Intuitionistic Fuzzy Set and Its Application in Selecting Specialization: A Case Study for Engineering Students

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1 International Journal of Mathematial nalysis and ppliations 2015; 2(6): Published online Deember 17, 2015 ( ISSN: Intuitionisti Fuzzy Set and Its ppliation in Seleting Speialization: Case Study for Engineering Students. M. Kozae 1, ssem Elshenawy 2, Manar Omran 2 Keywords Fuzzy Sets, Intuitionisti Fuzzy Sets, Speialization Choie, Speialization Determination Reeived: Otober 20, 2015 Revised: November 3, 2015 epted: November 5, Department of Mathematis, Faulty of Siene, Tanta University, Egypt, Tanta 2 Department of Physis and Engineering Mathematis, Faulty of Engineering, Tanta University, Egypt, Tanta address Omranmanar900@yahoo.om (M. Omran) Citation. M. Kozae, ssem Elshenawy, Manar Omran. Intuitionisti Fuzzy Set and Its ppliation in Seleting Speialization: Case Study for Engineering Students. International Journal of Mathematial nalysis and ppliations. Vol. 2, No. 6, 2015, pp bstrat Intuitionisti fuzzy set (IFS) is useful in providing a flexible model to elaborate unertainty and vagueness involved in deision making. In this paper, we reviewed the onept of IFS and proposed its appliation in seleting speialization. ase study for engineering students is explained in details. We give a method for omputing degrees of membership and non-membership using given data. The suggested method an be applied in many similar real life situations. 1. Introdution The onept of fuzzy sets (FS) introdued by L..Zadeh [12] has showed meaningful appliations. Fuzzy set handles unertainty and vagueness whih Cantorian set ould not address. The membership of an element to a fuzzy set is a single value between zero and one. However in reality, it may not always be true that the degree of non-membership of an element in a fuzzy set is equal to 1 minus the membership degree beause there may be some hesitation degree. Therefore, a generalization of fuzzy sets was proposed by K.T.tanassov [9], [7] as intuitionisti fuzzy sets (IFS) whih inorporate the degree of hesitation alled hesitation margin (and is defined as 1 minus the sum of membership and non-membership degrees respetively). The notion of defining intuitionisti fuzzy set as generalized fuzzy set is quite interesting and useful in many appliation areas. The knowledge and semanti representation of intuitionisti fuzzy set beome more meaningful, resoureful and appliable sine it inludes the degree of belongingness, degree of non-belongingness and the hesitation margin K.T.tanassov [10],[8].E.Szmidt et al. [4] showed that intuitionisti fuzzy sets are pretty useful in situations when desription of a problem by a linguisti variable given in terms of a membership funtion only seems too rough. Due to the flexibility of IFS in handling unertainty, they are tool for a more human onsistent reasoning under imperfetly defined fats and impreise knowledge E.Szmidt et al. [5]. S.K.De et al. [14] gave an intuitionisti fuzzy sets approah in medial diagnosis. So, Intuitionisti fuzzy set is used in modeling real life problems like sale analysis, new produt marketing, finanial servies, deision making, psyhologial investigations and in many useful appliation in the life.. Sine there is a fair hane of the existene of a non-null hesitation part at eah moment of evaluation of an unknown objet E.Szmidt et al. [6], [4]. K.T.tanassov [8], [11] arried out rigorous researh based

