Discriminating Fuzzy Preference Relations Based on Heuristic Possibilistic Clustering

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1 Mutcrtera Orderng and ankng: Parta Orders, Ambgutes and Apped Issues Jan W. Owsńsk and aner Brüggemann, Edtors Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng Dmtr A. Vattchenn Unted Insttute o Inormatcs Probems o the NAS o Bearus 6 Surganov St., 00 Mnsk, Bearus (vattchenn@ma.ru) The paper deas n a premnary way wth the probem o the seecton o a unque uzzy preerence reaton rom the set o uzzy preerence reatons. A drect agorthm o possbstc custerng s the bass o the method o uzzy preerence reaton dscrmnatng. A method o decson-makng based on a uzzy preerence reaton s descrbed and basc concepts o the heurstc method o possbstc custerng are consdered. An ustratve exampe s gven and some premnary concusons are made, Keywords: uzzy preerence reaton, aternatve, uzzy toerance, custerng, aotment. Introducton The rst subsecton o ths ntroducton provdes some premnary remarks. A method o decson-makng wth a uzzy preerence reaton s presented n the second subsecton... Fuzzy orderngs and group decson theory Fuzzy orderngs were ntroduced by Zadeh (97). Fuzzy orderngs have been apped eectvey n normaton processng, decson-makng and contro. In genera, a uzzy orderng s a transtve or negatvey transtve uzzy bnary reaton. Fuzzy preerence reatons can be consdered as a case o uzzy orderngs. The act was demonstrated by Ovchnnkov (99). A uzzy subset o most approprate aternatves corresponds to a uzzy preerence reaton and a probem o choosng o a unque aternatve s nvestgated by derent researchers. However, a set o uzzy preerence reatons can be obtaned rom severa decson makers. A uzzy subset o most approprate

2 Dmtr A. VIATTCHENIN aternatves corresponds to each uzzy preerence reaton n ths case and a unque aternatve must be seected rom the set o uzzy subsets. So, a unque uzzy preerence reaton must be obtaned and the constructon o approprate decson s a undamenta probem n group decson theory. A ew approaches can be used or the purpose. In the rst pace, the constructon o an approprate consensus measure s very useu approach, as consdered by Fedrzz et a. (999). A smar approach was deveoped by Kuzmn (98). On the other hand, a unque uzzy preerence reaton can be seected rom a set o uzzy preerence reatons whch were proposed by members o a team. The probem was outned by Nor (979). The man goa o the present paper s to consder the method o choosng a subset o most approprate weak uzzy preerence reatons rom the set o a weak uzzy preerence reatons. A drect agorthm o possbstc custerng consttutes the bass o the method. For ths purpose, a short consderaton o a method o the constructon o a set o non-domnated aternatves based on a uzzy preerence reaton s presented and basc concepts o a heurstc method o possbstc custerng are consdered. A method o data preprocessng s descrbed and a genera pan o the procedure o determnng a unque weak uzzy preerence reaton s proposed. An ustratve exampe s presented and resuts obtaned rom the proposed method are compared wth the resuts obtaned rom the Orovsky s method o decson-makng. Some concusons are ormuated... Fuzzy preerence reatons and non-domnated aternatves An eectve method o decson-makng based on a uzzy preerence reaton was proposed by Orovsky (978). A unque aternatve can be seected rom the set o aternatves usng ths method. Moreover, the method s used n the proposed approach. That s why the method must be outned n the rst pace. Let X = { x, K, x n } be a nte set o aternatves and : X X [0,] some bnary uzzy reaton on X wth μ ( x, x ), x, x X beng ts membershp uncton. The weak uzzy preerence reaton s the uzzy bnary reaton whch possesses the reexvty property μ ( x, x ) =, x X. () Let us consder the strong uzzy preerence reaton concept. The strong uzzy preerence reaton s used or the constructon o a uzzy set o non-domnated 98

