# Matrix Theory for Neutrosophic Hypersoft Set and Applications in Multiattributive Multicriteria Decision-Making Problems

Authors: Muhammad Naveed Jafar; Muhammad Saeed
Journal: Journal of Mathematics
 (2022)

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## Abstract

Decision-making is a complex issue due to the vague, imprecise, and indeterminate environment especially when attributes are more than one and further bifurcated. To solve such types of problems, the concept of neutrosophic hypersoft set is proposed by Smaranndache. In this paper, the primary focus is to extend the concept of neutrosophic hypersoft sets (NHSs) to the neutrosophic hypersoft matrices (NHSMs) with the essential study of matrices with suitable examples. Then, the analytical study of some common operations for NHSM has been created. Lastly, decision-making issues have been presented by establishing a new algorithm based on a score function, and it has been interpreted with the help of numerical example for the selection of teachers at the college level. In this study, NHSM algorithm is elaborated efficiently and conveniently for optimal choice selection to solve decision-making problems.


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## Body

## 1. Introduction

In decision-making, among the multiattributive and multiobjective problems, in uncertain and vague environments, it is difficult to differentiate valid from invalid and logical from illogical. In these cases, decision makers get more confused and uncertain. Zadeh developed fuzzy sets [1] to deal with such type of information. Another issue in information is vagueness. Likewise, it is the type of uncertainty where the investigators cannot’ separate between two unique things, and to deal with vagueness, intuitionistic fuzzy sets [2] are used. Later, Molodtsov [3] presents soft sets to manage uncertainties and vagueness, and this research was effectively applied in numerous applications such as game theory, activity research, and probability [4]. Maji et al. [5, 6] exhibited a logical study of the soft sets, which incorporates every essential operators and property. The study was extended to fuzzy soft set [7] and intuitionistic softsets [8] to deal uncertainity and vagueness. As a result, Smarandache [9, 10] has presented the idea of neutrosophic sets, which is a generalization of the crisp set, fuzzy set, and intuitionistic fuzzy set.In any case, from the philosophical perspective, truthness, indeterminacy, and falsity of neutrosophic set always lies in [0,1]. Maji [11] has extended the concept of a soft set to neutrosophic soft set. The matrix representation and aggregate operators of this idea were presented by Deli and Broumi in [12]. Multicriteria decision-making MCDM problems were solved by utilizing a neutrosophic soft set, and many mathematicians have proposed their examination work in various scientific fields by proposing TOPSIS, VIKOR, etc. techniques, and this idea is likewise utilized in advancing decision-making theories along with application in the neutrosophic environment [13–17]. Akram et al. [18–20] established group decision-making methods based on hesitant N-soft sets, Pythagorean fuzzy TOPSIS, and ELECTRIC I method in Pythagorean fuzzy information. Garg [21, 22] had carried out lot of work related to decision-making problems using different tools relating to fuzzy, intuitionistic, and neutrosophic theories. Mehmood et al. [23, 24] used bipolar soft sets and spherical fuzzy sets for decision-making problems. Sabbir and Naz [25] also worked on bipolar soft sets.Smarandache [26] displayed another strategy to manage uncertainty by providing the extension of the soft set to the hypersoft set and its hybrids, such as a fuzzy hypersoft set, intuitionistic hypersoft set, and neutrosophic hypersoft set, by changing the function into a multiargument function.


### 1.1. Motivation

(1)
Multicriteria decision problems (MCDM) consist of several attributes and indeterminacy. To deal with such types, neutrosophic sets (NSs) are used because (NSs) fully deal with indeterminacy, whereas to deal with vagueness and uncertainty, neutrosophic soft sets (NS’s) are used. However, when attributes are more than one and further bifurcated, the concept of neutrosophic soft set (NSs) cannot be used to tackle such issues. There was a dire need to define the new environment. For this purpose, the concept of neutrosophic hypersoft set (NHSS) was proposed by [27]. Matrices are more reliable, logical, and practical for the decision makers and play an important role in understanding, modeling, and solving the MCDM problems.(2)
how MCDM problems can be represented in the matrices’ form consisting of more than one attribute, which is further bifurcated? The answer to this question leads us to develop the matrix theory by combining the concept of NHSS and soft matrix theory and, hence, the motivation of the present study.(3)
In this exploration, the primary focus is to extend the neutrosophic hypersoft set (NHSS) concept to the neutrosophic hypersoft matrices (NHSM) by the essential study of matrices. This study helps us apply all the definitions, operators, and properties of matrices to NHSS and decision-making problems, especially when attributes are more than one and further subdivided.Section1 contains an introduction about soft set, neutrosophic soft set, hypersoft set, and neutrosophic hypersoft sets. Section 2 deals with mathematical preliminaries, which will be used in the rest of the paper. In Section 3 the concept of NHSM has been discussed broadly with definitions and suitable examples. In Section 4 basic operators of NHSM are proposed along with their properties. In Section 5, a decision-making algorithm has been developed with the help of score function and it is applied in the selection for the hiring of teachers. This algorithm is briefer and more accurate rather than others, and Section 6 contains some comparison in Table 7 with the existing techniques of Hashmi et al. [28], and finally, we will discuss the conclusion of the research paper.


## 1.1. Motivation

(1)
Multicriteria decision problems (MCDM) consist of several attributes and indeterminacy. To deal with such types, neutrosophic sets (NSs) are used because (NSs) fully deal with indeterminacy, whereas to deal with vagueness and uncertainty, neutrosophic soft sets (NS’s) are used. However, when attributes are more than one and further bifurcated, the concept of neutrosophic soft set (NSs) cannot be used to tackle such issues. There was a dire need to define the new environment. For this purpose, the concept of neutrosophic hypersoft set (NHSS) was proposed by [27]. Matrices are more reliable, logical, and practical for the decision makers and play an important role in understanding, modeling, and solving the MCDM problems.(2)
how MCDM problems can be represented in the matrices’ form consisting of more than one attribute, which is further bifurcated? The answer to this question leads us to develop the matrix theory by combining the concept of NHSS and soft matrix theory and, hence, the motivation of the present study.(3)
In this exploration, the primary focus is to extend the neutrosophic hypersoft set (NHSS) concept to the neutrosophic hypersoft matrices (NHSM) by the essential study of matrices. This study helps us apply all the definitions, operators, and properties of matrices to NHSS and decision-making problems, especially when attributes are more than one and further subdivided.Section1 contains an introduction about soft set, neutrosophic soft set, hypersoft set, and neutrosophic hypersoft sets. Section 2 deals with mathematical preliminaries, which will be used in the rest of the paper. In Section 3 the concept of NHSM has been discussed broadly with definitions and suitable examples. In Section 4 basic operators of NHSM are proposed along with their properties. In Section 5, a decision-making algorithm has been developed with the help of score function and it is applied in the selection for the hiring of teachers. This algorithm is briefer and more accurate rather than others, and Section 6 contains some comparison in Table 7 with the existing techniques of Hashmi et al. [28], and finally, we will discuss the conclusion of the research paper.


## 2. Preliminaries

In this section, we present some definitions which will help understand the rest of the article.


### 2.1. Soft Set [6]

LetU be the universal set and E be the set of attributes with respect to U. Let PU be the power set of U and A⊆E. A pair ℱ,A is called a soft set over U, and its mapping is given as(1)ℱ:A⟶PU.It is also defined as(2)ℱ,A=ℱe∈PUe∈E,ℱe=∅ife≠A.


### 2.2. Neutrosophic Soft Set [11]

LetU be the universal set and ℰ be the set of attributes with respect to U. Let PU be the set of neutrosophic values of U and A⊆E. A pair ℱ,A is called a neutrosophic soft set over U, and its mapping is given as(3)ℱ:A⟶PU.


### 2.3. Hypersoft Set [21]

LetU be the universal set and PUbe the power set of U. Consider ℓ1,ℓ2,ℓ3,…,ℓn, for n≥1, and let n be well-defined attributes, whose corresponding attributive values are, respectively, the set ℒ1,ℒ2,ℒ3,…,ℒn with ℒi∩ℒj=∅, for i≠j and i,jɛ1,2,3,…,n; then, the pair ℱ,ℒ1×ℒ2×ℒ3,…,ℒn is said to be hypersoft set over U, where(4)ℱ:ℒ1×ℒ2×ℒ3,…,ℒn⟶PU.


### 2.4. Neutrosophic Hypersoft Set [23]

LetU be the universal set and PU be the power set of U. Consider ℓ1,ℓ2,ℓ3,…,ℓn, for n≥1; let n be well-defined attributes, whose corresponding attributive values are, respectively, the set ℒ1,ℒ2,ℒ3,…,ℒn with ℒi∩ℒj=∅, for i≠j and i,jɛ1,2,3,…,n, and their relation ℒ1×ℒ2×ℒ3,…,ℒn=S; then, the pair ℱ,S is said to be neutrosophic hypersoft set (NHSS) over U, where(5)ℱ:ℒ1×ℒ2×ℒ3…,ℒn⟶PU,ℱℒ1×ℒ2×ℒ3,…,ℒn=x,TℱS,ℐℱS,ℱℱS,x∈U,where T is the membership value of truthiness, J is the membership value of indeterminacy, and ℱ is the membership value of falsity such that T,J,ℱ:U⟶0,1 also 0≤TℱS+JℱS+ℱℱS≤3.


