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Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx
DOI:10.3233/IFS-131031
IOS Press
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Bipolar fuzzy soft sets and its applications in
decision making problem
3
Saleem Abdullah∗ , Muhammad Aslam and Kifayat Ullah
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a Department
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b Department
or
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6
of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
of Mathematics, King Khalid University, Abha, Saudi Arabia
c Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
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Keywords: Keyword Soft set, bipolar fuzzy set, fuzzy soft set and bipolar fuzzy soft set
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1. Introduction
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30
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et al. [21] studied some new concepts of a soft set. Sezgin and Atagün [22] studied some new theoretical soft
set operations. Majumdar and Samanta, worked on soft
mappings [24]. 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 [7], Aktas. and C.ağman 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[8].
Recently, Acar studied soft rings [9]. Jun et. al, applied
the concept of soft set to BCK/BCI-algebras [10–12].
Sezgin and Atagün initiated the concept of normalistic
soft groups [13]. Zhan et al. worked on soft ideal of BLalgebras [15]. In [16], Kazancı et. al, used the concept
of soft set to BCH-algebras. Sezgin et al. studied soft
nearrings [17]. C.ağman et al. considered two types of
notions of a soft set with group, which is called group
Soft intersection group soft union groups of a group
[20]. see [14].
Fuzzy set originally proposed by Zadeh in [1] of
1965. After semblance of the concept of fuzzy set,
researcher given much attention to developed fuzzy set
theory. Maji et al. [36] introduced the concept of fuzzy
soft sets. Afterwards, many researchers have worked on
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Complicated problems in different field like engineering, economics, environmental science, medicine
and social sciences, which arising due to classical mathematical modelling and manipulating of various type
of uncertainty. While some of traditional mathematical tool fail to solve these complicated problems. We
used some mathematical modelling like fuzzy set theory
[1], rough set theory [2], interval mathematics [12] 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 [4].
Molodtsov, [4] 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. [6] defined new notions on soft sets. Ali
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co
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Un
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uth
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Abstract. 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 extended 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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∗ Corresponding author.
Saleem Abdullah, Department of Mathematics Quaid-i-Azam University Islamabad 45320, Pakistan. E-mail:
[email protected] (S. Abdullah).
1064-1246/13/$27.50 © 2013 – IOS Press and the authors. All rights reserved
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1. A ⊂ B
2. ∀ a ∈ A, F (a) is a subset of G(a).
Similarly, (F, A) is called a superset of (G, B) if
(G, B) is a soft subset of (F, A). This relation is denoted
˜
by (F, A)⊃(G,
B).
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In this section we provide previous concept of bipolar
fuzzy sets, soft sets and fuzzy soft sets.
Definition 2.1. [35] A bipolar fuzzy set A in a universe
−
U is an object having the form, A = {(x, +
A (x), A (x)) :
+
−
x ∈ U} where µA : U → [0, 1], µA : U → [−1, 0]. So
−
µ+
A denote for positive information and µA denote for
negative information.
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58
Definition 2.4. [6] If (F, A) and (G, B) are two soft sets
over a common universe U. The union of (F, A) and
(G, B) is defined to be the soft set (H, C) satisfying the
following conditions: (i) C = A ∪ B: (ii) for all c ∈ C,
or
P
2. Preliminaries
57
101
102
103
104
105
106
107
108
109
110
111
112
H(c) = F (c) if c ∈ A\B
113
= G(c) if c ∈ B\A
114
= F (c) ∪ G(c) if c ∈ A ∩ B
115
˜
This relation is denoted by (H, C) = (F, A)∪(G,
B).
uth
87
56
˜
soft subset of (G, B), denoted by (F, A)⊂(G,
B), if it
satisfies.
Definition 2.5. [21] Let (F, A) and (G, B) be two soft
sets over a common universe U such that A ∩ B =
/ ∅.
The restricted intersection of (F, A) and (G, B) is
defined to be the soft set (H, C), C = A ∩ B and ∀ c ∈
C, H(c) = F (c) ∩ G(c). We write (H, C) = (F, A) (G, B).
Definition 2.6. [36] Let U be an initial universe, E be the
set of all parameters, A ⊂ E and P̃(U) is the collection
of all fuzzy subsets of U. Then (F, A) is called fuzzy
soft set, where F : A → P̃(U).
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this concept. Roy and Maji [28] 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. bipolar-valued fuzzy set is another an extension of fuzzy
set whose membership degree range is different from
the above extensions. In 2000, Lee [35] 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 [0, 1] to [−1, 1]. In a bi-polar-valued fuzzy
set, the membership degree 0 indicate that elements
are irrelevant to the corresponding property, the membership degrees on (0, 1] assign that elements some
what satisfy the property, and the membership degrees
on [−1, 0) assign that elements somewhat satisfy the
implicit counter-property [35].
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 extended 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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Definition 2.7. [36] If (F, A) and (G, B) are two fuzzy
soft sets over a common universe U, then the union
of (F, A) and (G, B) is defined to be the fuzzy soft
set (H, C) satisfying the following conditions: (i) C =
A ∪ B: (ii) c ∈ C,
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121
122
123
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H(c) = F (c) if c ∈ A\B
133
= G(c) if c ∈ B\A
134
= F (c) ∪ G(c) if c ∈ A ∩ B
135
˜
This relation is denoted by(H, C) = (F, A)∪(G,
B).
Definition 2.2. [4] Let U be an initial universe, E be
the set of parameters, A ⊂ E and P(U) is the power set
of U. Then (F, A) is called a soft set, where F : A →
P(U).
3. Bipolar fuzzy soft sets
Definition 2.3. [21] For two soft sets (F, A) and (G, B)
over a common universe U,we say that (F, A) is a
In this section we introduce the concept of bipolar fuzzy soft set, absolute bipolar fuzzy soft set, null
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S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
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152
153
Definition 3.1. Let U be a universe, E a set of parameters and A ⊂ E. Define F : A → BF U , where BF U is
the collection of all bipolar fuzzy subsets of U. Then
(F, A) is said to be a bipolar fuzzy soft set over a universe U. It is defined by
Example 3.2. Let U = {c1 , c2 , c3 , c4 } be the set of
four cars under consideration and E = {e1 = Costly,
e2 = Beautiful, e3 = Fuel Efficient, e4 = Modern Technology } be the set of parameters and A =
{e1 , e2 , e3 }⊆ E. Then,
The complement of the bipolar fuzzy soft set (F , A)
is
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Definition 3.3. Let U be a universe and E a set of
attributes. Then, (U, E) is the collection of all bipolar
fuzzy soft sets on U with attributes from E and is said
to be bipolar fuzzy soft class.
