GENERALIZED Z-FUZZY SOFT 𝜷 -COVERING BASED ROUGH MATRICES AND ITS APPLICATION TO MAGDM PROBLEM BASED ON AHP METHOD

: Fuzzy sets, rough sets and soft sets are different mathematical tools mainly developed to deal with uncertainty. The combination of these theories has a wide range of applications in decision analysis. In this paper, we defined a generalized Z-fuzzy soft 𝛽 -covering-based rough matrices. Some algebraic properties are explored for this newly constructed matrix. The main aim of this paper is to propose a novel MAGDM model using generalized Z-fuzzy soft 𝛽 - covering-based rough matrices. A MAGDM algorithm based on AHP method is created to recruit the best candidate for an assistant professor job in an institute and a numerical example is presented to demonstrate the created method.


Introduction
Different theories have been proposed to deal with uncertainty in data mining when conventional mathematics fails. To handle problems containing uncertainty, many theories, such as probability, fuzzy sets, soft sets, rough sets, and a combination of these theories have been utilized. Each of these concepts appears to have its own set of constraints and limitations. Zadeh (1965) introduced fuzzy sets to handle uncertainties in datasets. Pawlak (1982) proposed the concept of a rough set (RS). Rough set is used in data analysis to discover basic patterns in data, eliminate redundancies, and develop decision rules. Sharma et al. (2018Sharma et al. ( , 2021Sharma et al. ( , 2022 developed various hybrid methods using rough set theory to create decision rules useful for realworld problems. Several scholars are working on this concept, which is being applied to a variety of fields such as in (Greco et al., 2001). Covering rough sets (CRS) is an important study topic as an extension of rough sets. CRS is a useful tool that allows academics to examine uncertainty in a broader context. Because of its broad variety of applications, CRS has attracted a large number of researchers. Wang (2007, 2012) proposed a variety of CRS models. Dubois and Prade (1990) have proposed two novel models such as fuzzy rough sets (FRS) and rough fuzzy sets (RFS). Molodtsov (1999) proposed a soft set theory to describe a useful mathematical tool to deal with multi-attribute uncertainty. Soft set theory describes a wide range of information and computational activities. Maji et al. (2002Maji et al. ( , 2003 developed a number of operations on soft sets. Bipolar soft sets are capable of dealing with ambiguity. Tufail et al. (2022) combined bipolar soft set with rough set and proposed a novel MCGDM algorithm. Ali (2011) examined the relationships between rough sets, fuzzy sets and soft sets. Enginoglu (2010, 2012) presented soft matrix and fuzzy soft matrix and explored their algebraic properties. Vijayabalaji (2014) introduced the generalized matrix representation for soft rough matrices. Muthukumar and Krishnan (2018) presented a novel way of representing the fuzzy soft rough matrix in a generalized manner. Yüksel et al. (2014) proposed soft covering-based rough sets (SCRS) and developed an application that combined rough sets and covering soft sets. Zhan and Wang (2019) defined five forms of SCRS and described the relationship between SRS and SCRS. Soft rough fuzzy-covering (SRFC) and soft fuzzy rough covering (SFRC) models are presented by Zhan and Sun (2019). Zhang and Zhan (2019) contributed fuzzy soft -coverings and defined four different fuzzy softcovering based fuzzy rough sets (FS CFRS). Yang (2022) proposed fuzzy covering based rough set over dual universe as an extension of single universe concept.
A multiple attributes group decision-making (MAGDM) problem is one in which a group of qualified specialists examine and select the best alternative from a collection of objects based on their attributes. MAGDM is used in variety of fields. Research in MAGDM problems using fuzzy sets, rough sets and soft sets has increased in recent years (Gurmani et al., 2022). Due to the ambiguity in choice objects and decisionmakers desire, it is not easy to explicit the decision maker's idea in precise value. To efficiently deal with vagueness information, a generalized Z-fuzzy soft -covering based rough matrix is introduced into the MAGDM problem. Saaty (1980) introduced Analytical Hierarchy Process (AHP), which is a well-known MCDM technique. In MCDM issues, AHP establishes a hierarchy of components and determines values for each of those elements via pairwise comparisons. AHP is a technique for guiding decision-making processes. The AHP framework assists firms in making various decisions by assessing and evaluating criteria. By following the phases of our proposed technique, we can make better-informed decisions.

