A Novel Approach towards Bipolar Soft Sets and Their Applications

Mahmood, Tahir

doi:https://doi.org/10.1155/2020/4690808

Journal of Mathematics

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Decision Making Based on Intuitionistic Fuzzy Sets and their Generalizations

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Research Article | Open Access

Volume 2020 | Article ID 4690808 | https://doi.org/10.1155/2020/4690808

A Novel Approach towards Bipolar Soft Sets and Their Applications

Tahir Mahmood¹

Academic Editor: Ali Jaballah

Received09 Aug 2020

Revised14 Sept 2020

Accepted24 Sept 2020

Published24 Oct 2020

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.

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 Remark 1). 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 Section 2 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 Section 3 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 Section 4, some algebraic structures are discussed associated with new defined T-BSSs.(4)In Section 5, some applications of T-BSSs towards decision making are discussed.(5)In Section 6, 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]). Let be a nonempty set. Then, a fuzzy set in is characterized by a membership function .

Definition 2 (see [3]). Let be a nonempty set of parameters and be an initial universe. Then, a soft set over is characterized by a set valued function .

Definition 3 (see [46]). Let be a nonempty set. Then, an intuitionistic fuzzy set in is characterized by two functions and , where is called a membership function and is called nonmembership function. The condition that the sum of the values of and must belong to is the part of the definition of intuitionistic fuzzy set.

Definition 4 (see [47]). Let be a nonempty set of parameters and be an initial universe. Then, a double framed soft set over is characterized by two set valued functions and .

Definition 5 (see [35]). Let be a nonempty set. Then, a bipolar-valued fuzzy set in is characterized by two functions and , where for some denotes the satisfaction degree of the element to the property corresponding to the bipolar-valued fuzzy set, which we denote by , and further denotes the satisfaction degree of to some implicit counterproperty of the bipolar-valued fuzzy set .

Definition 6 (see [37]). Let be a nonempty set of parameters, denotes the NOT set of , and let be an initial universe. Then, a bipolar soft set, denoted by over is characterized by two set valued functions and such that for all (empty set).

Definition 7 (see [39]). Let be a parameter set and and be two nonempty subsets of such that and . Then, the triplet is thought to be a bipolar soft set over , where and are set valued mappings given by and such that , where 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, will denote a set of parameters, , and will denote an initial universe. Further, the set of all soft sets over will be denoted by .

Definition 8 (see [14]). Let and Then, is called a soft subset of if(i)(ii)For all Then, we write . and are said to be soft equal if and only if and . Then, we write .

Definition 9 (see [14]). Let . Then,(i)Complement of is designated and specified by where , for all (ii) is said to be null if and only if for all (iii) is said to be absolute if and only if for all

Definition 10 (see [4]). Let and . Then,(i)“AND” product of and is designated and demarcated by where for all (ii)“OR” product of and is designated and demarcated by where for all

Definition 11 (see [4]). Let and . Then, “union” (which we may also call extended union) of and is designated and demarcated by , where

Definition 12 (see [5]). Let and . Then, “extended intersection” of and is designated and demarcated by , where

Definition 13 (see [5]). Let and such that is nonempty. Then,(i)“Restricted union” of and is designated and demarcated by , where (ii)“Restricted intersection” of and is designated and demarcated by , where

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 articles (on homological algebra), (on fuzzy sets), (on soft sets), and (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 of some proposed referees as well as a set of some referees to oppose. Hence, in this case, he has under consideration the set of all referees. For each of his article, he selects some referees from to propose and selects some referees from to oppose. Keeping in view all the aspects for the article(i) (on homological algebra), he decided to propose and he opposed (ii) (on fuzzy sets), he decided to propose and he opposed (iii) (on soft sets), he decided to propose and he opposed (iv) (on rough sets), he decided to propose and he opposed Note that all this information can be modeled mathematically as follows: let , , and . Define as , and as .
Here, it can be perceived that both the functions and have common domain , and codomains of and have nothing in common except the empty set . Further we notice that same is the case in bipolar-valued fuzzy sets. Hence, we have the following definition.

Definition 14. Let be a set of parameters, , and be an initial universe, and . Then, a triplet is said to be a T-BSS over , where and are set valued mappings given by and . In this case, we write or simply . The collection of all T-BSSs over is denoted by .

Remark 2. Let , , and be corresponding T-BSS. Then, we can represent as follows (Table 1).

