PICTURE FUZZY CROSS-ENTROPY FOR MULTIPLE ATTRIBUTE DECISION MAKING PROBLEMS

Transcription

1 Joural of Busess Ecoomcs ad Maagemet ISSN / eissn Volume 7(4): do:0.3846/ PICTURE FUZZY CROSS-ENTROPY FOR MULTIPLE ATTRIBUTE DECISION MAKING PROBLEMS Guwu WEI School of Busess, Schua Normal Uversty, Chegdu, 600, P.R. Cha E-mal: weguwu@63.com Receved 8 Aprl 206; accepted 3 May 206 Abstract. I ths paper, we vestgate the multple attrbute decso makg problems wth pcture fuzzy formato. The advatage of pcture fuzzy set s easly reflectg the ambguous ature of subectve udgmets because the pcture fuzzy sets are sutable for capturg mprecse, ucerta, ad cosstet formato the multple attrbute decso makg aalyss. Thus, the cross etropy of pcture fuzzy sets, called pcture fuzzy cross etropy, s proposed as a exteso of the cross etropy of fuzzy sets. The, a multple attrbute decso makg method based o the proposed pcture fuzzy cross etropy s establshed whch attrbute values for alteratves are pcture fuzzy umbers. I decso makg process, we utlze the pcture fuzzy weghted cross etropy betwee the deal alteratve ad a alteratve to rak the alteratves correspodg to the cross etropy values ad to select the most desrable oe(s). Fally, a practcal example for eterprse resource plag system selecto s gve to verfy the developed approach ad to demostrate ts practcalty ad effectveess. Keywords: multple attrbute decso makg, fuzzy set, pcture fuzzy set, pcture fuzzy cross-etropy, pcture fuzzy weghted cross-etropy, eterprse resource plag, system selecto. JEL Classfcato: C6. Itroducto Etropy s very mportat ad effectve tool for measurg ucerta formato. Frstly, Zadeh (965, 968) troduced the fuzzy etropy. The startg pot for the crossetropy approach s formato theory as developed by Shao (948). Kullback ad Lebler (95) proposed the cross-etropy dstace measure betwee two probablty dstrbutos. Later, L (99) proposed a modfed cross-etropy measure. Shag ad Jag (997) proposed a fuzzy cross-etropy measure ad a symmetrc dscrmato formato measure betwee fuzzy sets. Vlachos ad Sergads (2007) developed the tutostc fuzzy cross-etropy based o the De Luca-Term o-probablstc etropy (Deluca, Feru 972). Zhag ad Jag (2008) defed the cross-etropy betwee vague sets. Accordg to the cross-etropy of vague sets, Ye (2009a) has vestgated the fault dagoss problem of turbe. Ye (2009b) has appled the tutostc fuzzy cross- Copyrght 206 Vlus Gedmas Techcal Uversty (VGTU) Press