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Definition 3.4. A bipolar fuzzy soft set (F, A) is said to
be a null bipolar fuzzy soft set denoted by empty set ∅,
if for all e ∈ A, F (e) = ∅.
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Definition 3.5. A bipolar fuzzy soft set (F, A) is said
to be an absolute bipolar fuzzy soft set. If for all e ∈ A,
F (e) = BF U
Definition 3.6. The complement of a bipolar fuzzy soft
set (F , A) is denoted (F , A)c and is defined by
−
(F , A)c = {(x, 1 − µ+
A (x) , −1 − µA (x)) : x ∈ U}.
167
168
⎧
⎧
⎫⎫
(b1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.3, −0.6) ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
F (e1 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(b
,
0.4,
−0.2)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
(b4 , 0.7, −0.2) ⎬
(F, A) =
⎧
⎫
⎪
⎪
(b1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.4, −0.2) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
F (e2 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(b
,
0.5,
−0.2)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(b4 , 0.4, −0.2)
(F, A) = F (ei )
F (ei ) = (ci , µ+ (ci ), µ− (ci )) : ∀ci ∈ U, ∀ei ∈ A
⎧
⎧
⎫⎫
(c1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
(c3 , 0.4, −0.2) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(c
,
0.7,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
⎪
(c
,
0.3,
−0.5)
,
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
⎨ (c , 0.4, −0.2) , ⎬ ⎪
2
(F, A) =
F (e2 ) =
⎪
⎪
⎪
(c3 , 0.5, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(c
,
0.4,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(c
,
0.8,
−0.11)
,
1
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.6) , ⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
(c3 , 0.4, −0.3) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(c4 , 0.6, −0.2)
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169
170
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142
165
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Definition 3.7. Let U = {b1 , b2 , b3 , b4 } be the set of
four bikes under consideration and E = { e1 = Stylish,
e2 = Heavy duty, e3 = Light, e4 = Steel } be the set
of parameters and A = {e1 , e2 } be subset of E. Then,
or
P
bipolar fuzzy soft set and complement of bipolar fuzzy
soft set
⎧
⎧
⎫⎫
(b1 , 0.9, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.7, −0.4) ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
(b
,
0.6,
−0.8)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
⎬
(b
,
0.3,
−0.8)
4
c
(F, A) =
⎧
⎫
⎪
(b1 , 0.7, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬⎪
⎪
⎪
(b
,
0.6,
−0.8)
,
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
(b
,
0.5,
−0.8)
,
3
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(b4 , 0.6, −0.8)
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140
4. Bipolar fuzzy soft subsets
Definition 4.1. Let (F, A) and (G, B) be two bipolar
fuzzy soft sets over a common universe U. We say that
(F, A) is a bipolar fuzzy soft subset of (G, B), if (1)
A ⊆ B and (2) ∀ e ∈ A, F (e) is a bipolar fuzzy subset
¯ (G, B).
of G (e). We write (F, A) ⊂
Definition 4.2. Every element of (F, A) is presented in
(G, B) and do not depend on its membership or nonmembership.
Example 4.3. Let U = {m1 , m2 , m3 , m4 } be the set
of four men under consideration and E = { e1 =
Educated, e2 = Government employee, e3 = Businessman, e4 = Smart } be the set of parameters and
A = {e1 , e2 }, B = { e1 , e2 , e3 } be subsets of E. Then,
171
172
173
174
175
176
177
178
179
180
181
182
183
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S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
⎧
⎧
⎫⎫
(m1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.3, −0.6) ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
(m
,
0.4,
−0.2)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
(m4 , 0.7, −0.2) ⎬
(F, A) =
⎧
⎫
⎪
(m1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.4, −0.2) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
(m
,
0.5,
−0.2)
,
⎪
⎪
⎪ 3
⎪⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(m4 , 0.4, −0.2)
and
and
uth
A ⊆ B and for all e ∈ A, F (e) ≤ G (e). Then
¯ (G, B) .
(F, A) ⊂
192
5. Operations on bipolar fuzzy soft sets
190
193
194
195
196
197
198
199
200
201
202
co
189
Definition 5.1. An intersection of two bipolar fuzzy
soft sets (F, A) and (G, B) is a bipolar fuzzy soft set
(H, C), where C = A ∩ B =
/ ∅ and H : C → BF U is
defined by H (e) = F (e) ∩ G (e) ∀ e ∈ C and denoted
¯ (G, B).
by (H, C) = (F, A) ∩
Un
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191
Definition 4.4. Let (F, A) and (G, B) be two bipolar
fuzzy soft sets over a common universe U. We say that
(F, A) and (G, B) are bipolar fuzzy soft equal sets if
(F, A) is a bipolar fuzzy soft subset of (G, B) and (G, B)
is a bipolar fuzzy soft subset of (F, A).
187
⎧
⎧
⎫⎫
(b1 , 0.2, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.2, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
G
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
(b
,
0.2,
−0.3)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
⎪
(b4 , 0.7, −0.1)
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(b1 , 0.3, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨
⎨ (b , 0.2, −0.5) , ⎪
⎬⎪
2
(G, B) =
G (e2 ) =
⎪
⎪
⎪
(b3 , 0.5, −0.3) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(b4 , 0.5, −0.2)
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(b1 , 0.8, −0.01) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎨
⎬
⎪
⎪
(b
,
0.4,
−0.6)
,
2
⎪
⎪
⎪
⎪
(e
)
G
=
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
(b
,
0.2,
−0.3)
,
3
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎭
⎩
⎩
⎭⎪
(b4 , 0.7, −0.2)
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186
203
or
P
⎧
⎧
⎫⎫
(m1 , 0.2, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.2, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
G
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
(m
,
0.2,
−0.3)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
⎪
(m4 , 0.7, −0.1)
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(m1 , 0.3, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
⎨ (m , 0.2, −0.5) , ⎪
⎬⎪
2
(G, B) =
G (e2 ) =
⎪
⎪
⎪
(m3 , 0.5, −0.3) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(m4 , 0.5, −0.2)
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
⎪
(m
,
0.8,
−0.01)
,
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
⎪
⎪
(m
,
0.4,
−0.6)
,
2
⎪
⎪
⎪
⎪
(e
)
G
=
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
(m3 , 0.2, −0.3) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(m4 , 0.7, −0.2)
185
roo
f
⎧
⎧
⎫⎫
(b1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.3, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
(b3 , 0.4, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
(b4 , 0.7, −0.2) ⎬
(F, A) =
⎧
⎫
⎪
(b1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.4, −0.2) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
(b3 , 0.4, −0.4) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(b4 , 0.4, −0.2)
dA
4
Example 5.2. Let U = {b1 , b2 , b3 , b4 } be the set of
four bikes under consideration and E = {e1 = Light,
e2 = Beautiful, e3 = Good millage, e4 = Modern Technology } be the set of parameters and A = {e1 , e2
}⊆ E, B = {e1 , e2 , e3 }⊆ E. Then,
¯ (G, B), where C = A ∩
Then (H, C) = (F, A) ∩
B ={e1 , e2 }
⎧
⎧
⎫ ⎫
(b1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (b , 0.2, −0.6) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
H
,
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
(b3 , 0.2, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
⎬
(b4 , 0.7, −0.1)
(H, C) =
⎧
⎫
⎪
(b1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬⎪
⎪
⎪
(b
,
0.2,
−0.2)
,
2
⎪
⎪
⎪
⎪
(e
)
H
=
⎪
⎪
2
⎪
⎪
⎪
⎪ (b3 , 0.4, −0.3) , ⎪
⎪⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(b4 , 0.4, −0.2)
Definition 5.3. Union of two bipolar fuzzy soft sets over
a common universe U is a bipolar fuzzy soft set (H, C),
where C = A ∪ B and H : C → BF U is defined by
H (e) = F (e) if e ∈ A \ B
204
205
206
207
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
5
208
211
212
213
214
215
216
¯ (G, B).
