# A Generalized Decision-Making Technique Based on Bipolar-Valued Multivague Soft Sets

Authors: Hanan H. Sakr; Abdisalam Hassan Muse; Ramy Aldallal
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The decision-making technique, launched by Roy and Maji, is considered an effective method to overcome uncertainty and fuzziness in decision-making problems, though, adapting it to reflect the problem parameters’ vagueness, as well as multibipolarity, is very difficult. So, in this article, the bipolarity is interpolated into the multivague soft set of ordern. This gives a new more generalized, flexible, and applicable extension than the fuzzy soft model, or any previous hybrid model, which is the bipolar-valued multivague soft model of dimension n. Moreover, types of bipolar-valued multivague soft sets of dimension n, as well as some new associated concepts and operations, are investigated with examples. Furthermore, properties of bipolar-valued multivague soft sets of dimension n including absorption, commutative, associative, and distributive properties, as well as De Morgan’s laws, are provided in detail. Finally, a bipolar-valued multivague soft set-designed decision-making algorithm, as well as a real-life example, are discussed generalizing the Roy and Maji method.


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## Body

## 1. Introduction, Motivation, and Related Works

Nowadays, making decisions has risen to be one of the most significant areas of human endeavor, because of its strong connection to efficiency and sustainability. Making wise and effective judgments is how successful individuals accomplish their personal and professional aims. Individual perspectives, attitudes, and beliefs, along with ideas, are regularly utilized to guide decision-making. While a person can make judgments based on a variety of concepts, he should be very careful to choose one that is effective and adds to great achievement. The difficulty of making decisions in an uncertain environment has attracted attention recently. The most regular sources of complication while trying to make judgments in the actual world are uncertainty and confusion. Uncertain data is frequent and implicit in many basic applications, particularly where decision-making is needed. This ambiguous information is indeed a consequence of missing or delayed retrieval of information, information unpredictability, calculation tool restrictions, etc. One can find several kinds of research and uses of various specialized mathematical methods in the literature.In 1937, Black [1] introduced the vagueness of a term as it is shown by producing “borderline cases” to model uncertain data in many important applications. In 1965, Zadeh [2] introduced an extension of ordinary (crisp) sets for describing uncertainty and dealing with it, namely the theory of the fuzzy sets. Just as an ordinary set on the initial universal set X is determined by its membership function from X to 0,1, in fuzzy set theory, an element’s membership degree is determined by its membership percentage (the characteristic function from the domain X to the interval 0,1). The fuzzy set is considered a useful mathematical approach for working with ambiguity, but it has the following limitation. The particular number (membership percentage) includes advice for and against object belonging without indicating how much more of each there really is, i.e., this single number gives us very little information about its accuracy. In 1986, Atanassov [3] gave the notion “intuitionistic fuzzy set,” which is a generalization of the word “fuzzy set,” along with an example.In 1993, Gau and Buehrer [4] developed a new extension of the set theory called vague set theory, based on Black’s idea of vagueness in 1937 [1], to overcome the difficulties posed by fuzzy set theory. Instead of just a single value, they gave each object a membership degree, which is a subinterval of 0,1. The soft set idea was first suggested by Molodtsov [5], in 1999, when it was claimed that one of the causes of the aforementioned problems might be the inadequate parametrization tools of the aforementioned theories. The softness concept or the soft set concept is a mathematical tool, free from those above difficulties, for dealing easier with uncertainties. In 2002, Maji et al. ([6, 7]) looked into and explored the Molodtsov-initiated soft set theory. They examined a variety of concepts related to soft sets, created a comprehensive theoretical overview of soft set theory, and then used it to solve a problem involving decision-making.Numerous attempts were made to create the soft version of all known crisp concepts and many more results in the soft set environment, as the following: Ali et al. [8] introduced many new definitions and concepts in the soft set theory. In addition, Sezgin and Atagu¨n [9] established several novel theoretical operations in the soft set theory. In addition, Majumdar and Samanta [10] established the soft mappings concept, and also, Choudhure et al. worked on the soft relation concept and consequently used it for solving various decision-making issues. Moreover, Aktas and Cag˘man [11] extended the softness concept to group theory and defined the soft group concept. Furthermore, Feng et al. [12] applied and extended the soft set concept to semirings, Acar [13] initiated the soft rings, and Jun et al. extended the softness concept to BCK/BCI-algebras ([14–16]). Also, Sezgin and Atagu¨n [17] introduced the normalistic soft groups concept, Zhan et al. [18] defined the concept of soft ideal of BL-algebras and Kazanci et al. [19] applied the softness concept to BCH-algebras. In addition, Sezgin et al. [20] worked on soft near-rings and Cag˘man et al. [21] defined the group soft union and group soft intersection of a group (for more details, one can refer also to [22]). In addition, many other researchers introduced new extended concepts based on soft sets in recent years, providing examples and studying their properties, such as soft point [23], soft real numbers ([24, 25]), soft complex numbers [26], soft metric spaces [27], soft normed spaces [28], soft inner product spaces [29], and soft Hilbert spaces [30]. After that, Cag˘man et al. [31] established the soft matrix theory to facilitate dealing with soft sets and organized a model of soft decision-making.However, it appeared that it is very important to consider the bipolarity of knowledge in decision-making problems because it is a very useful factor when developing a mathematical framework for most situations in decision-making problems. Within the bipolarity idea, it is believed that bipolar subjectivity beliefs include a variety of individual decision-making processes. Examples of several decision analysis directions involve happiness and sadness, positive effects and negative effects, sweet and salty, and hunger and satisfaction. For instance, an inefficient medication might not have any negative effects. So, in 2000, Lee [32] introduced the bipolar-valued fuzzy set concept as a novel extension or generalization of the fuzzy set concept. In this instance, the element’s membership interval range value is extended from 0,1 to −1,1. For more information about appearing bipolarity in fuzzy sets and soft sets, one can refer to [33–36].After that, Maji et al. [37] defined the fuzzy soft set theory, in 2001, by entering the fuzzy sets ideas in the softness concept. In addition, Roy and Maji [38] proposed a decision-making technique to determine the optimal (best) choice of an object to buy among many objects based on fuzzy soft set. Based on fuzzy soft set concept, Yang et al. [39] introduced a fuzzy soft set matrix representation and Cag˘man et al. [40] studied the fuzzy soft matrices, several algebraic operations and made a theoretical study in fuzzy soft settings. Basu et al. [41] and Kumar and Kaur [42] studied fuzzy soft matrices and established some new notions and operations on them. Later on, Faried et al. developed a fuzzy soft version of functional analysis by introducing a series of results as follows: FS inner product space [43], FS Hilbert space [44], FS linear operators [45], and FS spectral theory ([46–50]).Maji et al. [51] introduced intuitionistic fuzzy soft sets, in 2004, as a new extension of soft sets. On intuitionistic fuzzy soft sets, additional operations were also provided, and several of their characteristics were identified. In addition, a simple example was given to demonstrate how such a mathematical tool can be used. Then, Chetia et al. [52] initiated the intuitionistic fuzzy soft matrices concepts to represent the intuitionistic fuzzy soft sets easily. In order to perform theoretical study in the intuitionistic fuzzy soft set environment and generate some findings, they, additionally, specified their higher functional operations.In 2010, Xu et al. [53] proposed the vague soft set concept and presented its general properties. In reality, the vague soft set theory makes element world characteristics more precise, useful, and genuine, trying to make it, in particular in some instances, a flexible tool. Recently, Wang [54] introduced many results on vague soft set theory and studied its associated properties and potential applications. Moreover, Alhazaymeh and Hassan [55] initiated the vague soft set relations and functions concepts. Also, Varol et al. [56] defined vague soft groups and Yin et al. [57] studied vague soft hemirings. Furthermore, Selvachandran and Salleh ([58, 59]) introduced rings and ideals in the vague soft sets settings and established some algebraic hyper structures of the vague soft set theory related to hyperrings and hyperideals. At present, Inthumathi and Pavithra [60], as well as Faried et al. [61], introduced the vague soft matrix concept, examined its general properties, and investigated its unique uses. Furthermore, Abdullah et al. [62] investigated the bipolar fuzzy soft set concept, in 2014, and introduced their basic characteristics. Basic operations of bipolar fuzzy soft sets were also established. Furthermore, they used the bipolar fuzzy soft set to overcome decision-making difficulties. Contrary to appearances, there is a distinction between the intuitionistic fuzzy soft sets and the bipolar fuzzy soft sets.After that, Sebastian and Ramakrishnan [63] initiated the multifuzzy set concept with multimembership function, which is an ordered sequence of membership functions stated above. In addition, Zhou et al. [64] introduced the multivague set concept and the multivague soft set concept with their properties. In fact, multifuzzy set and multivague set can overcome some specific problems that are very difficult to be represented by other fuzzy set extensions or vague set extensions. For instance, the color of pixels in a two-dimensional image cannot be described by a characteristic function of a regular fuzzy set, but it can be described by a three-dimensional characteristic function, whose components are the characteristic functions representing the primary known colors: red, green, and blue. Consequently, one can roughly represent any image by a collection of ordered pixels with a multicharacteristic function.Furthermore, Santhi and Shyamala [65] defined the bipolar-valued multifuzzy set and made some notes on bipolar-valued multifuzzy subgroups of a group. Moreover, Yang et al. [66] proposed the bipolar-valued multifuzzy soft set concept and introduced some decision-making applications using it. Finally, Sakr et al. [67] used the soft decision-making technique in bipolarity problems containing vague information by proposing bipolar vague soft sets. They introduced a real-life application to choose the optimal object based on bipolar vague soft information using decision-making techniques.Despite this progress, there is still another limitation: how we can deal with a multibipolarity problem involving too vague information by applying soft decision-making techniques. From here, the need to introduce another novel extension arose. The chief motivation of this article is to overcome the above-stated limitations of the previously used tools in decision-making problems. So, in this paper, the bipolar-valued multivague soft set of dimensionn, its types, operations, properties, and applications are introduced with an illustrative example of each. A bipolar-valued multivague soft set-based decision-making technique is designed, which extends Roy and Maji’s strategy and helps us to make more successful conclusions when picking the correct choice.The rest of this research is constructed as follows: Section2 is set up to give the main preliminary needed definitions. After that, in Section 3, the bipolar-valued multivague soft set is defined with its types and some new related concepts and operations. In addition, Section 3 gives general properties, absorption properties, commutative properties, associative properties, distributive laws, and De Morgan’s laws. Furthermore, the aim of Section 4 is to establish a generalized algorithm for Roy and Maji method based on the bipolar-valued multivague soft sets to determine the optimal alternative among others given in a decision-making problem. Finally, Section 5 gives open questions for further investigations and concluding remarks.


