 # Bipolar Fuzzy Soft sets and its applications in decision making problem

In this article, we combine the concept of a bipolar fuzzy set and a soft set. We introduce the notion of bipolar fuzzy soft set and study fundamental properties. We study basic operations on bipolar fuzzy soft set. We define exdended union, intersection of two bipolar fuzzy soft set. We also give an application of bipolar fuzzy soft set into decision making problem. We give a general algorithm to solve decision making problems by using bipolar fuzzy soft set.

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## 1. Introduction

Complicated problems in different field like engineering, economics, environmental science, medicine and social sciences, which arising due to classical mathematical modelling and manipolating of varouis type of uncertainty. While some of traditional mathematical tool fail to solve these complicated problems. We used some mathematical modelling like fuzzy set theory , rough set theory 

, interval mathematics  and probability theory are well-known and operative tools for handling with vagueness and uncertainty, each of them has its own inherent limitations; one major fault shared by these mathematical methodologies may be due to the inadequacy of parametrization tools

.

Molodtsov,  adopted the notion of soft sets. Soft set is a new mathematical tool to describe the uncertainties. Soft set theory is powerful tool to describe uncertainties. Recently, researcher are engaged in soft set theory. Maji et al.  defined new notions on soft sets. Ali et al.  studied some new concepts of a soft set. Sezgin and Atagün  studied some new theoretical soft set operations. Majumdar and Samanta, worked on soft mappings . Choudhure et al. defined the concept of soft relation and fuzzy soft relation and then applied them to solve a number of decision- making problems. In , Aktaṣ and C̣ağman applied the concept of soft set to groups theory and adopted soft group of a group. Feng et.al, studied and applied softness to semirings. Recently, Acar studied soft rings . Jun et. al, applied the concept of soft set to BCK/BCI-algebras [10, 11, 12]. Sezgin and Atagün initiated th concept of normalistic soft groups . Zhan et.al, worked on soft ideal of BL-algebras . In , Kazancı et. al, used the concept of soft set to BCH-algebras. Sezgin et. al, studied soft nearrings . C̣ağman et al. considered two types of notions of a soft set with group, which is called group Soft intersection group softunion groups of a group . see .

Fuzzy set originally proposed by Zadeh in  of 1965. After semblance of the concept of fuzzy set, researcher given much atttention to developed fuzzy set theory. Maji et al.  introduced the concept of fuzzy soft sets. Afterwards, many researchers have worked on this concept. Roy and Maji  provided some results on an application of fuzzy soft sets in decision making problems. F. Feng et al. give application in decision making problem [31, 32]

Fuzzy set is a type of important mathematical structure to represent a collection of objects whose boundary is vague. There are several types of fuzzy set extensions in the fuzzy set theory, for example, intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, etc. bi-polar-valued fuzzy set is another an extension of fuzzy set whose membership degree range is different from the above extensions. In , Lee  initiated an extension of fuzzy set named bi-polar-valued fuzzy set. He gave two kinds of representations of the notion of ni-polar-valued fuzzy sets. In case of Bi-polar-valued fuzzy sets membership degree range is enlarged from the interval to . In a bi-polar-valued fuzzy set, the membership degree 0 indicate that elements are irrelevant to the corresponding property, the membership degrees on assigne that elements some what satisfy the property, and the membership degrees on assigne that elements somewhat satisfy the implicit counter-property .

In this article, we combine the concept of a bipolar fuzzy set and a soft set. We introduce the notion of bipolar fuzzy soft set and study fundamental properties. We study basic operations on bipolar fuzzy soft set. We define exdended union, intersection of two bipolar fuzzy soft set. We also give an application of bipolar fuzzy soft set into decision making problem. We give a general algorithm to solve decision making problems by using bipolar fuzzy soft set.

## 2. Preliminaries

In this section we provide previous concept of bipolar fuzzy sets, soft sets and fuzzy soft sets.

###### Definition 1.

A bipolar fuzzy set in a universe is an object having the form, where , . So denote for positive information and denote for negative information.

###### Definition 2.

Let be an initial universe, be the set of parameters, and is the power set of . Then is called a soft set, where .

###### Definition 3.

For two soft sets and over a common universe ,we say that is a soft subset of , denoted by , if it satisfies.

1. , is a subset of .

Similarly, is called a superset of if is a soft subset of . This relation is denoted by .

###### Definition 4.

 If and are two soft sets over a common universe U. The union of and is defined to be the soft set satisfying the following conditions: (i) : (ii) for all

 H(c) = F(c) if c∈A∖B = G(c) if c∈B∖A = F(c)∪G(c) if c∈A∩B

This relation is denoted by .

###### Definition 5.

Let and be two soft sets over a common universe such that . The restricted intersection of and is defined to be the soft set and , . We write .

