# Decision-Making Framework Based on Multineutrosophic Soft Rough Sets

Authors: Kholood Mohammad Alsager
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The uncertain data appear in majority of real-world phenomena regarding business, engineering, manufacturing, industries, medical, and other social fields. The neutrosophic sets (NSs), soft sets (SSs), and rough sets (RSs) are distinct mathematical models to deal with different types of uncertainties in data and information. Nowadays, several hybrid models are emerging as their combination or their further generalizations. In this paper, the concept of multineutrosophic fuzzy soft rough sets (MNFSRSs) by combining multineutrosophic fuzzy soft sets (MNSSs) with RSs is presented. Certain eminent features of the proposed structure are investigated. Moreover, a general decision-making (DM) algorithm is described for DM problems to indicate the effectiveness of the presented model.


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## Body

## 1. Introduction

The notion of fuzzy set (FS) [1] is a useful tool to capture vague information. Atanassov [2] put forward the excellent concept of intuitionistic fuzzy sets (IFSs) to modify the structure of traditional FS. Pawlak [3] is acknowledged to set up the theory of RSs as a mathematical tool to deal the inexact information through approximations. The prominent feature of RSs was to provide formal approximations of conventional sets by the means of two sets, namely, lower and upper approximations. In this RS theory, the approximation sets are established in accordance to the equivalence relation defined on that set and thus is of prime importance in DM with the respective model. Molodtsov [4] offered his contribution for the development of SS that serve vigorously to capture obscure information. The significant characteristics of SS distinguish it from the other traditional DM tools, such as the theories of FSs [1], probability, and RSs [3]. According to Molodtsov [4], the SS theory finds frequent applications in certain fields, including function’s smoothness, theory of measurement, and game theory. Owing to its extensive applications, SS theory has become a captivating research field for many quality researchers. Maji and Roy [5] were first to fuse the ideas of SS and RS to efficiently manipulate the DM problem for beneficial decisions. Akram et al. [6] worked on intuitionistic fuzzy N-soft rough sets (IFNSRSs). Ali and Akram [7] provided fuzzy N-soft expert sets. Feng [8] gave a framework to combine FSs, RSs, and SSs altogether. Many others have also developed various types of models for handing complexity of vague information in an efficient way [9–13].Zhang et al. [14] considered idea of soft rough intuitionistic fuzzy sets (SRIFSs) and intuitionistic fuzzy soft rough sets (IFSRSs) that present the generalized extensions of soft RSs. On the other hand, Smarandache [15] studied the notion of NSs. NS can independently express T−degree, I−degree, and F−degree for the sake of compiling the incomplete and inconsistent information completely. The factors discussed by the NS fit more compatibly to express the human-thinking in the real life scenarios with imperfect knowledge. Sebastian and Ramakrishnan [16] made the way to establish multifuzzy sets (MFSs), which possess more general framework in FS theory using ordinary FSs as its building blocks. Its membership function is an ordered sequence of ordinary fuzzy membership functions. As a generalization of multifuzzy soft set (MFSS) [17], multineutrosophic soft set (MNSS) is an extension of the concept. This notion aids to represent those problems that are harder to illustrate through other extensions of FS theory.The combination of MNSSs and the RSs in our work proves to be an improved model of neutrosophic rough approaches, that concerns both areas of theoretical and practical applications. In this paper, we propose the model of multineutrosophic soft rough set (MNSRS) by combining MNSSs with RSs. We provide both the general definitions as well as some fundamental properties of the proposed model. Then, by utilizing the MNSRSs, we explore a new approach to DM.The presentation of the article is set as follows. In section2, we recall some basic concepts of RSs, SSs, and NSs. Section 3 presents MNSSs and discuses some properties. Further, section 4 introduces a RS model based on multineutrosophic soft (MNS) relation and examines some properties of this model. Section 5 aims to establish a general approach to DM based on MNSRSs and illustrates the significant steps of the proposed DM approach along with an illustrating example. Finally, in section 6, we conclude the work with a comprehensive summary and describe the goals for further research.


## 2. Preliminaries

We recall here some basic notions and definitions.Definition 1.
(see [3]). Let ℜ be an equivalence relation on the universe of discourse U˜. We use U˜/ℜ to represent family of all equivalence classes of ℜ and qℜ represent the equivalence class of ℜ containing the element q∈U˜. The pair U˜,ℜ is known as approximation space (AS). For any X˜⊆U˜, we can define the lower and upper approximations of X˜ as follows:(1)ℜ¯X˜=q∈U˜:qℜ⊆X˜,ℜ¯X˜=q∈U˜:qℜ∩X˜≠φ.
The pairℜ¯X˜,ℜ¯X˜ is known as the RS of X˜. The RS ℜ¯X˜,ℜ¯X˜ gives rise to a description of X˜ under the present knowledge, i.e., the classification of U˜. Furthermore, the positive, negative, and boundary regions of X˜ about the AS U˜,ℜ are defined, respectively, as follows:(2)posX˜=ℜ¯X˜,negX˜=∼ℜ¯X˜,bnX˜=ℜ¯X˜−ℜ¯X˜,where ∼X˜ denotes complementation of the set X˜.The idea of NS was proposed by Smarandache [15] as a generalization of FSs [1] that employs three components to represent the indeterminate and incompatible knowledge.Definition 2.
(see [15]). A NS Γ for the universe U˜ is defined as(3)Γ=q,TΓq,IΓq,FΓq:q∈U˜,where TΓ,IΓ,FΓ:U˜⟶0−,1+ and 0−≤TΓ+IΓ+FΓ≤3+.Definition 3.
(see [4]). Let E˜ be the set of parameters defined for the elements of the universe U˜. A pair F,E˜ is called a soft set over U˜, where F is a mapping given by F:E˜⟶PU˜, with PU˜ being the set of all subsets of U˜. This definition shows that a SS over U˜ is a parameterized family of subsets of the universe U˜. For any t∈E˜, Ft is regarded as the set of t-approximate elements of the SS F,E˜.Definition 4.
(see [18]). Let U˜ be a universe, NU˜ be the set of all NSs on U˜, E˜ be a set of parameters, and G⊆E˜. Then, a neutrosophic SS N over U˜ is a SS defined by a set-valued function fN representing a mapping:(4)fN:E˜⟶NU˜,such that fNq≠φ if q∈E˜−G, where fN is called approximate function of the neutrosophic SS N. In other words, the neutrosophic SS is a parameterized family of some elements of the set PU˜ and it can be written as follows:(5)N=q,fNq:q∈E˜,fNq≠φifq∈E˜−G.


