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: ncagman@gop.edu.tr

*Corresponding author

Filiz Çıtak

Department of Mathematics, Faculty of Arts and Sciences

Gaziosmanpasa University, 60250 Tokat, Turkey

E-mail: filizcitak@gop.edu.tr

Serdar Enginoğlu

Department of Mathematics, Faculty of Arts and Sciences

Gaziosmanpasa University, 60250 Tokat, Turkey

E-mail: serdarenginoglu@gop.edu.tr

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.

References

Acar, U., Koyuncu, F., Tanay, B., Soft sets and soft rings, Computers and Mathematics

with Applications, 59, 3458-3463, 2010.

Ahmad, B., Kharal, A., On fuzzy soft sets, Advances in Fuzzy Systems, 1-6, 2009.

Aktaş, H., Çagman, N., Soft sets and soft groups, Information Sciences, 1(77), 2726-

2735, 2007.

Ali, M.I., Feng, F., Liu, X., Min, W.K., Shabir, M., On some new operations in soft set

theory, Computers and Mathematics with Applications, 57, 1547-1553, 2009.

Aygünoglu, A. Aygün, H., Introduction to fuzzy soft groups, Computers and

Mathematics with Applications 58, 1279-1286, 2009.

Cagman, N., Enginoglu, S., Soft set theory and uni-int decision making, European

Journal of Operational Research, DOI: 10.16/j.ejor.2010.05.004, 2010a.

Cagman, N., Enginoglu, S., Soft matrices and its decision makings. Computers and

Mathematics with Applications, 59, 3308-3314, 2010b.

Cagman, N., Enginoglu, S., Citak, F., Fuzzy soft set theory and its applications,

(Submitted), 2010a,

Cagman, N., Citak, F., Enginoglu, S., Fuzzy parameterized soft set theory and its

applications, (Submitted), 2010b.

Chen, D., Tsang, E.C.C., Yeung, D.S., Wang, X., The parameterization reduction of soft

sets and its applications. Computers and Mathematics with Applications, 49, 757-763,

32

2005.

Deschrijver, G., Arithmetic operators in interval-valued fuzzy set theory, Information

Sciences, 177, 2906–2924, 2007.

Dubois, D. and Prade, H., Fuzzy Sets and Systems: Theory and Applications, Academic

Press, 1980.

Feng F., Jun, Y. B., Liu, X., Li, L., An adjustable approach to fuzzy soft set based

decision making, Journal of Computational and Applied Mathematics 234, 10-20, 2010.

Feng, F., Jun, Y. B., Zhao, X., Soft semirings, Fuzzy Sets and Systems: Theory and

Applications, 56 (10), 2621-2628, 2008.

Herawan T., Deris M. M., A direct proof of every rough set is a soft set, in 3rd asia

international conference on modelling and simulation, Vol. 1 and 2, Bundang,

INDONESIA, pp. 119-124, 2009a.

Herawan T., Deris M. M., On multi-soft sets construction in information systems, in

Emergıng Intellıgent Computıng Technology And Applıcatıons: Wıth Aspects Of

Artıfıcıal Intellıgence, Book series: Lecture Notes in Artificial Intelligence, Vol. 5755,

pp. 101-110, 2009b.

Herawan, T., Rose

, A. N. M., Deris, M. M., Soft set theoretic approach for

dimensionality reduction, in database theory and application 64, Springer Berlin

Heidelberg, pp.171-178, 2009.

Jiang, Y., Tang, Y., Chen, Q., Wang, J., Tang, s., Extending soft sets with description

logics, Computers and Mathematics with Applications 59, 2087-2096, 2010.

Jun, Y. B., Soft BCK/BCI-algebras, Computers and Mathematics with Applications, 56,

1408–1413, 2008.

Jun, Y. B., Generalizations of ( , q)-fuzzy subalgebras in BCK/BCI-algebras,

Computers and Mathematics with Applications 58,1383-1390, 2009.

Jun, Y. B., Park, C. H., Applications of soft sets in ideal theory of BCK/BCI-algebras.

Information Sciences, 178, 2466-2475, 2008.

Jun, Y. B., Park, C. H., Applications of soft sets in Hilbert algebras, Iranian Journal

Fuzzy Systems, 6 (2), 55-86, 2009.

