# A Novel Approach towards Bipolar Soft Sets and Their Applications

Authors: Tahir Mahmood
Journal: Journal of Mathematics
 (2020)

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## Abstract

The notion of bipolar soft sets has already been defined, but in this article, the notion of bipolar soft sets has been redefined, called T-bipolar soft sets. It is shown that the new approach is more close to the concept of bipolarity as compared to the previous ones, and further it is discussed that so far in the study of soft sets and their generalizations, the concept introduced in this manuscript has never been discussed earlier. We have also discussed the operational laws of T-bipolar soft sets and their basic properties. In the end, we have deliberated the algebraic structures associated with T-bipolar soft sets and the applications of T-bipolar soft sets in decision-making problems.


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## Body

## 1. Introduction

To handle the uncertainty has always been a problem for the researchers and decision makers as it appears in almost every field of real life and all sciences including basic sciences, management sciences, social sciences, and information sciences. Many efforts have been made to cope with this concern. The first compact attempt in this direction was made by Zadeh [1] when he familiarized the notion of fuzzy sets in 1965. In 1982, Pawlak [2] familiarized the notion of rough sets. Although these theories have their own advantages and these theories proved their effectiveness, the theory of soft sets by Molodtsov [3] in 1999 did shovel work as it generalizes both the theories. Maji et al. [4] furnished some operations to soft sets. Later on, Ali et al. [5] piercing out some inadequacies in the operations defined in [4] bequeathed some new operations to soft sets like extended union, restricted union, restricted intersection, and the restricted difference of two soft sets. In [6], Ali et al. deliberated some algebraic structures associated with the new defined operations on soft sets. Aktaş and Çağman [7] evidenced that soft sets generalize both fuzzy sets and rough sets and they are pragmatic soft sets in group theory. After the remarkable start of the era of soft sets, many researchers put their share in the progress of the theory of soft sets, for example, Acar et al. [8] presented the notion of soft rings, Sezer and Atagün [9] originated soft vector spaces, Ali et al. [10] represented graphs based on neighbourhoods and soft sets, Shabir and Naz [11] opened the notion of soft topological spaces, Sezer et al. [12] worked on soft intersection semigroups, Ali et al. [13] initiated the notion of lattice ordered soft sets, and Cagman [14] initiated a new approach in soft set theory.The applications of soft sets in decision making were initiated by Maji et al. [15] in 2002. Since then many other authors contributed in this direction, for example, Cagman and Enginoglu [16, 17] and Kong et al. [18] did copious work in the applications of soft sets in decision making. For more studies and applications of soft sets, one may study [19–23].The notion of fuzzy soft sets was introduced by Maji et al. [24]. Deng and Wang [25] espoused object parameter methodology for predicting unknown data in incomplete fuzzy soft sets. Naz and Shabir [26] instigated the study of algebraic structures associated with fuzzy soft sets. Roy and Maji [27] toiled on fuzzy soft set theoretic approach to decision-making problems. For more applications of fuzzy soft sets in decision making and other fields, one may study [28–34].The notion of bipolar-valued fuzzy sets was instigated by Lee [35] in 2000. Abdullah et al. [36] commenced the perception of bipolar fuzzy soft sets and applied this perception in a decision-making problem. In 2013, Shabir and Naz [37] instigated the idea of bipolar soft sets, and then keeping this concept in view, Naz and Shabir [38] familiarized the idea of fuzzy bipolar soft sets and studied their algebraic structures and their applications. In 2014, Karaaslan and Karatas [39] espoused a different methodology to introduce bipolar soft sets, and later on, Karaaslan et al. [40] toiled on bipolar soft groups. For additional work and applications of the impression of bipolarity in soft sets and allied topics, one may study [41–45].If we sum up all the above debate, then we noticed that keeping in view the association between fuzzy sets and soft sets and keeping in view the significance of bipolar-valued fuzzy sets, two attempts have been made to define bipolar soft sets: one by Shabir and Naz and the other by Karaaslan and Karatas. But if we notice, then we come to know that in both approaches, the conception of bipolar soft sets has some shortcomings, which we will discuss in our upcoming sections of the article (see Remark1). So keeping this downside of the defined bipolar soft sets, in this article, we have embraced a new approach to define bipolar soft set and we named it T-bipolar soft set. Rest of the article is organized as follows:(1)
In Section2 of the article, we have given some basic definitions to make the article self-contained and to justify redefining the notion of bipolar soft set.(2)
In Section3 of the article, the notion of T-bipolar soft sets is familiarized, its basic operational laws are given, and related results are conferred.(3)
In Section4, some algebraic structures are discussed associated with new defined T-BSSs.(4)
In Section5, some applications of T-BSSs towards decision making are discussed.(5)
In Section6, conclusion of the work presented is drawn and some future directions are discussed.


