Soft Rough q-Rung Orthopair m-Polar Fuzzy Sets and q-Rung Orthopair m-Polar Fuzzy Soft Rough Sets and Their Applications

The notion of a q-rung orthopair fuzzy soft rough set (<inline-formula> <tex-math notation="LaTeX">$^{q}ROFSRS$ </tex-math></inline-formula>) appeared as an extension of q-rung orthopair fuzzy set (<inline-formula> <tex-math notation="LaTeX">$^{q}ROFS$ </tex-math></inline-formula>) and q-rung orthopair fuzzy soft set (<inline-formula> <tex-math notation="LaTeX">$^{q}ROFSS$ </tex-math></inline-formula>) with the aid of rough set (RS) definition. Thus, <inline-formula> <tex-math notation="LaTeX">$^{q}ROFSRS$ </tex-math></inline-formula> and m-polar fuzzy set (<inline-formula> <tex-math notation="LaTeX">$_{m}PFS$ </tex-math></inline-formula>) are convenient to deal with uncertain knowledge which helps us to solve many problems in decision making. In this paper, we define the soft rough q-rung orthopair m-polar fuzzy sets (<inline-formula> <tex-math notation="LaTeX">$^{\text {q}}RO_{\text {m}}PFS$ </tex-math></inline-formula>) and q-rung orthopair m-polar fuzzy soft rough sets (<inline-formula> <tex-math notation="LaTeX">$^{q}RO_{\text {m}}PFSRS$ </tex-math></inline-formula>) through crisp soft and q- rung orthopair (q-RO) m-polar fuzzy soft approximation space. The related characteristics of these models are also studied. Then, we construct two new algorithms for these models to solve MADM issues. The successful application and corresponding comparative analyses proves that our proposed models are rational and effective.


I. INTRODUCTION
The rapid of research articles become very huge, especially in mathematics. Numerous suggestions were made to solve realworld problems using mathematical techniques by way of appropriate equations or formulas in helping decision makers to make their best decisions. To solve problems involving uncertainty, fuzzy sets (FS) was introduced by Zadeh [1] in 1965.
Later in 1982, Pawlak introduced the notion called Rough Sets (RS) [2], [3]. The beauty of RS is it is able to divide the area into three parts (Lower, Upper, and Boundary region). This idea comes from the meaning of the topology concept. Eight years later, Dubois and Prade [4] combine the notion of RS and FS, to form rough fuzzy sets and fuzzy rough sets. Since then, many researchers studied further on RS and FS as in the following published articles [5]- [15].
The associate editor coordinating the review of this manuscript and approving it for publication was Geng-Ming Jiang .
In 1994, as an extension of FS whose membership grade range is [−1, 1], bipolar fuzzy sets (BFS) was proposed by Zhang [46]. In a BFS, the membership grade 0 of a variable means that the variable is irrelevant to the corresponding property, the membership grade (0, 1] of a variable points out that the variable somewhat fulfills the property, while the membership grade [−1, 0) of a variable point out that the variable somewhat satisfies the implicit counter-property. The idea which lies behind such description is connected with the existence of ''bipolar information'' (e.g., plus information and minus information) about the given set. Plus information represents what is granted to be possible, while minus information represents what is considered to be impossible. Then to generalize the BFS to help experts to deal with uncertainty, the meaning of m-polar fuzzy sets ( m PS) was mooted by Chen et al. [47]. They proved that bipolar fuzzy sets and 2-polar fuzzy sets are cryptomorphic mathematical tools. In many real-life complicated problems, data sometimes comes from an employee (n ≥ 2), that is, multipolar information (not just bipolar information, which corresponds to two-valued logic) exists. There are many applications of m-polar fuzzy sets to decision-making problems when it is compulsory to make assessments with a group of agreements. Akram et al. [48], [49] proposed the soft rough m-polar fuzzy and m-polar fuzzy soft rough sets. By merging the concepts of SRS, PFS, and m PS, Riaz and Hashmi [50] investigated the Pythagorean m-polar fuzzy sets (P m PFS), soft rough Pythagorean m-polar fuzzy sets (SRP m PFS) and Pythagorean m-polar fuzzy soft rough sets (P m PFSRS). The concept of q-rung orthopair m-polar fuzzy sets ( q RO m PFS) was then defined by Riaz et al. [51].
Using the notions of SS, SRS, and q ROFS, Hussain et al. [52] proposed the q-rung orthopair fuzzy soft sets and their application. Wang et al. [53] explained the q ROF soft rough sets( q ROFSRS) with a few applications. Riaz et al. [54] introduced the notion of soft rough q-rung orthopair fuzzy sets and some of their properties were discussed. Thereafter, many researchers studied SRS, SS, and their applications such as [55]- [60], [64].
From these interesting studies, we intend to develop a hybrid of SRS and q RO m PFS and put forward a new model called q-rung orthopair m-polar fuzzy soft rough sets ( q RO m PFSRS) and soft rough q-rung orthopair fuzzy sets (SR q RO m PFS). These combinations provide us with the property of q RO m PFS and soft rough sets together which maximize the handling of uncertain data. Thus our proposed methods are generalized extensions of Akram et al. [48], Riaz and Hashmi [50] and Riaz et al. [54]. When q = 1, the presented formula reduces to those methods in [48] and [49] and if q = 2, it reduces to those methods in [50]. Our proposed method will cater for m sets which make our studies are reliable, compared to [54] which catered for only a single set. Their relevant properties will be investigated, a few definitions and theorems will be promulgated along with illustrative examples. We will then proceed to construct two algorithms along with their applications. Finally, we will run comparative analyses on the outcomes of those two algorithms.
The structure of this paper is as follows. The preliminary of basic notions will be introduced in Section 2. Section 3 will discuss the novel concept of SR q RO m PFS and the related characteristics. The hybrid concept of q RO m PFSRS will be proposed and its associated properties are discussed in Section 4. In Section 5, we give an illustrative example to show the applicability of the proposed constructed algorithms along with the comparative analyses, followed by the conclusion in Section 6.

