INTUITIONISTIC FUZZY SOFT MATRIX THEORY IN MEDICAL DIAGNOSIS USING MAX-MIN AVERAGE COMPOSITION METHOD

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1 Journal of Theoretial and lied Information Tehnology 10 th Setember 014. Vol. 67 No JTIT & LLS. ll rights reserved. ISSN: E-ISSN: INTUITIONISTI FUZZY SOFT MTRIX THEORY IN MEDIL DIGNOSIS USING MX-MIN VERGE OMPOSITION METHOD P. SHNMUGSUNDRM *1,.V. SESHIH, K.RTHI 3 1,3 Deartment of Mathematis, Velalar ollege of Engineering &Tehnology, Erode,Tamil Nadu,India. Deartment of Mathematis, Sri Ramakrishna Engineering ollege, oimbatore,tamil Nadu, India. 1 sserode@gmail.om, vseshaiah@gmail.om, 3 rathi.erode@gmail.om, BSTRT In this aer a new tehnique named as Intuitionisti fuzzy max-min average omosition method is roosed to onstrut the deision method for Medial Diagnosis using different tyes of Intuitionisti fuzzy soft matries and its oerations. Sanhez s aroah for deision making is studied and the onet is generalized by the aliation of Intuitionisti fuzzy soft set theory. Through a survey the relations between the symtoms and diseases are disussed and the roosed method is omared with the existing method. Keywords: Fuzzy Soft Sets, Intuitionisti Fuzzy Soft Sets, Intuitionisti Fuzzy Soft Matrix. Intuitionisti Fuzzy Max Min verage omosition Method. 1. INTRODUTION Soft set theory was initiated by Russian researher Molodtov [1]; he roosed soft set as a omletely generi mathematial tool for modeling unertainties. Mai et al. [,3] alied this theory to several diretions for dealing with the roblems in unertainty and imreision. Pei and Miao [4] and hen et al. [5] imroved the work of Mai et al. Yong et al [6] initiated a matrix reresentation of a fuzzy soft set and alied it in deision making roblems. Borah et al [7] and in Neog et al [8] extended fuzzy soft matrix theory and its aliation. hetia et al[9] roosed Intuitionisti fuzzy soft matrix theory Raaraeswari et al [10,11,1] roosed new definitions for Intuitionisti fuzzy soft matries and its tyes. In real life most of the existing mathematial tools for formal modeling, reasoning and omuting are ris, deterministi and reise in nature. The lassial ris mathematial tools are not aable of dealing with roblems in unertainty and imreision. There are many mathematial tools available for modeling omlex systems suh as robability theory, fuzzy set theory, interval mathematis et. Probability theory is aliable only for a stohastially stable system. Interval mathematis is not suffiiently adatable for roblems with different unertainties. Setting the membershi funtion value is always been a roblem in fuzzy set theory. Intuitionisti Fuzzy Soft Set theory (IFSS) may be more aliable in unertainty and imreision.the arameterization tool of fuzzy soft set theory enhanes the flexibility of its aliations In this aer, a new aroah is roosed to onstrut the deision method for medial diagnosis by using Intuitionisti fuzzy soft matries.in order to make this union, intersetion and the omlement of a Intuitionisti Fuzzy soft matries are alied. The result is obtained based on the maximum value in the sore matrix. We aly Intuitionisti fuzzy soft set theory to develo a 186

2 Journal of Theoretial and lied Information Tehnology 10 th Setember 014. Vol. 67 No JTIT & LLS. ll rights reserved. ISSN: E-ISSN: tehnique through Sanhez s method [13,14] to diagnose whih atient is suffering from what disease.. PRELIMINRIES The basi definitions of Intuitionisti fuzzy soft set theory that are useful for subsequent disussions are given. Definition.1. Suose that U is an initial Universe of disourse and E is a set of arameters, let P (U) denotes the ower set of U. air (F, E) is alled a soft set over U where F is a maing given by F: E P (U).learly, a soft set is a maing from arameters