉 \\ 〈 0.5, 0.3, 0.4 〉 \\ 〈 0.7, 0.2, 0.3 〉 \end{matrix}) \\ = & (\begin{matrix} 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.4, 0.3, 0.6 〉 \\ 〈 0.6, 0.5, 0.5 〉 \\ 〈 0.5, 0.3, 0.6 〉 \\ 〈 0.6, 0.5, 0.5 〉 \end{matrix}), \end{matrix}

and

\begin{matrix} \underset{\sim}{C} (A) \\ = & (M_{\hat{C}}^{β} * M_{\hat{C}}^{T}) ⋄ A \\ = & (\begin{matrix} 〈 0.6, 0.2, 0.5 〉 & 〈 0.5, 0.3, 0.8 〉 & 〈 0.2, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.3, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.5, 0.3, 0.8 〉 & 〈 0.2, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.3, 0.3, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.7 〉 & 〈 0.5, 0.3, 0.4 〉 & 〈 0.3, 0.6, 0.5 〉 & 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.5, 0.3, 0.7 〉 & 〈 0.4, 0.5, 0.4 〉 & 〈 0.5, 0.3, 0.7 〉 & 〈 0.3, 0.2, 0.6 〉 \\ 〈 0.1, 0.5, 0.6 〉 & 〈 0.4, 0.5, 0.8 〉 & 〈 0.2, 0.3, 0.6 〉 & 〈 0.3, 0.6, 0.7 〉 & 〈 0.6, 0.3, 0.5 〉 \end{matrix}) ⋄ (\begin{matrix} 〈 0.6, 0.3, 0.5 〉 \\ 〈 0.4, 0.5, 0.1 〉 \\ 〈 0.3, 0.2, 0.6 〉 \\ 〈 0.5, 0.3, 0.4 〉 \\ 〈 0.7, 0.2, 0.3 〉 \end{matrix}) \\ = & (\begin{matrix} 〈 0.6, 0.5, 0.5 〉 \\ 〈 0.6, 0.5, 0.4 〉 \\ 〈 0.4, 0.4, 0.5 〉 \\ 〈 0.4, 0.5, 0.6 〉 \\ 〈 0.6, 0.4, 0.3 〉 \end{matrix}) . \end{matrix}

Two operations of matrices are defined in [28]. We can use them to study the matrix representations of

\underset{̲}{C} (X)

and

\bar{C} (X)

of every crisp subset

X \in P (U)

Definition 11

([28]). Let

A = {(a_{i k})}_{n \times m}

and

B = {(b_{k j})}_{m \times l}

be two matrices. We define

C = A \cdot B = {(c_{i j})}_{n \times l}

and

D = A ⊙ B = {(d_{i j})}_{n \times l}

as follows:

\begin{matrix} c_{i j} = \lor_{k = 1}^{m} (a_{i k} \land b_{k j}), \\ d_{i j} = \land_{k = 1}^{m} [(1 - a_{i k}) \lor b_{k j}], f o r a n y i = 1, 2, \dots, n, a n d j = 1, 2, \dots, l . \end{matrix}

(11)

Let

U = {x_{1}, \dots, x_{n}}

and

X \in P (U)

. Then, the characteristic function of the crisp subset X is defined as

χ_{X}

, where

χ_{X} (x_{i}) = \{\begin{matrix} 1, & x_{i} \in X; \\ 0, & o t h e r w i s e . \end{matrix}

Proposition 7.

Let

\hat{C}

be a SVN β-covering of U with

U = {x_{1}, x_{2}, \dots, x_{n}}

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then,

M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T} = {(χ_{{\bar{N}}_{x_{i}}^{β}} (x_{j}))}_{1 \leq i \leq n, 1 \leq j \leq n},

(12)

Proof.

Suppose

M_{\hat{C}}^{β} = {(s_{i k})}_{n \times m}

and

M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T} = {(t_{i j})}_{n \times n}

. Since

\hat{C}

is a SVN β-covering of U, for each i (

1 \leq i \leq n

) there exists k (

1 \leq k \leq m

) such that

s_{i k} = 1

. If

t_{i j} = 1

, then

\land_{k = 1}^{m} [(1 - s_{i k}) \lor s_{j k}] = 1

It implies that if

s_{i k} = 1

, then

s_{j k} = 1

. Hence,

C_{k} (x_{i}) \geq β

implies

C_{k} (x_{j}) \geq β

. Therefore,

x_{j} \in {\bar{N}}_{x_{i}}^{β}

, i.e.,

χ_{{\bar{N}}_{x_{i}}^{β}} (x_{j}) = 1 = t_{i j}

t_{i j} = 0

, then

\land_{k = 1}^{m} [(1 - s_{i k}) \lor s_{j k}] = 0

. This implies that if

s_{i k} = 1

, then

s_{j k} = 0

. Hence,

C_{k} (x_{i}) \geq β

implies

C_{k} (x_{j}) < β

. Thus, we have

x_{j} \notin {\bar{N}}_{x_{i}}^{β}

, i.e.,

χ_{{\bar{N}}_{x_{i}}^{β}} (x_{j}) = 1 = t_{i j}

. □

Example 8

(Continued from Example 2). According to

M_{\hat{C}}^{β}

in Example 5, we have the following result.