3 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng aternatves. The strong uzzy preerence reaton P, whch corresponds to the weak uzzy preerence reaton, s the bnary uzzy reaton on X wth the membershp uncton μ ( x, x ), x, x X dened as P μ ( x, x ) μ ( x, x ), μ ( x, x ) > μ ( x, x ) μ P ( x, x ) =. () 0, μ ( x, x ) μ ( x, x ) The uzzy set ~ o non-domnated aternatves s dened as oows: μ ~ ( x ) = sup μ ( x, x ), x X. () x X P So, aternatves x X wth maxma vaues o the membershp uncton max μ ~ ( x ) are most approprate aternatves. x X. Outne o the approach The basc concepts o the heurstc method o possbstc custerng, based on the aotment concept are consdered rst. The probem o data preprocessng s then consdered n the second subsecton. The probem o seecton o the unque uzzy preerence reaton s dscussed n the thrd subsecton... Basc deas o the custerng method An outne or a new heurstc method o uzzy custerng was presented by Vattchenn (00), where concepts o uzzy -custer and aotment among uzzy -custers were ntroduced and a basc verson o drect uzzy custerng agorthm was descrbed. The basc verson o the agorthm s caed D-AFC(c)-agorthm and ths agorthm requres that the number c o uzzy -custers be xed. The aotment o eements o the set o cassed obects among uzzy custers can be consdered as a speca case o possbstc partton. That s why the D-AFC(c)- agorthm can be consdered as a drect agorthm o possbstc custerng. The act was demonstrated by Vattchenn (007). Let us remnd the basc concepts o the uzzy custerng method based on the concept o aotment among uzzy custers, whch was proposed by Vattchenn (00). The noton o uzzy toerance consttuted the bass or the concept o uzzy 99

4 Dmtr A. VIATTCHENIN -custer. That s why denton o uzzy toerance must be consdered n the rst pace. Let X = { x,..., x n } be the nta set o eements and T : X X [0,] some bnary uzzy reaton on X = { x,..., x n } wth μ T ( x, x ) [0,], x, x X beng ts membershp uncton. Denton.. Fuzzy toerance s the uzzy bnary ntranstve reaton whch possesses the symmetrcty property μ ( x, x ) = μ ( x, x ), x, x X, () T and the reexvty property (). T The notons o poweru uzzy toerance, eebe uzzy toerance and strct eebe uzzy toerance were consdered by Vattchenn (00), as we. In ths context the cassca uzzy toerance n the sense o Denton. was caed usua uzzy toerance and ths knd o uzzy toerance was denoted by T. The knd o the uzzy toerance mposed determnes the nature o the reveaed structure o data. So, the notons o poweru uzzy toerance, eebe uzzy toerance and strct eebe uzzy toerance must be consdered, as we. Denton.. The eebe uzzy toerance s the uzzy bnary ntranstve reaton whch possesses the symmetrcty property () and the eebe reexvty property μ ( x, x ) μ ( x, x ), x, x X. (5) T T Ths knd o uzzy toerance s denoted by T. Denton.. The strct eebe uzzy toerance s the eebe uzzy toerance wth strct nequaty n (5): μ ( x, x ) < μ ( x, x ), x, x X. (6) T T Ths knd o uzzy toerance s denoted by T 0. Denton.. The poweru uzzy toerance s the uzzy bnary ntranstve reaton whch possesses the symmetrcty property () and the poweru reexvty 00

5 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng property. The poweru reexvty property s dened as the condton o reexvty () together wth the condton μ ( x, x ) <, x, x X, x x. (7) T Ths knd o uzzy toerance s denoted by T. Notabe that uzzy toerances T and T 0 are subnorma bnary uzzy reatons the condton μ ( x, x ) <, x X (8) T s met. However, the essence o the method here consdered does not depend on the knd o uzzy toerance. That s why the method heren s descrbed or any uzzy toerance T. Let us consder the genera denton o uzzy custer, the concept o the uzzy custer's typca pont and the concept o the uzzy aotment o obects. The number c o uzzy custers can be equa to the number o obects, n. Ths s taken nto account n urther consderatons. Let X = { x,..., x n } be the nta set o obects. Let T be a uzzy toerance on X and be -eve vaue o T, (0,]. Coumns or nes o the uzzy toerance matrx are uzzy sets { A,..., A n }. Let { A,..., A n } be uzzy sets on X, whch are generated by some uzzy toerance T. ( ) = A A Denton.5. The -eve uzzy set A {( x, μ ( x )) μ ( x ) } s uzzy -custer. So, A( ) A, (0,], A { A, K, A } and μ s the membershp degree o the eement x X or some uzzy custer A ( ), (0,], {, K, n}. Vaue o s the toerance threshod o uzzy custers eements. The membershp degree o the eement ( ), (0,], {, K, n} can be dened as a A n x X or some uzzy custer 0

