Journal of Quality Measurement and Analysis Jurnal Pengukuran Kualiti dan Analisis JQMA 9(2) 20 7-8 INTUITIONISTIC FUZZY PARAMETERISED FUZZY SOFT SET (Set Lembut Kabur Berparameter Kabur Berintuisi) ENTISAR EL-YAGUBI & ABDUL RAZAK SALLEH ABSTRACT In this paper the definition of intuitionistic fuzzy parameterised fuzzy soft set (ifpfs-sets) is introduced with their properties. Two operations on ifpfs-set namely union and intersection are introduced. Also some examples for these operations are given. The definition of ifpfsaggregation operator which forms ifpfs-decision making method is introduced. Keywords: soft set; fuzzy soft set; fuzzy parameterised fuzzy soft set; ifpfs-sets; ifpfsaggregation operator; ifpfs-decision making method ABSTRAK Dalam makalah ini diperkenalkan takrif set lembut kabur berparameter kabur berintuisi (setlkbki) beserta sifat-sifatnya. Turut diperkenalkan dua operasi ke atas set-lkbki iaitu kesatuan dan persilangan. Hasil-hasil yang berkaitan dan contoh untuk operasi ini diberikan. Diperkenalkan juga takrif pengoperasi pengagregatan-lkbki yang membentuk kaedah pembuatan keputusan-lkbki. Kata kunci: set lembut; set lembut kabur; set lembut kabur berparameter kabur; set-lkbki; pengoperasi pengagregatan-lkbki; kaedah pembuatan keputusan-lkbki. Introduction In most of the real life problems in social sciences engineering medical sciences economics the data involved are imprecise in nature. The solutions of such problems involve the use of mathematical principles based on uncertainty and imprecision. A number of theories have been proposed for dealing with uncertainties in an efficient way such as fuzzy set (Zadeh 965) intuitionistic fuzzy set (Atanassov 986) vague set (Gau & Buchrer 99) and theory of interval mathematics (Atanassov 994). In 999 Molodtsov initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. He has shown several applications of this theory in solving many practical problems in economics engineering social science and medical science. Maji et al. (200) have further studied the theory of soft sets and used this theory to solve some decision making problems. They also introduced the concept of fuzzy soft set (Maji et al. 200a) a more general concepts which is a combination of fuzzy set and soft set. Moreover they introduced the idea of an intuitionistic fuzzy soft set (Maji et al. 200b) which is a combination of intuitionistic fuzzy set and soft set. Cağman et al. (200) introduced and studied the fuzzy parameterised fuzzy soft sets (fpfs-sets) and their operations. Alkhazaleh et al. (20) introduced the notion of fuzzy parameterised interval-valued fuzzy soft set and Bashir and Salleh (202) introduced the notion of fuzzy parameterised soft expert set. In this paper we use the notion of fuzzy parameterised fuzzy soft sets of Cağman et al. to define intuitionistic fuzzy parameterised fuzzy soft sets (ifpfs-sets). We also define their operations namely union and intersection and discuss their properties.
Entisar El-Yagubi & Abdul Razak Salleh 2. Preliminary In this section we recall some definitions which are required in this paper. Definition 2.. (Molodtsov 999). A pair (F E) is called a soft set over U where F is a mapping given by F : E P U ( ). In other words a soft set over U is a parameterised family of subsets of the universe U. Definition 2.2. (Maji et al. 200a). Let U be an initial universal set and let E be a set of parameters. Let I U denote the power set of all fuzzy subsets of U. Let A E. A pair(f E) is called a fuzzy soft set over U where