2 75. M. Kozae et al.: Intuitionisti Fuzzy Set and Its ppliation in Seleting Speialization: Case Study for Engineering Students on appliations of intuitionisti fuzzy sets. Many appliations of IFS are arried out using distane measures approah. Distane measure between intuitionisti fuzzy sets is an important onept in fuzzy mathematis beause of its wide appliations in real world, suh as pattern reognition, mahine learning, deision making and market predition. Many distane measures between intuitionisti fuzzy sets have been proposed and researhed in reent years E.Szmidt et al. [6], [3], W.Wang et al. [16] and used by E.Szmidt et al. [4], [5] in medial diagnosis. We show a novel appliation of intuitionisti fuzzy set in a more hallenging area of deision making (i.e. department hoie). n example of department determination will be presented, assuming there is a database (i.e. a desription of a set of subjets 5, and a set of department D). We will desribe the state of students knowing the results of their performane. The problem desription uses the onept of IFS that makes it possible to render two important fats. First, values of eah subjet performane hanges for eah student. Seond, in a department determination database desribing department for different students, it should be taken into aount that for different students aiming for the same department, values of the same subjet performane an be different. We use the Hamming distane method given in E.Szmidt et al.[6], [3], [2] to measure the distane between eah student and eah department. The smallest obtained value, points out a proper department determination based on aademi performane. 2. Conept of Intuitionisti Fuzzy Sets Definition 1 L.. Zadeh [12]: Let be a nonempty set. fuzzy set drawn from X is defined as = { x, µ ( x) :x, where µ ( x ): X [0, 1] is the membership funtion of the fuzzy set. Fuzzy set is a olletion of objets with graded membership i.e. having degrees of membership. Definition 2 K. T. tanassov [8]: Let X is a nonempty set. n intuitionisti fuzzy set in X is an objet having the form = { x, µ ( x), ν (x) :x, where the funtions µ ( x), ν (x): X [0, 1] define respetively, the degree of membership and degree of non-membership of the element x X to the set, whih is a subset of X, and for every element x X,0 µ ( x) + ν(x) 1 Furthermore, we have π ( x) = 1 µ ( x) ν(x) alled the intuitionisti fuzzy set index or hesitation margin of x in. π ( x ) is the degree of indeterminay of x X to the IFS and π ( ) [0,1] i.e., π ( ):X [0,1] and 0 1 x x for every x X. π( x) expresses the lak of knowledge of whether x belongs to IFS or not. For example, let is an intuitionisti fuzzy set with µ ( x ) = 0.5 and ν ( x ) = 0.3 π ( x) = 1- ( ) = 0.2. It an be interpreted as "The degree that the objet x belongs to IFS is 0.5, the degree that the objet does not belong to IFS is 0.3 and the degree of hesitany is 0.2". π 3. asi Relations and Operations on Intuitionisti Fuzzy Sets IF, be IFS in X, then: 1. [inlusion] µ ( x) µ ( x)and ν ( x) ν ( x) x X 2. [equality] = µ ( x) = µ ( x)and ν ( x) = ν ( x) x X 3. [omplement] = { x, ν ( x), µ ( x) :x 4. [union] = { x,max( µ ( x), µ ( x)),min( ν ( x), ν ( x)) :x 5. [intersetion] = { x,min( µ ( x), µ ( x)),max( ν( x), ν( x)) :x 6. [addition] = { x, µ ( x ) + µ ( x ) µ ( x ) µ ( x ), ν ( x ) ν ( x ) :x 7. [multipliation] = { x, µ ( x) µ ( x), ν( x) + ν( x) ν( x) ν( x) :x 8. [differene] = { x,min( µ ( x), ν( x)),max( ν ( x), µ ( x)) : x 9. [symmetri differene] = { x,max[min( µ, ν ),min( ν, µ )],min[max( ν, µ ),max( µ, ν )] : x 10. [Cartesian produt] = { µ (x) µ (x), ν( x) ν( x) : x Theorem 1: Let and be two IFS in a nonempty set X, then; (i) - = (ii) - = - iff = (iii) - = -. Proof: (i) Let ={ x, µ ( x), ν ( x) x and ={ x, µ ( x), ν ( x) x for, X, then = { x,min( µ ( x), ν ( x)),max( ν ( x), µ ( x)) : x but = { x, ν ( x), µ ( x) x,