to P (U), and it is not a set, but a arameterized family of subsets of the Universe. Definition. Let U be an initial Universe ot disourse and E be the set of arameters. Let E. air (F, ) is alled fuzzy soft set over U where F is a maing given by F: I U, where I U denotes the olletion of all fuzzy subsets of U. Definition.3. Let U be an initial universe set and E be the set of arameters. Let IF U denote the olletion of all Intuitionisti fuzzy subsets of U. Let E. air (F; ) is alled an Intuitionisti fuzzy soft set over U where F is a maing given by F: IF U. Definition.4 Let U = {1,, 3,..., m} be the Universal set and E be the set of arameters given by E = {e1, e, e3...., en}.let }.Let E and (F,) be a fuzzy soft set in the fuzzy soft lass (U,E).Then fuzzy soft set (F,) in a matrix form as m n =[ai ] m n or = [ai ], i = 1,,..., m, = 1,, 3,..., n, where a i ( ( ), ν ( )) = (0, 1) ) ( i i i if if e e reresents the membershi of i in the Intuitionisti fuzzy set F(e ). ν ) reresents the non-membershi of i in ( i the Intuitionisti fuzzy set F(e ). Definition.5. If = [a i ] IFSM mxn, B = [b i ] IFSM mxn, then we define the addition and subtration of Intuitionisti Fuzzy Soft Matries of and B as +B = { max[μ (a i), μ B(b i)], min[ν (a i), ν B(b i)] } i, B = { min[μ (a i), μ B(b i)], max[ν (a i), ν B(b i)] } i, Definition.6 Let = [ai] IFSM mxn,where a = ), ν ( ) i,. Then is i ( ) ( i i alled a Intuitionisti Fuzzy Soft omlement Matrix if = [ d i ] mxn, where d i = ν ), ( ) i,. ( ) ( i i Definition.7 If = [a i ] IFSM mxn, B = [b k ] IFSM nx, then the max min omosition fuzzy soft matrix relation of and B is defined as *B = [ ik ] mx, where ik = { Max{ Min[ (ai), B (bk )]}, Min{ Max[ ν(ai), νb (bk )]}} Definition.8. If = [a i ] IFSM mxn, B = [b k ] IFSM nx, then a new oeration named as Intuitionisti fuzzy max-min average omosition for fuzzy soft matrix relation is defined as Φ B (ai) + B (bk) ν(ai) + νb (bk) = { Max{ }, Min{ }} i, Examle.9 onsider (0.8,0.1) (0.4,0.5) = and (0.7,0.3) (0.4,0.6) (0.6,0.3) (0.8,0.) B = be the two (0.7,0.3) (0.5,0.5) Intuitionisti fuzzy soft matries, then the addition, subtration, omlement, Max Min omosition and Max Min verage omosition of fuzzy soft matrix relations are (0.8, 0.1) (0.8, 0.) + B = (0.7, 0.3) (0.5, 0.5) (0.6, 0.3) - B = (0.7, 0.3) (0.4, 0.5) (0.4, 0.6) 187

3 Journal of Theoretial and lied Information Tehnology 10 th Setember 014. Vol. 67 No JTIT & LLS. ll rights reserved. ISSN: E-ISSN: = (0.1, 0.8) (0.3, 0.7) (0.6, 0.3) * B = (0.6, 0.3) Φ B = (0.70, 0.0) (0.65, 0.3) (0.5, 0.4) (0.6, 0.4) (0.8, 0.) (0.7, 0.3) (0.80, 0.15) (0.75, 0.5) Definition.10. If = [a i ] IFSM mxn, B = [b i ] IFSM mxn, and, B are the omlement then the sore matrix of and B is defined as S(,B) [ V W ] = where V is the matrix defined as Φ Φ V = [ ( B) ν ( B) ] and W is the matrix Φ defined as W = ( B ) ν ( B ) 3. INTUITIONISTI FUZZY MX-MIN VERGE OMPOSITION METHOD FOR DEISION MKING: In this setion an aliation of Intuitionisti Fuzzy set theory using Max-min average omosition method for deision making is resented. In a given set of system, let P = {P 1, P,...,P m } be the set of m atients and S= {S 1, S,., S n } be the set of n symtoms and D ={D 1, D,., D k } be the set of k diseases. onstrut an IFSS relation matrix alled atient symtom matrix (F,S) over P where F is a maing F : S IF P, IF P is the olletion of all Intuitionisti Fuzzy subsets of P. Then onstrut another IFSS relation matrix (weighted matrix) B, alled symtom-disease matrix, whih is a olletion of an aroximate desrition of atient symtoms in the hosital (G, D) over S, where G is a maing G: D IF S, IF S is the olletion of all Intuitionisti Fuzzy subsets of S. in whih eah element denotes the weight of the symtoms for a ertain disease. Form the matries and B