An Appliaction of Generalized Fuzzy Soft Matrices in Decision Making Problem

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn:9-765x Volume 0, Issue Ver. II. (Feb. 04), PP -4 n ppliaction of Generalized Fuzzy Soft Matrices in Decision Making Problem B.K. Saikia, H. Boruah and P.K. Das Principal, L.T.K. College, zad, North Lakhimpur-7870, ssam Dept. of Mathematics, Genius cademy, Khelmati, North Lakhimpur-7870, ssam Dept. of Mathematics, NERIST, Niruli-7909, runachal Pradesh bstract: fter development of fuzzy soft matrices, it has been applying in many fields of real life scenarios. The problems which are unable to solve by ordinary matrices can be solved by fuzzy soft matrices. In this paper our main aim is to define generalized fuzzy soft matrices and to study a few of its properties. Finally, we presented a decision making problem based on one of the operation of generalized fuzzy soft matrices. Keywords: Fuzzy Soft Matrices, Generalized Fuzzy Soft Set, Generalized Fuzzy Soft Matrices. I. Introduction In 999, Molodtsov [9 introduced the theory of soft sets, which is a new approach to vagueness. In 00, Mai et al. [ studied the theory of soft sets initiated by Molodtsov and developed several basic notions of Soft Set Theory. Till now, researchers are contributing a lot on the extension of soft set theory. In 005, Pei and Miao [6 and Hai Long et al. [6 studied and improved the Mai et al. [. Recently Cagman et al. [, introduced soft matrices and applied it in decision making problems. Researchers published several papers on fuzzy soft matrices and it has been applying in many fields of real life scenarios. B. Chetia and P.K. Das [ defined Intuitionistic Fuzzy Soft Matrices with different products and properties on these products. In 00, Maumdar et al. [0 generalized the concept of fuzzy soft set introduced by Mai et al. [ to generalized fuzzy soft set and applied it in decision making problem[8. Keeping in view, in 00 Dinda et al. [5 gives the notion of Generalized Intuitionistic Fuzzy Soft Set.They continually work in this field and applied it in decision making problems [4. In this paper we define generalized fuzzy soft matrix and a few of their operations. Then we study some of its properties. Subsequently based on the operation intersection we have applied generalized fuzzy soft matrices in a decision making problem in medical diagnosis. II. Preliminaries Definition. [0 Let U be an initial universe, P(U ) be the power set of U, E be the set of all parameters and E. pair ( f, E) is called a soft set over U is defined as the set of ordered pairs ( f, E) { ( e, f ( e)) : e E, f ( e) P( U) }, where f is a mapping given by f : E P( U) such that f () e if e. Here f is called approximate function of the soft set ( f, E ). The set f() e is called e- approximate value set which consists of related obects of the parameter e E. In the other words, a soft set over U is a parameterized family of subsets of the universe U. Definition. Let ( f, E ) be a fuzzy soft set over U, where U { u, u,..., u m } and E { e, e,, e }, u U, e E there exists the membership degree a f ( u ) and then we can present all membership degrees by a table as follows: i e i n i Page

2 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem e e e u a a a n u a a a n n Then the fuzzy matrix u a a a m m m mn = [ m n ai m n = a a a a a a a a a n n m m m n is called the fuzzy soft matrix of ( f, E)over U. ccording to this definition, a fuzzy soft set is uniquely characterized by the fuzzy matrix and vice versa. Example. Let U u, u, u and { e, e, e}. Then a fuzzy soft set ( f, E ) over U given as below: f ( ) e = { u 0., u 0.5, u 0.} f( e ) u 0.5, u 0. 7, u 0.6 f ( e ) { u 0.4, u 0.6, u 0.5} Hence the corresponding fuzzy soft matrix [ ai m n is written by: Definition.4 [5 Let U { u, u,..., u m } be the universal set of elements and pair ( UE, ) will be called soft universe. Let F : E I : EI [0,, U I is the collection of all fuzzy subsets of U. U E { e, e,..., e n } and be a fuzzy subset of E i.e., U Let F : E I I be a function define as F ( e) ( F( e), ( e)), where F() e Then F is called the fuzzy soft set over the soft universe ( UE, ). e, F ( e ) ( F( e ), ( e )) be the set of parameters. The U I. Here for each parameters i i i i indicates not only the degree of belongingness of the Fe but also the degree of possibility of such belongingness which is represented by elements of U in ( i ) ( ). e i Example.5 Let U { s, s, s, s4} be the set of students under consideration and E { e, e, e, e4, e5} be the set of parameters where e = expert in Mathematics, e =expert in Chemistry, e =expert in Physics, 4 Page