## 3. Multineutrosophic Soft Sets

The notion of MNSS is a generalization of MFSS presented in [17].Definition 5.
LetU˜ be a universe, I be a unit interval, and p be a positive integer. A multi NS Γ in U˜ is the set of order sequences:(6)Γ=q,TΓgq,IΓgq,FΓgq:q∈U˜,∀g=1,2,…,p,where(7)TΓg,IΓg,FΓg:U˜⟶Ip,∀g=1,2,…,p are, respectively, denoting the T−degree, I−degrees, and F−degrees. The set of all multi NSs of dimension p over U˜ is denoted by MpNFSU˜.Remark 1.
Clearly, a multi NS of dimensionp=1 with 0−≤TΓ+IΓ+FΓ≤3+ is a NS.Definition 6.
LetU˜ be universe. Let Γ=TΓg,IΓg,FΓg and Λ=TΛg,IΛg,FΛg be two multineutrosophic numbers (MNNs), then the operations on MNNs can be defined as follows:(1)
Γc=q,FΓgq,1−IΓgq,TΓgq,q∈U˜,g=1,2,…,p(2)
Γ∪Λ=q,TΓgq∨TΛgq,IΓgq∧IΛgq,FΓgq∧FΛgq,q∈U˜,g=1,2,…,p(3)
Γ∩Λ=q,TΓgq∧TΛgq,IΓgq∨IΛgq,FΓgq∨FΛgq,q∈U˜,g=1,2,…,p(4)
Γ⊕Λ=q,TΓgq+TΛgq−TΓgqTΛgq,IΓgq.IΛgq,FΓgq.FΛgq,q∈U˜,g=1,2,…,p(5)
Γ⊗Λ=q,TΓgq.TΛgq,IΓgq+IΛgq−IΓgqIΛgq,FΓgq+FΛgq−FΓgqFΛgq,q∈U˜,g=1,2,…,p}Definition 7.
LetU˜ be a universe. Let Γ and Λ be the multi NSs, then Γ is said to be a multineutrosophic subset of Λ if the conditions(8)TΓgq≤TΛgq,IΓgq≥IΛgq,FΓgq≥FΛgq,hold for all g=1,2,…,p and it is denoted by Γ⊆Λ.Definition 8.
A pairΓ,G is called MNSS of dimension p over U˜, where Γ is a mapping given by(9)Γ:G⟶MpNFSU˜.
A MNSS is a parameterized family of multi neutrosophicsubsets ofU˜. For t∈G, Γt is considered as the set of t-approximate elements of the MNSS Γ,G.Example 1.
Suppose the universeU˜=q1,q2,q3 is a set of cell phones under consideration, in two different shops, and E˜=t1=price,t2=color,t3=brands the set of parameters, where t1 stands for price which can be high, medium, and low, t2 stands for colour which consist of red, white, and black, and t3 represents brands from the companies A, B, and C, then we define the MNSSs of dimension 2 as follows:(10)Γt1=q1,0.2,0.3,0.40.2,0.3,0.6,q2,0.1,0.4,0.80.1,0.3,0.5,q3,0.6,0.5,0.90.2,0.4,0.1,Γt2=q1,0.1,0.8,0.50.4,0.5,0.7,q2,0.1,0.6,0.20.2,0.1,0.6,q3,0.5,0.3,0.70.1,0.7,0.9,Γt3=q1,0.3,0.2,0.10.7,0.3,0.5,q2,0.3,0.2,0.90.3,0.1,0.7,q3,0.2,0.4,0.90.5,0.3,0.4.Definition 9.
LetG,H∈E˜. Suppose, Γ,G and Λ,H are two MNSS. Γ,G is said to be MNS subset of Λ,H if G⊆H, and Γt⊆Λt∀e∈E˜ and is denoted by Γ,G⊆Λ,H.Definition 10.
A MNSSΓ,G of dimension p over U˜ is called the null MNSS if Γt=φp for all e∈G and it is denoted by φGp.Definition 11.
A MNSSΓ,G of dimension p over U˜ is called the absolute MNSS if Γt=1p for all e∈G and it is denoted by U˜Gp.Definition 12.
LetΓ,G be a MNSS of dimension p over U˜. Then, the complement of Γ,G is denoted by Γ,Gc and defined by Γ,Gc  =  Γc,G, where Γc:G⟶MpNFSU˜ is mapping given by Γct=Γtc for all t∈G.Remark 2.
Clearly,Γ,Gcc=Γ,G, φGpc=U˜Gp, and U˜Gpc=φGp.Definition 13.
The union of two MNSSsΓ,G and Λ,H of dimension p over U˜ is the MNSS η,J, where J=G∪H, and for all t∈J,(11)ηt=Γt∪Λt.
We denote it asΓ,G∪Λ,H=η,J.Definition 14.
The intersection of two MNSSs of dimensionp over U˜Γ,G and Λ,H with G∩H≠φ is the MNSS η,J, where J=G∩H, and for all t∈J,(12)ηt=Γtfort∈G−H,Λtfort∈G−H,Λtfort∈H−G,Γt∪Λtfort∈G∩H.
In this case, we writeΓ,G∩Λ,H=η,J.Theorem 1.
LetΓ,GandΛ,Hbe two MNSSs of dimensionpoverU˜. Then,(1)
Γ,G∪Γ,G=Γ,G(2)
Γ,G∩Γ,G=Γ,G(3)
Γ,G∪φGp=Γ,G(4)
Γ,G∩φGp=φGp(5)
Γ,G∪U˜Gp=U˜Gp(6)
Γ,G∩U˜Gp=Γ,G(7)
Γ,G∪Λ,H=Λ,H∪Γ,G(8)
Γ,G∩Λ,H=Λ,H∩Γ,G


