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

Authors: Xiongwei Zhang; Mohammed Atef; Ahmed Mostafa Khalil
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

This paper aims to propose the notion of Type-1 single-valued neutrosophic complementaryβ-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.


---

## Body

## 1. Introduction

In 1982, the world-known new notion called rough sets (briefly, RSs) dealt with uncertain data on the hand of Pawlak [1, 2]. This notion helped researchers in several areas of research to develop these areas through RS, for instance, there are many published papers (see [3–19]). The famed generalization of RS is covering rough sets (briefly, CRSs). The CRSs were studied by many specialists from different views which made an evolution in many fields such as computer science, mathematics, and chemistry. Some of the relevant studies helped scholars to solve many life problems [20–33]. Consequently, in 1990, the notions of fuzzy rough sets (briefly, FRSs) and rough fuzzy sets (briefly, RFSs) are defined by Dubois and Prade [34] from the merging between the CRS and the fuzzy sets (briefly, FSs) which appeared by Zadeh [35]. From this point of view, the new kinds of covering fuzzy rough sets (briefly, CFRSs) through the fuzzy β-neighborhoods were called fuzzy β covering rough sets (briefly, FβCRSs) (see [36]). To complete this study, Yang et al. [37, 38] defined several basic notions of fuzzy complementary β-neighborhoods, fuzzy β-minimal description, and fuzzy β-maximal description to establish new classes of FβCRS.The 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]. The notions of neutrosophic soft rough sets and its generalizations are presented in [48–53].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.The organization of this article is as follows. In Section2, 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. Thus, 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. Then, the relevant properties are also discussed. The 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.


## 2. Preliminaries

In this section, we review some basic terminologies about the subject of this study.Definition 1 (Cf. [26]).
Assume thatΩ 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 2 (Cf. [54, 55]).
Assume thatΩ is a universe. We say Γ˜=C˜1,C˜2,…,C˜m, with C˜i∈ℱΩi=1,2,…,m, a fuzzy covering (briefly, FC) of Ω if ∪i=1mC˜ix=1, for each x∈Ω.
The notion of fuzzyβ-covering was discovered by Ma [36] 0<β≤1. This notion is considered as a generalization of FC. If Γ˜=C˜1,C˜2,…,C˜m, with C˜i∈ℱΩi=1,2,…,m, then Ω,Γ˜ is called a fuzzy β-covering approximation space (briefly, FβCAS).Definition 3 (Cf. [54, 55]).
Assume thatΩ is not an empty set. For each x∈Ω, define the SVN set A⊆Ω as the following formula:(1)A=x,TAx,ℐAx,ℱAx.where TA:Ω⟶0,1 is the degree of truth membership of the element x to A, ℐA:Ω⟶0,1 is the degree of indeterminacy membership of the element x to A, and ℱA:Ω⟶0,1 is the degree of falsity membership. These variables satisfy 0≤TA+ℐA+ℱA≤3.
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]).
LetΩ be a universe and SVN Ω be the SVN power set of Ω. For a SVN number β=a,b,c, we call Γ˜=C˜1,C˜2,…,C˜m, with C˜i∈SVNΩi=1,2,…,m, a Type-1 SVN β-covering (a Type-2 SVN β-covering) of Ω if C˜ix≥βC˜ix≽β, for each x∈Ω. Moreover, Ω,Γ˜ is called a Type-1 SVN β-covering approximation space (a Type-2 SVN β-covering approximation space) (briefly, 1-SVN β CAS (2-SVNβCAS)).
IfA=a1,b1,c1 and ℬ=a2,b2,c2 are two SVN numbers, then(i)
A≤ℬ⇔a1≤a2,b1≥b2,c1≥c2(ii)
A≥ℬ⇔a1≥a2,b1≤b2,c1≤c2(iii)
A≺¯ℬ⇔a1≤a2,b1≤b2,c1≥c2(iv)
A≽ℬ⇔a1≥a2,b1≥b2,c1≤c2
Here,∀A,ℬ∈SVNΩ, and we have the following relation, union, and intersection operations.
For Type-1,(1)
A⊆ℬ⇔TA≤Tℬ,ℐℬ≤ℐA,ℱA≤ℱℬ∀x∈Ω(2)
A∩ℬ=x,TA∧Tℬ,ℐA∨ℐℬ,ℱA∨ℱℬ(3)
A∪ℬ=x,TA∨Tℬ,ℐA∧ℐℬ,ℱA∧ℱℬ
For Type-2,(1)
A⊑ ℬ⇔TA≤Tℬ,ℐℬ≤ℐA,ℱA≤ℱℬ∀x∈Ω(2)
A⊓ ℬ=x,TA∧Tℬ,ℐA∧ℐℬ,ℱA∨ℱℬ(3)
A⊔ ℬ=x,TA∨Tℬ,ℐA∨ℐℬ,ℱA∧ℱℬDefinition 5 (Cf. [46, 47]).
LetΩ,Γ˜ be a 1-SVN β CAS with Γ˜=C˜1,C˜2,…,C˜m for some β=a,b,c. Then, for each x∈Ω, define the Type-1 SVN β-neighborhood (the Type-2 SVN β-neighborhood) of x as follows:(2)N1˜xβ=∩C˜i∈Γ˜:C˜ix≥β=∩C˜i∈Γ˜:TC˜i≥a,ℐC˜i≤b,ℱC˜i≤c,N2˜xβ=⊓C˜i∈Γ˜:C˜ix≽β=⊓C˜i∈Γ˜:TC˜i≥a,ℐC˜i≥b,ℱC˜i≤c.Definition 6 (Cf. [46, 47]).
LetΩ,Γ˜ be a 1-SVNβCAS (resp., 2-SVNβCAS) for some β=a,b,c. For each x∈Ω and A∈SVNΩ, define the first type of Type-1 SVN lower approximation (1-1-SVNLA) ℒ11A (resp., the first type of Type-2 SVN lower approximation (1-2-SVNLA) ℒ12A) and the first type of Type-1 SVN upper approximation (1-1-SVNUA) U11A (resp., the first type of Type-2 SVN upper approximation (1-2-SVNUA) U12A) as follows:(3)ℒ11A=x,∧y∈ΩℱN˜1xβy∨TAy,∧y∈Ω1−ℐN˜1xβy∨ℐAy,∨y∈ΩTN˜1xβy∧ℱAy,U11A=x,∨y∈ΩTN˜1xβy∧TAy,∨y∈ΩℐN˜1xβy∧ℐAy,∧y∈ΩℱN˜1xβy∨ℱAy,ℒ12A=x,∧y∈ΩℱN˜2xβy∨TAy,∧y∈Ω1−ℐN˜2xβy∨ℐAy,∨y∈ΩTN˜2xβy∧ℱAy,U12A=x,∨y∈ΩTN˜2xβy∧TAy,∨y∈ΩℐN˜2xβy∧ℐAy,∧y∈ΩℱN˜2xβy∨ℱAy.
Ifℒ11Aresp.,ℒ12A≠U11A (resp., U12A), then A is called the first type of Type-1 SVN β-covering rough sets (resp., the first type of Type-2 SVN β-covering rough sets) (briefly, 1-1-SVNβCRSs (resp., 1-2-SVNβCRSs)).