3 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem e 4 =expert in Biology and e 5 =expert in English. { e, e, e, e } E. Let : I [0, be given as: Let 4 ( e ) 0.8, ( e ) 0.6, ( e ) 0.5 and ( e4 ) 0.7. We define F as follows: F ( e ) ({ s 0.5, s 0.7, s 0.4, s 0.},0.6) 4 F ( e ) ({ s 0., s 0., s 0.5, s 0.},0.5) F ( e ) ({ s 0., s 0., S 0., s 0.4},0.7) which is the generalized fuzzy soft set representing overall expert of the students. Definition. III. Generalized Fuzzy Soft Matrix Let U be an initial universe, E be a set of parameters and E. Let ( F, E) be a generalized fuzzy soft set over ( UE, ). Then a subset of U E is uniquely defined by R {( u, e) : e, u F ( e)} which called a relation form of ( F, E). The membership function R and the function R are written by R : UE [0, and R : UE [0,, where R : ( ue, ) [0, is the membership value of u U for each e Eand : ( ue, ) [0,. R If [, ( ( u, e ), ( u, e )) [,, we define a matrix i m n R i i i mn (, ) (, ) ( n, n) (, ) (, ) ( n, n) ( m, ) ( m, ) ( mn, n ) which is called an m n generalized fuzzy soft matrix (GFSM) of generalized fuzzy soft set(gfss) ( F, E) Therefore, we can say that a generalized fuzzy soft set ( F, E) is uniquely characterized by the matrix. [ i, m n and both concepts are interchangeable. The set of all m n fuzzy soft matrices over U will be denoted by GFSM m n. Example. U { u, u, u } be the set of the students under consideration for the best academic year with respect the Let given parameters E { e, e, e, e4} where e = result, e = conduct, e = attendance and e 4 = games and sports. Consider 4 ( e ) 0.6. and 4 We define F as follows: { e, e, e } E. Let : [0, be given as ( e ) 0.8, ( e ) 0.7 F ( e ) ({ u 0.4, u 0.5, u 0.6},0.8), F ( e ) ({ u 0.5, u 0.7, u 0.},0.7) 4 4 F ( e ) ({ u 0., u 0., u 0.5},0.6) 4 We can write the above set as follows: 5 Page

4 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem R e e e e 4 u ( u, ) ( u, ) ( u, ) ( u, ) 4 4 u ( u, ) ( u, ) ( u, ) ( u, ) 4 4 u ( u, ) ( u, ) ( u, ) ( u, ) 4 4 Then the generalized fuzzy soft matrix can be written as follows: (0.4, 0.8) (0.5, 0.7) (0, 0) (0., 0.6) [ i, 4 (0.5,0.8) (0.7,0.7) (0,0) (0.,0.6) (0.6, 0.8) (0., 0.7) (0, 0) (0.5, 0.6) Definition. Let[ i, GFSM m n. Then [ i, is called Definition.4 (a) a zero matrix, denoted by [0, if (b) a universal GFSM, denoted by [, Let [ a,, [ b, GFSM i i m n Definition.5 i=0, =0 for all i and. if i. Then GFSM [ c, i is called (a) union of [ a, and [ b, denoted i ' min{, } for all i and. i i =, = for all i and. [ ai, [ bi,, if c min{ a, b } and i i i (b) intersection of [ a, and[ b,,denoted [ a, [ b,, if c min{ a, b } i max{, } for all i and. i i i i i i i and i i i (c) complement of [ a,, denoted by [ a, o, if c a and for all i and..then [ ai, and [ bi, are disoint, if Let [ a,, [ b, GFSM i i m n for all i and. Example.6 (0.4,0.5) (0.5,0.7) (0.,0.6) Let [ ai, (0.5,0.5) (0.7,0.7) (0.,0.6) (0.6,0.5) (0.,0.7) (0.4,0.6) and [ ai, [ bi, [0 (0.7,0.) (0.,0.5) (0.,0.4) (0.5,0.) (0.6,0.5) (0.5,0.4) [ bi, (0.4,0.) (0.,0.5) (0.,0.4) Then (0.7,0.) (0.5,0.5) (0.,0.4) (0.6,0.) (0.6,0.5) (0.5,0.4) [ ai, [ bi, (0.5,0.) (0.7,0.5) (0.,0.4) (0.4,0.5) (0.,0.7) (0.,0.6) (0.5,0.5) (0.,0.7) (0.4,0.6) [ ai, [ bi, (0.4,0.5) (0.,0.7) (0.,0.6) 6 Page