## 4. Rough Set Model Based on MNSSs

### 4.1. Rough MNSSs

Definition 15.
LetU˜ be the universe of discourse, and let E˜ be a set of parameters. For any crisp SS ℵ⊆U˜×E˜, a set-valued function ℵ:U˜⟶PE˜ is given by(13)ℵq=t∈E˜:q,t∈ℵ,q∈U˜.
Here,ℵ is referred to as serial if for all q∈U˜, ℵq≠φ. The pair U˜,E˜,ℵ is said to be a CSAS. For any G⊆E˜, the lower and upper soft approximations of G with respect to U˜,E˜,ℵ denoted by ℵ¯G and ℵ¯G, respectively, are defined as follows:(14)ℵ¯G=q∈U˜,ℵq⊆G,ℵ¯G=q∈U˜,ℵq∩G≠φ.
The pairℵ¯G,ℵ¯G is said to be a crisp soft RS and ℵ¯,ℵ¯:PE˜⟶PU˜ are, respectively, called lower and upper crisp soft rough approximation operators. Furthermore, if ℵ¯G=ℵ¯G, then G is called a definable set.Definition 16.
LetU˜ be a universe, E˜ be the set of parameters, and ℵ be the crisp soft relation over U˜×E˜, the pair U˜,E˜,ℵ is called crisp soft approximation space (CSAS). For any G∈MpNFSE˜, the lower and upper soft approximations of G with respect to U˜,E˜,ℵ denoted by ℵ¯G and ℵ¯G, respectively, are defined as follows:(15)ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,where(16)Tℵ¯Ggq=∧t∈ℵqTGgt,Tℵ¯Ggq=∨e∈ℵqTGgt,Iℵ¯Ggq=∨t∈ℵqIGgt,Iℵ¯Ggq=∧e∈ℵqIGgt,Fℵ¯Ggq=∨t∈ℵqFGgt,Fℵ¯Ggq=∧t∈ℵqFGgt.
The pairℵ¯G,ℵ¯G is called the rough MNSS of G with respect to U˜,E˜,ℵ. Moreover, if ℵ¯G = ℵ¯G, then G is called definable.Example 2.
LetU˜=q1,q2,q3,q4,q5,q6 be a finite universe and E˜=t1,t2,t3,t4 be the set of parameters. The representation of the soft set Γ,G is given in Table 1.
Then, a crisp soft relationℵ over U˜×E˜ is given by(17)ℵ=q1,t1,q1,t3,q2,t1,q2,t4,q3,t1,q3,t2,q3,t3,q4,t2,q4,t4,q5,t2,q5,t3,q6,t1q6,t2,t6,t3,t6,t4.
Consider a multineutrosophicG∈MpNFSE˜ with dimension p = 2 as follows:(18)G=t1,0.2,0.3,0.40.1,0.2,0.6,t2,0.1,0.4,0.70.2,0.4,0.5,t3,0.6,0.7,0.90.4,0.6,0.8,t4,0.1,0.3,0.60.4,0.5,0.7.
According to Definition16, we obtain the lower and upper soft approximations of G with respect to U˜,E˜,ℵ:(19)ℵ¯G=q1,0.2,0.7,0.90.1,0.6,0.8,q2,0.1,0.3,0.60.2,0.5,0.7,q3,0.1,0.7,0.90.1,0.6,0.8,q4,0.1,0.4,0.70.2,0.5,0.7,q5,0.1,0.7,0.90.2,0.6,0.8,q6,0.1,0.7,0.90.1,0.6,0.8,ℵ¯G=q1,0.6,0.3,0.40.4,0.2,0.6,q2,0.2,0.3,0.40.4,0.2,0.6,q3,0.6,0.3,0.40.4,0.2,0.5,q4,0.1,0.3,0.60.4,0.4,0.5,q5,0.6,0.4,0.70.4,0.4,0.5,q6,0.6,0.3,0.40.4,0.2,0.5.
The pairℵ¯G,ℵ¯G is called a soft rough multi NS of dimension 2.Table 1
The soft setΓ,G.
Γ,Gq1q2q3q4q5q6t1111000t2001111t3101011t4010100Theorem 2.
LetU˜,E˜,ℵbe a CSAS. The lower and upper soft rough multineutrosophic approximations operatorsℵ¯Gandℵ¯G, respectively, for anyG,H∈MpNFE˜satisfy the following properties:(1)
ℵ¯Gc=ℵ¯Gc,ℵ¯Gc=ℵ¯Gc(2)
G⊆H⇒ℵ¯G⊆ℵ¯H,G⊆H⇒ℵ¯G⊆ℵ¯G(3)
ℵ¯G∩H=ℵ¯G∩ℵ¯H,ℵ¯G∪H=ℵ¯G∪ℵ¯H(4)
ℵ¯G∪H⊇ℵ¯G∪ℵ¯H,ℵ¯G∩H⊆ℵ¯G∩ℵ¯HProof.
(1)
By Definition16, we have(20)ℵ¯Gc=q,Tℵ¯∼Ggq,Iℵ¯∼Ggq,Fℵ¯∼Ggq:q∈U˜Tℵ¯∼Gg=q,∧t∈ℵqT∼Ggt:g=1,2,…,p=q,∼∨e∈ℵqTGgt:g=1,2,…p=q,∼Tℵ¯Ggq:g=1,2,…,p,Iℵ¯∼Gg=q,∨t∈ℵqI∼Ggt:g=1,2,…,p=q,∼∧t∈ℵq∨IGgt:g=1,2,…,p=q,∼Iℵ¯Ggq:g=1,2,…,p,Fℵ¯∼Gg=q,∨t∈ℵqF∼Ggt:g=1,2,…,p=q,∼∧t∈ℵqFGgt:g=1,2,…,p=q,∼Fℵ¯Ggq:g=1,2,…,p=ℵ¯Gc.Similarly, we can obtain thatℵ¯Gc=ℵ¯Gc.(2)
It can be easily proved by Definition16.(3)
ℵ¯G∩H=q,Tℵ¯G∩Hgq,Iℵ¯G∩Hgq,Fℵ¯G∩Hgq:q∈U˜,g=1,2,…,p.(21)Tℵ¯G∩Hgq=q,∧t∈ℵqTG∩Hgt:g=1,2,…,p=q,∧t∈ℵqTGgt∧THgt:g=1,2,…,p=q,∨t∈ℵqTGgt:g=1,2,…,p∧∧t∈ℵqTHgt:g=1,2,…,p,Iℵ¯G∩Hgq=q,∨t∈ℵqIG∩Hgt:g=1,2,…,p=q,∨t∈ℵqIGgt∨IHgt:g=1,2,…,p=q,∨t∈ℵqIGgt:g=1,2,…,p∨∨t∈ℵqIHgt:g=1,2,…,p(22)Fℵ¯G∩Hgq=q,∨t∈E˜FG∩Hgt:g=1,2,…,p=q,∨t∈ℵqFGgt∨FHgt:g=1,2,…,p=q,∨t∈ℵqFGgt:g=1,2,…,p∨q,∨e∈ℵqFHgt:g=1,2,…,p.Hence,ℵ¯G∩ℵ¯H.(4)
ℵ¯G∪H=q,Tℵ¯G∪Hgq,Iℵ¯G∪Hgq,Fℵ¯G∪Hgq:q∈U˜,g=1,2,…,p.(23)Tℵ¯G∪Hgq=q,∧t∈ℵqTG∪Hgt:g=1,2,…,p=q,∧t∈ℵqTGgt∨THgt:g=1,2,…,p=q,∨t∈ℵqTGgt:g=1,2,…,p∨∧t∈ℵqTHgt:g=1,2,…,p,Iℵ¯G∪Hgq=q,∨t∈ℵqIG∪Hgt:g=1,2,…,p=q,∨t∈ℵqIGgt∧IHgt:g=1,2,…,p=q,∨t∈ℵqIGgt:g=1,2,…,pg=1,2,…p∧∨t∈ℵqIHgt:g=1,2,…,p,(24)Fℵ¯G∪Hgq=q,∨t∈E˜FG∪Hgt:g=1,2,…,p=q,∨t∈ℵqFGgt∧FHgt:g=1,2,…,p=q,∨t∈ℵqFGgt:g=1,2,…,p∧q,∨e∈ℵqFHgt:g=1,2,…,p.Hence,ℵ¯G∪ℵ¯H.Proposition 1.
LetU˜,E˜,ℵbe a CSAS, then for anyG∈MpNFE˜:(1)
ℵ¯G=∼ℵ¯∼G(2)
ℵ¯G=∼ℵ¯∼GProof.
(1)
∼ℵ¯∼Gq,Fℵ¯∼Ggq,1−Iℵ¯∼Ggq,Tℵ¯∼Ggq:q∈U˜=q,∧t∈ℵqF∼Ggt,1−∧e∈ℵqI∼Ggt,∨e∈ℵqT∼Ggt:g=1,2,…,p=q,∨t∈ℵqTGgt,∧e∈ℵqIGgt,∧e∈ℵqFGgt:g=1,2,…,p=ℵ¯G(2)
Similar to part 1.