## 2. Preliminaries

In this section of the article, we will provide and deliberate some basic definitions of fuzzy sets, intuitionistic fuzzy sets, bipolar-valued fuzzy sets, soft sets, double framed soft sets, and bipolar-valued soft sets to make the article self-contained and also to justify the need to define T-bipolar soft sets. We will also debate the motivation to define T-bipolar soft sets.Definition 1 (see [1]).
LetA be a nonempty set. Then, a fuzzy set in A is characterized by a membership function f:A⟶0,1.Definition 2 (see [3]).
LetA be a nonempty set of parameters and U be an initial universe. Then, a soft set F,A over U is characterized by a set valued function F:A⟶PU.Definition 3 (see [46]).
LetA be a nonempty set. Then, an intuitionistic fuzzy set in A is characterized by two functions f:A⟶0,1 and g:A⟶0,1, where f is called a membership function and g is called nonmembership function. The condition that the sum of the values of f and g must belong to 0,1 is the part of the definition of intuitionistic fuzzy set.Definition 4 (see [47]).
LetA be a nonempty set of parameters and U be an initial universe. Then, a double framed soft set over U is characterized by two set valued functions F:A⟶PU and G:A⟶PU.Definition 5 (see [35]).
LetA be a nonempty set. Then, a bipolar-valued fuzzy set in A is characterized by two functions f:A⟶0,1 and g:A⟶−1,0, where for some x∈A,fx denotes the satisfaction degree of the element x to the property corresponding to the bipolar-valued fuzzy set, which we denote by f,g,A, and further gx denotes the satisfaction degree of x to some implicit counterproperty of the bipolar-valued fuzzy set f,g,A.Definition 6 (see [37]).
LetA be a nonempty set of parameters, ¬A=¬x:x∈A denotes the NOT set of A, and let U be an initial universe. Then, a bipolar soft set, denoted by F,G,A, over U is characterized by two set valued functions F:A⟶PU and G:¬A⟶PU such that for all x∈A,Fx∩G¬x=∅ (empty set).Definition 7 (see [39]).
LetA be a parameter set and A1 and A2 be two nonempty subsets of A such that A1∪A2=A and A1∩A2=∅. Then, the triplet F,G,A is thought to be a bipolar soft set over U, where F and G are set valued mappings given by F:A1⟶PU and G:A2⟶PU such that Fx∩Gfx=∅, where f:A1⟶A2 is a bijective function.Remark 1.
From above definitions, we note that(1)
The definitions of fuzzy sets and that of soft sets have same characteristics in the sense that(i)
Both are characterized by a single function(ii)
Both have a single set as domain set(iii)
Both have a single set, which is a lattice in either case, as codomain set(2)
The definitions of intuitionistic fuzzy sets and that of double framed soft sets have same characteristics in the sense that(i)
Both are characterized by two functions(ii)
Both have a single set as domain set for both the functions(iii)
Both have a single set, which is a lattice in either case, as codomain set for both the functions(3)
But this is not the case for bipolar-valued fuzzy sets as compared to the definitions of bipolar soft sets defined in [37, 39]
All this dialogue demonstrates that the space to define bipolar soft set has not yet been filled. As every definition in mathematics has its own importance and it does not mean that the already existing definitions of bipolar soft sets are of no use and the proposed definition of bipolar soft set will nullify the existing definitions, but the purpose to redefine the notion of bipolar soft set is, one to elaborate the notion of bipolarity in soft sets more affectively and the other is, as there is, up to the best of our knowledge, no such type of situation is discussed in soft sets earlier.
Now, we deliberate some basic definitions connected to soft sets. In this section, from now onwards,E will denote a set of parameters, A,B,C…⊆E, and U will denote an initial universe. Further, the set of all soft sets over U will be denoted by SSU.Definition 8 (see [14]).
LetF1,A and F2,B∈SSU. Then, F1,A is called a soft subset of F2,B if(i)
A⊆B(ii)
For alla∈A,F1a⊆F2a
Then, we writeF1,A⊆F2,B. F1,A and F2,B are said to be soft equal if and only if F1,A⊆F2,B and F2,B⊆F1,A. Then, we write F1,A=F2,B.Definition 9 (see [14]).
LetF,A∈SSU. Then,(i)
Complement ofF,A is designated and specified by F,Ac=Fc,A where Fca=U−Fa, for all a∈A(ii)
F,A is said to be null if and only if for all a∈A,Fa=∅(iii)
F,A is said to be absolute if and only if for all a∈A,Fa=UDefinition 10 (see [4]).
LetF1,A and F2,B∈SSU. Then,(i)
“AND” product ofF1,A and F2,B is designated and demarcated by F1,A∧F2,B=F3,A×B where F3a,b=F1a∩F2b for all a,b∈A×B(ii)
“OR” product ofF1,A and F2,B is designated and demarcated by F1,A∨F2,B=F3,A×B where F3a,b=F1a∪F2b for all a,b∈A×BDefinition 11 (see [4]).
LetF1,A and F2,B∈SSU. Then, “union” (which we may also call extended union) of F1,A and F2,B is designated and demarcated by F1,A∪EF2,B=H,A∪B, where(1)He=F1e,ife∈A−B,F2e,ife∈B−A,F1e∪F2e,ife∈A∩B.Definition 12 (see [5]).
LetF1,A and F2,B∈SSU. Then, “extended intersection” of F1,A and F2,B is designated and demarcated by F1,A∩EF2,B=H,A∪B, where(2)He=F1e,ife∈A−B,F2e,ife∈B−A,F1e∩F2e,ife∈A∩B.Definition 13 (see [5]).
LetF1,A and F2,B∈SSU such that A∩B is nonempty. Then,(i)
“Restricted union” ofF1,A and F2,B is designated and demarcated by F1,A∪RF2,B=H,A∩B, where He=F1e∪F2e(ii)
“Restricted intersection” ofF1,A and F2,B is designated and demarcated by F1,A∩RF2,B=H,A∩B, where He=F1e∩F2e