orresonding to the Intuitionisti Fuzzy soft sets (F,E) and (G,E) and omute the omlements (F,E) and (G,E) and Φ their matries and B orresonding to (F,E) and (G,E) resetively. omute Φ B and Φ B whih is the maximum membershi and minimum non membershi of Symtoms of the diseases using definition (.8), omute S ( Φ B, Φ B, Φ B and the Sore matrix Φ B ) using Definition.10. Finally find the maximum sore for eah student P i in the sore matrix, and then onlude that the atient i is suffering from disease D. 3.1 LGORITHM Ste1: Inut the Intuitionisti fuzzy soft set (F,S), (G,D) and obtain the Intuitionisti fuzzy soft matries, B orresonding to (F,S) and (G,D) resetively. Ste: Using Definition.6, obtain the Intuitionisti fuzzy soft omlement matries, B. Ste3: Using Definition.8, omute the Intuitionisti fuzzy max-min average omosition Φ B and Φ B. Ste4: omute the matries V, W and obtain the sore matrix S ( Φ B, Φ B ) using Definition.10. Ste5: Identify the maximum sore S i, for eah atient P i Then we onlude that the atient P i is suffering from disease D. 4. SE STUDY Suose the test results of four atients P = {P 1, P, P 3, P 4 } as the universal set where P 1, P, P 3 and P 4 reresents atients mity, John, Peter, and Ram with symtoms S = {s 1, s, s 3, s 4, s 5 } as the set of symtoms where s 1, s, s 3, s 4, s 5 reresents symtoms temerature, headahe, ough, stomah roblem and body ain resetively for the ase study. Let the ossible diseases relating to the above symtoms D = {D 1, D, D 3 } be viral fever, tyhoid and malaria. Suose that IFSS (F, S) over P, where F is a maing F: S IF P, gives a olletion of an aroximate desrition of atient symtoms in the hosital. 188

4 Journal of Theoretial and lied Information Tehnology 10 th Setember 014. Vol. 67 No JTIT & LLS. ll rights reserved. ISSN: E-ISSN: (F, S) ={ F(s 1 ) = {( 1, 0.8, 0.1), (, 0.0, 0.8), ( 3, 0.8, 0.1), ( 4, 0.6, 0.1)} F(s ) = {( 1, 0.6, 0.1), (, 0.4, 0.4), ( 3, 0.8, 0.1), ( 4, 0.5, 0.4)} F(s 3 ) = {( 1, 0., 0.8), (, 0.6, 0.1), ( 3, 0.0, 0.6), ( 4, 0.3, 0.4)} F(s 4 ) = {( 1, 0.6, 0.1), (, 0.1, 0.7), ( 3, 0., 0.7), ( 4, 0.7, 0.)} F(s 5 ) = {( 1, 0.1, 0.6), (, 0.1, 0.8), ( 3, 0.0, 0.5), ( 4, 0.3, 0.4)} } This Intuitionisti fuzzy soft set is reresented by the following Intuitionisti fuzzy soft matrix s s s s s 1 1 (0.8, 0.1) (0.6, 0.1) (0., 0.8) (0.6, 0.1) (0.1, 0.6) = (0.0, 0.8) (0.4, 0.4) (0.6, 0.1) (0.1, 0.7) (0.1, 0.8) 3 (0.8, 0.1) (0.8, 0.1) (0.0, 0.6) (0., 0.7) (0.0, 0.5) 4 (0.6, 0.1) (0.5, 0.4) (0.3, 0.4) (0.7, 0.) (0.3, 0.4) Then the Intuitionisti fuzzy soft omlement matrix s1 s s3 s4 s5 1 (0.1, 0.8) (0.1, 0.6) (0.8,0.) (0.1, 0.6) (0.6, 0.1) = (0.8, 0.0) (0.4, 0.4) (0.1, 0.6) (0.7, 0.1) (0.8, 0.1) 3 (0.1, 0.8) (0.1, 0.8) (0.6, 0.0) (0.7, 0.) (0.5, 0.0) 4 (0.1, 0.6) (0., 40.5) (0.4, 0.3) (0., 0.7) (0.4, 0.3) gain the set S = {s 1, s, s 3, s 4, s 5 } as universal set where s 1, s, s 3, s 4, s 5 reresents symtoms temerature, headahe, ough, stomah roblem and body ain with the set D = {D1, D, D3} where D1, D and D3 reresent the diseases viral fever, tyhoid and malaria resetively. Suose that IFSS (G, D) over S, where G is a maing G: D IF S, gives an aroximate desrition of Intuitionisti fuzzy soft medial knowledge of the three diseases and their symtoms. Let (G,D) = {G (D 1 ) = {(s 1, 0.6, 0.), (s, 0.3, 0.5), (s 3, 0.1, 0.8), (s 4, 0.4, 0.5), (s 5, 0.1, 0.7)} G (D ) = {(s 1, 0.6, 0.), (s, 0., 0.6), (s 3, 0., 0.7), (s 4, 0.7, 0.), (s 5, 0.1, 0.8)} G (D 3 ) = {(s 1, 0.3, 0.4), (s, 0.7, 0.), (s 3, 0.7, 0.), (s 4, 0.3, 0.4), (s 5, 0., 0.7)}} This Intuitionisti fuzzy soft set is reresented by the following Intuitionisti fuzzy soft matrix B = s1 (0.6, 0.) s (0.3, 0.5) s3 (0.1, 0.8) s4 (0.4, 0.5) s 5 (0.1, 0.7) (0.6, 0.) (0., 0.6) (0., 0.7) (0.7, 0.) (0.1, 0.8) (0.3, 0.4) (0.7, 0.) (0.7, 