II. PRELIMINARIES
Now, we give some basic notions on IFS, PFS and q ROF before defining soft rough q-rung orthopair m-polar fuzzy sets SR q RO m PFS in the next section.
Definition 6 [47]: If is the origin set, where φ : → [0, 1] m is the set of all m-polar fuzzy sets on .
Riaz and Hashmi [50] extended it to a Pythagorean form below.
Definition 7 [50]: If is the origin set. For everyĤ ∈ , if we have a membership grade ϑ r E : → [0, 1] and a nonmembership grade r E : → [0, 1]. Define the Pythagorean m-polar fuzzy sets (P m PFS) E as indicated below. [51] further extended m-polar fuzzy sets of Chen et. al. [47] to q-rung orthopair form below.
Definition 8 [51]: If is the origin set. For everyĤ ∈ , if we have a membership grade ϑ r E : → [0, 1] and a non-membership grade r E : → [0, 1]. Define the q-rung orthopair m-polar fuzzy sets ( q RO m PFS) E as indicated below.
Molodtsov [16] defined the soft set as below. Definition 10 [16]: If is the origin set, and let E ⊆ . So,Ŝ = (F , A ) is a soft set over , when A ⊆ E and F : A → P( ).
Lately, the notion of q-rung orthopair fuzzy soft set ( q ROFSS) was investigated as follows.
Definition 11 [52]: If is the origin set. For everyĤ ∈ , let A ⊆ E and F : A → q ROFSS( ). Then define the q ROFSS E as indicated below.
Wang et al. [53] defined q ROFSS from blow. Definition 12 [53]: If is the origin set. For everyĤ ∈ and let (F,A ) be a q ROFSS. Then for E ⊆ × A is q ROFSS relation is defined as follows.

III. SOFT ROUGH q-RUNG ORTHOPAIR m-POLAR FUZZY SETS
In this section, we will define and illustrate the notion of soft rough q-rung orthopair m-polar fuzzy sets SR q RO m PFS and also discuss their relevant properties. Definition 13: If is the origin set, is the provisory features, and σ is the crisp soft relation, then ( , , σ ) is a CSAS. For anyÊ ∈ q RO m PFS( ), the soft rough q RO m PFS-lower and soft rough q RO m PFS-upper approximations (SR q RO m PFSLA, SR q RO m PFSUA), which are denoted by K and K ,respectively, are as follows.