## 3. T-Bipolar Soft Set

In this section, we will familiarize the perception of T-bipolar soft set (T-BSS), we will delineate binary operations for T-BSSs, and we will also deliberate some basic properties and some results concomitant with these concepts. First, we contemplate the succeeding example.Example 1.
Let us consider the case, where a researcher Dr. Shabir wants to submit his four research articlesa1 (on homological algebra), a2 (on fuzzy sets), a3 (on soft sets), and a4 (on rough sets) in some research journals. For the purpose, he has to propose some potential referees and also he has the option to oppose some referees. Keeping in view all the aspects, he prepared a set X=x1,x2,x3,x4,x5,x6 of some proposed referees as well as a set Y=y1,y2,y3,y4,y5 of some referees to oppose. Hence, in this case, he has under consideration the set U=x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5 of all referees. For each of his article, he selects some referees from X to propose and selects some referees from Y to oppose. Keeping in view all the aspects for the article(i)
a1 (on homological algebra), he decided to propose x2,x3,x4 and he opposed y1,y2(ii)
a2 (on fuzzy sets), he decided to propose x1,x3,x6 and he opposed y2,y4(iii)
a3 (on soft sets), he decided to propose x4,x5 and he opposed y3(iv)
a4 (on rough sets), he decided to propose x1,x5,x6 and he opposed y1,y4,y5
Note that all this information can be modeled mathematically as follows: letA=a1,a2,a3,a4, X=x1,x2,x3,x4,x5,x6, and Y=y1,y2,y3,y4,y5. Define F:A⟶PX as Fa1=x2,x3,x4,Fa2=x1,x3,x6,Fa3=x4,x5,Fa4=x1,x5,x6, and G:A⟶PY as Ga1=y1,y2,Ga2=y2,y4,Ga3=y3,Ga4=y1,y4,y5.
Here, it can be perceived that both the functionsF and G have common domain A, and codomains of F and G have nothing in common except the empty set ϕ. Further we notice that same is the case in bipolar-valued fuzzy sets. Hence, we have the following definition.Definition 14.
LetE be a set of parameters, A⊆E, and U be an initial universe, X⊂U and Y=U−X. Then, a triplet F,G,A is said to be a T-BSS over U, where F and G are set valued mappings given by F:A⟶PX and G:A⟶PY. In this case, we write F,G,A=a,Fa,Ga:Fa∈PXandGa∈PY or simply F,G,A=a,Fa,Ga. The collection of all T-BSSs over U is denoted by T−BSSU.Remark 2.
LetA=a1,a2,a3,…,al⊆E,X=x1,x2,x3,…,xm, Y=y1,y2,y3,…,yn, and F,G,A be corresponding T-BSS. Then, we can represent F,G,A as follows (Table 1).Table 1
Tabular form of a T-bipolar soft set.
F,G,Ax1,y1x1,y2…x1,ynx2,y1x2,y2…x2,yn…xm,y1xm,y2…xm,yna1ζ111ζ112…ζ11nζ121ζ122…ζ12n…ζ1m1ζ1m2…ζ1mna2ζ211ζ212…ζ21nζ221ζ222…ζ22n…ζ2m1ζ2m2…ζ2mna3ζ311ζ312…ζ31nζ321ζ322…ζ32n…ζ3m1ζ3m2…ζ3mn…………………………………………………………………………alζl11ζl12…ζl1nζl21ζl22…ζl2n…ζlm1ζlm2…ζlmn(3)ζijk=μj,νk=0,0ifxj∉Faiandyk∉Fai,1,0ifxj∈Faiandyk∉Fai,0,1ifxj∉Faiandyk∈Fai,1,1ifxj∈Faiandyk∈Fai,ζijk∗=μj,ζijk°=νk.Example 2.