M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T} = (\begin{matrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) = {(χ_{{\bar{N}}_{x_{i}}^{β}} (x_{j}))}_{1 \leq i \leq 5, 1 \leq j \leq 5}

For any

X \in P (U)

, we also denote

χ_{X} = {(a_{i})}_{n \times 1}

with

a_{i} = 1

iff

x_{i} \in X

; otherwise,

a_{i} = 0

Then, the set representations of

\bar{C} (X)

and

\underset{̲}{C} (X)

(for any

X \in P (U)

) can be converted to matrix representations.

Theorem 4.

Let

\hat{C}

be a SVN β-covering of U with

U = {x_{1}, x_{2}, \dots, x_{n}}

and

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

. Then, for any

X \in P (U)

\begin{matrix} χ_{\bar{C} (X)} = (M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}) \cdot χ_{X}, \\ χ_{\underset{̲}{C} (X)} = (M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}) ⊙ χ_{X} . \end{matrix}

(13)

Proof.

Suppose

(M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}) \cdot χ_{X} = {(a_{i})}_{n \times 1}

and

(M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}) ⊙ χ_{X} = {(b_{i})}_{n \times 1}

. For any

x_{i} \in U

(

i = 1, 2, \dots, n

\begin{matrix} x_{i} \in \bar{C} (X) & \Leftrightarrow χ_{\bar{C} (X)} (x_{i}) = 1 \\ \Leftrightarrow a_{i} = 1 \\ \Leftrightarrow \lor_{k = 1}^{n} [χ_{{\bar{N}}_{x_{i}}^{β}} (x_{k}) \land χ_{X} (x_{k})] = 1 \\ \Leftrightarrow \exists k \in {1, 2, \dots, n}, s . t ., χ_{{\bar{N}}_{x_{i}}^{β}} (x_{k}) = χ_{X} (x_{k}) = 1 \\ \Leftrightarrow \exists k \in {1, 2, \dots, n}, s . t ., x_{k} \in {\bar{N}}_{x_{i}}^{β} \cap X \\ \Leftrightarrow {\bar{N}}_{x_{i}}^{β} \cap X \neq \emptyset, \end{matrix}

and

\begin{matrix} x_{i} \in \underset{̲}{C} (X) & \Leftrightarrow χ_{\underset{̲}{C} (X)} (x_{i}) = 1 \\ \Leftrightarrow b_{i} = 1 \\ \Leftrightarrow \land_{k = 1}^{n} [(1 - χ_{{\bar{N}}_{x_{i}}^{β}} (x_{k})) \lor χ_{X} (x_{k})] = 1 \\ \Leftrightarrow χ_{{\bar{N}}_{x_{i}}^{β}} (x_{k}) = 1 \to χ_{X} (x_{k}) = 1, k = 1, 2, \dots, n \\ \Leftrightarrow x_{k} \in {\bar{N}}_{x_{i}}^{β} \to x_{k} \in X, k = 1, 2, \dots, n \\ \Leftrightarrow {\bar{N}}_{x_{i}}^{β} \subseteq X . \end{matrix}

□

Example 9

(Continued from Example 4). Let

X = {x_{1}, x_{2}}

. By

M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}

in Example 8, we have

\begin{matrix} (M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}) \cdot χ_{X} = (\begin{matrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 1 \\ 0 \\ 1 \\ 0 \end{matrix}) = χ_{\bar{C} (X)}, \\ (M_{\hat{C}}^{β} ⊙ {(M_{\hat{C}}^{β})}^{T}) ⊙ χ_{X} = (\begin{matrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) ⊙ (\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}) = χ_{\underset{̲}{C} (X)} . \end{matrix}

6. An Application to Decision Making Problems

In this section, we present a novel approach to DM problems based on the SVN covering rough set model. Then, a comparative study with other methods is shown.