## 2. Preliminary Definitions

The purpose of this section is to introduce the basic preliminary definitions needed in the following investigations.Definition 1. (see [2]).
LetΠ be a universal set (space of points or objects). A fuzzy set (class) Ϝ over Π is a set described by a function ηϜ:Π⟶0,1. ηϜ is said to be the characteristic, indicator, or membership function of the fuzzy set Ϝ, and ηϜu is said to be the membership grade value of π∈Π in Ϝ. A fuzzy set Ϝ over a universal set Π can be written as Ϝ=ηϜπ/π:π∈Π,ηϜπ∈0,1 or Ϝ=π,ηϜπ:π∈Π,ηϜπ∈0,1.Definition 2. (see [32]).
A bipolar-valued fuzzy setA on Π can be represented as the following formula: A=π,ηA+π,ηA−π:π∈Π, taking into account that ηA+:Π⟶0,1 expresses the positive characteristic function describing the satisfaction amount of π to the feature corresponding to A, and ηA−:Π⟶−1,0 expresses the negative characteristic function describing the satisfaction amount of π to the counter-feature of A.Definition 3. (see [63]).
A multifuzzy setN of dimension n over Π is represented by a set of ordered sequences as follows: N=π,η1Nπ,η2Nπ,⋯,ηnNπ:π∈Π, where for i=1,2,⋯,n, we have ηiN:Π⟶0,1 representing the membership functions. The function ηN=η1N,η2N,⋯,ηnN is said to be a fuzzy multimembership function of a multifuzzy set N of dimension n.Definition 4. (see [65]).
A bipolar-valued multifuzzy setB of dimension n over Π is given by the following structure: B=π,η1B+π,η2B+π,⋯,ηnB+π,η1B−π,η2B−π,⋯,ηnB−π:π∈Π, where for i=1,2,⋯,n, we have ηiB+:Π⟶0,1 representing the positive membership degrees denoting the satisfaction degrees of π to some properties corresponding to B and ηiB−:Π⟶−1,0 representing the negative membership degrees denoting the satisfaction degrees of π to some implicit counter-properties of B.Definition 5. (see [4]).
Given the universal setΠ=π1,π2,⋯,πn, a vague set V over Π is a set determined by a truth membership function τV and a false membership function ηV. The exact grade of membership of π∈Π(μVπ) belongs to an interval τVπ,1−ηVπ⊆0,1, i.e., μVπ may be unknown, but it is bounded by τVπ≤μVπ≤1−ηVπ, where τVπ+ηVπ≤1 and τV,ηV:Π⟶0,1. V can be written as V=τVπ,1−ηVπ/π:π∈Π,τVπ,ηVπ∈0,1, or V=π,τVπ,1−ηVπ:π∈Π,τVπ,ηVπ∈0,1.Definition 6. (see [68]).
A bipolar-valued vague setM over the universal set Π is defined by M=π,τM+π,1−ηM+π,−1−ηM−π,τM−π:π∈Π, where τM+,ηM+:Π⟶0,1 are the positive truth and false membership functions denoting the satisfaction degree of an element π to the property corresponding to M, such that τM++ηM+≤1 and τM−,ηM−:Π⟶−1,0 are the negative truth and false membership functions denoting the satisfaction degree of π to some implicit counter-property of M, such that τM−+ηM−≥−1, i.e., the intervals τM+π,1−ηM+π and −1−ηM−π,τM−π denote the satisfaction region of π to the property corresponding to M and to some implicit counter-property of M, respectively.Definition 7. (see [64]).
A multivague setD of dimension n over Π is represented by a set of ordered sequences as follows: D=π,τ1Dπ,1−η1Dπ,τ2Dπ,1−η2Dπ,⋯,τnDπ,1−ηnDπ:π∈Π, where for i=1,2,⋯,n, we have τiD and ηiD:Π⟶0,1, the truth and false membership functions, respectively, such that τiD+ηiD≤1, i.e., the intervals τiDπ,1−ηiDπ denote the satisfaction intervals of π to some properties corresponding to D. The function μD=τD,1−ηD=τ1D,1−η1D,τ2D,1−η2D,⋯,τnD,1−ηnD is said to be a vague multimembership function of a multivague set D of dimension n.Definition 8. (see [5]).
Suppose thatΠ is a universal set, Y is a set of attributes (or parameters) and Λ⊆Y. The power set of Π is given by PΠ=2Π. A pair Γ,Λ or ΓΛ is said to be a soft set over Π, where Γ is a mapping characterized by Γ:Λ⟶PΠ. Also, ΓΛ may be written as a set of ordered pairs ΓΛ=λ,ΓΛλ:λ∈Λ,ΓΛλ∈PΠ. Λ is called the support of ΓΛ, and we have ΓΛλ≠ϕ for all λ∈Λ and ΓΛλ=ϕ for all λ∉Λ. In other words, a soft set Γ,Λ over Π is a parameterized family of subsets of the set Π.Definition 9. (see [64]).
A multivague soft set of dimensionn over the universal set Π is a parameterized family of multivague sets of the universal set Π, stated above in Definition 7.