and denoted by (H, C) = (F, A) ∪
Example 5.4. Let U = {c1 , c2 , c3 , c4 } be the set of four
cars under consideration and E = {e1 = Costly, e2 =
Beautiful, e3 = Fuel Efficient, e4 = Modern Technology } be the set of parameters and A = {e1 , e2 , e3 }⊆ E,
B = {e1 , e2 , e3 , e4 }⊆ E. Then
⎧
⎧
⎫⎫
(c1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
(c
, 0.4, −0.2) , ⎪ ⎪
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(c
,
0.7,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(c1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨
⎨ (c , 0.4, −0.2) , ⎪
⎬⎪
2
(F, A) = F (e2 ) =
⎪
⎪
⎪
(c3 , 0.5, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(c
,
0.4,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(c1 , 0.8, −0.1) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
F (e3 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(c
,
0.4,
−0.3)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
⎩
⎭⎪
(c4 , 0.6, −0.2)
and
Definition 5.5. Let T ={(Fi , Ai ) : i ∈ I} be a family
of bipolar fuzzy soft sets in a bipolar fuzzy soft class
(U, E). Then the intersection of bipolar fuzzy soft sets in
T is a bipolar fuzzy soft set (H, C), where C = ∩Ai =
/ ∅
for all i ∈ I, H (e) = ∩Fi (e) for all e ∈ C.
Definition 5.6. Let T ={(Fi , Ai ) : i ∈ I} be a family
of bipolar fuzzy soft sets in a bipolar fuzzy soft class
(U, E). Then the union of bipolar fuzzy soft sets in T is
a bipolar fuzzy soft set(H, C), where C = ∪Ai for all
i ∈ I.
cte
217
roo
f
= F (e) ∪ G (e) if e ∈ A ∩ B
or
P
210
¯ (G, B), where C = A ∪ B =
Then (H, C) = (F, A) ∪
{e1 , e2 , e3 , e4 }
⎧
⎧
⎫⎫
(c1 , 0.2, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
H
=
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
(c3 , 0.4, −0.3) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
(c4 , 0.7, −0.2) ⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫⎪
⎪
⎪
⎪
⎪
(c
,
0.3,
−0.6)
,
⎪
⎪
⎪
⎪
⎪
⎪ 1
⎪⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
⎪
⎪
(c2 , 0.4, −0.5) , ⎪
⎪
⎪
⎪
⎪
H (e2 ) =
⎪
⎪
⎪
⎪
⎪
⎪ (c3 , 0.5, −0.3) , ⎪
⎪⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎨
(c4 , 0.5, −0.2) ⎬
(H, C) =
⎧
⎫
⎪
⎪
⎪
⎪ (c1 , 0.8, −0.1) , ⎪
⎪⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎨
⎪
⎪
(c2 , 0.4, −0.6) , ⎬ ⎪
⎪
⎪
⎪
⎪
⎪
⎪ H (e3 ) = ⎪ (c , 0.4, −0.3) , ⎪ ⎪
⎪
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭⎪
⎪
⎪
⎪
⎪
(c
,
0.7,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
⎪
(c
,
0.1,
−0.6)
,
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
⎪
⎪
(c2 , 0.3, −0.4) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪ H (e4 ) = ⎪ (c , 0.1, −0.6) , ⎪ ⎪
⎪
⎪
⎪
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(c4 , 0.0, −0.2)
uth
= G (e) if e ∈ B \ A
dA
209
Un
co
rre
⎧
⎧
⎫⎫
(c1 , 0.2, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.2, −0.6) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
G (e1 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(c
,
0.2,
−0.3)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
⎪
(c
,
0.7,
−0.1)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(c1 , 0.3, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.2, −0.5) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
G (e2 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(c
,
0.5,
−0.3)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎬
⎨
(c4 , 0.5, −0.2)
(G, B) =
⎧
⎫
⎪
(c1 , 0.8, −0.01) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.4, −0.6) ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
G (e3 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(c
,
,
0.2,
−0.3)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
⎪
(c4 , 0.7, −0.2)
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(c1 , 0.1, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.4) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
G (e4 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(c
,
0.1,
−0.6)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎭
⎩
⎩
⎭
(c4 , 0.0, −0.2)
218
H (e) = Fi (e) if e ∈ Ai
= ∅ if e ∈
/ Ai
Definition 5.7. Let (F , A) and (G, B) be two bipolar
fuzzy soft sets over a common universe U. The extended
intersection of (F ,A) and (G, B) is defined t o be the
bipolar fuzzy soft set (H,C), where C = A ∪ B and for
all e ∈ C.
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
H (e) = F (e) if e ∈ A\B
236
= G (e) if e ∈ B\A
237
= F (e) ∩ G (e) if e ∈ A ∩ B
238
This intersection is denoted by (H,C) = (F ,A) (G,B).
239
240
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
Definition 5.8. Let (F , A) and (G,B) be two bipolar fuzzy soft sets over a common universe U. The
restricted union of (F , A) and (G, B) is defined to be
the bipolar fuzzy soft set (H, C), where C = A ∩ B =
/
∅ and for all e ∈ C
H (e) = F (e) ∪ G (e)
243
244
245
246
247
248
249
250
251
Proof. (1)
¯ (F, A) = (F, A).
(F, A) ∪
A bipolar fuzzy soft set (H, C) is union of two bipolar
fuzzy soft sets (F, A) and (F, A) which is
¯ (F, A)
(H, C) = (F, A) ∪
(1)
Define by
H (e) = F (e) if e ∈ A\A
254
= F (e) if e ∈ A\A
255
= F (e) ∪ F (e) if e ∈ A ∩ A
256
L.H.S. There are three cases.