Jun, Y. B., Kim, H. S., Neggers, J., Pseudo d-algebras, Information Sciences 179, 1751-

1759, 2009a.

Jun, Y. B., Lee, K. J., Khan, A., Soft ordered semigroups, Mathematical Logic

Quarterly. 56 (1), 42-50, 2010a.

33

Jun, Y. B., Lee, K. J., Park, C. H., Soft set theory applied to commutative ideals, in

BCK--algebras, Journal of Applied Mathematics & Informatics, 26 (3-4), 707-720,

2008.

Jun, Y. B., Lee, K. J., Park, C. H., Soft set theory applied to ideals in d-algebras,

Computers and Mathematics with Applications 57, 367-378, 2009b.

Jun, Y. B., Lee, K. J., Park, C. H., Fuzzy soft set theory applied to BCK/BCI-algebras,

Computers and Mathematics with Applications 59, 3180-3192, 2010b.

Jun, Y. B., Lee, K. J., Zhan, J,, Soft p-ideals of soft BCI-algebras, Computers and

Mathematics with Applications, 58, 2060-2068, 2009c.

Kalayathankal, S. J., Singh, G. S., A fuzzy soft flood alarm model, Mathematics and

Computers in Simulation 80, 887–893, 2010.

Kharal, A., Ahmad, B., Mappings on fuzzy soft classes, Advances in Fuzzy Systems, 1-

1, 2009.

Klir, G.J., and Folger, T. A., Fuzzy Sets, uncertainty, and Information, New Jersey,

1988.

Kong, Z., Gao, L., Wang, L., Li, S., The normal parameter reduction of soft sets and its

algorithm, Computers and Mathematics with Applications, 56, 3029-3037, 2008.

Kong, Z., Gao, L., Wang, L., Comment on ―A fuzzy soft set theoretic approach to

decision making problems‖ Journal of Computational and Applied Mathematics, 223

(2), 540-542, 2009.

Kovkov, D.V., Kolbanov, V. M., Molodtsov, D. A., Soft sets theory-based optimization.

Journal of Computer and Systems Sciences International, 46 (6), 872-880, 2007.

Liu, J.H., Yan, R.X., Yao, B.X., Fuzzy soft sets and fuzzy soft groups, In: Control and

Decision Conference (CCDC 2008), pp. 2626-2629, 2008.

Maji, P.K., Biswas, R., Roy, A.R., Fuzzy soft sets, Journal of Fuzzy Mathematics, 9 (3),

589-602, 2001a.

Maji, P.K., Biswas, R., Roy, A.R., Intuitionistic fuzzy soft sets, Journal of Fuzzy

Mathematics, 9 (3), 677-691, 2001b.

Maji, P.K., Biswas, R., Roy, A.R., Soft set theory, Computers and Mathematics with

Applications, 45, 555-562, 2003.

Maji, P.K., Roy, A.R., Biswas, R., An application of soft sets in a decision making

problem, Computers and Mathematics with Applications, 44, 1077-1083, 2002.

Majumdar, P., Samanta, S. K., Similarity measure of soft sets, New Mathematics and

34

Natural Computation, 4 (1), 1-12, 2008.

Majumdar, P., Samanta, S. K., Generalised fuzzy soft sets, Computers and Mathematics

with Applications 59, 1425-1432, 2010.

Min, W. K., A note on ―Interval valued soft set‖, Journal of Fuzzy Logic and Intelligent

Systems, 18 (3), 412-415, 2008.

Molodtsov, D.A., Soft set theory-first results, Computers and Mathematics with

Applications, 37, 19-31, 1999.

Molodtsov, D.A., Describing dependences using soft sets, Journal of Computer and

Systems Scıences International, 40 (6), 975-982, 2001.

Molodtsov, D., The Theory of Soft Sets (in Russian), URSS Publishers, Moscow, 2004.

Molodtsov, D., Leonov, V. Y., Kovkov, D. V., Soft sets technique and its application,

Nechetkie Sistemy i Myagkie Vychisleniya, 1 (1), 8–39 2006.

Mukherjee, A., Conditional topology & some algebraic structures formed by

intuitionistic fuzzy soft set relations & their applications, Pre-ICM International

Convention on Mathematical Sciences, University of Delhi, 18-20 December, Delhi,

India, 2008.