## 3. Operations and Properties on Bipolar-Valued Multivague Soft Sets

The purpose of this section is to introduce the definition of the bipolar-valued multivague soft sets of dimensionn and many related new concepts and operations on them with illustrative examples on each item. In addition, some important properties of bipolar-valued multivague soft sets of dimension n are introduced with proofs such as absorption properties, commutative properties, associative properties, distributive laws, and De Morgan’s laws.Definition 10.
A bipolar-valued multivague setV of dimension n over the universal set Π is defined by V=π,τ1V+π,1−η1V+π,τ2V+π,1−η2V+π,⋯,τnV+π,1−ηnV+π,−1−η1V−π,τ1V−π,−1−η2V−π,τ2V−π,⋯,−1−ηnV−π,τnV−π:π∈Π, where for i=1,2,⋯,n, we have τiV+,ηiV+:Π⟶0,1, the positive truth and false membership functions denoting the satisfaction degrees of an element π to some properties corresponding to V, such that τiV++ηiV+≤1 and τiV−,ηiV−:Π⟶−1,0 are the negative truth and false membership functions denoting the satisfaction degrees of π to some implicit counter-properties of V, such that τiV−+ηiV−≥−1, i.e., the intervals τiV+π,1−ηiV+π and −1−ηiV−π,τiV−π denote the satisfaction regions of π to some properties corresponding to V and to some implicit counter-properties of V, respectively.Definition 11.
Assume thatΠ is a universal set, Y is a parameter set and Λ⊆Y. Then, a pair ℶ,Λ given by ℶΛ=λ,ℶΛλ:λ∈Λ,ℶΛλ∈BVnΠ is said to be a bipolar-valued multivague soft set of dimension n over the universal set Π, taking into account that ℶ is a mapping defined as ℶ:Λ⟶BVnΠ, where BVnΠ is the power set of bipolar-valued multivague sets of dimension n on Π (i.e., the family of all bipolar-valued multivague subsets of dimension n of Π) and the bipolar-valued multivague subset of dimension n of Π is as stated above in Definition 10.Definition 12.
Any bipolar-valued multivague soft setℶ,Λ of dimension n on the universal set Π is said to be complete (or absolute), which stands for CΛn, if for all λ∈Λ, we have ℶΛλ=BVnΠ. That is to say that, for i=1,2,⋯,n, we have τiℶΛλ+π=1, 1−ηiℶΛλ+π=1 (i.e., ηiℶΛλ+π=0), −1−ηiℶΛλ−π=−1 (i.e., ηiℶΛλ−π=0), and τiℶΛλ−π=−1, for all λ∈Λ and for all π∈Π. According to those assumptions, we have CΛn=λ,π,1,1,⋯n−times,1,1,−1,−1,⋯n−times,−1,−1:λ∈Λ,π∈Π.Definition 13.
Any bipolar-valued multivague soft setℶ,Λ of dimension n on the universal set Π is said to be null (or empty), which stands for ϕΛn, if for all λ∈Λ, we have ℶΛλ=ϕ. That is to say that, for i=1,2,⋯,n, we have τiℶΛλ+π=0, 1−ηiℶΛλ+π=0 (i.e., ηiℶΛλ+π=1), −1−ηiℶΛλ−π=0 (i.e., ηiℶΛλ−π=−1), and τiℶΛλ−π=0, for all λ∈Λ and for all π∈Π. According to those assumptions, we have ϕΛn=λ,π,0,0,⋯2n−times,0,0:λ∈Λ,π∈Π.Definition 14.
The complement of a bipolar-valued multivague soft setℶ,Λ of dimension n is defined by ℶ,Λc=ℶc,Λ, where ℶc:Λ⟶BVnΠ is a mapping given, for i=1,2,⋯,n, by the following: τiℶcλ+π=ηiℶλ+π, ηiℶcλ+π=τiℶλ+π, τiℶcλ−π=ηiℶλ−π, and ηiℶcλ−π=τiℶλ−π, for all λ∈Λ and for all π∈Π. Then, we have
(1)ℶΛc=λ,π,η1ℶΛλ+π,1−τ1ℶΛλ+π,η2ℶΛλ+π,1−τ2ℶΛλ+π,⋯,ηnℶΛλ+π,1−τnℶΛλ+π,(2)−1−τ1ℶΛλ−π,η1ℶΛλ−π,−1−τ2ℶΛλ−π,η2ℶΛλ−π,⋯,−1−τnℶΛλ−π,ηnℶΛλ−π:λ∈Λ,π∈Π.Definition 15.
Assume thatΓ,Λ and Ψ,Θ are two bipolar-valued multivague soft sets of dimension n on a universal set Π. Then, Γ,Λ is called a bipolar-valued multivague soft subset of Ψ,Θ if Λ⊆Θ, and Γλ⊆Ψλ for all λ∈Λ, that is to say that, for i=1,2,⋯,n, we have τiΓλ+π≤τiΨλ+π, 1−ηiΓλ+π≤1−ηiΨλ+π, −1−ηiΓλ−π≥−1−ηiΨλ−π, and τiΓλ−π≥τiΨλ−π, i.e., for i=1,2,⋯,n, we have τiΓλ+π≤τiΨλ+π, ηiΓλ+π≥ηiΨλ+π, ηiΓλ−π≤ηiΨλ−π, and τiΓλ−π≥τiΨλ−π, for all λ∈Λ and for all π∈Π. One can write Γ,Λ⊆~Ψ,Θ. In this case, Ψ,Θ is called a bipolar-valued multivague soft superset of Γ,Λ, denoted by Ψ,Θ⊇~Γ,Λ.Definition 16.
Two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension n on a common universal set Π are called bipolar-valued multivague soft equal if they are bipolar-valued multivague soft subsets of each other, i.e., Γ,Λ⊆~Ψ,Θ and Ψ,Θ⊆~Γ,Λ.Definition 17.
The union of two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension n on a common universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶ,Δ=Γ,Λ∪~Ψ,Θ, where Δ=Λ∪Θ and for all δ∈Δ:
(3)ℶ,Δ=δ,π,τ1Γδ+π,1−η1Γδ+π,τ2Γδ+π,1−η2Γδ+π,⋯,τnΓδ+π,1−ηnΓδ+π,−1−η1Γδ−π,τ1Γδ−π,−1−η2Γδ−π,τ2Γδ−π,⋯,−1−ηnΓδ−π,τnΓδ−π,π∈Π,ifδ∈Λ−Θ,(4)ℶ,Δ=δ,π,τ1Ψδ+π,1−η1Ψδ+π,τ2Ψδ+π,1−η2Ψδ+π,⋯,τnΨδ+π,1−ηnΨδ+π,−1−η1Ψδ−π,τ1Ψδ−π,−1−η2Ψδ−π,τ2Ψδ−π,⋯,−1−ηnΨδ−π,τnΨδ−π,π∈Π,ifδ∈Θ−Λ,(5)ℶ,Δ=δ,π,maxτ1Γδ+π,τ1Ψδ+π,max1−η1Γδ+π,1−η1Ψδ+πmaxτ2Γδ+π,τ2Ψδ+π,max1−η2Γδ+π,1−η2Ψδ+π,⋯,maxτnΓδ+π,τnΨδ+π,max1−ηnΓδ+π,1−ηnΨδ+π,min−1−η1Γδ−π,−1−η1Ψδ−π,minτ1Γδ−π,τ1Ψδ−π,min−1−η2Γδ−π,−1−η2Ψδ−π,minτ2Γδ−π,τ2Ψδ−π,⋯,min−1−ηnΓδ−π,−1−ηnΨδ−π,minτnΓδ−π,τnΨδ−π,π∈Π,ifδ∈Λ∩Θ.Definition 18.
The union of a familyΓi,Λi:i∈I of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶΔ=ℶ,Δ=∪~Γi,Λi, where Δ=∪Λi, for all i∈I defined as follows, for all δ∈Δ:
(6)ℶΔδ=Γiδifδ∈Λi−∪j≠iΛj,foralli∈I,∪i∈IΓiδ,ifδ∈∩i∈IΛi.Definition 19.
The restricted union of two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension n on a common universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶ,Δ=Γ,Λ∪~RΨ,Θ, where Δ=Λ∩Θ≠ϕ and for all δ∈Δ:
(7)ℶ,Δ=δ,π,maxτ1Γδ+π,τ1Ψδ+π,max1−η1Γδ+π,1−η1Ψδ+π,maxτ2Γδ+π,τ2Ψδ+π,max1−η2Γδ+π,1−η2Ψδ+π,⋯,maxτnΓδ+π,τnΨδ+π,max1−ηnΓδ+π,1−ηnΨδ+π,min−1−η1Γδ−π,−1−η1Ψδ−π,minτ1Γδ−π,τ1Ψδ−π,min−1−η2Γδ−π,−1−η2Ψδ−π,minτ2Γδ−π,τ2Ψδ−π,⋯,min−1−ηnΓδ−π,−1−ηnΨδ−π,minτnΓδ−π,τnΨδ−π:δ∈Λ∩Θ,π∈Π.Definition 20.
The restricted union of a familyΓi,Λi:i∈I of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶΔ=ℶ,Δ=∪~RΓi,Λi, where Δ=∩Λi, for all i∈I defined by ℶΔδ=∪Γiδ, δ∈Δ=∩Λi.Definition 21.
The intersection of two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension n over a common universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶ,Δ=Γ,Λ∩~Ψ,Θ, where Δ=Λ∪Θ and for all δ∈Δ:
(8)ℶ,Δ=δ,π,τ1Γδ+π,1−η1Γδ+π,τ2Γδ+π,1−η2Γδ+π,⋯,τnΓδ+π,1−ηnΓδ+π,−1−η1Γδ−π,τ1Γδ−π,−1−η2Γδ−π,τ2Γδ−π,⋯,−1−ηnΓδ−π,τnΓδ−π,π∈Π,ifδ∈Λ−Θ,(9)ℶ,Δ=δ,π,τ1Ψδ+π,1−η1Ψδ+π,τ2Ψδ+π,1−η2Ψδ+π,⋯,τnΨδ+π,1−ηnΨδ+π,−1−η1Ψδ−π,τ1Ψδ−π,−1−η2Ψδ−π,τ2Ψδ−π,⋯,−1−ηnΨδ−π,τnΨδ−π,π∈Π,ifδ∈Θ−Λ,(10)ℶ,Δ=δ,π,minτ1Γδ+π,τ1Ψδ+π,min1−η1Γδ+π,1−η1Ψδ+π,minτ2Γδ+π,τ2Ψδ+π,min1−η2Γδ+π,1−η2Ψδ+π,⋯,minτnΓδ+π,τnΨδ+π,min1−ηnΓδ+π,1−ηnΨδ+π,max−1−η1Γδ−π,−1−η1Ψδ−π,maxτ1Γδ−π,τ1Ψδ−π,max−1−η2Γδ−π,−1−η2Ψδ−π,maxτ2Γδ−π,τ2Ψδ−π,⋯,max−1−ηnΓδ−π,−1−ηnΨδ−π,maxτnΓδ−π,τnΨδ−π,π∈Π,ifδ∈Λ∩Θ.Definition 22.
The intersection of a familyΓi,Λi:i∈I of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶΔ=ℶ,Δ=∩~Γi,Λi, where Δ=∪Λi, for all i∈I defined as follows, for all δ∈Δ:
(11)ℶΔδ=Γiδ,ifδ∈Λi−⋃j≠iΛj,foralli∈I,⋂i∈IΓiδ,ifδ∈⋂i∈IΛi.Definition 23.
The restricted intersection of two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension n on a common universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶ,Δ=Γ,Λ∩~RΨ,Θ, where Δ=Λ∩Θ≠ϕ and for all δ∈Δ:
