Max - Min Composition of Linguistic Intuitionistic Fuzzy Relations and Application in Medical Diagnosis 1

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1 VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) Max - Min Compoition of Linguitic Intuitionitic Fuzzy elation and Application in Medical Diagnoi Bui Cong Cuong Pham Hong Phong Intitute of Mathematic Vietnam Academy of Science and Technology Vietnam Faculty of Information Technology National Univerity of Civil Engineering Vietnam Abtract In thi paper we firt introduce the notion of linguitic intuitionitic fuzzy relation Thi notion i ueful in ituation when each correpondence of object i preented a two label uch that the firt expree the degree of memberhip and the econd expree the degree of non-memberhip a in the intuitionitic fuzzy theory Sanchez' approach for medical diagnoi i extended uing the linguitic intuitionitic fuzzy relation 04 Publihed by VNU Journal of Science Manucript communication: received 0 December 0 revied 09 September 04; accepted 9 September 04 Correponding author: Bui Cong Cuong bccuong@gmailcom Keyword: Fuzzy et Intuitionitic fuzzy et Fuzzy relation Intuitionitic fuzzy relation Linguitic aggregation operator Max - min compoition Medical diagnoi Introduction * The correpondence between object can be uitably decribed a relation A traditional crip relation repreent the atifaction or the diatifaction of relationhip connection or correpondence between the object of two or more et Thi concept can be extended to allow for variou degree or trength of relationhip or connection between object Degree of relationhip can be repreented by memberhip grade in a fuzzy relation [] in the ame way a degree of memberhip are repreented in the fuzzy et [] However there i a heitancy or a doubtfulne about the grade aigned to the relationhip between object In fuzzy et theory there i no mean to Thi reearch i funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number deal with that heitancy in the memberhip grade A reaonable approach i to ue intuitionitic fuzzy et defined by Atanaov in 98 [-4] Motivated by intuitionitic fuzzy et theory in 995 [5] Burillo and Butince firt propoed intuitionitic fuzzy relation Further reearche of thi type of relation can be found in [6-9] There are many ituation due to the natural apect of the information the information cannot be given preciely in a quantitative form but in a qualitative one [0] Thu in uch ituation a more realitic approach i to ue linguitic aement intead of numerical value by mean of linguitic label which are not number but word or entence in a natural or artificial language []

2 58 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) One of the main concept in relational calculu i the compoition of relation Thi make a new relation uing two relation For example relation between patient and illnee can be obtained from relation between patient and ymptom and relation between ymptom and illnee (ee medical diagnoi [8 -]) In thi paper we define linguitic intuitionitic fuzzy relation which i an extenion of intuitionitic fuzzy relation uing linguitic label Then we propoe max - min compoition of the linguitic intuitionitic fuzzy relation Finally an application in medical diagnoi i introduced Preliminarie In thi ection we give ome baic definition ued in next ection Intuitionitic fuzzy et Intuitionitic fuzzy et a ignificant generalization of fuzzy et can be ueful in ituation when decription of a problem by a linguitic variable given in term of a memberhip function only eem too rough For example in deciion making problem particularly in medical diagnoi ale analyi new product marketing financial ervice