Intuitionistic Fuzzy Soft Expert Sets and its Application in Decision Making

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1 Received: Accepted: Year: 05, Nmber:, Pages: Original Article ** Intitionistic Fzzy Soft Expert Sets and its Application in Decision Making Said Bromi,* Florentin Smarandache Faclty of letters and Hmanities, Hay El Baraka Ben M'sik Casablanca B.P. 795, University of Hassan II -Casablanca, Morocco. Department of Mathematics, University of New Mexico, 705 Grley Avene, Gallp, NM 870, USA. Abstract - In this paper, we first introdced the concept of intitionistic fzzy soft expert sets (IFSESs for short which combines intitionistic fzzy sets and soft expert sets. We also define its basic operations, namely complement, nion, intersection, AND and OR, and stdy some of their properties. This concept is a generalization of fzzy soft expert sets (FSESs. Finally, an approach for solving MCDM problems is explored by applying intitionistic fzzy soft expert sets, and an example is provided to illstrate the application of the proposed method. Keywords - Intitionistic fzzy sets, soft expert sets, intitionistic fzzy soft expert sets, decision making.. Introdction Intitionistic fzzy set (IFS in short on a niverse was introdced by Atanassov [7] in 98, as a generalization of fzzy set []. The conception of IFS can be viewed as an appropriate /alternative approach in case where available information is not sfficient to define the impreciseness by the conventional fzzy set. In fzzy sets the degree of acceptance is considered only bt IFS is characterized by a membership fnction and a non-membership fnction so that the sm of both vales is less than one. A detailed theoretical stdy may be fond in [7]. Soft set theory was originally introdced by Molodtsov [] as a general mathematical tool for dealing with ncertainties which traditional mathematical tools cannot handle and how soft set theory is free from the parameterization inadeqacy syndrome of fzzy set theory, ** Edited by Irfan Deli (Area Editor and Naim Çağman (Editor-in-Chief. *Corresponding Athor.

2 Jornal of New Theory ( rogh set theory, and probability theory. A soft set is in fact a set-valed map which gives an approximation description of objects nder consideration based on some parameters. After Molodtsov s work, Maji et al. [6] introdced the concept of fzzy soft set, a more generalized concept, which is a combination of fzzy set and soft set and stdied its properties and also discssed their properties. Also, Maji et al. [7] devoted the concept of intitionistic fzzy soft sets by combining intitionistic fzzy sets with soft sets. Then, many interesting reslts of soft set theory have been stdied on fzzy soft sets [9, 0, 4, 5], on intitionistic fzzy soft set theory [,,, 7], on possibility fzzy soft set [], on generalized fzzy soft sets [5,9], on generalized intitionistic fzzy soft [, 8], on possibility intitionistic fzzy soft set [4], on possibility vage soft set [8] and so on. All these research aim to solve most of or real life problems in medical sciences, engineering, management, environment and social science which involve data that are not crisp and precise. Moreover all the models created will deal only with one expert. To redefine this one expert opinion, Alkhazaleh and Salleh in 0 [9] defined the concept of soft expert set in which the ser can know the opinion of all the experts in one model and give an application of this concept in decision making problem. Also, they introdced the concept of the fzzy soft expert set [0] as a combination between the soft experts set and the fzzy set. After Alkhazaleh s work, many researchers have worked with the concept of soft expert sets [,, 4, 6, 9, 0,, 5, 6, 8, ]. Until now, there is no stdy on soft experts in intitionistic fzzy environment, so there is a need to develop a new mathematical tool called intitionistic fzzy soft expert sets. The paper is organized as follows. In Section, we first recall the necessary backgrond on intitionistic fzzy sets, soft set, intitionistic fzzy soft sets, soft expert sets, fzzy soft expert sets. Section reviews varios proposals for the definition of intitionistic fzzy soft expert sets and derive their respective properties. Section 4 presents basic operations on intitionistic fzzy soft expert sets. Section 5 presents an application of this concept in solving a decision making problem. Finally, we conclde the paper.. Preliminaries In this section, we will briefly recall the basic concepts of intitionistic fzzy sets, soft set, soft expert sets and fzzy soft expert sets. Let U be an initial niverse set of objects and E the set of parameters in relation to objects in U. Parameters are often attribtes, characteristics or properties of objects. Let P (U denote the power set of U and A E... Intitionistic Fzzy Set Definition. [7 ]: Let U be an niverse of discorse then the intitionistic fzzy set A is an object having the form A = {< x,, >,x U},where the fnctions, : U [0,] define respectively the degree of membership, and the degree of nonmembership of the element x X to the set A with the condition. 0 +.