(12)ℶ,Δ=δ,π,minτ1Γδ+π,τ1Ψδ+π,min1−η1Γδ+π,1−η1Ψδ+π,minτ2Γδ+π,τ2Ψδ+π,min1−η2Γδ+π,1−η2Ψδ+π,⋯,minτnΓδ+π,τnΨδ+π,min1−ηnΓδ+π,1−ηnΨδ+π,max−1−η1Γδ−π,−1−η1Ψδ−π,maxτ1Γδ−π,τ1Ψδ−π,max−1−η2Γδ−π,−1−η2Ψδ−π,maxτ2Γδ−π,τ2Ψδ−π,⋯,max−1−ηnΓδ−π,−1−ηnΨδ−π,maxτnΓδ−π,τnΨδ−π:δ∈Λ∩Θ,π∈Π.Definition 24.
The restricted intersection of a familyΓi,Λi:i∈I of bipolar-valued multivague soft sets of dimension n over a universal set Π is a bipolar-valued multivague soft set ℶ,Δ of dimension n, written as ℶΔ=ℶ,Δ=∩~RΓi,Λi, where Δ=∩Λi, for all i∈I defined by ℶΔδ=∩Γiδ, δ∈Δ=∩Λi.Example 1.
Suppose that the bipolar-valued multivague soft setℶ,Y of dimension (order) 3 describes some features of five employees nominated for a position in a certain organization, denoted by π1,π2,π3,π4,π5 in the universal set Π, i.e., Π=π1,π2,π3,π4,π5. Let the two major sets of attributes be Y=λ1,λ2,λ3, λii=1,2,3 and its opposite (counter) set −Y=−λ1,−λ2,−λ3, −λii=1,2,3 stand for the properties and the counter-properties, respectively. These attributes are represented in the following main parameters: mental abilities, leadership or social skills, and professional or organizational talents, respectively. Mental abilities and their counter-abilities are as follows: “intelligence” and “stupidity”; “memory” and “forgetfulness”; and “problem-solving” and “puzzlement.” Leadership skills and their counter-skills are as follows: “patience” and “boredom”; “fairness” and “injustice”; and “cooperation” and “competition.” Professional talents and their counter-talents are as follows: “flexibility” and “strictness”; “precision” and “negligence”; and “experience” and “rawness.” After doing several oral and written tests for them as well as subjecting them to some personal interviews and tests for intelligence and psychological aspects by a selection committee, the following bipolar-valued multivague soft set of dimension 3, namely ℶ,Y is obtained:
ℶ,Y=λ1,π1,0.8,0.9,0.2,0.3,0.4,0.5,−0.2,−0.1,−0.8,−0.6,−0.3,−0.2,π2,0.5,0.7,0.4,0.6,0.9,1,−0.3,−0.1,−0.5,−0.4,−1,−0.8,π3,0.4,0.6,0.7,0.8,0.2,0.3,−0.5,−0.4,−0.8,−0.7,−0.9,−0.6,π4,0.9,1,0.1,0.3,0.5,0.6,−0.2,−0.1,−0.6,−0.5,−0.8,−0.7,π5,0.5,0.6,0,0.1,0.6,0.9,−1,−0.8,−0.5,−0.4,−1,−0.9,λ2,π1,0.1,0.2,0.7,0.8,0.5,0.6,−0.3,−0.2,−0.9,−0.8,−0.2,−0.1,π2,0.8,0.9,0.6,0.7,0.5,0.6,−0.4,−0.3,−0.8,−0.7,−0.9,−0.8,π3,0.3,0.4,0.9,1,0.2,0.3,−1,−0.9,−0.2,−0.1,−0.5,−0.4,π4,0.8,0.9,0.2,0.4,0.7,0.8,−0.6,−0.5,−0.7,−0.6,−0.2,−0.1,π5,0.7,0.8,0.5,0.6,0.8,0.9,−0.9,−0.8,−0.5,−0.4,−0.7,−0.6,λ3,π1,0.2,0.3,0.8,0.9,0.8,1,−0.8,−0.7,−0.9,−0.8,−0.6,−0.5,π2,0.8,1,0.5,0.7,0.8,0.9,−0.2,−0.1,−0.4,−0.3,−0.9,−0.7,π3,0.6,0.7,0.2,0.4,0.9,1,−1,−0.9,−0.2,−0.1,−0.4,−0.3,π4,0.4,0.5,0.6,0.8,0.3,0.4,−0.7,−0.6,−0.3,−0.2,−0.8,−0.7,π5,0.3,0.4,0.5,0.6,0.8,0.9,−1,−0.9,−0.6,−0.5,−0.3,−0.2Thus, the complement of ℶ,Y, which is again of dimension 3, is defined as following:
(13)ℶc,Y=λ1,π1,0.1,0.2,0.7,0.8,0.5,0.6,−0.9,−0.8,−0.4,−0.2,−0.8,−0.7,π2,0.3,0.5,0.4,0.6,0,0.1,−0.9,−0.7,−0.6,−0.5,−0.2,0,π3,0.4,0.6,0.2,0.3,0.7,0.8,−0.6,−0.5,−0.3,−0.2,−0.4,−0.1,π4,0,0.1,0.7,0.9,0.4,0.5,−0.9,−0.8,−0.5,−0.4,−0.3,−0.2,π5,0.4,0.5,0.9,1,0.1,0.4,−0.2,0,−0.6,−0.5,−0.1,0,λ2,π1,0.8,0.9,0.2,0.3,0.4,0.5,−0.8,−0.7,−0.2,−0.1,−0.9,−0.8,π2,0.1,0.2,0.3,0.4,0.4,0.5,−0.7,−0.6,−0.3,−0.2,−0.2,−0.1,π3,0.6,0.7,0,0.1,0.7,0.8,−0.1,0,−0.9,−0.8,−0.6,−0.5,π4,0.1,0.2,0.6,0.8,0.2,0.3,−0.5,−0.4,−0.4,−0.3,−0.9,−0.8,π5,0.2,0.3,0.4,0.5,0.1,0.2,−0.2,−0.1,−0.6,−0.5,−0.4,−0.3,λ3,π1,0.7,0.8,0.1,0.2,0,0.2,−0.3,−0.2,−0.2,−0.1,−0.5,−0.4,π2,0,0.2,0.3,0.5,0.1,0.2,−0.9,−0.8,−0.7,−0.6,−0.3,−0.1,π3,0.3,0.4,0.6,0.8,0,0.1,−0.1,0,−0.9,−0.8,−0.7,−0.6,π4,0.5,0.6,0.2,0.4,0.6,0.7,−0.4,−0.3,−0.8,−0.7,−0.3,−0.2,π5,0.6,0.7,0.4,0.5,0.1,0.2,−0.1,0,−0.5,−0.4,−0.8,−0.7.
In addition, we can define two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension 3 on Π, respectively, as follows:
(14)Γ,Λ=λ1,π1,0,0.1,0.2,0.3,0.1,0.2,−0.4,−0.3,−0.2,−0.1,−0.6,−0.5,π2,0.2,0.3,0.4,0.6,0,0,−0.8,−0.7,−0.4,−0.3,−0.1,0,π3,0.3,0.4,0.1,0.2,0.6,0.7,−0.5,−0.4,−0.2,−0.1,−0.2,−0.1,π4,0,0.1,0.6,0.8,0.1,0.2,−0.5,−0.5,−0.3,−0.2,0,0,π5,0.1,0.2,0.3,0.5,0,0.2,0,0,−0.4,−0.3,−0.1,0,
It is clear thatΓ,Λ⊆~Ψ,Θ.
Moreover, we have the union and intersection of bipolar-valued multivague soft sets of dimension3, namely ℶ,Δ=Γ,Λ∪~Ψ,Θ, and Σ,Δ=Γ,Λ∩~Ψ,Θ, which are again of dimension 3, where Δ=Λ∪Θ=λ1,λ2,λ3, respectively, as follows:
(15)ℶ,Δ=λ1,π1,0.1,0.2,0.7,0.8,0.5,0.6,−0.9,−0.8,−0.4,−0.2,−0.8,−0.7,π2,0.3,0.5,0.4,0.6,0,0.1,−0.9,−0.7,−0.6,−0.5,−0.2,0,π3,0.4,0.6,0.2,0.3,0.7,0.8,−0.6,−0.5,−0.3,−0.2,−0.4,−0.1,π4,0,0.1,0.7,0.9,0.4,0.5,−0.9,−0.8,−0.5,−0.4,−0.3,−0.2,π5,0.4,0.5,0.9,1,0.1,0.4,−0.2,0,−0.6,−0.5,−0.1,0,λ2,π1,0.8,0.9,0.2,0.3,0.4,0.5,−0.8,−0.7,−0.2,−0.1,−0.9,−0.8,π2,0.1,0.2,0.3,0.4,0.4,0.5,−0.7,−0.6,−0.3,−0.2,−0.2,−0.1,π3,0.6,0.7,0,0.1,0.7,0.8,−0.1,0,−0.9,−0.8,−0.6,−0.5,π4,0.1,0.2,0.6,0.8,0.2,0.3,−0.5,−0.4,−0.4,−0.3,−0.9,−0.8,π5,0.2,0.3,0.4,0.5,0.1,0.2,−0.2,−0.1,−0.6,−0.5,−0.4,−0.3,(16)Σ,Δ=λ1,π1,0,0.1,0.2,0.3,0.1,0.2,−0.4,−0.3,−0.2,−0.1,−0.6,−0.5,π2,0.2,0.3,0.4,0.6,0,0,−0.8,−0.7,−0.4,−0.3,−0.1,0,π3,0.3,0.4,0.1,0.2,0.6,0.7,−0.5,−0.4,−0.2,−0.1,−0.2,−0.1,π4,0,0.1,0.6,0.8,0.1,0.2,−0.5,−0.5,−0.3,−0.2,0,0,π5,0.1,0.2,0.3,0.5,0,0.2,0,0,−0.4,−0.3,−0.1,0λ2,π1,0.8,0.9,0.2,0.3,0.4,0.5,−0.8,−0.7,−0.2,−0.1,−0.9,−0.8,π2,0.1,0.2,0.3,0.4,0.4,0.5,−0.7,−0.6,−0.3,−0.2,−0.2,−0.1,π3,0.6,0.7,0,0.1,0.7,0.8,−0.1,0,−0.9,−0.8,−0.6,−0.5,π4,0.1,0.2,0.6,0.8,0.2,0.3,−0.5,−0.4,−0.4,−0.3,−0.9,−0.8,π5,0.2,0.3,0.4,0.5,0.1,0.2,−0.2,−0.1,−0.6,−0.5,−0.4,−0.3.
Finally, we have the restricted union and the restricted intersection of bipolar-valued multivague soft sets of dimension3, namely ℸ,Ω=Γ,Λ∪~RΨ,Θ and Φ,Ω=Γ,Λ∩~RΨ,Θ, which are again of dimension 3, where Ω=Λ∩Θ=λ1,λ2, respectively, as follows:
(17)ℸ,Ω=λ1,π1,0.1,0.2,0.7,0.8,0.5,0.6,−0.9,−0.8,−0.4,−0.2,−0.8,−0.7,π2,0.3,0.5,0.4,0.6,0,0.1,−0.9,−0.7,−0.6,−0.5,−0.2,0,π3,0.4,0.6,0.2,0.3,0.7,0.8,−0.6,−0.5,−0.3,−0.2,−0.4,−0.1π4,0,0.1,0.7,0.9,0.4,0.5,−0.9,−0.8,−0.5,−0.4,−0.3,−0.2,π5,0.4,0.5,0.9,1,0.1,0.4,−0.2,0,−0.6,−0.5,−0.1,0,Φ,Ω=λ1,π1,0,0.1,0.2,0.3,0.1,0.2,−0.4,−0.3,−0.2,−0.1,−0.6,−0.5,π2,0.2,0.3,0.4,0.6,0,0,−0.8,−0.7,−0.4,−0.3,−0.1,0,π3,0.3,0.4,0.1,0.2,0.6,0.7,−0.5,−0.4,−0.2,−0.1,−0.2,−0.1π4,0,0.1,0.6,0.8,0.1,0.2,−0.5,−0.5,−0.3,−0.2,0,0,π5,0.1,0.2,0.3,0.5,0,0.2,0,0,−0.4,−0.3,−0.1,0,
Assume the following throughout the remainder of the paper, except otherwise indicated:(18)Γ,Λ=λ,π,τ1Γλ+π,1−η1Γλ+π,τ2Γλ+π,1−η2Γλ+π,⋯,τnΓλ+π,1−ηnΓλ+π,−1−η1Γλ−π,τ1Γλ−π,−1−η2Γλ−π,τ2Γλ−π,⋯,−1−ηnΓλ−π,τnΓλ−π:λ∈Λ,π∈Π,Ψ,Θ=θ,π,τ1Ψθ+π,1−η1Ψθ+π,τ2Ψθ+π,1−η2Ψθ+π,⋯,τnΨθ+π,1−ηnΨθ+π,−1−η1Ψθ−π,τ1Ψθ−π,−1−η2Ψθ−π,τ2Ψθ−π,⋯,−1−ηnΨθ−π,τnΨθ−π:θ∈Θ,π∈Π,Σ,Ω=ω,π,τ1Σω+π,1−η1Σω+π,τ2Σω+π,1−η2Σω+π,⋯,τnΣω+π,1−ηnΣω+π,−1−η1Σω−π,τ1Σω−π,−1−η2Σω−π,τ2Σω−π,⋯,−1−ηnΣω−π,τnΣω−π:ω∈Ω,π∈Π.Theorem 25.
LetΓ,Λ be any bipolar-valued multivague soft set of dimension n, CΛn be the complete bipolar-valued multivague soft set of dimension n, and ϕΛn be the null bipolar-valued multivague soft set of dimension n on a common universal set Π, then:
(19)1Γ,Λ∪~Γ,Λ=Γ,Λ,18pt4Γ,Λ∩~ϕΛn=ϕΛn,7CΛn∪~ϕΛn=CΛn,2Γ,Λ∩~Γ,Λ=Γ,Λ,18pt5Γ,Λ∪~CΛn=CΛn,8CΛn∩~ϕΛn=ϕΛn.3Γ,Λ∪~ϕΛn=Γ,Λ,6Γ,Λ∩~CΛn=Γ,Λ.Proof.
We prove (8), and from (1) to (12) follow similarly. From Definitions 12 and 13, CΛn=λ,π,1,1,⋯n−times,1,1,−1,−1,⋯n−times,−1,−1:λ∈Λ,π∈Π and ϕΛn=λ,π,0,0,⋯2n−times,0,0:λ∈Λ,π∈Π, respectively. Assume, for Δ=Λ∪Λ=Λ, that
(20)CΛn∩~ϕΛn=ℶ,Δ=δ,π,τ1ℶδ+π,1−η1ℶδ+π,τ2ℶδ+π,1−η2ℶδ+π,⋯,τnℶδ+π,1−ηnℶδ+π,−1−η1ℶδ−π,τ1ℶδ−π,−1−η2ℶδ−π,τ2ℶδ−π,⋯,−1−ηnℶδ−π,τnℶδ−π:18ptδ∈Λ,π∈Π=δ,π,min1,0,min1,0,⋯n−times,min1,0,min1,0,max−1,0,max−1,0,⋯n−times,max−1,0,max−1,0:δ∈Λ,π∈Π=δ,π,0,0,⋯n−times,0,0,0,0,⋯n−times,0,0:δ∈Λ,π∈Π=δ,π,0,0,⋯2n−times,0,0:δ∈Λ,π∈Π=ϕΛn.
This result is true for the third case (whenδ∈Λ∩Λ=Λ) of Definition 21 of intersection of two bipolar-valued multivague soft sets of dimension n. For the first and second case, we have no parameters, since δ∈Λ−Λ=ϕ.Theorem 26.
Assume thatΓ,Λ and Ψ,Θ are two bipolar-valued multivague soft sets of dimension n on a universal set Π; then we have the absorption properties satisfied for them as follows:
(21)1Γ,Λ∪~Γ,Λ∩~RΨ,Θ=Γ,Λ,2Γ,Λ∩~RΓ,Λ∪~Ψ,Θ=Γ,Λ.Proof.
We just prove (1). The same steps can be followed to prove (5). Let
(22)Φ,Ξ=Γ,Λ∩~RΨ,Θ,Ξ=Λ∩Θ=ξ,π,τ1Φξ+π,1−η1Φξ+π,τ2Φξ+π,1−η2Φξ+π,⋯,τnΦξ+π,1−ηnΦξ+π,−1−η1Φξ−π,τ1Φξ−π,−1−η2Φξ−π,τ2Φξ−π,⋯,−1−ηnΦξ−π,τnΦξ−π:ξ∈Ξ,π∈Π,ℶ,Δ=Γ,Λ∪~Φ,Ξ,Δ=Λ∪Ξ=δ,π,τ1ℶδ+π,1−η1ℶδ+π,τ2ℶδ+π,1−η2ℶδ+π,⋯,τnℶδ+π,1−ηnℶδ+π,−1−η1ℶδ−π,τ1ℶδ−π,−1−η2ℶδ−π,τ2ℶδ−π,⋯,−1−ηnℶδ−π,τnℶδ−π:δ∈Δ,π∈Π.
According to Definition17, we have to show that (1) holds for all following three cases:
(i)
Ifδ∈Λ−Θ, then we obtain from Definition 23 that:(23)Φ,Ξ=Γ,Λ∩~RΨ,Θ=ξ,π,τ1Φξ+π,1−η1Φξ+π,τ2Φξ+π,1−η2Φξ+π,⋯,τnΦξ+π,1−ηnΦξ+π,−1−η1Φξ−π,τ1Φξ−π,−1−η2Φξ−π,τ2Φξ−π,⋯,−1−ηnΦξ−π,τnΦξ−π:ξ∈Λ−Θ,π∈Π=ϕ.
Then, by using (6) from Theorem 25, we have:
(24)ℶ,Δ=Γ,Λ∪~Φ,Ξ=Γ,Λ∪~ϕ=Γ,Λ.(ii)
Ifδ∈Θ−Λ, then we obtain from Definition 23 that:(25)Φ,Ξ=Γ,Λ∩~RΨ,Θ=ξ,π,τ1Φξ+π,1−η1Φξ+π,τ2Φξ+π,1−η2Φξ+π,⋯,τnΦξ+π,1−ηnΦξ+π,−1−η1Φξ−π,τ1Φξ−π,−1−η2Φξ−π,τ2Φξ−π,⋯,−1−ηnΦξ−π,τnΦξ−π:ξ∈Θ−Λ,π∈Π=ϕ.
Then, by using (6) from Theorem 25, we have:
(26)ℶ,Δ=Γ,Λ∪~Φ,Ξ=Γ,Λ∪~ϕ=Γ,Λ.(iii)
Ifδ∈Λ∩Θ, then we obtain from Definition 23 that:(27)Φ,Ξ=Γ,Λ∩~RΨ,Θ=ξ,π,τ1Φξ+π,1−η1Φξ+π,τ2Φξ+π,1−η2Φξ+π,⋯,τnΦξ+π,1−ηnΦξ+π,−1−η1Φξ−π,τ1Φξ−π,−1−η2Φξ−π,τ2Φξ−π,⋯,−1−ηnΦξ−π,τnΦξ−π:ξ∈Λ∩Θ,π∈Π=ξ,π,minτ1Γξ+π,τ1Ψξ+π,min1−η1Γξ+π,1−η1Ψξ+π,minτ2Γξ+π,τ2Ψξ+π,min1−η2Γξ+π,1−η2Ψξ+π,⋯,minτnΓξ+π,τnΨξ+π,min1−ηnΓξ+π,1−ηnΨξ+π,max−1−η1Γξ−π,−1−η1Ψξ−π,maxτ1Γξ−π,τ1Ψξ−π,max−1−η2Γξ−π,−1−η2Ψξ−π,maxτ2Γξ−π,τ2Ψξ−π,⋯,max−1−ηnΓξ−π,−1−ηnΨξ−π,maxτnΓξ−π,τnΨξ−π:ξ∈Λ∩Θ,π∈Π.
Therefore, we have from Definition17 that:
(28)ℶ,Δ=Γ,Λ∪~Φ,Ξ=δ,π,τ1ℶδ+π,1−η1ℶδ+π,τ2ℶδ+π,1−η2ℶδ+π,⋯,τnℶδ+π,1−ηnℶδ+π,−1−η1ℶδ−π,τ1ℶδ−π,−1−η2ℶδ−π,τ2ℶδ−π,⋯,−1−ηnℶδ−π,τnℶδ−π:δ∈Λ∩Θ,π∈Π=δ,π,maxτ1Γδ+π,minτ1Γδ+π,τ1Ψδ+π,max1−η1Γδ+π,min1−η1Γδ+π,1−η1Ψδ+π,maxτ2Γδ+π,minτ2Γδ+π,τ2Ψδ+π,max1−η2Γδ+π,min1−η2Γδ+π,1−η2Ψδ+π,⋯,maxτnΓδ+π,minτnΓδ+π,τnΨδ+π,max1−ηnΓδ+π,min1−ηnΓδ+π,1−ηnΨδ+π,min−1−η1Γδ−π,max−1−η1Γδ−π,−1−η1Ψδ−π,minτ1Γδ−π,maxτ1Γδ−π,τ1Ψδ−π,min−1−η2Γδ−π,max−1−η2Γδ−π,−1−η2Ψδ−π,minτ2Γδ−π,maxτ2Γδ−π,τ2Ψδ−π,⋯,min−1−ηnΓδ−π,max−1−ηnΓδ−π,−1−ηnΨδ−π,minτnΓδ−π,maxτnΓδ−π,τnΨδ−π:δ∈Λ∩Θ,π∈Π=δ,π,τ1Γδ+π,1−η1Γδ+π,τ2Γδ+π,1−η2Γδ+π,⋯,τnΓδ+π,1−ηnΓδ+π,−1−η1Γδ−π,τ1Γδ−π,−1−η2Γδ−π,τ2Γδ−π,⋯,−1−ηnΓδ−π,τnΓδ−π:δ∈Λ∩Θ,π∈Π=Γ,Λ.Corollary 27.
For two bipolar-valued multivague soft sets of dimensionn on a common universal set Γ,Λ and Ψ,Θ:
(29)Γ,Λ∪~Γ,Λ∩~RΨ,Θ=Γ,Λ∩~RΓ,Λ∪~Ψ,Θ=Γ,Λ.Proof.
One can prove this result directly with the help of the above Theorem26.Theorem 28.
Suppose thatΓ,Λ and Ψ,Θ are two bipolar-valued multivague soft sets of dimension n on a universal set Π; then, we have the commutative (Abelian) property satisfied for them as follows:
(30)1Γ,Λ∩~Ψ,Θ=Ψ,Θ∩~Γ,Λ,2Γ,Λ∪~Ψ,Θ=Ψ,Θ∪~Γ,Λ.Proof.
By applying similar techniques used to prove the above theorems with the help of Definitions21 and 17 of the intersection and union of two bipolar-valued multivague soft sets of dimension n on a universal set Π, respectively, we can show that (1) and (5) hold.Proposition 29.
IfΓ,Λ and Ψ,Θ are two bipolar-valued multivague soft sets of dimension n, Γ,Λ⊆~Ψ,Θ, then
(31)1Γ,Λ∩~RΨ,Θ=Γ,Λ,2Γ,Λ∪~Ψ,Θ=Ψ,Θ.Proof.
Apply similar methods but by using Definitions 23 and19 of restricted intersection and restricted union of two bipolar-valued multivague soft sets of dimension n, respectively, to prove (1) and (5).Theorem 30.
Suppose thatΓ,Λ, Ψ,Θ, and Σ,Ω are bipolar-valued multivague soft sets of dimension n on a common universal set Π; then associative and distributive laws, respectively, hold for them as follows:
(32)Γ,Λ∩~Ψ,Θ∩~Σ,Ω=Γ,Λ∩~Ψ,Θ∩~Σ,Ω,Γ,Λ∪~Ψ,Θ∪~Σ,Ω=Γ,Λ∪~Ψ,Θ∪~Σ,Ω,Γ,Λ∩~Ψ,Θ∪~Σ,Ω=Γ,Λ∩~Ψ,Θ∪~Γ,Λ∩~Σ,Ω,Γ,Λ∪~Ψ,Θ∩~Σ,Ω=Γ,Λ∪~Ψ,Θ∩~Γ,Λ∪~Σ,Ω.Proof.
Using Definitions21 and 17 of the intersection and union of two bipolar-valued multivague soft sets of dimension n on a universal set Π, respectively, one can prove the theorem, similarly as the above theorems.Theorem 31.
De Morgan’s laws are valid for any two bipolar-valued multivague soft setsΓ,Λ and Ψ,Θ of dimension n on a common universal set Π as follows:
(33)1Γ,Λ∪~Ψ,Θc=Γ,Λc∩~Ψ,Θc,2Γ,Λ∩~Ψ,Θc=Γ,Λc∪~Ψ,Θc.Proof.
To prove (1), let δ∈Δ=Λ∪Θ. We must show that (1) is true for all three cases according to Definitions 17 and 21. Assume that δ∈Λ∩Θ; therefore, we obtain from Definitions 14, 17, and 21 that:
(34)Γ,Λ∪~Ψ,Θc=δ,π,maxτ1Γδ+π,τ1Ψδ+π,max1−η1Γδ+π,1−η1Ψδ+π,maxτ2Γδ+π,τ2Ψδ+π,max1−η2Γδ+π,1−η2Ψδ+π,⋯,maxτnΓδ+π,τnΨδ+π,max1−ηnΓδ+π,1−ηnΨδ+π,min−1−η1Γδ−π,−1−η1Ψδ−π,minτ1Γδ−π,τ1Ψδ−π,min−1−η2Γδ−π,−1−η2Ψδ−π,minτ2Γδ−π,τ2Ψδ−π,⋯,min−1−ηnΓδ−π,−1−ηnΨδ−π,minτnΓδ−π,τnΨδ−π:δ∈Λ∩Θ,π∈Πc,Γ,Λ∪~Ψ,Θc=δ,π,maxτ1Γδ+π,τ1Ψδ+π,1−minη1Γδ+π,η1Ψδ+π,maxτ2Γδ+π,τ2Ψδ+π,1−minη2Γδ+π,η2Ψδ+π,⋯,maxτnΓδ+π,τnΨδ+π,1−minηnΓδ+π,ηnΨδ+π,−1−maxη1Γδ−π,η1Ψδ−π,minτ1Γδ−π,τ1Ψδ−π,−1−maxη2Γδ−π,η2Ψδ−π,minτ2Γδ−π,τ2Ψδ−π,⋯,−1−maxηnΓδ−π,ηnΨδ−π,minτnΓδ−π,τnΨδ−π:δ∈Λ∩Θ,π∈Πc,Γ,Λ∪~Ψ,Θc=δ,π,minη1Γδ+π,η1Ψδ+π,1−maxτ1Γδ+π,τ1Ψδ+π,minη2Γδ+π,η2Ψδ+π,1−maxτ2Γδ+π,τ2Ψδ+π,⋯,minηnΓδ+π,ηnΨδ+π,1−maxτnΓδ+π,τnΨδ+π,−1−minτ1Γδ−π,τ1Ψδ−π,maxη1Γδ−π,η1Ψδ−π,−1−minτ2Γδ−π,τ2Ψδ−π,maxη2Γδ−π,η2Ψδ−π,⋯,−1−minτnΓδ−π,τnΨδ−π,maxηnΓδ−π,ηnΨδ−π:δ∈Λ∩Θ,π∈Π,Γ,Λ∪~Ψ,Θc=δ,π,minη1Γδ+π,η1Ψδ+π,min1−τ1Γδ+π,1−τ1Ψδ+π,minη2Γδ+π,η2Ψδ+π,min1−τ2Γδ+π,1−τ2Ψδ+π,⋯,minηnΓδ+π,ηnΨδ+π,min1−τnΓδ+π,1−τnΨδ+π,max−1−τ1Γδ−π,−1−τ1Ψδ−π,maxη1Γδ−π,η1Ψδ−π,max−1−τ2Γδ−π,−1−τ2Ψδ−π,maxη2Γδ−π,η2Ψδ−π,⋯,max−1−τnΓδ−π,−1−τnΨδ−π,maxηnΓδ−π,ηnΨδ−π:δ∈Λ∩Θ,π∈Π,Γ,Λ∪~Ψ,Θc=δ,π,η1Γδ+π,1−τ1Γδ+π,η2Γδ+π,1−τ2Γδ+π,⋯,ηnΓδ+π,1−τnΓδ+π,−1−τ1Γδ−π,η1Γδ−π,−1−τ2Γδ−π,η2Γδ−π,⋯,−1−τnΓδ−π,ηnΓδ−π:∈Λ∩Θ,π∈Π∩~δ,π,η1Ψδ+π,1−τ1Ψδ+π,η2Ψδ+π,1−τ2Ψδ+π,⋯,ηnΨδ+π,1−τnΨδ+π,−1−τ1Ψδ−π,η1Ψδ−π,−1−τ2Ψδ−π,η2Ψδ−π,⋯,−1−τnΨδ−π,ηnΨδ−π:δ∈Λ∩Θ,π∈Π,Γ,Λ∪~Ψ,Θc=δ,π,τ1Γδ+π,1−η1Γδ+π,τ2Γδ+π,1−η2Γδ+π,⋯,τnΓδ+π,1−ηnΓδ+π,−1−η1Γδ−π,τ1Γδ−π,−1−η2Γδ−π,τ2Γδ−π,⋯,−1−ηnΓδ−π,τnΓδ−π:δ∈Λ,π∈Πc∩~δ,π,τ1Ψδ+π,1−η1Ψδ+π,τ1Ψδ+π,1−η1Ψδ+π,⋯,τ1Ψδ+π,1−η1Ψδ+π,−1−η1Ψδ−π,τ1Ψδ−π,−1−η1Ψδ−π,τ1Ψδ−π,⋯,−1−η1Ψδ−π,τ1Ψδ−π:δ∈Θ,π∈Πc=Γ,Λc∩~Ψ,Θc.
This proves the third case. The first and second cases are trivial and follow similarly as above. In addition, (5) can be proved similarly as (1).


