OPERATIONS ON INTUITIONISTIC FUZZY VALUES IN MULTIPLE CRITERIA DECISION MAKING

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1 Please cite this article as: Ludmila Dymova, Pavel Sevastjaov, Operatios o ituitioistic fuzzy values i multiple criteria decisio makig, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 20, Volume 0, Issue, pages The website: Scietific Research of the Istitute of Mathematics ad Computer Sciece, (0) 20, 4-48 OPERTIONS ON INTUITIONISTIC FUZZY VLUES IN MULTIPLE CRITERI DECISION MKING Ludmila Dymova, Pavel Sevastjaov Istitute of Computer ad Iformatio Scieces, Czestochowa Uiversity of Techology, Częstochowa, Polad dymowa@icis.pcz.pl, sevast@icis.pcz.pl, aa.tikhoeko@gmail.com bstract. This paper presets a aalysis of the basic defiitios of the theory of ituitioistic fuzzy sets. Some udesirable properties of commoly used operatios o ituitioistic fuzzy values are revealed ad the ways to approve the properties of ituitioistic fuzzy arithmetic are proposed. The aim of the aalysis preseted i the paper is to propose a set of operatios o ituitioistic fuzzy values, which provides o-cotroversial results of the solutio of multiple criteria decisio makig problems i the ituitioistic fuzzy settig. The theoretical aalysis is illustrated with umerical examples. Itroductio The ituitioistic fuzzy set itroduced by taassov [] may be treated as a geeralizatio of fuzzy sets theory which curretly is used maily for solvig multiple criteria decisio makig problems (MCDM) [2-8] ad group decisio makig problems [9-] whe the values of local criteria (attributes) of alteratives ad/or their weights are ituitioistic fuzzy values ( IFV ). s the so-called ituitioistic fuzzy set theory was idepedetly itroduced by Takeuti ad Titai [2], there are some termiological difficulties i fuzzy set theory. Dubois et al. [3] oted that Takeuti-Titai s ituitioitic fuzzy logic is simply a extesio of ituitioistic logic [4], i.e., all formulas provable i the ituitioistic logic are provable i their logic. Ituitioistic fuzzy set theory by Takeuti ad Titai is a absolutely legitimate approach, i the scope of ituitioistic logic, but it has othig to do with taassov s ituitioistic fuzzy sets. Therefore, to avoid a misuderstadig, i this paper, taassov s ituitioistic fuzzy sets is abbreviated as -IFS. Geerally, taassov s model (-IFS) may be treated as a classificatio model subject to a valuatio space with three classes ad defiig certai structure [5]. The cocept of -IFS is based o the simultaeous cosideratio of membership µ ad o-membership ν of a elemet of a set i the set itself []. It is postulated that 0 µ + ν. similar approach, the so-called vague sets, proposed by Gau ad uehrer i [6] is proved to be equivalet to -IFS (see [7]). Sice vague sets were proposed later tha -IFS, i this paper, we shall

2 42 L. Dymova, P. Sevastjaov always speak of -IFS. To make the basic defiitios of -IFS clearer ad more trasparet, cosider a illustrative example from [8]. Let E be the set of all coutries with elective govermets. ssume that we ow for every coutry x E the percetage of the electorate that has voted for the correspodig govermet. Deote it by M(x) ad let µ ( x) = M ( x ) / 00 (degree of membership, validity, etc.). Let ν ( x) = µ ( x ). This umber correspods to the part of the electorate who has ot voted for the govermet. Usig fuzzy set theory aloe, we caot cosider this value i more detail. However, if we defie ν (degree of o-membership, o-validity, etc.) as the umber of votes give to parties or persos outside the govermet, the we ca show the part of electorate who has ot voted at all or has spoiled their