TJFS: Turkish Journal of Fuzzy Systems (eISSN: 1309–1190)
An Official Journal of Turkish Fuzzy Systems Association
Vol.1, No.1, pp. 21-35, 2010.
21
Fuzzy parameterized fuzzy soft set
theory and its applications
Naim Çağman*
Department of Mathematics, Faculty of Arts and Sciences
Gaziosmanpasa University, 60250 Tokat, Turkey
E-mail: [email protected]
*Corresponding author
Filiz Çıtak
Department of Mathematics, Faculty of Arts and Sciences
Gaziosmanpasa University, 60250 Tokat, Turkey
E-mail: [email protected]
Serdar Enginoğlu
Department of Mathematics, Faculty of Arts and Sciences
Gaziosmanpasa University, 60250 Tokat, Turkey
E-mail: [email protected]
Received: February 24, 2010 - Revised: May 24, 2010 – Accepted: June 4, 2010
Abstract
In this work, we give definition of fuzzy parameterized fuzzy soft (fpfs) sets and their
operations. We then define fpfs-aggregation operator to form fpfs-decision making
method that allows constructing more efficient decision processes.
Keywords: Fuzzy soft sets, fuzzy parameterized soft sets, fpfs-sets, fpfs -aggregation
operator, fpfs-decision making.
1. Introduction
Many fields deal daily with the uncertain data that may not be successfully modeled by
the classical mathematics. There are some mathematical tools for dealing with
uncertainties; two of them are fuzzy set theory, developed by Zadeh (1965), and soft set
theory, introduced by Molodtsov (1999), that are related to this work.
At present, work on the soft set theory is progressing rapidly. Maji et al (2003) defined
operations of soft sets to make a detailed theoretical study on the soft sets. By using
these definitions, the applications of soft set theory have been studied increasingly. Soft
decision making (Cagman & Enginoglu 2010a, 2010b, Cagman et al 2010a, 2010b,
Chen et al 2005, Feng et al 2010, Herewan & Deris 2009b, Herawan et al 2009, Kong et
al 2008 2009, Maji et al 2002, Xiao et al 2003, Majumdar & Samanta 2008, 2010, Min
22
2008, Mushrif et al 2006, Roy & Maji 2007, Son 2007, Xiao et al 2009, Yang et al 2007
2009, Zou & Xiao 2008, Zou & Chen 2008) and the theoretical study of soft sets (Ali et
al 2009, Cagman & Enginoglu 2010a, 2010b, Cagman et al 2010a 2010b, Herawan &
Deris 2009a, Kovkov et al 2007, Maji et al 2003, Molodtsov 2001 2004, Molodtsov et
al 2006, Pei & Miao 2005, Yang 2008, Xiao et al 2005 2010, Xu et al 2010) have also
been studied in more detail. The algebraic structure of soft set theory has also been
studied (Acar et al 2010, Aktas&Cagman 2007, Feng et al 2008, Jun 2008 2009, Jun &
Park 2008 2009, Jun et al 2008, 2009a, 2009b, 2009c, 2010a, Mukherjee 2008, Park et
al 2008, Sun et al 2008, Zhan & Jun 2010, Jiang et al 2010, Qin & Hong 2010). Cagman
& Enginoglu (2010a) redefine the operations of soft sets to make them more functional
for improving several new results. By using these new definitions they also construct a
uni-int decision making method which selects a set of optimum elements from the
alternatives. Cagman & Enginoglu (2010b) introduced a matrix representation of this
work that gives several advantages to compute applications of the soft set theory.
By using fuzzy sets, Maji et al (2001a) defined the fuzzy soft set theory and many
scholars study the properties and applications of it (Ahmad & Kharal 2009, Aygunoglu
& Aygun 2009, Cagman et al 2010a 2010b, Deschrijver 2007, Feng et al 2010, Jun
2009, Jun et al 2010b, Kalayathankal&Singh 2010, Kharal & Ahmad 2009, Kong et al
2009, Liu et al 2008, Maji et al 2001b, Majumdar&Samanta 2010, Mukherjee &
Chakraborty 2008, Roy & Maji 2007, Son 2007, Xiao et al 2009, Yang et al 2008
2009).