IV. q-RUNG ORTHOPAIR m-POLAR FUZZY SOFT ROUGH SETS
Below, we construct the concept of q-rung orthopair m-polar fuzzy soft rough sets q RO m PFSRS, and will discuss their properties. Henceforth, the notions of I, J and (I, J )-cut sets will be proposed and their characteristics will be put forward.
Definition 14: Suppose is the origin set and is the provisory features for someÊ ⊆ . If we have a mapping µ : E → q RO m PFS( ), then (µ,Ê) is called q-rung orthopair m-polar fuzzy sets ( q RO m PFS), where q RO m PFS( ) is the set of all q-rung orthopair m-polar fuzzy subsets of the origin set .
Definition 15: If (µ,Ê) is a q RO m PFSS, then a q-rung orthopair m-polar fuzzy subset ν of × is called a q-rung orthopair m-polar fuzzy soft relation as below. 1] are the membership and non-membership scale, respectively, under the term of This relation can be viewed as the following, ν, as shown at the bottom of the next page. VOLUME 9, 2021 Definition 16: If is the origin set, is the provisory features, and ν is the q RO m PFSRS relation, then ( , , ν) is a q RO m PFS-approximation space. For anyÊ ∈ q RO m PFS( )), the q RO m PF soft rough-lower and q RO m PF soft rough-upper approximations, which are denoted by S and S , respectively, are as follows.

A. SOME PROPERTIES
In this segment we will propose a few definitions, propositions and illustrative examples to describe a few properties of our proposed notion on q RO m PFSRS.
(2) The proof is similar to the proof of (1).
, as shown at the bottom of the next page.

Proposition 5:
If we haveÊ is q RO m PFSRS through and H ∈ , then the following characteristics hold. (1)

VOLUME 9, 2021
Proof: (1) From Definition 20 we have, The proof is similar to the proof of (1).
The proof is similar to the proof of (3).
The proof is similar to the proof of (5). Proposition 6: If we haveÊ 1 andÊ 2 are q RO m PFSRS through andĤ ∈ , then the following characteristics hold. ( (3)-(4) The proofs are similar to the proofs of (1) and (2). Proposition 7: If we haveÊ 1 andÊ 2 are q RO m PFSRS through andĤ ∈ , then the following characteristics hold. (1) Proof: The proofs follow from Propositions 5 and 6. Proposition 8: If we haveÊ 1 andÊ 2 are q RO m PFSRS through andĤ ∈ , then the following characteristics hold. ( . Proof: The proofs follow from Proposition 5. Proposition 9: If we haveÊ 1 andÊ 2 are q RO m PFSRS through andĤ ∈ , then the following characteristics hold.

V. APPLICATIONS
Here, we construct two algorithms to solve MCDM issues via soft rough q-rung orthopair m-polar fuzzy sets (SR q RO m PFS) and q-RO m-polar fuzzy soft rough sets ( q RO m PFSRS).
These algorithms will aid managers to make decisions using our proposed models via the lower and upper approximations.

Let
= {Ĥ 1 ,Ĥ 2 , . . . ,Ĥ t } be t number of computer programmers and = { 1 , 2 , . . . , r } be r features required of these programmers by the institution which placed the advertisement. The institution establishes several criteria to best choose desirable candidates with the following features: Communication Skill 1 , Personality 2 , Experience 3 , Self-Dependability 4 . We will build a crisp soft relation for the first method σ over × and q-RO m-polar fuzzy soft relation for the second method ν : → . Therefore, through the proposed methods SR q RO m PFS and q RO m PFSRS, we introduce the following two subsections to aid with the managerial decision..

B. SR q RO m PFS APPROACH
The following steps in Algorithm 1 establishes our new approach using the q-ROF m-polar fuzzy sets and crisp soft approximation space.

Algorithm 1 Algorithm for SR q RO m PFS
Input: is the origin set and is the provisory features. Finally, we rank the alternatives as follows.
If ∇ = 1.Ĥ C. q RO m PFSRS APPROACH The following steps in Algorithm 2 establishes our new approach using the q-ROF m-polar fuzzy soft rough sets and crisp soft approximation space. Now, we give the following illustrated example using the proposed approach.
Hence, we have the q-RO 3-polar fuzzy soft relation as in the following matrix.
Then we set the q-RO 3-polar fuzzy subsets of as follows.    Finally, we rank the alternatives as follows.

