A Novel Approach towards Roughness of Bipolar Soft Sets and Their Applications in MCGDM

The uncertainty in the data is an obstacle in decision-making (DM) problems. In order to solve problems with a variety of uncertainties a number of useful mathematical approaches together with fuzzy sets, rough sets, soft sets, bipolar soft sets have been developed. The rough set theory is an effective technique to study the uncertainty in data, while bipolar soft sets have the ability to handle the vagueness, as well as bipolarity of the data in a variety of situations. This study develops a new methodology, which we call the theory of dominance-based bipolar soft rough sets (DB-BSRSs), which will be used to propose a new technique to solve decision-making problems. The idea introduced in this study has never been discussed earlier. Furthermore, this concept has been explored by means of a detailed study of the structural properties. Moreover, some important measures like the accuracy measure , the measure of precision , and the measure of quality for DB-BSRSs are also provided. Finally, an application of the DB-BSRSs in multi-criteria group decision-making (MCGDM) problem is presented and an algorithm for this application is proposed, supported by an example, which yields the best decision, as well as, the worst decision between some objects. In comparison with some existing results, we also present some advantages of our proposed method.


I. INTRODUCTION
Cantor's set theory is essential for the entire of mathematics.But one issue related to the notion of a set is the concept of uncertainty.This uncertainty has been an issue for a long time for researchers and mathematicians to solve complicated problems in various fields, like social sciences, economics, medical sciences, management sciences, engineering, decision making, and artificial intelligence, etc.
To get rid of this uncertainty, researchers in mathematics, computer science, and many other related fields have proposed various theories like Probability theory, Fuzzy set theory [67], Rough set (RS) theory [51], Vague set theory, Graph theory, Decision-Making theory, etc.But, all of these theories have their own internal problems, which might be due to the inadequacy of the parameterization tools of the theories as discussed in [47].
There has been a growing interest in SS theory.Maji et al. [41] introduced some operations on soft sets and made a theoretical study of soft sets.Based on [41], Ali et al. [4] proposed a number of new operations on soft sets and improve the concept of compliment of the soft set.Feng et al. ( [16], [17]), proposed the relationships among soft sets, rough sets, and fuzzy sets, obtaining three types of hybrid models: rough soft sets, soft rough sets, and softrough fuzzy sets.Shabir et al. [56] reclassified a model of a soft rough set known as a modified soft rough set (MRSset).Majumdar and Samanta [45] introduced the idea of soft mapping and studied some of its properties.They applied soft mapping in medical diagnosis.Maji et al. [42] proposed the notion of fuzzy soft sets by combining fuzzy sets and soft sets.Babitha and Sunil [11] introduced the concept of soft relations and soft functions.Then Qin et al. [53] pointed out that Babitha and Sunil's definition contradicts Cantor's set theory.Therefore they redefined the notion of soft relations and soft functions.
SS theory [47] and RS theory [51] are regarded as effective mathematical approaches to address uncertainty.In 2011, Feng et al. [16] established a relationship among these two theories and introduced the concept of a new hybrid version of the soft rough sets (SRSs), that can give better approximations over Pawlak's RS theory in some cases ( [16], Example 4.7).This approach can be viewed as a generalization of RS theory.
The idea of dominance-based rough sets was proposed by Greco, Matarazzo, and Słowinski ( [20]- [23], [64]).It is an extension of RS theory.Du and Hu [15] presented the notion of a dominance-based rough fuzzy set.In 2019, Shaheen et al. [63] put forth the idea of dominance-based SRSs and discuss their application in decision analysis.Jun and Ahn [29] discussed the double-framed soft sets with applications in BCK/BCI-algebras.