The approximate function of a soft set is defined from a crisp parameters set to a crisp
subsets of universal set. In this paper, we define fpfs-sets in which the approximate
functions are defined from fuzzy parameters set to the fuzzy subsets of universal set.
We also defined their operations and soft aggregation operator to form fpfs-decision
making method that allows constructing more efficient decision processes. We finally
present examples which show that the methods can be successfully applied to many
problems that contain uncertainties.
2. Preliminary
In this section we summarize the preliminary definitions, and results that are required
later in this paper.
2.1. Soft sets
In this subsection, we give definition of the soft set theory. More detailed theoritical
study of this concept is given in (Çağman & Enginoglu 2010a).
Definition Let U be an initial universe, )(UP be the power set of ,U E be the set of
all parameters and .EA Then, a soft set AF over U is a set defined by a function Af
representing a mapping
)(: UPEfA such that =)(xfA if Ax .
23
Here, Af is called approximate function of the soft set AF , and the value Af (x) is a set
called x-element of the soft set for all xE. It is worth noting that the sets Af (x) may
be arbitrary. Some of them may be empty, some may have nonempty intersection. Thus,
a soft set AF over U can be represented by the set of ordered pairs
)}()(,:))(,{(= UPxfExxfxF AAA
Note that the set of all soft sets over U will be denoted by )(US .
Example Let },,,,{= 54321 uuuuuU be a universal set and },,,{= 4321 xxxxE be a set of
parameters.
If },,{= 432 xxxA and },{=)( 422 uuxf A , =)( 3xf A , Uxf A =)( 4 , then the soft set
AF is written by
)},(}),,{,{(= 4422 UxuuxFA
2.2. Fuzzy sets
In this subsection, we present the basic definitions of fuzzy set theroy (Zadeh, 1965)
that is useful for subsequent discussions. More detailed explanations related to this
theory may be found in earlier studies (Dubois & Prade 1980, Klir & Folger 1988,
Zimmermann 1991).
Definition Let U be a universe. A fuzzy set X over U is a set defined by a function
X representing a mapping
[0,1]: UX
Here, X called membership function of X , and the value )(uX is called the grade of
membership of Uu . The value represents the degree of u belonging to the fuzzy set
.X Thus, a fuzzy set X over U can be represented as follows,
[0,1]})(,:))/({(= uUuuuX XX
Note that the set of all the fuzzy sets over U will be denoted by )(UF .
2.3. Fuzzy soft sets
By using fuzzy sets, Maji et al (2001a) defined fuzzy soft sets (fs-sets). In this
subsection, by using the definition, given in (Çağman & Enginoglu 2010a), we redefine
the fs-sets and their operations. More detailed theoritical study of this concept is given
in (Çağman et al (2010a).
24
In subsection 2.1, the approximate function of a soft set is defined from a crisp
parameters set to a crisp subsets of universal set. But the approximate functions of fs-
sets are defined from crisp parameters set to the fuzzy subsets of universal set. To avoid
the confusion we will use A , B , C ,..., etc. for fs -sets and A , B , C ,..., etc. for
their fuzzy approximate functions, respectively.
Definition Let U be an initial universe, E be the set of all parameters, EA and
)(xA be a fuzzy set over U for all xE. Then, an fs-set A over U is a set defined
by a function A representing a mapping
A : E )(UF such that =)(xA if Ax .
Here, A is called fuzzy approximate function of the fs-set A , and the value )(xA is a
fuzzy set called x-element of the fs-set for all xE. Thus, an fs-set A over U can be
represented by the set of ordered pairs
)}()(,:))(,{(= UFxExxx AAA
Note that from now on the sets of all fs -sets over U will be denoted by )(UFS .
Example Assume that },,,,{= 54321 uuuuuU is a universal set and },,,{= 4321 xxxxE is
a set of all parameters.