2 492 G. We. Pcture fuzzy cross-etropy for multple attrbute decso makg problems etropy to multcrtera fuzzy decso-makg problems. Ye (20) proposed a tervalvalued tutostc fuzzy cross-etropy for multple attrbute decso makg problems o the bass of the vague cross-etropy. Xa ad Xu (202) proposed some cross-etropy ad etropy formulas for tutostc fuzzy sets ad appled them to group decsomakg. For terval-valued tutostc fuzzy sets, Zhag ad Jag (200) proposed the etropy ad cross-etropy cocepts ad dscussed the coectos amog some mportat formato measures. Xu ad Xa (202) troduced the cocepts of etropy ad cross-etropy for hestat fuzzy formato, ad dscuss ther desrable propertes. Recetly, Cuog (203) proposed pcture fuzzy set (PFS) ad vestgated the some basc operatos ad propertes of PFS. The pcture fuzzy set s characterzed by three fuctos expressg the degree of membershp, the degree of eutral membershp ad the degree of o-membershp. The oly costrat s that the sum of the three degrees must ot exceed. Bascally, PFS based models ca be appled to stuatos requrg huma opos volvg more aswers of types: yes, absta, o, refusal, whch ca t be accurately expressed the tradtoal FS ad IFS. Utl ow, some progress has bee made the research of the PFS theory. Sgh (204) vestgated the correlato coeffcets for pcture fuzzy set ad apply the correlato coeffcet to clusterg aalyss wth pcture fuzzy formato. So (205) ad Thog ad So (205) troduced several ovel fuzzy clusterg algorthms o the bass of pcture fuzzy sets ad applcatos to tme seres forecastg ad weather forecastg. Thog (205) developed a ovel hybrd model betwee pcture fuzzy clusterg ad tutostc fuzzy recommeder systems for medcal dagoss ad applcato to health care support systems. Due to the advatage of pcture fuzzy set s easly reflectg the ambguous ature of subectve udgmets because the pcture fuzzy sets are sutable for capturg mprecse, ucerta, ad cosstet formato actual multple attrbute decso makg aalyss as metoed above, t s ecessary to develop some cross-etropy measures for pcture fuzzy set. I order to do so, the remader of ths paper s set out as follows. I the ext secto, we troduce some basc cocepts related to tutostc fuzzy set ad pcture fuzzy sets. I Secto 2, we shall propose the pcture fuzzy cross-etropy ad pcture fuzzy weghted cross-etropy. I Secto 3, based o pcture fuzzy weghted cross-etropy, we shall preset the model for multple attrbute decso makg problems wth pcture fuzzy formato. I Secto 4, we shall preset a umercal example for eterprse resource plag (ERP) system selecto wth pcture fuzzy formato order to llustrate the method proposed ths paper. The last secto cocludes the paper wth some remarks.. Prelmares I the followg, we troduce some basc cocepts related to tutostc fuzzy sets. Defto (Ataassov 986, 989). A IFS A X s gve by: = {, ma, A Î } where ma : X [ 0,] ad : [ 0,] A X, where, A A A x x x x X, () 0 m x x, xîx. The

3 umber m A ( x ) ad Joural of Busess Ecoomcs ad Maagemet, 206, 7(4): A x represets, respectvely, the membershp degree ad omembershp degree of the elemet x to the set A. The tutostc fuzzy sets have receved more ad more atteto sce ts appearace. Kosareva ad Krylovas (203) establshed the types of tutostc fuzzy umbers ad the expoet values of the geeralzed weghted averagg operator havg the least error probabltes cosderg alteratves rakg. Razav Haagha et al. (203) defed a complex proportoal assessmet method for group decso makg a terval-valued tutostc fuzzy evromet. Krohlg et al. (203) also exteded the tutostc fuzzy TODIM to mult-crtera decso makg. Che (204) proposed the terval-valued tutostc fuzzy QUALIFLEX method wth a lkelhood-based comparso approach for multple crtera decso aalyss. Lourezutt ad Krohlg (203) developed the TODIM method a tutostc fuzzy ad radom evromet. Wu et al. (203) researched o the AHP wth terval-valued tutostc fuzzy sets ad ts applcato mult-crtera decso makg problems. Zavadskas et al. (204) proposed the exteso of weghted aggregated sum product assessmet wth tervalvalued tutostc fuzzy umbers. Wag ad Lu (204) developed some hestat fuzzy geometrc operators for multple attrbute group decso makg. Razav Haagha et al. (205) evolved a lear programmg techque for MAGDM problems wth terval valued tutostc fuzzy formato. Wu (205) proposed a SD-IITFOWA operator ad TOPSIS based approach for MAGDM problems wth tutostc trapezodal fuzzy umbers. Abdullah ad Nab (206) developed a ew preferece scale MCDM method based o terval-valued tutostc fuzzy sets ad the aalytc herarchy process. Although, tutostc fuzzy set theory (Ataassov 986, 989) has bee successfully appled dfferet areas, but there are stuatos real lfe whch ca t be represeted by tutostc fuzzy sets. Pcture fuzzy sets are exteso of tutostc fuzzy sets. Pcture fuzzy set (Cuog 203) based models may be adequate stuatos whe we face huma opos volvg more aswers of types: yes, absta, o, refusal. It ca be cosdered as a powerful tool represet the ucerta formato the process of patters recogto ad cluster aalyss. Defto 2 (Cuog 203). A pcture fuzzy set (PFS) A o the uverse X s a obect of the form: x sat- x X, x x x x could be called the degree of refusal membershp of x A. Cuog (203) also defed some operatos as follows. {, A, A, A } where ma Î[ 0,] ha ( x ) Î[ 0,] s called the degree of eutral membershp of A ad A ( x ) Î [ 0, ] s called the degree of egatve membershp of A, ad m A ( x ), h A ( x ), A sfy the followg codto: 0 m A( x) h A( x) A( x ), xîx. The for Î π A = ( m A h A A) A= x m x h x x xîx, (2) x s called the degree of postve membershp of A, Defto 3 (Cuog 203). Gve two PFEs represeted by A ad B o uverse X, the cluso, uo, tersecto ad complemet operatos are defed as follows: 493