###### Definition 6.

 Let be an initial universe, be the set of all parameters, and is the collection of all fuzzy subsets of . Then is called fuzzy soft set, where .

###### Definition 7.

 If and are two fuzzy soft sets over a common universe , then the union of and is defined to be the fuzzy soft set satisfying the following conditions: (i) : (ii) ,

 H(c) = F(c) if c∈A∖B \ = G(c) if c∈B∖A = F(c)∪G(c) if c∈A∩B

This relation is denoted by.

## 3. Bipolar Fuzzy Soft Sets.

In this section we introduce the concept of bipolar fuzzy soft set, absolute bipolar fuzzy soft set, null bipolar fuzzy soft set and complement of bipolar fuzzy soft set

###### Definition 8.

Let be a universe, a set of parameters and . Define , where is the collection of all bipolar fuzzy subsets of . Then is said to be a bipolar fuzzy soft set over a universe . It is defined by

 (F,A)={(x, μ+e(x), μ−e(x) ): for all x∈U and e∈A}
###### Example 1.

Let {, , , } be the set of four cars under consideration and {=Costly, =Beautiful, =Fuel Efficient, =Modern Technology } be the set of parameters and {}. Then,

###### Definition 9.

Let be a universe and a set of attributes.Then, is the collection of all bipolar fuzzy soft sets on with attributes from and is said to be bipolar fuzzy soft class.

###### Definition 10.

A bipolar fuzzy soft set is said to be a null bipolar fuzzy soft set denoted by empty set , if for all ,

###### Definition 11.

A bipolar fuzzy soft set is said to be an absolute bipolar fuzzy soft set. If for all ,

###### Definition 12.

The complement of a bipolar fuzzy soft set is denoted and is defined by .

###### Example 2.

Let {, , , } be the set of four bikes under consideration and { Stylish, Heavy duty, Light, Steel } be the set of parameters and {, } be subset of . Then,

The complement of the bipolar fuzzy soft set is

## 4. Bipolar Fuzzy Soft Subsets

###### Definition 13.

Let and be two bipolar fuzzy soft sets over a common universe . We say that is a bipolar fuzzy soft subset of , if and   , is a bipolar fuzzy subset of . We write .

###### Remark 1.

Every element of is presented in and do not depend on its membership or non-membership.

###### Example 3.

Let {, , , } be the set of four men under consideration and { Educated, Government employee, Businessman, Smart } be the set of parameters and {, }, { , , } be subsets of . Then,

and for all , . Then

###### Definition 14.

Let and be two bipolar fuzzy soft sets over a common universe . We say that and are bipolar fuzzy soft  equal sets if is a bipolar fuzzy soft subset of and is a bipolar fuzzy soft subset of .

## 5. Operations on Bipolar Fuzzy Soft Sets

###### Definition 15.

An intersection of two bipolar fuzzy soft sets and is a bipolar fuzzy soft set , where and is defined by and  denoted by .

###### Example 4.

Let {, , , } be the set of four bikes under consideration and {=Light, =Beautiful, =Good millage, =Modern Technology } be the set of parameters and {, }, {, , }. Then,

Then , where {, }

###### Definition 16.

Union of two bipolar fuzzy soft sets over a common universe is a bipolar fuzzy soft set , where and is defined by

 H(e) = F(e) if e∈A∖B = G(e) if e∈B∖A = F(e)∪G(e) if e∈A∩B

and  denoted by .

###### Example 5.

Let {, , , } be the set of four cars under consideration and {Costly, Beautiful, Fuel Efficient, Modern Technology } be the set of parameters and {, , }, {, , , }. Then

Then ,where {, , , }

###### Definition 17.

Let { : } be a family of bipolar fuzzy soft sets in a bipolar fuzzy soft class . Then the intersection of bipolar fuzzy soft sets in is a bipolar fuzzy soft set , where for all ,  for all

###### Definition 18.

Let { : } be a family of bipolar fuzzy soft sets in a bipolar fuzzy soft class . Then the union of bipolar fuzzy soft sets in is a bipolar fuzzy soft set, where for all

 H(e) = Fi(e) if e∈Ai = ∅ if e∉Ai
###### Definition 19.

Let  and be two bipolar fuzzy soft sets over a common universe . The extended intersection of and is defined t o be the bipolar fuzzy soft set , where and for all

 H(e) = F(e) if e∈A∖B = G(e) if e∈B∖A = F(e)∩G(e) if e∈A∩B

This intersection is denoted by .

###### Definition 20.

Let  and be two bipolar fuzzy soft sets over a common universe . The restricted union of and is defined to be the bipolar fuzzy soft set , where and for all

 H(e)=F(e)∪G(e)

This union is denoted by .

###### Proposition 1.

Let be bipolar fuzzy soft set over a common universe . Then,

1. , where is a null bipolar fuzzy soft set.