### 4.2. Multineutrosophic Soft Rough Sets

Definition 17.
LetΓ,E˜ be a MNSS of dimension p over U˜. A multineutrosophic subset ℵ of U˜×E˜ is called a MNS relation MpNFSR from U˜ to E˜ given by(25)ℵ=q,t,Tℵgq,t,Iℵgq,t,Fℵgq,t:q∈U˜,e∈E˜,∀g=1,2,…,p,where Tℵ,Iℵ,Fℵ:U˜×E˜⟶0,1p denote the true membership degrees, indeterminacy-membership degrees, and falsity-membership of the relationships between q and t, respectively.Definition 18.
LetU˜ be a nonempty universe set, E˜ be the set of parameters, and ℵ be the MNS relation ℵ∈MpNFSRU˜×E˜, then the triple U˜,E˜,ℵ is called multineutrosophic soft approximation space (MNSAS). For any G∈MpNFSE˜, the lower and upper approximations of G with respect to U˜,E˜,ℵ denoted by ℵ¯G and ℵ¯G are two MNSSs, respectively, defined as follows:(26)ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,where(27)Tℵ¯Ggq=∧t∈E˜Fℵgq,tFℵgq,t∨TGgt:q∈U˜,g=1,2,…,p,Iℵ¯Ggq=∨t∈E˜1−Iℵgq,t∧IGgt:q∈U˜,g=1,2,…,p,Fℵ¯Ggq=∨t∈E˜Tℵgq,tTℵgq,t∧FGgt:q∈U˜,g=1,2,…,p,(28)Tℵ¯Ggq=∨t∈E˜Tℵgq,tTℵgq,t∧TGgt:q∈U˜,g=1,2,…,p,Iℵ¯Ggq=∧t∈E˜Iℵgq,tIℵgq,t∨IGgt:q∈U˜,g=1,2,…,p,Fℵ¯Ggq=∧t∈E˜Fℵgq,tFℵgq,t∨FGgt:q∈U˜,g=1,2,…,p.
ℵ¯G and ℵ¯G are, respectively, called the lower and upper MNS rough approximations operators. The pair ℵ¯G,ℵ¯G is called the MNSRS of G with respect to U˜,E˜,ℵ. Moreover, if ℵ¯G=ℵ¯G, then G is called definable.Theorem 3.
LetU˜,E˜,ℵbe the MNSAS. The lower and upper neutrosophic soft rough approximations operatorsℵ¯Gandℵ¯G, respectively, for anyG,H∈MpNFE˜satisfy the following properties:(1)
ℵ¯Gc=ℵ¯Gc,ℵ¯Gc=ℵ¯Gc(2)
G⊆H⇒ℵ¯G⊆ℵ¯H,G⊆H⇒ℵ¯G⊆ℵ¯G(3)
ℵ¯G∩H=ℵ¯G∩ℵ¯H, ℵ¯G∪H=ℵ¯G∪ℵ¯H(4)
ℵ¯G∪H⊇ℵ¯G∪ℵ¯H,ℵ¯G∩H⊆ℵ¯G∩ℵ¯HProof.
(1)
By Definition16, we have(29)ℵ¯Gc=q,Tℵ¯∼Ggq,Iℵ¯∼Ggq,Fℵ¯∼Ggq:q∈U˜,Tℵ¯∼Gg=q,∧t∈E˜Fℵgq,t∨T∼Ggt:q∈U˜,g=1,2,…,p=q,∼∨t∈E˜Tℵgq,t∧TGgt:q∈U˜,g=1,2,…,p=q,∼Tℵ¯Ggq:q∈U˜,Iℵ¯∼Gg=q,∨t∈E˜1−Iℵgq,t∧I∼Ggt:q∈U˜,g=1,2,…,p=q,∼∧t∈E˜Iℵgq,t∨IGgt:q∈U˜,g=1,2,…,p=q,∼Iℵ¯Ggq:q∈U˜,Fℵ¯∼Gg=q,∨t∈E˜Tℵgq,t∧F∼Ggt:q∈U˜,g=1,2,…,p=q,∼∧t∈E˜Fℵgq,t∨FGgt:q∈U˜,g=1,2,…,p=q,∼Fℵ¯Ggq:q∈U˜=ℵ¯Gc.Similarly, we can obtain thatℵ¯Gc=ℵ¯Gc.(2)
IfG⊆H, by Definition 7, TGgq≤THgq, IGgq≥IHgq, and FGgq≥FHgq for all q∈U˜; therefore,(30)∧t∈E˜Fℵgq,t∨TGgt≤∧e∈E˜Fℵgq,t∨THgt,∨t∈E˜1−Igℵq,t∧IGgt≥∨e∈E˜≥∨t∈E˜1−Igℵq,t∧IHgt,(31)∨t∈E˜Tℵgq,t∧FGgt≥∨e∈E˜Tℵgq,t∧FHgt.Thus,(32)Tℵ¯Ggq,∨t∈E˜1−Iℵgq,t∧IGgt∧∨e∈E˜1−Iℵgq,t∧IHgtq≤Tℵ¯Hgq,Iℵ¯Ggq≥Iℵ¯Hgq,(33)Fℵ¯Ggq≥Fℵ¯Hgq.It follows that(34)ℵ¯G⊆ℵ¯H.Similarly, we can obtain thatℵ¯G⊆ℵ¯H.(3)
ℵ¯G∩H=q,Tℵ¯G∩Hgq,Iℵ¯G∩Hgq,Fℵ¯G∩Hgq:q∈U˜,g=1,2,…,p.(35)Tℵ¯G∩Hgq=q,∧t∈E˜Fℵgq,t∨TG∩Hgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∧THgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgtTGgt∧∧t∈E˜Fℵgq,t∨THgt:q∈U˜,g=1,2,…,p,Iℵ¯G∩Hgq=q,∨t∈E˜1−Iℵgq,t∧IG∩Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∨IHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∨∨e∈E˜1−Iℵgq,t∧IHgt:q,t∧IHgt:q∈U˜,g=1,2,…,p,(36)Fℵ¯G∩Hgq=q,∨t∈E˜Tℵgq,t∧FG∩Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∨FHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∨∨e∈E˜Tℵgq,t∧FHgt:q∈U˜,g=1,2,…,p.Hence,ℵ¯G∩H=ℵ¯G∩ℵ¯H. Similarly, we can obtain that ℵ¯G∩H=ℵ¯G∩ℵ¯H.(4)
ℵ¯G∪H=q,Tℵ¯G∪Hgq,Iℵ¯G∪Hgq,Fℵ¯G∪Hgq:q∈U˜,g=1,2,…,p.(37)Tℵ¯G∩Hgq=q,∧t∈E˜Fℵgq,t∨TG∪Hgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∨THgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∨∧e∈E˜Fℵgq,t∨THgt:q∈U˜,g=1,2,…,p,Iℵ¯G∪Hgq=q,∨t∈E˜1−Iℵgq,t∧IG∪Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∧IHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∧∨e∈E˜1−Iℵgq,t∧IHgt:q,t∧IHgt:q∈U˜,g=1,2,…,p(38)Fℵ¯G∪Hgq=q,∨t∈E˜Tℵgq,t∧FG∪Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgtFGgt∧FHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∧∨e∈E˜Tℵgq,t∧FHgt:q∈U˜,g=1,2,…,p.Hence,ℵ¯G∪H=ℵ¯G∪ℵ¯H. Similarly, we can obtain that ℵ¯G∩H⊆ℵ¯G∩ℵ¯H.
This completes the proof.Theorem 4.
LetU˜,E˜,ℵbe a MNSAS. Suppose thatℵandℜare MNS relations overU˜, ifℵ⊆ℜ, for anyG∈MpNFE˜, then(1)
ℵ¯G⊇ℜ¯G(2)
ℵ¯G⊆ℜ¯GProof.
(1)
Ifℵ⊆ℜ, then by Definition 7, we have Tℵgq,t≤Tℜgq,t, Tℜgq,t, Iℵgq,t≥, and Iℵgq,t≥Fℵgq,t≥q∈U˜, then(39)ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,Tℵ¯Ggq=q,∧t∈E˜Fℵgq,t∨TGgt:q∈U˜,g=1,2,…,p≥q,∧t∈E˜Fℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=q,Tℜ¯Ggq:q∈U˜,Iℵ¯Ggq=q,∨t∈E˜1−Iℵgq,t∧IGgt:q∈U˜,g=1,2,…,p≤q,∨t∈E˜1−Iℜgq,t∧IGgt:q∈U˜,g=1,2,…,p=q,Iℜ¯Ggq:q∈U˜,Fℵ¯Ggq=q,∨t∈E˜Tℵgq,t∧FGgt:q∈U˜,g=1,2,…,p≤q,∨t∈E˜Tℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=q,Fℜ¯Ggq:q∈U˜=ℵ¯G⊇ℜ¯G.(2)
It can be obtained similarly as part 1.Theorem 5.
LetU˜,E˜,ℵandU˜,E˜,ℜbe two multi neutrosophic soft ASs overU˜for anyG∈MpNFE˜, then(1)
ℵ∪ℜ¯G  =  ℵ¯G∩ℜ¯G(2)
ℵ∪ℜ¯G  =  ℵ¯G∪ℜ¯GProof.
(1)
ℵ∪ℜ¯G=q,Tℵ∪ℜ¯Ggq,Iℵ∪ℜ¯Ggq,Fℵ∪ℜ¯Ggq:q∈U˜,g=1,2,…,p.(40)Tℵ∪ℜ¯Ggq=q,∧t∈E˜Fℵ∪ℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∧Fℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∧∧e∈E˜Fℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=Tℵ¯Ggq∩Tℜ¯Ggq,Iℵ∪ℜ¯Ggq=q,∨t∈E˜1−Iℵ∪ℜgq,t∧IGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∨1−Iℜgq,t∧IGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∨∨e∈E˜1−Iℜgq,t∧IGgt:q,t∧IGgt:q∈U˜,g=1,2,…,p=Iℵ¯Ggq∩Iℜ¯Ggq,(41)Fℵ∪ℜ¯Ggq=q,∨t∈E˜Tℵ∪ℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∨Tℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∨∨e∈E˜Tℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=Fℵ¯Ggq∩Fℜ¯Ggq.Hence,ℵ∪ℜ¯G=ℵ¯G∩ℜ¯G.(2)
It can be obtained similarly as part 1.