## 4. Generalized Decision-Making Technique

This section focuses on a practical, real-world example of how to handle a socialist economic decision-making issue. Generalizing the Roy and Maji method [38], one can apply their technique in the bipolar-valued multivague soft set of dimension n environment rather than the fuzzy soft set environment. This makes it possible to pick up the best option for issues with uncertain or ambiguous information more effectively than with the prior models.Definition 32. (comparison table).
The comparison table is a square table; its rows and columns are labeled by the object name of the universe such asπ1,π2,π3,⋯,πn, and the entries dij, where dij is the number of parameters for which, the value of di (the membership value of πi) exceeds or equal to ≥ the value of dj (the membership value of πj).


### 4.1. Algorithm

Step (1): construct the tabular representations of then positive poles membership intervalsStep (2): compute the comparison tables, as stated in Definition32, for the tables obtained in step (1) of the n positive poles membership intervals.Step (3): for each comparison table, compute the sum of every rowR1, R2, ⋯, Rn, and the sum of every column C1, C2, ⋯, Cn, then calculate their membership scores R1-C1, R2-C2, ⋯, Rn-Cn, to put them in the n positive poles score tablesStep (4): construct the final positive score table by calculatingF+ for each alternative πi obtained by adding its n positive pole scoresStep (5): repeat steps (1), (2), (3), and (4) but for negative informationStep (6): compute the final scoreF+−F− for each element πi by subtracting its final negative information score F− from its final positive information score F+, then put these final scores in a table called the final score tableStep (7): determinek under which πk=maxπi; therefore, the best alternative is πkStep (8): obtain the ranking of alternativesπi by ranking their final scores in descending order (ordinal ranking)Example 2.
Consider the same data of Example1. If the manager of this organization wants to select the most suitable employee among the five nominated employees to become the assistant director of his organization according to the ℶ,Y description.