6.1. The Problem of Decision Making

Let

U = {x_{k} : k = 1, 2, \dots, l}

be the set of patients and

V = {y_{i} | i = 1, 2, \dots, m}

be the m main symptoms (for example, cough, fever, and so on) for a Disease B. Assume that Doctor R evaluates every Patient

x_{k}

(

k = 1, 2, \dots, l

Assume that Doctor R believes each Patient

x_{k} \in U

(

k = 1, 2, \dots, l

) has a symptom value

C_{i}

(

i = 1, 2, \dots, m

), denoted by

C_{i} (x_{k}) = 〈 T_{C_{i}} (x_{k}), I_{C_{i}} (x_{k}, F_{C_{i}} (x_{k}) 〉

, where

T_{C_{i}} (x_{k}) \in [0, 1]

is the degree that Doctor R confirms Patient

x_{k}

has symptom

y_{i}

I_{C_{i}} (x_{k}) \in [0, 1]

is the degree that Doctor R is not sure Patient

x_{k}

has symptom

y_{i}

F_{C_{i}} (x_{k}) \in [0, 1]

is the degree that Doctor R confirms Patient

x_{k}

does not have symptom

y_{i}

, and

T_{C_{i}} (x_{k}) + I_{C_{i}} (x_{k}) + F_{C_{i}} (x_{k}) \leq 3

Let

β = 〈 a, b, c 〉

be the critical value. If any Patient

x_{k} \in U

, there is at least one symptom

y_{i} \in V

such that the symptom value

C_{i}

for Patient

x_{k}

is not less than β, respectively, then

\hat{C} = {C_{1}, C_{2}, \dots, C_{m}}

is a SVN β-covering of U for some SVN number β.

If d is a possible degree, e is an indeterminacy degree and f is an impossible degree of Disease B of every Patient

x_{k} \in U

that is diagnosed by Doctor R, denoted by

A (x_{k}) = 〈 d, e, f 〉

, then the decision maker (Doctor R) for the decision making problem needs to know how to evaluate whether Patients

x_{k} \in U

have Disease B.

6.2. The Decision Making Algorithm

In this subsection, we give an approach for the problem of DM with the above characterizations by means of the first type of SVN covering rough set model. According to the characterizations of the DM problem in Section 6.1, we construct the SVN decision information system and present the Algorithm 1 of DM under the framework of the first type of SVN covering rough set model.

Algorithm 1 The decision making algorithm based on the SVN covering rough set model.

Input: SVN decision information system (

U, \hat{C}, β, A

).
Output: The score ordering for all alternatives.

1: Compute the SVN β-neighborhood

{\tilde{N}}_{x}^{β}

of x induced by

\hat{C}

, for all

x \in U

according to Definition 4;
2: Compute the SVN covering upper approximation

\tilde{C} (A)

and lower approximation

\underset{\sim}{C} (A)

of A, according to Definition 6;
3: Compute

{\tilde{R}}_{A} = \tilde{C} (A) \oplus \underset{\sim}{C} (A)

according to

(6)

in the basic operations on

S V N (U)

;
4: Compute

\begin{matrix} s (x) = \frac{T_{{\tilde{R}}_{A}} (x)}{\sqrt{{(T_{{\tilde{R}}_{A}} (x))}^{2} + {(I_{{\tilde{R}}_{A}} (x))}^{2} + {(F_{{\tilde{R}}_{A}} (x))}^{2}}}; \end{matrix}

5: Rank all the alternatives

s (x)

by using the principle of numerical size and select the most possible patient.

According to the above process, we can get the decision making according to the ranking. In Step 4,

S (x)

is the cosine similarity measure between

{\tilde{R}}_{A} (x)

and the ideal solution

(1, 0, 0)

, which was proposed by Ye [44].

6.3. An Applied Example

Example 10.

Assume that

U = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}

is a set of patients. According to the patients’ symptoms, we write

V = {y_{1}, y_{2}, y_{3}, y_{4}}

to be four main symptoms (cough, fever, sore and headache) for Disease B. Assume that Doctor R evaluates every Patient

x_{k}

(

k = 1, 2, \dots, 5

) as shown in Table 1.

Let

β = 〈 0.5, 0.3, 0.8 〉

be the critical value. Then,

\hat{C} = {C_{1}, C_{2}, C_{3}, C_{4}}

is a SVN β-coverings of U.

{\tilde{N}}_{x_{k}}^{β}

(

k = 1, 2, 3, 4, 5

) are shown in Table 2.