## 4.1. Rough MNSSs

Definition 15.
LetU˜ be the universe of discourse, and let E˜ be a set of parameters. For any crisp SS ℵ⊆U˜×E˜, a set-valued function ℵ:U˜⟶PE˜ is given by(13)ℵq=t∈E˜:q,t∈ℵ,q∈U˜.
Here,ℵ is referred to as serial if for all q∈U˜, ℵq≠φ. The pair U˜,E˜,ℵ is said to be a CSAS. For any G⊆E˜, the lower and upper soft approximations of G with respect to U˜,E˜,ℵ denoted by ℵ¯G and ℵ¯G, respectively, are defined as follows:(14)ℵ¯G=q∈U˜,ℵq⊆G,ℵ¯G=q∈U˜,ℵq∩G≠φ.
The pairℵ¯G,ℵ¯G is said to be a crisp soft RS and ℵ¯,ℵ¯:PE˜⟶PU˜ are, respectively, called lower and upper crisp soft rough approximation operators. Furthermore, if ℵ¯G=ℵ¯G, then G is called a definable set.Definition 16.
LetU˜ be a universe, E˜ be the set of parameters, and ℵ be the crisp soft relation over U˜×E˜, the pair U˜,E˜,ℵ is called crisp soft approximation space (CSAS). For any G∈MpNFSE˜, the lower and upper soft approximations of G with respect to U˜,E˜,ℵ denoted by ℵ¯G and ℵ¯G, respectively, are defined as follows:(15)ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,where(16)Tℵ¯Ggq=∧t∈ℵqTGgt,Tℵ¯Ggq=∨e∈ℵqTGgt,Iℵ¯Ggq=∨t∈ℵqIGgt,Iℵ¯Ggq=∧e∈ℵqIGgt,Fℵ¯Ggq=∨t∈ℵqFGgt,Fℵ¯Ggq=∧t∈ℵqFGgt.
The pairℵ¯G,ℵ¯G is called the rough MNSS of G with respect to U˜,E˜,ℵ. Moreover, if ℵ¯G = ℵ¯G, then G is called definable.Example 2.
LetU˜=q1,q2,q3,q4,q5,q6 be a finite universe and E˜=t1,t2,t3,t4 be the set of parameters. The representation of the soft set Γ,G is given in Table 1.
Then, a crisp soft relationℵ over U˜×E˜ is given by(17)ℵ=q1,t1,q1,t3,q2,t1,q2,t4,q3,t1,q3,t2,q3,t3,q4,t2,q4,t4,q5,t2,q5,t3,q6,t1q6,t2,t6,t3,t6,t4.
Consider a multineutrosophicG∈MpNFSE˜ with dimension p = 2 as follows:(18)G=t1,0.2,0.3,0.40.1,0.2,0.6,t2,0.1,0.4,0.70.2,0.4,0.5,t3,0.6,0.7,0.90.4,0.6,0.8,t4,0.1,0.3,0.60.4,0.5,0.7.
According to Definition16, we obtain the lower and upper soft approximations of G with respect to U˜,E˜,ℵ:(19)ℵ¯G=q1,0.2,0.7,0.90.1,0.6,0.8,q2,0.1,0.3,0.60.2,0.5,0.7,q3,0.1,0.7,0.90.1,0.6,0.8,q4,0.1,0.4,0.70.2,0.5,0.7,q5,0.1,0.7,0.90.2,0.6,0.8,q6,0.1,0.7,0.90.1,0.6,0.8,ℵ¯G=q1,0.6,0.3,0.40.4,0.2,0.6,q2,0.2,0.3,0.40.4,0.2,0.6,q3,0.6,0.3,0.40.4,0.2,0.5,q4,0.1,0.3,0.60.4,0.4,0.5,q5,0.6,0.4,0.70.4,0.4,0.5,q6,0.6,0.3,0.40.4,0.2,0.5.
The pairℵ¯G,ℵ¯G is called a soft rough multi NS of dimension 2.Table 1
The soft setΓ,G.
Γ,Gq1q2q3q4q5q6t1111000t2001111t3101011t4010100Theorem 2.
LetU˜,E˜,ℵbe a CSAS. The lower and upper soft rough multineutrosophic approximations operatorsℵ¯Gandℵ¯G, respectively, for anyG,H∈MpNFE˜satisfy the following properties:(1)
ℵ¯Gc=ℵ¯Gc,ℵ¯Gc=ℵ¯Gc(2)
G⊆H⇒ℵ¯G⊆ℵ¯H,G⊆H⇒ℵ¯G⊆ℵ¯G(3)
ℵ¯G∩H=ℵ¯G∩ℵ¯H,ℵ¯G∪H=ℵ¯G∪ℵ¯H(4)
ℵ¯G∪H⊇ℵ¯G∪ℵ¯H,ℵ¯G∩H⊆ℵ¯G∩ℵ¯HProof.
(1)
By Definition16, we have(20)ℵ¯Gc=q,Tℵ¯∼Ggq,Iℵ¯∼Ggq,Fℵ¯∼Ggq:q∈U˜Tℵ¯∼Gg=q,∧t∈ℵqT∼Ggt:g=1,2,…,p=q,∼∨e∈ℵqTGgt:g=1,2,…p=q,∼Tℵ¯Ggq:g=1,2,…,p,Iℵ¯∼Gg=q,∨t∈ℵqI∼Ggt:g=1,2,…,p=q,∼∧t∈ℵq∨IGgt:g=1,2,…,p=q,∼Iℵ¯Ggq:g=1,2,…,p,Fℵ¯∼Gg=q,∨t∈ℵqF∼Ggt:g=1,2,…,p=q,∼∧t∈ℵqFGgt:g=1,2,…,p=q,∼Fℵ¯Ggq:g=1,2,…,p=ℵ¯Gc.Similarly, we can obtain thatℵ¯Gc=ℵ¯Gc.(2)
It can be easily proved by Definition16.(3)
ℵ¯G∩H=q,Tℵ¯G∩Hgq,Iℵ¯G∩Hgq,Fℵ¯G∩Hgq:q∈U˜,g=1,2,…,p.(21)Tℵ¯G∩Hgq=q,∧t∈ℵqTG∩Hgt:g=1,2,…,p=q,∧t∈ℵqTGgt∧THgt:g=1,2,…,p=q,∨t∈ℵqTGgt:g=1,2,…,p∧∧t∈ℵqTHgt:g=1,2,…,p,Iℵ¯G∩Hgq=q,∨t∈ℵqIG∩Hgt:g=1,2,…,p=q,∨t∈ℵqIGgt∨IHgt:g=1,2,…,p=q,∨t∈ℵqIGgt:g=1,2,…,p∨∨t∈ℵqIHgt:g=1,2,…,p(22)Fℵ¯G∩Hgq=q,∨t∈E˜FG∩Hgt:g=1,2,…,p=q,∨t∈ℵqFGgt∨FHgt:g=1,2,…,p=q,∨t∈ℵqFGgt:g=1,2,…,p∨q,∨e∈ℵqFHgt:g=1,2,…,p.Hence,ℵ¯G∩ℵ¯H.(4)
ℵ¯G∪H=q,Tℵ¯G∪Hgq,Iℵ¯G∪Hgq,Fℵ¯G∪Hgq:q∈U˜,g=1,2,…,p.(23)Tℵ¯G∪Hgq=q,∧t∈ℵqTG∪Hgt:g=1,2,…,p=q,∧t∈ℵqTGgt∨THgt:g=1,2,…,p=q,∨t∈ℵqTGgt:g=1,2,…,p∨∧t∈ℵqTHgt:g=1,2,…,p,Iℵ¯G∪Hgq=q,∨t∈ℵqIG∪Hgt:g=1,2,…,p=q,∨t∈ℵqIGgt∧IHgt:g=1,2,…,p=q,∨t∈ℵqIGgt:g=1,2,…,pg=1,2,…p∧∨t∈ℵqIHgt:g=1,2,…,p,(24)Fℵ¯G∪Hgq=q,∨t∈E˜FG∪Hgt:g=1,2,…,p=q,∨t∈ℵqFGgt∧FHgt:g=1,2,…,p=q,∨t∈ℵqFGgt:g=1,2,…,p∧q,∨e∈ℵqFHgt:g=1,2,…,p.Hence,ℵ¯G∪ℵ¯H.Proposition 1.
LetU˜,E˜,ℵbe a CSAS, then for anyG∈MpNFE˜:(1)
ℵ¯G=∼ℵ¯∼G(2)
ℵ¯G=∼ℵ¯∼GProof.
(1)
∼ℵ¯∼Gq,Fℵ¯∼Ggq,1−Iℵ¯∼Ggq,Tℵ¯∼Ggq:q∈U˜=q,∧t∈ℵqF∼Ggt,1−∧e∈ℵqI∼Ggt,∨e∈ℵqT∼Ggt:g=1,2,…,p=q,∨t∈ℵqTGgt,∧e∈ℵqIGgt,∧e∈ℵqFGgt:g=1,2,…,p=ℵ¯G(2)
Similar to part 1.