## 2.1. Soft Set [6]

LetU be the universal set and E be the set of attributes with respect to U. Let PU be the power set of U and A⊆E. A pair ℱ,A is called a soft set over U, and its mapping is given as(1)ℱ:A⟶PU.It is also defined as(2)ℱ,A=ℱe∈PUe∈E,ℱe=∅ife≠A.


## 2.2. Neutrosophic Soft Set [11]

LetU be the universal set and ℰ be the set of attributes with respect to U. Let PU be the set of neutrosophic values of U and A⊆E. A pair ℱ,A is called a neutrosophic soft set over U, and its mapping is given as(3)ℱ:A⟶PU.


## 2.3. Hypersoft Set [21]

LetU be the universal set and PUbe the power set of U. Consider ℓ1,ℓ2,ℓ3,…,ℓn, for n≥1, and let n be well-defined attributes, whose corresponding attributive values are, respectively, the set ℒ1,ℒ2,ℒ3,…,ℒn with ℒi∩ℒj=∅, for i≠j and i,jɛ1,2,3,…,n; then, the pair ℱ,ℒ1×ℒ2×ℒ3,…,ℒn is said to be hypersoft set over U, where(4)ℱ:ℒ1×ℒ2×ℒ3,…,ℒn⟶PU.


## 2.4. Neutrosophic Hypersoft Set [23]

LetU be the universal set and PU be the power set of U. Consider ℓ1,ℓ2,ℓ3,…,ℓn, for n≥1; let n be well-defined attributes, whose corresponding attributive values are, respectively, the set ℒ1,ℒ2,ℒ3,…,ℒn with ℒi∩ℒj=∅, for i≠j and i,jɛ1,2,3,…,n, and their relation ℒ1×ℒ2×ℒ3,…,ℒn=S; then, the pair ℱ,S is said to be neutrosophic hypersoft set (NHSS) over U, where(5)ℱ:ℒ1×ℒ2×ℒ3…,ℒn⟶PU,ℱℒ1×ℒ2×ℒ3,…,ℒn=x,TℱS,ℐℱS,ℱℱS,x∈U,where T is the membership value of truthiness, J is the membership value of indeterminacy, and ℱ is the membership value of falsity such that T,J,ℱ:U⟶0,1 also 0≤TℱS+JℱS+ℱℱS≤3.


## 3. Neutrosophic Hypersoft Matrix (NHSM)

In this section, we have introduced some definition with suitable examples.


### 3.1. NHSM

LetU=u1,u2,…,uα and PU be the universal set and power set of universal set, respectively; also, consider ℒ1,ℒ2,…,ℒβ, for β≥1, where β is well-defined attributes, whose corresponding attributive values are, respectively, the set ℒ1a,ℒ2b,…,ℒβz and their relation ℒ1a×ℒ2b×⋯×ℒβz, where a,b,c,…,z=1,2,…,n; then, the pair ℱ,ℒ1a×ℒ2b×⋯×ℒβz is said to be neutrosophic hypersoft set over U, where ℱ:ℒ1a×ℒ2b×⋯×ℒβz⟶PU and it is defined as ℱℒ1a×ℒ2b×⋯×ℒβz=u,TLu,ILu,FLuu∈U,L∈ℒ1a×ℒ2b×⋯×ℒβz. Table 1 represents the tabular form of NHSS ℛL.Table 1
Matrix representation of NHSS.
ℒ1aℒ2b…ℒβzu1XℛLu1,ℒ1aXℛLu1,ℒ2b…XℛLu1,ℒβzu2XℛLu2,ℒ1aXℛLu2,ℒ2b…XℛLu2,ℒβz⋮⋮⋮⋱⋮uαXℛLuα,ℒ1aXℛLuα,ℒ2b…XℛLuα,ℒβzIfOij=XℛLui,ℒjk, where i=1,2,3,…,α,j=1,2,3,…,β,andk=a,b,c,…,z, then a matrix is defined as(6)Oijα×β=O11O12…O1βO21O22…O2β⋮⋮⋱⋮Oα1Oα2…Oαβ,where Oij=Tℒjkui,Iℒjkui,Fℒjkui,ui∈U,,ℒjk∈ℒ1a×ℒ2b×⋯×ℒβz=Tijko,Jijko,ℱijko.Thus, we can represent any neutrosophic hypersoft set in terms of a neutrosophic hypersoft matrix (NHSM), and it means that they are interchangeable.Example 1.
Teachers’ recruitment problem (TRP) is the most complex and absurd task. There is no fixed and fabricated design to know their subject knowledge or pedagogical skills. Therefore, decision makers find themselves in a blind alley. Consequently, based on their own knowledge and experience, they select a person who does not meet the institutional requirement. Thus, TRP is typically a multicriteria decision-making MCDM problem.
Assumptions:(i)
Independent attributes are considered(ii)
Everyone attends the interview(iii)
Hesitant environment is not yet considered
Formulation of the Problem. Let us consider an institute that wants to hire a teacher appropriate to its requirements, and they received the following statistics-based CVs. Let U be the set of candidates for the teaching at the college level:(7)U=T1,T2,T3,T4,T5.
Also, consider the set of attributes as(8)A1=Qualification,A2=Experience,A3=Gender,A4=Publications.
Parameters:(i)
Ti = universal set of teachers, wherei=1,2,3,4,5(ii)
Ai = attributes, wherei=1,2,3,4 that are further categorized into the following:(iii)
A1a=qualification(iv)
A1a=BS Hons.,MS/Mphill,Phd,Post Doctorate(v)
A2b=experience=5yr,8yr,10yr,15yr(vi)
A3c=gender=Male,Female(vii)
A4d=publications=3,5,8,10+
Let the function beℱ:A1a×A2b×A3c×A4d⟶PU
Below are Tables2–5 of their neutrosophic values assigned by different decision makers.
The neutrosophic hypersoft set is defined as(9)ℱ:A1a×A2b×A3c×A4d⟶PU.
Let us assume(10)ℱA1a×A2b×A3c×A4d=ℱMphill,5yr,male,3=T1,T2,T4,T5,ℱA1a×A2b×A3c×A4d=ℱMphill,5yr,male,3=≪T1,Mphill0.5,0.3,0.6,5yr0.3,0.4,0.7,male0.5,0.6,0.9,30.6,0.4,0.5≫,≪T2,Mphill0.3,0.2,0.1,5yr0.6,0.5,0.3,male0.7,0.8,0.3,30.7,0.5,0.3≫,≪T4Mphill0.7,0.3,0.6,5yr0.6,0.4,0.8,male0.8,0.5,0.4,30.6,0.2,0.1≫,≪T5Mphill0.5,0.4,0.5,5yr0.3,0.6,0.7,male0.9,0.2,0.1,30.4,0.5,0.3≫,.
Then, a neutrosophic hypersoft set of above-assumed relation in the tabular form is represented in Table6.
And, its matrix is defined as(11)O4×4=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3.Table 2
Decision makers will assign neutrosophic numbers to each candidateTi against qualification.
A1aqualificationT1T2T3T4T5BS Hons.(0.4,0.5,0.8)(0.7,0.6,0.4)(0.4,0.5,0.7)(0.5,0.3,0.7)(0.5,0.3,0.8)MS/MPhil.(0.5,0.3,0.6)(0.3,0.2,0.1)(0.3,0.6,0.2)(0.7,0.3,0.6)(0.5,0.4,0.5)Ph.D.(0.8,0.2,0.4)(0.9,0.5,0.3)(0.9,0.4,0.1)(0.6,0.3,0.2)(0.6,0.1,0.2)Post doctorate(0.9,0.3,0.1)(0.5,0.2,0.1)(0.8,0.5,0.2)(0.8,0.2,0.1)(0.7,0.4,0.2)Table 3
Decision makers will assign neutrosophic numbers to each candidateTi against experience.
A2bexperienceT1T2T3T4T55 yr.(0.3,0.4,0.7)(0.6,0.5,0.3)(0.5,0.6,0.8)(0.6,0.4,0.8)(0.3,0.6,0.7)8 yr.(0.4,0.2,0.5)(0.8,0.1,0.2)(0.4,0.7,0.3)(0.4,0.8,0.7)(0.7,0.5,0.6)10 yr.(0.7,0.2,0.3)(0.9,0.3,0.1)(0.8,0.3,0.2)(0.5,0.4,0.3)(0.5,0.2,0.1)15 yr.(0.8,0.2,0.1)(0.6,0.4,0.3)(0.9,0.4,0.1)(0.6,0.2,0.3)(0.5,0.3,0.2)Table 4
Decision makers will assign neutrosophic numbers to each candidateTi against gender.
A3cGenderT1T2T3T4T5Male(0.5, 0.6, 0.9)(0.7, 0.8, 0.3)(0.6, 0.4, 0.3)(0.8, 0.5, 0.4)(0.9, 0.2, 0.1)Female(0.6, 0.4, 0.7)(0.3, 0.6, 0.4)(0.8, 0.2, 0.1)(0.4, 0.5, 0.6)(0.8, 0.4, 0.2)Table 5
Decision makers will assign neutrosophic numbers to each candidateTi against publication.
A4dpublicationzT1T2T3T4T53—(0.6, 0.4, 0.5)(0.7, 0.5, 0.3)(0.6, 0.4, 0.3)(0.6, 0.2, 0.1)(0.4, 0.5, 0.3)5—(0.8, 0.2, 0.4)(0.7, 0.3, 0.2)(0.8, 0.3, 0.1)(0.3, 0.4, 0.5)(0.3, 0.5, 0.8)8—(0.5, 0.3, 0.4)(0.6, 0.3, 0.4)(0.5, 0.7, 0.2)(0.8, 0.4, 0.1)(0.7, 0.4, 0.3)10+—(0.4, 0.9, 0.6)(0.8, 0.4, 0.2)(0.2, 0.6, 0.5)(0.7, 0.5, 0.2)(0.6, 0.4, 0.7)Table 6
The tabular form of the above relation.
A1aA2bA3cA4dT1Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5T2Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3T4Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1T5Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3