ballots, ad the correspodig umber will be π ( x) = µ ( x) ν ( x ) (degree of idetermiacy, ucertaity, hesitatio degree, etc.). Thus we ca costruct the set { x, µ ( x), ν ( x) x E } ad obviously, 0 µ + ν. It is clear that for every ordiary fuzzy set π ( x ) = 0 for each x E ad these sets have the form {, µ ( ), µ ( ) } x x x x E. s the most importat applicatios of -IFS are decisio makig problems whe the values of local criteria (attributes) of alteratives ad/or their weights are IFVs, it seems quite atural that the resultig alterative evaluatio should be a IFV as well. Therefore, appropriate operatios o IFVs used for aggregatig local criteria should be properly defied. Obviously, if the fial scores of alteratives are IFVs, the appropriate methods for their compariso are eeded to select the best alterative. Sice there are may differet operatios o IFVs ad methods for their compariso ad aggregatio have bee proposed i the literature, the aim of this paper is to aalyze their merits ad drawbacks ad extract those which provide the results of operatios o IFVs ad aggregatio with acceptable properties. It is safe to say that curretly -IFS is a ope theory, sice there are may competig defiitios of operatios o IFVs which ofte lead to o-ituitive or o-acceptable results of the aggregatio of IFVs. Therefore, the rest of the paper is set out as follows. Sectio presets the basic defiitio of -IFS, the commoly used arithmetical operatios o IFVs ad the method for their compariso. I Sectio 2, we provide a critical aalysis of the operatios preseted i Sectio 2 to elicit their disadvatages ad propose a compromisig set of operatios with at least satisfactory algebraic properties ad reasoable results of IFVs aggregatio. The last sectio cocludes with some remarks.. asic defiitios of ituitioistic fuzzy set theory I [], taassov defied -IFS as follows. Defiitio. Let = { x, x2,..., x } be a fiite uiversal set. ituitioistic fuzzy set i X is a object havig the followig form: = x, µ ( x ), ν ( x ) x X, where fuctios µ : X [0,], x X µ ( x ) [0.] ={ j j j j } j j

3 Operatios o ituitioistic fuzzy values i multiple criteria decisio makig 43 ad ν : X [0,], x j X ν ( x j ) [0.] defie the degree of membership ad degree of omembership of elemet x j X to set X, respectively, ad for every x j X we have 0 µ ( x j) + ν( x j). Followig [], we call π x ) = µ ( x ) + ν ( x ) the ituitioistic idex (or ( j j j the hesitatio degree) of elemet x j i set. It is obvious that for every x j X we have 0 π. s we oted above, -IFS is a geeralizatio of the stadard fuzzy set. Therefore, all the results which are typical for ordiary fuzzy sets theory ca be trasformed i the framework of -IFS as well as. Moreover, ay research based o fuzzy sets ca be described i terms of -IFS. O the other had, i the framework of -IFS, there are ot oly operatios similar to ordiary fuzzy set oes, but also operators that caot be defied i the case of ordiary fuzzy sets. The operatios of additio ad multiplicatio o IFVs were defied by taassov [9] as follows. Let = µ, ν ad = µ, ν be IFVs. The = µ + µ µ µ, ν ν, () = µ µ, ν + ν ν ν. (2) These operatios were costructed i such a way that they produce IFVs sice it is easy to prove that 0 µ + µ µ µ + νν ad 0 µ µ + ν + ν νν. Usig expressios () ad (2), i [20] the followig equatios were obtaied for ay iteger =,2,..: =... = ( µ ), ν, =... = µ, ( ν ) It was proved later that these operatios produce IFVs ot oly for iteger, but also for all real values λ > 0, i.e. λ λ λ = ( µ ), ν, (3) λ = µ λ, ( ν ) λ. (4) Operatios ()-(4) have the followig algebraic properties [2]: Theorem. Let = µ, ν ad = µ, ν be IFVs ad λ > 0 be a real value. The =, (5) =, (6) λ( ) = λ λ, (7)