4 494 G. We. Pcture fuzzy cross-etropy for multple attrbute decso makg problems () A B, f ma( x) mb( x ), ha( x) hb( x) ad A( x) B( x ), xîx, (2),max ( A B) ( A B) ( A B) (3),m ( A B) ( A B) ( A B) (4) A= x, ( x), h ( x), m ( x) xîx. = {( m m h h ) Î } = {( m m h h ) Î } {( A A A ) } A B x x x x x x x x X A B x x x x x x x x X For coveece, we call = (,, ) a m h a pcture fuzzy umber (PFN), where: a a a [ 0, ], [ 0, ], [ 0,] m Î h Î Î, m h. a a a 2. Cross-etropy betwee pcture fuzzy sets a a a I ths secto, we shall develop the cross-etropy ad dscrmato formato measures betwee two PFSs based o the exteso of the cocept of cross-etropy betwee two fuzzy sets. To do ths, we frstly troduce the cocepts of cross-etropy ad symmetrc dscrmato formato measures betwee two fuzzy sets whch were proposed by Shag ad Jag (997). Defto 4 (Shag, Jag 997). Assume that a= ( a( x), a( x2),, a( x )) ad b= ( b( x), b( x2),, b( x )) are two fuzzy sets the uverse of dscourse ( x, x2,, x ). The fuzzy cross-etropy of a from b s defed as follows: a a ( a, b ) = x x h a ( x) l ( a( x), (3) = ( a ( x ) b) ( a b) x x x whch dcates the degree of dscrmato of a from b. However, h( ab, ) s ot symmetrc wth respect to ts argumets. Shag ad Jag (997) proposed a symmetrc dscrmato formato measure: ( ab, ) = ( ab, ) ( ba, ) Moreover, there are I ( ab, ) 0ad ( ab, ) = 0 I h h. (4) I f ad oly f a = b. The, the cross-etropy ad symmetrc dscrmato formato measures betwee two fuzzy sets are exteded to these measures betwee PFSs. I order to do so, let us cosder two group of pcture fuzzy umbers a = ( m,, ) a h a a ad b= ( m,, ) b h b b, =, 2,,. Thus based o Eq. (3), the amout of formato for dscrmato of ma from m b ( =, 2,, ) ca be gve by: ma ma m C ( a, b, m ) =m a l ( ma, (5) ( m a mb ) ( m a m b ) 2 2,,

5 Joural of Busess Ecoomcs ad Maagemet, 206, 7(4): Therefore, the expected formato based o the sgle degree of postve membershp for dscrmato of a agast b s expressed by: C m ma m a a, b = m a l ( ma = ( m m ) m m a b a b. (6) Smlarly, cosderg the degree of eutral membershp ad the degree of egatve membershp, we have the followg amouts of formato: C C h ha h a a, b = h a l ( ha = ( h h ) h h a b a b va a a, b = a l ( a = ( ) a b a b ; (7). (8) Hece, a ovel pcture fuzzy cross-etropy measure betwee a ad b s obtaed as the sum of the three amouts: ma m a C ( a, b ) = m a l ( m a = ( m a mb ) ( m a mb ) ha h a h a l ( h a = ( h a hb ) ( h a hb ) = va a a l ( a ( ) a b a b, (9) whch also dcates dscrmato degree of a from b. Accordg to Shao s (948) equalty, oe ca easly prove that C ( ab, ) 0ad C ( ab, ) = 0f ad oly f a = b. The, C ( ab, ) s ot symmetrc. So t should be modfed to a symmetrc dscrmato formato measure for PFSs as: ( ab, ) = ( ab, ) ( ba, ) The larger the dfferece betwee a ad b s, the larger ( ab, ) D C C. (0) D s. 495