### 3.2. Square NHSM

LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko. Then, O is said to be square NHSM if α=β. It means that if an NHSM has the same number of rows (attributes) and columns (alternatives), it is a square NHSM.Example 2.
Above defined Example1 is also the example of square NHSM.


### 3.3. Transpose of Square NHSM

LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko; then, Ot is said to be transpose of square NHSM if rows and columns of O are interchanged. It is denoted as(12)Ot=Oijt=Tijko,Jijko,ℱijkot=Tjkio,Jjkio,ℱjkio=Oji.Example 3.
Transpose of the matrix define in Example1 is given as(13)O4×4t=Mphill,0.5,0.3,0.6Mphill,0.3,0.2,0.1Mphill,0.7,0.3,0.6Mphill,0.5,0.4,0.55yr,0.3,0.4,0.75yr,0.6,0.5,0.35yr,0.6,0.4,0.85yr,0.3,0.6,0.7male,0.5,0.6,0.9male,0.7,0.8,0.3male,0.8,0.5,0.4male,0.9,0.2,0.13,0.6,0.4,0.53,0.7,0.5,0.33,0.6,0.2,0.13,0.4,0.5,0.3.


### 3.4. Symmetric NHSM

LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko; then, O is said to be symmetric NHSM if Ot=O, i.e., Tijko,Jijko,ℱijko=Tjkio,Jjkio,ℱjkio.


### 3.5. Scalar Multiplication of NHSM

LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko and s be any scalar then the product of matrix O and a scalar s is a matrix formed by multiplying each element of matrix O by s. It is denoted as sO=sOij, where 0≤s≤1.Example 4.
Let us consider a NHSMO4×4:(14)O4×4=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3.
And,0.1 is the scalar; then, scalar multiplication of NHSM O4×4 is given as(15)0.1O4×4=Mphill,0.05,0.03,0.065yr,0.03,0.04,0.07male,0.05,0.06,0.093,0.06,0.04,0.05Mphill,0.03,0.02,0.015yr,0.06,0.05,0.03male,0.07,0.08,0.033,0.07,0.05,0.03Mphill,0.07,0.03,0.065yr,0.06,0.04,0.08male,0.08,0.05,0.043,0.06,0.02,0.01Mphill,0.05,0.04,0.055yr,0.03,0.06,0.07male,0.09,0.02,0.013,0.04,0.05,0.03.Proposition 1.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.For two scalarss,t∈0,1, then(i)
stO=stO(ii)
Ifs<t, then sO<tO(iii)
IfO⊆ℳ, then sO⊆sℳProof.
(i)
stO=stOij=stTijko,tJijko,tℱijko=stTijko,stJijko,stℱijko=stTijko,Jijko,ℱijko=stOij=stO(ii)
SinceTijko,Jijko,ℱijko∈0,1, so sTijko≤tTijko,sJijko≤tJijko,sℱijko≤tℱijko(iii)
Now,sO=sOij=sTijko,sJijko,sℱijko≤tTijko,tJijko,tℱijko=tOij=tO(iv)
O⊆ℳ⇒Oij⊆ℳij(16)⇒Tijko≤Tijkℳ,Jijko≤Jijkℳ,ℱijko≥ℱijkℳ⇒sTijko≤sTijkℳ,sJijko≤sJijkℳ,sℱijko≥sℱijkℳ⇒sOij⊆sℳij⇒sO⊆sℳ.Theorem 1.
LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko. Then,(i)
sOt=sOt, where s∈0,1(ii)
Ott=O(iii)
IfO=Oij is the upper triangular NHSM, then Ot is lower triangular NHSM and vice versaProof.
(i)
Here,sOt,sOt∈NHSMα×β, so(17)sOt=sTijko,sJijko,sℱijkot=sTjkio,sJjkio,sℱjkio=sTjkio,Jjkio,ℱjkio=sTijko,Jijko,ℱijkot=sOt.(ii)
SinceOt∈NHSMα×β,soOtt∈NHSMα×β. Now,(18)Ott=Tijko,Jijko,ℱijkott=Tjkio,Jjkio,ℱjkiot=Tijko,Jijko,ℱijko=O.(iii)
proved with the help of example.


### 3.6. Trace of NHSM

LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko and α=β. Then, trace of NHSM is denoted as trO and is defined as trO=∑i=1,k=aα,zTiiko−Jiiko+ℱiiko.Example 5.
Let us consider a NHSMO4×4:(19)O4×4=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3.
Then,trO=0.5−0.3−0.6+0.6−0.5−0.3+0.8−0.5−0.4+0.4−0.5−0.3=−1.1.Proposition 2.
LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko andα=β. s be any scalar then trsO=strO.Proof.
(20)trsO=∑i=1,k=aα,zsTiiko−sJiiko+sℱiiko=s∑i=1,k=aα,zTiiko−Jiiko+ℱiiko=strO.


### 3.7. Max-Min Product of NHSM

LetO=Oij and ℳ=ℳjm be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳjm=Tjkmℳ,Jjkmℳ,ℱjkmℳ. Then, O and ℳ are said to be conformable if their dimensions are equal to each other (number of columns of O is equal to number of rows of ℳ). If O=Oijα×β and ℳ=ℳjmβ×γ, then O⊗ℳ=Simα×γ, where(21)Sim=maxjkminTijko,Jjkmℳ,minjkmaxℐijko,Jjkmℳ,minjkmaxℱijko,ℱjkmℳ.Theorem 2.
LetO=Oijα×βand ℳ=ℳjmβ×γ be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳjm=Tjkmℳ,Jjkmℳ,ℱjkmℳ. Then,(22)O⊗ℳt=ℳt⊗Ot.Proof.
LetO⊗ℳ=Simα×γ; then, O⊗ℳt=Smiγ×α,Ot=Ojiβ×α,andℳt=ℳmjγ×β.
Now,(23)O⊗ℳt=TkmiS,JkmiS,ℱkmiSγ×α=maxjkminTmjkℳ,Tjkio,minjkmaxJmjkℳ,Jjkio,minjkmaxℱmjkℳ,ℱjki0γ×α=Tmjkℳ,Jmjkℳ,ℱmjkℳγ×β⊗Tjkio,Jjkio,ℱjki0β×α=ℳt⊗Ot.