## 3. Type-1 SVN Complementaryβ-Neighborhood and Three New Kinds of Type-1 SVNβ-CRS

We will propose the concept of a type-1 SVN complementaryβ-neighborhood and three new kinds of Type-1 SVNβ-CRS and introduce several definitions, propositions, and examples as indicated below.Definition 7.
LetΩ,Γ˜ be a 1-SVNβCAS with Γ˜=C˜1,C˜2,…,C˜m, for some β=a,b,c. Then, for each x∈Ω, define the type-1 SVN complementary β-neighborhood of x as follows:(4)ℳ˜1xβy=N˜1yβx,∀y∈Ω.Example 1.
LetΩ,Γ˜ be a 1-SVNβCAS, Ω=x1,x2,x3,x4,x5 and Γ˜=C˜1,C˜2,C˜3,C˜4, where β=0.5,0.3,0.8 are summarized in Table 1.
In Table1, N1˜x1β=C˜1∩C˜2,N1˜x2β=C˜1∩C˜2∩C˜4,N1˜x3β=C˜3∩C˜4,N1˜x4β=C˜1∩C˜4, and N1˜x5β=C˜2∩C˜3∩C˜4.
Table2 contains the results of type-1 SVN β-neighborhood.
Thus, we can obtain the values of type-1 SVN complementaryβ-neighborhood as in Table 3.
Hence, we can mergeN˜1xβ and ℳ˜1xβ and compute N˜1xβ∩ℳ˜1xβ as Table 4.
Also, we can computeN˜1xβ∪ℳ˜1xβ as set in Table 5.Table 1
Ω,Γ˜.
C˜1C˜2C˜3C˜4x10.7,0.2,0.50.6,0.2,0.40.4,0.1,0.50.1,0.5,0.6x20.5,0.3,0.20.5,0.2,0.80.4,0.5,0.40.6,0.1,0.7x30.4,0.5,0.20.2,0.3,0.60.5,0.2,0.40.6,0.3,0.4x40.6,0.1,0.70.4,0.5,0.70.3,0.6,0.50.5,0.3,0.2x50.3,0.2,0.60.7,0.3,0.50.6,0.3,0.50.8,0.1,0.2Table 2
N1˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5N1˜x1β0.6,0.2,0.50.5,0.3,0.80.2,0.5,0.60.4,0.5,0.70.3,0.3,0.6N1˜x2β0.1,0.5,0.60.5,0.3,0.80.2,0.5,0.60.4,0.5,0.70.3,0.3,0.6N1˜x3β0.1,0.5,0.60.4,0.5,0.70.5,0.3,0.40.3,0.6,0.50.6,0.3,0.5N1˜x4β0.1,0.5,0.60.5,0.3,0.70.4,0.5,0.40.5,0.3,0.70.3,0.2,0.6N1˜x5β0.1,0.5,0.60.4,0.5,0.80.2,0.3,0.60.3,0.6,0.70.6,0.3,0.5Table 3
ℳ1˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5ℳ1˜x1β0.5,0.8,0.60.8,0.7,0.50.6,0.5,0.20.7,0.5,0.40.6,0.7,0.3ℳ1˜x2β0.6,0.5,0.10.8,0.7,0.50.6,0.5,0.20.7,0.5,0.40.6,0.7,0.3ℳ1˜x3β0.6,0.5,0.10.7,0.5,0.40.4,0.7,0.50.5,0.4,0.30.5,0.7,0.6ℳ1˜x4β0.6,0.5,0.10.7,0.7,0.50.4,0.5,0.40.7,0.7,0.50.6,0.8,0.3ℳ1˜x5β0.6,0.5,0.10.8,0.5,0.40.6,0.7,0.20.7,0.4,0.30.5,0.7,0.6Table 4
N1˜xiβ∩ℳ1˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5N1˜x1β∩ℳ1˜x1β0.5,0.8,0.60.5,0.7,0.80.2,0.5,0.60.4,0.5,0.70.3,0.7,0.6N1˜x2β∩ℳ1˜x2β0.1,0.5,0.60.5,0.7,0.80.2,0.5,0.60.4,0.5,0.70.3,0.7,0.6N1˜x3β∩ℳ1˜x3β0.1,0.5,0.60.4,0.5,0.70.4,0.7,0.50.3,0.6,0.50.5,0.7,0.6N1˜x4β∩ℳ1˜x4β0.1,0.5,0.60.5,0.7,0.70.4,0.5,0.40.5,0.7,0.70.3,0.8,0.6N1˜x5β∩ℳ1˜x5β0.1,0.5,0.60.4,0.5,0.80.2,0.7,0.60.3,0.6,0.70.5,0.7,0.6Table 5
N1˜xiβ∪ℳ1˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5N1˜x1β∪ℳ1˜x1β0.6,0.2,0.50.8,0.3,0.50.6,0.5,0.20.7,0.5,0.40.6,0.3,0.3N1˜x2β∪ℳ1˜x2β0.6,0.5,0.10.8,0.3,0.50.6,0.5,0.20.7,0.5,0.40.6,0.3,0.3N1˜x3β∪ℳ1˜x3β0.6,0.5,0.10.7,0.5,0.40.5,0.3,0.40.5,0.4,0.30.6,0.3,0.5N1˜x4β∪ℳ1˜x4β0.6,0.5,0.10.7,0.3,0.50.4,0.5,0.40.7,0.3,0.50.6,0.2,0.3N1˜x5β∪ℳ1˜x5β0.6,0.5,0.10.8,0.5,0.40.6,0.3,0.20.7,0.4,0.30.6,0.3,0.5Proposition 1.
LetΩ,Γ˜ be a 1-SVNβCAS, for some β=a,b,c and for each x,y,z∈Ω. Then, the following statements hold:(1)
ℳ˜1xβx≥β(2)
Ifℳ˜1xβy≥β and ℳ˜1yβz≥β, then ℳ˜1xβz≥β(3)
0<β1≤β2≤β, then ℳ1˜xβ1⊆ℳ1˜xβ2Proof.
(1)
It follows directly from Definitions5 and 7.(2)
Sinceℳ˜1xβy≥β, then N˜1yβx≥β. If C˜ix≥β, then C˜iy≥β, and since ℳ1˜yβz≥β, then N1˜zβy≥β. If C˜iy≥β, then C˜iz≥β. Therefore, ℳ˜1xβz≥β.(3)
For eachx∈Ω,0<β1≤β2≤β, then ∩C˜i∈Γ˜:C˜ix≥β1⊇∩C˜i∈Γ˜:C˜ix≥β2. Thus, by Definition 7, we have ℳ1˜xβ1⊆ℳ1˜xβ2.Proposition 2.
LetΩ,Γ˜ be a 1-SVNβCAS, for some β=a,b,c. For each x,y∈Ω,(5)ℳ˜1xβy≥β⇔ℳ˜1yβ⊆ℳ˜1xβ.Proof.
Letℳ˜1xβy≥β, Tℳ˜1xβy=T∩C˜iy=∧TC˜iy≥a,ℐ1ℳ˜xβy=ℐ∩C˜iy=∨ℐC˜iy≤b, and ℱℳ˜1xβy=ℱ∩C˜iy=∨ℱC˜iy≤c. Then, C˜i∈Γ˜:TC˜ix≥a,ℐC˜ix≤b,ℱC˜ix≤c⊆C˜i∈Γ˜:TC˜iy≥a,ℐC˜iy≤b,ℱC˜iy≤c. Thus, ℳ1˜yβ⊆ℳ˜1xβ. On the contrary, let ℳ1˜yβ⊆ℳ˜1xβ. Then, Tℳ˜1xβy≥Tℳ˜1yβy≥a,ℐℳ˜1xβy≤ℐℳ˜1yβy≤b,andℱℳ˜1xβy≤ℱℳ˜1yβy≤c. Hence, ℳ˜1xβy≥β.Now, we present the three new types of 1-1-SVNβCRSs based on Definitions 5 and 7 as indicated below.Definition 8.