etc there i a fair chance of the exitence of a nonnull heitation part at each moment of evaluation of an unknown object Definition [] An intuitionitic fuzzy et A on a univere X i an object of the form { µ A ν A A = x x x x X where ( x A ) [ 0] memberhip of x in A ( x ) [ 0] µ i called the degree of ν i called the degree of non-memberhip of x in A and the following condition i atified µ A ( x) + ν A ( x) x X A Some development of the intuitionitic fuzzy et theory with application for example can be een in [ ] Linguitic Label In many real world problem the information aociated with an outcome and tate of nature i at bet expreed in term of linguitic label [9-] One of the approache i to let expert give their opinion uing linguitic label In order to deploy the above approach they have been uing a finite and totally ordered dicrete linguitic label et S = { K n Where n i an odd poitive integer repreent a poible value for a i linguitic variable and it require that []: - The et i ordered: i j iff i j ; - The negation operator i defined a: neg ( ) = uch that j = n + i i j For example a et of even linguitic label S could be defined a follow [0]: { S = = none = verylow = low 4 = medium = high = veryhigh = perfect An overview of linguitic aggregation operator which handle linguitic label i given in [] Intuitionitic fuzzy relation ) Intuitionitic fuzzy relation Intuitionitic fuzzy relation an extenion of fuzzy relation wa firt introduced by Burillo and Butince in 995 Definition [5] Let X Y be ordinary finite non-empty et an intuitionitic fuzzy relation ( IF ) between X and Y i defined a an intuitionitic fuzzy et on X Y that i i given by:

3 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) where { ( ) µ ( ) ν ( ) ( ) µ ν : [ 0] atify the = x y x y x y x y X Y condition X Y ( ) ( ) ( ) µ x y + ν x y x y X Y The et of all IF between X and Y i IF X Y denoted by ) Compoition of intuitionitic fuzzy relation Triangular norm and triangular co-norm are notion ued in the framework of probabilitic metric pace and in multi-valued logic pecifically in fuzzy logic Definition A triangular norm ( t -norm) i a commutative aociative increaing 0 0 T x = x [ ] [ ] mapping T atifying for all x [ 0] A triangular conorm ( t -conorm) i a commutative aociative increaing 0 0 mapping S atifying S ( 0 x) = x [ ] [ ] for all x [ 0] In 995 [5] Burillo and Butince introduced concept of intuitionitic fuzzy relation and compoition of intuitionitic fuzzy relation uing four triangular norm or conorm Definition 4 [5] Let α β λ ρ be four t - norm or t -conorm IF ( X Y ) α β P IF( Y Z ) elation P o IF ( X Z ) i defined a follow: where λ ρ α β P o = ( x z) µ α β ( x z) ν α β ( x z) ( x z) X Z λ ρ P o P o λ ρ λ ρ α β P o λ ρ { P ( x z) ( x y) ( y z) µ = α β µ µ α β P o λ ρ y { P ( x z) ( x y) ( y z) ν = λ ρ ν ν y whenever ( ) α β ( ) ( ) µ x z + ν x z x z X Z α β P o P o λ ρ λ ρ Conider the et L and the operation * L defined by: { [ ] 0 and L = x x x x x + x * x x y y x y and x y L ( x x ) ( ) y y L * Then L * i a complete lattice [] L Uing the relation * the minimum and the L maximum are defined They are denoted by 0 = ( 0) and = ( 0 ) repectively In the L L following intuitionitic fuzzy triangular norm and intuitionitic triangular conorm an extenion of fuzzy relation are recalled Definition 5 [4] An intuitionitic fuzzy triangular norm ( it -norm) i a commutative aociative increaing L L mapping T atifying ( x ) T L = x for all x L An intuitionitic fuzzy triangular conorm ( it -conorm) i a commutative aociative increaing L L mapping S atifying S ( x0 L ) = x for all x L In [7] we defined a new compoition of intuitionitic fuzzy relation uing two it - norm or it -conorm Uing the new compoition if we make a change in nonmemberhip component of two relation the memberhip component of the reult may change which i more realitic We alo proved the Burillo and Butince' notion i a pecial cae of our notion tated many propertie Linguitic Intuitionitic Fuzzy elation A Linguitic Intuitionitic Label