3 Jornal of New Theory ( For two IFS, = {<x,, > } and = {<x,, > } Then,. if and only if., =, = for any.. The complement of is denoted by and is defined by = {<x, > } 4. A B = {<x, min{ } max{ }> } 5. A B = {<x, max{ } min{ }> } As an illstration, let s consider the following example. Example.. Assme that the niverse of discorse U={x,x,x, }. It may be frther assmed that the vales of x, and are in [0, ] Then, A is an intitionistic fzzy set (IFS of U, sch that, A= {< x, 0.4, 0.6>, < x, 0., 0.7>, < x, 0.,0.8>,<, 0.,0.8>}.. Soft Set Definition.. [] Let U be an initial niverse set and E be a set of parameters. Let P(U denote the power set of U. Consider a nonempty set A, A E. A pair (K, A is called a soft set over U, where K is a mapping given by K : A P(U. As an illstration, let s consider the following example. Example.4.Sppose that U is the set of hoses nder consideration, say U = {h, h,..., h 5 }. Let E be the set of some attribtes of sch hoses, say E = {e, e,..., e 8 }, where e, e,..., e 8 stand for the attribtes beatifl, costly, in the green srrondings, moderate, respectively.

4 Jornal of New Theory ( In this case, to define a soft set means to point ot expensive hoses, beatifl hoses, and so on. For example, the soft set (K, A that describes the attractiveness of the hoses in the opinion of a byer, says Thomas, and may be defined like this: A={e,e,e,e 4,e 5 }; K(e = {h, h, h 5 }, K(e = {h, h 4 }, K(e = {h }, K(e 4 = U, K(e 5 = {h, h 5 }... Intitionistic Fzzy Soft Sets Definition.5 [7] Let be an initial niverse set and be a set of parameters. Let IFS(U denotes the set of all intitionistic fzzy sbsets of. The collection is termed to be the intitionistic fzzy soft set over, where is a mapping given by. Example.6 Let U be the set of hoses nder consideration and E is the set of parameters. Each parameter is a word or sentence involving intitionistic fzzy words. Consider {beatifl, wooden, costly, very costly, moderate, green srrondings, in good repair, in bad repair, cheap, expensive}. In this case, to define a intitionistic fzzy soft set means to point ot beatifl hoses, wooden hoses, hoses in the green srrondings and so on. Sppose that, there are five hoses in the niverse given by and the set of parameters,where stands for the parameter `beatifl', stands for the parameter `wooden', stands for the parameter `costly' and the parameter stands for `moderate'. Then the intitionistic fzzy set is defined as follows: ( { } ( { } ( { } ( { } { }.5. Soft Expert Sets Definition.7 [9] Let U be a niverse set, E be a set of parameters and X be a set of experts (agents. Let O= {=agree, 0=disagree} be a set of opinions. Let Z= E X O and A Z A pair (F, E is called a soft expert set over U, where F is a mapping given by F : A P(U and P(U denote the power set of U.