6 Dmtr A. VIATTCHENIN μ μ = A 0, ( x ), x A otherwse, (9) = A where an -eve A { x X μ ( x ) } o a uzzy set the uzzy custer A ( ). So, the -eve condton A = Supp A ) s met or each uzzy custer ( ( ) A o a uzzy set A ( ). A s the support o A s a crsp set and Membershp degree can be nterpreted as a degree o typcaty o an eement wth respect to a uzzy custer. The vaue o a membershp uncton o each eement o the uzzy custer n the sense o Denton.5 s the degree o smarty o the obect to some typca obect o uzzy custer. In other words, coumns or nes o a uzzy toerance T matrx are uzzy sets {,..., n n A A } on X then uzzy custers { A( ),..., A( )} are uzzy subsets o uzzy sets { A,..., A n } or some vaue, (0,]. The vaue zero or a uzzy set membershp uncton means that an eement does not beong to a uzzy set. That s why vaues o toerance threshod are consdered n the nterva ( 0,]. Denton.6. I T s a uzzy toerance on X, where X s the set o eements, n and { A( ),..., A( )} s the amy o uzzy custers or some (0,], then the pont τ, or whch e A τ e x = arg max μ, x A (0) s caed typca pont o the uzzy custer, (0,], {, K, } A ( ) n. Obvousy, a typca pont o a uzzy custer does not depend on the vaue o toerance threshod. Moreover, a uzzy custer can have severa typca ponts. That s why symbo e s the ndex o the typca pont. Denton.7. Let z ( X ) = { A( ) =, c, c n, (0,]} be a amy o uzzy custers or some vaue o toerance threshod, (0,], whch are 0

7 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng generated by some uzzy toerance T on the nta set o eements X = { x,..., x n }. I condton c = μ > 0, X () x s met or a uzzy custers A ( ), =, c, c n, then the amy s the aotment o eements o the set X = { x,..., x n } among uzzy custers { A ( ) =, c, c n} or some vaue o the toerance threshod, (0,]. It shoud be noted that severa aotments (X ) can exst or some toerance threshod, (0,]. That s why symbo z s the ndex o an aotment. The condton () requres that every obect x, =, K, n, be assgned to at east one uzzy custer A ( ), =, K, c, c n wth the membershp degree hgher than zero. The condton c n requres that the number o uzzy custers n z (X ) be at east two. Otherwse, the unque uzzy custer w contan a obects, possby wth derent postve membershp degrees. The concept o aotment s the centra pont o the method. But the next concept ntroduced shoud be pad attenton to, as we. Denton.8. Aotment I ( X ) = { A( ) =, n, (0,]} o the set o obects among n uzzy custers or some toerance threshod, (0,] s the nta aotment o the set X = x,..., x }. { n In other words, nta data are represented by a matrx o some uzzy T, then nes or coumns o the matrx are uzzy sets A X, =, n and eve uzzy sets A ( ), =, n, (0,] are uzzy custers. These uzzy custers consttute an nta aotment or some toerance threshod and they can be consdered as custerng components. Thus, the probem o uzzy custer anayss can be dened n genera as the probem o dscoverng the unque aotment (X ), resutng rom the casscaton process, whch corresponds to ether most natura aocaton o obects 0 z

8 Dmtr A. VIATTCHENIN among uzzy custers or to the researcher's opnon about casscaton. In the rst case, the number o uzzy custers c s not xed. In the second case, the researcher's opnon determnes the knd o the aotment sought and the number o uzzy custers c can be xed. I some aotment z ( X ) = { A( ) =, c, c n, (0,]} corresponds to the ormuaton o a concrete probem, then ths aotment s an adequate aotment. In partcuar, condtons and c U = A = X, () card m m ( A A ) ) = 0, A( ), A(, m, (0,] () are met or a the uzzy custers A ( ), =, c o some aotment z ( X ) = { A( ) =, c, c n, (0,]}, then the aotment s the aotment among uy separate uzzy custers. Fuzzy custers n the sense o Denton.5 can have a non-empty ntersecton, as shown n Vattchenn (007). I the ntersecton area o any par o derent uzzy custer s an empty set, then the condton () s met and uzzy custers are caed uy separate uzzy custers. Otherwse, uzzy custers are caed partcuary separate uzzy custers and w = { 0, K, n} s the maxmum number o eements n the ntersecton area o derent uzzy custers. Obvousy, or w = 0 uzzy custers are uy separate uzzy custers. The condtons () and () can be generazed or a case o partcuary separate uzzy custers. Condtons and c = ( ( ) z card A ) card( X ), A ( X ), (0,], card( ( X )) = c, () card m m ( A A ) ) w, A( ), A(, m, (0,], (5) z 0