6 G. We. Pcture fuzzy cross-etropy for multple attrbute decso makg problems If we cosder the weghts of a ad b, a pcture fuzzy weghted cross-etropy measure betwee a ad b s proposed as follows: C m m a b = w m ( m ) a a w(, ) a l a l = m a mb m a m b = = ha h a w h a l ( h a ( h a hb ) ( h a hb ) va a w a l ( a ( ) a b a b, () where w= ( w, w,, w ) T s the weght vector of ab, ( =, 2,, ), wth w Î[ 0,] =, 2,,, 2 = w =,, whch also dcates dscrmato degree of a from b. Accordg to Shao s equalty (5), oe ca easly prove that C w ( ab, ) 0 ad C (, ) w ab = 0 f ad oly f a = b. The, C w ( ab, ) s ot symmetrc. So t should be modfed to a symmetrc dscrmato formato measure for PFSs as w( ab, ) = w( ab, ) w( ba, ) The larger the dfferece betwee a ad b s, the larger ( ab, ) D C C. (2) D s. 3. Models for multple attrbute decso makg based o cross-etropy wth pcture fuzzy formato Based o cross-etropy wth pcture fuzzy formato, ths secto, we shall propose the correspodg model for multple attrbute decso makg wth pcture fuzzy formato. Let A= { A, A2,, Am } be a dscrete set of alteratves, ad G = { G, G2,, G} be the set of attrbutes, w= ( w, w2,, w ) s the weghtg G =, 2,, w Î 0, w =. Suppose that vector of the attrbute ( ), where [ ] ( r ) (,, ) w, = R = = m h s the pcture fuzzy decso matrx, where m m m dcates the degree of postve membershp that the alteratve A satsfes the attrbute G gve by the decso maker, h dcates the degree of eutral membershp that the alteratve A does t satsfy the attrbute G, dcates the degree that the alteratve A does t satsfy the attrbute G gve by the decso maker, m Î[ 0,], h Î[ 0,], Î[ 0,] m h, π = ( m h ) =, 2,, m, =, 2,,., 496

7 Joural of Busess Ecoomcs ad Maagemet, 206, 7(4): I multple attrbute decso-makg evromets, the cocept of deal pot has bee used to help detfy the best alteratve the decso set. Although the deal alteratve does ot exst real world, t does provde a useful theoretcal costruct agast whch to evaluate alteratves. Hece, we ca defe a deal attrbute value the deal alteratve A., where Let r = ( m, h, )( =, 2,, ) { } { } { }, { } { } { } max m m m = max m, h = m h, = m The we call: π = m h = m h (, 2,, ) =, =, 2,,. (3) A r r r, (4) the relatve pcture fuzzy deal alteratve. I the followg, we apply the cross-etropy to the MADM problems wth hestat fuzzy formato. Step. Defe the alteratves A ad A : ((,, ),( 2, 2, 2),(,, )) A = m h m h m h, =, 2,, m; (5) ((,, ), ( 2, 2, 2), (,, ) ) { } { } { } = A = m h m h m h ; (6) m = max m, h = m h, = m,, 2,,. (7) Step 2. Calculate t pcture fuzzy weghted cross-etropy betwee alteratves A ad A : m (, m C A A ) = w m l ( m = m m m m h h w h l ( h = ( h h ) ( h h ) w l ( = ( ) ( ) =, 2,, m. (8) ad select the best oe(s) accordace wth C( A, A ) ( =, 2,, m). The smaller the value of C( A, A ) s, the better the alteratve A s. I ths case, the alteratve A s close to the deal alteratve A. Through the weghted cross-etropy C( A, A ) ( =, 2,, m) betwee each alteratve ad the deal alteratve, the rakg order of all alteratves ca be determed ad the best oe ca be easly detfed as well. Step 3. Rak all the alteratves A ( =, 2,, m) 497