Example 2. A university wants to appoint a permanent faculty member from the set of visiting faculty members. For the purpose, the university authorities constitute two panels and of experts, where panel consists of members from outside the university and panel consists of members from inside the university. Further each member of the panel 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 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 will have to select a candidate by keeping in view his/her positive points while each member of the panel has to reject a candidate by keeping in view his/her negative points. According to the decisions taken by the members of the panels , the experts and are in favor to select the candidate while and decided to remain neutral for the candidate Similarly the decisions taken by the members of the panels the member is not in favor to select the candidate while and decided to remain neutral for the candidate . Hence, for the candidate , the situation can be modeled as . Now keeping under consideration the decisions taken by all the members from the panels and the result can be modeled mathematically as given in the following T-BSS:.
Tabular form of the is given as follows (Table 2).

Definition 15. Let . Then, is said to be T-bipolar soft subset of if(i)(ii)For all and .Then, we write . and are said to be equal if and only if and . Then, we write .

Definition 16. Let . Then,(i)Complement of is denoted and given by .(ii) is said to be null if and only if for all and . In our study, it will further be designated by , that is, .(iii) is said to be absolute if and only if for all and . In our study, it will further be designated by that is, .

Definition 17. Let . Then,(i)“AND” product of and is designated and demarcated by(ii)“OR” product of and is designated and demarcated by

Proposition 1. Let , . Then,(i)(ii)

Proof. (i)(ii)Similar to part (i).

Definition 18. Let , . Then, “extended union” of and is designated and demarcated by , where

Definition 19. Let , Then, “extended intersection” of and is designated and demarcated bywhere

Proposition 2. For any T-BSSs , , and ,(i)(ii)(iii)(iv)(v)(vi)(vii)

Proof. We prove (iv) and (vii); rest are straightforward.(iv)When , and , then When , and then When , and then When , and then result is obvious. Hence, it concludes that . (vii)When and , then When and , then When and then the result is a trivial case. Hence, in either case , .

Remark 3. For any arbitrary , , it is not necessary that(i)(ii)

Example 3. Let , , and .
Now letNowNextNowNext

Definition 20. Let with . Then,(i)“Restricted union” of and is designated and demarcated by(ii)“Restricted intersection” of and is designated and demarcated by

Proposition 3. For any T-BSSs , , and ,(i)(ii)(iii) (iv), (v), (vi)(vii)(viii)

Proof. Straightforward.

4. Algebraic Structures Associated with T-BSSs

In this section, we will discuss some algebraic structures associated with T-BSSs. Recall that denotes the collection of all T-BSSs over . Now in this section, denotes the collection of all T-BSSs over with domain

Proposition 4. For any , is a commutative semigroup whose every element is idempotent.

Proof. The proof is straightforward by using the Proposition 2 and Proposition 3.

Proposition 5. 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 Proposition 3.

Remark 4. It follows from Proposition 2 and Remark 3 that is not a semiring.

Proposition 6. 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 (Remark 3), if all the T-BSSs have same domain A.

Proposition 7. , ^c, is a bounded lattice.

Proof. The result follows from conditions (i)–(v) of Proposition 2.

Proposition 8. , ^c, is a bounded distributive lattice.

Proof. The result follows from Proposition 3.

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. Let , , and be corresponding T-BSS. Then, score of is denoted and defined as , where and .

Definition 22. Let , , and be corresponding T-BSS. Then, is said to be optimal if and only if , for all .

Example 4. Consider Example 2 with , with tabular form as in Table 3.
Then, the score values are given in Table 4.
Then, according to the Algorithm 1, the candidate “” will be selected.

(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 .
Step 3. Put .
Step 4. 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. Let , and such that each has weight and each has weight with . Then, for all , , where and .

Remark 6. The above stated algorithm (Algorithm 1) also works in the present case.

Example 5. Consider Example 4, with Table 5 representing weight values and Table 6 representing score values.
Then, according to the new criteria, the candidate “” will be selected.

Remark 7. According to Xu [48], the weight vector can also be calculated as

where

In this case, Example 5 takes the following form (Tables 7 and 8).

Then, in this case, the candidate “” will be selected.

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.

Data Availability

The data used in this article are artificial and hypothetical, and anyone can use these data before prior permission by just citing this article.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Copyright © 2020 Tahir Mahmood. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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