9 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng are generazatons o condtons () and (). Obvousy, w = 0 n condtons () and (5), then condtons () and () are met. The adequate aotment z (X ) or some vaue o toerance threshod, (0,] s a amy o uzzy custers whch are eements o the nta aotment I (X ) or the vaue o, and the amy o uzzy custers shoud satsy the condtons () and (5). Hence, constructon o adequate aotments z ( X ) = { A( ) =, c, c n} or every consttutes a trva probem o combnatorcs. Severa adequate aotments can exst. Thus, the probem conssts n the seecton o the unque adequate aotment (X ) rom the set B o adequate aotments, B = { z ( X )}, whch s the cass o possbe soutons o the concrete casscaton probem and B = { z ( X )} depends on the parameters o the casscaton probem. The seecton o the unque adequate aotment (X ) rom the set B = { z ( X )} o adequate aotments must be made on the bass o evauaton o aotments. The crteron = c n F ( z ( X ), ) μ c, (6) n = = where c s the number o uzzy custers n the aotment z (X ) and n = card A ), A ( ) s the number o eements n the support o the uzzy custer F ( ( ) z X A ( ), can be used or evauaton o aotments. The crteron = c n ( z ( X ), ) ( μ ) = =, (7) can aso be used or evauaton o aotments. Both crtera were consdered by Vattchenn (007). 05

10 Dmtr A. VIATTCHENIN The maxma o crtera (0) or () correspond to the best aotments o obects among c uzzy custers. So, the casscaton probem can be characterzed ormay as determnaton o a souton (X ) satsyng z ( X ) B z ( X ) = arg max F( ( X ), ), (8) where B = { z ( X )} s the set o adequate aotments correspondng to the ormuaton o a concrete casscaton probem and crtera (6) and (7) are denoted by F( z ( X ), ). The condton (8) must be met or the some unque aotment z ( X ) B( c). Otherwse, the number c o uzzy custers n the aotment sought (X ) s suboptma. Identcaton o c, the number o uzzy custers, can be consdered as the am o casscaton. The custerng procedure s consdered, or nstance, by Vattchenn (007)... A technque o data preprocessng Let X = { x, K, x n } be a nte set o aternatves and, K, } a nte set o weak uzzy preerence reatons on X. Ater appcaton o dstance k d(, { g ) = μ ( x, x ) μ ( x, x ), k, =, K, g, (9) ( x, x ) k to the set, K, } o weak uzzy preerence reatons a matrx o par-wse { g k dssmarty coecents D g g = [ d(, )], k, =, K, g s obtaned. The dstance (9) was proposed by Kuzmn (98). The matrx D g g = [ d(, )] can be normazed as oows: k d(, ) μ I (, ) =, (0) k max d(, ) k, k k 06

11 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng or a k, =, K, g. Actuay, the matrx o normazed par-wse dssmarty k coecents I = [ μ I (, )], k, =, K, g s the matrx o uzzy ntoerance reaton on the set o weak uzzy preerence reatons {, K, g }. A matrx o k uzzy toerance T = [ μt (, )], k, =, K, g s obtaned ater appcaton o the compement operaton k k T I, μ (, ) = μ (, ), k, =, K g, () to the matrx o uzzy ntoerance I = [ μ I (, )]. The D-AFC(c)-agorthm can k be apped drecty to the matrx o uzzy toerance T = [ μ (, )]... A methodoogy o choosng a unque uzzy preerence reaton The D-AFC(c)-agorthm can be apped to sove the probem o seecton o approprate weak uzzy preerence reaton. The basc dea o the approach s that weak uzzy preerence reatons can be cassed and a typca pont o each uzzy custer can be consdered as an eement o a subset o approprate weak uzzy preerence reatons. So, a methodoogy o sovng the probem o seecton o most approprate aternatve can be descrbed as oows:. The matrx o uzzy toerance correaton must be constructed or weak uzzy preerence reatons;. The D-AFC(c)-agorthm can be apped drecty to the matrx o the uzzy toerance reaton or a gven number c o casses;. Typca ponts o uzzy custers o the receved aotment (X ) can be seected as eements o a subset o approprate weak uzzy preerence reatons;. Eements o the subset o approprate weak uzzy preerence reatons must be ordered on the bass o the evauaton and approprate weak uzzy preerence reaton must be seected.. An ustratve exampe We provde, rst, the descrpton o the data set. The resuts o processng o these data by the D-AFC(c)-agorthm are presented n the second subsecton and 07 k T