### 3.8. Operators of NHSMs

LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ. Then,(i)
Union:(24)O∪ℳ=S,where Tijks=maxTijko,Tijkℳ, Jijks=Jijko+Jijkℳ/2, and ℱijks=minℱijko,ℱijkℳ.(ii)
Intersection:(25)O∩ℳ=S,where(26)Tijks=minTijko,Tijkℳ,Jijks=Jijko+Jijkℳ2,ℱijks=maxℱijko,ℱijkℳ.(iii)
Arithmetic mean:(27)O⊕ℳ=S,where(28)Tijks=Tijko+Tijkℳ2,Jijks=Jijko+Jijkℳ2,ℱijks=ℱijko+ℱijkℳ2.(iv)
Weighted arithmetic mean:(29)O⊙wℳ=S,where(30)Tijks=w1Tijko+w2Tijkℳw1+w2,Jijks=w1Jijko+w2Jijkℳw1+w2,ℱijks=w1ℱijko+w2ℱijkℳw1+w2⋅w1,w2>0.(v)
Geometric mean:(31)O⊙ℳ=S,where(32)Tijks=Tijko⋅Tijkℳ,Jijks=Jijko⋅Jijkℳ,ℱijks=ℱijko⋅ℱijkℳ.(vi)
Weighted geometric mean:(33)O⊙wℳ=S,where(34)Tijks=Tijkow1⋅Tijkℳw2w1+w2,Jijks=Jijkow1⋅Jijkℳw2w1+w2,ℱijks=ℱijkow1⋅ℱijkℳw2w1+w2,w1,w2>0.(vii)
Harmonic mean:(35)O⊘ℳ=S,where(36)Tijks=2TijkoTijkℳTijko+Tijkℳ,Jijks=2JijkoJijkℳJijko+Jijkℳ,ℱijks=2ℱijkoℱijkℳℱijko+ℱijkℳ.(viii)
Weighted harmonic mean:(37)O⊘wℳ=S,where(38)Tijks=w1+w2w1/Tijko+w2/Tijkℳ,Jijks=w1+w2w1/Jijko+w2/Jijkℳ,ℱijks=w1+w2w1/ℱijko+w2/ℱijkℳ,Proposition 3.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.
Then,(i)
O∪ℳt=Ot∪ℳt(ii)
O∩ℳt=Ot∩ℳt(iii)
O⊕ℳt=Ot⊕ℳt(iv)
O⊕wℳt=Ot⊕ℳt(v)
O⊙ℳt=Ot⊙ℳt(vi)
O⊙wℳt=Ot⊙wℳt(vii)
O⊘ℳt=Ot⊘ℳt(viii)
O⊘wℳt=Ot⊘wℳtProof.
(i)(39)O∪ℳt=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijkoℱijkℳt=maxTjkio,Tjkiℳ,Jjkio+Jjkiℳ2,minℱjkio,ℱjkiℳ=Tjkio,Jjkio,ℱjkio∪Tjkiℳ,Jjkiℳ,ℱjkiℳ=Tijko,Jijko,ℱijkot∪Tijkℳ,Jijkℳ,ℱijkℳt=Ot∪ℳt.
Remaining parts are proved in a similar way.Proposition 4.
LetO=Oij and ℳ=ℳij be two upper triangular NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ. Then, O∪ℳ, O∩ℳ, O⊕ℳ, O⊕wℳ, O⊙ℳ, and O⊙wℳ are all upper triangular NHSM and vice versa.Theorem 3.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.
Then,(i)
O∪ℳ⋄=O⋄∩ℳ⋄(ii)
O∩ℳ⋄=O⋄∪ℳ⋄(iii)
O⊕ℳ⋄=O⋄⊕ℳ⋄(iv)
O⊕wℳ⋄=O⋄⊕ℳ⋄(v)
O⊙ℳ⋄=O⋄⊙ℳ⋄(vi)
O⊙wℳ⋄=O⋄⊙wℳ⋄(vii)
O⊘ℳ⋄=O⋄⊘ℳ⋄(viii)
O⊘wℳ⋄=O⋄⊘wℳ⋄Proof.
(i)(40)O∪ℳ⋄=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijko,ℱijkℳ⋄=minℱijko,ℱijkℳ,Jijko+Jijkℳ2,maxTijko,Tijkℳ=ℱijko,Jijko,Tijko∩ℱijkℳ,Jijkℳ,Tijkℳ=Tijko,Jijko,ℱijko⋄∩Tijkℳ,Jijkℳ,ℱijkℳ⋄=O⋄∩ℳ⋄.
Remaining parts are proved in a similar way.Theorem 4.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.
Then,(i)
O∪ℳ=ℳ∪O(ii)
O∩ℳ=ℳ∩O(iii)
O⊕ℳ=ℳ⊕O(iv)
O⊕wℳ=ℳ⊕wO(v)
O⊙ℳ=ℳ⊙O(vi)
O⊙wℳ=ℳ⊙wO(vii)
O⊘ℳ=ℳ⊘O(viii)
O⊘wℳ=ℳ⊘wOProof.
(i)(41)O∪ℳ=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijko,ℱijkℳmaxTijkℳ,Tijko,Jijkℳ+Jijko2,minℱijkℳ,ℱijko=Tijkℳ,Jijkℳ,ℱijkℳ∪Tijko,Jijko,ℱijko=ℳ∪O.
Remaining parts are proved in a similar way.Theorem 5.
Let=Oij, ℳ=ℳij, and N=ℳijbe NHSM, where Oij=Tijko,Jijko,ℱijko, ℳij=Tijkℳ,Jijkℳ,ℱijkℳ, and Nij=TijkN,JijkN,ℱijkN. Then,(i)
O∪ℳ∪N=O∪ℳ∪N(ii)
O∩ℳ∩N=O∩ℳ∩N(iii)
(O⊕ℳ⊕N≠O⊕ℳ⊕N(iv)
O⊙ℳ⊙N≠O⊙ℳ⊙N(v)
O⊘ℳ⊘N≠O⊘ℳ⊘NProof.
(i)(42)O∪ℳ∪N=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijko,ℱijkℳ∪TijkN,JijkN,ℱijkN=maxTijko,Jijkℳ,TijkN,Jijko+Jijkℳ+JijkN3,minℱijko,ℱijkℳ,ℱijkN=maxTijkO,TijkM,TijkN,JijkO+Jijkℳ+JijkN3,minℱijkO,ℱijkℳ,ℱijkN=TijkO,JijkO,ℱijkO∪maxTijkℳ,TijkN,Jijkℳ+JijkN2,minℱijkℳ,ℱijkN=TijkO,JijkO,ℱijkO∪Tijkℳ,Jijkℳ,ℱijkℳ∪TijkN,JijkN,ℱijkN=O∪ℳ∪N.
Remaining parts are proved in a similar way.Theorem 6.
Let=Oij, ℳ=ℳij, and N=ℳijbe NHSM, where Oij=Tijko,Jijko,ℱijko, ℳij=Tijkℳ,Jijkℳ,ℱijkℳ, and Nij=TijkN,JijkN,ℱijkN. Then,(i)
O∩ℳ⊕N=O∩ℳ⊕O∩N(ii)
O⊕ℳ∩N=O∩N⊕ℳ∩N(iii)
O∪ℳ⊕N=O∪ℳ⊕O∪N(iv)
O⊕ℳ∪N=O∪N⊕ℳ∪NProof.
(i)(43)O∩ℳ⊕N=Tijko,Jijko,ℱijko∩Tijkℳ+TijkN2,Jijkℳ+JijkN2,ℱijkℳ+ℱijkN2=minTijko,Tijkℳ+TijkN2,Jijko+Jijkℳ+JijkN/22,maxℱijko,ℱijkℳ+ℱijkN2=minTijkO+Tijkℳ2,TijkO+TijkN2,JijkO+Jijkℳ/2+JijkO+JijkN/22,maxℱijkO+ℱijkℳ2,ℱijkO+ℱijkN2=minTijko,Tijkℳ,Jijko+Jijkℳ2,maxℱijko,ℱijkℳ⊕minTijko,TijkN,Jijko+JijkN2,maxℱijko,ℱijkN=Tijko,Jijko,ℱijko∩Tijkℳ,Jijkℳ,ℱijkℳ⊕Tijko,Jijko,ℱijko∩TijkN,JijkN,ℱijkN=O∩ℳ⊕O∩N.
The remaining parts are proved in a similar way.