Mukherjee, A., Chakraborty, S.B., On intuitionistic fuzzy soft relations. Bulletin of

Kerala Mathematics Association, 5 (1), 35-42, 2008.

Mushrif, M.M., Sengupta, S., Ray, A.K., Texture classification using a novel, Soft-Set

Theory Based Classification, Algorithm. Lecture Notes In Computer Science. 3851,

246-254, 2006.

Park, C.H., Jun, Y.B., Öztürk, M.A., Soft WS-algebras. Communications of the Korean

Mathematical Society., 23 (3), 313-324, 2008.

Pei, D., Miao D., From soft sets to information systems, In: Proceedings of Granular

Computing (Eds: X. Hu, Q. Liu, A. Skowron, T.Y. Lin, R.R. Yager, B.Zhang) IEEE

2005, Volume: 2, pp: 617- 621.

Qin K., Hong, Z., On soft equality, Journal of Computational and Applied Mathematics

234, 1347-1355, 2010.

Roy, A.R., Maji, P.K., A fuzzy soft set theoretic approach to decision making problems.

Journal of Computational and Applied Mathematics, 203, 412-418, 2007.

Son, M.J., Interval-valued fuzzy soft sets, Journal of Fuzzy Logic and Intelligent

Systems, 17 (4), 557-562, 2007.

35

Sun, Q.M., Zhang, Z.L, Liu, J., Soft sets and soft modules, Proceedings of Rough Sets

and Knowledge Technology, Third International Conference, RSKT 2008, 17-19 May,

Chengdu, China, pp. 403-409, 2008.

Xiao, Z., Chen, L., Zhong, B., Ye, S., Recognition for soft information based on the

theory of soft sets, In Proceedings of ICSSSM-05 (Ed: J. Chen), IEEE 2005, Volume:

2, pp: 1104- 1106.

Xiao, Z., Gong, K., Zou, Y., A combined forecasting approach based on fuzzy soft sets,

Journal of Computational and Applied Mathematics 228, 326-333, 2009.

Xiao, Z., Gong, K., Xia, S., Zou, Y., Exclusive disjunctive soft sets, Computers and

Mathematics with Applications 59, 2128-2137, 2010.

Xiao, Z., Li, Y., Zhong, B., Yang, X., Research on synthetically evaluating method for

business competitive capacity based on soft set, Statistical Research, 52-54. 2003.

Xu, W., Ma, J., Wang, S., Hao, G., Vague soft sets and their properties, Computers and

Mathematics with Applications 59, 787-794, 2010.

Yang, C.F, A note on ―Soft Set Theory‖ [Comput. Math. Appl. 45 (4-5) (2003) 555–

562], Computers and Mathematics with Applications 56, 1899–1900, 2008.

Yang, X., Yu, D, Yang, J., Wu, C., ―Generalization of Soft Set Theory: From Crisp to

Fuzzy Case, Proceedings of the Second International Conference of Fuzzy Information

and Engineering (ICFIE-2007)‖, 27-Apr., pp. 345-355, 2007.

Yang, X., Yu, D, Yang, J., Wu, C., Fuzzy soft set and soft fuzzy set, Fourth

International Conference on Natural Computation (ICNC '08), Vol. 6, pp: 252-255,

IEEE, 2008.

Yang, X., Lin, T. Y., Yang, J., Li, Y., Yu, D., Combination of interval-valued fuzzy set

and soft set, Computers and Mathematics with Applications, 58, 521-527, 2009.

Zadeh, L.A., Fuzzy Sets, Information and Control, 8, 338-353, 1965.

Zhan, J., Jun, Y.B., Soft BL-algebras based on fuzzy sets, Computers and Mathematics

with Applications 59, 2037—2046, 2010.

Zimmermann, H.J., Fuzzy Set Theory and Its Applications, Kluwer, 1991.

Zou, Y., Chen, Y., Research on soft set theory and parameters reduction based on

relational algebra, Second International Symposium, Intelligent Information

Technology Application, IITA '08, 20-22 Dec., Vol.1, pp.152 – 156, 2008.

Zou, Y., Xiao, Z., Data analysis approaches of soft sets under incomplete information.

Knowledge-Based Systems, 21, 941-945, 2008.