12 Dmtr A. VIATTCHENIN the resuts are compared wth the resuts obtaned rom appcaton o the Orovsky s method to weak uzzy preerence reatons... The data Let X = { x, x, x, x} be the nte set o aternatves under consderaton. Let 5 6 {,,,,, } be a set o weak uzzy preerence reatons on X. The set o reatons s presented beow. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x So, sx weak uzzy preerence reatons orm the set o obects or casscaton.

13 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng.. The resuts o casscaton The matrx o the uzzy toerance reaton was constructed or sx weak uzzy preerence reatons. By executng the D-AFC(c)-agorthm or two casses, we obtan the oowng: the rst cass s ormed by eements, and the second cass s composed o eements too. Membershp unctons o two casses are presented n Fg.. Fgure. Membershp unctons o two casses o the aotment Membershp vaues o the rst cass are represented n Fg. by, membershp vaues o the second cass are represented n Fg. by. The vaue o the membershp uncton o the uzzy custer whch corresponds to the rst cass s maxma or the rst obect and s equa one. So, the rst obect s the typca pont o the uzzy custer whch corresponds to the rst cass. The membershp vaue o the ourth obect s equa one or the uzzy custer whch corresponds to the second cass. Thus, the ourth obect s the typca pont o the uzzy custer whch corresponds to the second cass. That s why the set {, } s the subset o most approprate weak uzzy preerence reatons. Obvousy, the anayss o the set o weak uzzy preerence reatons s a smpe probem. 09

14 Dmtr A. VIATTCHENIN A moded near ndex o uzzness can be used or the evauaton o the resuts o casscaton. A near ndex o uzzness was dened by Kaumann (975) and the ndex can be moded or bnary uzzy reatons as oows ( ) = μ ( x, x ) ( x, x ), ( x, x ) X X n μ, () ( x, x ) where n = card(x ) s the number o aternatves and the crsp bnary reaton nearest to the uzzy preerence reaton The membershp uncton μ ( x, x ) o the crsp reaton can be dened as 0, μ ( x, x ) 0.5 μ ( x, x ) =, ( x, x ). (8), μ ( x, x ) > 0.5 A unque uzzy preerence reaton can be seected the condton mn ( ), =,,K, s met, where uzzy preerence reatons are typca ponts o uzzy custers obtaned rom the D-AFC(c)-agorthm. For exampe, ( ) = 0.5 and ( ) = So, the uzzy preerence reaton can be seected as a most approprate weak uzzy preerence reaton rom the set 5 6 ~ {,,,,, } o weak uzzy preerence reatons on X. A uzzy set o non-domnated aternatves can be constructed as oows: ~ =. (9) (( x,0.5), ( x,0.6), ( x,0.8), ( x,0.5)) So, the aternatve x can be seected as a most approprate aternatve rom the set X = { x, x, x, x} and the membershp degree μ ~ ( x ) = 0. 8 s a grade o the acceptance o the aternatve x. However, the method must be compared wth another eectve method. Let us consder the resuts obtaned rom the Orovsky s method. The method was apped 5 6 or each weak uzzy preerence reaton rom the set {,,,,, }. So, sx strong uzzy preerence reatons P, =, K, 6, and sx uzzy sets o 0

15 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng ~ non-domnated aternatves, =, K, 6, were constructed. Fuzzy sets o ~ on-domnated aternatves, =, K, 6 are presented n Tabe. The resut shows that the uzzy set o non-domnated aternatves s the most approprate uzzy set n terms o essenta postons. Then, obvousy, aternatve x can be seected as the most approprate aternatve or uzzy sets o non-domnated ~ aternatves ~, ~, ~, and 6. The aternatve x can be seected as the most ~ approprate aternatve or the uzzy set o non-domnated aternatves 5. Aternatves x and x can be seected as two most approprate aternatves or the ~ uzzy set o non-domnated aternatves. Tabe. Fuzzy sets o non-domnated aternatves ~ 6 Aternatves Fuzzy sets o non-domnated aternatves ~ ~ x x x x Let us consder vaues o the moded near ndex o uzzness () or a weak uzzy preerence reatons. These vaues are presented n Tabe. Tabe. Vaues o the ( ) or weak uzzy preerence reatons Fuzzy reatons Vaues o the ( ) So, the condton mn ( ), =, K, 6 s met or the reaton ~ ~ ~ 5 ~ and the resut corresponds to the resut o the evauaton o weak uzzy preerence reatons, whch are typca ponts o uzzy custers. Moreover, the moded near ndex o uzzness () s seems to be approprate ndex or the evauaton o the resut o the casscaton.