F is a mapping given by F : A I U. Definition 2.. (Cağman et al. 200). Let U be an initial universe E the set of all parameters and a fuzzy set over E with membership function : µ : E 0. and let γ be a fuzzy set over U for all x E. Then a fuzzy parameterised fuzzy soft set (fpfs-set) over U is a set defined by a function γ representing a mapping γ : E F U ( ) such that γ ( x) if µ ( x) 0. ( x) is a Here γ is called a fuzzy approximate function of the fpfs-set and the value γ set called x-element of the fpfs-set for all x E. Thus an fpfs-set over U can be represented by the set of ordered pairs {( µ ( x) / xγ ( x) ): x E γ ( x) F ( U )µ ( x) 0 }.. Intuitionistic fuzzy parameterised fuzzy soft sets In this section we introduce the definition of an intuitionistic fuzzy parameterised fuzzy soft set (ifpfs-set). Also we introduce some operations on the intuitionitstic fuzzy parameterised fuzzy soft sets namely union and intersection and discuss their properties. Definition.. Let U be an initial universe E the set of all parameters and an intuitionistic fuzzy set over E with membership function µ : E 0 non-membership function ( ) be a fuzzy set over U ν : E 0 and let γ x for all x E. Then an intuitionistic fuzzy parameterised fuzzy soft set (ifpfs-set) over U is a set defined by a function γ : E F ( U ) where F(U) is a set of fuzzy sets over U such that γ ( x) if µ ( x) 0 and ν ( x) 0. Here γ is called a fuzzy approximate function of the ifpfs-set and the value γ ( x) is a fuzzy set called an x-element of the ifpfs-set for all x E. Thus an ifpfs-set over U can be represented by the set of ordered pairs (µ (x) ν (x)) γ (x) : x E γ (x) F(U ) µ (x) ν (x) [0]. It must be noted that the set of all ifpfs-sets over U will be denoted by IFPFS(U ). 74
Intuitionistic fuzzy parameterised fuzzy soft set Example.. Let U { u u 2 u } be a universal set and E { x x 2 x } be a set of parameters. x Suppose (0.2 0.6) x 2 (0.5 0.) x and ( 0) γ ( x ) u 0.5 u 2 0 u 0. ( ) u γ x 2 Then the ifpfs-set is given as follows: 0. u 2 0. u 0.6 ( ) u γ x 0. u. 0.7 (0.2 0.6) u 0.5 u 2 0 u. 2 (0.5 0.) u 0. u 2 0. u.6 ( 0) u 0. u.7. Definition.2. Let IFPFS(U ). If γ (x) for all x E then is called an -empty ifpfs-set denoted by Γ φ. If then the -empty ifpfs-set (Γ φ ) is called an empty ifpfs-set denoted by Γ φ. Definition.. Let IFPFS(U ). If γ (x) U for all x then is called an -universal ifpfs-set denoted by Γ!. If E then the -universal ifpfs-set (Γ! ) is called a universal ifpfs-set denoted by Γ!E. Example.2. Assume that U u u 2 u all parameters. x Let ( 0) x 2 (0.2 0.6) x and γ (0.7 0.2) Then the -empty ifpfs-set { } is a universal set and E { x x 2 x } is a set of ( x ) γ ( x 2 ) γ x ( ) u 0 u. 0 Γ φ ( 0) u 0 u 2 (0.2 0.6) u 0 u (0.7 0.2) u 0 u. x Suppose (0 0) x 2 (0 0) x and γ (0 0) Then the empty ifpfs-set is ( x ) γ ( x 2 ) γ x ( ) u 0 u. 0 Γ φ (0 0) u 0 u 2 (0 0) u 0 u (0 0) u 0 u. x Let ( 0) x 2 (0.2 0.6) x and γ (0.7 0.2) ( x ) γ ( x 2 ) γ x ( ) u u 2 u. 75
Entisar El-Yagubi & Abdul Razak Salleh Then an -universal ifpfs-set Γ! ( 0) u u 2 u 2 (0.2 0.6) u u 2 u (0.7 0.2) u u 2 u. x Suppose ( 0) x 2 ( 0) x ( 0) and γ ( x ) γ ( x 2 ) γ x ( ) u u 2 u. Then the universal ifpfs-set is Γ!E ( 0) u u 2 u 2 ( 0) u u 2 u ( 0) u u 2 u. Definition.4. Let Γ Y IFPFS(U ). is called an intuitionistic fuzzy parameterised fuzzy soft subset of Γ Y denoted by Γ Y if: i. is an intuitionistic fuzzy subset of Y. ii. γ x x ( ) is a fuzzy subset of γ x Y ( ). Proposition.. Let Γ Y IFPFS(U ). Then i.! Γ!E ii. Γ φ! iii. Γ φ! iv. Γ φ! v.! vi.! Γ Y and Γ! Γ Γ Y Z! Γ Z Proof. The proof is straightforward. Definition.5. Let Γ Y IFPFS(U ). The union of and Γ Y denoted by is an ifpfs-set Γ Z given by Z S norm Y γ Z (x) γ (x) γ s norm Y (x) x E. Here S is an S-norm (Fathi 2007) and s is an s-norm (Lowen 996; Wang 997). { } be a universal set and let E { x x 2 x } be a set of Example.. Let U u u 2 u parameters. Consider the ifpfs-set (0. 0.4) u 0. u 2 0. u.5 2 (0.7 0.) u 0. u 2 0 u.4 (0.2 0.6) u 0.6 u 2 0.2 u 76