Assume that Doctor R diagnoses the value

A = \frac{(0.6, 0.3, 0.5)}{x_{1}} + \frac{(0.4, 0.5, 0.1)}{x_{2}} + \frac{(0.3, 0.2, 0.6)}{x_{3}} + \frac{(0.5, 0.3, 0.4)}{x_{4}} + \frac{(0.7, 0.2, 0.3)}{x_{5}}

of Disease B of every patient. Then,

\begin{matrix} \tilde{C} (A) = {〈 x_{1}, 0.6, 0.3, 0.5 〉, 〈 x_{2}, 0.4, 0.3, 0.6 〉, 〈 x_{3}, 0.6, 0.5, 0.5 〉, 〈 x_{4}, 0.5, 0.3, 0.6 〉, 〈 x_{5}, 0.6, 0.5, 0.5 〉}, \\ \underset{\sim}{C} (A) = {〈 x_{1}, 0.6, 0.5, 0.5 〉, 〈 x_{2}, 0.6, 0.5, 0.4 〉, 〈 x_{3}, 0.4, 0.4, 0.5 〉, 〈 x_{4}, 0.4, 0.5, 0.4 〉, 〈 x_{5}, 0.6, 0.4, 0.3 〉} . \end{matrix}

Then,

\begin{matrix} {\tilde{R}}_{A} \\ = & \tilde{C} (A) \oplus \underset{\sim}{C} (A) \\ = & {〈 x_{1}, 0.84, 0.15, 0.25 〉, 〈 x_{2}, 0.76, 0.15, 0.24 〉, 〈 x_{3}, 0.76, 0.2, 0.25 〉, 〈 x_{4}, 0.7, 0.15, 0.24 〉, 〈 x_{5}, 0.84, 0.2, 0.15 〉} . \end{matrix}

Hence, we can obtain

s (x_{k})

(

k = 1, 2, \dots, 5

) in Table 3.

According to the principle of numerical size, we have:

s (x_{4}) < s (x_{3}) < s (x_{2}) < s (x_{1}) < s (x_{5}) .

Therefore, Doctor R diagnoses Patient

x_{5}

as more likely to be sick with Disease B.

6.4. A Comparison Analysis

To validate the feasibility of the proposed decision making method, a comparative study was conducted with other methods. These methods, which were introduced by Liu [43], Yang et al. [32] and Ye [44], are compared with the proposed approach using SVN information system.

6.4.1. The Results of Liu’s Method

Liu’s method is shown in Algorithm 2.

Algorithm 2 The decision making algorithm [43].

Input: A SVN decision matrix D, a weight vector

w

and γ.
Output: The score ordering for all alternatives.

1: Compute

\begin{array}{l} n_{k} & = 〈 T_{n_{k}}, I_{n_{k}}, F_{n_{k}} 〉 [2 m m] \\ = H S V N N W A (n_{k 1}, n_{k 2}, \dots, n_{k m}) [2 m m] \\ = 〈 \frac{\prod_{i = 1}^{m} {(1 + (γ - 1) T_{k i})}^{w_{i}} - \prod_{i = 1}^{m} {(1 - T_{k i})}^{w_{i}}}{\prod_{i = 1}^{m} {(1 + (γ - 1) T_{k i})}^{w_{i}} + (γ - 1) \prod_{i = 1}^{m} {(1 - T_{k i})}^{w_{i}}}, [2 m m] \\ \frac{γ \prod_{i = 1}^{m} I_{k i}^{w_{i}}}{\prod_{i = 1}^{m} {(1 + (γ - 1) (1 - I_{k i}))}^{w_{i}} + (γ - 1) \prod_{i = 1}^{m} I_{k i}^{w_{i}}}, [2 m m] \\ \frac{γ \prod_{i = 1}^{m} F_{k i}^{w_{i}}}{\prod_{i = 1}^{m} {(1 + (γ - 1) (1 - F_{k i}))}^{w_{i}} + (γ - 1) \prod_{i = 1}^{m} F_{k i}^{w_{i}}} 〉 (k = 1, 2, \dots, l); \end{array}

2: Calculate

s (n_{k}) = \frac{T_{n_{k}}}{\sqrt{T_{n_{k}}^{2} + I_{n_{k}}^{2} + F_{n_{k}}^{2}}}

;
3: Obtain the ranking for all

s (n_{k})

by using the principle of numerical size and select the most possible patient.

