Theory of Fuzzy Soft Matrix and its Multi Criteria in Decision Making Based on Three Basic t-norm Operators

Transcription

1 Theory of Fuzzy Soft Matrix ad its Multi Criteria i Decisio Makig Based o Three Basic t-norm Operators Md. Jalilul Islam Modal 1, Dr. Tapa Kumar Roy 2 Research Scholar, Dept. of Mathematics, BESUS, Howrah , W.B., Idia 1 Professor, Dept. of Mathematics, BESUS, Howrah , W.B. Idia 2 ABSTRACT : The purpose of this paper is to put forward the otio of fuzzy soft matrix theory ad some basic results. I this paper, we defie fuzzy soft matrices ad some ew defiitios based o t-orms with appropriate examples.lastly we have give a applicatio i decisio makig based o differet operators of t-orms. Key words: Soft sets, fuzzy soft matrices, operators of t-orms. I. INTRODUCTION Most of our traditioal tools for formal modelig, reasoig, ad computig are crisp, determiistic, ad precise i character. However, i real life, there are may complicated problems i egieerig, ecoomics, eviromet, social scieces medical scieces etc. that ivolve data which are ot all always crisp, precise ad determiistic i character because of various ucertaities typical problems. Such ucertaities are beig dealig with the help of the theories, like theory of probability, theory of fuzzy sets, theory of ituitioistic fuzzy sets, theory of iterval mathematics ad theory of rough sets etc. Molodtsov [1] also described the cocept of Soft Set Theory havig parameterizatio tools for dealig with ucertaities. Researchers o soft set theory have received much attetio i recet years. Maji ad Roy [3,4] first itroduced soft set ito decisio makig problems. Maji et al.[2] itroduced the cocept of fuzzy soft sets by combiig soft sets ad fuzzy sets. Cagma ad Egioglu [5] defied soft matrices which were a matrix represetatio of the soft sets ad costructed a soft max-mi decisio makig method. Cagma ad Egioglu [6] defied fuzzy soft matrices ad costructed a decisio makig problem. Borah et al.[7] exteded fuzzy soft matrix theory ad its applicatio. Maji ad Roy [8] preseted a ovel method of object from a imprecise multi-observer data to deal with decisio makig based o fuzzy soft sets. Majumdar ad Samata[9] geeralized the cocept of fuzzy soft sets. I this paper, we have itroduced some operators of fuzzy soft matrix o the basis of t-orms. We have also discussed their properties. Fially we have give a applicatio i decisio makig problem o the basis of t-orms operators. II. DEFINITION AND PRELIMINARIES 2.1 Soft set [1] Let U be a iitial uiverse, P(U) be the power set of U, E be the set of all parameters ad A E. A soft set (f A,E) o the uiverse U is defied by the set of order pairs (f A,E) = {(e, f A (e)) : e E, f A (e) P(U) } where f A : E P(U) such that f A (e) = φ if e A. Here f A is called a approximate fuctio of the soft set(f A, E). The set f A (e) is called e-approximate value set or e-approximate set which cosists of related objects of the parameter e E. Copyright to IJIRSET