Case (2) If a ∈ A\A.
258
Case (3) If a ∈ A ∩ A.
260
H (a) = F (a) if a ∈ A
263
264
265
Un
= F (a) if a ∈ A
262
H (e) = F (e) ∩ F (e)
¯ (F, A) = (F, A)
(2) (F, A) ∩
= F (e) if e ∈ C = A
268
= F (e) if e ∈ A
269
270
(H, C) = (F, A) from Eq 2
271
¯ (F, A) = (F, A) from Eq 2
(F, A) ∩
272
¯ (F, A) = (F, A).
Hence (F, A) ∩
273
Lemma 5.10. Absorption property of bipolar fuzzy soft
sets (F, A) and (G, B).
¯ (F, A) ∩
¯ (G, B) = (F, A)
1. (F, A) ∪
¯ (F, A) ∪
¯ (G, B) = (F, A)
2. (F, A) ∩
¯ (G, B)
(H, C) = (F, A) ∩
274
275
276
277
278
279
(3)
where C = A ∩ B. Define if e ∈ C = A ∩ B
H (e) = F (e) ∩ G (e)
Let bipolar fuzzy soft set (K, M) is union of two
bipolar fuzzy soft sets (F, A) and (H, C) which is
¯
(K, M) = (F, A) ∪(H,
C)
Define by
280
281
(4)
282
283
(H, C) = (F, A) from Eq 1
¯ (F, A) = (F, A) from Eq 1
(F, A) ∪
It is satisfied in all three cases.
¯ (F, A) = (F, A).
(F, A) ∪
267
H (e) = F (e)
H (a) = F (a)∪F (a) if a ∈ A ∩ A = A
259
261
co
H (a) = F (a) if a ∈ A\A = ∅
rre
H (a) = F (a) if a ∈ A\A = ∅
266
Proof. (1) ¯ (F, A) ∩
¯ (G, B) = (F, A)
(F, A) ∪
Let bipolar fuzzy soft set (H, C) is an intersection of
two bipolar fuzzy soft sets (F, A) and (G, B) , where
C =A∩B
Case (1) If a ∈ A\A.
257
L.H.S. Let a ∈ C = A ∩ A.
dA
253
H (e) = F (e) ∩ F (e) if e ∈ C = A ∩ A
cte
252
Define by
Proposition 5.9. Let (F, A) be bipolar fuzzy soft set over
a common universe U. Then,
¯ (F, A) = (F, A)
1. (F, A) ∪
¯ (F, A) = (F, A)
2. (F, A) ∩
¯ ∅ = (F, A), where ∅ is a null bipolar
3. (F, A) ∪
fuzzy soft set.
¯ ∅ = ∅, where ∅ is a null bipolar fuzzy soft
4. (F, A) ∩
set.
(2)
or
P
242
¯ R (G, B).
This union is denoted by (H, C) = (F , A) ∪
¯ (F, A) where C = A ∩ A
(H, C) = (F, A) ∩
uth
241
A bipolar fuzzy soft set (H, C) is intersection of two
bipolar fuzzy soft sets (F, A) and (F, A) which is
roo
f
6
K (e) = F (e) if e ∈ A\C
Hence
284
= H (e) if e ∈ C\A
285
= F (e) ∪ H (e) if e ∈ A ∩ C
286
L.H.S. There are three cases.
287
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
291
292
293
294
= F (e) if e ∈ A
Cases (2) If e ∈ C\A = A ∩ B − A = 0.
K (e) = ∅ if e ∈ ∅
297
(K, M) = ∅ from Eq 4
300
Cases (3). If e ∈ A ∩ C.
= F (e)
304
K (e) = F (e)
323
(7)
324
H (e) = F (e) if e ∈ A\B
(8)
325
= G (e) if e ∈ B\A
(9)
326
(10)
327
(5)
A bipolar fuzzy soft set (K, D) is an intersection of
two bipolar fuzzy soft sets (G, B) and (F, A), where
D=B∩A
(6)
328
Case (1) If e ∈ A\B
cte
rre
A bipolar fuzzy soft set (H, C) is an intersection of
two bipolar fuzzy soft sets (F, A) and (G, B) , where
C =A∩B
K (e) = G (e) ∩ F (e) if e ∈ D = B ∩ A
322
H (e) = F (e) if e ∈ A\B from Eq 8
(11)
Case (2) If e ∈ B\A
H (e) = G (e) if e ∈ B\A from Eq 9
(12)
Case (3) If e ∈ A ∩ B
329
H (e) = F (e) ∪ G (e) if e ∈ A ∩ B = B ∩ A
= G (e) ∪ F (e) if e ∈ B ∩ A
¯ (G, B) = (G, B) ∩
¯ (F, A) .
(F, A) ∩
H (e) = F (e) ∩ G (e) if e ∈ C = A ∩ B
¯ (G, B) = (G, B) ∪
¯ (F, A).
(2) To show that (F, A) ∪
There are three cases.
co
¯ (G, B) = (G, B) ∩
¯ (F, A)
(1) (F, A) ∩
¯ (G, B) = (G, B) ∪
¯ (F, A)
(2) (F, A) ∪
320
321
= F (e) ∪ G (e) if e ∈ A ∩ B
Theorem 5.11. Commutative property of bipolar fuzzy
soft sets (F, A) and (G, B).
Proof.
(1) To show that
319
Define by
(2) same as above.
Un
311
318
¯ (G, B) where C = A ∪ B
(H, C) = (F, A) ∪
(K, M) = (F, A) from Eq 4
It is satisfied in three cases. Hence
¯ (F, A) ∩
¯ (G, B) = (F, A) .
(F, A) ∪
310
317
dA
303
309
= K (e) for all e ∈ B ∩ A = D
A bipolar fuzzy soft set (H, C) is union of two bipolar
fuzzy soft sets (F, A), (G, B) over a common universe U
C = A∩B
= F (e) since (F (e) ∩ G (e)) ⊂ F (e)
308
316
¯ (G, B) = (G, B) ∩
¯ (F, A)
Hence (F, A) ∩
L.H.S.