A university wants to appoint a permanent faculty member from the setA=a1,a2,a3,a4,a5 of visiting faculty members. For the purpose, the university authorities constitute two panels X=x1,x2,x3,x4 and Y=y1,y2,y3 of experts, where panel X consists of members from outside the university and panel Y consists of members from inside the university. Further each member of the panel X will decide about each candidate by considering his/her experience, number of research publications, number of conferences attended, etc., while each member of the panel Y will decide about each candidate by considering his/her regularity and punctuality, attitude towards other faculty members, and behavior with students during class. Now the university authorities decided the selection criteria that towards each candidate each member of the panel X will have to select a candidate by keeping in view his/her positive points while each member of the panel Y has to reject a candidate by keeping in view his/her negative points. According to the decisions taken by the members of the panels X, the experts x1 and x3 are in favor to select the candidate a1 while x2 and x4 decided to remain neutral for the candidate a1. Similarly the decisions taken by the members of the panels Y, the member y3 is not in favor to select the candidate a1 while y1 and y2 decided to remain neutral for the candidate a1. Hence, for the candidate a1, the situation can be modeled as a1,x1,x3,y3. Now keeping under consideration the decisions taken by all the members from the panels X and Y, the result can be modeled mathematically as given in the following T-BSS:F,G,A=a1,x1,x3,y3,a2,x1,x3,x4,y1,a3,x1,x4,y1,y3,a4,x2,x3,x4,y2,y3,a5,x2,x4,y2.
Tabular form of theF,G,A is given as follows (Table 2).Table 2
Tabular form of the T-bipolar soft setF,G,A.
F,G,Ax1,y1x1,y2x1,y3x2,y1x2,y2x2,y3x3,y1x3,y2x3,y3x4,y1x4,y2x4,y3a1(1, 0)(1, 0)(1, 1)(0, 0)(0, 0)(0, 1)(1, 0)(1, 0)(1, 1)(0, 0)(0, 0)(0, 1)a2(1, 1)(1, 0)(1, 0)(0, 1)(0, 0)(0, 0)(1, 1)(1, 0)(1, 0)(1, 1)(1, 0)(1, 0)a3(1, 1)(1, 0)(1, 1)(0, 1)(0, 0)(0, 1)(0, 1)(0, 0)(0, 1)(1, 1)(1, 0)(1, 1)a4(0, 0)(0, 1)(0, 1)(1, 0)(1, 1)(1, 1)(1, 0)(1, 1)(1, 1)(1, 0)(1, 1)(1, 1)a5(0, 0)(0, 1)(0, 0)(1, 0)(1, 1)(1, 0)(0, 0)(0, 1)(0, 0)(1, 0)(1, 1)(1, 0)Definition 15.
LetF1,G1,A∈T−BSSU. Then, F1,G1,A is said to be T-bipolar soft subset of F2,G2,B if(i)
A⊆B(ii)
For alla∈A,F1a⊆F2a and G2a⊆G1a.
Then, we writeF1,G1,A⊆F2,G2,B. F1,G1,A and F2,G2,B are said to be equal if and only if F1,G1,A⊆F2,G2,B and F2,G2,B⊆F1,G1,A. Then, we write F1,G1,A=F2,G2,B.Definition 16.
LetF,G,A∈T−BSSU. Then,(i)
Complement ofF,G,A is denoted and given by F,G,Ac=Fc,Gc,A=a,Fca=X−Fa,Gca=Y−Ga.(ii)