4 44 L. Dymova, P. Sevastjaov λ λ λ ( ) =, (8) λ λ2 = ( λ + λ2 ), λ, λ 2 > 0, (9) λ λ2 λ + λ2 =, λ, λ 2 > 0. (0) Operatios ()-(4) are used to aggregate local criteria for the solutio of MCDM problems i the ituitioistic fuzzy settig. Let,..., be IFVs represetig the values of local criteria ad w,..., w, i= w =, be their weights. i The the Ituitioistic Weighted rithmetic Mea (IWM) ca be obtaied usig operatios () ad (3) as follows: IWM = w wi w2 2 w = ( ), i i= i=... µ ν () wi i This aggregatig operator provides IFVs ad curretly is the most popular i the solutio of MCDM problems i the ituitioistic fuzzy settig. importat problem is the compariso of IFVs. This problem arises, e.g., whe we have to choose the best alterative i the framework of MCDM ad the fial scores of alteratives are preseted by IFVs, e.g., by IWM. ustice ad urillo [22] aalyzed the geeral properties of ituitioistic fuzzy relatios ad showed that the defiitio of these properties does ot always coicide with the defiitio of the properties of fuzzy relatios. Therefore, the specific methods which are rather of the heuristic ature were developed to compare IFVs. For this purpose, Che ad Ta [23] proposed to use the so-called score fuctio (or et membership) S( x) = µ ( x) ν ( x), where x is a IFV. Let a ad b be IFVs. It is ituitively appealig that if S ( a) > S( b) the a should be greater (better) tha b, but if S ( a) = S( b) this does ot always mea that a is equal to b. Therefore, Hog ad Choi [24] i additio to the above score fuctio itroduced the socalled accuracy fuctio H ( x) = µ ( x) + ν ( x) ad showed that the relatio betwee fuctios S ad H is similar to the relatio betwee mea ad variace i statistics. Xu [25] used fuctios S ad H to costruct order relatios betwee ay pair of ituitioistic fuzzy values as follows: If ( S( a) > S( b)), the b is smaller tha a; If ( S( a) = S( b)), the () If ( H ( a) = H ( b)), the a = b; (2) If ( H ( a) < H ( b)) the a is smaller tha b. (2)

5 Operatios o ituitioistic fuzzy values i multiple criteria decisio makig 45 asig o these relatios, Xu [25] itroduced the cocepts of ituitioistic preferece relatio, cosistet ituitioistic preferece relatio, icomplete ituitioistic preferece relatio ad acceptable ituitioistic preferece relatio. The method for IFVs compariso based o fuctios S ad H seems to be ituitively obvious ad this is its udeiable merit. Sice the approach described above is rather of a heuristic ature, there are some differet defiitios of the score fuctio proposed i literature. They were preseted i [26] as follows: S = µ ν, S 2 = µ νπ, S 3 = µ 0.5( µ + π ), S 4 = 0.5( µ + ν ) π, S5 = γµ + ( γ )( ν ), γ [0,]. I [26], these score fuctios were aalyzed ad compared, ad fially the author cocludes: The observed differeces amog the score fuctios will motivate further research o the questio of the justificatio of the five score fuctios i real-world decisio-makig problems. For routie or limited decisio-makig problems, S is suggested to be a appropriate score fuctio. The reasos provided for the superiority of this score fuctio are as follows: it is easily uderstadable, it takes little time to calculate the score value, ad it is ideal for dealig with MCD problems because of its high cosistecy. Therefore, i the followig, we shall use the score fuctio S = µ ν. 2. Limitatios of operatios o ituitioistic fuzzy values i cotext of multiple criteria decisio makig problem The problems with the above defied operatios are revealed whe they are used with order relatio (2). Hece additio () is ot a additio ivariat operatio. To show this, cosider the followig example: Example. Let = 0.5,0. 