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.

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

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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, P(U ) be the power set of U , E be the set of all parameters and A E. Then, a soft set FA over U is a set defined by a function f A representing a mapping

f A : E P(U ) such that f A (x) = if x A . 22

Here, f A is called approximate function of the soft set FA , and the value f A (x) is a set called x-element of the soft set for all x E. It is worth noting that the sets f A (x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. Thus, a soft set FA over U can be represented by the set of ordered pairs

FA = {( x, f A ( x)) : x E, f A ( x) P(U )} Note that the set of all soft sets over U will be denoted by S (U ) . Example Let U = {u1 , u2 , u3 , u4 , u5} be a universal set and E = {x1 , x2 , x3 , x4 } be a set of parameters. If A = {x2 , x3 , x4 } and f A ( x2 ) = {u 2 , u 4 } , f A ( x3 ) = , f A ( x4 ) = U , then the soft set FA is written by FA = {( x2 ,{u 2 , u 4 }), ( x4 ,U )}

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

X : U [0,1] Here, X called membership function of X , and the value X (u ) is called the grade of membership of u U . 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,

X = {( X (u)/u) : u U , X (u) [0,1]} Note that the set of all the fuzzy sets over U will be denoted by F (U ) .

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). 23

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 fssets 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, A E and A (x) be a fuzzy set over U for all x E. Then, an fs-set A over U is a set defined by a function A representing a mapping

A : E F (U ) such that A (x) = if x A . Here, A is called fuzzy approximate function of the fs-set A , and the value A (x) is a fuzzy set called x-element of the fs-set for all x E. Thus, an fs-set A over U can be represented by the set of ordered pairs

A = {( x, A ( x)) : x E, A ( x) F (U )} Note that from now on the sets of all fs -sets over U will be denoted by FS (U ) . Example Assume that U = {u1 , u2 , u3 , u4 , u5} is a universal set and E = {x1 , x2 , x3 , x4 } is a set of all parameters. If A = {x2 , x3 , x4 } , A ( x2 ) = {0.5/u2 ,0.9/u4 } , A ( x3 ) = and A ( x4 ) = U , then the fs set A is written by A = {( x2 ,{0.5/u2 ,0.9/u4 }), ( x4 ,U )}

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 X , Y , Z ,... , to avoid confusion and complexity of the symbols. Definition Let U be an initial universe, P(U ) 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 X : E [0,1] . Then, an fps-set FX over U is a set defined by a function f X representing a mapping 24

f X : E P(U ) such that f X (x) = if X (x) = 0 . Here, f X is called approximate function of the fps-set FX , and the value f X (x) is a set called x-element of the fps-set for all x E. Thus, an fps-set FX over U can be represented by the set of ordered pairs

FX = {( X ( x)/x, f X ( x)) : x E, f X ( x) P(U ), X ( x) [0,1]} Note that from now on the sets of all fps -sets over U will be denoted by FPS (U ) . Example Let U = {u1 , u2 , u3 , u4 , u5} be a universal set and E = {x1 , x2 , x3 , x4 } be a set of parameters. If X = {0.2/x2 ,0.5/x3 ,1/x4 } and f X ( x2 ) = {u 2 , u 4 } , f X ( x3 ) = , f X ( x4 ) = U , then the fps-set FX is written by FX = {(0.2/x2 ,{u 2 , u 4 }), (1/x4 ,U )}

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 X : E [0,1] and X (x) be a fuzzy set over U for all x E. Then, an fpfs-set X over U is a set defined by a function X (x) representing a mapping

X : E F (U ) such that X (x) = if X ( x) 0 . Here, X is called fuzzy approximate function of the fpfs-set X , and the value X (x) is a fuzzy set called x-element of the fpfs-set for all x E. Thus, an fpfs-set X over U can be represented by the set of ordered pairs

X = {( X ( x)/x, X ( x)) : x E, X ( x) F (U ), X ( x) [0,1]} It must be noted that the sets of all fpfs-sets over U will be denoted by FPFS (U ) .