If },,{= 432 xxxA , },0.9/{0.5/=)( 422 uuxA , =)( 3xA and UxA =)( 4 , then the fs -
set A is written by
)},(}),,0.9/{0.5/,{(= 4422 UxuuxA
2.4. Fuzzy parameterized soft sets
In this subsection, by using the definition, given in (Çağman & Enginoglu, 2010a), we
define the fuzzy parameterized soft sets (fps-sets) and their operations. More detailed
theoritical study of this concept is given by Çağman et al (2010b). In subsection 2.1, the
approximate function of a soft set is defined from a crisp parameters set to a crisp
subsets of universal set. But the approximate functions of fps-sets are defined from
fuzzy parameters set to the crisp subsets of universal set.
Throught this subsection, the fuzzy subsets of parameters set are denoted by the
letter ,...,, ZYX , to avoid confusion and complexity of the symbols.
Definition Let U be an initial universe, )(UP be the power set of U , E be the set of
all parameters and X be a fuzzy set over E with the membership function
[0,1]: EX . Then, an fps-set XF over U is a set defined by a function Xf
representing a mapping
25
)(: UPEfX such that =)(xfX if 0=)(xX .
Here, Xf is called approximate function of the fps-set XF , and the value )(xf X is a set
called x-element of the fps-set for all xE. Thus, an fps-set XF over U can be
represented by the set of ordered pairs
,:))(,)/({(= ExxfxxF XXX [0,1]})(),()( xUPxf XX
Note that from now on the sets of all fps -sets over U will be denoted by )(UFPS .
Example Let },,,,{= 54321 uuuuuU be a universal set and },,,{= 4321 xxxxE be a set of
parameters.
If },1/,0.5/{0.2/= 432 xxxX and },{=)( 422 uuxf X , =)( 3xf X , Uxf X =)( 4 , then the
fps-set XF is written by
)},(1/}),,{,{(0.2/= 4422 UxuuxFX
3. Fuzzy parameterized fuzzy soft sets
In this section, we define fuzzy parameterized fuzzy soft sets (fpfs-sets) and their
operations with examples. In subsection 2.1, the approximate function of a soft set is
defined from a crisp parameters set to a crisp subsets of universal set. But the
approximate functions of fpfs-set are defined from fuzzy parameters set to the fuzzy
subsets of universal set.
To avoid the confusion we will use X , Y , Z ,..., etc. for fpfs-sets and X , Y , Z ,...,
etc. for their fuzzy approximate functions, respectively.
Definition 3.1: Let U be an initial universe, E be the set of all parameters and X be a
fuzzy set over E with the membership function [0,1]: EX and )(xX be a fuzzy
set over U for all xE. Then, an fpfs-set X over U is a set defined by a function
)(xX representing a mapping
X : E )(UF such that =)(xX if 0)( xX .
Here, X is called fuzzy approximate function of the fpfs-set X , and the value )(xX
is a fuzzy set called x-element of the fpfs-set for all xE. Thus, an fpfs-set X over U
can be represented by the set of ordered pairs
,:))(,)/({(= Exxxx XXX [0,1]})(),()( xUFx XX
It must be noted that the sets of all fpfs-sets over U will be denoted by )(UFPFS .
26
Example: Assume that },,,,{= 54321 uuuuuU is a universal set and },,,{= 4321 xxxxE is
a set of parameters.
If },1/,0.5/{0.2/= 432 xxxX and },0.3/{0.5/=)( 512 uuxX , =)( 3xX , and UxX =)( 4 ,
then the fpfs-set X is written by
)},(1/}),,0.3/{0.5/,{(0.2/= 4512 UxuuxX
Definition 3.2: Let )(UFPFSX . If =)(xX for all Ex , then X is called an X-
empty fpfs-set, denoted by X .
If =X , then the X-empty fpfs-set (X ) is called empty fpfs-set, denoted by .
Definition 3.3: Let )(UFPFSX . If 1=)(xX and UxX =)( for all Xx , then X
is called X-universal fpfs-set, denoted byX~ .
If EX = , then the X-universal fpfs-set is called universal fpfs-set, denoted byE~ .
Example: Assume that },,,,{= 54321 uuuuuU is a universal set and },,,{= 4321 xxxxE is
a set of parameters.