3, = 0.4,0. ad C = 0.,0.. Sice S ( ) = 0.2 ad S ( ) = 0. 3 the accordig to (2) we have <. O the other had, C = 0.55,0. 03, C = 0.46,0. 0, S( C) = 0.52, S( C) = = 0.45 ad from S( C) > S( C) we get C > C. It is worth otig that these udesirable properties curretly caot be elimiated as there are o other defiitios of IFVs sum ad orderig proposed i literature. other udesirable property of orderig (2) is that it is ot preserved uder the multiplicatio by a scalar: < does ot ecessarily imply λ < λ, λ > 0. To illustrate this, cosider the followig example. Example 2. Let = 0.5,0. 4, = 0.4,0. 3 ad λ = The S () = S () = = 0., H () = 0.9, H () = 0.7 ad from (2) we get >. Usig (3) we obtai

6 46 L. Dymova, P. Sevastjaov λ = ,0.632, λ = 0.225, , S( λ ) = , S( λ ) = Sice S( λ ) < S( λ ) we get λ < λ. I [27], with the use of the Lukasiewicz t-coorm ad t-orm, the followig expressio was iferred: λ = λµ λ( ν ), λ [0,]. (3), It is easy to prove that the use of (3) guaratees that for IFVs ad the iequality < always implies λ < λ ( λ [0,] ), but ufortuately, properties (7) ad (9) with operatio (3) do ot hold. importat problem with aggregatio operatio () is that it is ot cosistet with the aggregatio operatio o ordiary fuzzy sets (whe µ = ν ). This ca be easily see from the followig example. Example 3. Let = 0.95,0. 0 ad = 0.,0. 6, w = w 0.5. The i the 2 = framework of ordiary fuzzy sets we get the Ordiary Weighted rithmetic Mea OWM = w w µ = = 0.48 ad i the framework of µ + 2 -IFS, from () we obtai IWM = 0.78, We ca see that the resultig value of µ obtaied usig IWM is cosiderably greater tha that obtaied from OWM. I [27], usig the correspodig t-orms ad t-coorms, the followig simple expressio was iferred for IWM : IWM = i= w i µ, wiν i. (4) i i= It is easy to show that this operator is cosistet with the aggregatio operatio o ordiary fuzzy sets. For Example 3 from (4) we obtai IWM = 0.48, other problem with aggregatio operatio () is that it is ot mootoic with respect to the orderig i (2). Cosider a illustrative example: Example 4. Let = 0,, = 0.5,0. 4 ad C = 0.3,0. 2. Sice S ( ) =, S ( ) = 0., S( C) = 0. ad H ( ) =, H ( ) = 0. 9, H ( C) = 0. 5, from (2) we obtai > C >. Suppose w = w The from () we obtai 2 = IWM (, C) = 0.634,0.4472, IWM (, ) = , The score fuctios of these results are as follows: S( IWM (, C)) = , S ( IWM (, )) = We ca see that S ( IWM (, C)) > S( IWM (, )). It is importat that this problem does ot arise whe we use aggregatio (4). Summarizig, we ca say that for the solutio of MCDM problems i the ituitioistic fuzzy settig, the set of operatios (), (2), (4), (3) ad (4) should be recommeded as they do ot provide cotroversial results. Nevertheless, this set of

7 Operatios o ituitioistic fuzzy values i multiple criteria decisio makig 47 operatios caot be cosidered as iheretly cosistet as aggregatio (4) caot be directly obtaied from () ad (3). Coclusio The aim of this paper is to preset the set of operatios o ituitioistic fuzzy values which provides o-cotroversial results of the solutio of multiple criteria decisio makig problems i the ituitioistic fuzzy settig. For this purpose, the properties of commoly used operatios o ituitioistic fuzzy values have bee aalyzed ad some of their udesirable properties were revealed. The ways to approve the properties of ituitioistic fuzzy arithmetic are proposed. Fially, the set of operatios providig o-cotroversial results of solvig multiple criteria decisio makig problems i the ituitioistic fuzzy settig is proposed. The aalysis is illustrated with umerical examples throughout the paper. Refereces [] taassov K.T., Ituitioistic fuzzy sets, Fuzzy Sets ad Systems 986, 20, [2] Che S.M., Ta J.M., Hadlig multicriteria fuzzy decisio-makig problems based o vague set theory, Fuzzy Sets ad Systems 994, 67, [3] Hog D.H., Choi C.