F,G,A is said to be null if and only if for all a∈A,Fa=∅ and Ga=Y. In our study, it will further be designated by ϕ, that is, ϕ=a,∅,Y.(iii)
F,G,A is said to be absolute if and only if for all a∈A,Fa=X and Ga=∅. In our study, it will further be designated by A that is, A=a,X,∅.Definition 17.
LetF1,G1,A,F2,G2,B∈T−BSSU. Then,(i)
“AND” product ofF1,G1,A and F2,G2,B is designated and demarcated by(4)F1,G1,A∧F2,G2,B=<a,b,F1a∩F2b,G1a∪G2b>:a,b∈A×B.(ii)
“OR” product ofF1,G1,A and F2,G2,B is designated and demarcated by(5)F1,G1,A∨F2,G2,B=<a,b,F1a∪F2b,G1a∩G2b>:a,b∈A×B.Proposition 1.
LetF1,G1,A, F2,G2,BF2,G2,B∈T−BSSU. Then,(i)
F1,G1,A∧F2,G2,Bc=F1,G1,Ac∨F2,G2,Bc(ii)
F1,G1,A∨F2,G2,Bc=F1,G1,Ac∧F2,G2,BcProof.
(i)
F1,G1,A∧F2,G2,Bc=a,b,F1a∩F2b,G1a∪G2bc=a,b,X−F1a∩F2b,Y−G1a∪G2b=a,b,X−F1a∪X−F2b,Y−G1a∩Y−G2b=<a,b,F1ca∪F2cb,G1ca∩G2cb=F1,G1,Ac∨F2,G2,Bc.(ii)
Similar to part (i).Definition 18.
LetF1,G1,A, F2,G2,B∈T−BSSU. Then, “extended union” of F1,G1,A and F2,G2,B is designated and demarcated by F1,G1,A∪EF2,G2,B=H,K,A∪B, where(6)He=F1e,ife∈A−B,F2e,ife∈B−A,F1e∪F2e,ife∈A∩B,Ke=G1e,ife∈A−B,G2e,ife∈B−A,G1e∩G2e,ife∈A∩B.Definition 19.
LetF1,G1,A, F2,G2,B∈T−BSSU Then, “extended intersection” of F1,G1,A and F2,G2,B is designated and demarcated by(7)F1,G1,A∩EF2,G2,B=H,K,A∪B,where(8)He=F1e,ife∈A−B,F2e,ife∈B−A,F1e∩F2e,ife∈A∩B,Ke=G1e,ife∈A−B,G2e,ife∈B−A,G1e∪G2e,ife∈A∩B.Proposition 2.
For any T-BSSsF1,G1,A, F2,G2,B, and F3,G3,C,(i)
F1,G1,A∩Eϕ=ϕ,F1,G1,A∪Eϕ=F1,G1,A,F1,G1,A∩EA=F1,G1,A,F1,G1,A∪EA=A(ii)
F1,G1,A∩EF1,G1,A=F1,G1,A,F1,G1,A∪EF1,G1,A=F1,G1,A(iii)
F1,G1,A∩EF2,G2,B=F2,G2,B∩EF1,G1,A,F1,G1,A∪EF2,G2,B=F2,G2,B∪EF1,G1,A(iv)
F1,G1,A∪EF2,G2,B∪EF3,G3,C=F1,G1,A∪EF2,G2,B∪EF3,G3,C(v)
F1,G1,A∩EF2,G2,B∪EF1,G1,A=F1,G1,A,F1,G1,A∪EF2,G2,B∩EF1,G1,A=F1,G1,A(vi)
F1,G1,Acc=F1,G1,A,F1,G1,A∩EF1,G1,Ac=ϕ,F1,G1,A∪EF1,G1,Ac=A(vii)
F1,G1,A∩EF2,G2,Bc=F1,G1,Ac∪EF2,G2,Bc,F1,G1,A∪EF2,G2,Bc=F1,G1,Ac∩EF2,G2,BcProof.
We prove (iv) and (vii); rest are straightforward.(iv)
Whenx∈A,x∉B, and x∈C, then(9)F1,G1,A∩EF2,G2,B∩EF3,G3,C=F1,G1,A∩EF3,G3,C=<x∈A∩C:F1x∩F3x,G1x∪G3x>,F1,G1,A∩EF2,G2,B∩EF3,G3,C=F1,G1,A∩EF3,G3,C=x∈A∩C:F1x∩F3x,G1x∪G3x.Whenx∈A,x∈B, and x∉C, then(10)F1,G1,A∩EF2,G2,B∩EF3,G3,C=F1,G1,A∩EF2,G2,B=x∈A∩B:F1x∩F2x,G1x∪G2x,F1,G1,A∩EF2,G2,B∩EF3,G3,C=F1,G1,A∩EF2,G2,B=x∈A∩B:F1x∩F2x,G1x∪G2x.Whenx∉A,x∈B, and x∈C, then(11)F1,G1,A∩EF2,G2,B∩EF3,G3,C=F2,G2,B∩EF3,G3,C=x∈B∩C:F2x∩F3x,G2x∪G3x,F1,G1,A∩EF2,G2,B∩EF3,G3,C=F2,G2,B∩EF3,G3,C=x∈B∩C:F2x∩F3x,G2x∪G3x.Whenx∉A,x∈B, and x∈C, then result is obvious. Hence, it concludes that F1,G1,A∩EF2,G2,B∩EF3,G3,C=F1,G1,A∩EF2,G2,B∩EF3,G3,C. SimilarlyF1,G1,A∪EF2,G2,B∪EF3,G3,C=F1,G1,A∪EF2,G2,B∪EF3,G3,C.(vii)