If },1/,0.5/{0.2/= 432 xxxX and },0.3/{0.5/=)( 512 uuxX , =)( 3xX , and UxX =)( 4 ,
then the fpfs-set X is written by
)},(1/}),,0.3/{0.5/,{(0.2/= 4512 UxuuxX
If },0.7/{1/= 41 xxY and =)( 1xX , =)( 4xX then the fpfs-set Y is a Y-empty fpfs-
set, i.e., YY = .
If },1/{1/= 21 xxZ , UxZ =)( 1 and UxZ =)( 2 , then the fpfs-set Z is Z -universal
fpfs-set, i.e., ZZ ~= .
If =X , then the fpfs -set X is an empty fpfs-set, i.e., =X .
If EX = and UxiX =)( , for all Exi ( 1,2,3,4=i ), then the fpfs-set X is a universal
fpfs-set, i.e., EX ~= .
Definition 3.4: Let )(, UFPFSYX . Then, X is an fpfs-subset of Y , denoted by
YX ~ , if )()( xx YX and )()( xx YX for all Ex .
27
Proposition 3.1: Let )(, UFPFSYX . Then,
i. EX ~
~
ii. XX
~
iii. X~
iv. XX ~
v. YX ~ and ZXZY ~~
They can be proved easily by using the fuzzy approximate and membership functions of
the fpfs-sets.
Definition 3.5: Let )(, UFPFSYX . Then, X and Y are fpfs-equal, written as
YX = , if and only if )(=)( xx YX and )(=)( xx YX for all Ex .
Proposition 3.2: Let )(,, UFPFSZYX . Then,
i. ( YX = and ZXZY =)=
ii. ( YX ~ and YXXY =)~
The proofs are trivial.
Definition 3.6: Let )(UFPFSX . Then, the complement of X , denoted by c
X
~
, is
defined by
)(1=)( xx XcX and )(=)(~ xx c
XcX
Exallfor , where )(xc
X is complement of the set )(xX , that is, )(\=)( xUx X
c
X
for every Ex .
Proposition 3.3: Let )(UFPFSX . Then,
i. X
cc
X =)(~~
ii. E
c~
~
=
By using the fuzzy approximate and membership functions of the fpfs-sets, the proof
can be straightforward.
Definition 3.7: Let )(, UFPFSYX . Then, union of X and Y , denoted by
YX ~ , is defined by
})(),({max=)(~ xxx YXYX
and
)()(=)(~ xxx YXYX
Exallfor .
Proposition 3.4: Let )(,, UFPFSZYX . Then,
i. XXX =~
28
ii. XXX
~
iii. XX =~
iv. EEX ~~ =~
v. XYYX ~=~
vi. )~(~=~)~( ZYXZYX
The proofs can be easily obtained from Definition 3.7.
Definition 3.8: Let )(, UFPFSYX . Then, intersection of X and Y , denoted by
YX ~ , is defined by
)}(),({min=)(~ xxx YXYX
and
)()(=)(~ xxx YXYX
Exallfor .
Proposition 3.5: Let )(,, UFPFSZYX . Then,
i. XXX =~
ii. XXX
~
iii. =~X
iv. XEX =~~
v. XYYX ~=~
vi. )~(~=~)~( ZYXZYX
The proofs can be easily obtained from Definition 3.8.
Remark 3.1: Let )(UFPFSX . If EX ~ or X , then
E
c
XX ~
~~ and
c
XX
~~ .
Proposition 3.6: Let )(, UFPFSYX . Then, De Morgan's laws are valid
i. c
Y
c
X
c
YX
~~~ ~=)~(
ii. c
Y
c
X
c
YX
~~~ ~=)~(
Proof: For all Ex ,
)(=
)}(),({min=
)}(),1({1min=
)}(),({max1=
)(1=)(.
~
~)~(
x
xx
xx
xx
xxi
cYcX
cYcX
YX
YX
YXcYX
and
29
).(=
)()(=
)()(=
))(())((=
))()((=
)(=)(
~~~
~~
~~)~(
x
xx
xx
xx
xx
xx
cYcX
cYcX
c
Y
c
X
c
Y
c
X
c
YX
c
YXcYX
Likewise, the proof of ii. can be made similarly.