2 Example 1 let U = { u 1, u 2, u 3, u 4 } be a set of four shirts ad E = { white( e 1 ), red (e 2 ),blue (e 3 ) } be a set of parameters. If A ={e 1, e 2 } E. Let f A (e 1 )= { u 1, u 2, u 3, u 4 } ad f A (e 2 )= { u 1, u 2, u 3 }, the we write the soft set (f A,E)= {(e 1,{ u 1, u 2, u 3, u 4 }), (e 2,{ u 1, u 2, u 3 })} over U which describe the colour of the shirts which Mr. X is goig to buy. 2.2 Fuzzy set [2 Let U be a iitial uiverse, E be the set of all parameters ad A E. A pair ( F, A ) is called a fuzzy set over U where F : A P(U) is a mappig from A ito P(U), where P(U) deotes the collectio of all subsets of U. Example 2. Cosider the example 1, here we ca ot express with oly two real umbers 0 ad 1, we ca characterized it by a membership fuctio istead of crisp umber 0 ad 1, which associate with each elemet a real umber i the iterval [0,1].The (f A,E) = { f A (e 1 ) = { ( u 1,.7),(u 2,.5),(u 3,.4),(u 4,.2) }, f A (e 2 ) = { (u 1,.5), (u 2,.1), (u 3,.5)} } is the fuzzy soft set represetig the colour of the shirts which Mr. X is goig to buy. 2.3 Fuzzy Soft Matrices (FSM) [5] Let (f A, E) be fuzzy soft set over U. The a subset of U x E is uiquely defied by R A = { ( u, e) : e A, u f A (e) }, which is called relatio form of (f A, E). The characteristic fuctio of R A is writte by µ RA : U x E [ 0, 1], where µ RA (u, e ) [ 0,1] is the membership value of u U for each e U. If µ 11 µ 12 µ 1 µ 21 µ 22 µ 2 µ ij = µ RA (u i,e j ), we ca defie a matrix [µ ij ] mx =, which is called a m x soft matrix µ m1 µ m2 µ m of the soft set (f A,E) over U. Therefore we ca say that a fuzzy soft set (f A,E) is uiquely characterized by the matrix [µ ij ] mx ad both cocepts are iterchageable. Example 3. Assume that U = { u 1, u 2, u 3, u 4, u 5, u 6 } is a uiversal set ad E = { e 1, e 2, e 3, e 4 } is a set of all parameters. If A E = { e 1, e 2, e 3 } ad f A (e 1 )= { (u 1,.3), (u 2,.4), (u 3,.6), (u 4,.1),( u 5,.6), (u 6,.5 }, f A (e 2 )= { (u 1,.2), (u 2,.5), (u 3,.7), (u 4,.3),( u 5,.7), (u 6,.1)}, f A (e 3 )= { (u 1,.5), (u 2,.2), (u 3,.5), (u 4,.6),( u 5,.7), (u 6,.3) } The the fuzzy soft set (f A,E) is a parameterized family { f A (e 1 ), f A (e 2 ), f A (e 3 )} of all fuzzy sets over U. Hece the fuzzy soft matrix [µ ij ] ca be writte as [µ ij ] = Zero Fuzzy Soft Matrix [6] Let [a ij ] FSM m X. The [a ij ] is called a Zero Fuzzy Soft Matrix deoted by [0], if a ij = 0 for all i ad j Uiversal Fuzzy Soft Matrix [6] Let [a ij ] FSM m X. The [a ij ] is called a Uiversal Fuzzy Soft Matrix deoted by [1], if a ij = 1 for all i ad j. Copyright to IJIRSET

3 2.6. Fuzzy Soft Sub Matrix [6] Let [a ij ], [b ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Sub Matrix of [b ij ] deoted by [a ij ] [b ij ] if a ij b ij for all i ad j Uio of Fuzzy Soft Matrices [6] Let [a ij ], [b ij ] FSM m X. The Uio of [a ij ] ad [b ij ], deoted by [a ij ] [b ij ] is defied as [a ij ] [b ij =max {a ij, b ij } for all i ad j Itersectio of Fuzzy Soft Matrices [6] Let [a ij ], [b ij ] FSM m X. The Itersectio of [a ij ] ad [b ij ], deoted by [a ij ] [b ij ] is defied as [a ij ] [b ij ] =mi {a ij, b ij } for all i ad j Complimet Fuzzy Soft Matrix [6] Let [a ij ] FSM m X. The Complemet of Fuzzy Soft Matrix [a ij ], deoted by [a ij ] 0 is defied as [a ij ] 0 = 1 a ij for all i ad j Fuzzy Soft Equal Matrices [6] Let [a ij ], [b ij ] FSM m X. The [a ij ] ad [b ij ] are said to be Fuzzy Soft Equal Matrices, deoted by [a ij ] =[b ij ] if a ij =b ij for all i ad j. Example 4. Let [a ij ], [b ij ] FSM 3 X4 where [a ij ] = [a ij ] [b ij ] = , [a ij ] [b ij ] = [a ij ] 0 = ad [b ij ] = The Propositio1. Let [a ij ] FSM m X. The i) [[a ij ] 0 ] 0 = [a ij ] iv) [a ij ] [a ij ] = [a ij ] ii) [a ij ] [a ij ] v) a ij ] [0] = [a ij ] iii) [a ij ] [a ij ] = [a ij ] vi) [a ij ] [0] = [0] Propositio2. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] = [b ij ] ad [b ij ] = [c ij ] [a ij ] = [c ij ] ii) a ij ] [b ij ] ad [b ij ] [c ij ] [a ij ] = [c ij ] Propositio3. Let [a ij ], [b ij ] FSM m X. The De Morga s type results are true. i) ([a ij ] [b ij ]) 0 = [a ij ] 0 [b ij ] 0 ii) ([a ij ] [b ij ]) 0 = [a ij ] 0 [b ij ] 0 Proof: For all i ad j, i) ([a ij ] [b ij ]) 0 = [max { a ij, b ij }] 0 = [ 1 max { a ij, b ij } ] = [ mi { 1 a ij, 1 b ij } ] =[a ij ] 0 [b ij ] 0 Copyright to IJIRSET