Whenx∈A and x∉B, then(12)F1,G1,A∩EF2,G2,Bc=F1,G1,Ac,F1,G1,Ac∪EF2,G2,Bc=F1,G1,Ac.Whenx∉A and x∈B, then(13)F1,G1,A∩EF2,G2,Bc=F2,G2,Bc,F1,G1,Ac∪EF2,G2,Bc=F2,G2,Bc.Whenx∈A and x∈B, then the result is a trivial case. Hence, in either case F1,G1,A∩EF2,G2,Bc=F1,G1,Ac∪EF2,G2,BcSimilarly, F1,G1,A∪EF2,G2,Bc=F1,G1,Ac∩EF2,G2,Bc.Remark 3.
For any arbitraryF1,G1,AF2,G2,B, F3,G3,C∈T−BSSU, it is not necessary that(i)
F1,G1,A∩EF2,G2,B∪EF3,G3,C=F1,G1,A∩EF2,G2,B∪EF1,G1,A∩EF3,G3,C(ii)
F1,G1,A∪EF2,G2,B∩EF3,G3,C=F1,G1,A∪EF2,G2,B∩EF1,G1,A∪EF3,G3,CExample 3.
LetE=e1,e2,e3,e4,e5,A=e1,e2,e3,B=e3,e4,C=e4,e5,U=x1,x2,x3,x4,x5, X=x1,x2,x3, and Y=x4,x5.
Now let(14)F1,G1,A=e1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x3,x4,F2,G2,B=e3,x2,x3,x5,e4,x1,x2,x4,x5,F3,G3,C=e4,x1,x2,x3,∅,e5,x1,x2,x5.
Now(15)F1,G1,A∩EF2,G2,B∪EF3,G3,C=F1,G1,A∩Ee3,x2,x3,x5,e4,x1,x2,x3,∅,e5,x1,x2,x5,=e1,x1,x2,x4,e2,x1,x4,x5,e3,x3,x4,x5,e4,x1,x2,x3,∅,e5,x1,x2,x5.
Next(16)F1,G1,A∩EF2,G2,B∪EF1,G1,A∩EF3,G3,Ce1,x1,x2,x4,e2,x1,x4,x5,e3,x3,x4,x5,e4,x1,x2,x4,x5∪Ee1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x3,x4,e4,x1,x2,x3,∅,e5,x1,x2,x5=e1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x3,x4,e4,x1,x2,x3,∅,e5,x1,x2,x5,⟹F1,G1,A∩EF2,G2,B∪EF3,G3,C≠F1,G1,A∩EF2,G2,B∪EF1,G1,A∩EF3,G3,C.
Now(17)F1,G1,A∪EF2,G2,B∩EF3,G3,C=F1,G1,A∪Ee3,x2,x3,x5,e4,x1,x2,x4,x5,e5,x1,x2,x5=e1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x2,x3,∅,e4,x1,x2,x4,x5,e5,x1,x2,x5.
Next(18)F1,G1,A∪EF2,G2,B∩EF1,G1,A∪EF3,G3,C=e1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x2,x3,∅,e4,x1,x2,x4,x5∩​Ee1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x3,x4,e4,x1,x2,x3,∅,e5,x1,x2,x5=e1,x1,x2,x4,e2,x1,x4,x5,e3,x1,x3,x4,e4,x1,x2,x4,x5,e5,x1,x2,x5⟹F1,G1,A∪EF2,G2,B∩EF3,G3,C≠F1,G1,A∪EF2,G2,B∩EF1,G1,A∪EF3,G3,C.Definition 20.
LetF1,G1,AF2,G2,B∈T−BSSU with A∩B≠∅. Then,(i)
“Restricted union” ofF1,G1,A and F2,G2,B is designated and demarcated by(19)F1,G1,A∪RF2,G2,B=c,F1c∪F2c,G1c∩G2c:c∈A∩B.(ii)
“Restricted intersection” ofF1,G1,A and F2,G2,B is designated and demarcated by(20)F1,G1,A∩RF2,G2,B=c,F1c∩F2c,G1c∪G2c:c∈A∩B.Proposition 3.
For any T-BSSsF1,G1,A, F2,G2,B, and F3,G3,C,(i)
F1,G1,A∩Rϕ=ϕ,F1,G1,A∪Rϕ=F1,G1,A,F1,G1,A∩RA=F1,G1,A,F1,G1,A∪RA=A(ii)
F1,G1,A∩RF1,G1,A=F1,G1,A,F1,G1,A∪RF1,G1,A=F1,G1,A(iii)
F1,G1,A∩RF2,G2,B=F2,G2,B∩RF1,G1,A,F1,G1,A∪RF2,G2,B=F2,G2,B∪RF1,G1,A,F1,G1,A∪RF2,G2,B=F2,G2,B∪RF1,G1,A,F1,G1,A∪RF2,G2,B=F2,G2,B∪RF1,G1,A(iv)
F1,G1,A∩RF2,G2,B∩RF3,G3,C=F1,G1,A∩RF2,G2,B∩RF3,G3,C, F1,G1,A∪RF2,G2,B∪RF3,G3,C=F1,G1,A∪RF2,G2,B∪RF3,G3,C(v)
F1,G1,A∩RF2,G2,B∪RF3,G3,C=F1,G1,A∩RF2,G2,B∪RF1,G1,A∩RF3,G3,C, F1,G1,A∪RF2,G2,B∩RF3,G3,C=F1,G1,A∪RF2,G2,B∩RF1,G1,A∪RF3,G3,C(vi)
F1,G1,A∩RF2,G2,B∪RF1,G1,A=F1,G1,A,F1,G1,A∪RF2,G2,B∩RF1,G1,A=F1,G1,A(vii)
F1,G1,Acc=F1,G1,A,F1,G1,A∩RF1,G1,Ac=ϕ,F1,G1,A∪RF1,G1,Ac=A(viii)
F1,G1,A∩RF2,G2,Bc=F1,G1,Ac∪RF2,G2,Bc,F1,G1,A∪RF2,G2,Bc=F1,G1,Ac∩RF2,G2,BcProof.
Straightforward.