3 International Journal of Mathematial nalysis and ppliations 2015; 2(6): = { x,min( µ ( x), ν ( x)),max( ν ( x), µ ( x)) :x sine = { x,min( µ ( x), µ ( x)),max( ν ( x), ν ( x)) :x. The result follows. (ii) = { x,min( µ ( x), ν ( x)),max( ν ( x), µ ( x)) : x If = µ ( x) = µ ( x) and ν ( x) = ν ( x) x X from this, it is ertain that - = - and the result follows. (iii) = { x,min( µ ( x), ν ( x)),max( ν ( x), µ ( x)) : x Given that, = { x, ν ( x), µ ( x) x and = { x, ν ( x), µ ( x) x it implies that, - = [x, min( ν ( x), µ ( x)),max( µ ( x), ν ( x)) x and the result is straightforward. Corollary1: Whenever =, =, X. Proof is straightforward from the proof of theorem 1 (ii). Theorem 2: Let and be two IFS in a nonempty set X, then; (i) - = φ (ii) -φ = (iii) - (iv) - =φ iff = (v) - = iff = φ (vi) - = iff =φ It is easy to prove the above results. Theorem 3: For IFS,,C in X and C, then we have; (i) - C - (ii) C Proof: (i) Given that,, C X and C " " is transitive i.e. and C C i.e. µ ( x) µ ( x) µ C( x) and ν( x) ν( x) νc( x) x X. Sine is the smallest of and C, subtrating from both side of C means nothing, i.e. - C -. The result follows. Sine " " is the extension of "-", the result of (ii) follows. Corollary 2: From the basi operations, we dedued the following relations: a.x =X b.( X ) X C = X ( X C). X ( C) =( X ) ( XC) d. X ( C) = ( X ) ( X C) f. X ( C) = ( X ) ( X C) e.x ( C) = ( X ) ( X C) 4. lgebra Laws in Intuitionisti Fuzzy Sets Let a, band be IFS in x,then the following algebra follow: i. Complementary Law: (. ) = ii. Idempotent Laws: (i) = (ii) = iii. Commutative Laws: (i) = (ii) = (iii) = (iv) = i. ssoiative Laws: (i) ( ) C = ( C) (ii) ( ) C = ( C) (iii) ( C)=( ) C(iv) ( C)= ( ) C ii. Distributive Laws: (i) ( C) = ( ) ( C) (ii) ( C)= ( ) ( C) (iii) ( C) =( ) ( C) (iv) ( C) =( ) ( C) (v) ( C) =( ) ( C) (vi) ( C) =( ) ( C) vi. De Morgan's laws: (i) ( ) = (ii) ( ) = (iii)( ) = (iv) ( ) = vii. bsorption Laws: (i) ( ) = (ii) ( ) = (i) φ = (ii) =φ (iii) φ = (iv) φ=φ (v) =φ (i) X =X (ii) =X (iii) X = Note: Distributive Laws hold for both right and left hands. The proofs follow from the basi operations. Theorem4: Let,,C be IFS in X and C, then we have; (i) C (ii) C (iii) C (iv) C. Proof:(i) Given that,, C X and C, it means µ () x µ () x C and ν () x ν () x C for every x X. If another IFS X is added to C, it is ertain that, C and the result follows. Results of (ii) -(iv) follow from the proof of (i) Theorem 5: Let and be IFS in X, then (i) = or =, (ii) = or = iff =. Proof: (i) For,, C X, it implies that X. If = and X. Sine = µ () x = µ () x ν() x = ν () x x, from idempotent laws, = or =. The result follows. Result of (ii) follows from (i). Definition 3 E. Szmidt [2]: The Hamming distane d (, ) n H between two IFS and is defined as n 1 = µ x µ x n H 2n i i i= 1 d (, ) ( ) ( ) + ν ( x) ν ( x) + π( x) π ( x) i i i i, = {,,..., } fori= 1,2,..., n. X x x x 1 2 n 5. ppliation of Intuitionisti Fuzzy Sets In Seleting Speialization In Egypt, For Example, IN Faulty Of Engineering, Tanta University, We suffer from Lak of speialization and How

4 77. M. Kozae et al.: Intuitionisti Fuzzy Set and Its ppliation in Seleting Speialization: Case Study for Engineering Students and Why hoosing the department to eah student where the famous in Egypt eah student hoose the department aording to that its famous and ignoring that if this student eligible to this department or not. So, we need hoosing the suitable department to eah student by its degree of eah objet where eah department needs to be exellent in speifi objets. For example: * In eletrial department need the student to be exellent in Mathematial, Physis and Computer (Logi). * In rhiteture department need the student to be exellent in drawing engineered and Computer (Logi). * In Mehanis department need the student to be exellent in drawing engineered, mehanis and Computer (Logi). * In ivil department need the student to be exellent in drawing engineered and Computer (Logi). We use intuitionisti fuzzy sets as tool sine it inorporate the membership degree (i.e. the marks of the questions answered by the student),the non-membership degree (i.e. the marks of the questions the student failed) and the hesitation