4 ii) ([a ij ] [b ij ]) 0 = [mi { a ij, b ij }] 0 = [ 1 mi { a ij, b ij } ] = [ max { 1 a ij, 1 b ij } ] = [a ij ] 0 [b ij ] 0 Propositio4. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] ( [b ij ] [c ij ] = ( [a ij ] [b ij ]) ( [a ij ] [c ij ]) ii) [a ij ] ( [b ij ] [c ij ] = ( [a ij ] [b ij ]) ( [a ij ] [c ij ]) Fuzzy Soft Rectagular Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Rectagular Matrix if m Fuzzy Soft Square Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Square Matrix if m = Fuzzy Soft Row Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Row Matrix if m= Fuzzy Soft Colum Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Colum Matrix if = Fuzzy Soft Diagoal Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Diagoal Matrix if m = ad a ij = 0 for all i j Fuzzy Soft Upper Triagular Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Upper Triagular Matrix if m = ad a ij = 0 for all i> j Fuzzy Soft Lower Triagular Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Lower Triagular Matrix if m = ad a ij = 0 for all i< j Fuzzy Soft Triagular Matrix [7] Let [a ij ] FSM m X. The [a ij ] is said to be a Fuzzy Soft Triagular Matrix if it is either fuzzy soft lower or fuzzy soft upper triagular matrix for all i ad j t-norm [10]: Let T : [0,1] x [0,1] be a fuctio satisfyig the followig axioms: i) T( a, 1 ) = a, a [0,1] (Idetity) ii) T(a, b) = T(b,a), a,b [0,1] iii) if b 1 b 2, the T( a, b 1 ) T( a, b 2 ), a, b 1, b 2 [0,1] iv) T ( a, T( b, c) ) = T(T (a, b), c), a,b,c [0,1] (Commutativity) (Mootoicity) (Associativity) Copyright to IJIRSET

5 The T is called t-orm. A t-orm is said to be cotiuous if T is cotiuous fuctio i [0,1]. A example of cotiuous t- Norm is a b. N.B : The fuctios used for itersectio of fuzzy sets are called t-orms t- Coorm [10]: Let S : [0,1] x [0,1] be a fuctio satisfyig the followig axioms: i) S( a, 0 ) = a, a [0,1] ( Idetity ) ii) S(a, b) = S(b, a), a,b [0,1] ( Commutativity) iii) if b 1 b 2, the S( a, b 1 ) S( a, b 2 ), a, b 1, b 2 [0,1] ( Mootoicity ) iv) S ( a, S( b, c) ) = S(S (a,b), c) a,b, c [0,1] (Associativity) The S is called t- coorm. A t- coorm is said to be cotiuous if S is cotiuous fuctio i [0,1]. N.B. :The fuctios used for uio of fuzzy sets are called t-coorms. A example of cotiuous t- Coorm is a + b a.b Uio of Fuzzy Soft Matrices o t-orm: Let [a ij ], [b ij ] FSM m X.The Uio of Fuzzy Soft Matrices [a ij ] ad [b ij ] o t-orm is defied by [a ij ] [b ij ] = [a ij + b ij a ij. b ij ] for all i ad j Itersectio of Fuzzy Soft Matrices o t-orm: Let [a ij ], [b ij ] FSM m X.The Itersectio of Fuzzy Soft Matrices [a ij ] ad [b ij ] o t-orm is defied by [a ij ] [b ij ] = [a ij. b ij ] for all i ad j. Propositio5. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] [0] = [a ij ] iii) [a ij ] [b ij ] = [b ij ] [a ij ] ii) [a ij ] [1] = [1] iv) ([a ij ] [b ij ] ) [c ij ] = [a ij ] ( [b ij ] [c ij ] ) Propositio6. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] [0] = [a ij ] iii) [a ij ] [b ij ] = [b ij ] [a ij ] ii) [a ij ] [1] = [a ij ] iv) ([a ij ] [b ij ] ) [c ij ] = [a ij ] ( [b ij ] [c ij ] ) Propositio7. Let [a ij ], [b ij ] FSM m X. The De Morga s type results are true : i) ([a ij ] [b ij ]) 0 = [a ij ] 0 [b ij ] 0 ii) ([a ij ] [b ij ]) 0 = [a ij ] 0 [b ij ] 0 Copyright to IJIRSET