## 4. Algebraic Structures Associated with T-BSSs

In this section, we will discuss some algebraic structures associated with T-BSSs. Recall thatT−BSSU denotes the collection of all T-BSSs over U. Now in this section, T−BSSUA denotes the collection of all T-BSSs over U with domain A.Proposition 4.
For anyΔ∈∩E,∪E,∩R,∪R, T−BSSU,Δ is a commutative semigroup whose every element is idempotent.Proof.
The proof is straightforward by using the Proposition2 and Proposition 3.Proposition 5.
T−BSSU,∩R,∪R is a commutative semiring.Proof.
The proof follows from the definitions of restricted intersection of T-BSSs, restricted union of T-BSSs, and parts (iv) and (v) of Proposition3.Remark 4.
It follows from Proposition2 and Remark 3 that T−BSSU,∩E,∪E is not a semiring.Proposition 6.
T−BSSUA,∩E,∪E is a commutative semiring.Proof.
This is straightforward as extended intersection of T-BSSs and extended union of T-BSSs satisfy the distributive laws, which usually do not hold (Remark3), if all the T-BSSs have same domain A.Proposition 7.
T−BSSU,∩E,∪E, c, ϕ,A is a bounded lattice.Proof.
The result follows from conditions (i)–(v) of Proposition2.Proposition 8.
T−BSSU,∩R,∪R, c, ϕ,A is a bounded distributive lattice.Proof.
The result follows from Proposition3.