0.) (0.3, 0.4) (0., 0.7) Then the Intuitionisti fuzzy soft omlement matrix s1 (0., 0.6) (0., 0.6) (0.4, 0.3) s (0.5, 0.3) (0.6, 0.) (0., 0.7) B = s3 (0.8, 0.1) s4 (0.5, 0.4) s 5 (0.7, 0.1) (0.7, 0.) (0., 0.7) (0.8, 0.1) (0., 0.7) (0.4, 0.3) (0.7, 0.) Then the max-min average omosition method matries are (Using Definition.8) Φ B = 1 (0.70, ( (0.70, 4 (0.60,, 0.45 ) 1 (0.80, 0.10 ) (0.75, 0.10 ) 3 (0.70, 0.05 ) 4 (0.60, 0.0 ) (0.70, (0.65, (0.40, 0.40) (0.65, (0.70, (0.75, (0.70, (0.60, 0.5 ) Φ B = (0.75, 0.10 ) (0.65, (0.80, 0.10 ) (0.75, (0.65, 0.05 ) (0.60, 0.10 ) (0.60, 0.0 ) (0.55, 0.5 ) Intuitionisti fuzzy max-min average omosition method is (Using.10) V = W =

5 Journal of Theoretial and lied Information Tehnology 10 th Setember 014. Vol. 67 No JTIT & LLS. ll rights reserved. ISSN: E-ISSN: S(, B) = It is lear from the above matrix that atients mity, John, Peter ( 1,, and 3 ) is suffering from malaria (D 3 ) and P 4 is suffering from tyhoid (D ). 5. ONLUSION It is seen that the max-min average omosition method and max min omosition method [10, 11] gives the same maximum sore in the sore matrix of the atients and the diseases. The dotors agree that mity, John and Peter are suffered from malaria (D 3 ) whereas the max sore of John is 0.55, 4 is suffering from tyhoid. omared with onventional tehniques, the roosed aroah in medial diagnosis effetively redues the reetition. For examle reetition ours in the fourth row of the sore matrix when the membershi value of a 44 in is 0.6, but in roosed method it does not ours. s a result, our aroah makes it ossible to introdue weights for all symtoms and redues the onfusion about the ossibility of two diseases in a atient and also it is an effiient tool for deision making roblem. REFERENES: [1] Molodtsov.D, Soft set theory-first results, omuters and mathematis with liations, 37 (1999) [] Mai,P.K,.Biswas.R,.Roy..R, Fuzzy soft set Journal of Fuzzy Mathematis, 9 (001) [3] Mai,P.K,.Biswas.R,.Roy..R, Intuitionisti Fuzzy soft sets, Journal of Fuzzy Mathematis, 1(004) [4] Pei.D and Mia.D, From soft sets to information systems, in Proeedings of the IEEE International onferene on Granular omuting, vol., , 005. [5] D. hen, E... Tsang, D.S. Yeung, and X.Wang, The arameterization redution of soft sets and its aliations, omuters & Mathematis with liations, vol. 49, no.5-6, , 005 [6] Y.Yang, henli Ji, Fuzzy Soft Matries and their liations, Leture notes in omuter Siene, 700 (011) [7] M.J.Borah, T.J.Neog, D.K.Sut, Fuzzy soft matrix theory and its Deision making, International Journal of Modern Engineering Researh, (01) [8] T.J.Neog, D.K.Sut, n liation of Fuzzy soft sets in Deision making roblems using Fuzzy SoftMatries, International Journal of Mathematial rhive, (01) [9] B.hetia, P.K.Das, Some results of Intuitionisti Fuzzy soft matrix theory, dvanes in lied Siene Researh, 3 (01) [10] P. Raaraeswari, P. Dhanalakshmi, Intuitionisti Fuzzy Soft Matrix Theory and its aliation in Deision Making, International Journal of Engineering Researh and Tehnology, vol.(4) , 013. [11] P. Raaraeswari, P.Dhanalakshmi, Intuitionisti fuzzy soft matrix theory and its aliation in medial diagnosis, nnals of Fuzzy Mathematis and Informatis, vol.,. 1-11, 013 [1] P.Raaraeswari, P.Dhanalakshmi, Similarity measures of intuitionisti fuzzy soft sets and their aliation in medial diagnosis, International ournal of mathematial arhive, vol.5(5), , 014. [13] Sanhez, E. (1979). Inverse of fuzzy relations, aliation to ossibility distributions and medial diagnosis, Fuzzy sets and Systems, vol.,. (1), [14] P. Shanmugasundaram,.V. Seshaiah, n liation of Intuitionisti Fuzzy Tehnique in Medial Diagnosis, ustralian Journal of Basi and lied Sienes, vol.8(9), , June 014. [15] Zadeh, L.. Fuzzy Sets, Information and ontrol, vol.8,,