6 Proof: for all i ad j, i) ([a ij ] [b ij ]) 0 = [a ij + b ij a ij. b ij ] 0 = [ 1 (a ij + b ij a ij. b ij ) ] = [ 1 a ij b ij + a ij. b ij ) ] = [ (1 a ij ) (1 b ij ) = [ 1 a ij ] [1 b ij ] = [a ij ] 0 [b ij ] 0 ii) The proof of (ii) is exactly similar to (i) Scalar Multiplicatio of Fuzzy Soft Matrix : Let [a ij ] FSM m X. The Scalar Multiplicatio of Fuzzy Soft Matrix [a ij ] by a scalar k deoted by k [a ij ] is defied as k [a ij ] = [ ka ij ], 0 k 1. Propositio8. Let [a ij ] FSM m X ad s ad t are two scalars such that 0 s, t 1. The i) s(t[a ij ]) = (st)[a ij ] ii) s t s[a ij ] t[a ij ] iii) [a ij ] [b ij ] s[a ij ] s[b ij ] Three Importat Operators of t- Norm : i) Miimum Operator : T M ( µ 1 ) = mi ( µ 1 ) ii) Product Operator : T P ( µ 1 ) = i=1 µ i iii) Operator Lukasiewicz t-orm ( Bouded t-orm) : T L ( µ 1 ) = max ( i=1 µ i + 1, 0 ) Propositio9. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] TM [0] = [a ij ] ii) [a ij ] TM [1] = [a ij ] iii)[a ij ] TM [b ij ] = [b ij ] TM [a ij ] iv) ([a ij ] TM [b ij ] ) TM [c ij ] = [a ij ] TM ( [b ij ] TM [c ij ] ) Propositio10. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] TP [0] = [0] ii) [a ij ] TP [1] = [a ij ] iii) [a ij ] TP [b ij ] = [b ij ] TP [a ij ] iv) ([a ij ] TP [b ij ] ) TP [c ij ] = [a ij ] TP ( [b ij ] TP [c ij ] ) Propositio11. Let [a ij ], [b ij ], [c ij ] FSM m X. The i) [a ij ] TL [0] = [0] ii) [a ij ] TL [1] = [a ij ] iii) [a ij ] TL [b ij ] = [b ij ] TL [a ij ] iv) ([a ij ] TL [b ij ] ) TL [c ij ] = [a ij ] TL ( [b ij ] TL [c ij ] ) Three Importat Operators of t-coorm : i) S M {µ 1 } = max { µ 1 } Copyright to IJIRSET

7 ii) S P { µ 1 } = 1 i=1 ( 1 µ i ) iii) S L { µ 1 } = mi { µ i i=1, 1 } Propositio12. Let [a ij ], [b ij ] FSM m X. The i) [a ij ] SM [0] = [a ij ] ii) [a ij ] SM [1] = [1] iii) [a ij ] SM [b ij ] = [b ij ] SM [a ij ] iv) ([a ij ] SM [b ij ] ) SM [c ij ] = [a ij ] SM ( [b ij ] SM [c ij ] ) Propositio13. Let [a ij ], [b ij ] FSM m X. The i) [a ij ] SP [0] = [a ij ] ii) [a ij ] SP [1] = [1] iii) [a ij ] SP [b ij ] = [b ij ] SP [a ij ] iv) ([a ij ] SP [b ij ] ) SP [c ij ] = [a ij ] SP ( [b ij ] SP [c ij ] ) Propositio14. Let [a ij ], [b ij ] FSM m X. The i) [a ij ] SL [0] = [a ij ] ii) [a ij ] SL [1] = [1] iii) [a ij ] SL [b ij ] = [b ij ] SL [a ij ] iv) ([a ij ] SL [b ij ] ) SL [c ij ] = [a ij ] SL ( [b ij ] SL [c ij ] ) Example 5. From the above Example 4, [a ij ], [b ij ] FSM 3 X4 where [a ij ] = T P ( [a ij ], [b ij ] ) = ad [b ij ] = The T M ( [a ij ], [b ij ] ) = ad T L ( [a ij ], [b ij ] ) = , Arithmetic Mea (A.M.) of Fuzzy Soft Matrix : Let A = [a ij ] FSM m X. The Arithmetic Mea of Fuzzy Soft Matrix of membership value deoted by A AM is defied as A AM = A j =1 μ ij. III. FUZZY SOFT MATRICES IN DECISION MAKING BASED ON T NORM OPERATORS I this sectio, we put forward fuzzy soft matrices i decisio makig by usig differet operators of t- orm. Iput: Fuzzy soft set of m objects, each of which has parameters. Output: A optimum result. ALGORITHM Step- 1: Choose the set of parameters. Copyright to IJIRSET