## 3.1. NHSM

LetU=u1,u2,…,uα and PU be the universal set and power set of universal set, respectively; also, consider ℒ1,ℒ2,…,ℒβ, for β≥1, where β is well-defined attributes, whose corresponding attributive values are, respectively, the set ℒ1a,ℒ2b,…,ℒβz and their relation ℒ1a×ℒ2b×⋯×ℒβz, where a,b,c,…,z=1,2,…,n; then, the pair ℱ,ℒ1a×ℒ2b×⋯×ℒβz is said to be neutrosophic hypersoft set over U, where ℱ:ℒ1a×ℒ2b×⋯×ℒβz⟶PU and it is defined as ℱℒ1a×ℒ2b×⋯×ℒβz=u,TLu,ILu,FLuu∈U,L∈ℒ1a×ℒ2b×⋯×ℒβz. Table 1 represents the tabular form of NHSS ℛL.Table 1
Matrix representation of NHSS.
ℒ1aℒ2b…ℒβzu1XℛLu1,ℒ1aXℛLu1,ℒ2b…XℛLu1,ℒβzu2XℛLu2,ℒ1aXℛLu2,ℒ2b…XℛLu2,ℒβz⋮⋮⋮⋱⋮uαXℛLuα,ℒ1aXℛLuα,ℒ2b…XℛLuα,ℒβzIfOij=XℛLui,ℒjk, where i=1,2,3,…,α,j=1,2,3,…,β,andk=a,b,c,…,z, then a matrix is defined as(6)Oijα×β=O11O12…O1βO21O22…O2β⋮⋮⋱⋮Oα1Oα2…Oαβ,where Oij=Tℒjkui,Iℒjkui,Fℒjkui,ui∈U,,ℒjk∈ℒ1a×ℒ2b×⋯×ℒβz=Tijko,Jijko,ℱijko.Thus, we can represent any neutrosophic hypersoft set in terms of a neutrosophic hypersoft matrix (NHSM), and it means that they are interchangeable.Example 1.
Teachers’ recruitment problem (TRP) is the most complex and absurd task. There is no fixed and fabricated design to know their subject knowledge or pedagogical skills. Therefore, decision makers find themselves in a blind alley. Consequently, based on their own knowledge and experience, they select a person who does not meet the institutional requirement. Thus, TRP is typically a multicriteria decision-making MCDM problem.
Assumptions:(i)
Independent attributes are considered(ii)
Everyone attends the interview(iii)
Hesitant environment is not yet considered
Formulation of the Problem. Let us consider an institute that wants to hire a teacher appropriate to its requirements, and they received the following statistics-based CVs. Let U be the set of candidates for the teaching at the college level:(7)U=T1,T2,T3,T4,T5.
Also, consider the set of attributes as(8)A1=Qualification,A2=Experience,A3=Gender,A4=Publications.
Parameters:(i)
Ti = universal set of teachers, wherei=1,2,3,4,5(ii)
Ai = attributes, wherei=1,2,3,4 that are further categorized into the following:(iii)
A1a=qualification(iv)
A1a=BS Hons.,MS/Mphill,Phd,Post Doctorate(v)
A2b=experience=5yr,8yr,10yr,15yr(vi)
A3c=gender=Male,Female(vii)
A4d=publications=3,5,8,10+
Let the function beℱ:A1a×A2b×A3c×A4d⟶PU
Below are Tables2–5 of their neutrosophic values assigned by different decision makers.
The neutrosophic hypersoft set is defined as(9)ℱ:A1a×A2b×A3c×A4d⟶PU.
Let us assume(10)ℱA1a×A2b×A3c×A4d=ℱMphill,5yr,male,3=T1,T2,T4,T5,ℱA1a×A2b×A3c×A4d=ℱMphill,5yr,male,3=≪T1,Mphill0.5,0.3,0.6,5yr0.3,0.4,0.7,male0.5,0.6,0.9,30.6,0.4,0.5≫,≪T2,Mphill0.3,0.2,0.1,5yr0.6,0.5,0.3,male0.7,0.8,0.3,30.7,0.5,0.3≫,≪T4Mphill0.7,0.3,0.6,5yr0.6,0.4,0.8,male0.8,0.5,0.4,30.6,0.2,0.1≫,≪T5Mphill0.5,0.4,0.5,5yr0.3,0.6,0.7,male0.9,0.2,0.1,30.4,0.5,0.3≫,.
Then, a neutrosophic hypersoft set of above-assumed relation in the tabular form is represented in Table6.
And, its matrix is defined as(11)O4×4=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3.Table 2
Decision makers will assign neutrosophic numbers to each candidateTi against qualification.
A1aqualificationT1T2T3T4T5BS Hons.(0.4,0.5,0.8)(0.7,0.6,0.4)(0.4,0.5,0.7)(0.5,0.3,0.7)(0.5,0.3,0.8)MS/MPhil.(0.5,0.3,0.6)(0.3,0.2,0.1)(0.3,0.6,0.2)(0.7,0.3,0.6)(0.5,0.4,0.5)Ph.D.(0.8,0.2,0.4)(0.9,0.5,0.3)(0.9,0.4,0.1)(0.6,0.3,0.2)(0.6,0.1,0.2)Post doctorate(0.9,0.3,0.1)(0.5,0.2,0.1)(0.8,0.5,0.2)(0.8,0.2,0.1)(0.7,0.4,0.2)Table 3
Decision makers will assign neutrosophic numbers to each candidateTi against experience.
A2bexperienceT1T2T3T4T55 yr.(0.3,0.4,0.7)(0.6,0.5,0.3)(0.5,0.6,0.8)(0.6,0.4,0.8)(0.3,0.6,0.7)8 yr.(0.4,0.2,0.5)(0.8,0.1,0.2)(0.4,0.7,0.3)(0.4,0.8,0.7)(0.7,0.5,0.6)10 yr.(0.7,0.2,0.3)(0.9,0.3,0.1)(0.8,0.3,0.2)(0.5,0.4,0.3)(0.5,0.2,0.1)15 yr.(0.8,0.2,0.1)(0.6,0.4,0.3)(0.9,0.4,0.1)(0.6,0.2,0.3)(0.5,0.3,0.2)Table 4
Decision makers will assign neutrosophic numbers to each candidateTi against gender.
A3cGenderT1T2T3T4T5Male(0.5, 0.6, 0.9)(0.7, 0.8, 0.3)(0.6, 0.4, 0.3)(0.8, 0.5, 0.4)(0.9, 0.2, 0.1)Female(0.6, 0.4, 0.7)(0.3, 0.6, 0.4)(0.8, 0.2, 0.1)(0.4, 0.5, 0.6)(0.8, 0.4, 0.2)Table 5
Decision makers will assign neutrosophic numbers to each candidateTi against publication.
A4dpublicationzT1T2T3T4T53—(0.6, 0.4, 0.5)(0.7, 0.5, 0.3)(0.6, 0.4, 0.3)(0.6, 0.2, 0.1)(0.4, 0.5, 0.3)5—(0.8, 0.2, 0.4)(0.7, 0.3, 0.2)(0.8, 0.3, 0.1)(0.3, 0.4, 0.5)(0.3, 0.5, 0.8)8—(0.5, 0.3, 0.4)(0.6, 0.3, 0.4)(0.5, 0.7, 0.2)(0.8, 0.4, 0.1)(0.7, 0.4, 0.3)10+—(0.4, 0.9, 0.6)(0.8, 0.4, 0.2)(0.2, 0.6, 0.5)(0.7, 0.5, 0.2)(0.6, 0.4, 0.7)Table 6
The tabular form of the above relation.
A1aA2bA3cA4dT1Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5T2Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3T4Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1T5Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3


## 3.2. Square NHSM

LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko. Then, O is said to be square NHSM if α=β. It means that if an NHSM has the same number of rows (attributes) and columns (alternatives), it is a square NHSM.Example 2.
Above defined Example1 is also the example of square NHSM.


## 3.3. Transpose of Square NHSM

LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko; then, Ot is said to be transpose of square NHSM if rows and columns of O are interchanged. It is denoted as(12)Ot=Oijt=Tijko,Jijko,ℱijkot=Tjkio,Jjkio,ℱjkio=Oji.Example 3.
Transpose of the matrix define in Example1 is given as(13)O4×4t=Mphill,0.5,0.3,0.6Mphill,0.3,0.2,0.1Mphill,0.7,0.3,0.6Mphill,0.5,0.4,0.55yr,0.3,0.4,0.75yr,0.6,0.5,0.35yr,0.6,0.4,0.85yr,0.3,0.6,0.7male,0.5,0.6,0.9male,0.7,0.8,0.3male,0.8,0.5,0.4male,0.9,0.2,0.13,0.6,0.4,0.53,0.7,0.5,0.33,0.6,0.2,0.13,0.4,0.5,0.3.


## 3.4. Symmetric NHSM

LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko; then, O is said to be symmetric NHSM if Ot=O, i.e., Tijko,Jijko,ℱijko=Tjkio,Jjkio,ℱjkio.


## 3.5. Scalar Multiplication of NHSM

LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko and s be any scalar then the product of matrix O and a scalar s is a matrix formed by multiplying each element of matrix O by s. It is denoted as sO=sOij, where 0≤s≤1.Example 4.
Let us consider a NHSMO4×4:(14)O4×4=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3.
And,0.1 is the scalar; then, scalar multiplication of NHSM O4×4 is given as(15)0.1O4×4=Mphill,0.05,0.03,0.065yr,0.03,0.04,0.07male,0.05,0.06,0.093,0.06,0.04,0.05Mphill,0.03,0.02,0.015yr,0.06,0.05,0.03male,0.07,0.08,0.033,0.07,0.05,0.03Mphill,0.07,0.03,0.065yr,0.06,0.04,0.08male,0.08,0.05,0.043,0.06,0.02,0.01Mphill,0.05,0.04,0.055yr,0.03,0.06,0.07male,0.09,0.02,0.013,0.04,0.05,0.03.Proposition 1.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.For two scalarss,t∈0,1, then(i)
stO=stO(ii)
Ifs<t, then sO<tO(iii)
IfO⊆ℳ, then sO⊆sℳProof.
(i)
stO=stOij=stTijko,tJijko,tℱijko=stTijko,stJijko,stℱijko=stTijko,Jijko,ℱijko=stOij=stO(ii)
SinceTijko,Jijko,ℱijko∈0,1, so sTijko≤tTijko,sJijko≤tJijko,sℱijko≤tℱijko(iii)
Now,sO=sOij=sTijko,sJijko,sℱijko≤tTijko,tJijko,tℱijko=tOij=tO(iv)
O⊆ℳ⇒Oij⊆ℳij(16)⇒Tijko≤Tijkℳ,Jijko≤Jijkℳ,ℱijko≥ℱijkℳ⇒sTijko≤sTijkℳ,sJijko≤sJijkℳ,sℱijko≥sℱijkℳ⇒sOij⊆sℳij⇒sO⊆sℳ.Theorem 1.
LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko. Then,(i)
sOt=sOt, where s∈0,1(ii)
Ott=O(iii)
IfO=Oij is the upper triangular NHSM, then Ot is lower triangular NHSM and vice versaProof.
(i)
Here,sOt,sOt∈NHSMα×β, so(17)sOt=sTijko,sJijko,sℱijkot=sTjkio,sJjkio,sℱjkio=sTjkio,Jjkio,ℱjkio=sTijko,Jijko,ℱijkot=sOt.(ii)
SinceOt∈NHSMα×β,soOtt∈NHSMα×β. Now,(18)Ott=Tijko,Jijko,ℱijkott=Tjkio,Jjkio,ℱjkiot=Tijko,Jijko,ℱijko=O.(iii)
proved with the help of example.