ConsiderΩ,Γ˜ is a 1-SVNβCAS with Γ˜=C˜1,C˜2,…,C˜m, for some β=a,b,c. For each x∈Ω and A∈SVNΩ, then we have the following paradigms:Paradigm 1: the second type of Type-1 SVN lower approximation (2-1-SVNLA)ℒ21A and the second type of Type-1 SVN upper approximation (2-1-SVNUA) U21A are as follows:(6)ℒ21A=x,∧y∈Ωℱℳ˜1xβy∨TAy,∧y∈Ω1−ℐℳ˜1xβy∨ℐAy,∨y∈ΩTℳ˜1xβy∧ℱAy,U21A=x,∨y∈ΩTℳ˜1xβy∧TAy,∨y∈Ωℐℳ˜1xβy∧ℐAy,∧y∈Ωℱℳ˜1xβy∨ℱAy.Ifℒ21A≠U21A, then A is called the second type of Type-1 SVN β-covering rough sets (briefly, 2-1-SVNβCRSs).Paradigm 2: the third type of Type-1 SVN lower approximation (3-1-SVNLA)ℒ31A and the third type of Type-1 SVN upper approximation (3-1-SVNUA) U31A are introduced as follows:(7)ℒ31A=x,∧y∈ΩℱN˜1xβ∩ℳ˜1xβy∨TAy,∧y∈Ω1−ℐN˜1xβ∩ℳ˜1xβy∨ℐAy,∨y∈ΩTN˜1xβ∩ℳ˜1xβy∧ℱAy,U31A=x,∨y∈ΩTN˜1xβ∩ℳ˜1xβy∧TAy,∨y∈ΩℐN˜1xβ∩ℳ˜1xβy∧ℐAy,∧y∈ΩℱN˜1xβ∩ℳ˜1xβy∨ℱAy.Ifℒ31A≠U31A, then A is called the third type of Type-1 SVN β-covering rough sets (briefly, 3-1-SVNβCRSs).Paradigm 3: the fourth type of Type-1 SVN lower approximation (4-1-SVNLA)ℒ41A and the fourth type of Type-1 SVN upper approximation (4-1-SVNUA) U41A are proposed as follows:(8)ℒ41A=x,∧y∈ΩℱN˜1xβ∪ℳ˜1xβy∨TAy,∧y∈Ω1−ℐN˜1xβ∪ℳ˜1xβy∨ℐAy,∨y∈ΩTN˜1xβ∪ℳ˜1xβy∧ℱAy,U41A=x,∨y∈ΩTN˜1xβ∪ℳ˜1xβy∧TAy,∨y∈ΩℐN˜1xβ∪ℳ˜1xβy∧ℐAy,∧y∈ΩℱN˜1xβ∪ℳ˜1xβy∨ℱAy.
Ifℒ41A≠U41A, then A is called the fourth type of Type-1 SVN β-covering rough sets (briefly, 4-1-SVNβCRSs).Example 2.
Consider Example1 if β=0.5,0.3,0.8 and A=0.5,0.3,0.6/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5; then, we have the following results:(9)ℒ11A=x1,0.6,0.5,0.5,x2,0.6,0.5,0.4,x3,0.4,0.4,0.5,x4,0.4,0.5,0.4,x5,0.6,0.4,0.3,U11A=x1,0.6,0.3,0.5,x2,0.4,0.3,0.6,x3,0.6,0.5,0.5,x4,0.5,0.3,0.6,x5,0.6,0.5,0.5,ℒ21A=x1,0.3,0.3,0.6,x2,0.3,0.3,0.6,x3,0.4,0.3,0.5,x4,0.4,0.2,0.4,x5,0.3,0.3,0.6,U21A=x1,0.6,0.5,0.3,x2,0.6,0.5,0.3,x3,0.6,0.5,0.4,x4,0.6,0.5,0.3,x5,0.6,0.5,0.4,ℒ31A=x1,0.6,0.3,0.5,x2,0.6,0.3,0.4,x3,0.5,0.3,0.4,x4,0.4,0.2,0.3,x5,0.6,0.3,0.3,U31A=x1,0.5,0.5,0.6,x2,0.4,0.5,0.6,x3,0.5,0.5,0.5,x4,0.5,0.5,0.6,x5,0.5,0.5,0.6,ℒ41A=x1,0.3,0.5,0.6,x2,0.3,0.5,0.6,x3,0.4,0.5,0.5,x4,0.4,0.5,0.5,x5,0.3,0.5,0.6,U41A=x1,0.6,0.3,0.3,x2,0.6,0.3,0.3,x3,0.6,0.5,0.4,x4,0.6,0.3,0.3,x5,0.6,0.5,0.4.
Next, we will present Proposition3 for the 2-1-SVNβCRS model; also, it satisfies in case of the 3-1-SVNβCRS and the 4-1-SVNβCRS models.Proposition 3.
LetΩ,Γ˜ be a 1-SVNβCAS, for some β=a,b,c. For each x,y,z∈Ω and A,ℬ∈SVNΩ. Then, the following statements hold:(1)
(SVNL1)ℒ21Ac=U21Ac.(SVNU1)U21Ac=ℒ21Ac.(2)
IfA⊆ℬ, then(SVNL2)ℒ21A⊆ℒ21ℬ.(SVNU2)U21A⊆U21ℬ.(3)
(SVNL3)ℒ21A∩ℬ=ℒ21A∩ℒ21ℬ.(SVNU3)U21A∩ℬ⊆U21A∩U21ℬ.(4)
(SVNL4)ℒ21A∪ℬ⊇ℒ21A∪ℒ21ℬ.(SVNU4)U21A∩ℬ=U21A∩U21ℬ.Proof.
We shall only prove (SVNL1), (SVNL2), (SVNL3), and (SVNL4).
(SVNL1):(10)ℒ21Ac=x,∧y∈Ωℱℳ˜1xβy∨TAcy,∧y∈Ω1−ℐℳ˜1xβy∨ℐAcy,∨y∈ΩTℳ˜1xβy∧ℱAcy=x,∧y∈Ωℱℳ˜1xβy∨ℱAy,∧y∈Ω1−ℐℳ˜1xβy∨1−ℐAy,∨y∈ΩTℳ˜1xβy∧TAy=U21Ac.
(SVNL2): letA,ℬ∈SVNΩ such that A⊆ℬ (i.e., TA≤Tℬ,ℐℬ≤ℐA and ℱℬ≤ℱA) and x∈Ω. Then, we get the following result:(11)ℒ21Ax=x,∧y∈Ωℱℳ˜1xβy∨TAy,∧y∈Ω1−ℐℳ˜1xβy∨ℐAy,∨y∈ΩTℳ˜1xβy∧ℱAy,Tℒ21A=∧y∈Ωℱℳ˜1xβy∨TAy≤∧y∈Ωℱℳ˜1xβy∨Tℬy=Tℒ21ℬ,ℐℒ21A=∧y∈Ω1−ℐℳ˜1xβy∨ℐAy≥∧y∈Ω1−ℐℳ˜1xβy∨ℐℬy=ℐℒ21ℬ,ℱℒ21A=∨y∈ΩTℳ˜1xβy∧ℱAy≥∨y∈ΩTℳ˜1xβy∧ℱℬy=ℱℒ21ℬ.
Therefore,ℒ21A⊆ℒ21ℬ.
(SVNL3): ifx∈Ω, then we have(12)ℒ21Ax=x,∧y∈Ωℱℳ˜1xβy∨TA∩ℬy,∧y∈Ω1−ℐℳ˜1xβy∨ℐA∩ℬy,∨y∈ΩTℳ˜1xβy∧ℱA∩ℬy=x,∧y∈Ωℱℳ˜1xβy∨TAy∧Tℬy,∧y∈Ω1−ℐℳ˜1xβy∨ℐAy∨ℐℬy,∨y∈ΩTℳ˜1xβy∧ℱAy∨ℱℬy=x,∧y∈Ωℱℳ˜1xβy∨TAy∧ℱℳ˜1xβy∨Tℬy,∧y∈Ω1−ℐℳ˜1xβy∨ℐAy∨1−ℐℳ˜1xβy∨ℐℬy,∨y∈ΩTℳ˜1xβy∧ℱAy∨Tℳ˜1xβy∧ℱℬy=ℒ21A∩ℒ21ℬ.
(SVNL4): sinceA∪ℬ⊇A, then, by (SVNL2), we have ℒ21A∪ℬ⊇ℒ21A. Similarly, A∪ℬ⊇ℬ; then, by (SVNL2), we have ℒ21A∪ℬ⊇ℒ21ℬ. Thus, ℒ21A∪ℬ⊇ℒ21A∪ℒ21ℬ.Now, we proceed to explain some relationships among these models.Proposition 4.
LetΩ,Γ˜ be a 1-SVNβCAS and A∈SVNΩ. Then, we have the following properties:(1)
ℒ41A≤ℒ21A≤ℒ31A(2)
ℒ41A≤ℒ11A≤ℒ31A(3)
U31A≤U21A≤ℒ41A(4)
U31A≤U11A≤U41AProof.
The proof is clear from Definition8.Proposition 5.
LetΩ,Γ˜ be a 1-SVNβCAS and A∈SVNΩ. Then, we have the following properties:(1)
ℒ31A≥ℒ11A∪ℒ21A(2)
U31A≤U11A∩U21A(3)
ℒ41A≤ℒ11A∩ℒ21A(4)
U41A≥U11A∪U21AProof.
(straightforward)