Then, Algorithm 2 can be used for Example 10. Let

n_{k i} = 〈 T_{k i}, I_{k i}, F_{k i} 〉

be the evaluation information of

x_{k}

C_{_{i}}

in Table 1. That is to say, Table 1 is the SVN decision matrix D. We suppose the weight vector of the criteria is

w = (0.35, 0, 25, 0.3, 0.1)

and

γ = 1

Step 1: Based on HSVNNWA operator, we get

\begin{matrix} n_{1} = 〈 0.557, 0.178, 0.482 〉, n_{2} = 〈 0.484, 0.283, 0.395 〉, \\ n_{3} = 〈 0.414, 0.318, 0.347 〉, n_{4} = 〈 0.465, 0.286, 0.558 〉, \\ n_{5} = 〈 0.578, 0.233, 0.486 〉 . \end{matrix}

Step 2: We get

s (n_{1}) = 0.735, s (n_{2}) = 0.706, s (n_{3}) = 0.660, s (n_{4}) = 0.596, s (n_{5}) = 0.734

Step 3: According to the cosine similarity degrees

s (n_{k})

(

k = 1, 2, \dots, 5

), we obtain

x_{4} < x_{3} < x_{2} < x_{5} < x_{1}

Therefore, Patient

x_{1}

is more likely to be sick with Disease B.

6.4.2. The Results of Yang’s Method

Yang’s method is shown in Algorithm 3.

Algorithm 3 The decision making algorithm [32].

Input: A generalized SVN approximation space (

U, V, \tilde{R}

B \in S V N (V)

.
Output: The score ordering for all alternatives.

1: Calculate the lower and upper approximations

\bar{\tilde{R}} (B)

and

\underset{̲}{\tilde{R}} (B)

;
2: Compute

n_{x_{k}} = (\bar{\tilde{R}} (B) ⨁ \underset{̲}{\tilde{R}} (B)) (x_{k})

(

k = 1, 2, \dots, l

);
3: Compute

\begin{matrix} s (n_{x_{k}}, n^{*}) = \frac{T_{n_{x_{k}}} \cdot T_{n^{*}} + I_{n_{x_{k}}} \cdot I_{n^{*}} + F_{n_{x_{k}}} \cdot F_{n^{*}}}{\sqrt{T_{n_{x_{k}}}^{2} + I_{n_{x_{k}}}^{2} + F_{n_{x_{k}}}^{2}} \cdot \sqrt{{(T_{n^{*}})}^{2} + {(I_{n^{*}})}^{2} + {(F_{n^{*}})}^{2}}} (k = 1, 2, \dots, l), \end{matrix}

where

n^{*} = 〈 T_{n^{*}}, I_{n^{*}}, F_{n^{*}} 〉 = 〈 1, 0, 0 〉

;
4: Obtain the ranking for all

s (n_{x_{k}}, n^{*})

by using the principle of numerical size and select the most possible patient.

For Example 10, we suppose Disease

B \in S V N (V)

and

B = \frac{(0.3, 0.6, 0.5)}{y_{1}} + \frac{(0.7, 0.2, 0.1)}{y_{2}} + \frac{(0.6, 0.4, 0.3)}{y_{3}} + \frac{(0.8, 0.4, 0.5)}{y_{4}}

. According to Table 1, the generalized SVN approximation space (

U, V, \tilde{R}

) can be obtained in Table 4, where

U = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}

and

V = {y_{1}, y_{2}, y_{3}, y_{4}}

Step 1: We get

\begin{matrix} \bar{\tilde{R}} (B) = {〈 x_{1}, 0.6, 0.2, 0.4 〉, 〈 x_{2}, 0.6, 0.2, 0.4 〉, 〈 x_{3}, 0.6, 0.3, 0.4 〉, 〈 x_{4}, 0.5, 0.4, 0.5 〉, 〈 x_{5}, 0.8, 0.3, 0.5 〉}, \\ \underset{̲}{\tilde{R}} (B) = {〈 x_{1}, 0.5, 0.6, 0.5 〉, 〈 x_{2}, 0.3, 0.6, 0.5 〉, 〈 x_{3}, 0.3, 0.5, 0.5 〉, 〈 x_{4}, 0.6, 0.6, 0.5 〉, 〈 x_{5}, 0.6, 0.6, 0.5 〉} . \end{matrix}

Step 2:

\begin{matrix} \bar{\tilde{R}} (B) ⨁ \underset{̲}{\tilde{R}} (B) = & {〈 x_{1}, 0.80, 0.12, 0.20 〉, 〈 x_{2}, 0.72, 0.12, 0.20 〉, 〈 x_{3}, 0.72, 0.15, 0.20 〉, 〈 x_{4}, 0.80, 0.24, 0.25 〉, \\ 〈 x_{5}, 0.92, 0.18, 0.25 〉} . \end{matrix}

Step 3: Let

n^{*} = 〈 1, 0, 0 〉

. Then,

s (n_{x_{1}}, n^{*}) = 0.960, s (n_{x_{2}}, n^{*}) = 0.951, s (n_{x_{3}}, n^{*}) = 0.945, s (n_{x_{4}}, n^{*}) = 0.918, s (n_{x_{5}}, n^{*}) = 0.948

Step 4:

s (n_{x_{4}}, n^{*}) < s (n_{x_{3}}, n^{*}) < s (n_{x_{5}}, n^{*}) < s (n_{x_{2}}, n^{*}) < s (n_{x_{1}}, n^{*})

Therefore, Patient

x_{1}

is more likely to be sick with Disease B.