16 Dmtr A. VIATTCHENIN. Fna remarks.. Dscusson An approprate weak uzzy preerence reaton was seected on the bass o possbstc custerng o weak uzzy preerence reatons and a unque aternatve was seected rom the correspondng uzzy set o non-domnated aternatves. The resuts were vered. The resuts obtaned wth the use o the Orovsky s method show that the proposed method seems to be correct. A near ndex o uzzness was moded or purposes o evauaton o the resuts o casscaton. Obvousy, some other ndex can be proposed or ths purpose. It shoud be noted that the vaues o the moded near ndex o uzzness () do not correspond to the resuts, whch are presented n Tabe. For exampe, ~ the uzzy set ~ s equa to the uzzy set, whe ( ) < ( ). So, equa uzzy sets o non-domnated aternatves can be obtaned rom derent weak uzzy preerence reatons. Moreover, the casscaton resuts do not depend on vaues o the moded near ndex o uzzness (). For exampe, the uzzy reaton s the typca pont o the second cass and membershp uncton vaues are ordered as oows μ > μ 6 > μ 5 or eements o the second uzzy custer. However, vaues 5 6 o the ndex () are ordered as oows ( ) < ( ) < ( ) or uzzy reatons whch are eements o correspondng cass. These acts must be expaned n urther nvestgatons. In genera, the proposed method o casscaton o weak uzzy preerence reatons seems to be satsactory or seectng a subset o weak uzzy preerence reatons or the detaed anayss... Concusons In concuson t shoud be sad that the concept o uzzy custer and aotment have an epstemoogca motvaton. That s why the resuts o appcaton o the heurstc possbstc custerng method based on the aotment concept can be very we nterpreted. Moreover, the possbstc custerng method based on the aotment concept depends on the set o adequate aotments ony. That s why the custerng resuts are stabe. The resuts o appcaton o the proposed agorthms to the expermenta data set show that the agorthm s a precse and eectve numerca procedure or sovng the casscaton probem n group decson process.

17 Dscrmnatng Fuzzy Preerence eatons Based on Heurstc Possbstc Custerng Acknowedgements I woud ke to thank the anonymous reerees and or ther constructve revews and useu remarks. I am grateu to Dr Jan W. Owsnsk and Proessor Janusz Kacprzyk or ther nterest n my nvestgatons and support. I aso thank Mr Aaksandr Damaratsk or eaboratng the expermenta sotware. eerences Fedrzz M., Fedrzz M., Marques Perera. A. (999) Sot consensus and network dynamcs n group decson makng. Internatona Journa o Integent Systems,,, Kaumann A. (975) Introducton to the Theory o Fuzzy Subsets. Academc Press, New York. Kuzmn V.B. (98) A reerence approach to obtanng uzzy preerence reatons and the probem o choce. In:.. Yager, ed., Fuzzy Set and Possbty Theory. Pergamon Press, New York. Nor H. (979) A mode o uzzy team decson. Fuzzy Sets and Systems,,, 0-. Orovsky S.A. (978) Decson-makng wth a uzzy preerence reaton. Fuzzy Sets and Systems,,, Ovchnnkov S. (99) Smarty reatons, uzzy parttons, and uzzy orderngs. Fuzzy Sets and Systems, 0,, Vattchenn D.A. (00) A new heurstc agorthm o uzzy custerng. Contro and Cybernetcs,,, -0. Vattchenn D.A. (007) A drect agorthm o possbstc custerng wth parta supervson. Journa o Automaton, Mobe obotcs and Integent Systems,,, 9-8. Zadeh L.A. (97) Smarty reatons and uzzy orderngs. Inormaton Scences,,,

18 Dmtr A. VIATTCHENIN