degree (whih is the mark alloated to the questions the student do not attempt). Let Table 1. Departments vs. Subjets. be the set of students, D= S = {s,s,s,s} (arhiteture, eletrial, ivil, mehanis} be the set of department and Su= (logi, Mathematis, drawing engineered, physis, Physis, mehanis} be the set of subjets related to the departments. We assume the above students sit for examinations(i.e. over 100 marks total) on the above mentioned subjets to determine their department plaements and hoies. The table below shows departments and related subjets requirements. Logi Mathematis Drawing engineered Physis Mehanis rhiteture (0.8,0.1,0.1) (0.5,0.3,0.2) (0.9,0.1,0) (0.5,0.5,0) (0.5,0.4,0.1) Eletrial (0.8,0.1,0.1) (0.9,0.1,0) (0.5,0.4,0.1) (0.8,0.1,0.1) (0.5,0.5,0) Civil (0.8,0.1,0.1) (0.7,0.2,0.1) (0.9,0.1,0) (0.5,0.2,0.3) (0.6,0.3,0.1) Mehanial (0.6,0.3,0.1) (0.5,0.5,0) (0.8,0.1,0.1) (0.5,0.5,0) (0.9,0.1,0) Eah performane is desribed by three numbers i.e. membership, non-membership and hesitation margin. fter the various examinations, the students obtained the following marks as shown in the table below. Table 2. Studentsvs Subjets. Logi Mathematis Drawing engineered Physis mehanis S1 (0.9,0.1,0) (0.9,0.1,0) (0.6,0.2,0.2) (0.9,0.1,0) (0.5,0.5,0) S2 (0.5,0.3,0.2) (0.6,0.2,0.2) (0.5,0.3,0.2) (0.4,0.5,0.1) (0.7,0.2,0.1) S3 (0.7,0.1,0.2) (0.6,0.3,0.1) (0.7,0.1,0.2) (0.5,0.4,0.1) (0.4,0.5,0.1) S4 (0.6,0.4,0.0) (0.8,0.1,0.1) (0.6,0.0,0.4) (0.6,0.3,0.1) (0.5,0.3,0.2) Using Def. 3 above to alulate the distane between eah studentand eah speialization with referene to the subjets, we get the table below. Table 3. Students vs. Careers. arhiteture Eletrial Civil Mehanis S S S S From the above table, the shortest distane gives the proper department determination. S 1 is to read eletrial (Eletrial Engineer), S 2 is to read mehanis (Mehanial Engineer), S 3 is to read arhiteture (arhiteture Engineer), ands 4 is to read ivil (ivil Engineer). 6. Conlusion The ase study presented in this mark an be applied in many real life appliations. For example, mediine, politial or soial ase. Referenes [1].M. Kozae, H.M. bodonia, Granular omputing, bi approximation spaes, 2005 IEEE international onferene on Granular omputing,1, [2] E. Szmidt, Distanes and similarities in intuitionisti fuzzy sets, Springer (2014). [3] E. Szmidt, J. Kaprzyk, Distanes between intuitionisti fuzzy Sets, Fuzzy Sets and Systems 114 (3) (2000) [4] E. Szmidt, J. Kaprzyk, Intuitionisti fuzzy sets in some medial appliations, Note on IFS 7 (4) (2001) [5] E. Szmidt, J. Kaprzyk, Medial diagnosti reasoning using a similarity measure for intuitionisti fuzzy sets, Note on IFS 10 (4) (2004) [6] E. Szmidt, J. Kaprzyk, On measuring distanes between intuitionisti fuzzy Sets, Notes on IFS 3 (4) (1997) 1-3. [7] K.T. tanassov, Intuitionisti fuzzy sets, Fuzzy Sets and Systems 20(1986) [8] K.T. tanassov, Intuitionisti fuzzy sets: theory and appliation, Springer (1999).

5 International Journal of Mathematial nalysis and ppliations 2015; 2(6): [9] K.T. tanassov, Intuitionisti fuzzy sets, VII ITKR's Session, Sofia, [10] K.T. tanassov, New operations defined over intuitionisti fuzzy sets,fuzzy Sets and Systems Vol. 61, 2 (1994) [11] K.T. tanassov, On Intuitionisti fuzzy sets, Springer (2012). [12] L.. Zadeh, Fuzzy sets, Information and Control 8 (1965) [13] ME El-Monsef, M Kozae, l El-Maghrabi, Some Semi topologial appliations on rough set, J.Egypt Math. So12, 45-53, [14] S. K. De, R. iswas,.r. Roy, n appliation of intuitionisti fuzzy sets in medial diagnosti, Fuzzy sets and systems 117 (2) (2001) [15] T. K. Shinoj, J. J. Sunil, Intuitionisti Fuzzy multisets and its appliation in medial diagnosis, International journal of mathematial and omputational sienes 6(2012) [16] W. Wang, X. Xin, Distane measure between intuitionisti fuzzy sets.pattern Reognition Letters 26 (2005)