6.4.3. The Results of Ye’s Methods

Ye presented two methods [44]. Thus, Algorithms 4 and 5 are presented for Example 10.

Algorithm 4 The decision making algorithm [44].

Input: A SVN decision matrix D and a weight vector

w

.
Output: The score ordering for all alternatives.

1: Compute

\begin{matrix} W_{k} (x_{k}, A^{*}) = \frac{\sum_{i = 1}^{m} w_{i} [a_{k i} \cdot a_{i}^{*} + b_{k i} \cdot b_{i}^{*} + c_{k i} \cdot c_{i}^{*}]}{\sqrt{\sum_{i = 1}^{m} w_{i} [a_{k i}^{2} + b_{k i}^{2} + c_{k i}^{2}]} \cdot \sqrt{\sum_{i = 1}^{m} w_{i} [{(a_{i}^{*})}^{2} + {(b_{i}^{*})}^{2} + {(c_{i}^{*})}^{2}]}} (k = 1, 2, \dots, l), \end{matrix}

where

α_{i}^{*} = 〈 a_{i}^{*}, b_{i}^{*}, c_{i}^{*} 〉 = 〈 1, 0, 0 〉

(

i = 1, 2, \dots, m

);
2: Obtain the ranking for all

W_{k} (x_{k}, A^{*})

by using the principle of numerical size and select the most possible patient.

For Example 10, Table 1 is the SVN decision matrix D. We suppose the weight vector of the criteria is

w = (0.35, 0, 25, 0.3, 0.1)

Step 1:

W_{1} (x_{1}, A^{*}) = 0.677, W_{2} (x_{2}, A^{*}) = 0.608, W_{3} (x_{3}, A^{*}) = 0.580, W_{4} (x_{4}, A^{*}) = 0.511, W_{5} (x_{5}, A^{*}) = 0.666 .

Step 2: The ranking order of

{x_{1}, x_{2}, \dots, x_{5}}

x_{4} < x_{3} < x_{2} < x_{5} < x_{1}

. Therefore, Patient

x_{1}

is more likely to be sick with Disease B.

Algorithm 5 The other decision making algorithm [44].

Input: A SVN decision matrix D and a weight vector

w

.
Output: The score ordering for all alternatives.

1: Compute

\begin{matrix} M_{k} (x_{k}, A^{*}) = \sum_{i = 1}^{m} w_{i} \frac{a_{k i} \cdot a_{i}^{*} + b_{k i} \cdot b_{i}^{*} + c_{k i} \cdot c_{i}^{*}}{\sqrt{a_{k i}^{2} + b_{k i}^{2} + c_{k i}^{2}} \cdot \sqrt{{(a_{i}^{*})}^{2} + {(b_{i}^{*})}^{2} + {(c_{i}^{*})}^{2}}} (k = 1, 2, \dots, l), \end{matrix}

where

α_{i}^{*} = 〈 a_{i}^{*}, b_{i}^{*}, c_{i}^{*} 〉 = 〈 1, 0, 0 〉

(

i = 1, 2, \dots, m

);
2: Obtain the ranking for all

M_{k} (x_{k}, A^{*})

by using the principle of numerical size and select the most possible patient.

By Algorithms 5, we have:

Step 1:

M_{1} (x_{1}, A^{*}) = 0.676, M_{2} (x_{2}, A^{*}) = 0.637, M_{3} (x_{3}, A^{*}) = 0.581, M_{4} (x_{4}, A^{*}) = 0.521, M_{5} (x_{5}, A^{*}) = 0.654 .

Step 2: The ranking order of

{x_{1}, x_{2}, \dots, x_{5}}

x_{4} < x_{3} < x_{2} < x_{5} < x_{1}

. Therefore, Patient

x_{1}

is more likely to be sick with Disease B.

All results are shown in Table 5, Figure 1 and Figure 2.

Liu [43] and Ye [44] presented the methods by SVN theory. In their methods, the ranking order would be changed by different

w

and γ. We as well as Yang et al. [32] used different rough set models to make the decision. Yang et al. present a SVN rough set model based on SVN relations, while we present a new SVN rough set model based on coverings. The results are different by Yang’s and our methods, although the methods are both based on an operator presented by Ye [44].