## 4. Type-2 SVN Complementaryβ-Neighborhood and Three New Kinds of Type-2 SVNβ-CRS

Definition 9.
LetΩ,Γ˜ be a 2-SVNβCAS with Γ˜=C˜1,C˜2,…,C˜m, for some β=a,b,c. Then, for each x∈Ω, define the type-2 SVN complementary β-neighborhood of x as follows:(13)ℳ˜2xβy=N˜2yβx,∀y∈Ω.Example 3.
Consider Example1, β=0.5,0.1,0.8 and A=0.6,0.3,0.5/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5. Then, the values of type-2 SVN β-neighborhood are seen in Table 6:(14)N2˜x1β=C˜1⊓C˜2,N2˜x2β=C˜1⊓C˜2⊓C˜4,N2˜x3β=C˜3⊓C˜4,N2˜x4β=C˜1⊓C˜4,N2˜x5β=C˜2⊓C˜3⊓C˜4.
Also, we compute type-2 SVN complementaryβ-neighborhood as in Table 7.
Thus, we can mergeℳ˜2xβ and ℳ˜2xβ and calculate N˜2xβ⊓ℳ˜2xβ as Table 8.
Furthermore, we can calculateN˜2xβ⊔ℳ˜2xβ, as set in Table 9.Table 6
N2˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5N2˜x1β0.6,0.2,0.50.5,0.2,0.80.2,0.3,0.60.4,0.1,0.70.3,0.2,0.6N2˜x2β0.1,0.2,0.60.5,0.1,0.80.2,0.3,0.60.4,0.1,0.70.3,0.1,0.6N2˜x3β0.1,0.1,0.60.4,0.1,0.70.5,0.2,0.40.3,0.3,0.50.6,0.1,0.5N2˜x4β0.1,0.2,0.60.5,0.1,0.70.4,0.3,0.40.5,0.1,0.70.3,0.1,0.6N2˜x5β0.1,0.1,0.60.4,0.1,0.80.2,0.2,0.60.3,0.3,0.70.6,0.1,0.5Table 7
ℳ2˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5ℳ2˜x1β0.5,0.8,0.60.8,0.8,0.50.6,0.7,0.20.7,0.9,0.40.6,0.8,0.3ℳ2˜x2β0.6,0.8,0.10.8,0.9,0.50.6,0.7,0.20.7,0.9,0.40.6,0.9,0.3ℳ2˜x3β0.6,0.9,0.10.7,0.9,0.40.4,0.8,0.50.5,0.7,0.30.5,0.9,0.6ℳ2˜x4β0.6,0.8,0.10.7,0.9,0.50.4,0.7,0.40.7,0.9,0.50.6,0.9,0.3ℳ2˜x5β0.6,0.9,0.10.8,0.9,0.40.6,0.8,0.20.7,0.7,0.30.5,0.9,0.6Table 8
N2˜xiβ⊓ℳ2˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5N2˜x1β⊓ℳ2˜x1β0.5,0.2,0.60.5,0.2,0.80.2,0.3,0.60.4,0.1,0.70.3,0.2,0.6N2˜x2β⊓ℳ2˜x2β0.1,0.2,0.60.5,0.1,0.80.2,0.3,0.60.4,0.1,0.70.3,0.1,0.6N2˜x3β⊓ℳ2˜x3β0.1,0.1,0.60.4,0.1,0.70.4,0.2,0.50.3,0.3,0.50.5,0.1,0.6N2˜x4β⊓ℳ2˜x4β0.1,0.2,0.60.5,0.1,0.70.4,0.3,0.40.5,0.1,0.70.3,0.1,0.6N2˜x5β⊓ℳ2˜x5β0.1,0.1,0.60.4,0.1,0.80.2,0.2,0.60.3,0.3,0.70.5,0.1,0.6Table 9
N2˜xiβ⊔ℳ2˜xiβ,∀i=1,2,…,5.
x1x2x3x4x5N2˜x1β⊔ℳ2˜x1β0.6,0.8,0.50.8,0.8,0.50.6,0.7,0.20.7,0.9,0.40.6,0.8,0.3N2˜x2β⊔ℳ2˜x2β0.6,0.8,0.10.8,0.9,0.50.6,0.7,0.20.7,0.9,0.40.6,0.9,0.3N2˜x3β⊔ℳ2˜x3β0.6,0.9,0.10.7,0.9,0.40.5,0.8,0.40.5,0.7,0.30.6,0.9,0.5N2˜x4β⊔ℳ2˜x4β0.6,0.8,0.10.7,0.9,0.50.4,0.7,0.40.7,0.9,0.50.6,0.9,0.3N2˜x5β⊔ℳ2˜x5β0.6,0.9,0.10.8,0.9,0.40.6,0.8,0.20.7,0.7,0.30.6,0.9,0.5Proposition 6.
LetΩ,Γ˜ be a 2-SVNβCAS, for some β=a,b,c and for each x,y,z∈Ω. Then, the following statements hold:(1)
ℳ˜2xβx≽β(2)
Ifℳ˜2xβy≽β and ℳ2˜yβz≽β, then ℳ˜2xβz≽β(3)
0≺β1≼¯β2≼¯β, then ℳ2˜xβ1⊑ℳ2˜xβ2Proof.
(1)
follows directly from Definitions5 and 9.(2)
Sinceℳ˜2xβy≽β, then N˜2yβx≽β. If C˜ix≽β, then C˜iy≽β, and since ℳ2˜yβz≽β, then N2˜zβy≽β. If C˜iy≽β, then C˜iz≽β. Therefore, ℳ˜2xβz≽β.(3)
For eachx∈Ω,0≺β1≺¯β2≺¯β, then ⊓C˜i∈Γ˜:C˜ix≽β1⊒⊓C˜i∈Γ˜:C˜ix≽β2. Thus, by Definition 9, we have ℳ2˜xβ1⊑ℳ2˜xβ2.Proposition 7.
LetΩ,Γ˜ be a 1-SVNβCAS, for some β=a,b,c. For each x,y∈Ω, ℳ˜2xβy≽β⇔ℳ2˜yβ⊑ℳ˜2xβ.Proof.
Letℳ˜2xβy≽β, Tℳ˜2xβy=T⊓C˜iy=∧TC˜iy≥a,ℐℳ˜2xβy=ℐ⊓C˜iy=∨ℐC˜iy≥b, and ℱℳ˜2xβy=ℱ⊓C˜iy=∨ℱC˜iy≤c. Then, C˜i∈Γ˜:TC˜ix≥a,ℐC˜ix≥b,ℱC˜ix≤c⊆C˜i∈Γ˜:TC˜iy≥a,ℐC˜iy≥b,ℱC˜iy≤c Thus, ℳ2˜yβ⊑ℳ˜2xβ. On the contrary, let ℳ2˜yβ⊑ℳ˜2xβ. Then, Tℳ˜2xβy≥Tℳ˜2yβy≥a,ℐℳ˜2xβy≥ℐℳ2˜yβy≥b,andℱℳ˜2xβy≤ℱℳ2˜yβy≤c. Hence, ℳ˜2xβy≽β.Here, we construct three new types of 2-1-SVNβCRSs based on Definitions 5 and 9 as seen below.Definition 10.
