International Journal of Pure and Applied Sciences and Technology

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1 It.. Pure ppl. Sc. Techol., ) 0), pp. -8 Iteratoal oural of Pure ad ppled Sceces ad Techology ISSN valable ole at Research Paper Solvg Itutostc Fuzzy ssgmet Problem by usg Smlarty easures ad Score Fuctos Sa ukheree,* ad Kala Basu Departmet of aematcs, Begal College of Egeerg & Techology, SS Baeree Sara, Bdhaagar, Durgapur-7,West Begal, Ida. E-mal: dgpsm_@yahoo.co. Departmet of aematcs, Natoal Isttute of Techology, ahatma Gadh veue, Durgapur-709, West Begal, Ida. E-mal: kala007@redffmal.com *Correspodg auor, E-mal: dgpsm_@yahoo.co. Receved: ; ccepted: ) bstract: Classcal ssgmet Problem P) s a well-kow topc world-wde. I s problem c deotes e cost for assgg e ob to e perso. Ths cost s usually determstc ature. But realstc stuatos, t may ot be practcable to kow e precse values of ese costs. I such ucerta stuatos, stead of exact values of costs, f we ca evaluate e prefereces for assgg e ob to e perso e form of composte relatve degree d ) of smlarty to deal soluto maxmum degree dcates most preferable combato), we ca replace c by d e classcal P e maxmzato form ad ca solve t by ay stadard procedure to get e optmal assgmet. I s paper e cost c has bee cosdered to be tutostc fuzzy umbers IFN) deoted by c ~ whch volves e postve ad e egatve evdece for e membershp of a elemet a set. It s a more realstc descrpto a usg e crsp ad fuzzy cocept. The smlarty measures of tutostc fuzzy sets have bee used s paper for determg e composte relatve degree of smlarty d. The oto of score fucto has also bee used for valdatg e soluto obtaed by e composte relatve smlarty degree meod. Numercal examples show e effectveess of e proposed meod for hadlg e Itutostc Fuzzy ssgmet Problem IFP). aematcal formulato of IFP has bee preseted s paper.

2 It.. Pure ppl. Sc. Techol., ) 0), -8. Keywords: Itutostc Fuzzy ssgmet Problem;Itutostc Fuzzy sets, Itutostc Fuzzy Number, Smlarty measures of Itutostc Fuzzy Sets, Score Fucto, aematcal formulato.. Itroducto: I recet years, fuzzy trasportato ad fuzzy assgmet problems have receved much atteto. L ad We solved e assgmet problem w costs e form of fuzzy terval umber by a labelg algorm [5] ). I e paper by Sakawa et al [6], e auors dealt w actual problems o producto ad work force assgmet a housg materal maufacturer ad a subcotract frm ad formulated two kds of two-level programmg problems. Che [0] proved some eorems ad proposed a fuzzy assgmet model at cosders all dvduals to have same sklls. Wag [44] solved a smlar model by graph eory. Dubos ad Fortemps [6] surveys refemets of e orderg of solutos suppled by e max m formulato. Dfferet kds of fuzzy trasportato problems are solved e papers [7], [8], [9], [4], [4]). oer ssgmet problem w restrctos o tme lmts for obs ca be foud e umercal example of e papers [], [4] ). The cocept of IFS ca be vewed as a approprate/alteratve approach to defe a fuzzy set case where avalable formato s ot suffcet for e defto of a mprecse cocept by meas of a covetoal fuzzy set. I fuzzy sets e degree of acceptace s cosdered oly but IFS s characterzed by a membershp fucto ad a o-membershp fucto so at e sum of bo values s less a oe. Presetly tutostc fuzzy sets are beg studed ad used dfferet felds of scece. mog e research works o ese sets we ca meto taassov [], [], [] ), Szmdt ad Kacprzyk [8]-[4] ), Buhaescu [4], Deschrver ad Kerre [8], Stoyaova [7]. W e best of our kowledge, Burllo et al.[5] proposed defto of tutostc fuzzy umber ad studed perturbatos of tutostc fuzzy umber ad e frst propertes of e correlato betwee ese umbers. tchell [] cosdered e problem of rakg a set of tutostc fuzzy umbers to defe a fuzzy rak ad a characterstc vagueess factor for each tutostc fuzzy umber. The tutostc fuzzy sets were frst troduced by K. taassov [] whch s a geeralzato of e cocept of fuzzy set [5]. The tutostc fuzzy set has receved much atteto sce ts appearace. Gau ad Buehrer [9] troduced e cocept of vague sets. But Bustce ad Burllo [5] showed at vague sets are tutostc fuzzy sets ad has bee appled to may felds sce