4 60 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) In deciion making problem particularly in medical diagnoi ale analyi new product marketing financial ervice etc there i a heitation part at each moment of the evaluation of an object In thi cae the information can be expreed in term of pair of label where one label repreent the degree of memberhip and the econd repreent the degree of non-memberhip For example in medical diagnoi an expert can ae the correpondence between patient p and ymptom q a a pair ( i j ) where i S i the degree of memberhip of the patient p in the et of all patient uffered from the ymptom q and j S i the degree of non-memberhip of the patient p in thi et In [6] we firt propoed the notion of intuitionitic label to preent expert' aement in thee ituation S Definition [6] A linguitic intuitionitic label i defined a a pair of linguitic label S uch that i + j n + where ( i j ) { S = K n i the linguitic label et i S repectively define the degree of j memberhip and the degree of non-memberhip of an object in a et The et of all linguitic intuitionitic label i denoted a IS ie {( ) i j IS = S i + j n + For example if the linguitic label et S contain = none = very low = low 4 = medium 5 = high 6 = very high and 7 = perfect the correponding linguitic intuitionitic label et IS i given a in table I In [6] we alo defined ome lexical order relation on IS : the memberhip-baed order relation and the non-memberhip-baed order relation TABLE I LINGUISTIC INTUITIONISTIC LABEL SET ( 7 ) ( 6 ) ( 6 ) ( 5 ) ( 5 ) ( 5 ) ( 4 ) ( 4 ) ( 4 ) ( 4 4 ) ( ) ( ) ( ) ( 4 ) ( 5 ) ( ) ( ) ( ) ( 4 ) ( 5 ) ( 6 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Definition [6] For all ( µ ν ) ( ) µ ν in IS memberhip-baed order relation M and non-memberhip- baed order relation are defined a following µ > µ µ ν M µ ν µ = µ ν ν ( ) ( ) ν < ν µ ν N µ ν ν = ν µ µ ( ) ( ) N k Some linguitic intuitionitic aggregation operator wa propoed by uing M N relation [6] Thee operator are the implet linguitic intuitionitic aggregation which could be ued to develop other operator for aggregating linguitic intuitionitic information In thi paper a new order relation on IS i propoed (a new relation i denoted by M N aigned to repectively) Thi implied from obervation that: how a linguitic intuitionitic label great may depend on:

5 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) How it memberhip component i greater than it non-memberhip one; - How much information i contained in it For each ( i j ) A = IS thee propertie can be meaured by i j i + j which repectively called core and confidence of A Definition For each A ( ai a j ) = in IS core and confidence of A ( SC ( A ) and CF ( A ) repectively) are define a follow SC ( A) = i j CF A = i + j Definition 4 For all A B in IS relation i defined a following SC ( B) CF ( B) SC A > A B SC A = SC B CF A Theorem elation i a total order relation Proof It i eaily een that i reflexive and aymmetric We now conider the tranitivity and totality Let A B C be arbitrary intuitionitic linguitic label we have: Tranitivity: let u aume that A B and B C Then SC A > SC B SC B > SC C SC A = SC B AND SC B = SC C CF A CF B CF B CF C SC ( B) SC ( C ) CF ( C ) SC ( B) CF ( B) = SC ( C) CF ( C ) SC A > SC ( A) > SC ( B) O SC B = SC ( B) > SC ( C ) CF B SC A = SC ( A) = SC ( B) CF A O CF ( A) CF ( B) O SC B SC ( B) > SC ( C ) CF B CF ( C) SC A = SC C SC ( A) > SC ( C ) O CF A A C Totality: There are four cae Cae SC ( A) SC ( B) A B Cae SC ( A) SC ( B) B A SC A Cae CF A > In thi cae < Thi implie = SC ( B) CF ( B) = SC ( B) < CF ( B) We have A B SC A Cae 4 Thi condition CF A implie B A + Uing thi relation we define max min operator a the following: where ( K ) max A A Am = B A A K A = B min m m Ai IS for all i B = A σ Bm = A σ ( m) σ i a permutation { K m { K m uch that A A L A σ σ σ ( m) In order to convert linguitic intuitionitic label to linguitic label we define CV : IS S uch that: - SC ( A) SC ( B) CF ( A) CF ( B) CV A CV B A B IS - CV map a linguitic label to itelf (linguitic label i i identified with linguitic + ): intuitionitic label (( )) i n i CV i n+ i = i i S Definition 5 For each A ( i j ) define = in IS we CV ( A) = p where p max{ i min { j n i j = + ;

6 6 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) In the following theorem we examine deiderative propertie of CV Theorem For all A B IS we have q () CV ( A) S ; SC ( A) SC ( B) () CV ( A) CV ( B) CF ( A) CF ( B) () A = ( ) CV ( A) = i n i+ i Proof Let u aume that p CV ( A) CV ( B) A = and B ( ) = where ( i j ) Then CV ( A) { { = and p = max i min j n + i j and { { q = max h min k n + h k h ; = () It i eaily een that p n then S () By SC ( A) SC ( B) i j h k () By SC ( A) SC ( B) and CF ( A) CF ( B) i j h k i h j k or i + j h + k i h k j So i h 0 Then ( ) ( ) i n + i j h n + h k ( i h) ( k j) ( i h) ( k j) k = 0 ( ) ( ) i n + i j h n + h k () By ()-() i min { j n + i j { i j i ( n i j) = max + { h k h ( n h k ) { max + = h min k n + h k CV ( B) CV A () If = ( ) and CV ( A) A i n i + p = { { p = max i min j n + i n + i { { = max i min j0 = i B Linguitic Intuitionitic Fuzzy elation Linguitic intuitionitic fuzzy relation i defined in a imilar way to intuitionitic fuzzy relation; however the correpondence of each pair of object i given a a linguitic intuitionitic label Definition 6 Let X and Y be finite nonempty et A linguitic intuitionitic fuzzy relation between X and Y i given by = { ( x y) µ ( x y) ν ( ) ( ) x y x y X Y where for each ( x y) X Y : - ( ( ) ( )) µ x y ν x y IS ; - µ ( x y ) and ( x y) ν define linguitic memberhip degree and linguitic non- x y in the relation memberhip degree of repectively The et of all linguitic intuitionitic fuzzy LIF X Y We denote relation i denoted by the pair ( µ ( x y) ν ( x y) ) by ( ) ( x y) ( x y) ( x y) ( x y) ( x y) x y So µ ν = There are ome way to define linguitic memberhip degree and linguitic nonmemberhip degree in linguitic intuitionitic fuzzy relation The following i an example: Example Expert ue linguitic label to acce the interconnection between two object x and y There are aement voting for atifaction of ( x y ) into the remainder vote for diatifaction of ( x y ) into Aggregating the firt group of aement we obtain linguitic memberhip degree; aggregating the econd one we obtain linguitic memberhip degree (for example ue fuzzy collective olution [0]) In the following max min compoition of two linguitic intuitionitic fuzzy relation i defined