A. LITERATURE REVIEW
In numerous sorts of data analysis, the bipolarity of the data is a key component to be taken into consideration while developing mathematical models for some issues.Bipolarity discusses the positive and negative aspects of the data.The positive data addresses what is assumed to be possible, while the negative data addresses what is not possible or certainly false.The concept that lies behind the presence of bipolar information is that a wide assortment of human decisionmaking depends on bipolar judgemental thinking.For example, sweetness and sourness of food, participation and rivalry, friendship and hostility, effects and side effects of drugs are the two different aspects of information in decision-making and coordination.The soft sets, the fuzzy sets, and the rough sets are not appropriate tools to handle this bipolarity.
Because of the significance of giving positive and negative aspects of data, Shabir and Naz [58] in 2013, initiated the concept of bipolar soft sets (BSSs) and its set-theoretic operations such as union, intersection, and complement and discuss its application to DM problems.After this research, BSSs have become increasingly popular with researchers.In 2015, Karaaslan and Karataş [32] reclassified a version of BSSs with another approximation giving opportunity to examine topological structures of BSSs.They also provide a DM methodology using BSSs with the help of an example.Later on, Karaaslan et al. [34] presented the notion of bipolar soft groups.In addition, Naz and Shabir [48] initiated the notion of fuzzy BSSs and studied their algebraic structures.In 2017, Shabir and Bakhtawar [59] first proposed the notion of bipolar soft connected, bipolar soft disconnected, and bipolar soft compact spaces.Then, Öztürk ' [50] further discussed the notions of interior and closure operators, basis, and subspaces in the bipolar soft topological spaces.Abdullah et al. [1] brought the idea of bipolar fuzzy soft sets by hybridizing the soft sets and the bipolar fuzzy sets and applied this idea in DM problem.Alkouri et al. [8] initiated the idea of bipolar complex fuzzy sets and discussed their application in DM problems.In 2018, Karaaslan and Çagman [35] proposed the concept of the bipolar soft rough sets (BSRSs) which is a combination of RS theory and BSSs.They also provide applications of BSRSs in DM.Shabir and Gul [60] introduced the notion of modified rough bipolar soft sets (MRBS-sets) and discussed their application in MCGDM.Mahmood et al. [40] introduced a novel complex fuzzy N-soft sets and their DM algorithm.Malik and Shabir [44] initiated the idea of rough fuzzy BSSs and applied this idea to solve DM problems.Also, Malik and Shabir [45] developed a consensus model based on rough bipolar fuzzy approximations.Mahmood [39] redefined a version of BSSs which is called T-bipolar soft sets and applied this idea in DM problems.Shabir et al. [61] initiated the idea of rough approximations of BSSs by soft relations and applied this idea in DM.Al-shami [9] came up with the concept of belong and nonbelong relations between a BSS and an ordinary point.Riaz and Tehrim [54] initiated the concept of bipolar fuzzy soft mappings and discuss their applications to bipolar disorders.Kamacı and Petchimuthu [31] proposed the idea of bipolar N-soft set, which is an extension of N-soft set, and discuss its applications in DM problems.Akram and Ali [2] developed a hybrid model for DM under rough Pythagorean fuzzy bipolar soft information.

B. MOTIVATION
If we summarize all the above arguments, then we have noticed that the BSSs have the capability to deal with the bipolarity of the information about certain objects with the help of two mappings.One mapping handles the positivity of the information, while the other mapping measures the negativity.Keeping in view the association between RSs and BSSs, two attempts have been made to study the roughness of BSSs: one by Karaaslan and Çagman [35], and the other by Shabir and Gul [60].This is the main motivation for us to introduce and study the novel approach of the roughness of BSSs by using dominance-based bipolar soft rough sets (DB-BSRSs) and discuss their application in decision-making.

C. AIM OF THE PROPOSED STUDY
The main goal of this study is to present another interesting and novel version of BSRSs by utilizing DB-BSRSs.
We highlight the article by the following pioneering work: • A novel concept known as DB-BSRSs is proposed.
• Some important structural properties of DB-BSRSs are investigated in detail.
• Some important measures associated with DB-BSRSs are proposed.• A comprehensive MAGDM method in the framework of DB-BSRSs is introduced and the validity of this approach is also verified by a practical example.• The detailed comparative analysis with other existing methods is done in order to show the advantages of the proposed methodology.
The article has been organized in the following manner.Section 2 gives an overview of some basic ideas, which is required for the understanding of our research work.Section 3 starts by characterizing some dominance-based bipolar soft operators.Further, we discuss the relationship between these operators and their properties.Moreover, based upon these operators, we proposed the notion of DB-BSRSs.The notion is further investigated by considering its important structural properties in detail.In section 4, we discuss some important measures associated with DB-BSRSs and examine their properties.In section 5, we present a general framework of the MCGDM technique based on DB-BSRSs to choose the best element among the alternatives.Section 6 states our proposed DM algorithm for selecting the best alternative.
After that, we give an illustrative example of the proposed DM technique to show that the technique can be effectively applied to some real-life problems in section 7.In section 8, a comparison analysis is made between the proposed model and some other well-known DM techniques.At the last, section 9 concludes with a summary of the present work and a suggestion for further research.

