Parameterized Soft Complex Fuzzy Sets

Transcription

1 Journal of Progressive Researc in Matematics(JPRM) IN: CITECH Volume Issue REERCH ORGNITION Publised online: June 7 05 Journal of Progressive Researc in Matematics Parameterized oft Comple Fuzzy ets Dr. P. Tirunavukarasu R. ures sst. Prof -P.G & Researc Department of Matematics Periyar E.V.R College (utonomous)tirucirappalli 60 0 Tamilnadu India. sst. Prof Department of Matematics Kings College of Engineering Punalkulam Pudukkottai(Dt) 6 0 Tamilnadu India. bstract In tis paper we presents a new concept of parameterized soft comple fuzzy set (pscf-set) wic is generalized from te innovative concept of a comple fuzzy set by adding pase term to te definition of fuzzy set. Various operators wit eamples of pscf-set provided in tis paper. We ten define a pscf-set aggregation operator Keywords Fuzzy set; Fuzzy soft set;fuzzy parameterized fuzzy soft set; Comple fuzzy set (CF) ;oft comple fuzzy set; Parameterized soft comple fuzzy set (pscf-set).. INTRODUCTION Many fields deal daily wit te uncertainity periodicity data tat may not be successfully modeled by te classical matematics. Tere are some matematical tools for dealing wit uncertainties periodicities; tree of tem are fuzzy set teory developed by Zade (965) soft set teory introduced by Molodtsov (999) comple fuzzy set developed by Ramot et al(00) tat are related to tis work. Fuzzy sets [9] provide a robust matematical framework for dealing wit real-world imprecision nonstatistical uncertainty. Qualitative linguistic variables allow one to represent a range of numerical values as a single descriptive term tat is described by a fuzzy set. Given tat te present day comple networks are dynamic tat tere is great uncertainty associated wit te input traffic oter environmental parameters tat tey are subject to unepected overloads failures perturbations tat tey defy accurate analytical modeling fuzzy logic appears to be a promising approac to address key aspects of networks. Te ability to model networks in te continuum matematics of fuzzy sets rater tan wit traditional discrete values coupled wit etensive simulation offers a reasonable compromise between rigorous analytical modeling purely qualitative simulation. oft set teory is a generalization of fuzzy set teory wic was proposed by Molodtsov [6] in 999 to deal wit uncertainty in a non-parametric manner. One of te most important steps for te teory of oft ets was to define mappings on soft sets tis was acieved in 009 by matematician tar Karal toug te results were publised in 0. oft sets ave also been applied to te problem of medical diagnosis for use in medical epert systems. Fuzzy soft sets ave also been introduced in []. Comple fuzzy set (CF) []-[5] is a new development in te teory of fuzzy systems. Te concept of CF is an etension of fuzzy set by wic te membersip for eac element of a comple fuzzy set is etended to comple-valued state. In tis paper we define parameterized soft comple fuzzy sets (pscf-sets) in wic te approimate functions are defined from comple fuzzy parameters set to te comple fuzzy subsets of universal set. We also defined teir operations soft aggregation operator. We also present eamples wic sow tat te metods can be successfully applied to many problems tat contain uncertainties periodicities.. PRELIMINRIE Definition:. Let U be a universe. fuzzy set over U is a set defined by a function representing a mapping Volume Issue available at 0