3 It.. Pure ppl. Sc. Techol., ) 0), -8. ts appearace. The eory of e IF set has bee foud to be more useful to deal w vagueess ad ucertaty decso stuatos a at of e fuzzy set []-[], [8], [8]- [4]). Over e last decades, e IF set eory has bee successfully appled to solve decso makg problems [], [0], [5]-[8], [8]-[4], [0], [5], [45]-[48]). ay auors have worked o e cocept of smlarty measures of fuzzy sets [], [4], [] ). Che et al. 995) [] examed e smlarty measures of fuzzy sets, whch are based o e geometrc model, set-eoretc approach, ad matchg fucto. Smlarty measures of Itutostc fuzzy sets has bee studed ad modfed by several auors. [], [5], [], [50], [47]). Che 988) [] ad Che et al. 995) [] troduced a matchg fucto to calculate e degree of smlarty betwee fuzzy sets. Later o, s has bee exteded by Zeshu Xu, 007 [47] to deal w e smlarty measure of IFSs whch has bee successfully appled solvg a varety of ult-ttrbute Decso akg Problem [47], [48] ). Oer applcatos of smlarty measures of IFSs are e felds, such as patter recogto [7], [9], [] ), descrpto ad classfcato of complex structured obects [], [6] ) etc. ssgmet Problem P) s a well-kow topc Operatos Research ad s used worldwde for solvg dfferet types of problems egeerg ad maagemet scece. Exstg P cost matrx cotas determstc ad fxed values. But a stuato, whe decso aker D) has doubt to decde ose costs, D may duce e dea of acceptace ad reecto boud o e costs. The costs may have a target value w degree of acceptace as well as degree of reecto. Ths fact seems to take e costs as a tutostc fuzzy set: a geeralzato of fuzzy set ad s a more realstc descrpto a e cocepto of crsp ad fuzzy sese. These types of P w tutostc fuzzy parameters are yet to be explored. I s paper, we developed a meodology for solvg e Itutostc Fuzzy ssgmet Problem IFP) by usg smlarty measures of IFSs. Ths ew soluto approach for IFP caot be foud e lterature so far. We have valdated our meod w score fucto meod rough a umercal example. We arrage e paper e followg way. I Secto, we descrbe some prelmary deas o IFSs ad er smlarty measures. I Secto we develop e maematcal model of e problem. I Secto 4, we descrbe e soluto procedure ad Secto 5 we llustrate e meod by sutable examples. Secto 6 cocludes e paper.. Prelmares o Itutostc Fuzzy Sets

4 It.. Pure ppl. Sc. Techol., ) 0), Itutostc Fuzzy Set IFS) Let X= { x, x,... x } be a uverse of dscourse. fuzzy set = { < x, µ ~ x ) > / x X}, defed by Zadeh 965) [5] s characterzed by a membershp fucto µ ~ : X [0,], where µ ~ ) deotes e degree of membershp of e elemet x x to e set. taassov 986) [] troduced a geeralzed fuzzy set called IFS, show as follows: IFS X s a obect havg e form: = { < x, µ x ), ν ~ x ~ ~ ) > / x X µ ad a o-membershp fucto ν, where } ~ whch s characterzed by a membershp fucto µ ~ : X [0,], x X µ ~ x ) [0,], ν ~ : X [0,], x X ν ~ x ) [0,], w e codto µ ~ x ) ν ~ x ), for all x X For each IFS X, f π ~ x ) = µ ~ x ) ν ~ x ), e π ~ ) s called e degree of x determacy or hestacy of x to. Especally, f π ~ x ) µ ~ x ) ν ~ x ) = 0, for each x X = e e IFS s reduced to a fuzzy set... Defto: Itutostc Fuzzy Number tutostc fuzzy umber ~ s defed as follows: ) a tutostc fuzzy sub set of e real le ) ormal.e. ere s ay x 0 R such at µ ~ x0 ) = so ~ x0 ) = 0) ν ) a covex set for e membershp fucto µ ~ x).e. µ ~ λx λ) x m µ ~ x ), µ ~ x )) x, x R, λ [0,] v) a cocave set for e o-membershp fucto ν ~ x).e. ν ~ λx λ) x max ν ~ x ), ν ~ x )) x, x R, λ [0,]