2. , where is a null bipolar fuzzy soft set.

###### Proof.

. .

A bipolar fuzzy soft set is union of two bipolar fuzzy soft sets and which is

 (5.1) (H,C)=(F,A)¯∪(F,A) where C=A∪A

Define by

 H(e) = F(e) if e∈A∖A = F(e) if e∈A∖A = F(e)∪F(e) if e∈A∩A

L.H.S. There are three cases.

Case (1).If .

 H(a)=F(a) if a∈A∖A=∅

Case (2) If .

 H(a)=F(a) if a∈A∖A=∅

Case (3) If .

 H(a) = F(a)∪F(a) if a∈A∩A=A = F(a) if a∈A H(a) = F(a) if a∈A (H,C) = (F,A) from equation ??? (F,A)¯∪(F,A) = (F,A) from equation ???

It is satisfied in all three cases. Hence .

2:

A bipolar fuzzy soft set is intersection of two bipolar fuzzy soft sets and which is

 (5.2) (H,C)=(F,A)¯∩(F,A) where C=A∩A

Define by

 H(e)=F(e)∩F(e) if e∈C=A∩A

L.H.S. Let aC=.

 H(e) = F(e)∩F(e) if e∈C=A∩A=A = F(e) if e∈C=A = F(e) if e∈A H(e) = F(e) (H,C) = (F,A) from equation ??? (F,A)¯∩(F,A) = (F,A) from equation ???

Hence .

###### Lemma 1.

Absorption property of bipolar fuzzy soft sets and .

###### Proof.

.

Let bipolar fuzzy soft set is an intersection of two bipolar fuzzy soft sets and where

 (5.3) (H,C)=(F,A)¯∩(G,B) where C=A∩B

Define by

 H(e)=F(e)∩G(e) if e∈C=A∩B

Let bipolar fuzzy soft set is union of two bipolar fuzzy soft sets and which is

 (5.4) (K,M)=(F,A)¯∪(H,C) where M=A∪C

Define by

 K(e) = F(e) if e∈A∖C = H(e) if e∈C∖A = F(e)∪H(e) if e∈A∩C

. There are three cases.

Cases (1). If .

 K(e) = F(e) if e∈A∖C = F(e) if e∈A K(e) = F(e) (K,M) = (F,A) from equation ???

Case (2). If .

 K(e) = H(e) if e∈C∖A=0 = ∅ if e∈∅ K(e) = ∅ if e∈∅ (K,M) = ∅ from equation ???

Case (3). If .

 K(e) = F(e)∪H(e) if e∈A∩C% and C=A∩B = F(e)∪(F(e)∩G(e)) from equation ??? = F(e) since (F(e)∩G(e))⊂F(e) = F(e) K(e) = F(e) (K,M) = (F,A) from equation ???

It is satisfied in three cases. Hence .

. same as above.

###### Theorem 1.

Commutative property of bipolar fuzzy soft sets and .

###### Proof.

To show that .

A bipolar fuzzy soft set is an intersection of two bipolar fuzzy soft sets and where

 (5.5) H(e)=F(e)∩G(e) if e∈C=A∩B

A bipolar fuzzy soft set is an intersection of two bipolar fuzzy soft sets and , where

 (5.6) K(e)=G(e)∩F(e) if e∈D=B∩A

To show that

L.H.S

 H(e) = F(e)∩G(e) for all e∈C=A∩B = G(e)∩F(e) since F(e)∩G(e)=G(e)∩F(e) = G(e)∩F(e) for all e∈C=A∩B=B∩A=D = K(e) for all e∈B∩A=D H(e) = K(e) (H,C) = (K,D) from equation ???, equation ??? (F,A)¯∩(G,B) = (G,B)¯∩(F,A) from equation % ???, equation ???

Hence

To show that .

L.H.S.

A bipolar fuzzy soft set  is union of two bipolar fuzzy soft sets , over a common universe

 (5.7) (H,C)=(F,A)¯∪(G,B) where C=A∪B

Define by

 (5.8) H(e) = F(e) if e∈A∖B (5.9) = G(e) if e∈B∖A (5.10) = F(e)∪G(e) if e∈A∩B

There are three cases.

Case (1). If

 (5.11) H(e)=F(e) if e∈A∖B from equation ???

Case (2). If

 (5.12) H(e)=G(e) if e∈B∖A from equation ???

Case (3). If

 (5.13) H(e) = F(e)∪G(e) if e∈A∩B=B∩A from equation ??? = G(e)∪F(e) if e∈B∩A

Combine equation 5.11, equation 5.12 and equation 5.13. We get

 H(e) = G(e) if e∈B∖A = F(e) if e∈A∖B = G(e)∪F(e) if e∈B∩A

becomes

 (H,C) = (G,B)¯∪(F,A) where C=B∪A = R.H.S

Hence .