K (e) = F (e) ∪ H (e) if e ∈ A∩C and
302
307
= G (e) ∩ F (e)
¯ (G, B) = (G, B) ∩
¯ (F, A)
(F, A) ∩
= F (e) ∪ (F (e) ∩ G) (e) from Eq 3
306
315
(H, C) = (K, D) using Eqs 5, 6
301
305
= G (e) ∩ F (e)
H (e) = K (e)
K (e) = H (e) if e ∈ C\A = 0
296
299
314
(K, M) = (F, A) from Eq 4
= ∅ if e ∈ ∅
298
H (e) = F (e) ∩ G (e)
K (e) = F (e)
295
313
roo
f
290
K (e) = F (e) if e ∈ A\C
312
or
P
289
To show that (H, C) = (K, D)
L.H.S
Cases (1) If e ∈ A\C.
uth
288
7
Combine Eq 11, Eq 12 and Eq 13. We get
(13)
330
331
332
H (e) = G (e) if e ∈ B\A
333
= F (e) if e ∈ A\B
334
= G (e) ∪ F (e) if e ∈ B ∩ A
335
(H, C) becomes
¯ (F, A) where C = B ∪ A
(H, C) = (G,B) ∪
= R.H.S
336
337
338
8
Hence
346
Theorem 5.12. Associative law of bipolar fuzzy soft
sets (F, A),(G, B) and (H, C).
¯ (G, B) ∩
¯ (H, C)
1. (F, A)
∩
¯ (G, B) ∩
¯ (H, C)
= (F,
A) ∩
¯ (G, B) ∪
¯ (H, C)
2. (F, A)
∪
¯ (G, B) ∪
¯ (H, C)
= (F, A) ∪
347
Proof. (1)
342
343
344
345
348
349
¯ (G, B) ∩
¯ (H, C)
(F, A) ∩
¯ (G, B) ∩
¯ (H, C)
= (F, A) ∩
L (e) = G ( ) ∩ H (e)
351
373
374
375
¯ (H, C) = (L, D) where D = B ∪ C
(G, B) ∪
Define by
(17)
376
378
= G (e) ∪ H (e) if e ∈ B ∩ C
379
(15)
(16)
A bipolar fuzzy soft set (M, V ) is an intersection of
two bipolar fuzzy soft sets (F, A) and (L, D).
¯ (L, D) = (M, V ) where V = A ∩ D
(F, A) ∩
(18)
Define by
cte
L.H.S:
M (e) = F (e) ∩ L (e) if e ∈ V = A ∩ D
M (e) = F (e) ∩ L (e)
355
= (F (e) ∩ G (e))∩H (e)
rre
= F (e) ∩ (G (e)∩H (e))
M (e) = (F (e)∩G (e)) ∩ H (e)
¯ (G, B)
(M, X) = (F, A) ∩
356
357
co
¯ (H, C)
∩
¯
¯ (G, B)
(F, A) ∩(L,
D) = (F, A) ∩
358
359
Un
¯ (H, C)
∩
¯ (G, B) ∩
¯ (H, C) = (F, A) ∩
¯ (G, B)
(F, A) ∩
¯ (H, C)
∩
362
Hence
¯ (G, B) ∩
¯ (H, C) = (F, A) ∩
¯ (G, B)
(F, A) ∩
365
366
372
= H (e) if e ∈ C\B
354
364
371
A bipolar fuzzy soft set (M, X) is an intersection of
two bipolar fuzzy soft sets (F, A) and (L, D) which is
353
363
370
377
M (e) = F (e) ∩ L (e) if e ∈ X = A ∩ D
361
Theorem 5.13. Distributive law of bipolar fuzzy soft
sets (F, A), (G, B) and (H, C).
¯ (G, B) ∪
¯ (H, C)
1. (F, A)
∩
¯ (G, B) ∪
¯ (F, A) ∩
¯ (H, C)
= (F,
A) ∩
¯ (G, B) ∩
¯ (H, C)
2. (F, A)
∪
¯ (G, B) ∩
¯ (F, A) ∪
¯ (H, C)
= (F, A) ∪
369
L (e) = G (e) if e ∈ B\C
Define by
360
(14)
¯
(F, A) ∩(L,
D) = (M, X)
352
368
Proof. (1) A bipolar fuzzy soft set (L, D) is union of
two bipolar fuzzy soft sets (G, B) and (H, C) over a
common universe U.
A bipolar fuzzy soft set (L, D) is an intersection of
two bipolar fuzzy soft sets (G, B) and (H, C) which is
¯ (H, C) = (L, D) where D = B ∩ C Define by
(G, B) ∩
350
¯ (H, C) .
∪
or
P
341
¯ (G, B) ∪
¯ (H, C) = (F, A) ∪
¯ (G, B)
(F, A) ∪
roo
f
¯ (G, B) = (G, B) ∪
¯ (F, A) .
(F, A) ∪
367
uth
340
Hence
dA
339
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
(2) Same as above.
¯ (H, C)
∩
(19)
L.H.S
380
M (e) = F (e) ∩ L (e)
M (e) = F (e) ∩ L (e) so e ∈ A, e ∈ D
381
(20)
If e ∈ D = B ∪ C from Eq 17. Then there are three
cases.
Case (1) If e ∈ B\C
L (e) = G (e) if e ∈ B\C
382
383
384
(21)
Case (2) If e ∈ C\B
L (e) = H (e) if e ∈ C\B
(22)
Case (3) If e ∈ C ∩ B
L (e) = G (e) ∪ H (e)
if e ∈ B ∩ C from Eq 17
385
(23)
386
(24)
387
388
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
389
Put Eq 21, Eq 22 and Eq 23 in Eq 20
390
M (e) = F (e) ∩ G(e)
for all e ∈ A, e ∈ B\C
= F (e) ∩ H (e)
392
= F (e) ∩ (G (e) ∪ H (e))
= (F (e) ∩ G (e)) ∪ (F (e) ∩ H (e))
396
397
398
399
400
401
M (e) = (F (e) ∩ G (e)) ∪ (F (e) ∩ H (e))
¯ (G, B)
(F, A) ∩
(M, V ) =
using Eq 18
¯ (F, A) ∩
¯ (H, C)
∪
¯ (G, B)
(F, A) ∩
¯ (L, D) =
(F, A) ∩
using Eq 18
¯ (F, A) ∩
¯ (H, C)
∪
¯ (G, B) ∪
¯ (H, C)
(F, A) ∩
¯ (G, B)
(F, A) ∩
=
using Eq 17
¯ (F, A) ∩
¯ (H, C)
∪
402
Hence proof.
403
(2). Same as above.
408
409
410
411
412
413
414
415
416
417
424
425
= F (e) ∩ G (e) if e ∈ A ∩ B
427
To show that
¯ (G, B) c = (F, A)c (G, B)c .
(F, A) ∪
428
429
L.H.S: There are three cases.
Case (i): If e ∈ A\B. Then e ∈ C. Such that
430
H (e) = F (e) if e ∈ A\B from 25
Taking complement of above. So
(H (e))c = (F (e))c if e ∈ A\B
(27)
Case (ii) If e ∈ B\A. Then e ∈ C. Such that
H (e) = G (e) if e ∈ B\A from 25
¯ (G, B) = (F, A)
1. (F, A) ⊂ (G, B) ⇒ (F, A) ∩
¯ (G, B) = (G, B).