## 4.2. Multineutrosophic Soft Rough Sets

Definition 17.
LetΓ,E˜ be a MNSS of dimension p over U˜. A multineutrosophic subset ℵ of U˜×E˜ is called a MNS relation MpNFSR from U˜ to E˜ given by(25)ℵ=q,t,Tℵgq,t,Iℵgq,t,Fℵgq,t:q∈U˜,e∈E˜,∀g=1,2,…,p,where Tℵ,Iℵ,Fℵ:U˜×E˜⟶0,1p denote the true membership degrees, indeterminacy-membership degrees, and falsity-membership of the relationships between q and t, respectively.Definition 18.
LetU˜ be a nonempty universe set, E˜ be the set of parameters, and ℵ be the MNS relation ℵ∈MpNFSRU˜×E˜, then the triple U˜,E˜,ℵ is called multineutrosophic soft approximation space (MNSAS). For any G∈MpNFSE˜, the lower and upper approximations of G with respect to U˜,E˜,ℵ denoted by ℵ¯G and ℵ¯G are two MNSSs, respectively, defined as follows:(26)ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,g=1,2,…,p,where(27)Tℵ¯Ggq=∧t∈E˜Fℵgq,tFℵgq,t∨TGgt:q∈U˜,g=1,2,…,p,Iℵ¯Ggq=∨t∈E˜1−Iℵgq,t∧IGgt:q∈U˜,g=1,2,…,p,Fℵ¯Ggq=∨t∈E˜Tℵgq,tTℵgq,t∧FGgt:q∈U˜,g=1,2,…,p,(28)Tℵ¯Ggq=∨t∈E˜Tℵgq,tTℵgq,t∧TGgt:q∈U˜,g=1,2,…,p,Iℵ¯Ggq=∧t∈E˜Iℵgq,tIℵgq,t∨IGgt:q∈U˜,g=1,2,…,p,Fℵ¯Ggq=∧t∈E˜Fℵgq,tFℵgq,t∨FGgt:q∈U˜,g=1,2,…,p.
ℵ¯G and ℵ¯G are, respectively, called the lower and upper MNS rough approximations operators. The pair ℵ¯G,ℵ¯G is called the MNSRS of G with respect to U˜,E˜,ℵ. Moreover, if ℵ¯G=ℵ¯G, then G is called definable.Theorem 3.
LetU˜,E˜,ℵbe the MNSAS. The lower and upper neutrosophic soft rough approximations operatorsℵ¯Gandℵ¯G, respectively, for anyG,H∈MpNFE˜satisfy the following properties:(1)
ℵ¯Gc=ℵ¯Gc,ℵ¯Gc=ℵ¯Gc(2)
G⊆H⇒ℵ¯G⊆ℵ¯H,G⊆H⇒ℵ¯G⊆ℵ¯G(3)
ℵ¯G∩H=ℵ¯G∩ℵ¯H, ℵ¯G∪H=ℵ¯G∪ℵ¯H(4)
ℵ¯G∪H⊇ℵ¯G∪ℵ¯H,ℵ¯G∩H⊆ℵ¯G∩ℵ¯HProof.
(1)
By Definition16, we have(29)ℵ¯Gc=q,Tℵ¯∼Ggq,Iℵ¯∼Ggq,Fℵ¯∼Ggq:q∈U˜,Tℵ¯∼Gg=q,∧t∈E˜Fℵgq,t∨T∼Ggt:q∈U˜,g=1,2,…,p=q,∼∨t∈E˜Tℵgq,t∧TGgt:q∈U˜,g=1,2,…,p=q,∼Tℵ¯Ggq:q∈U˜,Iℵ¯∼Gg=q,∨t∈E˜1−Iℵgq,t∧I∼Ggt:q∈U˜,g=1,2,…,p=q,∼∧t∈E˜Iℵgq,t∨IGgt:q∈U˜,g=1,2,…,p=q,∼Iℵ¯Ggq:q∈U˜,Fℵ¯∼Gg=q,∨t∈E˜Tℵgq,t∧F∼Ggt:q∈U˜,g=1,2,…,p=q,∼∧t∈E˜Fℵgq,t∨FGgt:q∈U˜,g=1,2,…,p=q,∼Fℵ¯Ggq:q∈U˜=ℵ¯Gc.Similarly, we can obtain thatℵ¯Gc=ℵ¯Gc.(2)
IfG⊆H, by Definition 7, TGgq≤THgq, IGgq≥IHgq, and FGgq≥FHgq for all q∈U˜; therefore,(30)∧t∈E˜Fℵgq,t∨TGgt≤∧e∈E˜Fℵgq,t∨THgt,∨t∈E˜1−Igℵq,t∧IGgt≥∨e∈E˜≥∨t∈E˜1−Igℵq,t∧IHgt,(31)∨t∈E˜Tℵgq,t∧FGgt≥∨e∈E˜Tℵgq,t∧FHgt.Thus,(32)Tℵ¯Ggq,∨t∈E˜1−Iℵgq,t∧IGgt∧∨e∈E˜1−Iℵgq,t∧IHgtq≤Tℵ¯Hgq,Iℵ¯Ggq≥Iℵ¯Hgq,(33)Fℵ¯Ggq≥Fℵ¯Hgq.It follows that(34)ℵ¯G⊆ℵ¯H.Similarly, we can obtain thatℵ¯G⊆ℵ¯H.(3)