5 It.. Pure ppl. Sc. Techol., ) 0), Fgure. embershp ad o-membershp fuctos of IFN.. Rakg of Itutostc Fuzzy Numbers Let a ~ = µ, ν ) be a tutostc fuzzy umber. Che ad Ta 994) troduced a score fucto S of a tutostc fuzzy value, whch s represeted as follows. The S a~ ) = µ ν be e score of a ~ where S a~ ) [, ]. The larger e score S a~ ), e greater e tutostc fuzzy value ~a.. Smlarty easures of Itutostc Fuzzy Sets Let ΦX ) be e set of all IFSs of X. Defto: Let s: Φ X ) [0,], e e degree of smlarty betwee ΦX ) ad B ΦX ) s defed as s, B), whch satsfes e followg propertes:. 0 s, B) ;. s, B) = ff = B. s, B) = s B, ); 4. s, C) s, B) ad s, C) s B, C), f B C, C Φ X ).. Smlarty measures based o matchg fucto Che 988) ad Che et al. 995) troduced a matchg fucto to calculate e degree of smlarty betwee fuzzy sets. I e followg, e matchg fucto has bee exteded to deal w e smlarty measure of IFSs. Let ΦX ) ad B ΦX ), e e degree of smlarty of ad B has bee defed based o e matchg fucto as: s, B) µ x ). µ x ) ν x ). ν x ) π x ). π x )) B B = = max µ x ) ν x ) π x )), = = Cosderg e weght w of each elemet x X, we get B µ x ) ν x ) π x )), B B B )

6 It.. Pure ppl. Sc. Techol., ) 0), s, B) w µ x ). µ x ) ν x ). ν x ) π x ). π x )) B B = = max w µ x ) ν x ) π x )), = = B w µ B x ) ν B x ) π B x )), ) If each elemet x X has e same mportace, e ) s reduced to ). The larger e value of s, B), e more s e smlarty betwee ad B. Here s, B) has all e propertes descrbed as lsted e Defto Secto... aematcal odel: Let ere be persos ad obs. Each ob must be doe by exactly oe perso ad oe perso ca do, at most, oe ob. The problem s to assg e persos to e obs so at e total cost of completg all obs becomes mmum. I s problem c deotes e cost for assgg e We troduce e 0- varable x, where ob to e perso., f e perso s assged e ob ;, =,,, x = 0, oerwse Correspodg to e ) evet of assgg perso to ob, e costrat = x =, =,,., meas at each ob must be doe by exactly oe perso, ad e costrat x =, =,, meas each perso must be assged at most oe ob. = Thus e model for crsp ssgmet Problem s gve by ODEL : z = c~ x ) = = Subect to x =, =,,., 4) = = x =, =,,, 5) x = 0 or,, =,,, 6)