2. (F, A) ⊂ (G, B) ⇒ (F, A) ∪
6. De Morgan’s law of bipolar fuzzy soft sets
Theorem 6.1. De Morgan’s law of bipolar fuzzy soft
sets (F, A) and (G, B).
¯ (G, B) c = (F, A)c (G, B)c .
1. (F, A) ∪
¯ (G, B)c .
2. ((F, A) (G, B))c = (F, A)c ∪
Proof. (1) Let (F, A) and (G, B) be a two bipolar fuzzy
soft sets over a common universe U. Then the union of
two bipolar fuzzy soft sets (F, A) and (G, B) is a bipolar
fuzzy soft set (H,C) where C = A ∪ B and (H,C) =
¯ (G, B) is define by
(F, A) ∪
418
H (e) = F (e) if e ∈ A\B
419
= G (e) if e ∈ B\A
420
= F (e) ∪ G (e) if e ∈ A ∩ B
(28)
Case (iii) If e ∈ A ∩ B. Then e ∈ C. Such that
cte
407
Lemma 5.14. (F, A) and (G, B) are two bipolar fuzzy
soft sets.
rre
406
423
426
(H (e))c = (G (e))c if e ∈ B\A
co
405
422
Taking complement of above. So
Un
404
421
= G (e) if e ∈ B\A
or
P
if e ∈ A, e ∈ B ∩ C
(26)
roo
f
if e ∈ A, e ∈ C\B
394
395
K (e) = F (e) if e ∈ A\B
uth
393
The extended intersection of two bipolar fuzzy soft
sets (F, A) and (G, B) is bipolar fuzzy soft set (K,D)
where D = A ∩ B and (K, D) = (F, A) (G, B) is
define by
dA
391
9
H (e) = F (e) ∪ G (e) if e ∈ A ∩ B from 25
Taking complement of above. So
(H (e))c = (F (e) ∪ G (e))c if e ∈ A ∩ B
(29)
We define F (e) and G (e) as
−
F (e) = {(u,µ+
A (u) ,µA (u) ) :u ∈ U}
(30)
−
G (e) = {(u,µ+
B (u) ,µB (u) ) :u ∈ U}
(31)
and
Putting Eq 30 , Eq 31 in Eq 29. We get
c
−
+
−
(H (e))c = (u, µ+
A (u), µA (u)) ∪ (u, µB (u)!, µB (u) )
if e ∈ A ∩ B.
431
432
433
434
(25)
(H (e))c = (F (e))c ∩ (G (e))c
if e ∈ A ∩ B and C = A ∩ B
(32)
435
436
437
Smart, e5 = Weak} be the set of parameters and A =
{e1 , e2 }⊆ E, B = {e4 , e5 }⊆ E. Then
⎧
⎧
⎫ ⎫
(m1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.3, −0.6) ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
F
=
,
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
(m3 , 0.4, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
⎬
(m4 , 0.7, −0.2)
(F, A) =
⎧
⎫
⎪
(m1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬⎪
⎪
⎪
(m
,
0.4,
−0.2)
,
2
⎪
⎪
⎪
⎪
(e
)
F
=
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
(m3 , 0.5, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(m4 , 0.4, −0.2)
From Eq 27, Eq 28 and Eq 32. We get
439
(H (e)) = (F (e)) if e ∈ A\B
440
= (G (e))c if e ∈ B\A
441
= (F (e))c ∩ (G (e))c
442
if e ∈ A ∩ B and C = A ∩ B
c
roo
f
c
Then
(H (e))c = (F (e))c (G (e))c
443
Hence it is proved.
444
(2). Same as above.
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
⎧
⎧
⎫ ⎫
(m1 , 0.1, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.3, −0.4) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
G
=
,
⎪
⎪
4
⎪
⎪
⎪
⎪
⎪
⎪
(m
,
0.1,
−0.6)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
⎬
(m4 , 0.0, −0.2)
(G, B) =
⎧
⎫
⎪
(m1 , 0.4, −0.1) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬⎪
⎪
⎪
(m
,
0.2,
−0.4)
,
2
⎪
⎪
⎪
⎪
(e
)
G
=
⎪
⎪
5
⎪
⎪
⎪
⎪
⎪
(m
,
0.6,
−0.4)
,
3
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(m4 , 0.7, −0.0)
Definition 7.1. Let (F, A) and (G, B) be two bipolar
fuzzy soft sets over a common universe U. Then,
˜ (G, B) is a bipolar fuzzy soft set
1. (F, A) ∧
˜ (G, B) = (H, A × B) where
defined by (F, A) ∧
H (a, b) = F (a) ∩ G (b) for all (a, b) , ∈ C =
A × B, where ∩ is the intersection operation of
sets.
˜ (G, B) is a bipolar fuzzy soft set
2. (F, A) ∨
˜ (G, B) = (H, A × B) where
defined by (F, A) ∨
H (a, b) = F (a) ∪ G (b) for all (a, b) ∈ C =
A × B, where ∪ is the intersection operation of
sets.
˜ (G, B) = (H, A × B) and C = A ×
Then (F, A) ∧
B ={e1 , e2 }×{e4 , e5 }={(e1 , e4 ), (e1 , e5 ), (e2 , e4 ),
(e2 , e5 )} define by H (a, b) = F (a) ∩ G (b) for all
(a, b) ∈ C
⎧
⎧
⎫ ⎫
(m1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.3, −0.4) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
H
,
e
=
,
⎪
⎪
1
4
⎪
⎪
⎪
⎪
⎪
(m3 , 0.1, −0.2) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭ ⎪
⎪
⎪
⎪
⎪
(m
,
0.0,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(m
,
0.1,
−0.1)
,
1
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.2, −0.4) , ⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
,
H
,
e
=
⎪
⎪
1
5
⎪
⎪
⎪
⎪
⎪
(m3 , 0.4, −0.2) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭ ⎪
⎨
⎬
(m4 , 0.7, −0.0)
(H, C) =
⎧
⎫
⎪
⎪
⎪
⎪ (m1 , 0.1, −0.5) , ⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎨
⎪
⎪
(m2 , 0.3, −0.2) , ⎬ ⎪
⎪
⎪
⎪
H (e2 , e4 ) =
,⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(m
,
0.1,
−0.2)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
⎪
(m
,
0.0,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(m1 , 0.3, −0.1) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (m , 0.2, −0.2) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪ H (e2 , e5 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(m
,
0.5,
−0.2)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎩
⎭
⎩
⎭
(m4 , 0.4, −0.0)
cte
448
rre
447
7. OR and AND operations on bipolar fuzzy
soft sets
Definition 7.2. Let T ={(Fi , Ai ) : i ∈ I} be a family
of bipolar fuzzy soft sets in a bipolar fuzzy soft class
(U, E). Then the intersection of bipolar fuzzy soft sets
in T is bipolar fuzzy soft set (H, C), where, C = ×Ai
˜ i (e) for all e ∈ C, (H, C) =
for all i ∈ I, H (e) = ∧F
˜ (Fi , Ai ) for all i ∈ I.