## 5. Applications of T-BSSs in Decision Making

In this section, we will discuss some decision-making problems by using T-BSSs. We will discuss decision-making problems in the absence of weights, in the presence of weights selected randomly, and in the presence of weights taken as discussed in [48].Definition 21.
LetA=a1,a2,a3,…,al⊆E,X=x1,x2,x3,…,xm, Y=y1,y2,y3,…,yn, and F,G,A be corresponding T-BSS. Then, score of ai,1≤i≤l is denoted and defined as Si=σi¯−σi¯, where σi¯=∑j,kζijk∗ and σi¯=∑j,kζijk°.Definition 22.
LetA=a1,a2,a3,…,al⊆E,X=x1,x2,x3,…,xm, Y=y1,y2,y3,…,yn, and F,G,A be corresponding T-BSS. Then, ai,1≤i≤l is said to be optimal if and only if Si>Si′, for all i′≠i.Example 4.
Consider Example 2 withF,G,A=a1,x1,x3,y3,a2,x1,x2,x4,y1,a3,x1,x4,y1,y3,a4,x2,x3,x4,y2,y3,a5,x2,x4,y2, with tabular form as in Table 3.
Then, the score values are given in Table4.
Then, according to the Algorithm1, the candidate “a2” will be selected.Table 3
Tabular expression of the T-bipolar soft setF,G,A.
F,G,Ax1,y1x1,y2x1,y3x2,y1x2,y2x2,y3x3,y1x3,y2x3,y3x4,y1x4,y2x4,y3a1(1, 0)(1, 0)(1, 1)(0, 0)(0, 0)(0, 1)(1, 0)(1, 0)(1, 1)(0, 0)(0, 0)(0, 1)a2(1, 1)(1, 0)(1, 0)(1, 1)(1, 0)(1, 0)(0, 1)(0, 0)(0, 0)(1, 1)(1, 0)(1, 0)a3(1, 1)(1, 0)(1, 1)(0, 1)(0, 0)(0, 1)(0, 1)(0, 0)(0, 1)(1, 1)(1, 0)(1, 1)a4(0, 0)(0, 1)(0, 1)(1, 0)(1, 1)(1, 1)(1, 0)(1, 1)(1, 1)(1, 0)(1, 1)(1, 1)a5(0, 0)(0, 1)(0, 0)(1, 0)(1, 1)(1, 0)(0, 0)(0, 1)(0, 0)(1, 0)(1, 1)(1, 0)Table 4
Scores ofa1,a2,a3,a4,a5.
F,G,Aσi¯σi¯Sia1642a2945a368−2a4981a5642Algorithm 1: Finding an optimal value for a given data.
(1
Here we state an algorithm for finding an optimal value for a given data.Step 1. Write given T-BSS in tabular form.Step 2. Calculate S1,S2,S3,…,Sl.Step 3. Put maxiSi=Sp.Step 4. Sp is optimal value.Remark 5.
Sometimes in decision making, some decision makers have less importance as compared to other decision makers, for example, to decide about admission policy of a school, a meeting was called in which four persons participated who were the school owner, school principal, school vice principal, and accountant of the school. Now here it is clear that all the decision makers have not the same weightage. So, in decision making, the weightage of a decision maker also matters a lot. So, now we establish an algorithm to handle a decision-making problem in the presence of weights.Definition 23.
LetA=a1,a2,a3,…,al⊆E,X=x1,x2,x3,…,xm, and Y=y1,y2,y3,…,yn such that each xj has weight wj and each yk has weight wk′ with ∑jwj=1&∑kwk′=1. Then, for all i, Si=σi¯−σi¯, where σi¯=∑j,kwjζijk∗ and σi¯=∑j,kwk′ζijk°.Remark 6.
The above stated algorithm (Algorithm1) also works in the present case.Example 5.
Consider Example4, with Table 5 representing weight values and Table 6 representing score values.
Then, according to the new criteria, the candidate “a1” will be selected.Table 5
Weight values.
xjWeight ofxjykWeight ofykx10.3y10.4x20.2y20.4x30.4y30.2x40.1——Table 6
Scores ofa1,a2,a3,a4,a5.
F,G,Aσi¯σi¯Sia12.10.81.3a21.81.60.2a31.22.4−1.2a42.12.4−0.3a50.91.6−0.7Remark 7.
According to Xu [48], the weight vector w=w1,w2,w3,…,wnT can also be calculated as(21)wi=e−j−μn2/2σn2∑j=1ne−j−μn2/2σn2,where(22)μn=1+n2,σn=1n∑i=1ni−μn2.In this case, Example5 takes the following form (Tables 7 and 8).Then, in this case, the candidate “a2” will be selected.Table 7
Weight values.
xjWeight ofxjykWeight ofykx10.1550y10.2429x20.3450y20.5142x30.3450y30.2429x40.1550——Table 8
Scores ofa1,a2,a3,a4,a5.
F,G,Aσi¯σi¯Sia11.50.97160.5284a21.9650.97160.9934a30.931.9432−1.0132a42.5353.0284−0.4934a51.52.0568−0.5568


## 6. Conclusion and Future Prospective

Keeping in view the shortcoming in predefined notions of BSSs, in this article, we have defined and discussed the notion of T-BSS. Then, rendering to new definition, we have defined different binary operations for T-BSSs and then we conferred some results associated with these binary operations. We evidenced the existence of bounded lattices and De Morgan algebras interrelated with these binary operations. We also established some algorithms to solve decision-making problems and then solved the problems from daily life by using these algorithms. In future, this work can be extended to its applications in algebraic structures and in rough set theory.



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