ConsiderΩ,Γ˜ be a 2-SVNβCAS with Γ˜=C˜1,C˜2,…,C˜m for some β=a,b,c. For each x∈Ω and A∈SVNΩ, then we have the following paradigms.Paradigm 1: The second type of Type-2 SVN lower approximation (2-2-SVNLA)ℒ22A and the second type of Type-2 SVN upper approximation (2-2-SVNUA) U22A are found as follows:(15)ℒ22A=x,∧y∈Ωℱℳ˜2xβy∨TAy,∧y∈Ω1−ℐℳ˜2xβy∨ℐAy,∨y∈ΩTℳ˜2xβy∧ℱAy,U22A=x,∨y∈ΩTℳ˜2xβy∧TAy,∨y∈Ωℐℳ˜2xβy∧ℐAy,∧y∈Ωℱℳ˜2xβy∨ℱAy.Ifℒ22A≠U22A, then A is called the second type of Type-2 SVN β-covering rough sets (briefly, 2-2-SVNβCRSs)Paradigm 2: the third type of Type-2 SVN lower approximation (3-2-SVNLA)ℒ32A and the third type of Type-2 SVN upper approximation (3-2-SVNUA) U32A are introduced as follows:(16)ℒ32A=x,∧y∈ΩℱN˜2xβ⊓ℳ˜2xβy∨TAy,∧y∈Ω1−ℐN˜2xβ⊓ℳ˜1xβy∨ℐAy,∨y∈ΩTN˜2xβ⊓ℳ˜2xβy∧ℱAy,U32A=x,∨y∈ΩTN˜2xβ⊓ℳ˜2xβy∧TAy,∨y∈ΩℐN˜2xβ⊓ℳ˜2xβy∧ℐAy,∧y∈Ωℱ2N˜xβ⊓ℳ˜2xβy∨ℱAy.Ifℒ32A≠U32A, then A is called the third type of Type-2 SVN β-covering rough sets (briefly, 3-2-SVNβCRSs).Paradigm 3: the fourth type of Type-2 SVN lower approximation (4-2-SVNLA)ℒ42A and the fourth type of Type-2 SVN upper approximation (4-2-SVNUA) U42A are proposed as follows:(17)ℒ42A=x,∧y∈ΩℱN˜2xβ⊔ℳ˜2xβy∨TAy,∧y∈Ω1−ℐN˜2xβ⊔ℳ˜2xβy∨ℐAy,∨y∈ΩTN˜2xβ⊔ℳ˜2xβy∧ℱAy,U42A=x,∨y∈ΩTN˜2xβ⊔ℳ˜2xβy∧TAy,∨y∈ΩℐN˜2xβ⊔ℳ˜1xβy∧ℐAy,∧y∈ΩℱN˜2xβ⊔ℳ˜2xβy∨ℱAy.
Ifℒ42A≠U42A, then A is called the fourth type of Type-2 SVN β-covering rough sets (briefly, 4-2-SVNβCRSs).Example 4.
Consider Example1 if β=0.5,0.1,0.8 and A=0.6,0.3,0.5/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5; then, we have the following results:(18)ℒ12A=x1,0.6,0.7,0.5,x2,0.6,0.7,0.4,x3,0.4,0.7,0.5,x4,0.4,0.7,0.4,x5,0.6,0.7,0.3,U12A=x1,0.6,0.2,0.5,x2,0.4,0.2,0.6,x3,0.6,0.3,0.5,x4,0.5,0.2,0.6,x5,0.6,0.3,0.5,ℒ22A=x1,0.3,0.2,0.6,x2,0.3,0.2,0.6,x3,0.4,0.2,0.5,x4,0.4,0.2,0.4,x5,0.3,0.2,0.6,U22A=x1,0.6,0.5,0.3,x2,0.6,0.5,0.3,x3,0.6,0.5,0.4,x4,0.6,0.5,0.3,x5,0.6,0.5,0.4,ℒ32A=x1,0.6,0.7,0.5,x2,0.6,0.7,0.4,x3,0.5,0.7,0.4,x4,0.4,0.7,0.3,x5,0.6,0.7,0.3,U32A=x1,0.5,0.2,0.6,x2,0.4,0.2,0.6,x3,0.5,0.3,0.5,x4,0.5,0.2,0.6,x5,0.5,0.3,0.6,ℒ42A=x1,0.3,0.2,0.6,x2,0.3,0.2,0.6,x3,0.4,0.2,0.5,x4,0.4,0.2,0.5,x5,0.3,0.2,0.6,U42A=x1,0.6,0.5,0.3,x2,0.6,0.5,0.3,x3,0.6,0.5,0.4,x4,0.6,0.5,0.3,x5,0.6,0.5,0.4.
In the following, we will propose Proposition8 for the 2-2-SVNβCRS model; also, it fulfills in case of the 3-2-SVNβCRS and the 4-2-SVNβCRS models.Proposition 8.
LetΩ,Γ˜ be a 1-SVNβCAS, for some β=a,b,c. For each x,y,z∈Ω and A,ℬ∈SVNΩ, then the following statements hold:(1)
(SVNL1):ℒ22Ac=U22Ac.(SVNU1):U22Ac=ℒ22Ac.(2)
IfA⊆ℬ, then(SVNL2):ℒ22A⊆ℒ22ℬ.(SVNU2):U22A⊆U22ℬ.(3)
(SVNL3):ℒ22A∩ℬ=ℒ22A∩ℒ22ℬ.(SVNU3):U22A∩ℬ⊆U22A∩U22ℬ.(4)
(SVNL4):ℒ22A∪ℬ⊇ℒ22A∪ℒ22ℬ.(SVNU4):U22A∩ℬ=U22A∩U21ℬ.(5)
(SVNL5):ℒ22Ω=Ω.(SVNU5):U22∅=∅.Proof.
We shall only prove (SVNL1), (SVNL2), (SVNL3), and (SVNL4).
(SVNL1):(19)ℒ22Ac=x,∧y∈Ωℱ2ℳ˜xβy∨TAcy,∧y∈Ω1−ℐ2ℳ˜xβy∨ℐAcy,∨y∈ΩT2ℳ˜xβy∧ℱAcy=x,∧y∈Ωℱ2ℳ˜xβy∨ℱAy,∧y∈Ω1−ℐ2ℳ˜xβy∨1−ℐAy,∨y∈ΩT2ℳ˜xβy∧TAy=U22Ac.
(SVNL2): letA,ℬ∈SVNΩ such that A⊑ℬ (i.e., TA≤Tℬ,ℐℬ≤ℐA and ℱℬ≤ℱA) and x∈Ω. Then, we get the following result:(20)ℒ22Ax=x,∧y∈Ωℱℳ˜2xβy∨TAy,∧y∈Ω1−ℐℳ˜2xβy∨ℐAy,∨y∈ΩTℳ˜2xβy∧ℱAy,Tℒ22A=∧y∈Ωℱℳ˜2xβy∨TAy≤∧y∈Ωℱℳ˜2xβy∨Tℬy=Tℒ22ℬ,ℐℒ22A=∧y∈Ω1−ℐℳ˜2xβy∨ℐAy≤∧y∈Ω1−ℐℳ˜2xβy∨ℐℬy=ℐℒ22ℬ,ℱℒ22A=∨y∈ΩTℳ˜2xβy∧ℱAy≥∨y∈ΩTℳ˜2xβy∧ℱℬy=ℱℒ22ℬ.
Therefore,ℒ22A⊑ℒ22ℬ.
(SVNL3): Ifx∈Ω, then we have(21)ℒ22Ax=x,∧y∈Ωℱℳ˜2xβy∨TA⊓ℬy,∧y∈Ω1−ℐℳ˜2xβy∨ℐA⊓ℬy,∨y∈ΩTℳ˜2xβy∧ℱA⊓ℬy=x,∧y∈Ωℱℳ˜2xβy∨TAy∧Tℬy,∧y∈Ω1−ℐℳ˜2xβy∨ℐAy∧ℐℬy,∨y∈ΩTℳ˜2xβy∧ℱAy∨ℱℬy=x,∧y∈Ωℱℳ˜2xβy∨TAy∧ℱℳ˜2xβy∨Tℬy,∧y∈Ω1−ℐℳ˜2xβy∨ℐAy∧1−ℐℳ˜2xβy∨ℐℬy,∨y∈ΩTℳ˜2xβy∧ℱAy∨Tℳ˜2xβy∧ℱℬy=ℒ22A⊓ℒ22ℬ.
(SVNL4): sinceA⊔ℬ⊒A, then by SVNL2 we have ℒ22A⊔ℬ⊒ℒ22A. Similarly, A⊔ℬ⊒ℬ; then, by SVNL2, we have ℒ22A⊔ℬ⊒ℒ22ℬ. Thus, ℒ22A⊔ℬ⊒ℒ22A⊔ℒ22ℬ.
(SVNL5): since SVN universe isΩ=x,1,1,0 and SVN empty set is ∅=x,0,0,1, then we have(22)ℒ22Ω=x,∧y∈Ωℱℳ˜2xβy∨TΩy,∧y∈Ω1−ℐℳ˜2xβy∨ℐΩy,∨y∈ΩTℳ˜2xβy∧ℱΩy=x,∧y∈Ωℱℳ˜2xβy∨1,∧y∈Ω1−ℐℳ˜2xβy∨1,∨y∈ΩTℳ˜2xβy∧0=x,1,1,0=Ω.In the following, we give some relationships among these models.Proposition 9.
LetΩ,Γ˜ be a 2-SVNβCAS and A∈SVNΩ. Then, we have the following properties:(1)
ℒ42A≼¯ℒ22A≼¯ℒ32A(2)
ℒ42A≼¯ℒ12A≼¯ℒ32A(3)
U32A≼¯U22A≼¯U42A(4)
U32A≼¯U12A≼¯U42AProof.
The proof is clear from Definition10.Proposition 10.
LetΩ,Γ˜ be a 2-SVNβCAS and A∈SVNΩ. Then, we have the following properties:(1)
ℒ32A≽ℒ12A⊔ℒ22A(2)
U32A≼¯U12A⊓U22A(3)
ℒ42A≽ℒ12A⊓ℒ22A(4)
U42A≼¯U12A⊔U22AProof.
(clear).