2 : U [0 ] Journal of Progressive Researc in Matematics(JPRM) IN: Here called membersip function of te value (u) is called te grade of membersip of u U value represents te degree of u belonging to te fuzzy set. Tus a fuzzy set over U can be represented as follow ( u) / u : u U ( u) [0] Note tat te set of all te fuzzy sets over U will be denoted by F(. [9] Definition:. Let U be an initial universe P( be te power set of U E be te set of all parameters Ten a soft set [6]F over U is a set defined by a function f representing a mapping f : E P( suc tat f ()= if.. Te E. Here f is called approimate function of te soft set F te value f () is a set called -element of te soft set for all E. It is wort noting tat te sets f () may be arbitrary. ome of tem may be empty some may ave nonempty intersection. Tus a soft set F over U can be represented by te set of ordered pairs F f ( ) : E f ( ) P( Note tat te set of all soft sets over U will be denoted by (. Definition:. Let U be an initial universe E be te set of all parameters E () for all E. Ten an Fuzzy oft (fs-set) over U is a set defined by a function :E F( suc tat () =Ф; if. be a fuzzy set over U representing a mapping Here is called fuzzy approimate function of te fs-set te value () is a fuzzy set called -element of te fs-set for all E. Tus an fs-set over U can be represented by te set of ordered pairs ( ) : E ( ) F( Note tat te set of all te fuzzy sets over U will be denoted by F( from now on te sets of all fs-sets over U will be denoted by F (. [][] Definition:. Let U be an initial universe E be te set of all parameters be a fuzzy set over E wit membersip function ( ) : E [0 ] () Ten an fuzzy be a fuzzy set over U for all E. parameterized fuzzy soft set (fpfs-set) Γ over U is a set defined by a function () representing a mapping : E F( suc tat () =Ф; if () =0. is called fuzzy approimate function of te fpfs-set Γ te value () E Here is a fuzzy set called - element of te fpfs-set[] for all. Tus an fpfs-set Γ over U can be represented by te set of ordered pairs ( ) / ( ) : E ( ) F( ( ) 0 It must be noted tat te sets of all fpfs-sets over U will be denoted by fpfs(. Definition:.5 Ramot et al. [] recently proposed an important etension of tese ideas te Comple Fuzzy ets[] is caracterized by a membersip function () is a comple-valued function tat assigns a grade of membersip of te form r ( ). e j ( ) ; ( j= ) to any element in te universe of discourse. Te Value of () is defined by te two variables r () () bot real valued wit r (). Parameterized oft Comple Fuzzy ets [0]. Volume Issue available at 0

3 Journal of Progressive Researc in Matematics(JPRM) IN: Definition:. [8] Let U be an initial universe E be te set of all parameters E () fuzzy set over U for all E. function representing a mapping be a comple Ten an oft comple fuzzy set over U is a set defined by a : E C( suc tat () =Ф;if. Here is called comple fuzzy approimate function of te soft comple fuzzy set comple fuzzy set called -element of te soft comple fuzzy set for all E. Tus an soft comple fuzzy set over U can be represented by te set of ordered pairs ( ) : E ( ) C( Note tat te set of all te comple fuzzy sets over U will be denoted by C(. Eample:. [7] Let U= { (India) (ustralia) (UK) (} be an initial set te value () consider E= {e (Inflation rate) e (population growt) e (Unemployment rate) e (sare market inde)} be an country s growt parameters set E={e e } j0.5 j0.6 j0.8 j e.0e ( e ) = 0.6e ( e ) = j e j0.9 j0.95 ten soft comple fuzzy set is written by 0.e e j0.5 j0.6 j0.8.0e 0.75 j0.75 j0.95 e 0.6e j e j0.9 j Definition:. Let U be an initial universe E be te set of all parameters be a fuzzy set over E wit membersip function ( ) : E [0 ] () is a comple fuzzy set over U for all E. Ten a Parameterized oft comple fuzzy set over U is a set defined by a function representing a mapping : E C( suc tat () =Ф if () =0 Here is called comple fuzzy approimate function of te Parameterized soft comple fuzzy set te value () is a comple fuzzy set called -element of te Parameterized soft comple fuzzy set for all E. Tus an Parameterized soft comple fuzzy set over U can be represented by te set of ordered pairs ( ) ( ) : E ( ) C( ( ) [0] Note tat te set of all te comple fuzzy sets over U will be denoted by C( from now on te sets of all Parameterized soft comple fuzzy sets over U will be denoted by pscf (. Eample:. j0.95 is a Let U= { } be an initial set consider E= {e e e e } is set of parameters If = e e 0.e 0.5 j0. ( e ) ( e 0.e ) 0. j0.7 Volume Issue available at 05