#### 4.1.1. Solution 1

Construct the tabular representations of the first, second, and third positive poles membership intervals, putting them in Tables1–3, respectively.Table 1
The first positive pole membership intervals.
Πλ1′λ2′λ3′π10.8,0.90.1,0.20.2,0.3π20.5,0.70.8,0.90.8,1π30.4,0.60.3,0.40.6,0.7π40.9,10.8,0.90.4,0.5Table 2
The second positive pole membership intervals.
Πλ1′′λ2′′λ3′′π10.2,0.30.7,0.80.8,0.9π20.4,0.60.6,0.70.5,0.7π30.7,0.80.9,10.2,0.4π40.1,0.30.2,0.40.6,0.8π50,0.10.5,0.60.5,0.6Table 3
The third positive pole membership intervals.
Πλ1′′′λ2′′′λ3′′′π10.4,0.50.5,0.60.8,1π20.9,10.5,0.60.8,0.9π30.2,0.30.2,0.30.9,1π40.5,0.60.7,0.80.3,0.4π50.6,0.90.8,0.90.8,0.9


#### 4.1.2. Solution 2

Construct the comparison Tables4–6, as stated in Definition 32, of Tables 1–3, respectively.Table 4
The comparison table of Table1.
Ππ1π2π3π4π5π131101π223323π320311π432233π520203Table 5
The comparison table of Table2.
Ππ1π2π3π4π5π132133π213123π322322π401132π500113Table 6
The comparison table of Table3.
Ππ1π2π3π4π5π132211π223222π311311π421230π522233


#### 4.1.3. Solution 3

For Tables4–6, compute the sum of every row (R1, R2, and R3, respectively) and the sum of every column (C1, C2, and C3, respectively), then calculate their membership scores (R1-C1, R2-C2, and R3-C3, respectively) to put them in the first, second, and third positive pole score tables, namely, Tables 7–9, respectively.Table 7
The first positive pole score table.
ΠR1C1R1-C1π1612-6π21367π3711-4π41367π5711-4Table 8
The second positive pole score table.
ΠR2C2R2-C2π11266π21082π31174π4711-4π5513-8Table 9
The third positive pole score table.
ΠR3C3R3-C3π1910-1π21192π3711-4π4810-2π51275


#### 4.1.4. Solution 4

Compute the final positive score (F+) for each alternative by adding its first, second, and third positive pole scores, putting them in Table 10.Table 10
The final positive score table.
ΠR1-C1R2-C2R3-C3F+π1-66-1-1π272211π3-44-4-4π47-4-21π5-4-85-7


#### 4.1.5. Solution 5

Repeat steps (1), (2), (3), and (4) for negative information as follows. Construct the tabular representations of the first, second, and third negative poles membership intervals, putting them in Tables11–13, respectively. Then, construct Tables 14–16 of Tables 11–13, as stated in Definition 32. After that, for Tables 14–16, compute the sum of every row (R4, R5, and R6, respectively) and the sum of every column (C4, C5, and C6, respectively), then calculate their membership scores (R4-C4, R5-C5, and R6-C6, respectively) to put them in the first, second, and third negative poles score tables, namely, Tables 17–19, respectively. Finally, compute the final negative score (F−) for each alternative by adding its first, second, and third negative pole scores, putting them in Table 20.Table 11
The first negative pole membership intervals.
Π−λ1′−λ2′−λ3′π1[-0.2,-0.1][-0.3,-0.2][-0.8,-0.7]π2[-0.3,-0.1][-0.4,-0.3][-0.2,-0.1]π3[-0.5,-0.4][-1,-0.9][-1,-0.9]π4[-0.2,-0.1][-0.6,-0.5][-0.7,-0.6]π5[-1,-0.8][-0.9,-0.8][-1,-0.9]Table 12
The second negative pole membership intervals.
Π−λ1′′−λ2′′−λ3′′π1[-0.8,-0.6][-0.9,-0.8][-0.9,-0.8]π2[-0.5,-0.4][-0.8,-0.7][-0.4,-0.3]π3[-0.8,-0.7][-0.2,-0.1][-0.2,-0.1]π4[-0.6,-0.5][-0.7,-0.6][-0.3,-0.2]π5[-0.5,-0.4][-0.5,-0.4][-0.6,-0.5]Table 13
The third negative pole membership intervals.
Π−λ1′′′−λ2′′′−λ3′′′π1[-0.3,-0.2][-0.2,-0.1][-0.6,-0.5]π2[-1,-0.8][-0.9,-0.8][-0.9,-0.7]π3[-0.9,-0.6][-0.5,-0.4][-0.4,-0.3]π4[-0.8,-0.7][-0.2,-0.1][-0.8,-0.7]π5[-1,-0.9][-0.7,-0.6][-0.3,-0.2]Table 14
The comparison table of Table11.
Ππ1π2π3π4π5π131020π223010π333332π422030π533333Table 15
The comparison table of Table12.
Ππ1π2π3π4π5π133233π203212π311311π401232π502213Table 16
The comparison table of Table13.
Ππ1π2π3π4π5π130111π233332π320321π430131π521223Table 17
The first negative pole score table.
ΠR4C4R4-C4π1613-7π2612-6π31468π4712-5π515510Table 18
The second negative pole score table.
ΠR5C5R5-C5π114410π2810-2π3711-4π489-1π5811-3Table 19
The third negative pole score table.
ΠR6C6R6-C6π1613-7π214410π3810-2π4811-3π51082Table 20
The final negative score table.
ΠR4-C4R5-C5R6-C6F−π1-710-7-4π2-6-2102π38-4-22π4-5-1-3-9π510-329


#### 4.1.6. Solution 6

Now, subtract each final negative score (from Table10) from its opposite final positive score (from Table 20) to calculate entries of the final score table F+−F−, then put all of them in Table 21.Table 21
The final score table.
ΠF+F−F+−F−π1-1-43π21129π3-42-6π41-910π5-79-16


#### 4.1.7. Solution 7

Since the fourth alternative (employee)π1 has the maximum final score (which is 10) among the others, then the decision is that the most suitable employee to be the manager assistant in this organization among the five employees is the fourth one π1.


#### 4.1.8. Solution 8

Consequently, we have the ranking of the alternatives (employees) as follows:(35)π4>π2>π1>π3>π5.


## 4.1. Algorithm

Step (1): construct the tabular representations of then positive poles membership intervalsStep (2): compute the comparison tables, as stated in Definition32, for the tables obtained in step (1) of the n positive poles membership intervals.Step (3): for each comparison table, compute the sum of every rowR1, R2, ⋯, Rn, and the sum of every column C1, C2, ⋯, Cn, then calculate their membership scores R1-C1, R2-C2, ⋯, Rn-Cn, to put them in the n positive poles score tablesStep (4): construct the final positive score table by calculatingF+ for each alternative πi obtained by adding its n positive pole scoresStep (5): repeat steps (1), (2), (3), and (4) but for negative informationStep (6): compute the final scoreF+−F− for each element πi by subtracting its final negative information score F− from its final positive information score F+, then put these final scores in a table called the final score tableStep (7): determinek under which πk=maxπi; therefore, the best alternative is πkStep (8): obtain the ranking of alternativesπi by ranking their final scores in descending order (ordinal ranking)Example 2.
Consider the same data of Example1. If the manager of this organization wants to select the most suitable employee among the five nominated employees to become the assistant director of his organization according to the ℶ,Y description.


### 4.1.1. Solution 1

Construct the tabular representations of the first, second, and third positive poles membership intervals, putting them in Tables1–3, respectively.Table 1
The first positive pole membership intervals.
Πλ1′λ2′λ3′π10.8,0.90.1,0.20.2,0.3π20.5,0.70.8,0.90.8,1π30.4,0.60.3,0.40.6,0.7π40.9,10.8,0.90.4,0.5Table 2
The second positive pole membership intervals.
Πλ1′′λ2′′λ3′′π10.2,0.30.7,0.80.8,0.9π20.4,0.60.6,0.70.5,0.7π30.7,0.80.9,10.2,0.4π40.1,0.30.2,0.40.6,0.8π50,0.10.5,0.60.5,0.6Table 3
The third positive pole membership intervals.
Πλ1′′′λ2′′′λ3′′′π10.4,0.50.5,0.60.8,1π20.9,10.5,0.60.8,0.9π30.2,0.30.2,0.30.9,1π40.5,0.60.7,0.80.3,0.4π50.6,0.90.8,0.90.8,0.9


### 4.1.2. Solution 2

Construct the comparison Tables4–6, as stated in Definition 32, of Tables 1–3, respectively.Table 4
The comparison table of Table1.
Ππ1π2π3π4π5π131101π223323π320311π432233π520203Table 5
The comparison table of Table2.
Ππ1π2π3π4π5π132133π213123π322322π401132π500113Table 6
The comparison table of Table3.
Ππ1π2π3π4π5π132211π223222π311311π421230π522233


### 4.1.3. Solution 3

For Tables4–6, compute the sum of every row (R1, R2, and R3, respectively) and the sum of every column (C1, C2, and C3, respectively), then calculate their membership scores (R1-C1, R2-C2, and R3-C3, respectively) to put them in the first, second, and third positive pole score tables, namely, Tables 7–9, respectively.Table 7
The first positive pole score table.
ΠR1C1R1-C1π1612-6π21367π3711-4π41367π5711-4Table 8
The second positive pole score table.
ΠR2C2R2-C2π11266π21082π31174π4711-4π5513-8Table 9
The third positive pole score table.
ΠR3C3R3-C3π1910-1π21192π3711-4π4810-2π51275


### 4.1.4. Solution 4

Compute the final positive score (F+) for each alternative by adding its first, second, and third positive pole scores, putting them in Table 10.Table 10
The final positive score table.
ΠR1-C1R2-C2R3-C3F+π1-66-1-1π272211π3-44-4-4π47-4-21π5-4-85-7