## 5. Decision-Making Approach to DM Based on SVNβCRSs

### 5.1. Description and Process

#### 5.1.1. Method I

Assume thatΩ=xr:r=1,…,k is the set of alternatives (patients), m is main attributes (symptoms) (e.g., cough and fever) V=yi:i=1,2,…,m of A disease, C˜ixr=TC˜ixr,ℐC˜ixr,ℱC˜ixr 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 TC˜ixr∈0,1 (i.e., the degree that doctor D confirms the patient xr has symptom yi), ℐC˜ixr∈0,1 (i.e., the degree that doctor D is not sure if the patient xr has symptom yi), ℱC˜ixr∈0,1 (i.e., the degree that doctor D confirms the patient xr does not have any symptom yi), and 0≤TC˜ixr+ℐC˜ixr+ℱC˜ixr≤3. According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps:Step 1: consider, for eachxr∈Ω, there is at least one yi∈V such that the symptom value Ci for patient xr is not less than β, where β is a critical value.Step 2: considerAxr=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 Definition8 and 3-1-SVNβCRSs model to calculate the lower and upper approximation of A.Step 4: calculateℜA by the following equation:(23)ℜA=U31A⊕ℒ31A,whereA⊕ℬ=x,TAx+Tℬx−TAx∗Tℬx,ℐAx∗ℐABx,ℱAx∗ℱℬx:x∈Ω.Step 5: calculate the decision method by the following formula:(24)Sx=TℜAxTℜAx2+ℐℜAx2+ℱℜAx2,hence, ranking the alternatives.Based on these steps, we give an algorithm to solve the decision-making problems based on Definition8. The steps corresponding to it are summarized in Algorithm 1.Algorithm 1:Algorithm for a 1-SVNβCRSs to make a decision.
Input: SVN decision information system Ω,Γ˜,β,A.Output: Decision-making.(1)
EnterA,β and Ω.(2)
From Definition5, compute the 1-SVN β-neighborhood 1N˜xβ.(3)
FromStep 2 and by Definition 7, compute 1-SVN complementary β-neighborhood 1ℳ˜xβ.(4)
FromSteps 2 and 3 and by Definition 8, compute 3-1-SVNβCRSs ℒ31A and U31A(5)
ComputeℜA.(6)
Compute the cosine similarity measureSx.(7)
Obtain the decision.


#### 5.1.2. Method II

Suppose thatΩ=xr:r=1,…,k is the set of alternatives (papers), m is main attributes (symptoms) (e.g., spot and steak) V=yi:i=1,2,…,m of A paper trouble, C˜ixr=TC˜ixr,ℐC˜ixr,ℱC˜ixr 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 TC˜ixr∈0,1 (i.e., the degree that the investigator I asserts the paper xr has symptom yi), ℐC˜ixr∈0,1 (i.e., the degree that the investigator I is not sure whether the paper xr has symptom yi), ℱC˜ixr∈0,1 (i.e., the degree that investigator I affirms paper xr does not have any symptom yi), and 0≤TC˜ixr+ℐC˜ixr+ℱC˜ixr≤3. According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps:Step 1: consider, for eachxr∈Ω, there is at least one yi∈V such that the symptom value Ci for paper xr is not less than β (i.e., C˜ix≽β), where β is a critical value.Step 2: considerAxr=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 Definition10 and 3-2-SVNβCRSs model to calculate the lower and upper approximation of A.Step 4: calculateℜA by the following equation:(25)ℜA=U32A⊕ℒ32A,whereA⊕ℬ=x,TAx+Tℬx−TAx∗Tℬx,ℐAx∗ℐℬx,ℱAx∗ℱℬx:x∈Ω.Step 5: calculate the decision method by the following formula.(26)Sx=TℜAxTℜAx2+ℐℜAx2+ℱℜAx2,hence, ranking the alternatives.Based on these steps, we give an algorithm to solve the decision-making problems based on Definition10. The steps corresponding to it are summarized in Algorithm 2.Algorithm 2:Algorithm for 2-SVNβCRSs to make a decision.
Input: SVN decision information system Ω,Γ˜,β,A.Output: Decision-making.(1)
EnterA,β, and Ω.(2)
From Definition5, compute the 2-SVN β-neighborhood N2˜xβ.(3)
FromStep 2 and by Definition 9, compute the 2-SVN complementary β-neighborhood ℳ2˜xβ.(4)
FromSteps 2 and 3 and by Definition 10, compute 3-2-SVNβCRSs ℒ32A and U32A(5)
ComputeℜA.(6)
Compute the cosine similarity measureSx.(7)
Obtain the decision.


### 5.2. Numerical Example

Example 5.
Diseased people form a setΩ=x1,x2,x3,x4,x5 and their relevant symptoms are collected by the attribute set V=coughy1,fevery2,sorey3,headachey4 for A disease. Here, the following steps of the algorithm described are implemented.Step 1: under the attribute set, doctorD estimates each patient and presents its decisions with suitable values which are summarized in Table 1.Step 2: considerβ=0.5,0.3,0.8 is a critical and Γ˜=C˜1,C˜2,C˜3,C˜4 is a Type-1 SVNβCRS. Then, we compute the Type-1 SVN β-neighborhood 1N˜xβ and the Type-1 SVN complementary β-neighborhood 1ℳ˜xβ, as shown in Tables 2 and 3.ConsiderA=0.6,0.3,0.5/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5.Step 3: by Definition8 and 3-1-SVNβRSs model, we have the following results:(27)ℒ31A=x1,0.6,0.3,0.5,x2,0.6,0.3,0.4,x3,0.5,0.3,0.4,x4,0.4,0.2,0.3,x5,0.6,0.3,0.3,U31A=x1,0.5,0.5,0.6,x2,0.4,0.5,0.6,x3,0.5,0.5,0.5,x4,0.5,0.5,0.6,x5,0.5,0.5,0.6.Step 4: computeℜA as follows:(28)ℜA=ℒ31A⊕U31A=x1,0.8,0.15,0.3,x2,0.76,0.15,0.24,x3,0.75,0.15,0.2,x4,0.7,0.1,0.18,x5,0.8,0.15,0.18.Step 5: according to the above information, we getSx as follows:(29)Sx1=0.923,Sx2=0.938,Sx3=0.949,Sx4=0.959,Sx5=0.964,and hence, we get the ranking order as(30)Sx5>Sx4>Sx3>Sx2>Sx1.
So, by the above computation, the verdict of the decision makerD is x5.Example 6.
LetΩ=x1,x2,x3,x4,x5 be the set of papers, and their relevant symptoms are collected by the attribute set V=spoty1,steaky2,cratery3,fracturey4 for A paper error. Here, the following steps of the algorithm described are implemented.Step 1: under the attribute set, investigatorI estimates each paper and presents its decisions with suitable values which are summarized in Table 1.Step 2: considerβ=0.5,0.1,0.8 is a critical and Γ˜=C˜1,C˜2,C˜3,C˜4 is a 2-SVNβCRS. Then, we compute the Type-2 SVN β-neighborhood N˜2xβ and the Type-2 SVN complementary β-neighborhood 2ℳ˜xβ, as shown in Tables 6 and 7. Consider A=0.6,0.3,0.5/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5.Step 3: by Definition10 and 3-2-SVNβRSs model, we have the following results:(31)ℒ32A=x1,0.6,0.7,0.5,x2,0.6,0.7,0.4,x3,0.5,0.7,0.4,x4,0.4,0.7,0.3,x5,0.6,0.7,0.3,U32A=x1,0.5,0.2,0.6,x2,0.4,0.2,0.6,x3,0.5,0.3,0.5,x4,0.5,0.2,0.6,x5,0.5,0.3,0.6.Step 4: computeℜA as follows:(32)ℜA=ℒ32A⊕U32A=x1,0.8,0.14,0.3,x2,0.76,0.14,0.24,x3,0.75,0.21,0.2,x4,0.7,0.14,0.18,x5,0.8,0.21,0.18.Step 5: according to above information, we getSx as follows:(33)Sx1=0.924,Sx2=0.939,Sx3=0.933,Sx4=0.951,Sx5=0.945,and hence, we get the ranking order as(34)Sx4>Sx5>Sx2>Sx3>Sx1.
So, by the above calculations, the verdict of the decision makerI is x4.