In any method, if there are more than one most possible patient, then each patient will be the optimal decision. In this case, we need other methods to make a further decision. By means of different methods, the obtained results may be different. To achieve the most accurate results, further diagnosis is necessary in combination with other hybrid methods.

7. Conclusions

This paper is a bridge, linking SVNSs and covering-based rough sets. By introducing some definitions and properties in SVN β-covering approximation spaces, we present two types of SVN covering rough set models. Then, their characterizations and matrix representations are investigated. Moreover, an application to the problem of DM is proposed. The main conclusions in this paper and the further work to do are listed as follows.

Two types of SVN covering rough set models are first presented, which combine SVNSs with covering-based rough sets. Some definitions and properties in covering-based rough set model, such as coverings and neighborhoods, are generalized to SVN covering rough set models. Neutrosophic sets and related algebraic structures [47,48,49] will be connected with the research content of this paper in further research.
It would be tedious and complicated to use set representations to calculate SVN covering approximation operators. Therefore, the matrix representations of these SVN covering approximation operators make it possible to calculate them through the new matrices and matrix operations. By these matrix representations, calculations will become algorithmic and can be easily implemented by computers.
We propose a method to DM problems under one of the SVN covering rough set models. It is a novel method based on approximation operators specific to SVN covering rough sets firstly. The comparison analysis is very interesting to show the difference between the proposed method and other methods.

Author Contributions

All authors contributed equally to this paper. The idea of this whole thesis was put forward by X.Z., and he also completed the preparatory work of the paper. J.W. analyzed the existing work of rough sets and SVNSs, and wrote the paper.

Funding

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61573240 and 61473239.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The first chat of different values of patient in utilizing different methods in Example 10.

Figure 2. The second chat of different values of patient in utilizing different methods in Example 10.

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

\hat{C}

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

\hat{C}

$U$	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$
$x_{1}$	$〈 0.7, 0.2, 0.5 〉$	$〈 0.6, 0.2, 0.4 〉$	$〈 0.4, 0.1, 0.5 〉$	$〈 0.1, 0.5, 0.6 〉$
$x_{2}$	$〈 0.5, 0.3, 0.2 〉$	$〈 0.5, 0.2, 0.8 〉$	$〈 0.4, 0.5, 0.4 〉$	$〈 0.6, 0.1, 0.7 〉$
$x_{3}$	$〈 0.4, 0.5, 0.2 〉$	$〈 0.2, 0.3, 0.6 〉$	$〈 0.5, 0.2, 0.4 〉$	$〈 0.6, 0.3, 0.4 〉$
$x_{4}$	$〈 0.6, 0.1, 0.7 〉$	$〈 0.4, 0.5, 0.7 〉$	$〈 0.3, 0.6, 0.5 〉$	$〈 0.5, 0.3, 0.2 〉$
$x_{5}$	$〈 0.3, 0.2, 0.6 〉$	$〈 0.7, 0.3, 0.5 〉$	$〈 0.6, 0.3, 0.5 〉$	$〈 0.8, 0.1, 0.2 〉$

Table 2. The tabular representation of

{\tilde{N}}_{x_{k}}^{β}

(

k = 1, 2, 3, 4, 5

Table 2. The tabular representation of

{\tilde{N}}_{x_{k}}^{β}

(

k = 1, 2, 3, 4, 5

${\tilde{N}}_{x_{k}}^{β}$	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$
${\tilde{N}}_{x_{1}}^{β}$	$〈 0.6, 0.2, 0.5 〉$	$〈 0.5, 0.3, 0.8 〉$	$〈 0.2, 0.5, 0.6 〉$	$〈 0.4, 0.5, 0.7 〉$	$〈 0.3, 0.3, 0.6 〉$
${\tilde{N}}_{x_{2}}^{β}$	$〈 0.1, 0.5, 0.6 〉$	$〈 0.5, 0.3, 0.8 〉$	$〈 0.2, 0.5, 0.6 〉$	$〈 0.4, 0.5, 0.7 〉$	$〈 0.3, 0.3, 0.6 〉$
${\tilde{N}}_{x_{3}}^{β}$	$〈 0.1, 0.5, 0.6 〉$	$〈 0.4, 0.5, 0.7 〉$	$〈 0.5, 0.3, 0.4 〉$	$〈 0.3, 0.6, 0.5 〉$	$〈 0.6, 0.3, 0.5 〉$
${\tilde{N}}_{x_{4}}^{β}$	$〈 0.1, 0.5, 0.6 〉$	$〈 0.5, 0.3, 0.7 〉$	$〈 0.4, 0.5, 0.4 〉$	$〈 0.5, 0.3, 0.7 〉$	$〈 0.3, 0.2, 0.6 〉$
${\tilde{N}}_{x_{5}}^{β}$	$〈 0.1, 0.5, 0.6 〉$	$〈 0.4, 0.5, 0.8 〉$	$〈 0.2, 0.3, 0.6 〉$	$〈 0.3, 0.6, 0.7 〉$	$〈 0.6, 0.3, 0.5 〉$