7 It.. Pure ppl. Sc. Techol., ) 0), Ths cost c s usually determstc ature. But real stuatos, t may ot be practcable to kow e precse values of ese costs. I such a ucerta stuato, stead of exact values of costs, f we kow e prefereces for assgg e ob to e perso e form of composte relatve degree d ) of smlarty to deal soluto maxmum degree dcates most preferable combato), we ca replace c by d e classcal assgmet problem e maxmzato form ad ca solve t by ay stadard procedurehugara meod or by ay software) to get e optmal assgmet. I at case e model for e preferece P becomes ODEL: ax z = d x = = 7) Subect to x =, =,,., 4) = = x =, =,,, 5) x = 0 or,, =,,, 6) The cost c may be ay form represetg ucerta data, lke terval umber, tragular or trapezodal fuzzy umber or eve tutostc fuzzy umber. I s paper c has bee cosdered to be tutostc fuzzy umbers deoted byc ~ whch volves e postve ad e egatve evdece for e membershp of a elemet a set. The tutostc fuzzy set s a geeralzato of fuzzy set ad s a more realstc descrpto volvg more ucertaty compared to e crsp ad fuzzy cocept. The cost of perso dog e ob s cosdered as a tutostc fuzzy umber c~ = { µ, ν )},, =,,. Here µ deotes e degree of acceptace ad ν deotes e degree of reecto of e cost of dog e ob by e perso. Ths problem s more realstc e sese at stead of cost we have used ts degree of acceptace ad reecto. The problem s to determe e composte relatve smlarty degree d to deal soluto, deotg e prefereces of e perso for dog e c gve e form of tutostc fuzzy umber. ob ad vce versa w e costs lteratvely, f we replace c ~ by c~ = { µ, ν )} e e equato ) becomes

8 It.. Pure ppl. Sc. Techol., ) 0), z = { µ, ν )} x 8) = = Our obectve s to maxmze acceptace degree µ ad to mmze e reecto degree ν. So e obectve fucto b) ca be wrtte as axmze mze z = = = x Hece e IFP becomes a mult-obectve LPP e form ODEL : axmze mze z = = = z = = = x z = = = µ 9) x ν 0) µ 9) ν 0) x Subect to µ ν ) x 0 ) µ x ν x ) ν 0 ) x = = x =, =,,., 4) x =, =,,, 5) x = 0 or,, =,,, 6) The above mult-obectve model for IFP ca aga be wrtte as a sgle obectve fucto LPP e form ODEL 4: axmze z = µ ν ) x 4) = = Subect to e codtos 4), 5), 6), ), ) ad ) 4. Soluto Procedure The cost matrx for e gve assgmet problem has bee cosdered. But t caot be solved by e tradtoal Hugara meod, sce e elemets of s matrx are e form of Itutostc Fuzzy umbers. So, e cocept of relatve degree of smlarty measures to e postve deal soluto of Itutostc Fuzzy Sets has bee appled for solvg s

9 It.. Pure ppl. Sc. Techol., ) 0), Itutostc Fuzzy ssgmet Problem IFP) w e cost matrx as e gve matrx. The lgorm for descrbg s meodology has bee stated. Ths meod ca be used for solvg assgmet problems w ay fte umber of persos or obs, e data for e selecto alteratves beg tutostc fuzzy sets. I e followg, we shall exted e meod for solvg tutostc fuzzy decso-makg problem for solvg Itutostc Fuzzy ssgmet Problem IFP). The algorm of e meod s as follows: For a Itutostc Fuzzy ssgmet problem, let =,,,..., } be a set of { m alteratves for a row or colum e ssgmet cost) atrx, ad let C be a attrbute lke cost or tme or proft etc. ) descrbg e selecto alteratve. ssume at e characterstcs of e alteratve are represeted by e IFS as: = { C, µ C), ν C) C beg e attrbute descrbg e selecto alteratve}, / =,,,.,m where µ C) dcates e degree at e alteratve satsfes e attrbute C, ν C) dcates e degree at e alteratve does ot satsfy e attrbute C, ad µ C) [0,], ν C) [0,], µ C) ν C) lgorm : Iput: Cost matrx w e data beg IFN. Output: Proft matrx w data beg e composte relatve degree of smlarty to e deal soluto, represetg e preferece or sutablty to offer ob to e perso or at e perso s chose for performg e ob ad hece e optmal assgmet. t frst e relatve degree of smlarty for e obs w respect to each perso are evaluated by applyg e cocept of smlarty measures of IFSs for solvg Itutostc Fuzzy ult- ttrbute Decso-akg Xu, 007). The data of e frst colum of e ssgmet cost) matrx are cosdered tally. Step: Let π C) = µ C) ν C), for all.=,,,..,m. Determe e postve-deal ad egatve-deal soluto based o tutostc fuzzy umbers, defed as follows, respectvely: = { µ C), ν C) } 5) ad = { C, µ C), ν C) } 6)