∧
co
446
= (F, A)c (G, B)c
Definition 7.3. Let T ={(Fi , Ai ) : i ∈ I} be a family
of bipolar fuzzy soft sets in a bipolar fuzzy soft class
(U, E). Then the union of bipolar fuzzy soft sets in T is
bipolar fuzzy soft set (H, C), where, C = ×Ai for all i ∈
˜ i (e) for all e ∈ C, (H, C) = ∨
˜ (Fi , Ai ) for
I,H (e) = ∨F
all i ∈ I.
Un
445
c
¯ (G, B)
(F, A) ∪
or
P
Thus
uth
438
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
dA
10
Example 7.4. Let U = {m1 , m2 , m3 , m4 } be the set of
four man under consideration and E ={e1 = Educated,
e2 = Government employee, e3 =Businessman, e4 =
Example 7.5. Let U ={h1 , h2 , h3 , h4 } be the set of four
houses under consideration and E ={e1 =Expensive,
471
472
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
⎧
⎧
⎫ ⎫
(h1 , 0.1, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (h , 0.3, −0.4) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
G (e4 ) =
,⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(h
,
0.1,
−0.6)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎨
⎬
(h4 , 0.0, −0.2)
(G, B) =
⎧
⎫
⎪
⎪
(h1 , 0.4, −0.1) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎪
⎪
⎪
⎨ (h , 0.2, −0.4) , ⎪
⎬⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
G (e5 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(h
,
0.6,
−0.4)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭⎪
(h4 , 0.7, −0.0)
⎧
⎧
⎫ ⎫
(h1 , 0.1, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (h , 0.3, −0.6) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
(e
)
H
=
,
e
,
⎪
⎪
1
4
⎪
⎪
⎪
⎪
⎪
(h3 , 0.4, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭ ⎪
⎪
⎪
⎪
⎪
(h
,
0.7,
−0.2)
4
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎫
⎪
⎪
⎪
(h
,
0.4,
−0.5)
,
1
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ (h , 0.3, −0.6) , ⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
)
(e
,
=
,
e
H
⎪
⎪
1
5
⎪
⎪
⎪
⎪
⎪
⎪
(h
,
0.6,
−0.4)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎩
⎭
⎬
⎨
(h4 , 0.7, −0.2)
(H, C) =
⎧
⎫
⎪
⎪
(h1 , 0.3, −0.6) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (h , 0.4, −0.4) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
,⎪
⎪
⎪ H (e2 , e4 ) =
⎪
⎪
⎪
⎪
⎪
⎪
(h
,
0.5,
−0.6)
,
⎪
⎪
⎪
⎪
⎪
⎪ 3
⎪ ⎪
⎪
⎪
⎩
⎭
⎪
⎪
⎪
⎪
(h4 , 0.4, −0.2)
⎪
⎪
⎪
⎪
⎪
⎪
⎫
⎧
⎪
⎪
⎪
(h1 , 0.4, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬⎪
⎨ (h , 0.4, −0.4) , ⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
H (e2 , e5 ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(h
,
0.6,
−0.4)
,
3
⎪
⎪
⎪
⎪
⎪
⎪⎪
⎪
⎭
⎩
⎭
⎩
(h4 , 0.7, −0.2)
480
481
482
483
˜ (F, A) = (F, A)
Proof. (1). (F, A) ∧
˜ (F, A) = (H, C), where C =
Suppose that (F, A) ∧
A × A, Let a ∈ A
H (a, a) = F (a) ∩ F (a)
484
485
486
487
since F (a) ∩ F (a) = F (a)
488
= F (a)
489
H (a, a) = F (a)
490
(H, C) = (F, A)
491
˜ (F, A) = (F, A)
(F, A) ∧
492
˜ (F, A) = (F, A)
Hence (F, A) ∧
493
˜ (F, A) = (F, A)
(2). (F, A) ∨
˜ (F, A) = (H, C), where C =
Suppose that (F, A) ∨
A × A, Let a ∈ A
cte
479
rre
478
co
477
˜ (G, B) = (H, A × B) and C = A ×
Then (F, A) ∨
B ={e1 , e2 }×{e4 , e5 }={(e1 , e4 ), (e1 , e5 ), (e2 , e4 ),
(e2 , e5 )} define by H (a) = F (a) ∪ G (a), for all a ∈
C =A×B
Un
476
˜ (F, A) = (F, A)
1. (F, A) ∧
˜ (F, A) = (F, A)
2. (F, A) ∨
roo
f
⎧
⎧
⎫ ⎫
(h1 , 0.1, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎪
⎪
⎪
⎨ (h , 0.3, −0.6) , ⎪
⎬ ⎪
⎪
⎪
2
⎪
⎪
⎪
(e
)
F
=
,⎪
⎪
⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
(h
,
0.4,
−0.2)
,
3
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎬
⎨
(h4 , 0.7, −0.2)
(F, A) =
⎧
⎫
⎪
(h1 , 0.3, −0.5) , ⎪ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬ ⎪
⎪
⎪
(h
,
0.4,
−0.2)
,
2
⎪
⎪
⎪
⎪
(e
)
,
F
=
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
(h
,
0.5,
−0.2)
,
⎪
⎪
⎪ 3
⎪ ⎪
⎪
⎪
⎪
⎭
⎩
⎩
⎭ ⎪
(h4 , 0.4, −0.2)
Proposition 7.6. Idempotent Property. If (F, A) ,
(G, B) are two bipolar fuzzy soft sets over U. Then,
or
P
475
e2 =Beautiful, e3 =Wooden, e4 =In the green surrounding, e5 =Convenient traffic} be the set of parameters and
A ={e1 , e2 }⊆ E, B ={e4 , e5 }⊆ E. Then,
uth
474
dA
473
11
H (a, a) = F (a) ∩ F (a)
494
495
496
497
since F (a) ∪ F (a) = F (a)
498
= F (a)
499
H (a, a) = F (a)
500
(H, C) = (F, A)
501
˜ (F, A) = (F, A)
(F, A) ∨
502
˜ (F, A) = (F, A).