II. PRELIMINARIES
This section is dedicated to recalling some essential ideas and notions that will be utilized in the coming sections.
All through this article, we will use U to denote the initial universe, A to denote the parameters set, and 2 U to denote the power set of U, except if expressed something else.Definition 1: [51] Assume that U be a non-empty finite universe, and Ω be an equivalence relation on U. Then P = (U, Ω) is named an approximation space.
For a non-empty set X ⊆ U, the lower and upper approximations of X with respect to P = (U, Ω) are respectively characterized as follows: Subsequently set X is called definable with respect to Ω if Ω * (X) = Ω * (X); equivalently Bnd Ω (X) = ∅.Furthermore, set X is called undefinable (rough) with respect to Ω if Ω * (X) = Ω * (X); equivalently Bnd Ω (X) = ∅.Definition 2: [47] Let U be a non-empty finite universe of objects and A be a non-empty set of parameters associated with the objects of U. Then a pair (f, A) is named a SS over U, where f is a set-valued mapping given by f : A −→ 2 U .Thus, a SS over the universe U offers a parameterized family of subsets of the universe U.For e ∈ A, f (e) could also be considered as the set of e-approximate elements of U by the SS (f, A).A SS over U may also be expressed by the set of ordered pairs: Subsequently, a BSS over the universe U offers a couple of parameterized families of subsets of U and f (e) ∩ g(¬e) = ∅ for all e ∈ A, ¬e ∈ A, is used as a consistency constraint.A BSS can also be represented through the set of triples as follows: (f, g : A) = e, f (e), g(¬e) : e ∈ A, ¬e ∈ A and f (e) ∩ g(¬e) = ∅ .
(6) From now onward, the set of all BSSs over the universe U are going to be represented by BPSS(U).Definition 5: [35] A BSS (f, g : A) ∈ BPSS(U) is said to be a full BSS if the following two conditions are satisfied: (1) Definition 6: [35] For (f, g : A) ∈ BPSS(U), the object of the form Υ = U, (f, g : A) is known as a BSAspace (bipolar soft approximation space).Based on Υ, the subsequent four operators are defined for any X ⊆ U as follows: ) are regarded as soft Υ-lower positive, soft Υ-lower negative, soft Υ-upper positive, and soft Υ-upper negative approximations of X, respectively.Generally, the ordered pairs are given as: VOLUME 4, 2016 are called bipolar soft rough approximations (BSRapproximations) of X with respect to BSA-space.Moreover, when BS Υ (X) = BS Υ (X), then X is referred to as bipolar soft rough set (BSRS); in any case, X is referred to as bipolar soft Υ-definable.Further, the boundary region of the BSRS is characterized as follows: The properties satisfied by BSR-approximations of X ⊆ U can be found in [35].
To preserve the underlying idea of the BSSs as a parameterization tool and the capacity of RS theory to accommodate with vague ideas because of indiscernibility in data, Shabir and Gul [60] offered the notions of modified rough bipolar soft sets (MRBS-sets) as follows: Definition 7: [60] Assume that (f, g : A) ∈ BPSS(U), where the mappings f and g are given by f : A −→ 2 U and g : A −→ 2 U .Let us characterize two different mappings Φ and Ψ as follows: Then Θ = U, (Φ, Ψ) is named as MRBSA-space (modified rough bipolar soft approximation space).
For any M ⊆ U, the lower and the upper modified bipolar pairs with respect to Θ are respectively characterized as follows: where are called Φ-lower positive, Φ-upper positive, Ψ-lower negative and Ψ-upper negative M RBS-approximations of M ⊆ U, respectively.Moreover, If MBS Θ (M ) = MBS Θ (M ), then M is stated to be M RBS-set; otherwise, M is called M RBS-definable.Furthermore, the boundary region of the M RBS-set is described as follows: The properties satisfied by M RBS-approximations can be found in [60].