8 G. We. Pcture fuzzy cross-etropy for multple attrbute decso makg problems 4. Numercal example Thus, ths secto we shall preset a umercal example for potetal evaluato of emergg techology commercalzato wth pcture fuzzy formato order to llustrate the method proposed ths paper. Let us suppose there s a problem to deal wth the potetal evaluato of emergg techology commercalzato whch s classcal multple attrbute decso makg problems. There s a pael wth fve possble emergg techology eterprses ( =,2,3,4,5) A to select. The experts selects sx attrbute to evaluate the fve possble emergg techology eterprses: ) G s the techcal advacemet; 2) G 2 s the potetal market ad market rsk; 3) G 3 s the dustralzato frastructure; 4) G 4 s the developmet of scece ad techology; 5) G 5 s the facal codtos; 6) G 6 s the employmet creato. I order to avod fluece each other, the decso makers are requred to evaluate the fve possble emergg techology eterprses ( =,2,3,4,5) R s preseted Table, where r ( =,2,3,4,5, =,2,3,4,5,6) The weght vector of x ( =, 2,, 6) s: ( 0.2, 0.25, 0.09, 0.6, 0.20, 0.8) T are the form of PFNs. w=. A uder the above sx attrbutes ad the decso matrx = ( r ) 5 6 Table. The pcture fuzzy decso matrx A A 2 A 3 A 4 A 5 x (0.53,0.33,0.09) (0.73,0.2,0.08) (0.9,0.03,0.02) (0.85,0.09,0.05) (0.90,0.05,0.02) x 2 (0.89,0.08,0.03) (0.3,0.64,0.2) (0.07,0.09,0.05) (0.74,0.6,0.0) (0.68,0.08,0.2) x 3 (0.42,0.35,0.8) (0.03,0.82,0.3) (0.04,0.85,0.0) (0.02,0.89,0.05) (0.05,0.87,0.06) x 4 (0.08,0.89,0.02) (0.73,0.5,0.08) (0.68,0.26,0.06) (0.08,0.84,0.06) (0.3,0.75,0.09) x 5 (0.33,0.5,0.2) (0.52,0.3,0.6) (0.5,0.76,0.07) (0.6,0.7,0.05) (0.5,0.73,0.08) x 6 (0.7,0.53,0.3) (0.5,0.24,0.2) (0.3,0.39,0.25) (0.8,0.5,0.09) (0.9,0.03,0.05) To get the most desrable emergg techology eterprses, the followg steps are volved: Step. Based o the Table, we deote the fve possble emergg techology eter- A =,2,3,4,5 by: prses A { = (0.53,0.33,0.09),(0.89,0.08,0.03),(0.42,0.35,0.8) (0.08,0.89,0.02),(0.33,0.5,0.2),(0.7,0.53,0.3) ; { A2 = (0.73,0.2,0.08),(0.3,0.64,0.2),(0.03,0.82,0.3) (0.73,0.5,0.08),(0.52,0.3,0.6),(0.5,0.24,0.2) ; { A3 = (0.9,0.03,0.02),(0.07,0.09,0.05),(0.04,0.85,0.0) (0.68,0.26,0.06),(0.5,0.76,0.07),(0.3,0.39,0.25) ; } } } 498

9 { Joural of Busess Ecoomcs ad Maagemet, 206, 7(4): A4 = (0.85,0.09,0.05),(0.74,0.6,0.0),(0.02,0.89,0.05) (0.08,0.84,0.06),(0.6,0.7,0.05),(0.8,0.5,0.09) ; { A5 = (0.90,0.05,0.02),(0.68,0.08,0.2),(0.05,0.87,0.06) (0.3,0.75,0.09),(0.5,0.73,0.08),(0.9,0.03,0.05) Step 2. Based o the Table ad Eqs (5) (7), we ca get the RPFIS A : A { = (0.9,0.03,0.02),(0.89,0.08,0.03),(0.42,0.35,0.05) (0.73,0.5,0.02),(0.52,0.3,0.05),(0.9,0.03,0.05) Step 3. Calculate the cross-etropy (, w ) RPFIS A by usg Eq. (8): ( ) ( 2 ) ( 3 ) ( 4, ) = 0.67, ( 5, w ) = } } }.. C A A betwee A ( =,2,3,4,5) C A, A = 0.202, C A, A = 0.92, C A, A = 0.225, w w w C A A C A A w Step 4. Rak the emergg techology eterprses A ( =,2,3,4,5) the cross-etropy C ( A, A )(,2,3,4,5) : w = A5 A4 A2 A A3. Thus, the most desrable emergg techology eterprse s A 5. Coclusos ad the accordace wth The proposed method dffers from prevous approaches for fuzzy multple attrbute decso makg ot oly due to the fact that the proposed method use the pcture fuzzy set theory, but also due to the cosderato of the degree of eutral membershp besdes degree of postve membershp ad degree of egatve membershp the evaluato of the alteratve wth respect to attrbute, whch makes t have more feasble ad practcal tha other tradtoal decso makg methods real decso makg problems. Therefore, ts advatage s easly reflectg the ambguous ature of subectve udgmets because the pcture fuzzy sets are sutable for capturg mprecse, ucerta, ad cosstet formato the multple attrbute decso makg aalyss. I ths paper, we vestgate the multple attrbute decso makg problems wth pcture fuzzy formato. The advatage of pcture fuzzy set s easly reflectg the ambguous ature of subectve udgmets because the pcture fuzzy sets are sutable for capturg mprecse, ucerta, ad cosstet formato the multple attrbute decso makg aalyss. Thus, the cross etropy of pcture fuzzy sets, called pcture fuzzy cross etropy, s proposed as a exteso of the cross etropy of fuzzy sets. The, a multple attrbute decso makg method based o the proposed pcture fuzzy cross etropy s establshed whch attrbute values for alteratves are pcture fuzzy umbers. I decso makg process, we utlze the pcture fuzzy weghted cross etropy betwee 499