5 and Proportion.7 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem (0.6,0.5) (0.5,0.) (0.8,0.4) (0.4,0.5) (0.8,0.) (0.6,0.4) 0 [ ai, (0.5,0.5) (0.,0.) (0.7,0.4) Let [ a, and [b, GFSM m n i Proof: (a) (b) (a) i 0 0 [[ ai, [ ai,. Then (b) ([ a, [b, ) [ a, [b, i i i i (c) ([ a, [b, ) [ a, [b, i i i i [[ ai, [ ai, [ ( ai ), ( ) [ ai, ([ a, [ b, ) [min{ a, b },max{, } 0 0 i i i i [ min{ ai, bi }, max{, } [max{ ai, bi }, min{, } [ ai, [ bi, [ a, [ b, 0 0 i i ( c)([ a, [ b, ) [max{ a, b },min{, } 0 0 i i i i [ max{ ai, bi}, min{, } [min{ ai, bi},max{, } [ ai, [ bi, [ a, [ b, 0 0 i i Proportion.8 Let [ a,,[ b, and [ c, GFSM m n i i i. Then ( i) [ a, [ b, [ b, [ a, i i i i (ii) [ a, [ b, [ b, [ a, i i i i (iii) [ a, ([ b, [ c, ) ([ a, [ b, ) [ c, i i i i i i (iv) [ a, ([ b, [ c, ) ([ a, [ b, ) [ c, i i i i i i (v) [ a, ([ b, [ c, ) ([ a, [ b, ) ([ a, [ c, ) i i i i i i i (vi) [ a, ([ b, [ c, ) ([ a, [ b, ) ([ a, [ c, ) i i i i i i i IV. Product of Generalized Fuzzy Soft Matrices In these section we define four different product of Generalized Intuitionistic Fuzzy Soft Matrices. Cagman et al. [ defined four different types of product of fozzy soft matrices.we extend these four products to Generalized Intuitionistic Fuzzy Soft Matrices and study some of its properties. Definition 4. Let [ a, and [ b, GFSM. Then nd product of [ a, and [ b, is defined by i ik k mn i ik k and [ a, [ b, [ c, : GFSM m n GFSM mn GFSM mn i ik k i p p 7 Page

6 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem where c min{ a, b } and max{, } such that p n( ) k. Definition 4. i p i ik p k Let [ a, and [ b, GFSM. Then Or product of [ a, and [ b, is defined by i ik k mn i ik k and [ a, [ b, [ c, : GFSM m n GFSM m n GFSM mn i ik k i p p where c max{ a, b } and min{, } such that p n( ) k. i p i ik p k Definition 4. Let [ a, and [ b, GFSM. Then nd Not product of [ a, and [ b, is defined by i ik k mn i ik k and [ a, [ b, [ c, : GFSM m n GFSM m n GFSM mn i ik k i p p where c min{ a, b } and max{, } such that p n( ) k. Definition 4.4 i p i ik p k Let [ a, and [ b, GFSM. Then Or Not product of [ a, and [ b, is defined by i ik k m n i ik k and [ a, [ b, [ c, : GFSM m n GFSM m n GFSM mn i ik k i p p where c max{ a, b } and min{, } such that p n( ) k. i p i ik p k Example 4.5 ssume that [ ai,, [ bik, k GFSM are given as follows: (0.4,0.5) (0.5,0.7) (0.,0.6) (0.7,0.) (0.,0.5) (0.,0.4) [ ai, (0.5,0.5) (0.7,0.7) (0.,0.6) and [ bik, k (0.4,0.) (0.,0.5) (0.,0.4) (0.6,0.5) (0.,0.7) (0.4,0.6) (0.5,0.) (0.6,0.5) (0.5,0.4) Then [ a, [ b, i i k (0.4,0.5) (0.,0.5) (0.,0.5) (0.5,0.7) (0.,0.7) (0.,0.7) (0.,0.6) (0.,0.6) (0.,0.6) (0.4,0.5) (0.,0.5) (0.,0.5) (0.4,0.7) (0.,0.7) (0.,0.7) (0.,0.6) (0.,0.6) (0.,0.6) (0.5,0.5) (0.6,0.5) (0.5,0.5) (0.,0.7) (0.,0.7) (0.,0.7) (0.4,0.6) (0.4,0.6) (0.4,0.6) Similarly, wecanalsofind the other products [ a, [ b,,[ a, [ b, and [ a, [ b, i i k i i k i i k generalized fuzzy soft matrices. Note that commutativity is not valid for the product of generalized fuzzy soft matrices. Proposition 4.6 Let [ ai, and [ bik, k GFSM mn. Then the following are true ( i)([ a, [ b, ) [ a, [ b, i i k i i k ( ii)([ a, [ b, ) [ a, [ b, i i k i i k ( iii)([ a, [ b, ) [ a, [ b, i i k i i k ( iv) ([ a, [ b, ) [ a, [ b, i i k i i k of 8 Page