## 3.6. Trace of NHSM

LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko and α=β. Then, trace of NHSM is denoted as trO and is defined as trO=∑i=1,k=aα,zTiiko−Jiiko+ℱiiko.Example 5.
Let us consider a NHSMO4×4:(19)O4×4=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3.
Then,trO=0.5−0.3−0.6+0.6−0.5−0.3+0.8−0.5−0.4+0.4−0.5−0.3=−1.1.Proposition 2.
LetO=Oij be the square NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko andα=β. s be any scalar then trsO=strO.Proof.
(20)trsO=∑i=1,k=aα,zsTiiko−sJiiko+sℱiiko=s∑i=1,k=aα,zTiiko−Jiiko+ℱiiko=strO.


## 3.7. Max-Min Product of NHSM

LetO=Oij and ℳ=ℳjm be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳjm=Tjkmℳ,Jjkmℳ,ℱjkmℳ. Then, O and ℳ are said to be conformable if their dimensions are equal to each other (number of columns of O is equal to number of rows of ℳ). If O=Oijα×β and ℳ=ℳjmβ×γ, then O⊗ℳ=Simα×γ, where(21)Sim=maxjkminTijko,Jjkmℳ,minjkmaxℐijko,Jjkmℳ,minjkmaxℱijko,ℱjkmℳ.Theorem 2.
LetO=Oijα×βand ℳ=ℳjmβ×γ be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳjm=Tjkmℳ,Jjkmℳ,ℱjkmℳ. Then,(22)O⊗ℳt=ℳt⊗Ot.Proof.
LetO⊗ℳ=Simα×γ; then, O⊗ℳt=Smiγ×α,Ot=Ojiβ×α,andℳt=ℳmjγ×β.
Now,(23)O⊗ℳt=TkmiS,JkmiS,ℱkmiSγ×α=maxjkminTmjkℳ,Tjkio,minjkmaxJmjkℳ,Jjkio,minjkmaxℱmjkℳ,ℱjki0γ×α=Tmjkℳ,Jmjkℳ,ℱmjkℳγ×β⊗Tjkio,Jjkio,ℱjki0β×α=ℳt⊗Ot.


## 3.8. Operators of NHSMs

LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ. Then,(i)
Union:(24)O∪ℳ=S,where Tijks=maxTijko,Tijkℳ, Jijks=Jijko+Jijkℳ/2, and ℱijks=minℱijko,ℱijkℳ.(ii)
Intersection:(25)O∩ℳ=S,where(26)Tijks=minTijko,Tijkℳ,Jijks=Jijko+Jijkℳ2,ℱijks=maxℱijko,ℱijkℳ.(iii)
Arithmetic mean:(27)O⊕ℳ=S,where(28)Tijks=Tijko+Tijkℳ2,Jijks=Jijko+Jijkℳ2,ℱijks=ℱijko+ℱijkℳ2.(iv)
Weighted arithmetic mean:(29)O⊙wℳ=S,where(30)Tijks=w1Tijko+w2Tijkℳw1+w2,Jijks=w1Jijko+w2Jijkℳw1+w2,ℱijks=w1ℱijko+w2ℱijkℳw1+w2⋅w1,w2>0.(v)
Geometric mean:(31)O⊙ℳ=S,where(32)Tijks=Tijko⋅Tijkℳ,Jijks=Jijko⋅Jijkℳ,ℱijks=ℱijko⋅ℱijkℳ.(vi)
Weighted geometric mean:(33)O⊙wℳ=S,where(34)Tijks=Tijkow1⋅Tijkℳw2w1+w2,Jijks=Jijkow1⋅Jijkℳw2w1+w2,ℱijks=ℱijkow1⋅ℱijkℳw2w1+w2,w1,w2>0.(vii)
Harmonic mean:(35)O⊘ℳ=S,where(36)Tijks=2TijkoTijkℳTijko+Tijkℳ,Jijks=2JijkoJijkℳJijko+Jijkℳ,ℱijks=2ℱijkoℱijkℳℱijko+ℱijkℳ.(viii)
Weighted harmonic mean:(37)O⊘wℳ=S,where(38)Tijks=w1+w2w1/Tijko+w2/Tijkℳ,Jijks=w1+w2w1/Jijko+w2/Jijkℳ,ℱijks=w1+w2w1/ℱijko+w2/ℱijkℳ,Proposition 3.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.
Then,(i)
O∪ℳt=Ot∪ℳt(ii)
O∩ℳt=Ot∩ℳt(iii)
O⊕ℳt=Ot⊕ℳt(iv)
O⊕wℳt=Ot⊕ℳt(v)
O⊙ℳt=Ot⊙ℳt(vi)
O⊙wℳt=Ot⊙wℳt(vii)
O⊘ℳt=Ot⊘ℳt(viii)
O⊘wℳt=Ot⊘wℳtProof.
(i)(39)O∪ℳt=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijkoℱijkℳt=maxTjkio,Tjkiℳ,Jjkio+Jjkiℳ2,minℱjkio,ℱjkiℳ=Tjkio,Jjkio,ℱjkio∪Tjkiℳ,Jjkiℳ,ℱjkiℳ=Tijko,Jijko,ℱijkot∪Tijkℳ,Jijkℳ,ℱijkℳt=Ot∪ℳt.
Remaining parts are proved in a similar way.Proposition 4.
LetO=Oij and ℳ=ℳij be two upper triangular NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ. Then, O∪ℳ, O∩ℳ, O⊕ℳ, O⊕wℳ, O⊙ℳ, and O⊙wℳ are all upper triangular NHSM and vice versa.Theorem 3.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.
Then,(i)
O∪ℳ⋄=O⋄∩ℳ⋄(ii)
O∩ℳ⋄=O⋄∪ℳ⋄(iii)
O⊕ℳ⋄=O⋄⊕ℳ⋄(iv)
O⊕wℳ⋄=O⋄⊕ℳ⋄(v)
O⊙ℳ⋄=O⋄⊙ℳ⋄(vi)
O⊙wℳ⋄=O⋄⊙wℳ⋄(vii)
O⊘ℳ⋄=O⋄⊘ℳ⋄(viii)
O⊘wℳ⋄=O⋄⊘wℳ⋄Proof.
(i)(40)O∪ℳ⋄=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijko,ℱijkℳ⋄=minℱijko,ℱijkℳ,Jijko+Jijkℳ2,maxTijko,Tijkℳ=ℱijko,Jijko,Tijko∩ℱijkℳ,Jijkℳ,Tijkℳ=Tijko,Jijko,ℱijko⋄∩Tijkℳ,Jijkℳ,ℱijkℳ⋄=O⋄∩ℳ⋄.
Remaining parts are proved in a similar way.Theorem 4.
LetO=Oij and ℳ=ℳij be two NHSM, where Oij=Tijko,Jijko,ℱijko and ℳij=Tijkℳ,Jijkℳ,ℱijkℳ.
Then,(i)
O∪ℳ=ℳ∪O(ii)
O∩ℳ=ℳ∩O(iii)
O⊕ℳ=ℳ⊕O(iv)
O⊕wℳ=ℳ⊕wO(v)
O⊙ℳ=ℳ⊙O(vi)
O⊙wℳ=ℳ⊙wO(vii)
O⊘ℳ=ℳ⊘O(viii)
O⊘wℳ=ℳ⊘wOProof.
(i)(41)O∪ℳ=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijko,ℱijkℳmaxTijkℳ,Tijko,Jijkℳ+Jijko2,minℱijkℳ,ℱijko=Tijkℳ,Jijkℳ,ℱijkℳ∪Tijko,Jijko,ℱijko=ℳ∪O.
Remaining parts are proved in a similar way.Theorem 5.
Let=Oij, ℳ=ℳij, and N=ℳijbe NHSM, where Oij=Tijko,Jijko,ℱijko, ℳij=Tijkℳ,Jijkℳ,ℱijkℳ, and Nij=TijkN,JijkN,ℱijkN. Then,(i)
O∪ℳ∪N=O∪ℳ∪N(ii)
O∩ℳ∩N=O∩ℳ∩N(iii)
(O⊕ℳ⊕N≠O⊕ℳ⊕N(iv)
O⊙ℳ⊙N≠O⊙ℳ⊙N(v)
O⊘ℳ⊘N≠O⊘ℳ⊘NProof.
(i)(42)O∪ℳ∪N=maxTijko,Tijkℳ,Jijko+Jijkℳ2,minℱijko,ℱijkℳ∪TijkN,JijkN,ℱijkN=maxTijko,Jijkℳ,TijkN,Jijko+Jijkℳ+JijkN3,minℱijko,ℱijkℳ,ℱijkN=maxTijkO,TijkM,TijkN,JijkO+Jijkℳ+JijkN3,minℱijkO,ℱijkℳ,ℱijkN=TijkO,JijkO,ℱijkO∪maxTijkℳ,TijkN,Jijkℳ+JijkN2,minℱijkℳ,ℱijkN=TijkO,JijkO,ℱijkO∪Tijkℳ,Jijkℳ,ℱijkℳ∪TijkN,JijkN,ℱijkN=O∪ℳ∪N.
Remaining parts are proved in a similar way.Theorem 6.
Let=Oij, ℳ=ℳij, and N=ℳijbe NHSM, where Oij=Tijko,Jijko,ℱijko, ℳij=Tijkℳ,Jijkℳ,ℱijkℳ, and Nij=TijkN,JijkN,ℱijkN. Then,(i)
O∩ℳ⊕N=O∩ℳ⊕O∩N(ii)
O⊕ℳ∩N=O∩N⊕ℳ∩N(iii)
O∪ℳ⊕N=O∪ℳ⊕O∪N(iv)
O⊕ℳ∪N=O∪N⊕ℳ∪NProof.
(i)(43)O∩ℳ⊕N=Tijko,Jijko,ℱijko∩Tijkℳ+TijkN2,Jijkℳ+JijkN2,ℱijkℳ+ℱijkN2=minTijko,Tijkℳ+TijkN2,Jijko+Jijkℳ+JijkN/22,maxℱijko,ℱijkℳ+ℱijkN2=minTijkO+Tijkℳ2,TijkO+TijkN2,JijkO+Jijkℳ/2+JijkO+JijkN/22,maxℱijkO+ℱijkℳ2,ℱijkO+ℱijkN2=minTijko,Tijkℳ,Jijko+Jijkℳ2,maxℱijko,ℱijkℳ⊕minTijko,TijkN,Jijko+JijkN2,maxℱijko,ℱijkN=Tijko,Jijko,ℱijko∩Tijkℳ,Jijkℳ,ℱijkℳ⊕Tijko,Jijko,ℱijko∩TijkN,JijkN,ℱijkN=O∩ℳ⊕O∩N.
The remaining parts are proved in a similar way.