5 Jornal of New Theory ( Definition.8 [9] An agree- soft expert set (,A defined as : over U, is a soft expert sbset of = {F( E X { Definition.9[9] A disagree- soft expert set (,A defined as : over U, is a soft expert sbset of = {F( E X {0.6. Fzzy Soft Expert Sets Definition.0 [0] A pair (F, A is called a fzzy soft expert set over U, where F is a mapping given by F : A,and denote the set of all fzzy sbsets of U.. Intitionistic Fzzy Soft Expert Sets In this section, we generalize the fzzy soft expert sets as introdced by Alkhazaleh and Salleh [0] to intitionistic fzzy soft expert sets and give the basic properties of this concept. Let U be niversal set of elements, E be a set of parameters, X be a set of experts (agents, O= {=agree, 0=disagree} be a set of opinions. Let Z= E X O and Definition. Let U= {,,,, n } be a niversal set of elements, E={ e, e, e,, e m } be a niversal set of parameters, X={ x, i } be a set of experts (agents and O= {=agree, 0=disagree} be a set of opinions. Let Z= { E X Q } and A Z. Then the pair (U, Z is called a soft niverse. Let where denotes the collection of all intitionistic fzzy sbsets of U. Sppose be a fnction defined as: F (z = F(z( i, for all i U. Then F (z is called an intitionistic fzzy soft expert set (IFSES in short over the soft niverse (U, Z. For each zi Z. F (z = F( z i ( i where F( z i represents the degree of belongingness and non-belongingness of the elements of U in F( z i. Hence F ( z i can be written as: F( z i = {( i, F z ( ( i i i }, for i=,,,,n F z ( ( i i

6 Jornal of New Theory ( where F( z i ( i = < F(z i, ( i F(z i > with ( i F(z i and ( i F(z i ( i representing the membership fnction and non-membership fnction of each of the elements i U respectively. Sometimes we write as (, Z. If A Z. we can also have IFSES (, A. Example. Let U={,, } be a set of elements, E={, } be a set of decision parameters, where ( i=,,} denotes the parameters E ={ = beatifl, = cheap} and X= {, } be a set of experts. Sppose that :Z is fnction defined as follows: (,, = { ( ( e, = { ( ( e, = { ( ( e, = { ( ( e,0 = { ( ( e,0 = { ( ( e,0 = { ( ( e,0 = { ( 0.,0.8 0.,0.6, 0.4,0.5 }, 0.5, , ,0.4 }, }, 0., ,0. 0.6,0. 0.,0.6 0.,0. 0.,0.5 }, }, 0.,0.4 0.,0.9 0.,0.5 }, 0.,0.4 0., ,0., ( } 0.,0.4 0., ,0., ( 0.4, ,0. 0.,0.4 Then we can view the intitionistic fzzy soft expert set (, Z as consisting of the following collection of approximations: (, Z ={ (,, = { ( {( e, = { ( {( e, = { ( {( e, = { ( {( e,0 = { ( {( e,0 = { ( {( e,0 = { ( ( e,0 = { ( 0.,0.8 0.,0.6, 0.4, , , ,0.4 0., ,0. 0.6,0. 0.,0.6 0.,0. 0.,0.5 0.,0.4 0.,0.9 0.,0.5 0.,0.4 0., ,0. 0.,0.4 0., ,0. 0.4, ,0. 0.,0.4 Then (, Z is an intitionistic fzzy soft expert set over the soft niverse ( U, Z. Definition.. For two intitionistic fzzy soft expert sets (,A and (,B over a soft niverse (U, Z. Then (, A is said to be an intitionistic fzzy soft expert sbset of (,B if i. B A ii. is an intitionistic fzzy sbset of, for all A

7 Jornal of New Theory ( This relationship is denoted as (, A (, B. In this case, B is called an intitionistic fzzy soft expert sperset (IFSES sperset of (, A. Definition.4. Two intitionistic fzzy soft expert sets (, A and (, B over soft niverse (U, Z are said to be eqal if (, A is a intitionistic fzzy soft expert sbset of (, B and (, B is an intitionistic fzzy soft expert sbset of (, A. Definition.5. An IFSES (, A is said to be a nll intitionistic fzzy soft expert sets denoted and defined as: = F( where Z. Where F( = < 0, >, that is =0 and = for all Z. Definition.6. An IFSES (, A is said to be an absolte intitionistic fzzy soft expert sets denoted and defined as: = F(, where Z. Where F( = <, 0>, that is = and = 0, for all Z. Definition.7. Let (, A be an IFSES over a soft niverse (U,Z. An agree- intitionistic fzzy soft expert set (agree- IFSES over U, denoted as is an intitionistic fzzy soft expert sbset of (, A which is defined as : = {F( E X { Definition.8. Let (, A be a IFSES over a soft niverse (U, Z. A disagree- intitionistic fzzy soft expert set (disagree- IFSES over U, denoted as is a intitionistic fzzy soft expert sbset of (, A which is defined as : = {F( E X {0 Example.9 consider Example..Then the agree- intitionistic fzzy soft soft expert set = {((,,,{ ( (( e,,{ ( (( e,,{ ( (( e,,{ ( }, 0.,0.8 0., , , , ,0.4 }, }, 0., ,0. 0.6,0. }} 0.,0.6 0.,0. 0.,0.5 And the disagree-intitionistic fzzy soft expert set over U ={ (( e,0,{ ( (( e,0, { ( }, 0.,0.4 0.,0.9 0.,0.5 }, 0.,0.4,0.6 0., ,0.