-H., Multicriteria fuzzy decisio-makig problems based o vague set theory, Fuzzy Sets ad Systems 2000, 4, [4] Li D-F., Multiattribute decisio makig models ad methods usig ituitioistic fuzzy sets, Joural of Computer ad System Scieces 2005, 70, [5] Li F., Lu., Cai L., Methods of multi-criteria fuzzy decisio makig based o vague sets, Joural of Huazhog Uiversity of Sciece ad Techology 200, 29, -3 (i Chiese). [6] Li F., Rao Y., Weighted methods of multi-criteria fuzzy decisio makig based o vague sets, Computer Sciece 200, 28, (i Chiese). [7] Li L., Yua X-H., Xia Z-Q, Multicriteria fuzzy decisio-makig methods based o ituitioistic fuzzy sets, Joural of Computer ad System Scieces 2007, 73, [8] Liu H.-W., Wag G.-J., Multi-criteria decisio-makig methods based o ituitioistic fuzzy sets, Europea Joural of Operatioal Research 2007, 79, [9] taassov K., Pasi G., Yager R., Ituitioistic fuzzy iterpretatios of multi-perso multicriteria decisio makig, Proc. of 2002 First Iteratioal IEEE Symposium Itelliget Systems 2002,, 5-9. [0] taassov K., Pasi G., Yager R., taassova V., Ituitioistic fuzzy group iterpretatios of multi-perso multi-criteria decisio makig, Proc. of the Third Coferece of the Europea Society for Fuzzy Logic ad Techology EUSFLT 2003, Zittau, 0-2 September 2003, [] Pasi G., taassov K., Melo Pito P., Yager R., taassova V., Multi-perso multi-criteria decisio makig: Ituitioistic fuzzy approach ad geeralized et model, Proc. of the 0th ISPE Iteratioal Coferece o Cocurret Egieerig dvaced Desig, Productio ad Maagemet Systems, July 2003, Madeira 2003, [2] Takeuti G., Titai S, Ituitioistic fuzzy Logic ad ituitioistic fuzzy set theory, J. Symbolic Logic 984, 49, [3] Dubois D., Gottwald S., Hajek P., Kacprzyk J., Prade H., Termiological difficulties i fuzzy set theory - The case of Ituitioistic Fuzzy Sets, Fuzzy Sets ad Systems 2005, 56,

8 48 L. Dymova, P. Sevastjaov [4] Dale Va D., Ituitioistic logic, Hadbook of Philosophical Logic, 2 d ed., 5, Kluwer, Dordrecht. [5] Motero J., Gmez D., ustice H., O the relevace of some families of fuzzy sets, Fuzzy Sets ad Systems 2007, 58, [6] Gau W.L., uehrer D.J., Vague sets, IEEE Tras. Systems Ma Cyberet 993, 23, [7] ustice H., urillo P., Vague sets are ituitioistic fuzzy sets, Fuzzy Sets ad Systems 996, 79, [8] taassov K., Ituitioistic Fuzzy Sets, Spriger Physica-Verlag, erli 999. [9] taassov K., New operatios defied over the ituitioistic fuzzy sets, Fuzzy Sets ad Systems, 994, 6, [20] Dey S.K., iswas R., Roy.R., Some operatios o ituitioistic fuzzy sets, Fuzzy Sets ad Systems 2000, 4, [2] Xu Z., Ituitioistic preferece relatios ad their applicatio i group decisio makig, Iformatio Scieces 2007, 77, [22] ustice H., urillo P., Structures o ituitioistic fuzzy relatios, Fuzzy Sets ad Systems 996, 78, [23] Che S.M., Ta J.M., Hadlig multicriteria fuzzy decisio-makig problems based o vague set theory, Fuzzy Sets ad Systems 994, 67, [24] Hog D.H., Choi C.-H., Multicriteria fuzzy decisio-makig problems based o vague set theory, Fuzzy Sets ad Systems 2000, 4, [25] Xu Z., Ituitioistic preferece relatios ad their applicatio i group decisio makig, Iformatio Scieces 2007, 77, [26] Che T.Y., comparative aalysis of score fuctios for multiple criteria decisio makig i ituitioistic fuzzy settigs, Iformatio Scieces 20, 8, [27] eliakov G., ustice H., Goswami D.P., Mukherjee U.K., Pal N.R., O averagig operators for taassov s ituitioistic fuzzy sets, Iformatio Scieces 20, 82, 6-24.