### 5.3. Comparative Analysis

The 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.Table 10
Sorting outcomes for 1-SVNβCAS.
Different methodsObtain a decisionWang and Zhang’s model [46]x5≻x1≻x2≻x3≻x4Our modelx5≻x4≻x3≻x2≻x1Table 11
Sorting outcomes for 2-SVNβCAS.
Different methodsObtain a decisionWang and Zhang’s model [47]x5≻x1≻x2≻x4≻x3Our modelx4≻x5≻x2≻x3≻x1An easy way to explain these outcomes, see Figures1 and 2 which simplify the comparisons between our presented method and the previous one.Figure 1
The representations of the results by using our model and Wang and Zhang’s model [46].Figure 2
The representations of the results by using our model and Wang and Zhang’s model [47].Figure1 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). Thus, there are slight differences among these distinct methods, and these variations made our model better than others.


## 5.1. Description and Process

### 5.1.1. Method I

Assume thatΩ=xr:r=1,…,k is the set of alternatives (patients), m is main attributes (symptoms) (e.g., cough and fever) V=yi:i=1,2,…,m of A disease, C˜ixr=TC˜ixr,ℐC˜ixr,ℱC˜ixr 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 TC˜ixr∈0,1 (i.e., the degree that doctor D confirms the patient xr has symptom yi), ℐC˜ixr∈0,1 (i.e., the degree that doctor D is not sure if the patient xr has symptom yi), ℱC˜ixr∈0,1 (i.e., the degree that doctor D confirms the patient xr does not have any symptom yi), and 0≤TC˜ixr+ℐC˜ixr+ℱC˜ixr≤3. According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps:Step 1: consider, for eachxr∈Ω, there is at least one yi∈V such that the symptom value Ci for patient xr is not less than β, where β is a critical value.Step 2: considerAxr=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 Definition8 and 3-1-SVNβCRSs model to calculate the lower and upper approximation of A.Step 4: calculateℜA by the following equation:(23)ℜA=U31A⊕ℒ31A,whereA⊕ℬ=x,TAx+Tℬx−TAx∗Tℬx,ℐAx∗ℐABx,ℱAx∗ℱℬx:x∈Ω.Step 5: calculate the decision method by the following formula:(24)Sx=TℜAxTℜAx2+ℐℜAx2+ℱℜAx2,hence, ranking the alternatives.Based on these steps, we give an algorithm to solve the decision-making problems based on Definition8. The steps corresponding to it are summarized in Algorithm 1.Algorithm 1:Algorithm for a 1-SVNβCRSs to make a decision.
Input: SVN decision information system Ω,Γ˜,β,A.Output: Decision-making.(1)
EnterA,β and Ω.(2)
From Definition5, compute the 1-SVN β-neighborhood 1N˜xβ.(3)
FromStep 2 and by Definition 7, compute 1-SVN complementary β-neighborhood 1ℳ˜xβ.(4)
FromSteps 2 and 3 and by Definition 8, compute 3-1-SVNβCRSs ℒ31A and U31A(5)
ComputeℜA.(6)
Compute the cosine similarity measureSx.(7)
Obtain the decision.


### 5.1.2. Method II

Suppose thatΩ=xr:r=1,…,k is the set of alternatives (papers), m is main attributes (symptoms) (e.g., spot and steak) V=yi:i=1,2,…,m of A paper trouble, C˜ixr=TC˜ixr,ℐC˜ixr,ℱC˜ixr 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 TC˜ixr∈0,1 (i.e., the degree that the investigator I asserts the paper xr has symptom yi), ℐC˜ixr∈0,1 (i.e., the degree that the investigator I is not sure whether the paper xr has symptom yi), ℱC˜ixr∈0,1 (i.e., the degree that investigator I affirms paper xr does not have any symptom yi), and 0≤TC˜ixr+ℐC˜ixr+ℱC˜ixr≤3. According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps:Step 1: consider, for eachxr∈Ω, there is at least one yi∈V such that the symptom value Ci for paper xr is not less than β (i.e., C˜ix≽β), where β is a critical value.Step 2: considerAxr=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 Definition10 and 3-2-SVNβCRSs model to calculate the lower and upper approximation of A.Step 4: calculateℜA by the following equation:(25)ℜA=U32A⊕ℒ32A,whereA⊕ℬ=x,TAx+Tℬx−TAx∗Tℬx,ℐAx∗ℐℬx,ℱAx∗ℱℬx:x∈Ω.Step 5: calculate the decision method by the following formula.(26)Sx=TℜAxTℜAx2+ℐℜAx2+ℱℜAx2,hence, ranking the alternatives.Based on these steps, we give an algorithm to solve the decision-making problems based on Definition10. The steps corresponding to it are summarized in Algorithm 2.Algorithm 2:Algorithm for 2-SVNβCRSs to make a decision.
Input: SVN decision information system Ω,Γ˜,β,A.Output: Decision-making.(1)
EnterA,β, and Ω.(2)
From Definition5, compute the 2-SVN β-neighborhood N2˜xβ.(3)
FromStep 2 and by Definition 9, compute the 2-SVN complementary β-neighborhood ℳ2˜xβ.(4)
FromSteps 2 and 3 and by Definition 10, compute 3-2-SVNβCRSs ℒ32A and U32A(5)
ComputeℜA.(6)
Compute the cosine similarity measureSx.(7)
Obtain the decision.


## 5.1.1. Method I

Assume thatΩ=xr:r=1,…,k is the set of alternatives (patients), m is main attributes (symptoms) (e.g., cough and fever) V=yi:i=1,2,…,m of A disease, C˜ixr=TC˜ixr,ℐC˜ixr,ℱC˜ixr 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 TC˜ixr∈0,1 (i.e., the degree that doctor D confirms the patient xr has symptom yi), ℐC˜ixr∈0,1 (i.e., the degree that doctor D is not sure if the patient xr has symptom yi), ℱC˜ixr∈0,1 (i.e., the degree that doctor D confirms the patient xr does not have any symptom yi), and 0≤TC˜ixr+ℐC˜ixr+ℱC˜ixr≤3. According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps:Step 1: consider, for eachxr∈Ω, there is at least one yi∈V such that the symptom value Ci for patient xr is not less than β, where β is a critical value.Step 2: considerAxr=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 Definition8 and 3-1-SVNβCRSs model to calculate the lower and upper approximation of A.Step 4: calculateℜA by the following equation:(23)ℜA=U31A⊕ℒ31A,whereA⊕ℬ=x,TAx+Tℬx−TAx∗Tℬx,ℐAx∗ℐABx,ℱAx∗ℱℬx:x∈Ω.Step 5: calculate the decision method by the following formula:(24)Sx=TℜAxTℜAx2+ℐℜAx2+ℱℜAx2,hence, ranking the alternatives.Based on these steps, we give an algorithm to solve the decision-making problems based on Definition8. The steps corresponding to it are summarized in Algorithm 1.Algorithm 1:Algorithm for a 1-SVNβCRSs to make a decision.
Input: SVN decision information system Ω,Γ˜,β,A.Output: Decision-making.(1)
EnterA,β and Ω.(2)
From Definition5, compute the 1-SVN β-neighborhood 1N˜xβ.(3)
FromStep 2 and by Definition 7, compute 1-SVN complementary β-neighborhood 1ℳ˜xβ.(4)
FromSteps 2 and 3 and by Definition 8, compute 3-1-SVNβCRSs ℒ31A and U31A(5)
ComputeℜA.(6)
Compute the cosine similarity measureSx.(7)
Obtain the decision.