Hence (F, A) ∨
Example 7.7. Let U = {c1 , c2 , c3 , c4 } be the set
of four cars under consideration, E = {e1 = Costly,
e2 = Beautiful, e3 = Fuel efficient,e4 = Luxurious} be
the set of parameters and A = {e1 , e2 , e3 }⊂ E then
F : A → BF U define by
⎫
⎧
(c , 0.2, −0.5) , ⎪
⎪
⎪
⎪ 1
⎪
⎬
⎨ (c , 0.3, −0.6) , ⎪
2
F (e1 ) =
⎪
⎪ (c3 , 0.4, −0.3) , ⎪
⎪
⎪
⎪
⎭
⎩
(c4 , 0.7, −0.2)
⎧
⎫
(c1 , 0.1, −0.6) , ⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.5) , ⎪
⎬
2
F (e2 ) =
⎪
(c3 , 0.6, −0.1) , ⎪
⎪
⎪
⎪
⎪
⎩
⎭
(c4 , 0.4, −0.4)
503
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
⎧
⎫
(c1 , 0.2, −0.8) , ⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.4, −0.3) , ⎪
⎬
2
F (e3 ) =
⎪
(c3 , 0.5, −0.3) , ⎪
⎪
⎪
⎪
⎪
⎩
⎭
(c4 , 0.7, −0.3)
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
Bipolar fuzzy soft set has several applications to deal
with uncertainties from our different kinds of daily life
problems. Here, we discuss such an application for solving a socialistic decision making problem. We apply
the concept of bipolar fuzzy soft set for modelling of a
socialistic decision making problem and then we give
an algorithm for the choice of optimal object based upon
the available sets of information.
Suppose that U = {c1 , c2 , c3 , c4 } be the set of four
cars under consideration say U is an initial universe and
E = {e1 =costly, e2 = Beautiful, e3 = Fuel efficient,
e4 =Modern technology, e5 =Luxurious} be a set of
parameters.
Suppose a man Mr. X is going to buy a car on the
basis of his wishing parameter among the listed above.
Our aim is to find out the attractive car for Mr. X.
Suppose the wishing parameters of Mr. X be A ⊂ E
where A = {e1 , e2 , e5 }. Consider the bipolar fuzzy soft
set as below.
⎧
⎫
(c1 , 0.4, −0.5) , ⎪
⎪
⎪
⎪
⎪
⎪
⎨
(c2 , 0.6, −0.3) , ⎬
F (e1 ) =
⎪
(c3 , 0.8, −0.2) , ⎪
⎪
⎪
⎪
⎪
⎩
⎭
(c4 , 0.5, −0.2)
Find the maximum score, if it occurs in i-th row, then
Mr. X will buy to di , 1 ≤ i ≤ 4.
Clearly the maximum score is 4 scored by the car c3 .
Decision: Mr. X will buy c3 . If he does not want to
buy c3 due to certain reason, his second choice will be
c2 or c4 , so the score of c2 or c4 are same.
Table 1
Tabular representation of positive information function.
Un
co
⎧
⎫
(c1 , 0.5, −0.5) , ⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.3, −0.1) , ⎪
⎬
2
F (e2 ) =
⎪ (c3 , 0.4, −0.4) , ⎪
⎪
⎪
⎪
⎪
⎩
⎭
(c4 , 0.7, −0.3)
rre
cte
523
8. An application of bipolar fuzzy soft sets in
decision making
1. Input the set A ⊂ E of choice of parameters of
Mr. X.
2. Consider the bipolar fuzzy soft set in tabular form.
3. Compute the comparison table of positive
information function and negative information
function.
4. Compute the positive information score and negative information score.
5. Compute the final score by subtracting positive information score from negative information
score.
roo
f
506
⎧
⎫
(c1 , 0.7, 0) , ⎪
⎪
⎪
⎪
⎪
⎨ (c , 0.5, −0.3) , ⎪
⎬
2
F (e5 ) =
⎪
(c3 , 0.6, −0.3) , ⎪
⎪
⎪
⎪
⎪
⎩
⎭
(c4 , 0.4, −0.4)
524
525
Definition 8.1. ( Comparison table). It is a square table
in which number of rows and number of columns are
.
e1
e2
e5
c1
c2
c3
c4
0.4
0.6
0.8
0.5
0.5
0.3
0.4
0.7
0.7
0.5
0.6
0.4
Table 2
Comparison table of the above table.
.
c1
c2
c3
c4
c1
c2
c3
c4
3
1
1
2
2
3
3
1
2
0
3
1
1
2
2
3
Table 3
Positive membership score table.
.
c1
c2
c3
c4
526
527
528
529
530
or
P
505
Algorithm.
Hence F (e1 ), F (e2 ) and F (e3 ) are the elements of
bipolar fuzzy soft set over a universe U.
uth
504
equal and both are labeled by the object name of the
universe such as c1 , c2 , c3 ,...,cn and the entries dij where
dij = the number of parameters for which the value of
di exceeds or equal to the value of dj .
dA
12
Row Column Membership
sum(a) sum(b)
score(a-b)
8
6
9
7
7
9
6
8
1
−3
3
−1
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
S. Abdullah et al. / Bipolar fuzzy soft sets and its applications in decision making problem
Table 4
Tabular representation of negative information function.
e1
e2
e5
c1
c2
c3
c4
−0.5
−0.3
−0.2
−0.2
−0.5
−0.1
−0.4
−0.3
0
−0.3
−0.3
−0.4
c2
c3
c4
c1
c2
c3
c4
3
1
1
1
2
3
2
2
2
2
3
2
2
1
2
3
References
Column
sum(d)
Non-membership
score(c-d)
c1
c2
c3
c4
9
7
8
8
6
9
9
8
3
−2
−1
0
dA
Row
sum(d)
uth
Table 6
Negative membership score table.
.
Table 7
Final score table.
551
552
553
554
555
556
557
558
559
560
561
562
563
c1
c2
c3
c4
1
−3
3
−1
3
−2
−1
0
−2
−1
4
−1
9. Conclusions
cte
Final score(m-n)
rre
550
Negative
score(n)
We combine the concept of bipolar fuzzy set and
soft set to introduced the concept of bipolar fuzzy soft
sets. We examine some operations on bipolar fuzzy soft.
We study basic operations on bipolar fuzzy soft set.
We define extended 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 solved decision making problems
by bipolar fuzzy soft set. Therefore, this paper gives an
idea for the beginning of a new study for approximations of data with uncertainties. We will focus on the
following problems : bipolar fuzzy soft relations, bipolar fuzzy soft matrix, bipolar fuzzy soft function and
bipolar fuzzy soft graphs, and applications in artificial
intelligence and general systems.
co
549
Positive
score(m)
Un
548
.
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We would like to express our warmest thanks to the
referees for their time to read the manuscript very carefully and their useful suggestions, and to the Assoc.
Editor of the journal Prof. Guiwu Wei for editing and
communicating the paper.
Table 5
Comparison table of the above table.
.
Acknowledgments
roo
f
.
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