III. DOMINANCE-BASED BIPOLAR SOFT ROUGH SETS (DB-BSRSS)
BSRSs were initially presented by Karaaslan and Çagman [35] to deal with the roughness through BSSs which were subsequently altered and upgraded by Shabir and Gul [60].In the current section, we introduce another novel version of BSRSs by using DB-BSRSs.Definition 8: Assume that (f, g : A) ∈ BPSS(U).For any object, u ∈ U, define the A-dominating set f + (u) and Adominating set g + (u) as follows: for all e ∈ A u∈f (e) for some e ∈ A, for all ¬e ∈ A u∈g(¬e) for some ¬e ∈ A.
Similarly, A-dominated set f − (u) and A-dominated set g − (u) are defined as follows: And, A-equivalent sets f ± (u) and A-equivalent set g ± (u) are defined as follows: Generally, the operators f + , g + , f − , g − , f ± and g ± are called A-dominating, A-dominating, A-dominated, Adominated, A-equivalent and A-equivalent bipolar soft operators, respectively.Remark 1: From the above definition, we have f + , f − , f ± : U −→ 2 U and g + , g − , g ± : U −→ 2 U .Remark 2: From the above definition, we can see that: (1) The sets f + (u)and g + (u) consists of all those elements that are not inferior to u subject to given parameters in A and A, respectively.(2) The sets f − (u)and g − (u) consists of all those elements that are not superior to u subject to given parameters in A and A.
(3) The sets f ± (u)and g ± (u), is the intersection of the preceding two, consists of all those elements that have the similar parameters as u.
(4) The objects that do have not any parameter from the given set of parameters are going to be mapped under the six operators to all those objects that have a similar behavior subject to the given parameter set.Definition 9: Let x, y ∈ U, then preference among the objects of U is taken in the following sense: • We say that x is preferred over y with respect to the parameter e, represented as x e y, if x is at least as good as y with respect to the parameter e. • Similarly, x is preferred over y with respect to the not parameter ¬e, represented as x ¬e y, if x is at least as bad as y with respect to the not parameter ¬e.Equivalently, and but y / ∈ g(¬e).
In the framework of BSSs, the relation x e y implies that either x and y both have the parameter e or x possesses this parameter but y does not, or x and y both do not have the parameter e.Similarly, the relation x ¬e y means that either x and y both have the not parameter ¬e or x possesses this not parameter but y does not or x and y both does not have the not parameter ¬e.Proposition 1: Suppose that (f, g : A) ∈ BPSS(U).Then for any object u ∈ U, we have (1) f ± (u) is the greatest set which is contained in both f + (u) and f − (u).
Similarly, the bipolar lower and upper approximations of X under the A-dominated bipolar soft operator f − and the A-dominated bipolar soft operator g − can be defined by the following two pairs: where, then X is called bipolar soft dominated rough set, otherwise, it is called bipolar soft dominated definable set.Moreover, the bipolar lower and upper approximations of X under the A-equivalent bipolar soft operator f ± and the A-equivalent bipolar soft operator g ± can be defined by the following two pairs: where, The approximations in Equations ( 16) to ( 18) are collectively called dominance-based bipolar soft rough approximations (DB-BSR-approximations) of X ⊆ U. Remark 3: In the framework of the decision-making process: (1) X f + may be considered as the collection of most favored alternatives in X under f + , while X g + may be considered as the collection of most favored alternatives in X c under g + .
(2) X f − may be considered as the collection of least favored alternatives in X under f − , while X g − may be considered as the collection of least favored alternatives in X c under g − .
(3) X f + may be considered as the collection of all possible alternatives that could be favored over the elements of X under f + , while X g + can be considered as the collection of all possible alternatives that could be favored over the elements of X c under g + .
(4) X f − may be interpreted as having elements over which the elements of X could possibly be favored under f − , while X g − may be interpreted as having elements over which the elements of X c could possibly be favored under g − .

Definition 11:
The regions listed below: (1) are called the bipolar positive dominating region, the bipolar boundary dominating region and the bipolar negative dominating region, respectively.The same terminologies can be defined for bipolar dominated region and bipolar equivalent region.
Corollary 1: From the above definition, we immediately have that X ⊆ U is a dominance-based bipolar soft definable set if and only if Example 2: To illustrate the notion of DB-BSRapproximations, let us revisit Example 1, where X = {u 1 , u 3 , u 5 } ⊆ U and X c = {u 2 , u 4 , u 6 }.Now using Definition 10, we get Thus, bipolar soft dominating rough approximations of X ⊆ U are: Similarly, Therefore, bipolar soft dominated rough approximations of X ⊆ U are: Also, So, bipolar soft equivalent rough approximations of X ⊆ U are: Moreover, the bipolar positive dominating region, the bipolar boundary dominating region and the bipolar negative dominating region can be calculated as: Similarly, the bipolar positive dominated region, the bipolar boundary dominated region and the bipolar negative dominated region can be calculated as: Also, the bipolar positive equivalent region, the bipolar boundary equivalent region and the bipolar negative equiv-alent region can be calculated as: Since BND (f + ,g + ) (X) = (∅, ∅), BND (f − ,g − ) (X) = (∅, ∅) and BND (f ± ,g ± ) (X) = (∅, ∅), so X is a bipolar soft dominating (as well as bipolar soft dominated) rough set and a bipolar soft equivalent definable set.Remark 4: Assume that (f, g : A) ∈ BPSS(U).Then: (1) For any u ∈ U such that u / ∈ f (e) for all e ∈ A, we have for all e ∈ A .
In this situation, we express the set by f (u).That is, (2) For any u ∈ U such that u / ∈ g(¬e) for all ¬e ∈ A, we have for all ¬e ∈ A .
In this situation, we express the set by g (u).That is, g + (u) = g − (u) = g ± (u) = g (u).For instance, in Example 1, u 4 , u 6 / ∈ f (e) for all e ∈ A, so f (u 4 ) = f (u 6 ) = {u 4 , u 6 }.Similarly, u 1 , u 5 / ∈ g(¬e) for all ¬e ∈ A, so g (u 1 ) = g (u 5 ) = {u 1 , u 5 }.Definition 12: For (f, g : A) ∈ BPSS(U), an object u ∈ U is said to be a redundant object with respect to f if u / ∈ f (e) for all e ∈ A. Similarly, an object u ∈ U is said to be a redundant object with respect to g if u / ∈ g(¬e) for all A. Proposition 3: Let (f, g : A) ∈ BPSS(U).Then (f, g : A) is a full BSS if and only if (f, g : A) has no redundant object with respect to f and g.Proof 3: We can observe that in Example 1, u 4 , u 6 ∈ U are redundant objects with respect to f and u 1 , u 5 ∈ U are redundant objects with respect to g.Thus, In order to discover the relationship between the DB-BSR-approximations of a single set and the DB-BSRapproximations of two sets under the operators f + , g + , f − and g − , the following properties are given.
Theorem 1: Assume that (f, g : A) ∈ BPSS(U).Then for any X, Y ⊆ U, the DB-BSR-approximations fulfill the following axioms: Proof 5: Follows from Definition 10.In order to discover the connection between the DB-BSR-approximations of X ⊆ U under the operators f + , g + , f − , g − , f ± and g ± , the subsequent results are given.Proposition 5: Suppose that (f, g : A) ∈ BPSS(U).Then the following properties hold for any X ⊆ U; (1) ) X g ± ⊆ X g + and X g + ⊆ X g ± .
Remark 5: If we compare the DB-BSR-approximations with BSR-approximations given in [35], we conclude the following points.
(1) To show S Υ P (X ∩ Y ) = S Υ P (X) ∩ S Υ P (Y ), there is a strong condition on the positive SS to be the intersection complete.However, in the case of DB-BSRapproximations, no such condition is needed to prove ) Similarly, to show S Υ P (U) = U = S Υ P (U), the BSS (f, g : A) must be full.However in DB-BSRapproximations no such condition is required for Karaaslan and Çagman imposed a constraint on the BSS (f, g : A) to be semi-intersection.But in DB-BSR-approximations, we have in any case.