Motivation
Vijayabalaji (2014) first introduced the idea of converting soft rough sets into soft rough matrices and they generalised the concept of soft rough matrices by redefining it. Muthukumar and Krishnan (2018) proposed the concept of generalized fuzzy soft rough matrices and developed a novel decision-making model using this technique. Inspired by these concepts, the idea of Generalized Z-fuzzy soft -covering based rough (Z-GFS CR) matrices is presented and applied to decision-making problems using a -level soft set. The purpose of this manuscript is to establish a foundation for Z-GFS CR matrices to handle uncertainty problems. We first proposed the theoretical concepts of Z-GFS CR matrices and their operations, which are more important for conducting theoretical studies in the extension of fuzzy soft rough set theory.
The summary of the present paper is systematized as follows. In section 2, all the preliminaries needed to understand the following sections are provided. In section 3, we define a new type of Z-fuzzy soft -covering based rough set (Z-FS CRS) using fuzzy soft -adhesion formed from -level soft set. We develop a matrix representation for Z-FS CRS. Further, we define a generalized matrix representation for Z-FS CRS known as generalized Z-fuzzy soft -covering based rough (Z-GFS CR) matrix and we examine some algebraic properties of Z-GFS CR matrices. In section 4, we create a decision making algorithm using Z-GFS CR matrices. In section 5, the conclusion is addressed.

Preliminaries
In this section, the preliminaries that are necessary to understand the below sections are described. Let be a finite universal set and ⊆ be the attribute set throughout this paper.
Definition 7 (Cagman and Enginoglu, 2012). Let ( , ) be a FSS over . Then, a subset of × is uniquely defined by = {( , ): ∈ , ∈ ( )} which is known as the relation form of ( , ). The indicator function of is given by , which is known as fuzzy soft matrix of ( , ) of order × over Ω.

11) From 1),
( ) ⊆ ( ( )). Now we have to prove The above theorem shows that the fuzzy soft -covering lower and upper approximations satisfy the Pawlak's rough lower and upper approximations properties. Hence, the Z-fuzzy soft -covering based rough set is significant. , where ∈ (0,1).

Proof.
The proof is obvious from Definition 24.
1) For all and , 2) It is similar to the proof 1). .

Proof.
The proof is obvious from Definitions 22, 25 and 27.
1) For each and , 2) It is similar to the proof 1).

A novel approach to MAGDM using Z-GFS CR matrices
In this section, a decision-making algorithm is created to select the best object from a list of possible objects Ω based on a decision maker's chosen parameters.
Step 2: Let ( , ) be a FS CS over Ω. Let = 0.5. Compute the -level soft set of ( , ) by the formula, . Then the ordered pair = (Ω, ) is the FS CAS with respect to -level soft set of each ( , ) are computed.
Step 3: Using -level soft set, calculate the fuzzy soft -adhesion of each ∈ Ω. Now compute the FS CLA and FS CUA by using the formula, Step 4: By using Definition 14, construct the Z-GFS CR matrices of order × over Ω.
Step 5: By using Saaty's (2008) nine-point scale, construct the pairwise comparison matrices for each criteria according to the three experts. Calculate the weight for each criteria using AHP method.
Step 6 Step 7: Select the value for and by means of max-min technique, determine the best alternative from the list of .

Algorithm
• Construct the fuzzy soft -covering set ( , ) based on the important parameters. • Compute the -level soft set. ] with the corresponding Z-GFS CR matrices. • By means of max-min technique, determine the best alternative from .

Illustrative Example
The steps mentioned in the algorithm are demonstrated in the following numerical example.
By using AHP method, the weights for each criteria are calculated and it is attained as  .
Step 6: Multiply the weight of each parameters with the corresponding Z-GFS CR matrix. Let a = 0.6. Step 7: By means of max-min technique,

Comparative analysis
In this section, we compare Z-GFS CR matrices with GFSR matrices developed in (Muthukumar and Krishnan, 2018) to demonstrate the importance of our model in decision making method. Both the Z-GFS CR matrices and GFSR matrices are applied to real-life problems of finding the best alternative from the set of candidates applied for the job interview. Using GFSR matrices, we obtain { 1 } as the best alternative. Similarly, by applying our model Z-GFS CR matrices, we obtain the same { 1 } as the best alternative. From the analysis, we can say that our model is effective and feasible.

Conclusion
In our work, we have defined Z-FS CRS with respect to -level soft set. The fuzzy soft -covering lower and upper approximations of Z-FS CRS satisfy the properties of Pawlak's rough approximations showing that the proposed Z-FS CRS is significant. A new type of matrix called Z-FS CR matrix is introduced and we re-defined the concept of Z-FS CR matrix by generalizing it. Each new definition is illustrated with examples for better understanding. Several algebraic properties and De Morgan's laws are investigated based on the study of Z-GFS CR matrices. A novel MAGDM model is developed using the Z-GFS CR matrices to recruit the best applicant for the assistant professor job. Using the proposed MAGDM model, we found that the candidate { 1 } is the suitable one. Our MAGDM algorithm can be applied to any real-world problem which will give effective results. In future work, a generalized intuitionistic fuzzy soft rough matrix could be explored to develop a novel MAGDM model.