10 G. We. Pcture fuzzy cross-etropy for multple attrbute decso makg problems the deal alteratve ad a alteratve to rak the alteratves correspodg to the cross etropy values ad to select the most desrable oe(s). I the future, we shall vestgate pcture fuzzy multple attrbute group decso makg problems ad apply the pcture fuzzy cross-etropy to solve practcal applcatos other areas such as expert system, formato fuso system, ad medcal dagoses. Ackowledgmets The work was supported by the Natoal Natural Scece Foudato of Cha uder Grat No ad ad the Humates ad Socal Sceces Foudato of Mstry of Educato of the People s Republc of Cha (No.4YJCZH082, 5XJA630006) ad the costructo pla of scetfc research ovato team for colleges ad uverstes Schua Provce (5TD0004). Refereces Abdullah, L.; Nab, L A ew preferece scale MCDM method based o terval-valued tutostc fuzzy sets ad the aalytc herarchy process, Soft Computg 20(2): Ataassov, K.986. Itutostc fuzzy sets, Fuzzy Sets ad Systems 20: Ataassov, K More o tutostc fuzzy sets, Fuzzy Sets ad Systems 33: Che, T. Y Iterval-valued tutostc fuzzy QUALIFLEX method wth a lkelhoodbased comparso approach for multple crtera decso aalyss, Iformato Sceces 26: Cuog, B Pcture fuzzy sets-frst results, part, Semar Neuro-Fuzzy Systems wth Applcatos. Isttute of Mathematcs, Hao. Deluca, A.; Feru, S A defto of o-probablstc etropy the settg of fuzzy sets theory, Iformato ad Cotrol 20: Kosareva, N.; Krylovas, A Comparso of accuracy rakg alteratves performg geeralzed fuzzy average fuctos, Techologcal ad Ecoomc Developmet of Ecoomy 9(): Krohlg, R. A.; Pacheco, A. G. C.; Svero, A. L. T IF-TODIM: a tutostc fuzzy TODIM to mult-crtera decso makg, Kowledge-Based Systems 53: Kullback, S.; Lebler, R. A. 95. O formato ad suffcecy, Aals of the Isttute of Statstcal Mathematcs 4: L, J. 99. Dvergece measures based o Shao etropy, IEEE Trasactos o Iformato Theory 37: Lourezutt, R.; Krohlg, R. A A study of TODIM a tutostc fuzzy ad radom evromet, Expert Systems wth Applcatos 40(6): Razav Haagha, S. H.; Hashem, S. S.; Zavadskas, E. K A complex proportoal assessmet method for group decso makg a terval-valued tutostc fuzzy evromet, Techologcal ad Ecoomc Developmet of Ecoomy 9():