Proposition 3.7: Let )(,, UFPFSZYX . Then,
i. )~(~)~(=)~(~ZXYXZYX
ii. )~(~)~(=)~(~ZXYXZYX
Proof: For all Ex ,
.i )()~(~ xZYX
)(=
)}(),({min=
)}}(),({max)},(),({max{min=
)}}(),({min),({max=
)}(),({max=
)~(~)~(
~~
~
x
xx
xxxx
xxx
xx
ZXYX
ZXYX
ZXYX
ZYX
ZYX
and
)()~(~ xZYX
)(=
)()(=
))()(())()((=
))()(()(=
)()(=
)~(~)~(
~~
~
x
xx
xxxx
xxx
xx
ZXYX
ZXYX
ZXYX
ZYX
ZYX
Likewise, the proof of ii. can be made in a similar way.
4. fpfs -aggregation operator
In this section, we define an aggregate fuzzy set of an fpfs-set. We also define fpfs-
aggregation operator that produce an aggregate fuzzy set from an fpfs-set and its fuzzy
parameter set.
Definition: Let )(UFPFSX . Then fpfs-aggregation operator, denoted by aggFPFS ,
is defined by
),()()(: UFUFPFSEFFPFS agg
30
*=),( XXagg XFPFS
where
}:)/({= *
* UuuuX
X
which is a fuzzy set over U . The value *
X is called aggregate fuzzy set of the X .
Here, the membership degree )(* uX
of u is defined as follows
)()(||
1=)( )(* ux
Eu xX
ExXX
where || E is the cardinality of E .
5. fpfs -set decision making method
The approximate functions of an fpfs-set are fuzzy. The aggFPFS on the fuzzy sets is an
operation by which several approximate functions of an fpfs -set are combined to
produce a single fuzzy set that is the aggregate fuzzy set of the fpfs-set. Once an
aggregate fuzzy set has been arrived at, it may be necessary to choose the best single
crisp alternative from this set. Therefore, we can construct an fpfs-decision making
method by the following algorithm.
Step 1 Construct an fpfs -set X over ,U
Step 2 Find the aggregate fuzzy set *
X of X ,
Step 3 Find the largest membership grade )(max * uX
.
Example: In this example, we have presented an application for the fpfs-decision
making method. Assume that a company wants to fill a position. There are eight
candidates who form the set of alternatives, },,,,,,,{= 87654321 uuuuuuuuU . The hiring
committee consider a set of parameters, },,,,{= 54321 xxxxxE . The parameters ix
( 1,2,3,4,5=i ) stand for "experience", "computer knowledge", "young age", "good
speaking" and "friendly", respectively.
After a serious discussion each candidate is evaluated from point of view of the goals
and the constraint according to a chosen subset },0.6/,0.9/{0.5/= 432 xxxX of E .
Finally, the committee constructs the following fpfs-set over .U
Step 1 Let the constructed fpfs-set, X , be as follows,
,,0.9/,0.1/,0.4/{0.3/,{(0.5/ 54322 uuuuxX })0.7/ 7u , ,,0.4/{0.4/,(0.9/ 213 uux
}),0.3/0.9/ 43 uu , ,,0.1/,0.5/{0.2/,(0.6/ 5214 uuux })},1/0.7/ 87 uu
31
Step 2 The aggregate fuzzy set can be found as,
,,0.064/,0.202/,0.162/{0.096/= 4321
* uuuuX },0.12/,0.154/0.102/ 875 uuu
Step 3: Finally, the largest membership grade can be chosen by 0.202=)(max * uX
which means that the candidate 3u has the largest membership grade, hence he is
selected for the job.
6. Conclusion
In this paper, we first defined fpfs-sets and their operations. We then presented the
decision making method on the fpfs-set theory. Finally, we provided an example that
demonstrated that this method can be successfully worked. It can be applied to
problems of many fields that contain uncertainty. However, the approach should be
more comprehensive in the future to solve the related problems.
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