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Example: Assume that U = {u1 , u2 , u3 , u4 , u5} is a universal set and E = {x1 , x2 , x3 , x4 } is a set of parameters. If X = {0.2/x2 ,0.5/x3 ,1/x4 } and X ( x2 ) = {0.5/u1 ,0.3/u5} , X ( x3 ) = , and X ( x4 ) = U , then the fpfs-set X is written by

X = {(0.2/x2 ,{0.5/u1 ,0.3/u5}), (1/x4 ,U )} Definition 3.2: Let X FPFS (U ) . If X (x) = for all x E , then X is called an Xempty 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 X FPFS (U ) . If X (x) = 1 and X ( x) = U for all x X , then X is called X-universal fpfs-set, denoted by X~ . If X = E , then the X-universal fpfs-set is called universal fpfs-set, denoted by E~ . Example: Assume that U = {u1 , u2 , u3 , u4 , u5} is a universal set and E = {x1 , x2 , x3 , x4 } is a set of parameters. If X = {0.2/x2 ,0.5/x3 ,1/x4 } and X ( x2 ) = {0.5/u1 ,0.3/u5} , X ( x3 ) = , and X ( x4 ) = U , then the fpfs-set X is written by

X = {(0.2/x2 ,{0.5/u1 ,0.3/u5}), (1/x4 ,U )} If Y = {1/x1 ,0.7/x4 } and X ( x1 ) = , X ( x4 ) = then the fpfs-set Y is a Y-empty fpfsset, i.e., Y = Y . If Z = {1/x1 ,1/x2 } , Z ( x1 ) = U and Z ( x2 ) = U , then the fpfs-set Z is Z -universal fpfs-set, i.e., Z = Z~ . If X = , then the fpfs -set X is an empty fpfs-set, i.e., X = . If X = E and X ( xi ) = U , for all xi E ( i = 1,2,3,4 ), then the fpfs-set X is a universal fpfs-set, i.e., X = E~ . Definition 3.4: Let X , Y FPFS (U ) . Then, X is an fpfs-subset of Y , denoted by ~ , if ( x) ( x) and ( x) ( x) for all x E . X Y X Y X Y

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Proposition 3.1: Let X , Y FPFS (U ) . Then, ~ ~ i. X E ~ ii. X

iii. iv. v.

X

~ X ~ X X ~ and ~ ~ X Y Y Z X Z

They can be proved easily by using the fuzzy approximate and membership functions of the fpfs-sets. Definition 3.5: Let X , Y FPFS (U ) . Then, X and Y are fpfs-equal, written as X = Y , if and only if X ( x) = Y ( x) and X ( x) = Y ( x) for all x E . Proposition 3.2: Let X , Y , Z FPFS (U ) . Then, i. ( X = Y and Y = Z ) X = Z ~ ) = ~ and ii. ( X Y X X Y Y The proofs are trivial. ~

Definition 3.6: Let X FPFS (U ) . Then, the complement of X , denoted by Xc , is defined by c ( x) = 1 X ( x) and c~ ( x) = Xc ( x) X

X

for all x E , where (x) is complement of the set X (x) , that is, Xc ( x) = U \ X ( x) for every x E . c X

Proposition 3.3: Let X FPFS (U ) . Then, ~ ~ i. (Xc ) c = X ~ ii. c = E~ By using the fuzzy approximate and membership functions of the fpfs-sets, the proof can be straightforward. Definition 3.7: Let X , Y FPFS (U ) . Then, union of X and Y , denoted by ~ , is defined by X Y X ~ Y ( x) = max{ X ( x), Y ( x)} and X ~ Y ( x) = X ( x) Y ( x) for all x E . Proposition 3.4: Let X , Y , Z FPFS (U ) . Then, ~ = i. X X X 27

iii. iv.

~ X X X ~ X = X ~ ~ = ~

v. vi.

~ = ~ X Y Y X ~ ~ ~ ( ~ ) (X Y ) Z = X Y Z

ii.

X

E

E

The proofs can be easily obtained from Definition 3.7. Definition 3.8: Let X , Y FPFS (U ) . Then, intersection of X and Y , denoted by ~ , is defined by X Y X ~ Y ( x) = min{ X ( x), Y ( x)} and X ~ Y ( x) = X ( x) Y ( x) for all x E . Proposition 3.5: Let X , Y , Z FPFS (U ) . Then, ~ = i. X X X ~ X X X ii. ~ = iii. X ~ ~ = iv. X

v. vi.