10 It.. Pure ppl. Sc. Techol., ) 0), where µ C) = max { µ C) }, ν C) = m { ν C) } 7) µ C) = m { µ C) }, ν C) = max { ν C) } 8) Step: Based o e Equato ), e followg smlarty measures of IFSs have bee defed. Calculate e degree of smlarty of e postve deal IFS ad e degree of smlarty of e egatve deal IFS ad e alteratve, ad e alteratve, usg e followg equatos respectvely. The degree of smlarty of each alteratve postve deal IFS s µ, ) = max{ µ s defed as: C). µ C) ν C). ν C) π C). π C) C) C) C)), C) ν π µ ν C) π C))} ad e 9) =,,.; =,,,., Smlarly, degree of smlarty of each alteratve as: s µ, ) = max{ µ C). µ C) ν ad e egatve deal IFS C). ν C) π C). π C) C) C) C)), C) ν π µ ν =,,.; =,,,., C) π C))} s defed Step : Based o 9) ad 0) calculate e relatve smlarty measure d correspodg to e alteratve as: 0) d s, ) =, =,,,..., ) s, ) s, ) Clearly, e bgger e value of d, e more smlar s hece better s e alteratve. to e postve deal IFS ad Step 4: Repeat Step to Step for e rest of e colums of e cost matrx ad fd e relatve smlarty measure d correspodg to e alteratve e obs w respect to e persos. for ese colums.e. for Step 5: W ese relatve smlarty measure d of e obs w respect to e persos, form e matrx R where [R ] = [ p ] x, p s e relatve smlarty measure represetg how much e perso prefers e ob cosderg all e tutostc fuzzy attrbutes. We

11 It.. Pure ppl. Sc. Techol., ) 0), -8. put ε >0, a very small umber degree of smlarty) e postos of e matrx R to deote e stuato at e perso caot be assged to e ob for ese postos, f e data e orgal problem cosders at opto. Step 6: Now fd e relatve smlarty measure d for e persos w respect to each ob. Cosder e data of e frst row of e cost matrx. Repeat Step to Step for s row ad also e rest of e rows of e cost matrx ad fd e relatve smlarty measure for ese rows.e. e relatve smlarty measure of e persos w respect to e obs. Step 7: W ese relatve smlarty measure d of e persos w respect to e obs, form e matrx R where [R ] = [ q ] x, q s e relatve smlarty measure represetg how much e ob s sutable for e perso cosderg all e tutostc fuzzy attrbutes. We put ε >0, a very small umber degree of smlarty) e postos of R to deote e stuato at e perso caot be assged to e ob for ese postos, f e orgal problem cosders s case. Step 8: The form e composte matrx CompR R ) = p q x d ) x ) = whose elemets are e composte relatve degree of smlarty represetg e preferece or sutablty to offer e ob to e perso or at e perso s chose for performg e ob. Step 9: The cosderg s matrx Comp R R ) as e tal table for a assgmet problem e maxmzato form ODEL4), t s solved by Hugara meod or by ay stadard software to fd e optmal assgmet whch maxmzes e total composte relatve degree of smlarty. Step 0: Ed. Thus by usg e above algorm e more realstc Itutostc Fuzzy ssgmet Problem IFP) ca be solved. 4. Soluto Procedure by usg e Score Fucto of IF costs lgorm : Iput: Cost matrx w e data gve e form of IFN. Output: Optmal ssgmet. Step : Fd e Score fucto matrx of e gve cost matrx w data e form of IFN w e help of e formula defed Secto 5... Step: Cosderg s Score fucto matrx as e proft matrx e maxmzato form, solve t by Hugara eod or by ay stadard software to fd e optmal assgmet.