7 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) Definition 7 Let LIF ( X Y ) P LIF ( Y Z ) Max min compoition o between and P i defined by P o = { ( x z) P o ( x z) ( x z) X Z { where ( x z ) max min ( x y ) P ( y z ) µ = P o y ( ) x z X Z C Application in Medical Diagnoi In thi ection we preent an application of linguitic intuitionitic fuzzy relation in Sanchez' approach for medical diagnoi [- ] In a given pathology uppoe that P i the et of patient S i the et of ymptom and D i the et of diagnoe Now let u dicu linguitic intuitionitic fuzzy medical diagnoi The methodology mainly involve with the following four tep: Step Determination of ymptom In thi tep the interconnection between each patient and each ymptom i given by a linguitic memberhip grade and a linguitic non-memberhip grade All uch interconnection form linguitic intuitionitic fuzzy relation Q between P and S Here the linguitic memberhip grade and the linguitic non-memberhip grade could be collected by examination of doctor Step Formulation of medical knowledge baed on linguitic intuitionitic fuzzy relation Analogou to the Sanchez' notion of "Medical Knowledge" we define "Linguitic Intuitionitic Medical Knowledge" a a linguitic intuitionitic fuzzy relation between the et of ymptom S and the et of diagnoe D which expree the memberhip grade and the non-memberhip grade between ymptom and diagnoi Thi relation can be obtained by from medical expert or ome training procee Step Determination of diagnoi uing the compoition of linguitic intuitionitic fuzzy relation In thi tep relation T i determined a compoition of the relation Q (tep ) and (tep ) So T i the relation between P and D Step 4 Uing the mapping CV (definition 5) converting T (tep ) into linguitic fuzzy relation S For each patient p and diagnoi d if S ( p d ) i greater than or equal to the median value of S it i tated that p uffer from d Let u conider a cae tudy adapted from De Biwa oy [8] where The et of patient i = { p p p p The et of ymptom i { S= = Temperature Headache P = 4 Stomach Pain Cough Chet Pain and the et of diagnoe i { D = Viral Fever Malaria Typhoid Stomach problem Heart problem In thi example intuitionitic label et IS i contructed uing label et: { S = = none = very low = low = lightly low = medium = lightly high = high = very high = perfect The linguitic intuitionitic fuzzy relation Q LIF( P Sp ) and LIF ( S D ) are hypothetical given a in table II and table III The linguitic intuitionitic fuzzy relation T (table IV) and linguitic fuzzy relation S (table V) are obtained a follow: T = o Q where o i max-min compoition (definition 7) For example T ( p Typhoid ) D

8 64 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) TABLE II LINGUISTIC INTUITIONISTIC ELATION BETWEEN PATIENTS AND SYMPTOMS Q TEMPEATUE HEADACHE STOMACH PAIN COUGH CHEST PAIN p ( 8 ) ( 6 ) ( 7 ) ( 6 ) ( 6 ) p ( 7 ) ( 4 4 ) ( 6 ) ( 6 ) ( 7 ) p ( 8 ) ( 8 ) ( 7 ) ( 7 ) ( 4 ) p 4 ( 5 ) ( ) ( ) ( ) ( ) TABLE III LINGUISTIC INTUITIONISTIC ELATION BETWEEN SYMPTOMS AND DIAGNOSES VIAL FEVE MALAIA TYPHOID STOMACH POBLEM CHEST POBLEM TEMPE-ATUE ( 5 ) ( 6 ) ( ) ( 6 ) ( 7 ) HEAD-ACHE ( 7 ) ( 4 4 ) ( 6 ) ( 6 ) ( 7 ) STOMACH PAIN ( ) ( ) ( ) ( ) ( ) COUGH ( ) ( ) ( ) ( ) ( ) CHEST PAIN ( ) ( ) ( ) ( ) ( ) TABLE IV LINGUISTIC INTUITIONISTIC ELATION BETWEEN PATIENTS AND DIAGNOSES T VIAL FEVE MALAIA TYPHOID STOMACH POBLEM CHEST POBLEM p ( 5 ) ( 6 ) ( 6 ) ( 6 ) ( 7 ) p ( 6 ) ( 4 4 ) ( 4 4 ) ( 6 ) ( 6 ) p ( 5 ) ( 6 ) ( 6 ) ( 6 ) ( 4 ) p 4 ( 5 ) 7 8 ( ) ( ) ( ) ( ) D { Q( p Temperature ) ( Temperature Typhoid ) Q( p Headache ) ( Headache Typhoid ) min min { = max min { Q( p Stomach Pain ) ( Stomach Pain Typhoid ) min { Q( p Cough ) ( Cough Typhoid ) min { Q( p Chet Pain ) ( Chet Pain Typhoid ) {( 7 ) ( 4 4 ) ( 5 ) ( 6 ) ( 7 ) ( 4 4 ) = max = Uing mapping CV (definition 5) T i converted to linguitic fuzzy relation S For example T ( Typhoid ) = V T ( p Typhoid ) S p C ( ( Typhoid )) ( 4 4 ) = CV T p = CV = { { = max 4 min max{ 4 min{ = max{ 4 = max{ = For each patient p and diagnoi d if ( ) 5 S p d ( 5 i the median value of the label et S ) p uffer from d From table V it i obviou that if the doctor agree p p and p 4 uffer from Malaria p and p uffer from Typhoid wherea p face Stomach problem 4 Concluion In thi paper linguitic intuitionitic fuzzy relation i introduced Max - min compoition of linguitic intuitionitic fuzzy relation i