E

X

~ = ~ X Y Y X ~ ~ ~ ( ~ ) (X Y ) Z = X Y Z

The proofs can be easily obtained from Definition 3.8.

~ c~ ~ and Remark 3.1: Let X FPFS (U ) . If X E~ or X , then X X E c~ ~ X X . Proposition 3.6: Let X , Y FPFS (U ) . Then, De Morgan's laws are valid ~ ) c~ = c~ ~ c~ i. (X Y X Y ~ ) c~ = c~ ~ c~ ii. (X Y X Y Proof: For all x E , i.

~ Y )c (X

( x) = 1 X ~ Y ( x) = 1 max{ X ( x), Y ( x)} = min{1 X ( x),1 Y ( x)} = min{ =

~Y Xc

Xc

( x), c ( x)} Y

c ( x)

and

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( X ~ Y )c~ ( x) = Xc ~ Y ( x)

= ( X ( x) Y ( x)) c = ( X ( x)) c ( Y ( x)) c = Xc ( x) Yc ( x) = =

~ Xc

( x)

~ ~ c~ Xc Y

~ Yc

( x)

( x).

Likewise, the proof of ii. can be made similarly. Proposition 3.7: Let X , Y , Z FPFS (U ) . Then, ~ ( ~ ) = ( ~ ) ~ ( ~ ) i. X Y Z X Y X Z ~ ( ~ ) = ( ~ ) ~ ( ~ ) ii. X Y Z X Y X Z Proof: For all x E , i. X ~ (Y ~ Z ) ( x) = max{ X ( x), Y ~ Z ( x)} = max{ X ( x), min{Y ( x), Z ( x)}} = min{max{ X ( x), Y ( x)}, max{ X ( x), Z ( x)}} = min{ X ~ Y ( x), X ~ Z ( x)} = ( X ~ Y )~ ( X ~ Z ) ( x) and X ~ (Y ~ Z ) ( x) = X ( x) Y ~ Z ( x) = X ( x) ( Y ( x) Z ( x)) = ( X ( x) Y ( x)) ( X ( x) Z ( x)) = X ~ Y ( x) X ~ Z ( x) = ( X ~ Y )~ ( X ~ Z ) ( x)

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 fpfsaggregation operator that produce an aggregate fuzzy set from an fpfs-set and its fuzzy parameter set. Definition: Let X FPFS (U ) . Then fpfs-aggregation operator, denoted by FPFS agg , is defined by FPFS agg : F ( E ) FPFS (U ) F (U ),

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FPFS agg ( X , X ) = X* where X* = {

* X

(u )/u : u U }

which is a fuzzy set over U . The value X* is called aggregate fuzzy set of the X . Here, the membership degree

*X

(u ) of u is defined as follows

* (u ) = X

1 X ( x) X ( x) (u) | E | xE

where | E | is the cardinality of E . 5. fpfs -set decision making method The approximate functions of an fpfs-set are fuzzy. The FPFS agg 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

*X

(u ) .

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, U = {u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8} . The hiring committee consider a set of parameters, E = {x1 , x2 , x3 , x4 , x5} . The parameters xi ( i = 1,2,3,4,5 ) 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 X = {0.5/x2 ,0.9/x3 ,0.6/x4 } of E . Finally, the committee constructs the following fpfs-set over U . Step 1 Let the constructed fpfs-set, X , be as follows,

X {(0.5/x2 ,{0.3/u 2 ,0.4/u3 ,0.1/u 4 ,0.9/u5 , 0.7/u7 }) , (0.9/x3 , {0.4/u1 ,0.4/u2 , 0.9/u3 ,0.3/u4 }) , (0.6/x4 ,{0.2/u1 ,0.5/u 2 ,0.1/u5 , 0.7/u7 ,1/u8})}

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Step 2 The aggregate fuzzy set can be found as, X* = {0.096/u1 ,0.162/u 2 ,0.202/u3 ,0.064/u 4 , 0.102/u5 ,0.154/u7 ,0.12/u8 }

Step 3: Finally, the largest membership grade can be chosen by max

* X

(u ) = 0.202

which means that the candidate u3 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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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.