IV. MEASURES ASSOCIATED WITH DB-BSRSS
Generally, the uncertainty of a set is because of the presence of the boundary region.The broader the boundary region of a set is, the lower the axactness of the set is.To present the concept accurately, in this section, we propose a few significant measures related with DB-BSRSs and examine their properties.
Pawlak [51] proposed the notions of accuracy measure (AM) and roughness measures (RM) related to RS approximations.The AM is the ratio of the cardinality of lower approximation to the cardinality of upper approximation while the RM is the complement of the AM.The aim of introducing these measures is to express the degree of completeness of information about set X or to communicate the quality of an approximation.The RM is considered as the degree of incompleteness of information about set X.As an extension of these measures, we introduce AM and RM using DB-BSRapproximations as follows: Definition 13: Let (f, g : A) ∈ BPSS(U), Υ = U, (f, g : A) be BSA-space and X ⊆ U. Then the accuracy measure for DB-BSRS with respect to X under dominating bipolar soft operators f + and g + denoted by AM(X + ) and is defined by an ordered pair: where Here |•| denote the cardinality of the set.Similarly, the roughness measure for DB-BSRS with respect to X under dominating bipolar soft operators f + and g + denoted by RM(X + ) and is defined as: Also the accuracy measures for DB-BSRS with respect to X under dominated bipolar soft operators f − and g − and equivalent bipolar soft operators f ± and g ± are respectively given as: where The corresponding roughness measures are: U, (f, g : A) be BSA-space.For any X ⊆ U, AM(X + ) under dominating bipolar soft operators f + and g + satisfies the following properties.
Similar results can be proved for AM(X − ) and AM(X ± ).Proposition 7: be the accuracy measures of X and Y under dominating bipolar soft operators f + and g In 2001, Gediga and Düntsch [18] described the idea of the measure of the precision for ∅ = X ⊆ U, which is the ratio of the cardinality of lower approximation of X to the cardinality of X.In the framework of DB-BSRapproximations it can be characterized in the following manner: Definition 14: Let (f, g : A) ∈ BPSS(U) and Υ = U, (f, g : A) be BSA-space.Then for any ∅ = X ⊆ U the measure of the precision for DB-BSRS with respect to X under dominating bipolar soft operators f + and g + is expressed as MP(X + ) and is characterized by an ordered pair: where Here X c = U − X.Clearly, MP(X + ) ≥ AM(X + ).It is noteworthy that MP(X + ) needs complete knowledge about the set X; while AM(X + ) does not.
Also, 0 ≤ π f + X ≤ 1 and 0 ≤ π g + X 1 for any ∅X ⊆ U.Moreover, it can be noticed that MP(X + ) = (1, 1) if and only if Same terminologies of measure of the precision can be defined for DB-BSRS with respect to X under dominated and equivalent bipolar soft operators.Proposition 8: Let X, Y ⊆ U and MP(X be the measures of the precision of X and Y under dominating bipolar soft operators f + and g In 2010, Yao [66] proposed another measure related to RS approximations known as the measure of quality, which is described in the following way: In the framework of the DB-BSR-approximations it can be characterized in the following manner: Definition 15: Let (f, g : A) ∈ BPSS(U) and Υ = U, (f, g : A) be BSA-space.Then for any ∅ = X ⊆ U the measure of quality for DB-BSRS with respect to X under dominating bipolar soft operators f + and g + is expressed as MQ(X + ) and is characterized by means of an ordered pair: where and Obviously, 0 ≤ α f + ≤ 1 and 0 ≤ β g + ≤ 1 for any ∅ = X ⊆ U.
The corresponding measure of roughness is defined as: Same terminologies of measure of quality can be defined for DB-BSRS with respect to X under dominated and equivalent bipolar soft operators.Proposition 9: Let (f, g : A) ∈ BPSS(U), Υ = U, (f, g : A) a BSA-space and ∅ = X ⊆ U. Then MQ(X) = (1, 1) if and only if X = U or X = ∅.Proof 10: Straightforward.
In order to understand the aforementioned terminologies, here we employ an example.Example 4: Let us consider (f, g : A) ∈ BPSS(U) as given in Example 1, where U = {u 1 , u 2 , u 3 , u 4 , u 5 , u 6 }, A = {e 1 , e 2 , e 3 }, A = {¬e 1 , ¬e 2 , ¬e 3 } and the maps f and g describe as: We already know that DB-BSR-approximations of X = {u 1 , u 3 , u 5 } ⊆ U under f + and g + are: So the values of accuracy measure, measure of precision and measure of quality are respectively given as:
Hence, A f + X and A g + X depicts the elements of U precisely up to the degree 0.75.While π f + X and π g + X depicts the elements of U precisely up to the degree 1.Moreover, α f + and β g + depicts the elements of U precisely up to the degree 0.666 and 0.833, respectively.