### 4.1.5. Solution 5

Repeat steps (1), (2), (3), and (4) for negative information as follows. Construct the tabular representations of the first, second, and third negative poles membership intervals, putting them in Tables11–13, respectively. Then, construct Tables 14–16 of Tables 11–13, as stated in Definition 32. After that, for Tables 14–16, compute the sum of every row (R4, R5, and R6, respectively) and the sum of every column (C4, C5, and C6, respectively), then calculate their membership scores (R4-C4, R5-C5, and R6-C6, respectively) to put them in the first, second, and third negative poles score tables, namely, Tables 17–19, respectively. Finally, compute the final negative score (F−) for each alternative by adding its first, second, and third negative pole scores, putting them in Table 20.Table 11
The first negative pole membership intervals.
Π−λ1′−λ2′−λ3′π1[-0.2,-0.1][-0.3,-0.2][-0.8,-0.7]π2[-0.3,-0.1][-0.4,-0.3][-0.2,-0.1]π3[-0.5,-0.4][-1,-0.9][-1,-0.9]π4[-0.2,-0.1][-0.6,-0.5][-0.7,-0.6]π5[-1,-0.8][-0.9,-0.8][-1,-0.9]Table 12
The second negative pole membership intervals.
Π−λ1′′−λ2′′−λ3′′π1[-0.8,-0.6][-0.9,-0.8][-0.9,-0.8]π2[-0.5,-0.4][-0.8,-0.7][-0.4,-0.3]π3[-0.8,-0.7][-0.2,-0.1][-0.2,-0.1]π4[-0.6,-0.5][-0.7,-0.6][-0.3,-0.2]π5[-0.5,-0.4][-0.5,-0.4][-0.6,-0.5]Table 13
The third negative pole membership intervals.
Π−λ1′′′−λ2′′′−λ3′′′π1[-0.3,-0.2][-0.2,-0.1][-0.6,-0.5]π2[-1,-0.8][-0.9,-0.8][-0.9,-0.7]π3[-0.9,-0.6][-0.5,-0.4][-0.4,-0.3]π4[-0.8,-0.7][-0.2,-0.1][-0.8,-0.7]π5[-1,-0.9][-0.7,-0.6][-0.3,-0.2]Table 14
The comparison table of Table11.
Ππ1π2π3π4π5π131020π223010π333332π422030π533333Table 15
The comparison table of Table12.
Ππ1π2π3π4π5π133233π203212π311311π401232π502213Table 16
The comparison table of Table13.
Ππ1π2π3π4π5π130111π233332π320321π430131π521223Table 17
The first negative pole score table.
ΠR4C4R4-C4π1613-7π2612-6π31468π4712-5π515510Table 18
The second negative pole score table.
ΠR5C5R5-C5π114410π2810-2π3711-4π489-1π5811-3Table 19
The third negative pole score table.
ΠR6C6R6-C6π1613-7π214410π3810-2π4811-3π51082Table 20
The final negative score table.
ΠR4-C4R5-C5R6-C6F−π1-710-7-4π2-6-2102π38-4-22π4-5-1-3-9π510-329


### 4.1.6. Solution 6

Now, subtract each final negative score (from Table10) from its opposite final positive score (from Table 20) to calculate entries of the final score table F+−F−, then put all of them in Table 21.Table 21
The final score table.
ΠF+F−F+−F−π1-1-43π21129π3-42-6π41-910π5-79-16


### 4.1.7. Solution 7

Since the fourth alternative (employee)π1 has the maximum final score (which is 10) among the others, then the decision is that the most suitable employee to be the manager assistant in this organization among the five employees is the fourth one π1.


### 4.1.8. Solution 8

Consequently, we have the ranking of the alternatives (employees) as follows:(35)π4>π2>π1>π3>π5.


## 4.1.1. Solution 1

Construct the tabular representations of the first, second, and third positive poles membership intervals, putting them in Tables1–3, respectively.Table 1
The first positive pole membership intervals.
Πλ1′λ2′λ3′π10.8,0.90.1,0.20.2,0.3π20.5,0.70.8,0.90.8,1π30.4,0.60.3,0.40.6,0.7π40.9,10.8,0.90.4,0.5Table 2
The second positive pole membership intervals.
Πλ1′′λ2′′λ3′′π10.2,0.30.7,0.80.8,0.9π20.4,0.60.6,0.70.5,0.7π30.7,0.80.9,10.2,0.4π40.1,0.30.2,0.40.6,0.8π50,0.10.5,0.60.5,0.6Table 3
The third positive pole membership intervals.
Πλ1′′′λ2′′′λ3′′′π10.4,0.50.5,0.60.8,1π20.9,10.5,0.60.8,0.9π30.2,0.30.2,0.30.9,1π40.5,0.60.7,0.80.3,0.4π50.6,0.90.8,0.90.8,0.9


## 4.1.2. Solution 2

Construct the comparison Tables4–6, as stated in Definition 32, of Tables 1–3, respectively.Table 4
The comparison table of Table1.
Ππ1π2π3π4π5π131101π223323π320311π432233π520203Table 5
The comparison table of Table2.
Ππ1π2π3π4π5π132133π213123π322322π401132π500113Table 6
The comparison table of Table3.
Ππ1π2π3π4π5π132211π223222π311311π421230π522233


## 4.1.3. Solution 3

For Tables4–6, compute the sum of every row (R1, R2, and R3, respectively) and the sum of every column (C1, C2, and C3, respectively), then calculate their membership scores (R1-C1, R2-C2, and R3-C3, respectively) to put them in the first, second, and third positive pole score tables, namely, Tables 7–9, respectively.Table 7
The first positive pole score table.
ΠR1C1R1-C1π1612-6π21367π3711-4π41367π5711-4Table 8
The second positive pole score table.
ΠR2C2R2-C2π11266π21082π31174π4711-4π5513-8Table 9
The third positive pole score table.
ΠR3C3R3-C3π1910-1π21192π3711-4π4810-2π51275


## 4.1.4. Solution 4

Compute the final positive score (F+) for each alternative by adding its first, second, and third positive pole scores, putting them in Table 10.Table 10
The final positive score table.
ΠR1-C1R2-C2R3-C3F+π1-66-1-1π272211π3-44-4-4π47-4-21π5-4-85-7


## 4.1.5. Solution 5

Repeat steps (1), (2), (3), and (4) for negative information as follows. Construct the tabular representations of the first, second, and third negative poles membership intervals, putting them in Tables11–13, respectively. Then, construct Tables 14–16 of Tables 11–13, as stated in Definition 32. After that, for Tables 14–16, compute the sum of every row (R4, R5, and R6, respectively) and the sum of every column (C4, C5, and C6, respectively), then calculate their membership scores (R4-C4, R5-C5, and R6-C6, respectively) to put them in the first, second, and third negative poles score tables, namely, Tables 17–19, respectively. Finally, compute the final negative score (F−) for each alternative by adding its first, second, and third negative pole scores, putting them in Table 20.Table 11
The first negative pole membership intervals.
Π−λ1′−λ2′−λ3′π1[-0.2,-0.1][-0.3,-0.2][-0.8,-0.7]π2[-0.3,-0.1][-0.4,-0.3][-0.2,-0.1]π3[-0.5,-0.4][-1,-0.9][-1,-0.9]π4[-0.2,-0.1][-0.6,-0.5][-0.7,-0.6]π5[-1,-0.8][-0.9,-0.8][-1,-0.9]Table 12
The second negative pole membership intervals.
Π−λ1′′−λ2′′−λ3′′π1[-0.8,-0.6][-0.9,-0.8][-0.9,-0.8]π2[-0.5,-0.4][-0.8,-0.7][-0.4,-0.3]π3[-0.8,-0.7][-0.2,-0.1][-0.2,-0.1]π4[-0.6,-0.5][-0.7,-0.6][-0.3,-0.2]π5[-0.5,-0.4][-0.5,-0.4][-0.6,-0.5]Table 13
The third negative pole membership intervals.
Π−λ1′′′−λ2′′′−λ3′′′π1[-0.3,-0.2][-0.2,-0.1][-0.6,-0.5]π2[-1,-0.8][-0.9,-0.8][-0.9,-0.7]π3[-0.9,-0.6][-0.5,-0.4][-0.4,-0.3]π4[-0.8,-0.7][-0.2,-0.1][-0.8,-0.7]π5[-1,-0.9][-0.7,-0.6][-0.3,-0.2]Table 14
The comparison table of Table11.
Ππ1π2π3π4π5π131020π223010π333332π422030π533333Table 15
The comparison table of Table12.
Ππ1π2π3π4π5π133233π203212π311311π401232π502213Table 16
The comparison table of Table13.
Ππ1π2π3π4π5π130111π233332π320321π430131π521223Table 17
The first negative pole score table.
ΠR4C4R4-C4π1613-7π2612-6π31468π4712-5π515510Table 18
The second negative pole score table.
ΠR5C5R5-C5π114410π2810-2π3711-4π489-1π5811-3Table 19
The third negative pole score table.
ΠR6C6R6-C6π1613-7π214410π3810-2π4811-3π51082Table 20
The final negative score table.
ΠR4-C4R5-C5R6-C6F−π1-710-7-4π2-6-2102π38-4-22π4-5-1-3-9π510-329


## 4.1.6. Solution 6

Now, subtract each final negative score (from Table10) from its opposite final positive score (from Table 20) to calculate entries of the final score table F+−F−, then put all of them in Table 21.Table 21
The final score table.
ΠF+F−F+−F−π1-1-43π21129π3-42-6π41-910π5-79-16


## 4.1.7. Solution 7

Since the fourth alternative (employee)π1 has the maximum final score (which is 10) among the others, then the decision is that the most suitable employee to be the manager assistant in this organization among the five employees is the fourth one π1.


## 4.1.8. Solution 8

Consequently, we have the ranking of the alternatives (employees) as follows:(35)π4>π2>π1>π3>π5.


## 5. Concluding Notes and Future Researches

Set theory extensions have been considered a vital issue nowadays because of their importance in decision-making methods. In fact, combining any two basic extensions of the set theory not only is more general and flexible than applying one only of them but also gives more accurate applicable extended results. Some mathematicians have studied a few types of these combined extensions, such as bipolar-valued multifuzzy soft sets, multivague sets, and multivague soft sets. They have presented their characteristics and have used them in various areas as an illustration of actual problem-solving in decision-making. It goes without saying that several of these extensions are extensions for one another, not just for set theory.In this article, the bipolar-valued multivague soft set of dimensionn has been introduced as a novel hybrid set. Its kinds and its related concepts, as well as its operations, have been established, reflected by an example on each. Moreover, absorption properties, commutative properties, associative properties, and distributive laws, as well as De Morgan’s laws, are investigated with detailed proof on each. Furthermore, Roy and Maji’s technique has been generalized by using the concept of the bipolar-valued multivague soft set of dimension n instead of the fuzzy soft set concept to make more effective decisions to choose the optimal object among others. A useful real-world example has also been provided to further illustrate and clarify the suggested approach. This type of research fills in some gaps in the literature.In future research work, authors can investigate some necessary comparison analysis between the proposed method and other existing approaches. The analysis will be valuable and interesting. Moreover, several new results with some more generalized measures can be introduced using analogous methods in this article. Furthermore, our future research ideas can be extended to spherical vague soft sets and Pythagorean vague soft sets. By using any of those combinatorial extensions, one can create numerous different applications in a variety of scientific fields.



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*Source: 9453172-2022-08-02.xml*