8 Jornal of New Theory ( (( e,0,{ ( (( e,0, { ( } 0.,0.4 0., ,0. }} 0.4, ,0. 0., Basic Operations on Intitionistic Fzzy Soft Expert Sets In this section, we introdce some basic operations on IFSES, namely the complement, AND, OR, nion and intersection of IFSES, derive their properties, and give some examples. Definition 4. Let ( F, A be an IFSES over a soft niverse (U, Z. Then the complement of ( F, A denoted by c ( F, A is defined as: c ( F, A = c ~ (F( for all U. where c ~ is an intitionistic fzzy complement. Example 4. Consider the IFSES ( F, Z over a soft niverse (U, Z as given in Example.. By sing the intitionistic fzzy complement for F(, we obtain defined as: c ( F, Z ={ (,, = { ( 0.8,0. 0.6,0., 0.5,0.4 {( e, = { ( {( e, = { ( {( e, = { ( {( e,0 = { ( {( e,0 = { ( {( e,0 = { ( ( e,0 = { ( 0.5, , , ,0. 0.,0.4 0., ,0. 0.,0. 0.5,0. 0.4,0. 0.9,0. 0.5,0. 0.4,0. 0.7,0. 0., ,0. 0.6,0. 0., ,0.4 0., ,0. c ( F, Z which is Proposition 4. If is an IFSES over a soft niverse (U, Z, then, =. Proof. Sppose that is an IFSES over a soft niverse (U, Z defined as = F(e. Now let IFSES =. Then by Definition 4., = G(e sch that G(e = (F(, Ths it follows that: Therefore = (G( =( ( (F( F(e=.

9 Jornal of New Theory ( = =.Hence it is proven that =. Definition 4.4 Let and be any two IFSES s over a soft niverse (U, Z. Then the nion of and, denoted by is an IFSES defined as =, where C= A B and H( = F( G(, for all C where H( = { ( Where is a s- norm. Proposition 4.5 Let, and be any three IFSES over a soft niverse (U, Z.Then the following properties hold tre. (i (ii (iii (iv = = (, A = (, A Proof (i Let =. Then by definition 4.4, for all C, we have = H( Where H( = F( G( However H( = F( G( = G( F( since the nion of these sets are commtative by definition 4.4. Therfore =. Ths the nion of two IFSES are commtative i.e =. (ii (iii (iv The proof is similar to proof of part(i and is therefore omitted The proof is straightforward and is therefore omitted. The proof is straightforward and is therefore omitted. Definition 4.6 Let and be any two IFSES over a soft niverse (U, Z. Then the intersection of and, denoted by is an IFSES defined as = where C= A B and H( = F( G(, for all C where H( = { Where is a t-norm (