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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

21

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, P(U ) be the power set of U , E be the set of all parameters and A E. Then, a soft set FA over U is a set defined by a function f A representing a mapping

f A : E P(U ) such that f A (x) = if x A . 22

Here, f A is called approximate function of the soft set FA , and the value f A (x) is a set called x-element of the soft set for all x E. It is worth noting that the sets f A (x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. Thus, a soft set FA over U can be represented by the set of ordered pairs

FA = {( x, f A ( x)) : x E, f A ( x) P(U )} Note that the set of all soft sets over U will be denoted by S (U ) . Example Let U = {u1 , u2 , u3 , u4 , u5} be a universal set and E = {x1 , x2 , x3 , x4 } be a set of parameters. If A = {x2 , x3 , x4 } and f A ( x2 ) = {u 2 , u 4 } , f A ( x3 ) = , f A ( x4 ) = U , then the soft set FA is written by FA = {( x2 ,{u 2 , u 4 }), ( x4 ,U )}

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

X : U [0,1] Here, X called membership function of X , and the value X (u ) is called the grade of membership of u U . 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,

X = {( X (u)/u) : u U , X (u) [0,1]} Note that the set of all the fuzzy sets over U will be denoted by F (U ) .

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). 23

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 fssets 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, A E and A (x) be a fuzzy set over U for all x E. Then, an fs-set A over U is a set defined by a function A representing a mapping

A : E F (U ) such that A (x) = if x A . Here, A is called fuzzy approximate function of the fs-set A , and the value A (x) is a fuzzy set called x-element of the fs-set for all x E. Thus, an fs-set A over U can be represented by the set of ordered pairs

A = {( x, A ( x)) : x E, A ( x) F (U )} Note that from now on the sets of all fs -sets over U will be denoted by FS (U ) . Example Assume that U = {u1 , u2 , u3 , u4 , u5} is a universal set and E = {x1 , x2 , x3 , x4 } is a set of all parameters. If A = {x2 , x3 , x4 } , A ( x2 ) = {0.5/u2 ,0.9/u4 } , A ( x3 ) = and A ( x4 ) = U , then the fs set A is written by A = {( x2 ,{0.5/u2 ,0.9/u4 }), ( x4 ,U )}

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 X , Y , Z ,... , to avoid confusion and complexity of the symbols. Definition Let U be an initial universe, P(U ) 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 X : E [0,1] . Then, an fps-set FX over U is a set defined by a function f X representing a mapping 24

f X : E P(U ) such that f X (x) = if X (x) = 0 . Here, f X is called approximate function of the fps-set FX , and the value f X (x) is a set called x-element of the fps-set for all x E. Thus, an fps-set FX over U can be represented by the set of ordered pairs

FX = {( X ( x)/x, f X ( x)) : x E, f X ( x) P(U ), X ( x) [0,1]} Note that from now on the sets of all fps -sets over U will be denoted by FPS (U ) . Example Let U = {u1 , u2 , u3 , u4 , u5} be a universal set and E = {x1 , x2 , x3 , x4 } be a set of parameters. If X = {0.2/x2 ,0.5/x3 ,1/x4 } and f X ( x2 ) = {u 2 , u 4 } , f X ( x3 ) = , f X ( x4 ) = U , then the fps-set FX is written by FX = {(0.2/x2 ,{u 2 , u 4 }), (1/x4 ,U )}

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 X : E [0,1] and X (x) be a fuzzy set over U for all x E. Then, an fpfs-set X over U is a set defined by a function X (x) representing a mapping

X : E F (U ) such that X (x) = if X ( x) 0 . Here, X is called fuzzy approximate function of the fpfs-set X , and the value X (x) is a fuzzy set called x-element of the fpfs-set for all x E. Thus, an fpfs-set X over U can be represented by the set of ordered pairs

X = {( X ( x)/x, X ( x)) : x E, X ( x) F (U ), X ( x) [0,1]} It must be noted that the sets of all fpfs-sets over U will be denoted by FPFS (U ) .