7 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem V. Generalized Fuzzy Matrices Based Decision Making Problem Let more than one decision makers want to take a decision or to select an obect ointly from the m number of obects which have n number of features i.e., parameters (E ). Suppose that each decision maker has freedom to take his decision of inclusion and evaluation of parameters associated with the selected obect or may choice the same set of parameters i.e., each decision maker has his own choice parameters belonging to the parameter set E and has his own view of evaluation. Here it is assumed that the parameter evaluation of the obects by the decision makers must be generalized fuzzy and may be presented in linguistic form or generalized fuzzy soft set format, alternatively, in the form of generalized fuzzy soft matrices. Now the aim of the decision makers is to find out the obect out of m obect ointly as far as possible. Definition 5. Let [ c, B where and B are two generalized fuzzy soft matrices. Then the set i W( u ) { u U c } is called weight for each ui U. i i i Procedure of solving problem: real life problem can be solved by using different Mathematical methods. The decision maker can choose the easiest method from the alternative. fter taking decision by the decision makers that, they solve a particular problem by using the operation of generalized soft matrices, the decision makers go through the following algorithm. lgorithm: Step : To construct the generalized fuzzy soft matrices with respect to their own choice parameters of the decision makers. Step : To compute the union or intersection of generalised fuzzy soft matrices. Step : To compute the weight of each obect ( O ) by adding the membership values of the entries of its th concerned row ( i -row) of the union or intersection of generalised fuzzy soft matrices. Step 4: The obect having the highest weight becomes the optimal choice obect. i To illustrate the basic idea of the GFSM-algorithm, now we apply it to the following generalized fuzzy soft set (or generalized fuzzy soft matrix) based decision making problems. Case Study: Generally in medical science a patient suffering from a disease may have multiple symptoms. gain it is also observed that there are certain symptoms which may be common to more than one diseases leading to diagnostic dilemma. Sometimes the doctor has to face many problems when an area is largely affected new disease. Then the doctor has detected the disease commencing the common symptoms of the patients. be the set of patients and E { e, e, e, e4, e5, e6, e7, e8} be the set of U { p, p, p, p } Let 4 parameters of symptoms of dengue where e high fever (04 F, 40 C), e headache, e extreme fatigue e4 red eyes and pain in the eyes, e5 enlarged lymph nodes, e6 deep muscle and oint pains (during first hours of illness) e7 nausea and vomiting, e8 low pressure and heart rate. Suppose two Dr. X and Dr. Y examines the patients on based on the same set of parameters. Let : E [0,. They consider the function on the parameters as follows: Dr. X grade on the parameters as 9 Page