10 Jornal of New Theory ( Proposition 4.7 If, and are three IFSES over a soft niverse (U, Z, then, (i = (ii = (iii (iv ( F, A (, A = (, A Proof (i (ii (iii (iv The proof is similar to that of Propositio 4.5 (i and is therefore omitted The prof is similar to the prof of part (i and is therefore omitted The proof is straightforward and is therefore omitted. The proof is straightforward and is therefore omitted. Proposition 4.8. If then,, and are three IFSES over a soft niverse (U, Z, (i ( = ( ( (ii ( = ( ( Proof. The proof is straightforward by definitions 4.4 and 4.6 and is therefore omitted. Proposition 4.9 If, are two IFSES over a soft niverse (U, Z, then, i. =. ii. =. Proof. (i sppose that and be IFSES over a soft niverse (U, Z defined as: = F( for all A Z and = G( for all B Z. Now, de to the commtative and associative properties of IFSES, it follows that: by Definition 4.0 and 4., it follows that: = = ( (F( ( (G( = ( (F( G( =. (ii The proof is similar to the proof of part (i and is therefore omitted. Definition 4.0 Let and be any two IFSES over a soft niverse (U, Z. Then AND denoted is a defined by: = (

11 Jornal of New Theory ( Where ( = H(, sch that H( = F( G(, for all (. and represent the basic intersection. Definition 4. Let and be any two IFSES over a soft niverse (U, Z. Then OR denoted is a defined by: = ( Where ( = H( sch that H( = F( G(, for all (. and represent the basic nion. Proposition 4. If then,, and are three IFSES over a soft niverse (U, Z, i. ( = ( ii. ( = ( iii. ( = ( ( iv. ( = ( ( Proof. The proofs are straightforward by Definitions 4.0 and 4. and are therefore omitted. Note: The AND and OR operations are not commtative since generally A B B A. Proposition 4.. If and are two IFSES over a soft niverse (U, Z, then, i. =. ii. =. Proof. (i sppose that and be IFSES over a soft niverse (U, Z defined as: ( = (F( for all A Z and = G( for all B Z. Then by Definition 4.0 and 4., it follows that: = = = ( (F( G( = ( (F( (G( = =. (ii the proof is similar to that of part (i and is therefore omitted.

12 Jornal of New Theory ( Application of Intitionistic Fzzy Soft Expert Sets in a Decision Making Problem. In this section, we introdce a generalized algorithm which will be applied to the IFSES model introdced in Section and sed to solve a hypothetical decision making problem. Sppose that company Y is looking to hire a person to fill in the vacancy for a position in their company. Ot of all the people who applied for the position, three candidates were shortlisted and these three candidates form the niverse of elements, U= {,, } The hiring committee consists of the hiring manager, head of department and the HR director of the company and this committee is represented by the set {p, q, r }(a set of experts while the set Q= {=agree, 0=disagree } represents the set of opinions of the hiring committee members. The hiring committee considers a set of parameters, E={ e, e, e, e 4 } where the parameters ei represent the characteristics or qalities that the candidates are assessed on, namely relevant job experience, excellent academic qalifications in the relevant field, attitde and level of professionalism and technical knowledge respectively. After interviewing all the three candidates and going throgh their certificates and other spporting docments, the hiring committee constrcts the following IFSES. (, Z ={ (,, = { ( (,, = { ( (,, = { ( (,, = { ( (,, = { ( (,, = { ( (,, ={ ( (,, ={ ( (,, = { ( (,, = { (,, = { ( (,, 0 = { ( (,, 0 = { ( (,, 0 = { (,, 0 = { ( (,, 0 = { ( (,, 0 = { ( (,, 0 = { ( (,, 0 = { ( 0.,0.4 0.,0.4 0.,0.7 0.,0., 0.5,0. 0.,0.6 0., ,0. 0.,0.6 0.,0.6 0.,0. 0.,0. 0.4,0.6 0.,0.,0.,0. 0.,0. 0.9,0. 0.,0. 0., ,0. 0., ,0. 0.8,0. 0., , ,0.4 0.,0.4 ( 0., ,0. 0.,0. 0.5,0. 0.,0.6 0.,0. 0.,0.4 0.,0. 0.,0.4 0.,0. 0.,0.4 0.,0. (, ( 0.,0. 0.6,0.4, 0.4,0.5 0.,0.4 0.,0.9 0.,0., ( 0.,0.4 0.,0.7 0.,0.5 0.,0.8 0.,0. 0.6,0. 0., ,0. 0.,0.4 0.,0.4 0., ,0.