25

Example: Assume that U = {u1 , u2 , u3 , u4 , u5} is a universal set and E = {x1 , x2 , x3 , x4 } is a set of parameters. If X = {0.2/x2 ,0.5/x3 ,1/x4 } and X ( x2 ) = {0.5/u1 ,0.3/u5} , X ( x3 ) = , and X ( x4 ) = U , then the fpfs-set X is written by

X = {(0.2/x2 ,{0.5/u1 ,0.3/u5}), (1/x4 ,U )} Definition 3.2: Let X FPFS (U ) . If X (x) = for all x E , then X is called an Xempty 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 X FPFS (U ) . If X (x) = 1 and X ( x) = U for all x X , then X is called X-universal fpfs-set, denoted by X~ . If X = E , then the X-universal fpfs-set is called universal fpfs-set, denoted by E~ . Example: Assume that U = {u1 , u2 , u3 , u4 , u5} is a universal set and E = {x1 , x2 , x3 , x4 } is a set of parameters. If X = {0.2/x2 ,0.5/x3 ,1/x4 } and X ( x2 ) = {0.5/u1 ,0.3/u5} , X ( x3 ) = , and X ( x4 ) = U , then the fpfs-set X is written by

X = {(0.2/x2 ,{0.5/u1 ,0.3/u5}), (1/x4 ,U )} If Y = {1/x1 ,0.7/x4 } and X ( x1 ) = , X ( x4 ) = then the fpfs-set Y is a Y-empty fpfsset, i.e., Y = Y . If Z = {1/x1 ,1/x2 } , Z ( x1 ) = U and Z ( x2 ) = U , then the fpfs-set Z is Z -universal fpfs-set, i.e., Z = Z~ . If X = , then the fpfs -set X is an empty fpfs-set, i.e., X = . If X = E and X ( xi ) = U , for all xi E ( i = 1,2,3,4 ), then the fpfs-set X is a universal fpfs-set, i.e., X = E~ . Definition 3.4: Let X , Y FPFS (U ) . Then, X is an fpfs-subset of Y , denoted by ~ , if ( x) ( x) and ( x) ( x) for all x E . X Y X Y X Y

26

Proposition 3.1: Let X , Y FPFS (U ) . Then, ~ ~ i. X E ~ ii. X

iii. iv. v.

X

~ X ~ X X ~ and ~ ~ X Y Y Z X Z

They can be proved easily by using the fuzzy approximate and membership functions of the fpfs-sets. Definition 3.5: Let X , Y FPFS (U ) . Then, X and Y are fpfs-equal, written as X = Y , if and only if X ( x) = Y ( x) and X ( x) = Y ( x) for all x E . Proposition 3.2: Let X , Y , Z FPFS (U ) . Then, i. ( X = Y and Y = Z ) X = Z ~ ) = ~ and ii. ( X Y X X Y Y The proofs are trivial. ~

Definition 3.6: Let X FPFS (U ) . Then, the complement of X , denoted by Xc , is defined by c ( x) = 1 X ( x) and c~ ( x) = Xc ( x) X

X

for all x E , where (x) is complement of the set X (x) , that is, Xc ( x) = U \ X ( x) for every x E . c X

Proposition 3.3: Let X FPFS (U ) . Then, ~ ~ i. (Xc ) c = X ~ ii. c = E~ By using the fuzzy approximate and membership functions of the fpfs-sets, the proof can be straightforward. Definition 3.7: Let X , Y FPFS (U ) . Then, union of X and Y , denoted by ~ , is defined by X Y X ~ Y ( x) = max{ X ( x), Y ( x)} and X ~ Y ( x) = X ( x) Y ( x) for all x E . Proposition 3.4: Let X , Y , Z FPFS (U ) . Then, ~ = i. X X X 27

iii. iv.

~ X X X ~ X = X ~ ~ = ~

v. vi.

~ = ~ X Y Y X ~ ~ ~ ( ~ ) (X Y ) Z = X Y Z

ii.

X

E

E

The proofs can be easily obtained from Definition 3.7. Definition 3.8: Let X , Y FPFS (U ) . Then, intersection of X and Y , denoted by ~ , is defined by X Y X ~ Y ( x) = min{ X ( x), Y ( x)} and X ~ Y ( x) = X ( x) Y ( x) for all x E . Proposition 3.5: Let X , Y , Z FPFS (U ) . Then, ~ = i. X X X ~ X X X ii. ~ = iii. X ~ ~ = iv. X

v. vi.