## 4. Neutrosophic Hypersoft Matrix (NHSM) in Decision-Making Using Score Function

Suppose that some decision makers wish to select fromα number of objects. Each object is further characterized by β number of attributes, whose respective attributes form a relation just like NHSM. Each decision makes different neutrosophic values to these respective attributes. Corresponding to these neutrosophic values for the required relation, we get a NHSM of order α×β. From this NHSM, we calculate values’ matrices, which help to obtain a score matrix. And, finally, we calculate the total score of each object from the score matrix.Value matrices are the real matrices that obey all the properties of real matrices. Score function is also a real matrix which is obtained from two or more value matrices.Definition 1.
LetO=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko; then, the value of matrix O is denoted as VO, and it is defined as VO=VijO of order α×β, whereVijO=Tijko−Jijko,−ℱijko. The score of two NHSM O=Oij and ℳ=ℳij of order α×β is given as SO,ℳ=VO+Vℳ and SO,ℳ=Sij, where Sij=VijO+Vijℳ. The total score of each object in universal set is ∑j=1nSij.
Algorithm is graphically represented with Figure 1.Step 1: construct a NHSM as defined in Section 3.1.Step 2: calculate the value matrix from NHSM. Let O=Oij be the NHSM of order α×β, where Oij=Tijko,Jijko,ℱijko; then, the value of matrix O is denoted as VO, and it is defined as VO=VijO of order α×β, whereVijO=Tijko−Jijko,−ℱijko.Step 3: compute the score matrix with the help of value matrices. The score of two NHSM O=Oij and ℳ=ℳij of order and α×β is given as SO,ℳ=VO+Vℳ and SO,ℳ=Sij, where Sij=VijO+Vijℳ.Step 4: compute the total score from the score matrix. The total score of each object in the universal set is ∑j=1j=1Sij.Step 5: find the optimal solution by selecting an object of maximum score from the total score matrix.Figure 1
Flowchart of the proposed algorithm.


### 4.1. Numerical Example

Teachers’ recruitment problem (TRP) is the most complex and absurd task. There is no fixed and fabricated design to know their subject knowledge or pedagogical skills. Therefore, decision makers find themselves in a blind alley. Consequently, based on their own knowledge and experience, they select a person who does not meet the institutional requirement; thus, TRP is typically a multicriteria decision-making MCDM problem.Assumptions:(i)
Independent attributes are considered(ii)
Everyone attends the interview(iii)
Hesitant environment is not yet consideredFormulation of the Problem. Let us consider an institute that wants to hire a teacher appropriate to its requirements, and he received the following statistics-based CVs. Let U be the set of candidates for the teaching at the college level:(44)U=T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T12,T13,T14,T15.Also, consider the set of attributes as(45)A1=Qualification,A2=Experience,A3=Gender,A4=Publications.Parameters:Ti = universal set of teachers,wherei=1,2,3,4,5Ai = attributes, wherei=1,2,3,4 that are further categorized into the following:(i)
A1a=Qualification(ii)
A1a=BS Hons.,MS/Mphill,Phd,Post Doctorate(iii)
A2b=Experience=5yr,8yr,10yr,15yr(iv)
A3c=Gender=Male,Female(v)
A4d=Publications=3,5,8,10+The functionℱ:A1a×A2b×A3c×A4d⟶PU.Let us assume the relationℱA1a×A2b×A3c×A4d=ℱMphill,5yr,male,3 which is the actual requirement of college for the selection of candidates.Four candidatesT2,T6,T8,T14 are shortlisted on the basis of assumed relation, i.e., Mphill,5yr,male,3.A jury of two membersA,B is set for the selection of shortlisted candidates. These jury members give their valuable opinion in the form of NHSSs separately as(46)A=ℱMphill,5yr,male,3=≪T2,Mphill0.5,0.3,0.6,5yr0.3,0.4,0.7,male0.5,0.6,0.9,30.6,0.4,0.5≫,≪T6,Mphill0.3,0.2,0.1,5yr0.6,0.5,0.3,male0.7,0.8,0.3,30.7,0.5,0.3≫,≪T8Mphill0.7,0.3,0.6,5yr0.6,0.4,0.8,male0.8,0.5,0.4,30.6,0.2,0.1≫,≪T14,Mphill0.5,0.4,0.5,5yr0.3,0.6,0.7,male0.9,0.2,0.1,30.4,0.5,0.3≫,,B=ℱMphill,5yr,male,3=≪T2,Mphill0.8,0.1,0.2,5yr0.7,0.4,0.3,male0.4,0.6,0.3,30.5,0.3,0.5≫,≪T6,Mphill0.8,0.2,0.1,5yr0.7,0.4,0.3,male0.8,0.2,0.1,30.9,0.3,0.2≫,≪T8Mphill0.5,0.3,0.4,5yr0.7,0.3,0.2,male0.9,0.2,0.1,30.4,0.2,0.7≫,≪T14,Mphill0.7,0.4,0.2,5yr0.2,0.4,0.7,male0.7,0.2,0.1,30.6,0.3,0.4≫,.Let us apply the above define algorithm for the calculation of total score.Step I (construction of NHSM):the above two NHSSs are given in the form of NHSMs as(47)A=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3,B=Mphill,0.8,0.1,0.25yr,0.7,0.4,0.3male,0.4,0.6,0.33,0.5,0.3,0.5Mphill,0.8,0.2,0.15yr,0.7,0.4,0.3male,0.8,0.2,0.13,0.9,0.3,0.2Mphill,0.5,0.3,0.45yr,0.7,0.3,0.2male,0.9,0.2,0.13,0.4,0.2,0.7Mphill,0.7,0.4,0.25yr,0.2,0.4,0.7male,0.7,0.2,0.13,0.6,0.3,0.4.Step II: calculation of the value matrices of NHSMs defined in Step I:(48)VA=Mphill,−0.45yr,−0.8male,−13,−0.3Mphill,05yr,−0.2male,−0.43,−0.1Mphill,−0.25yr,−0.6male,−0.13,0.3Mphill,−0.45yr,−1male,0.63,−0.4,VB=Mphill,0.55yr,0male,−0.53,−0.3Mphill,0.55yr,0male,0.53,0.4Mphill,−0.25yr,0.2male,0.63,−0.5Mphill,0.15yr,−0.9male,0.43,−0.1.Step III: computation of the score matrix by adding value matrices obtained in Step II:(49)SA,B=Mphill,0.15yr,−0.8male,−1.53,−0.6Mphill,0.55yr,−0.2male,0.13,0.3Mphill,−0.45yr,−0.4male,0.53,−0.2Mphill,−0.35yr,−1.9male,13,−0.5.Step IV: calculation of the score matrix:(50)Total score=2.80.10.51.7.Step V: the candidate T2 will be selected for teaching at the college level as the total score of T2 is highest among the rest of the total score of candidates.