12 It.. Pure ppl. Sc. Techol., ) 0), Illustratve Examples I s secto, e applcato of e proposed approach has bee demostrated by evaluatg e optmal assgmet of proects to teams based o certa attrbutes whch are represeted by tutostc fuzzy umbers. Example : Let us cosder a Itutostc Fuzzy ssgmet Problem IFP ) havg ree maches ad ree obs where e cost matrx cotas tutostc fuzzy elemets deotg tme for completg e ob by e mache. The cost matrx s gve Table. It s requred to fd e optmal assgmet of obs to maches. Soluto: Ths problem has bee solved by usg e lgorm, sce e data of e cost matrx are IFNs. The results are show Table, Table ad Table 4. Table: Data for e Itutostc Fuzzy ssgmet Problem IFP ) wout restrctos obs ache 0.4,0.5) 0.6,0.) 0.5,0.) 0.,0.8) 0.8,0.) 0.6,0.4) 0.7,0.) 0.,0.6) 0.4,0.) W e Table 4 as e proft matrx of e assgmet problem e maxmzato form sce our obectve s to maxmze e composte relatve degree of smlarty to e PIIFS), t has bee solved by Hugara meod or by usg ay stadard software. The optmal assgmet s st ob s assged to e rd ache d ob s assged to e d ache ad rd ob s assged to e st ache. The above problem has also bee solved by applyg lgorm or by wrtg t e form of e ODEL 4 ad hece solvg t. I all ese cases e optmal assgmet was foud to be e same. The score fucto matrx for applyg lgorm s show Table 5.

13 It.. Pure ppl. Sc. Techol., ) 0), -8. Table : Values of s, ), s, ) ad values of d R for aches w respect to e obs colum wse) Values of s, ) for aches w respect to e obs colum wse) s, ) Values of s, ) for Persos w respect to e obs colum wse) s, ) atrx R cotag e values of d for aches w respect to e obs colum wse) d or p

14 It.. Pure ppl. Sc. Techol., ) 0), Table : Values of s, ), s, ), d e matrx R for obs w respect to e aches row wse) Values of s, ) for obs w respect to e aches row wse) s, ) Values of s, ) for obs w respect to e aches row wse) s, ) atrx R cotag e values of d for obs w respect to e aches row wse) d or q Table 4: Composte atrx represetg e Composte relatve degree of smlarty prefereces) of obs ad aches Comp R R ) = pq ) x = d ) x = p q

15 It.. Pure ppl. Sc. Techol., ) 0), Table 5: Scores of IF umbers of Table 5. S Coclusos I s paper, e Itutostc Fuzzy ssgmet Problems IFP) ad ts soluto procedure has bee troduced. The problem has bee formulated ad depcted by varous maematcal models. The procedure for solvg t has bee descrbed whch uses e cocept of relatve degree of smlarty to e PIIFS uder IF evromet. oreover, here all e parameters are cosdered as IFNs whch makes e problem more geeral ad realstc ature e sese at t cosders bo e degree of acceptace ad e degree of reecto. Ths meod ca be used for solvg assgmet problems w ay fte umber of persos or obs, e data for e selecto alteratves beg IFN. Illustratve example shows practcalty, effectveess ad mportace of e developed approach. Sce, o oer algorms are avalable at preset to solve IFP, e results obtaed by applyg e lgorm5.. are compared w at obtaed by applyg lgorm 5.. or by solvg ODEL The result obtaed by e proposed meod s valdated w e same result obtaed by solvg e IFP cosderg e score fucto matrx as e proft matrx. The umercal examples show at for a partcular problem, all e meods result smlar optmal assgmets. The results reveal at e proposed meodologes ca effectvely solve e IFP. type of smlarty measures of IFSs has bee used s paper. Oer types of smlarty measures may be costructed ad used for solvg IFP. Refereces: [] K. taassov, Itutostc fuzzy sets. Fuzzy Sets ad Systems, 0986), [] K. taassov, ore o tutostc fuzzy sets, Fuzzy Sets ad Systems, 989), 7-46.