E

X

~ = ~ X Y Y X ~ ~ ~ ( ~ ) (X Y ) Z = X Y Z

The proofs can be easily obtained from Definition 3.8.

~ c~ ~ and Remark 3.1: Let X FPFS (U ) . If X E~ or X , then X X E c~ ~ X X . Proposition 3.6: Let X , Y FPFS (U ) . Then, De Morgan's laws are valid ~ ) c~ = c~ ~ c~ i. (X Y X Y ~ ) c~ = c~ ~ c~ ii. (X Y X Y Proof: For all x E , i.

~ Y )c (X

( x) = 1 X ~ Y ( x) = 1 max{ X ( x), Y ( x)} = min{1 X ( x),1 Y ( x)} = min{ =

~Y Xc

Xc

( x), c ( x)} Y

c ( x)

and

28

( X ~ Y )c~ ( x) = Xc ~ Y ( x)

= ( X ( x) Y ( x)) c = ( X ( x)) c ( Y ( x)) c = Xc ( x) Yc ( x) = =

~ Xc

( x)

~ ~ c~ Xc Y

~ Yc

( x)

( x).

Likewise, the proof of ii. can be made similarly. Proposition 3.7: Let X , Y , Z FPFS (U ) . Then, ~ ( ~ ) = ( ~ ) ~ ( ~ ) i. X Y Z X Y X Z ~ ( ~ ) = ( ~ ) ~ ( ~ ) ii. X Y Z X Y X Z Proof: For all x E , i. X ~ (Y ~ Z ) ( x) = max{ X ( x), Y ~ Z ( x)} = max{ X ( x), min{Y ( x), Z ( x)}} = min{max{ X ( x), Y ( x)}, max{ X ( x), Z ( x)}} = min{ X ~ Y ( x), X ~ Z ( x)} = ( X ~ Y )~ ( X ~ Z ) ( x) and X ~ (Y ~ Z ) ( x) = X ( x) Y ~ Z ( x) = X ( x) ( Y ( x) Z ( x)) = ( X ( x) Y ( x)) ( X ( x) Z ( x)) = X ~ Y ( x) X ~ Z ( x) = ( X ~ Y )~ ( X ~ Z ) ( x)

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 fpfsaggregation operator that produce an aggregate fuzzy set from an fpfs-set and its fuzzy parameter set. Definition: Let X FPFS (U ) . Then fpfs-aggregation operator, denoted by FPFS agg , is defined by FPFS agg : F ( E ) FPFS (U ) F (U ),

29

FPFS agg ( X , X ) = X* where X* = {

* X

(u )/u : u U }

which is a fuzzy set over U . The value X* is called aggregate fuzzy set of the X . Here, the membership degree

*X

(u ) of u is defined as follows

* (u ) = X

1 X ( x) X ( x) (u) | E | xE

where | E | is the cardinality of E . 5. fpfs -set decision making method The approximate functions of an fpfs-set are fuzzy. The FPFS agg 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

*X

(u ) .

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, U = {u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8} . The hiring committee consider a set of parameters, E = {x1 , x2 , x3 , x4 , x5} . The parameters xi ( i = 1,2,3,4,5 ) 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 X = {0.5/x2 ,0.9/x3 ,0.6/x4 } of E . Finally, the committee constructs the following fpfs-set over U . Step 1 Let the constructed fpfs-set, X , be as follows,

X {(0.5/x2 ,{0.3/u 2 ,0.4/u3 ,0.1/u 4 ,0.9/u5 , 0.7/u7 }) , (0.9/x3 , {0.4/u1 ,0.4/u2 , 0.9/u3 ,0.3/u4 }) , (0.6/x4 ,{0.2/u1 ,0.5/u 2 ,0.1/u5 , 0.7/u7 ,1/u8})}

30

Step 2 The aggregate fuzzy set can be found as, X* = {0.096/u1 ,0.162/u 2 ,0.202/u3 ,0.064/u 4 , 0.102/u5 ,0.154/u7 ,0.12/u8 }

Step 3: Finally, the largest membership grade can be chosen by max

* X

(u ) = 0.202

which means that the candidate u3 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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