## 4.1. Numerical Example

Teachers’ recruitment problem (TRP) is the most complex and absurd task. There is no fixed and fabricated design to know their subject knowledge or pedagogical skills. Therefore, decision makers find themselves in a blind alley. Consequently, based on their own knowledge and experience, they select a person who does not meet the institutional requirement; thus, TRP is typically a multicriteria decision-making MCDM problem.Assumptions:(i)
Independent attributes are considered(ii)
Everyone attends the interview(iii)
Hesitant environment is not yet consideredFormulation of the Problem. Let us consider an institute that wants to hire a teacher appropriate to its requirements, and he received the following statistics-based CVs. Let U be the set of candidates for the teaching at the college level:(44)U=T1,T2,T3,T4,T5,T6,T7,T8,T9,T10,T11,T12,T13,T14,T15.Also, consider the set of attributes as(45)A1=Qualification,A2=Experience,A3=Gender,A4=Publications.Parameters:Ti = universal set of teachers,wherei=1,2,3,4,5Ai = attributes, wherei=1,2,3,4 that are further categorized into the following:(i)
A1a=Qualification(ii)
A1a=BS Hons.,MS/Mphill,Phd,Post Doctorate(iii)
A2b=Experience=5yr,8yr,10yr,15yr(iv)
A3c=Gender=Male,Female(v)
A4d=Publications=3,5,8,10+The functionℱ:A1a×A2b×A3c×A4d⟶PU.Let us assume the relationℱA1a×A2b×A3c×A4d=ℱMphill,5yr,male,3 which is the actual requirement of college for the selection of candidates.Four candidatesT2,T6,T8,T14 are shortlisted on the basis of assumed relation, i.e., Mphill,5yr,male,3.A jury of two membersA,B is set for the selection of shortlisted candidates. These jury members give their valuable opinion in the form of NHSSs separately as(46)A=ℱMphill,5yr,male,3=≪T2,Mphill0.5,0.3,0.6,5yr0.3,0.4,0.7,male0.5,0.6,0.9,30.6,0.4,0.5≫,≪T6,Mphill0.3,0.2,0.1,5yr0.6,0.5,0.3,male0.7,0.8,0.3,30.7,0.5,0.3≫,≪T8Mphill0.7,0.3,0.6,5yr0.6,0.4,0.8,male0.8,0.5,0.4,30.6,0.2,0.1≫,≪T14,Mphill0.5,0.4,0.5,5yr0.3,0.6,0.7,male0.9,0.2,0.1,30.4,0.5,0.3≫,,B=ℱMphill,5yr,male,3=≪T2,Mphill0.8,0.1,0.2,5yr0.7,0.4,0.3,male0.4,0.6,0.3,30.5,0.3,0.5≫,≪T6,Mphill0.8,0.2,0.1,5yr0.7,0.4,0.3,male0.8,0.2,0.1,30.9,0.3,0.2≫,≪T8Mphill0.5,0.3,0.4,5yr0.7,0.3,0.2,male0.9,0.2,0.1,30.4,0.2,0.7≫,≪T14,Mphill0.7,0.4,0.2,5yr0.2,0.4,0.7,male0.7,0.2,0.1,30.6,0.3,0.4≫,.Let us apply the above define algorithm for the calculation of total score.Step I (construction of NHSM):the above two NHSSs are given in the form of NHSMs as(47)A=Mphill,0.5,0.3,0.65yr,0.3,0.4,0.7male,0.5,0.6,0.93,0.6,0.4,0.5Mphill,0.3,0.2,0.15yr,0.6,0.5,0.3male,0.7,0.8,0.33,0.7,0.5,0.3Mphill,0.7,0.3,0.65yr,0.6,0.4,0.8male,0.8,0.5,0.43,0.6,0.2,0.1Mphill,0.5,0.4,0.55yr,0.3,0.6,0.7male,0.9,0.2,0.13,0.4,0.5,0.3,B=Mphill,0.8,0.1,0.25yr,0.7,0.4,0.3male,0.4,0.6,0.33,0.5,0.3,0.5Mphill,0.8,0.2,0.15yr,0.7,0.4,0.3male,0.8,0.2,0.13,0.9,0.3,0.2Mphill,0.5,0.3,0.45yr,0.7,0.3,0.2male,0.9,0.2,0.13,0.4,0.2,0.7Mphill,0.7,0.4,0.25yr,0.2,0.4,0.7male,0.7,0.2,0.13,0.6,0.3,0.4.Step II: calculation of the value matrices of NHSMs defined in Step I:(48)VA=Mphill,−0.45yr,−0.8male,−13,−0.3Mphill,05yr,−0.2male,−0.43,−0.1Mphill,−0.25yr,−0.6male,−0.13,0.3Mphill,−0.45yr,−1male,0.63,−0.4,VB=Mphill,0.55yr,0male,−0.53,−0.3Mphill,0.55yr,0male,0.53,0.4Mphill,−0.25yr,0.2male,0.63,−0.5Mphill,0.15yr,−0.9male,0.43,−0.1.Step III: computation of the score matrix by adding value matrices obtained in Step II:(49)SA,B=Mphill,0.15yr,−0.8male,−1.53,−0.6Mphill,0.55yr,−0.2male,0.13,0.3Mphill,−0.45yr,−0.4male,0.53,−0.2Mphill,−0.35yr,−1.9male,13,−0.5.Step IV: calculation of the score matrix:(50)Total score=2.80.10.51.7.Step V: the candidate T2 will be selected for teaching at the college level as the total score of T2 is highest among the rest of the total score of candidates.


## 5. Result and Comparison Analysis

We propose an algorithm for NHSM of the real-world problems, and results are compared with the algorithms on NSM already established. Graphical representations of the ranking of the proposed algorithm are given in Figure1. The proposed algorithm is valid and practical. As it could be observed in the comparison Table 7, the proposed method’s best selection is comparable with the already established method, which is expressive in itself and approve the reliability and validity of the proposed method. According to the refinement of the philosophy of neutrosophy, it could be a more efficient technique.Table 7
Alternative rank comparison using NHSM and NSM techniques.
MethodAlternative final rankingOptimal choiceProposed in this paperT2>T6>T8>T14T2Hashmi et al. [28]T2>T8>T14>T6T2


### 5.1. Limitations and Advantages of Proposed Matrix Theory

The neutrosophic soft set theory is not very efficient in selecting the optimal object of a decision-making problem that possesses some attributes which are further divided, whereas neutrosophic hypersoft matrix theory can be applied.The advantages of the proposed theory are(1)
Firstly, this new method’s specialty is that it may solve any MCDM problem involving a huge number of decision makers very easily along with a simple computational procedure(2)
Secondly, when compared with existing methods for MCDM problems under a neutrosophic environment, the proposed operators are consistent and accurate, which illustrate their application’s practicability(3)
Thirdly, the proposed method considers the interrelationships of attributes in practical application, while existing approaches cannot(4)
Lastly, the proposed algorithm for MCDM problems in this paper can further consider more correlations between attributes, which means that they have higher accuracy and greater reference value(5)
The matrix is useful for storing (neutrosophic hypersoft set) in the computer memory, which is very useful and applicable


## 5.1. Limitations and Advantages of Proposed Matrix Theory

The neutrosophic soft set theory is not very efficient in selecting the optimal object of a decision-making problem that possesses some attributes which are further divided, whereas neutrosophic hypersoft matrix theory can be applied.The advantages of the proposed theory are(1)
Firstly, this new method’s specialty is that it may solve any MCDM problem involving a huge number of decision makers very easily along with a simple computational procedure(2)
Secondly, when compared with existing methods for MCDM problems under a neutrosophic environment, the proposed operators are consistent and accurate, which illustrate their application’s practicability(3)
Thirdly, the proposed method considers the interrelationships of attributes in practical application, while existing approaches cannot(4)
Lastly, the proposed algorithm for MCDM problems in this paper can further consider more correlations between attributes, which means that they have higher accuracy and greater reference value(5)
The matrix is useful for storing (neutrosophic hypersoft set) in the computer memory, which is very useful and applicable


## 6. Conclusion

This paper has first defined NHSM theory and then introduced some aggregate operators that are more functional to make theoretical studies in the neutrosophic soft set arena. Moreover, we have proposed the concept of the score function. Additionally, the utilization of NHSM in the decision-making problem (teacher recruitment problem (TRP)) has been made with the score matrix’s assistance. At the end, we compared the result with existing techniques and showed that the purposed technique is more efficient and refined. We expect, this paper will advance the future investigation on various calculations such as TOPSIS, VIKOR, and AHP in other decision-making problems. Also, in future, it can be linked with Pythagorean fuzzy interactive Hamacher power aggression operators, interval-valued q-rung orthopair fuzzy sets in decision-making, CN-q-ROFS, connection number-based q-rung orthopair fuzzy set and their application to the decision-making process, and average operators based on the spherical cubic fuzzy number.



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*Source: 6666408-2022-06-25.xml*