13 Jornal of New Theory ( (,, 0 = { ( (,, 0 = { ( 0.4, ,0. 0.4,0. 0.,0. 0., ,0. Next the IFSES ( F, Z is sed together with a generalized algorithm to solve the decision making problem stated at the beginning of this section. The algorithm given below is employed by the hiring committee to determine the best or most sitable candidate to be hired for the position. This algorithm is a generalization of the algorithm introdced by Alkhazaleh and Salleh (see [0] which is sed in the context of the IFSES model that is introdced in this paper. The generalized algorithm is as follows: Algorithm. Inpt the IFSES ( F, Z.. Find the vales of - for each element i U where, and are the membership fnction and non-membership fnction of each of the elements i U respectively.. Find the highest nmerical grade for the agree- IFSES and disagree- IFSES. 4. Compte the score of each element i U by taking the sm of the prodcts of the nmerical grade of each element for the agree- IFSES and disagree IFSES, denoted by A and D respectively. i i 5. Find the vales of the score r i = A - D for each element i i i U. Table I. Vales of - for all i U. (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,, (,,

14 Jornal of New Theory ( Determine the vale of the highest score, s= max i { r i }. Then the decision is to choose element as the optimal or best soltion to the problem. If there is more than one element with the highest r i score, then any one of those elements can be chosen as the optimal soltion. Then we can conclde that the optimal choice for the hiring committee is to hire candidate to fill the vacant position i Table I gives the vales of - for each element i U The notation a,b gives the vales of -. In Table II and Table III, we gives the highest nmerical grade for the elements in the agree- IFSES and disagree IFSES respectively. Table II. Nmerical Grade for Agree- IFSES (,, -0. (,, 0. (,, 0. (,, 0. (,, 0. (,, 0.8 (,, 0.4 (,, 0. 6 (,, 0. (,, 0. (,, 0. Highest Nmeric Grade Score ( = = 0. Score ( = =. Score ( =0.+0. = 0. Table III. Nmerical Grade for Disagree-IFSES Highest Nmeric Grade (,, 0 0. (,, 0 0. (,, 0 0. (,, 0-0. (,, 0-0. (,, 0 0. (,, (,, (,, 0 0. (,, 0 0.4

15 Jornal of New Theory ( Score ( = =0 Score ( = = 0.7 Score ( = = 0.85 Let A i and D i represent the score of each nmerical grade for the agree- IFSES and disagree- IFSES respectively. These vales are given in Table IV. Table IV. The score = - Score ( = 0. Score ( = 0 0. Score ( =. Score ( = Score ( = 0. Score ( = Then s= max i vacant position { r i } =, the hiring committee shold hire candidate to fill in the 6. Conclsion In this paper we have introdced the concept of intitionistic fzzy soft expert soft set and stdied some of its properties. The complement, nion, intersection, AND or OR operations have been defined on the intitionistic fzzy soft expert set. Finally, an application of this concept is given in solving a decision making problem. This new extension will provide a significant addition to existing theories for handling ncertainties, and lead to potential areas of frther research and pertinent applications. References [] A. Arokia Lancy, C. Tamilarasi and I. Arockiarani, Fzzy parameterization for decision making in risk management system via soft expert set, International Jornal of Innovative Research and stdies, Vol isse , from [] A. Arokia Lancy, I. Arockiarani, A Fsion of soft expert set and matrix models, International Jornal of Research in Engineering and Technology, Vol 0, isse , from [] D. Molodtsov, Soft set theory-first reslt, Compters and Mathematics with Applications, 7( [4] G. Selvachandran, Possibility Vage Soft Expert Set Theory.(04 Sbmitted. [5] H. L. Yang, Notes On Generalized Fzzy Soft Sets, Jornal of Mathematical Research and Exposition, / ( [6] I. Arockiarani and A. A. Arokia Lancy, Mlti criteria decision making problem with soft expert set. International jornal of Compter Applications, Vol 78- No.5,(0-4, from [7] K.T. Atanassov, Intitionistic Fzzy Sets, Fzzy Sets and Systems 0(,(

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17 Jornal of New Theory ( [] T. A. Albinaa and I. Arockiarani, SOFT EXPERT *pg SET, Jornal of Global Research in Mathematical Archives, Vol, No.,(