8 Step -2: Costruct the fuzzy soft matrix for the set of parameters. Step -3: Compute T M. Step- 4: Compute the arithmetic mea of membership value of fuzzy soft matrix as A AM (T M ). Step-5: Fid the decisio with highest membership value. Example 6: Suppose a compay produces five types of water purifier p 1, p 2, p 3, p 4, p 5 such that U = { p 1, p 2, p 3, p 4, p 5 } ad E = { e 1 ( oly filter available ), e 2 (oly UV available), e 3 ( UV+ RO available ) } be the set of parameters. Suppose Mr. X is goig to buy a purifier. O the basis of the parameters, three experts Mr. A, Mr. B ad Mr. C give their valuable commets o the water purifier ad the followig fuzzy soft matrices are costructed as follows: A = The T M =, B = ad C = ad A AM (T M ) = From the above result (3.1),it is obvious that p 5 water purifier will be preferred (3.1) Note: If T P ad T L are used istead of T M,the we havet P = T L = ad A AM (T L ) = ad A AM (T P )= (3.3) (3.2) From the above results (3.2) ad (3.3), it is clear that p 1 water purifier ad p 3 water purifier will be selected respectively by Mr. X. IV. CONCLUSION I this paper, we proposed fuzzy soft matrices ad defied differet types of fuzzy soft matrices. We have also give some defiitios based o t-orm with examples ad some properties with proof. Some operators o t-orm are also Copyright to IJIRSET

9 give. Fially, we exted our approach o t-orms i applicatio of decisio makig problems. We have show that decisios are differet for differet methods o same applicatio. Our future work i this regard is to fid other methods whether the otios i this paper yield fruitful result. REFERENCES [1] D. Molodtsov, Soft set theory first result, Computers ad Mathematics with Applicatios 37(1999), pp [2] P. K. Maji, R. Biswas ad A. R. Roy, Fuzzy Soft Sets, Joural of Fuzzy Mathematics, 9(3), ( 2001), pp [3] P.K.Maji, R. Biswas ad A.R.Roy, A applicatio of soft sets i a decisio makig problems, Computer ad Mathematics with Applicatios, 44(2002), pp [4] P.K.Maji, R. Biswas ad A.R.Roy, Soft Set Theory, Computer ad Mathematics with Applicatios, 45(2003), pp [5] Naim Cagma, Serdar Egioglu, Soft matrix theory ad its decisio makig, Computers ad Mathematics with Applicatios 59(2010), pp [6] N. Cagma ad S. Egioglu, Fuzzy soft matrix theory ad its applicatio i decisio makig, Iraia Joural of Fuzzy Systems, 9(1), (2012), pp [7] Maas Jyoti Borah, Tridiv Jyoti Neog, Dusmata Kumar Sut, Fuzzy soft matrix theory ad its decisio makig, IJMER, 2(2) (2012), pp [8] P. K.Maji, A. R. Roy, A fuzzy soft set theoretic approach to decisio makig problems, Joural of Computatioal ad Applied Mathematics, 203(2007), pp [9] P. Majumdar, S.K.Samata, Geeralized fuzzy soft sets, Computers ad Mathematics with Applicatios, 59(2010), pp [10] James J. Buckley, Esfadiar Eslami, A Itroductio to Fuzzy Logic ad Fuzzy Sets, Physica-Verlag, Heidelberg,New York(2002). Copyright to IJIRSET