D. COMPARATIVE ANALYSES
In this section, we will explain the merits of the proposed methods by comparisons between ours, that is, SR q RO m PFS and q RO m PFSRS, and the previous methods, that is, soft rough m-polar fuzzy sets and m-polar fuzzy soft rough sets by Akram et al. [48], soft rough Pythagorean fuzzy set and Pythagorean fuzzy soft rough set by Riaz and Hashmi [50] and soft rough q-rung orthopair fuzzy sets and q-rung orthopair fuzzy soft rough sets by Riaz et al. [54]. The novel approaches to solve MADM issues can be seen as illustrated in Tables 1 and 2.  Table 1 shows the ordering outcomes for different ∇ (i.e., Akram et al. [48], Riaz and Hashmi [50] and our   proposed methods) for SR q RO m PFS. The best selection of the proposed different approaches is by hiring programmerĤ 1 . This means that our model is reliable and rational. Table 2 shows the ordering outcomes for different ∇ (i.e., Akram et al. [48], Riaz and Hashmi [50] and our proposed methods) for q RO m PFSRS. The best selection of the proposed different approaches is by hiring programmerĤ 1 . This means that our model is reasonable and effective.
We can also show the differences between different ∇ (i.e., Akram et al. [48], Riaz and Hashmi [50] and our proposed methods) using the following two figures, Figure 1 and Figure 2. Figure 1 illustrates the comparisons on the outcomes for ∇ = 1, 2, 3, 5 for SR q RO m PFS, which means that theĤ 1 alternative is the best choice for this institution under the given requirements. Figure 2 illustrates the comparisons on the outcomes for ∇ = 1, 2, 3, 5 for q RO m PFSRS, which means that theĤ 1 VOLUME 9, 2021 alternative is the best choice for this institution under the given requirements. Figure 2 illustrates the comparisons on the outcomes for ∇ = 1, 2, 3, 5 (i.e., Akram et al. [48], Riaz and Hashmi [50] and our proposed methods) for q RO m PFSRS, which means that theĤ 1 alternative is the best choice for this institution under the given requirements. Note that the data used here cannot be processed by the methods of Riaz et al. [54] which can only handle a single set. Hence, our proposed methods have overcome the hurdle of set limitations of the previous existing methods of Akram et al. [48], Riaz and Hashmi [50] and Riaz et al. [54].

VI. CONCLUSION
We have constructed new algorithms using soft rough q-RO m-polar fuzzy sets (SR q RO m PFS) and q-RO m-polar fuzzy soft rough sets ( q RO m PFSRS) to provide us with novel approaches to help make a decision on managerial problems. These new models proved their effectiveness and reliability, as can be seen in Tables 1 and 2, and displayed on Figures 1 and 2. The characteristics related to these models have also been discussed. We have established two different groups of steps for these new models according to the crisp soft and q-RO m-polar fuzzy soft approximation space to solve MADM problems. The comparative analyses indicated that the proposed approaches yield consistent results. In future, we shall extend the proposed methods to a variety of other environments such as the T-spherical power Muirhead operators [62], multi-objective programming [64], neurogenetics [65] and polynomial zeros [66]- [68]. MUHAMMAD RIAZ received the M.Sc., M.Phil., and Ph.D. degrees in mathematics from the University of Punjab, Lahore. He has published more than 85 research articles in international peerreviewed SCIE and ESCI journals with more than 1300 citations. His research interests include pure mathematics, fuzzy mathematics, topology, algebra, fuzzy systems, soft set theory, rough set theory with applications in decision-making, medical diagnosis, artificial intelligence, computational intelligence, information measures, information aggregation, and pattern recognition.
NASRUDDIN HASSAN received the B.Sc. degree in mathematics from Western Illinois University, USA, the M.Sc. degree in applied mathematics from Western Michigan University, USA, and the Ph.D. degree in applied mathematics from Universiti Putra Malaysia, Malaysia. He is currently an Associate Professor with the School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Malaysia. He has published more than a 100 articles of which 153 are currently listed in the SCOPUS database with H-index of 31. His research interests include decision-making, operations research, fuzzy sets, and numerical convergence.