9 BC Cuong PH Phong / VNU Journal of Science: Comp Science & Com Eng Vol 0 No 4 (04) defined uing a new order relation on intuitionitic label et New notion are applied in medical diagnoi Thi give a flexible and imple olution for medical diagnoi problem in linguitic and intuitionitic environment eference [] LA Zadeh Toward a theory of fuzzy ytem In: NASA Contractor eport - 4 Electronic eearch Laboratory Univerity of California Berkeley 969 [] LA Zadeh Fuzzy Set Information and Control vol [] KT Atanaov Intuitionitic fuzzy et Fuzzy Set and Sytem vol 0 pp [4] KT Atanaov S Stoeva Intuitionitic L- fuzzy et In: Cybernetic and Sytem eearch (Ed Trappl) Elevier Science Pub Amterdam vol pp [5] P Burillo H Butince Intuitionitic fuzzy relation (Part I) Mathware Soft Computing vol pp [6] H Butince P Burillo Structure on intuitionitic fuzzy relation Fuzzy Set and Sytem vol 78 pp [7] BC Cuong P H Phong New compoition of intuitionitic fuzzy relation In: Proc The Fifth International Conference KSE 0 vol pp 6 0 [8] SK De Biwa A oy An application of intuitionitic fuzzy et in medical diagnoi Fuzzy Set and Sytem vol 7 pp [9] G Dechrijver EE Kerre On the compoition of intuitionitic fuzzy relation Fuzzy Set and Sytem vol 6 no pp [0] F Herrera E Herrera-Viedma Linguitic Deciion Analyi: Step for Solving Deciion Problem under Linguitic Information Fuzzy Set and Sytem vol 5 pp [] LA Zadeh J Kacprzy Computing with Word in Information/Intelligent Sytem - part : foundation: part : application Phyica- Verlag Heidenberg vol 999 [] E Sanchez Solution in compoite fuzzy relation equation Application to Medical diagnoi in Brouwerian Logic In: Fuzzy Automata and Deciion Proce (ed MM Gupta GN Saridi B Gaine) Elevier North-Holland pp [] E Sanchez eolution of compoition fuzzy relation equation Inform Control vol 0 pp [4] P Burillo H Butince V Mohedano Some definition of intuitionitic fuzzy number In: workhop on Fuzzy baed expert ytem Sofia Bulgaria pp [5] BC Cuong TH Anh BD Hai Some operation on type- intuitionitic fuzzy et Journal of Computer Science and Cybernetic vol 8 no [6] BC Cuong PH Phong Some intuitionitic linguitic aggregation operator Journal of Computer Science and Cybernetic vol 0 no pp [7] Parvathi C Malathi Arithmetic operation on ymmetric trapezoidal intuitionitic fuzzy number International Journal of Soft Computing and Engineering vol no pp [8] LH Son BC Cuong PL Lanzi NT Thong A novel intuitionitic fuzzy clutering method for geodemographic analyi Expert Sytem with application vol9 no0 pp [9] G Bordogna G Pai A fuzzy linguitic approach generalizing Boolean information retrieval: a model and it evaluation Journal of the American Society for Information Science vol 44 pp [0] BC Cuong Fuzzy Aggregation Operator and Application In: Proceeding of the Sixth International Conference on Fuzzy Sytem AFSS 004 Hanoi Vietnam Vietnam Academy of Science and Technology Pub [] F Herrera E Herrera-Viedma JL Verdegay A equential election proce in group deciion making with a linguitic aement approach Information Science vol 85 pp [] ZS Xu Linguitic Aggregation Operator: An Overview In: Fuzzy Set and Their Extenion: epreentation Aggregation and Model (Ed H Butince F Herrera J Montero) Heidelberg: Springer pp [] G Dechrijver EE Kerre On the relationhip between ome extenion of fuzzy et theory Fuzzy Set and Sytem vol no pp [4] G Dechrijver C Corneli EE Kerre On the repreentation of intuitionitic fuzzy t-norm and t- conorm IEEE Tran Fuzzy Sytem vol pp