## 5.1.2. Method II

Suppose thatΩ=xr:r=1,…,k is the set of alternatives (papers), m is main attributes (symptoms) (e.g., spot and steak) V=yi:i=1,2,…,m of A paper trouble, C˜ixr=TC˜ixr,ℐC˜ixr,ℱC˜ixr 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 TC˜ixr∈0,1 (i.e., the degree that the investigator I asserts the paper xr has symptom yi), ℐC˜ixr∈0,1 (i.e., the degree that the investigator I is not sure whether the paper xr has symptom yi), ℱC˜ixr∈0,1 (i.e., the degree that investigator I affirms paper xr does not have any symptom yi), and 0≤TC˜ixr+ℐC˜ixr+ℱC˜ixr≤3. According to the presented covering methods, we propose a decision-making algorithm to obtain the result by the following steps:Step 1: consider, for eachxr∈Ω, there is at least one yi∈V such that the symptom value Ci for paper xr is not less than β (i.e., C˜ix≽β), where β is a critical value.Step 2: considerAxr=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 Definition10 and 3-2-SVNβCRSs model to calculate the lower and upper approximation of A.Step 4: calculateℜA by the following equation:(25)ℜA=U32A⊕ℒ32A,whereA⊕ℬ=x,TAx+Tℬx−TAx∗Tℬx,ℐAx∗ℐℬx,ℱAx∗ℱℬx:x∈Ω.Step 5: calculate the decision method by the following formula.(26)Sx=TℜAxTℜAx2+ℐℜAx2+ℱℜAx2,hence, ranking the alternatives.Based on these steps, we give an algorithm to solve the decision-making problems based on Definition10. The steps corresponding to it are summarized in Algorithm 2.Algorithm 2:Algorithm for 2-SVNβCRSs to make a decision.
Input: SVN decision information system Ω,Γ˜,β,A.Output: Decision-making.(1)
EnterA,β, and Ω.(2)
From Definition5, compute the 2-SVN β-neighborhood N2˜xβ.(3)
FromStep 2 and by Definition 9, compute the 2-SVN complementary β-neighborhood ℳ2˜xβ.(4)
FromSteps 2 and 3 and by Definition 10, compute 3-2-SVNβCRSs ℒ32A and U32A(5)
ComputeℜA.(6)
Compute the cosine similarity measureSx.(7)
Obtain the decision.


## 5.2. Numerical Example

Example 5.
Diseased people form a setΩ=x1,x2,x3,x4,x5 and their relevant symptoms are collected by the attribute set V=coughy1,fevery2,sorey3,headachey4 for A disease. Here, the following steps of the algorithm described are implemented.Step 1: under the attribute set, doctorD estimates each patient and presents its decisions with suitable values which are summarized in Table 1.Step 2: considerβ=0.5,0.3,0.8 is a critical and Γ˜=C˜1,C˜2,C˜3,C˜4 is a Type-1 SVNβCRS. Then, we compute the Type-1 SVN β-neighborhood 1N˜xβ and the Type-1 SVN complementary β-neighborhood 1ℳ˜xβ, as shown in Tables 2 and 3.ConsiderA=0.6,0.3,0.5/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5.Step 3: by Definition8 and 3-1-SVNβRSs model, we have the following results:(27)ℒ31A=x1,0.6,0.3,0.5,x2,0.6,0.3,0.4,x3,0.5,0.3,0.4,x4,0.4,0.2,0.3,x5,0.6,0.3,0.3,U31A=x1,0.5,0.5,0.6,x2,0.4,0.5,0.6,x3,0.5,0.5,0.5,x4,0.5,0.5,0.6,x5,0.5,0.5,0.6.Step 4: computeℜA as follows:(28)ℜA=ℒ31A⊕U31A=x1,0.8,0.15,0.3,x2,0.76,0.15,0.24,x3,0.75,0.15,0.2,x4,0.7,0.1,0.18,x5,0.8,0.15,0.18.Step 5: according to the above information, we getSx as follows:(29)Sx1=0.923,Sx2=0.938,Sx3=0.949,Sx4=0.959,Sx5=0.964,and hence, we get the ranking order as(30)Sx5>Sx4>Sx3>Sx2>Sx1.
So, by the above computation, the verdict of the decision makerD is x5.Example 6.
LetΩ=x1,x2,x3,x4,x5 be the set of papers, and their relevant symptoms are collected by the attribute set V=spoty1,steaky2,cratery3,fracturey4 for A paper error. Here, the following steps of the algorithm described are implemented.Step 1: under the attribute set, investigatorI estimates each paper and presents its decisions with suitable values which are summarized in Table 1.Step 2: considerβ=0.5,0.1,0.8 is a critical and Γ˜=C˜1,C˜2,C˜3,C˜4 is a 2-SVNβCRS. Then, we compute the Type-2 SVN β-neighborhood N˜2xβ and the Type-2 SVN complementary β-neighborhood 2ℳ˜xβ, as shown in Tables 6 and 7. Consider A=0.6,0.3,0.5/x1+0.4,0.5,0.1/x2+0.3,0.2,0.6/x3+0.5,0.3,0.4/x4+0.7,0.2,0.3/x5.Step 3: by Definition10 and 3-2-SVNβRSs model, we have the following results:(31)ℒ32A=x1,0.6,0.7,0.5,x2,0.6,0.7,0.4,x3,0.5,0.7,0.4,x4,0.4,0.7,0.3,x5,0.6,0.7,0.3,U32A=x1,0.5,0.2,0.6,x2,0.4,0.2,0.6,x3,0.5,0.3,0.5,x4,0.5,0.2,0.6,x5,0.5,0.3,0.6.Step 4: computeℜA as follows:(32)ℜA=ℒ32A⊕U32A=x1,0.8,0.14,0.3,x2,0.76,0.14,0.24,x3,0.75,0.21,0.2,x4,0.7,0.14,0.18,x5,0.8,0.21,0.18.Step 5: according to above information, we getSx as follows:(33)Sx1=0.924,Sx2=0.939,Sx3=0.933,Sx4=0.951,Sx5=0.945,and hence, we get the ranking order as(34)Sx4>Sx5>Sx2>Sx3>Sx1.
So, by the above calculations, the verdict of the decision makerI is x4.


## 5.3. Comparative Analysis

The 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.Table 10
Sorting outcomes for 1-SVNβCAS.
Different methodsObtain a decisionWang and Zhang’s model [46]x5≻x1≻x2≻x3≻x4Our modelx5≻x4≻x3≻x2≻x1Table 11
Sorting outcomes for 2-SVNβCAS.
Different methodsObtain a decisionWang and Zhang’s model [47]x5≻x1≻x2≻x4≻x3Our modelx4≻x5≻x2≻x3≻x1An easy way to explain these outcomes, see Figures1 and 2 which simplify the comparisons between our presented method and the previous one.Figure 1
The representations of the results by using our model and Wang and Zhang’s model [46].Figure 2
The representations of the results by using our model and Wang and Zhang’s model [47].Figure1 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). Thus, there are slight differences among these distinct methods, and these variations made our model better than others.


## 6. Conclusion

This 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. Thus, 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. There are some issues in these two covering methods:(1)
Ifβ=0.5,0.1,0.8 in Example 2, then Γ˜ is not 1-SVNβCRSs, but it is applicable in 2-SVNβCRSs(2)
Ifβ=0.5,0.3,0.8 in Example 4, then Γ˜ is not 2-SVNβCRSs, but it is applicable in 1-SVNβCRSsIn 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) complementaryβ-neighborhoods and published papers (see [50–55]). 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.



---

*Source: 7362006-2021-07-31.xml*