V. MCGDM USING DB-BSRSS
The growing complexity of the socio-economic environment, operational research, and industrial engineering force humans to address problems crossing many fields.Group decision-making (GDM), as one of the successful techniques to cope with complex DM problems, is characterized as a decision problem in which numerous experts give their judgment over a set of alternatives.The purpose is to reconcile (or compromise) variations of opinion expressed by individual experts to discover an alternative (or set of alternatives) that is most acceptable by the group of experts as a whole.In a complex society, GDM methods must inevitably take many criteria (or factors) under consideration.Thus, research on GDM that explicitly incorporates multiple criteria has been a major perspective and has made significant advancement with the rapid development of operations research, management science, systems engineering, and other fields.Hwang and Lin [27] first look at to investigate systematically how multiple criteria could be used in GDM.
In general, MCGDM is a procedure in which a group of experts (decision-makers) collaborates to select the best alternative from a set of feasible alternatives that are classified according to their attributes in a given situation.In this section, we develop a novel MCGDM approach using the DB-BSRSs.We give a brief description of a MCGDM problem under the environment of the DB-BSRSs, and afterward, give a general DM methodology for the MCGDM problem by using the DB-BSRSs.

A. PROBLEM DESCRIPTION
We firstly provide the basic description of the considered MCGDM problem in this section.Let U = {u 1 , u 2 , . . ., u n } be the non-empty finite universe of n objects (alternatives) and A = {e 1 , e 2 , . . ., e m } be the finite collection of all possible parameters of objects.Suppose that H = {p 1 , p 2 , . . ., p k } is a set of k independent experts (decision-makers), X 1 , X 2 , . . ., X k are non-empty subsets of U, represent results of primary assessments of experts p 1 , p 2 , . . ., p k , respectively and T 1 , T 2 , . . ., T r ∈ BPSS(U) are the actual results that previously received for problems in various periods or various locations.Then the decision-making for this MCGDM problem is: "how to obtain the evaluation of these particular experts so that the selected object (alternative) is optimal for all criteria".

B. METHODOLOGY OF DECISION-MAKING
In this subsection, we propose the mathematical formulation and the strategy of the MCGDM approach based on the DB-BSRSs.Definition 16: Assume that DBS + Tq (X j ) = X j f + , X j g + and DBS + Tq (X j ) = X j f + , X j g + be lower and upper dominating soft rough approximations of X j ; (j = 1, 2, . . ., k) related to T q = (f q , g q : A) ∈ BPSS(U); (q = 1, 2, . . ., r). Then, In the current section, we give an algorithm for the established strategy of the MCGDM problem being considered in section V.The corresponding steps of the algorithm are given as follows: Step 1: Take primary assessments X 1 , X 2 , . . ., X k of experts p 1 , p 2 , . . ., p k .
Step 5: Calculate (V ) Step 6: Compute decision vector V d by using Definition 18.
Step 7: calculate max i v i ; i = 1, 2, . . ., n.An alternative with maximum WN should be chosen for the final decision.
Flow chart portrayal of the above algorithm is displayed in Figure 1.