8 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem ( e ) 0.7, ( e ) 0.4, ( e ) 0., ( e ) 0.5, ( e ) 0.6, ( e ) 0.5, ( e ) 0.7, ( e ) and the Dr. Y grade on the parameters as ( e ) 0.7, ( e ) 0.5, ( e ) 0.4, ( e ) 0., ( e ) 0.6, ( e ) 0.4, ( e ) 0.7, ( e ) Step : Based on these functions the two doctors construct generalized matrices for the patients and B respectively given as follows: p (0.7,0.8) (0.6,0.6) (0.8,0.) (0.6,0.5) (0.7,0.4) (0.5,0.5) (0.7,0.7) (0.4,0.) p (0.6,0.8) (0.5,0.6) (0.,0.) (0.,0.5) (0.,0.4) (0.6,0.5) (0.,0.7) (0.5,0.) p (0.5,0.8) (0.4,0.6) (0.,0.) (0.,0.5) (0.,0.4) (0.,0.5) (0.4,0.7) (0.,0.) p4 (0.,0.8) (0.7,0.6) (0.5,0.) (0.,0.5) (0.4,0.4) (0.,0.5) (0.,0.7) (0.5,0.) p (0.7,0.7) (0.6,0.5) (0.8,0.4) (0.6,0.) (0.7,0.6) (0.6,0.4) (0.8,0.7) (0.5,0.) p (0.5,0.7) (0.,0.5) (0.4,0.4) (0.4,0.) (0.,0.6) (0.5,0.4) (0.,0.7) (0.4,0.) B p (0.4,0.7) (0.,0.5) (0.5,0.4) (0.,0.) (0.,0.6) (0.,0.4) (0.,0.7) (0.,0.) p4 (0.,0.7) (0.4,0.5) (0.,0.4) (0.,0.) (0.,0.6) (0.,0.4) (0.4,0.7) (0.,0.) (0.7,0.8) (0.6,0.6) (0.8,0.4) (0.6,0.5) (0.7,0.6) (0.5,0.5) (0.7,0.7) (0.4,0.) (0.5,0.8) (0.,0.6) (0.,0.4) (0.,0.5) (0.,0.6) (0.5,0.5) (0.,0.7) (0.4,0.) B (0.4,0.8) (0.,0.6) (0.,0.4) (0.,0.5) (0.,0.6) (0.,0.5) (0.,0.7) (0.,0.) (0.,0.8) (0.4,0.6) (0.,0.4) (0.,0.5) (0.,0.6) (0.,0.5) (0.,0.7) (0.,0.) Step : Now the weights of the patients are respectively, W( p ) W( p ) W( p ) W( p ) Step 4: The maximum weight (5.0) obtains by patient is p. Hence the patient p has suffered in dengue. VI. Conclusion In this paper, we have introduced generalized fuzzy soft matrices and some of their properties. We also defined product of generalized fuzzy soft matrices and some of their properties. Then we defined weighted generalized fuzzy soft fuzzy soft sets. Finally we presented an application of generalized soft matrices in medical diagnosis. References [ Chetia, B. and Das, P. K. : Some Results of Intuitionistic Fuzzy Soft matrix Theory, Pelagia Research Library, dvances in pplied Science Research, 0,Vol. ()4-4. [ Cagman, N. and Enginoglu, S. : Fuzzy soft Matrix Theory and Its pplication in Decision Making, Iranian Journal of Fuzzy systems, Vol.9(), 0. [ Cagman, N., Enginoglu, S. : Soft Matrix Theory and It s Decision Making, Computers and Mathematics with applications, 59(00), [4 Dinda, B., Bera, T. and Samanta, T. K. : Generalized Intuitionistic Fuzzy Soft Sets and n dustable pproach to Decision Making, nnals of Fuzzy Mathematics and Informatics, Vol. 4, No., 0, pp [5 Dinda, B., Bera, T. and Samanta, T. K. : Generalized Intuitionistic Fuzzy Soft Sets and Its pplication in Decision Making, [math.gm, 00. [6 D. Pei and D. Miao: From Soft Sets to Information Systems, Proceedings of the IEEE International Conference on Granular Computing (005), Page

9 n ppliaction Of Generalized Fuzzy Soft Matrices In Decision Making Problem [7 Hai Long YNG: Notes on Generalised Fuzzy Soft Sets, Journal of Mathematical Research & Exposition, 0, Vol., No., pp [8 Maumdar, P., Samanta, T. K.: Generalized Fuzzy Soft Set Based Student Ranking System, International Journal dvance Soft Computer pplication, Vol., 0. [9 Molodtsov, D.: Soft Set Theory - First Result, Computers and Mathematics with pplications 7 (999), 9-. [0 P. Maumdar, S. K. Samanta : Generalized Fuzzy Soft Sets, Computers and Mathematics with pplications, 59(00), pp [ P. K. Mai and. R. Roy: Soft Set Theory, Computers and Mathematics with pplications, 45 (00), cknowledgments: The work is supported by the University Grants Commission (UGC) as Maor Research Proect No. F (SR) Dt. 7 July, Page