Table 3.

s (x_{k})

(

k = 1, 2, \dots, 5

Table 3.

s (x_{k})

(

k = 1, 2, \dots, 5

$U$	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$
$s (x_{_{k}})$	$0.945$	$0.937$	$0.922$	$0.909$	$0.958$

Table 4. The generalized SVN approximation space (

U, V, \tilde{R}

Table 4. The generalized SVN approximation space (

U, V, \tilde{R}

$\tilde{R}$	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$
$y_{1}$	$〈 0.7, 0.2, 0.5 〉$	$〈 0.5, 0.3, 0.2 〉$	$〈 0.4, 0.5, 0.2 〉$	$〈 0.6, 0.1, 0.7 〉$	$〈 0.3, 0.2, 0.6 〉$
$y_{2}$	$〈 0.6, 0.2, 0.4 〉$	$〈 0.5, 0.2, 0.8 〉$	$〈 0.2, 0.3, 0.6 〉$	$〈 0.4, 0.5, 0.7 〉$	$〈 0.7, 0.3, 0.5 〉$
$y_{3}$	$〈 0.4, 0.1, 0.5 〉$	$〈 0.4, 0.5, 0.4 〉$	$〈 0.5, 0.2, 0.4 〉$	$〈 0.3, 0.6, 0.5 〉$	$〈 0.6, 0.3, 0.5 〉$
$y_{4}$	$〈 0.1, 0.5, 0.6 〉$	$〈 0.6, 0.1, 0.7 〉$	$〈 0.6, 0.3, 0.4 〉$	$〈 0.5, 0.3, 0.2 〉$	$〈 0.8, 0.1, 0.2 〉$

Table 5. The results utilizing the different methods of Example 10.

Methods	The Final Ranking	The Patient Is Most Sick With the Disease B
Algorithm 2 in Liu [43]	$x_{4}, x_{3}, x_{2}, x_{5}, x_{1}$	$x_{1}$
Algorithm 3 in Yang et al. [32]	$x_{4}, x_{3}, x_{5}, x_{2}, x_{1}$	$x_{1}$
Algorithm 4 in Ye [44]	$x_{4}, x_{3}, x_{2}, x_{5}, x_{1}$	$x_{1}$
Algorithm 5 in Ye [44]	$x_{4}, x_{3}, x_{2}, x_{5}, x_{1}$	$x_{1}$
Algorithm 1 in this paper	$x_{4}, x_{3}, x_{2}, x_{1}, x_{5}$	$x_{5}$

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Wang, J.; Zhang, X. Two Types of Single Valued Neutrosophic Covering Rough Sets and an Application to Decision Making. Symmetry 2018, 10, 710. https://doi.org/10.3390/sym10120710

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Two Types of Single Valued Neutrosophic Covering Rough Sets and an Application to Decision Making

Abstract

1. Introduction

2. Basic Definitions

3. Single Valued Neutrosophic $β$ -Covering Approximation Space

4. Two Types of Single Valued Neutrosophic Covering Rough Set Models

5. Matrix Representations of These Single Valued Neutrosophic Covering Rough Set Models

6. An Application to Decision Making Problems

6.1. The Problem of Decision Making

6.2. The Decision Making Algorithm

6.3. An Applied Example

6.4. A Comparison Analysis

6.4.1. The Results of Liu’s Method

6.4.2. The Results of Yang’s Method

6.4.3. The Results of Ye’s Methods

7. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

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Two Types of Single Valued Neutrosophic Covering Rough Sets and an Application to Decision Making

Abstract

1. Introduction

2. Basic Definitions

3. Single Valued Neutrosophic β -Covering Approximation Space

4. Two Types of Single Valued Neutrosophic Covering Rough Set Models

5. Matrix Representations of These Single Valued Neutrosophic Covering Rough Set Models

6. An Application to Decision Making Problems

6.1. The Problem of Decision Making

6.2. The Decision Making Algorithm

6.3. An Applied Example

6.4. A Comparison Analysis

6.4.1. The Results of Liu’s Method

6.4.2. The Results of Yang’s Method

6.4.3. The Results of Ye’s Methods

7. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

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3. Single Valued Neutrosophic $β$ -Covering Approximation Space