4 Ten Parameterized soft comple fuzzy set is written by = e e 0.5 j0. Definition:.5 Let (U ) pscf-set denoted by e e Journal of Progressive Researc in Matematics(JPRM) IN: j0.7 for all E ten is called an -empty pscf. If ( ) 0 ( ) If =ø ten te -empty pscf-set is called empty pscf-set denoted by Definition:.6 Let (U ) pscf-set denoted by pscf. If () = ( ) U for all ten is called an -Universal ~. If =E ten te -universal pscf-set is called universal pscf-set denoted by E ~ Eample:.7 Let U= { } be an initial set consider E= {e e e e } is set of parameters If = e e 0.e 0.5 j0. ( e ) ( e Ten Parameterized soft comple fuzzy set is written by = If = If = e e e e e e 0.5 j e e ( e ) ø ( e ) ( e ) U ( e ) 0. 0.e ) j j0.7 ø ten te pscf-set is a -empty pscf-set U ten te pscf-set is a -Universal pscf-set. Operations on Parameterized oft Comple Fuzzy ets Definition:.Let T PCF (. Ten ( ) ( ) ( ) ( ) for all E T Definition:.Let PCF (. Ten union of () ma( ( ) ( )) () are defined in [] U is defined as follows. T a) (um) U = b) (Ma) U c) (Min) U T te pscf-equal written as denoted by j ( ) U = ma ) = if only if r ( ) r ( ) e for all E ( = min ) ( d) (Winner takes all) U = if... r r if.. r r T is defined by ; r ( ) r ( ) Volume Issue available at 06

5 Journal of Progressive Researc in Matematics(JPRM) IN: Definition:. Let PCF (. Ten Intersection of denoted by () min( ( ) ( )) () r ( ) r ( ) by are defined in [] j ( ) U is defined as in Definition. r ( ) r ( ) e for all E ; is defined Definition:. Let PCF(U ). Ten pscf-aggregation operator denoted by PCF agg is defined by PCF agg : F( E) PCF ( C( PCF agg ( ) were ( u) / u u U wic is a comple fuzzy set over U. Te value = : is called aggregate comple fuzzy set of te follows (u) = ( ). ( ) ( ) u E CONCLUION: E.Here te membersip degree (u) were E is te cardinality of E. of u is defined as soft set is a mapping from parameter to te crisp subset of universe. In soft comple fuzzy sets te soft set teory is etended to a comple fuzzy set. Tis paper presented a new concept of parameterized soft comple fuzzy set wic is generalized from te innovative concept of a soft comple fuzzy set. To develop te teory in tis work we define pscf-set wit teir operations. Finally we define pscf-set ggregation operator for develop decision making problems. REFERENCE: [] Cagman N. Citak F. Enginoglu. Fuzzy Parameterized Fuzzy oft ets- Teory Its pplications Turkis Journal of Fuzzy ystems Vol. No. (00) pp. 5. [] Cagman N. Citak F. Enginoglu. FP-oft set Teory its pplications nnals of Fuzzy Matematics Informatics Vol. No. (0) pp [] Cagman N. Enginoglu. Citak F. Fuzzy oft et Teory Its pplications Iranian Journal of Fuzzy ystems Vol.8 No. (0) pp [] Ramot D. Milo R. Friedman M.. Kel Comple fuzzy sets IEEE Trans. Fuzzy yst. vol. 0 No. pp pr.00. [5] Ramot D. Friedman M. Langolz G. Kel. Comple fuzzy logic IEEE Trans Fuzzy yst. vol. no. pp ug. 00. [6] Molodtsov D.. oft set teory First result Computers Mat. ppl. 7 (/5)9-999 [7] Tirunavukarasu P. ures R. Tamilmani P. pplications of Comple Fuzzy ets JP Journal of pplied Matematics Vol.6 Issues & pp [8] Tirunavukarasu P. ures R. oft Comple Fuzzy ets IMRF Journal Vol: Issue: pp [9] Zade L.. Fuzzy ets Inform. nd Control 8 (965) pp utors iograpy Dr. P. Tirunavukarasu received te received te.c. M.c. M.Pil degree in Matematics from te aratidasan University Tamilnadu out India.. He completed is P.D degree from aratidasan University/Regional Engineering College. He as publised many papers in International National level conferences. He also publised many books. He is te Life member of ITE TeMatematicsTeacer/JM/ooks/official Journal of te ssociation of Matematics Teacers of India. Volume Issue available at 07

6 Journal of Progressive Researc in Matematics(JPRM) IN: Mr. R. ures received te.c. M.c. M.Pil degree in Matematics from te aratidasan University Tamilnadu out India in respectively. His ongoing researc focusing on te subject of comple fuzzy sets e as publised papers in International National level conferences. Volume Issue available at 08