ℵ¯G∩H=q,Tℵ¯G∩Hgq,Iℵ¯G∩Hgq,Fℵ¯G∩Hgq:q∈U˜,g=1,2,…,p.(35)Tℵ¯G∩Hgq=q,∧t∈E˜Fℵgq,t∨TG∩Hgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∧THgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgtTGgt∧∧t∈E˜Fℵgq,t∨THgt:q∈U˜,g=1,2,…,p,Iℵ¯G∩Hgq=q,∨t∈E˜1−Iℵgq,t∧IG∩Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∨IHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∨∨e∈E˜1−Iℵgq,t∧IHgt:q,t∧IHgt:q∈U˜,g=1,2,…,p,(36)Fℵ¯G∩Hgq=q,∨t∈E˜Tℵgq,t∧FG∩Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∨FHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∨∨e∈E˜Tℵgq,t∧FHgt:q∈U˜,g=1,2,…,p.Hence,ℵ¯G∩H=ℵ¯G∩ℵ¯H. Similarly, we can obtain that ℵ¯G∩H=ℵ¯G∩ℵ¯H.(4)
ℵ¯G∪H=q,Tℵ¯G∪Hgq,Iℵ¯G∪Hgq,Fℵ¯G∪Hgq:q∈U˜,g=1,2,…,p.(37)Tℵ¯G∩Hgq=q,∧t∈E˜Fℵgq,t∨TG∪Hgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∨THgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∨∧e∈E˜Fℵgq,t∨THgt:q∈U˜,g=1,2,…,p,Iℵ¯G∪Hgq=q,∨t∈E˜1−Iℵgq,t∧IG∪Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∧IHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∧∨e∈E˜1−Iℵgq,t∧IHgt:q,t∧IHgt:q∈U˜,g=1,2,…,p(38)Fℵ¯G∪Hgq=q,∨t∈E˜Tℵgq,t∧FG∪Hgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgtFGgt∧FHgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∧∨e∈E˜Tℵgq,t∧FHgt:q∈U˜,g=1,2,…,p.Hence,ℵ¯G∪H=ℵ¯G∪ℵ¯H. Similarly, we can obtain that ℵ¯G∩H⊆ℵ¯G∩ℵ¯H.
This completes the proof.Theorem 4.
LetU˜,E˜,ℵbe a MNSAS. Suppose thatℵandℜare MNS relations overU˜, ifℵ⊆ℜ, for anyG∈MpNFE˜, then(1)
ℵ¯G⊇ℜ¯G(2)
ℵ¯G⊆ℜ¯GProof.
(1)
Ifℵ⊆ℜ, then by Definition 7, we have Tℵgq,t≤Tℜgq,t, Tℜgq,t, Iℵgq,t≥, and Iℵgq,t≥Fℵgq,t≥q∈U˜, then(39)ℵ¯G=q,Tℵ¯Ggq,Iℵ¯Ggq,Fℵ¯Ggq:q∈U˜,Tℵ¯Ggq=q,∧t∈E˜Fℵgq,t∨TGgt:q∈U˜,g=1,2,…,p≥q,∧t∈E˜Fℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=q,Tℜ¯Ggq:q∈U˜,Iℵ¯Ggq=q,∨t∈E˜1−Iℵgq,t∧IGgt:q∈U˜,g=1,2,…,p≤q,∨t∈E˜1−Iℜgq,t∧IGgt:q∈U˜,g=1,2,…,p=q,Iℜ¯Ggq:q∈U˜,Fℵ¯Ggq=q,∨t∈E˜Tℵgq,t∧FGgt:q∈U˜,g=1,2,…,p≤q,∨t∈E˜Tℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=q,Fℜ¯Ggq:q∈U˜=ℵ¯G⊇ℜ¯G.(2)
It can be obtained similarly as part 1.Theorem 5.
LetU˜,E˜,ℵandU˜,E˜,ℜbe two multi neutrosophic soft ASs overU˜for anyG∈MpNFE˜, then(1)
ℵ∪ℜ¯G  =  ℵ¯G∩ℜ¯G(2)
ℵ∪ℜ¯G  =  ℵ¯G∪ℜ¯GProof.
(1)
ℵ∪ℜ¯G=q,Tℵ∪ℜ¯Ggq,Iℵ∪ℜ¯Ggq,Fℵ∪ℜ¯Ggq:q∈U˜,g=1,2,…,p.(40)Tℵ∪ℜ¯Ggq=q,∧t∈E˜Fℵ∪ℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∧Fℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=q,∧t∈E˜Fℵgq,t∨TGgt∧∧e∈E˜Fℜgq,t∨TGgt:q∈U˜,g=1,2,…,p=Tℵ¯Ggq∩Tℜ¯Ggq,Iℵ∪ℜ¯Ggq=q,∨t∈E˜1−Iℵ∪ℜgq,t∧IGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∨1−Iℜgq,t∧IGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜1−Iℵgq,t∧IGgt∨∨e∈E˜1−Iℜgq,t∧IGgt:q,t∧IGgt:q∈U˜,g=1,2,…,p=Iℵ¯Ggq∩Iℜ¯Ggq,(41)Fℵ∪ℜ¯Ggq=q,∨t∈E˜Tℵ∪ℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∨Tℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=q,∨t∈E˜Tℵgq,t∧FGgt∨∨e∈E˜Tℜgq,t∧FGgt:q∈U˜,g=1,2,…,p=Fℵ¯Ggq∩Fℜ¯Ggq.Hence,ℵ∪ℜ¯G=ℵ¯G∩ℜ¯G.(2)
It can be obtained similarly as part 1.