11 Joural of Busess Ecoomcs ad Maagemet, 206, 7(4): Razav Haagha, S. H.; Mahdra, H. A.; Hashem, S. A.; Zavadskas, E. K Evolvg a lear programmg techque for MAGDM problems wth terval valued tutostc fuzzy formato, Expert Systems wth Applcatos 42(23): Shag, X. G.; Jag, W. S A ote o fuzzy formato measures, Patter Recogto Letters 8: Shao, C. E A mathematcal theory of commucato, Bell System Techcal Joural 27: Sgh, P Correlato coeffcets for pcture fuzzy sets, Joural of Itellget & Fuzzy Systems 27: So, L. H DPFCM: a ovel dstrbuted pcture fuzzy clusterg method o pcture fuzzy sets, Expert System wth Applcatos 2: Thog, N. T HIFCF: a effectve hybrd model betwee pcture fuzzy clusterg ad tutostc fuzzy recommeder systems for medcal dagoss, Expert Systems wth Applcatos 42(7): Thog, P. H.; So, L. H A ew approach to mult-varables fuzzy forecastg usg pcture fuzzy clusterg ad pcture fuzzy rules terpolato method, 6th Iteratoal Coferece o Kowledge ad Systems Egeerg (KSE 204), 9 0 October 204, Hao, Vetam, Vlachos, I. K.; Sergads, G. D Itutostc fuzzy formato-applcatos to patter recogto, Patter Recogto Letters 28: Wag, W. Z.; Lu, X. W Some hestat fuzzy geometrc operators ad ther applcato to multple attrbute group decso makg, Techologcal ad Ecoomc Developmet of Ecoomy 20(3): Wu, J A SD-IITFOWA operator ad TOPSIS based approach for MAGDM problems wth tutostc trapezodal fuzzy umbers, Techologcal ad Ecoomc Developmet of Ecoomy 2(): Wu, J.; Huag, H. B.; Cao, Q. W Research o AHP wth terval-valued tutostc fuzzy sets ad ts applcato mult-crtera decso makg problems, Appled Mathematcal Modellg 37(24): Xa, M. M.; Xu, Z. S Etropy/cross etropy-based group decso makg uder tutostc fuzzy evromet, Iformato Fuso 3: Xu, Z. S.; Xa, M. M Hestat fuzzy etropy ad cross-etropy ad ther use multattrbute decso-makg, Iteratoal Joural of Itellget Systems 27: Ye, J. 2009a. Fault dagoss of turbe based o fuzzy cross etropy of vague sets, Expert Systems wth Applcatos 36: Ye, J. 2009b. Multcrtera fuzzy decso-makg method based o the tutostc fuzzy crossetropy, Y. C. Tag, J. Lawry, V. N. Huyh (Eds.). Proceedgs Iteratoal Coferece o Itellget Huma-Mache Systems ad Cyberetcs, : IEEE Computer Socety, Ye, J. 20. Fuzzy cross etropy of terval-valued tutostc fuzzy sets ad ts optmal decso-makg method based o the weghts of alteratves, Expert Systems wth Applcatos 38: Zadeh, L. A Fuzzy sets, Iformato Cotrol 8:

12 G. We. Pcture fuzzy cross-etropy for multple attrbute decso makg problems Zadeh, L. A Probablty measures of fuzzy evets, Joural of Mathematcal Aalyss ad Applcatos 23: Zavadskas, E. K.; Atuchevcee, J.; Razav Haagha, S. H.; Hashem, S. S Exteso of weghted aggregated sum product assessmet wth terval-valued tutostc fuzzy umbers (WASPAS-IVIF), Appled Soft Computg 24: Zhag, Q. S.; Jag, S. Y A ote o formato etropy measures for vague sets, Iformato Sceces 78: Zhag, Q. S.; Jag, S. Y Relatoshps betwee etropy ad smlarty measure of tervalvalued tutostc fuzzy sets, Iteratoal Joural of Itellget Systems 25: Guwu WEI has a MSc ad a PhD degree appled mathematcs from SouthWest Petroleum Uversty, Busess Admstrato from School of Ecoomcs ad Maagemet at SouthWest Jaotog Uversty, Cha, respectvely. From May 200 to Aprl 202, he was a Postdoctoral Researcher wth the School of Ecoomcs ad Maagemet, Tsghua Uversty, Beg, Cha. He s a Professor the School of Busess at Schua Normal Uversty. He has publshed more tha 90 papers ourals, books ad coferece proceedgs cludg ourals such as Omega, Decso Support Systems, Expert Systems wth Applcatos, Appled Soft Computg, Kowledge ad Iformato Systems, Computers & Idustral Egeerg, Kowledge-based Systems, Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systems, Iteratoal Joural of Computatoal Itellgece Systems ad Iformato: A Iteratoal Iterdscplary Joural. He has publshed book. He has partcpated several scetfc commttees ad serves as a revewer a wde rage of ourals cludg Computers & Idustral Egeerg, Iteratoal Joural of Iformato Techology ad Decso Makg, Kowledge-based Systems, Iformato Sceces, Iteratoal Joural of Computatoal Itellgece Systems ad Europea Joural of Operatoal Research. He s curretly terested Aggregato Operators, Decso Makg ad Computg wth Words. 502