Intuitionistic fuzzy parameterised fuzzy soft set and Γ Y (0. 0.4) u 0.4 u 2 0 u. 2 (0. 0.) u 0.5 u 2 0. u. (0. 0.5) u 0.2 u 2 0.6 u. By applying the Atanassov union (basic intuitionistic fuzzy union) for S-norm and the basic fuzzy union max for s-norm we have Γ Atan Y (0. 0.4) u 0.4 u 2 0. u 0.5 max 2 (0.7 0.) u 0.5 u 2 0. u.4 (0.2 0.5) u 0.6 u 2 0.6 u. Proposition.2. Let Γ Y Γ Z IFPFS(U). Then i.! ii. Γ φ! iii.! Γ φ iv.! Γ!E Γ!E v. Γ Y! vi. ( )! Γ Z! (Γ Y! Γ Z ) Proof. The proof can be easily obtained from Definition.5. Definition.6. Let Γ Y IFPFS(U ). The intersection of and Γ Y denoted by! Γ Y is an ifpfs-set Γ Z where Z Y T norm γ Z (x) γ (x) t norm γ Y (x) x E Here T is a T-norm (Fathi 2007) and t is a t-norm (Lowen 996; Wang 997). Example.4. Let U u u 2 u parameters. Consider the ifpfs-set { } be a universal set and let E { x x 2 x } be a set of (0.7 0.) u 0. u 2 0 u.4 2 (0. 0.2) u 0.5 u 2 0.2 u (0.4 0.) u 0.5 u 2 0. u. 77
Entisar El-Yagubi & Abdul Razak Salleh Γ Y (0. 0.6) u 0. u.4 2 (0. 0.2) u 0.2 u 2 0.5 u. (0.8 0) u 0.6 u 2 0.2 u.. By applying the Atanassov intersection (basic intuitionistic fuzzy intersection) for T-norm and the basic fuzzy intersection min for t-norm we have Γ Atan Y (0. 0.6) u 0 u 0.4 min 2 (0. 0.2) u 0.2 u 2 0.2 u (0.4 0.) u 0.5 u 2 0. u.. Proposition.. Let Γ Y Γ Z IFPFS(U). Then the following results hold: i.! ii. Γ φ! iii.! Γ φ iv.! Γ! E Γ!E v.! Γ Y Γ Y! vi. (! Γ Y )! Γ Z! (Γ Y! Γ Z ) Proof. The proof can be easily obtained from Definition.6. Proposition.4. Let Γ Y Γ Z IFPFS(U). Then the following results hold: i! (Γ Y! Γ Z ) ( )! (! Γ Z ) ii! (Γ Y! Γ Z ) (! Γ Y )! (! Γ Z ) Proof. (i) From the fact that the union and the intersection of fuzzy set and intuitionistic fuzzy set satisfy distributive law we have (Y Z) ( Y ) ( Z) Atan Atan Atan Atan Atan γ! (Y! Z ) (x) γ (! Y )! (! Z ) (x). Therefore! (Γ Y! Γ Z ) (Y Z ) Γ ( Y ) ( Z ) ( )! (! Γ Z ). Likewise the proof of (ii) can be made similarly. 4. ifpfs-aggregation Operator In this section we define an aggregate fuzzy set of an ifpfs-set. We also define ifpfs-aggregation operator that produces an aggregate fuzzy set from an ifpfs-set and its fuzzy parameter set. Definition 4.. Let Γ Χ IFPFSS(U ). Then an ifpfs-aggregation operator denoted by IFPFS agg is defined by 78
Intuitionistic fuzzy parameterised fuzzy soft set IFPFS agg : F(E) IFPFSS(U ) F(U ) IFPFS agg ( ) where Γ u µ Γ (u) : u U which is a fuzzy set over U. The value is called an aggregate fuzzy set of the.here the membership degree µ (u) of u is defined as follows: µ Γ (u) E x E where E is the cardinality of E. 5. ifpfs-decision Making Method ( µ (x) ν (x)) µ (u) γ ( x) The approximate functions of an ifpfs-set are fuzzy. The IFPFS agg on the fuzzy sets is an operation by which several approximate functions of an ifpfs-set are combined to produce a single fuzzy set that is the aggregate fuzzy set of the ifpfs-set. Then it may be necessary to choose the best single crisp alternative from this set. Therefore we can construct an ifpfs-decision making method by the following algorithm. Step : Construct an ifpfs-set Γ Χ over U. Step 2: Find the aggregate fuzzy set Γ Χ of Γ Χ. Step : Find the largest membership grade max µ ΓΧ (u). Example 5.. Assume that a company wants to fill a position. There are seven candidates who form the set of alternatives U {u u 2 u u 4 u 5 u 6 u 7 }. The hiring committee considers a set of parameters E {x x 2 x x 4 x 5 }. The parameters x i (i 2 4 5) stand for experience computer knowledge young age good speaking and friendly respectively. After a serious discussion each candidate is evaluated from the point of view of the goals and the constraint according to a chosen subset x (0.7 0.) x 2 (0. 0) x (0. 0.) x 4 (0.6 0.) x 5 of E. (0.2 0.4) Finally the committee constructs the following ifpfs-set over U. Step : Let the constructed ifpfs-set be as follows: 79
Entisar El-Yagubi & Abdul Razak Salleh (0.7 0.) u 0. u 0.4 u 4 0. u 5 0.9 u 6 0 u 7 0.7 2 (0. 0) u 0. u 2 0.2 u 0.5 u 4 0 u 5 0. u 6 0.6 u 7 0 (0. 0.) u 0.4 u 2 0.4 u 0.9 u 4 0. u 5 0.6 u 6 0 u 7 0. 4 (0.6 0.) u 0.2 u 2 0.5 u 0. u 4 0.7 u 5 0 u 6 u 7 0 5 (0.2 0.4) u u 0.6 u 4 0.2 u 5 0. u 6 0. u 7 0.4. Step 2: The aggregate fuzzy set can be found as u Γ 0.042 u 2 0.074 u 0. u 4 0.078 u 5 0.8 u 6 0.2 u 7. 0.08 Step : Finally the largest membership grade can be chosen by max µ (u 5 ) 0.8. This means that the candidate u 5 has the largest membership grade hence he is selected for the job. 6. Conclusion In this paper we have introduced the concept of intuitionistic fuzzy parameterised fuzzy soft set and their properties. Moreover we have defined some operations on ifpfs-set which are union and intersection with some propositions on ifpfs-set. The definition of ifpfs-aggregation operator is introduced with application of this operator in decision making problems. Acknowledgement The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant AP-20-009. References Alkhazaleh S. Salleh A.R. & Hassan N. 20. Fuzzy parameterized interval-valued fuzzy soft set. Applied Mathematical Sciences 5(67): 5-46. Atanassov K.T. 986. Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20: 87-96. Atanassov K.T. 994. Operators over interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems 64: 59-74. Bashir M. & Salleh A.R. 202. Fuzzy parameterized soft expert set. Abstract and Applied Analysis. Article ID 2586 6 pages. Cağman N. Citak F. & Enginoglu S. 200. Fuzzy parameterised fuzzy soft set theory and its applications. Turkish Journal of Fuzzy Systems : 2-5. Fathi M. 2007. On intutionistic fuzzy sets. MSc Research Project Faculty of Science and Technology Universiti Kebangsaan Malaysia. Gau W.L. & Buchrer D.J.99. Vague sets. IEEE Trans. Systems Man Cybernet 2(2): 60-64. Lowen R. 996. Fuzzy Set Theory. Dordrecht: Kluwer Academic Publishers. 80
Intuitionistic fuzzy parameterised fuzzy soft set Maji P.K. Biswas R. & Roy A.R. 200a. Fuzzy soft set. The Journal of Fuzzy Mathematics 9(): 589-602. Maji P.K. Biswas R. & Roy A.R. 200b. Intuitionistic fuzzy soft set. The Journal of Fuzzy Mathematics 9: 677-692. Maji P.K. Biswas R. & Roy A.R. 200. Soft set theory. Computers and Mathematics with Applications 45: 555-562. Molodtsov D. 999. Soft set theory - First result. Computers and Mathematics with Applications 7: 9-. Wang L.. 997. A Course in Fuzzy Systems and Control. Upper Saddle River NJ: Prentice-Hall. Zadeh L.A. 965. Fuzzy sets. Information and Control 8: 8-5. School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 4600 UKM Bangi Selangor DE MALAYSIA E-mail: entisar_e980@yahoo.com aras@ukm.my Corresponding author 8