## 5. MNSRSs to Decision-Making Problem

RS theory was developed by Pawlak in 1982 to deal with the inconsistent and incomplete information through approximation. It has wide applications in many practical problems, especially the use of RSs in decision making. The concept of neutrosophic soft fuzzy set (NSFS), which is a generalization of soft FS, was introduced to solve different DM problems. However, it is a pretty hard work to apply the NSFS in real scientific problems and other areas without the complete specification. Therefore, we will use the proposed notion of MNSRSs in DM situations. The proposed model can competently take over this difficulty more objectively rather than other existing set techniques.Based on the definition proposed in [14], we define the cosine similarity measure (CSM) for multi neutrosophic elements, in order to rank MkNFS numbers in DM process.Definition 19.
LetΓ be MkNFS over U˜ and Γ∗ be the ideal MkNFS, then the CSM is defined as(42)JMkNFSΓ,Γ∗=∑g=1pTΓg∗qTΓgq+IΓgqIΓg∗q+FΓgqFΓg∗q∑g=1pTΓgq2+IΓgq2+FΓgq2×TΓg∗q2+IΓg∗q2+FΓg∗q2.
The concept of ideal point has been used to identify the best alternative in the decision set. Although the ideal alternative does not exist in real world, it does provide a useful theoretical construct against which we may evaluate alternatives. Hence, we can compute the CSM between each multineutrosophic element and the idealMkNFS. The bigger the value of the CSM JMpNFS is, the better the alternative is, because the alternative is close to the ideal alternative.(43)Γ∗=q,TΓg∗q,IΓg∗q,FΓg∗q:q∈U˜,∀g=1,2,…,p=1,0,0,…,1,0,0.
By comparing the CSM value, the ranking of alternatives can be determined, and we can obtain the optimal alternative.
Now, we present an approach to the DM based onMpNFS RSs.
LetU˜,E˜,ℵ be the MNSAS, where U˜ is the universe of discourse, E˜ is the parameters set, and ℵ is the MNS relation on U˜×E˜. Then, we give algorithm 1 based on MNSRSs as follows:Algorithm 1: Algorithm by using MpNFS RSs.
(1)
Input the universal setU˜.(2)
Input the set of parametersE˜.(3)
Construct multineutrosophic soft relation according to decision makers.(4)
Give the testing setG∈MpNFE˜.(5)
Compute theMkNFS operators ℵ¯G,ℵ¯G.(6)
Calculateℵ¯G⊕ℵ¯G.(7)
Compute the cosine similarity measureJMpNFS(8)
The optimal decision is to select the maximum value of the cosine similarity measureAn Illustrative Example. For illustrating the efficiency of the proposed algorithm, let us consider the example as follows.Suppose a group of engineers are considered the best energy source for industrial city. Assume that the set of energy sources isU˜=q1,q2,q3,q4,q5,q6, where qg stands for nuclear, wind, natural gas, solar, oil, coal, respectively, and Ε˜=t1,t2,t3,t4,t5 be the set of evaluation factors, where tg stands for cost, pollution, availability, efficiency, and simplicity. Each factor is shown in three level that could be high, medium, and low. The evaluation process is carried out in two phases, one in summer and the other in winter to be sure of the effectiveness of each type of energy source at different seasons and weather conditions. The knowledge base of the selection process with MkNFS information with dimension p=2 is presented in Table 2.Table 2
Multineutrosophic fuzzy soft relationℵ.
ℵt1t2t3t4t5q1{(0.1, 0.7, 0.5), (0.2, 0.3, 0.7)}{(0.9, 0.6, 0.2), (0.3, 0.4, 0.5)}{(0.5, 0.4, 0.9), (0.1, 0.8, 0.2)}{(0.3, 0.6, 0.4), (0.6, 0.8, 0.4)}{(0.8, 0.5, 0.9), (0.5, 0.7, 0.3)}q2{(0.6, 0.5, 0.9), (0.5, 0.4, 0.5)}{(0.8, 0.1, 0.5), (0.2, 0.8, 0.2)}{(0.1, 0.8, 0.7), (0.4, 0.7, 0.6)}{(0.8, 0.3, 0.2), (0.1, 0.2, 0.5)}{(0.2, 0.6, 0.5), (0.3, 0.4, 0.5)}q3{(0.7, 0.1, 0.3), (0.2, 0.9, 0.6)}{(0.4, 0.7, 0.8), (0.4, 0.5, 0.1)}{(0.6, 0.9, 0.1), (0.7, 0.5, 0.4)}{(0.3, 0.4, 0.7), (0.1, 0.2, 0.3)}{(0.3, 0.7, 0.2), (0.2, 0.3, 0.6)}q4{(0.1, 0.2, 0.4), (0.3, 0.5, 0.8)}{(0.5, 0.2, 0.3), (0.2, 0.5, 0.7)}{(0.9, 0.4, 0.5), (0.1, 0.8, 0.3)}{(0.7, 0.4, 0.9), (0.5, 0.3, 0.7)}{(0.7, 0.8, 0.3), (0.5, 0.4, 0.7)}q5{(0.3, 0.8, 0.6), (0.1, 0.4, 0.5)}{(0.6, 0.3, 0.9), (0.6, 0.9, 0.3)}{(0.8, 0.6, 0.2), (0.3, 0.5, 0.5)}{(0.1, 0.6, 0.5), (0.7, 0.8, 0.5)}{(0.3, 0.2, 0.9), (0.1, 0.2, 0.3)}q6{(0.6, 0.7, 0.5), (0.2, 0.4, 0.3)}{(0.8, 0.9, 0.5), (0.3, 0.6, 0.8)}{(0.4, 0.3, 0.6), (0.2, 0.5, 0.9)}{(0.9, 0.8, 0.1), (0.1, 0.6, 0.3)}{(0.2, 0.5, 0.6), (0.7, 0.5, 0.5)}Now, suppose that decision makers from the government give the optimum normal decision objectG, which is presented by the following multineutrosophic information:(44)G=t1,0.2,0.4,0.6,0.5,0.7,0.8,t2,0.1,0.6,0.8,0.4,0.7,0.9,t3,0.2,0.5,0.4,0.3,0.6,0.7,t4,0.3,0.6,0.4,0.5,0.7,0.3,t5,0.4,0.5,0.7,0.5,0.2,0.4.Now, by applying algorithm 1, that we mentioned above, we first calculate the lower and upper approximations of MNSRSs ofG with respect to U˜,E˜,ℵ, respectively.(45)ℵ¯G=q1,0.2,0.6,0.80.3,0.7,0.4,q2,0.3,0.5,0.80.4,0.7,0.5,q3,0.2,0.6,0.60.4,0.7,0.7,q4,0.3,0.5,0.70.3,0.7,0.4,q5,0.2,0.6,0.60.4,0.7,0.6,q6,0.3,0.5,0.80.5,0.7,0.4,ℵ¯G=q1,0.4,0.5,0.40.5,0.7,0.4q1,0.4,0.5,0.40.5,0.7,0.4,q2,0.3,0.5,0.40.5,0.4,0.5,q3,0.3,0.4,0.40.4,0.3,0.3,q4,0.4,0.4,0.50.5,0.4,0.7,q5,0.3,0.5,0.40.5,0.2,0.4,q6,0.3,0.5,0.40.5,0.5,0.3.Then, by Definition6, we obtain ℵ¯G⊕ℵ¯G as follows:(46)ℵ¯G⊕ℵ¯G=q1,0.52,0.3,0.320.65,0.49,0.16,q2,0.51,0.25,0.320.7,0.28,0.25q3,0.44,0.24,0.240.64,0.21,0.21,q4,0.58,0.2,0.350.65,0.28,0.28,q5,0.44,0.3,0.240.7,0.14,0.24,q6,0.51,0.25,0.320.75,0.35,0.12.According to Definition19, we can compute the CSM JMpNFSΓ,Γ∗ between the MNN and the multi ideal neutrosophic number as follows:(47)JMpNFSΓq1,Γ∗≃0.77,JMpNFSΓq2,Γ∗≃0.84,JMpNFSΓq3,Γ∗≃0.86,JMpNFSΓq4,Γ∗≃0.84,JMpNFSΓq5,Γ∗≃0.85,JMpNFSΓq6,Γ∗≃0.85.So, the optimal decision is to selectq3. Therefore, we can conclude that the decision makers should select natural gas as the best energy source for industrial city.


## 6. Conclusion

This article has presented new hybrid models that cumulate the noticeable features of the preceding theories to present more broader, eminent, and practical framework. We have discussed the concept of MNSRS, which is a combination of two powerful models: MNSS and RS. We have definedMpNFS rough approximation operators in terms of MpNFS relations. We have also explored the pivotal characteristics of lower and upper MpNFS rough approximation operators. We have demonstrated a general DM framework to capture the complicated situation carrying the obscure data using the MNSRS. Finally, we have indicated the principle steps of DM methodology by a numerical example. Further, our future research plan is to develop more literatures in the field of FS theory to establish more generalized mathematical models of (1) uncertain measures and knowledge reductions of the MpNFS RSs, (2) complex MNSSs, and (3) MNS TOPSIS method.[19–23].



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