VII. APPLICATION: A DESCRIPTIVE EXAMPLE
To illustrate the potential of the above-proposed technique of DM, here we consider a real-life example Example 5: (Selection of a best company for investment) Suppose there are three investors (decision-makers) p 1 , p 2 , and p 3 who want to make an investment in a specific company.Assume that U = {u 1 , u 2 , u 3 , u 4 , u 5 } be the universe consisting five different companies, where u 1 = Automobile company, u 2 = Food company, u 3 = Computer company, u 4 = Arms company and u 5 = Medicine company.Consider A = {e 1 , e 2 , e 3 } is the set of parameters (qualities of companies), where e 1 = appreciation, e 2 = economical growth and e 3 = yearly benefit.Then A = {¬e 1 = depreciation, ¬e 2 = economical decay, ¬e 3 = ayearly loss}.
Step 7: As max i∈In u i = u 5 = 19.5.So u 5 (Medicine company) is the optimal element and min i∈In u i = u 4 = −4.5, so u 4 (Arms company) is the worst element.Moreover the ranking among the elements of the universe U is given as: u 5 u 1 u 2 u 3 u 4 .The pictorial portrayal for the ranking of the companies is shown in Figure 2. In this section, we discuss the validity of the proposed method, its advantages, and disadvantages, and finally a comparison of the proposed technique with some existing methodologies.

A. VALIDITY OF THE PROPOSED MODEL
As we know that, the preferences or opinions of all decisionmakers, aggregation is the crucial step for the classical GDM methods.In our proposed DM approach, every decisionmaker is expressed their opinion as a BSS, and afterward, all opinions given by decision-makers are aggregated through the usage the DB-BSR-approximations, and then a compromise optimal proposal is acquired.So, the DB-BSRSs approach to MCGDM provides a different strategy to aggregate the preferences of decision-makers.Therefore, the proposed DM approach is valid and offers a novel technique and perspective to investigate GDM problems in real life.

B. ADVANTAGES OF THE PROPOSED TECHNIQUE
In general, real-world MCDM and MCGDM problems arise in a complicated environment under uncertain and imprecise data, which is hard to address.The proposed technique is exceptionally appropriate for the scenario when the data is complex, vague, and uncertain.Especially, when the existing data is depending on the bipolar information by decisionmakers.A few benefits of the proposed technique are listed below: (i) The proposed approach considers positive and negative aspects of each individual alternative in the form of a BSS.This hybrid model is more generalized and appropriate to deal with aggressive DM problems.(ii) This technique is additionally ideal on the grounds that in this strategy the decision-makers are liberated from any external conditions and prerequisites.
(iii) Our proposed technique is effective in solving MCGDM problems when the weights information of criteria is completely unknown.The proposed MCGDM technique is more effective for discrete data problems.(v) The method not only considers the opinions of key decision-makers but also incorporates the past experiences by DB-BSR-approximations in actual scenarios.
Hence it is a more comprehensive method for a better interpretation of available information and thus making decisions using artificial intelligence.(vi) The proposed MCGDM approach is easy to understand and can be applied to DM problems in real life.

C. DISADVANTAGES OF THE PROPOSED TECHNIQUE
Some minor flaws are there in the proposed technique, inclusive of its complicated structure, the large data in the form of bipolar information.Such large data is hard to deal with, due to massive calculations, which are not so natural to perform.However, one could create a MATLAB programming code to make these complicated calculations simpler.

D. COMPARISON WITH SOME EXISTING TECHNIQUES
There are numerous techniques in the literature helpful for tackling MCGDM problems.All these techniques of MCGDM have their own merits and demerits.The capability of every technique relies on the problem under consideration.
In this subsection, we make a set-based comparison of the proposed MCGDM technique with some current MCGDM techniques in the fuzzy and bipolar fuzzy environments and see the significance of the proposed MCGDM strategy.We talk about comparative analysis of proposed strategy with dominance-based soft rough sets [63], fuzzy soft set [3], dominance-based rough fuzzy set [15], intuitionistic fuzzy rough sets [65], picture fuzzy set [10], generalized hesitant fuzzy rough sets [62] and generalized intuitionistic fuzzy soft sets [36].All these techniques have their own value in the literature.If we compare all these techniques with our proposed strategy, we investigate the following points.
(i) The previously-mentioned techniques cannot catch bipolarity in DM which is a fundamental aspect of human thinking and behavior.(ii) Besides, these techniques do not ensure harmony in the opinions of decision-makers.(iii) If we compare our proposed model with the methods presented in [10], [36], we have seen that these methods requires a weight vector that represents the weight of each alternative, but in our proposed model there is no need for a weight vector.(iv) It is well known that the models presented in [10], [36], [65] can manage some DM problems to describe the idea of decision-makers through a crisp number.However, because of the uncertainty of the objective world and the complexity of the decision-making problems, they fail to handle some GDM problems.For example, several experts argue the membership degree of an element to a set and cannot compromise each other.One wants to assign 0.3, but the other tends to assign 0.7.In this case, DB-BSRSs can be a very good solution to this problem.(v) If we compare our proposed result with the method presented in [9], we have seen that in this method the optimal alternative is obtained just by using the tabular form of BSSs, while in our proposed model the optimal alternative is obtained by using the DB-BSRapproximations. (vi) In [39], Mahmood uses the tabular form of the T-BSSs and presented two types of algorithms with the help of score function to obtain the optimal alternative.But in our proposed model there is no need for the score function.(vii) If we apply the recent approaches proposed in [35] and [60] to our Example 5, we get the following ranking among the alternatives (shown in Table 1) and the corresponding pictorial depiction is given in Figure 3.

