#### 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.

#### 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.Section 1 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]

Let be the universal set and be the set of attributes with respect to . Let be the power set of and . A pair is called a soft set over and its mapping is given as

It is also defined as

##### 2.2. Neutrosophic Soft Set [11]

Let be the universal set and be the set of attributes with respect to . Let be the set of neutrosophic values of and . A pair is called a neutrosophic soft set over , and its mapping is given as

##### 2.3. Hypersoft Set [21]

Let be the universal set and be the power set of . Consider , for , and let be well-defined attributes, whose corresponding attributive values are, respectively, the set with , for and ; then, the pair is said to be hypersoft set over , where

##### 2.4. Neutrosophic Hypersoft Set [23]

Let be the universal set and be the power set of . Consider , for ; let be well-defined attributes, whose corresponding attributive values are, respectively, the set with , for and , and their relation ; then, the pair is said to be neutrosophic hypersoft set (NHSS) over , wherewhere is the membership value of truthiness, is the membership value of indeterminacy, and is the membership value of falsity such that also .

#### 3. Neutrosophic Hypersoft Matrix (NHSM)

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

##### 3.1. NHSM

Let and be the universal set and power set of universal set, respectively; also, consider , for , where is well-defined attributes, whose corresponding attributive values are, respectively, the set and their relation , where ; then, the pair is said to be neutrosophic hypersoft set over , where and it is defined as . Table 1 represents the tabular form of NHSS .

If , where , then a matrix is defined aswhere .

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 be the set of candidates for the teaching at the college level:Also, consider the set of attributes as*Parameters*:(i) *=* universal set of teachers, (ii) = attributes, that are further categorized into the following:(iii)(iv)(v)(vi)(vii)Let the function be

Below are Tables 2–5 of their neutrosophic values assigned by different decision makers.

The neutrosophic hypersoft set is defined asLet us assumeThen, a neutrosophic hypersoft set of above-assumed relation in the tabular form is represented in Table 6.

And, its matrix is defined as

##### 3.2. Square NHSM

Let be the NHSM of order , where . Then, is said to be square NHSM if . 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 Example 1 is also the example of square NHSM.

##### 3.3. Transpose of Square NHSM

Let be the square NHSM of order , where ; then, is said to be transpose of square NHSM if rows and columns of are interchanged. It is denoted as

*Example 3. *Transpose of the matrix define in Example 1 is given as

##### 3.4. Symmetric NHSM

Let be the square NHSM of order , where ; then, is said to be symmetric NHSM if , i.e., .

##### 3.5. Scalar Multiplication of NHSM

Let be the NHSM of order , where and be any scalar then the product of matrix and a scalar is a matrix formed by multiplying each element of matrix by . It is denoted as , where .

*Example 4. *Let us consider a NHSM :And, **0.1** is the scalar; then, scalar multiplication of NHSM is given as

Proposition 1. *Let and be two NHSM, where and .*

For two scalars , then(i)(ii)If , then (iii)If , then

*Proof. *(i)(ii)Since , so (iii)Now, (iv)

Theorem 1. *Let be the NHSM of order , where . Then,*(i)*, where *(ii)*(iii)**If is the upper triangular NHSM, then is lower triangular NHSM and vice versa*

*Proof. *(i)Here, , so(ii)Since . Now,(iii)proved with the help of example.

##### 3.6. Trace of NHSM

Let be the square NHSM of order , where and . Then, trace of NHSM is denoted as and is defined as .

*Example 5. *Let us consider a NHSM :Then, .

Proposition 2. *Let be the square NHSM of order , where and. be any scalar then .*

*Proof. *

##### 3.7. Max-Min Product of NHSM

Let and be two NHSM, where and . Then, and are said to be conformable if their dimensions are equal to each other (number of columns of is equal to number of rows of ). If and , then , where

Theorem 2. *Let and be two NHSM, where and . Then,*

*Proof. *Let ; then, .

Now,

##### 3.8. Operators of NHSMs

Let and be two NHSM, where and . Then,(i)Union:where , , and .(ii)Intersection: where(iii)Arithmetic mean: where(iv)Weighted arithmetic mean: where(v)Geometric mean: where(vi)Weighted geometric mean: where(vii)Harmonic mean: where(viii)Weighted harmonic mean: where

Proposition 3. *Let and be two NHSM, where and .**Then,*(i)*(ii)**(iii)**(iv)**(v)**(vi)**(vii)**(viii)*

*Proof. *(i)Remaining parts are proved in a similar way.

Proposition 4. *Let and be two upper triangular NHSM, where and . Then, , , , , , and are all upper triangular NHSM and vice versa.*

Theorem 3. *Let and be two NHSM, where and .**Then,*(i)*(ii)**(iii)**(iv)**(v)**(vi)**(vii)**(viii)*

*Proof. *(i)Remaining parts are proved in a similar way.

Theorem 4. *Let and be two NHSM, where and .**Then,*(i)*(ii)**(iii)**(iv)**(v)**(vi)**(vii)**(viii)*

*Proof. *(i)Remaining parts are proved in a similar way.

Theorem 5. *Let , , and be NHSM, where , , and . Then,*(i)*(ii)**(iii)**(*(iv)*(v)*

*Proof. *(i)Remaining parts are proved in a similar way.

Theorem 6. *Let , , and be NHSM, where , , and . Then,*(i)*(ii)**(iii)**(iv)*