###### Theorem 2.

Associative law of bipolar fuzzy soft sets , and .

###### Proof.

.

A bipolar fuzzy soft set is an intersection of two bipolar fuzzy soft sets and which is

 (5.14) (G,B)¯∩(H,C)=(L,D) where D=B∩C

Define by

 (5.15) L(e)=G(e)∩H(e) if e∈D=B∩C

A bipolar fuzzy soft set is an intersection of two bipolar fuzzy soft sets and which is

 (5.16) (F,A)¯∩(L,D)=(M,X) where X=A∩D

Define by

 (5.17) M(e)=F(e)∩L(e) if e∈X=A∩D

L.H.S:

 M(e) = F(e)∩L(e) for all e∈X=A∩D from equation ??? = F(e)∩(G(e)∩H(e)) from equation ??? = (F(e)∩G(e))∩H(e) for all e∈X=A∩D=A∩(B∩C) M(e) = (F(e)∩G(e))∩H(e) for all e∈A∩(B∩C) (M,X) = ((F,A)¯∩(G,B))¯∩(H,C) from equation ??? (F,A)¯∩(L,D) = ((F,A)¯∩(G,B))¯∩(H,C) from equation ??? (F,A)¯∩((G,B)¯∩(H,C)) = ((F,A)¯∩(G,B))¯∩(H,C) from equation ???

Hence .

. same as above.

Hence .

###### Theorem 3.

. Distributive law of bipolar fuzzy soft sets , and .

###### Proof.

A bipolar fuzzy soft set  is union of two bipolar fuzzy soft sets and over a common universe .

 (5.18) (G,B)¯∪(H,C)=(L,D) where D=B∪C

Define by

 L(e) = G(e) if e∈B∖C = H(e) if e∈C∖B = G(e)∪H(e) if e∈B∩C

A bipolar fuzzy soft set is an intersection of two bipolar fuzzy soft sets and .

 (5.19) (F,A)¯∩(L,D)=(M,V) where V=A∩D

Define by

 (5.20) M(e)=F(e)∩L(e) if e∈V=A∩D

L.H.S

 M(e) = F(e)∩L(e) for all e∈V=A∩D from equation ??? (5.21) M(e) = F(e)∩L(e) for all e∈A∩D so e∈A, e∈D

If from equation 5.18. Then there are three cases.

Case (1) If

 (5.22) L(e)=G(e) if e∈B∖C from equation ???

Case (2)  If

 (5.23) L(e)=H(e) if e∈C∖B from equation ???

Case (3)  If

 (5.24) L(e)=G(e)∪H(e) if e∈B∩C from equation ???

Put equation 5.22,equation 5.23 and equation 5.24 in equation 5.21

 M(e) = F(e)∩G(e) for all e∈A, e∈B∖C = F(e)∩H(e) if e∈A, e∈C∖B = F(e)∩(G(e)∪H(e)) if e∈A, e∈B∩C = (F(e)∩G(e))∪(F(e)∩H(e)) M(e) = (F(e)∩G(e))∪(F(e)∩H(e)) (M,V) = ((F,A)¯∩(G,B))¯∪((F,A)¯∩(H,C)) from equation ??? (F,A)¯∩(L,D) = ((F,A)¯∩(G,B))¯∪((F,A)¯∩(H,C)) from equation % ??? (F,A)¯∩((G,B)¯∪(H,C)) = ((F,A)¯∩(G,B))¯∪((F,A)¯∩(H,C)) from equation % ???

Hence .

. same as above.

###### Lemma 2.

: and are two bipolar fuzzy soft sets.

1. .

## 6. De Morgan’s Law of Bipolar Fuzzy Soft Sets

###### Theorem 4.

. De Morgan’s law of bipolar fuzzy soft sets and .

1. .

2. .

###### Proof.

.

Let and be a two bipolar fuzzy soft sets over a common universe . Then the union of two bipolar fuzzy soft sets and is a bipolar fuzzy soft set where and is define by

 H(e) = F(e) if e∈A∖B = G(e) if e∈B∖A = F(e)∪G(e) if e∈A∩B

The extended intersection of two bipolar fuzzy soft sets and is bipolar fuzzy soft set where and is define by

 K(e) = F(e) if e∈A∖B = G(e) if e∈B∖A = F(e)∩G(e) if e∈A∩B

To show that .

L.H.S: There are three cases.

Case (i): If . Then . Such that

 H(e)=F(e) if e∈A∖B from equation ???

Taking complement of above. So

 (6.3) (H(e))c=(F(e))c if e∈A∖B

Case (ii) If . Then . Such that

 H(e)=G(e) if e∈B∖A from ???

Taking complement of above. So

 (6.4) (H(e))c=(G(e))c if e∈B∖A

Case (iii) If . Then . Such that

 H(e