IX. CONCLUSION AND FUTURE DIRECTIONS
The RS theory is arising as an incredible theory and has different applications in numerous fields.On the other hand, the BSSs are the appropriate mathematical model to deal with the uncertainty as well as the bipolarity of the data.In this study, we have initiated a novel technique of roughness of BSSs known as "dominance-based bipolar soft rough sets (DB-BSRSs)".For a given BSS, we first define some dominancebased bipolar soft operators.Using these operators, DB-BSR-approximations have been defined.Some important structural properties of DB-BSR-approximations have also been studied in detail with examples.Additionally, some significant measures related to DB-BSRSs like the accuracy measure, the measure of precision, and the measure of quality are also provided.In the meantime, we provide a general methodology of the MCGDM approach based on the DB-BSRSs.An algorithm of decision-making is also proposed which has the following two main advantages.Firstly, it manipulates the bipolarity of the data, endowed with uncertainty.Secondly, this algorithm accommodates the opinions of any (finite) number of decision-makers about any (finite) number of alternatives.Furthermore, the legitimacy of this methodology is illustrated by a practical application.Lastly, a comparison analysis of the proposed model is performed.
In the future, on the basis of the characterized ideas and operations in this paper, researchers may additionally look at the algebraic structures of DB-BSRSs.Modeling the supported natural phenomenon is our subsequent objective.Another perspective route is to have a look at the topological axioms and similarity measures of DB-BSRSs in order to seek a strong basis of the research studies and enhancement of working techniques.Also, the idea of DB-BSRSs could also be extended to dominance-based bipolar fuzzy soft rough set and successful decision-making strategies can be developed.The notions of the DB-BSRSs can also be extended to multi-granulation DB-BSRSs.For more complicated decision-making problems, the technique requires further investigation.The different types of correlation coefficients can also be studied in the framework of DB-BSRSs.Some important aggregation operators such as Hammy mean operators, weighted aggregation operators, arithmetic and harmonic aggregation operators, power aggregation operators, etc. can also be developed in the followup work.Also, the technique is more practical for discrete data problems.For continuous scaled data sets, this approach requires further research.Moreover, we will concentrate on the implementation of the proposed strategy in tackling a more extensive scope of selection problems, like TOPSIS, VIKOR, ELECTRE, AHP, and PROMETHEE.We may also be searching on the possible fuzzification of the proposed strategy to obtain more accuracy in outcomes and applying these procedures to certifiable problems with large data sets.In this way, we can acquire and demonstrate the usefulness of our proposed model.
e) : e ∈ A, f (e) ∈ 2 U .(5) Definition 3: [41] By a NOT set of parameters of A, we mean a set of the form A = {¬e : e ∈ A} in which ¬e = not e for e ∈ A. Definition 4: [58] Let f and g are two mappings, given by f : A −→ 2 U and g : A −→ 2 U such that f (e) ∩ g(¬e) = ∅ for all e ∈ A. Then, a triplet (f, g : A) is called a BSS over the universe U.
Gul et al.: A Novel Approach towards Roughness of Bipolar Soft Sets and Their Applications in MCGDM This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/This article has been accepted for publication in a future issue of this journal, but has not been fully edited.Content may change prior to final publication.Citation information: DOI 10.1109/ACCESS.2021.3116097,IEEE Access Gul et al.: A Novel Approach towards Roughness of Bipolar Soft Sets and Their Applications in MCGDM This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/This article has been accepted for publication in a future issue of this journal, but has not been fully edited.Content may change prior to final publication.Citation information: DOI 10.1109/ACCESS.2021.3116097,IEEE Access Gul et al.: A Novel Approach towards Roughness of Bipolar Soft Sets and Their Applications in MCGDM AM(X + ) = A f + X , A g +