16 It.. Pure ppl. Sc. Techol., ) 0), [] K. T. taassov, G. Pas ad R.R. Yager, Itutostc fuzzy terpretatos of mult-crtera mult- perso ad mult-measuremet tool decso makg, Iteratoal oural of Systems Scece 6 005) [4] T. Buhaesku, O e covexty of tutostc fuzzy sets, Iterat semar o fuctoal equatos, approxmato ad covexty, Clu-Napoca, 988),7-44. [5] H. Buste ad P. Burllo, Vague sets are tutostc fuzzy sets, Fuzzy sets ad Systems 79996) [6] Chakraborty Chada ad Chakraborty Deba, decso scheme based o OW operator for a evaluato programme: a approxmate reasog approach, ppled Soft Computg, Volume 5, Issue, December 004, Pages 45-5 [7] S Chaas, W. Kolodzeczyk ad. acha, fuzzy approach to e trasportato problem, Fuzzy Sets ad Systems, 984) [8] S. Chaas ad D. Kuchta, cocept of e optmal soluto of e trasportato problem w fuzzy cost coeffcets, Fuzzy Sets ad Systems, 8 996) [9] S. Chaas ad D. Kuchta, Fuzzy teger trasportato problem, Fuzzy Sets ad Systems, ), [0]. S. Che, O a fuzzy assgmet problem, Tamkag., 985), [] S.. Che, ew approach to hadlg fuzzy decso makg problems., IEEE Trasactos o Systems, a, ad Cyberetcs-8, 988), [] S.. Che ad.. Ta, Hadlg multcrtera fuzzy decso-makg problems based o vague set eory., Fuzzy Sets ad Systems, 67994), 6 7. [] S.. Che ad P. H. Hsao, comparso of smlarty measures of fuzzy values., Fuzzy Sets ad Systems, 7995), [4] C. H. Cheg ad Y. L, Evaluatg e best ma battle tak usg fuzzy decso eory w lgustc crtera evaluato, Europea oural of operatoal research, vol. 4)00), [5] Ch-e L, Ue-Pyg We, labelg algorm for e fuzzy assgmet problem, Fuzzy Sets ad Systems, 4004), 7 9 [6] D. Dubos ad P. Fortemps, Computg mproved optmal solutos to max m flexble costrat satsfacto problems, Europea. Operatos. Research., 8999), 95 6 [7] Deg Yog, Sh Wekag, Du Feg, Lu Q, ew smlarty measure of geeralzed fuzzy umbers ad ts applcato to patter recogto, Patter Recogto Letters, Volume 5, Issue 8, ue 004, Pages [8] G. Deschrver ad E. E. Kerre, O e posto of tutostc fuzzy set eory e framework of eores modelg mprecso., Iformato Sceces, ), [9] W. L. Gau ad D.. Buehrer, Vague Sets, IEEE Trasactos o Systems, a ad Cyberetcs ) 99), [0] D.H. Hog ad C. H. Cho, 000. ultcrtera fuzzy decso-makg problems based o vague set eory., Fuzzy Sets ad Systems, 4, 0. [] Hog Hu Dug, Chul Km, ote o smlarty measures betwee vague sets ad betwee elemets,

17 It.. Pure ppl. Sc. Techol., ) 0), Iformato Sceces, Volume 5, Issues -4, prl 999, Pages 8-96, [] Hońko Potr, Descrpto ad classfcato of complex structured obects by applyg smlarty measures, Iteratoal oural of pproxmate Reasog, Volume 49, Issue, November 008, Pages [] Huag Log-sheg ad Zhag L-pu, Soluto meod for Fuzzy ssgmet problem w Restcto of Qualfcato, Proceedgs of e Sx Iteratoal Coferece o Itellget Systems Desg ad pplcatos ISD 06 ), 006. [4] D.H. Hug ad S. Y. Hwag, ote o e value smlarty of fuzzy systems varables., Fuzzy Sets ad Systems, 66994), [5] D. F. L, ultattrbute decso makg models ad meods usg tutostc fuzzy sets., oural of Computer ad System Scece, 70005), [6] D. F. L ad Y. C. Wag, aematcal programmg approach to multattrbute decso makg uder tutostc fuzzy evromets., Iteratoal oural of Ucertaty Fuzzess ad Kowledge-Based Systems, 64)008), [7] D. F. L, Exteso of e LINP for multattrbute decso makg uder taassov s tutostc fuzzy evromet., Fuzzy Optmzato ad Decso akg, 008), 7-4. [8] D. F. L, Y. C. Wag, S. Lu ad F. Sha., Fractoal programmg meodology for multattrbute group decso-makg usg IFS., ppled Soft Computg, 9)009), 9-5. [9] L Degfeg ad Cheg Chuta, New smlarty measures to Itutostc Fuzzy Sets ad pplcatos to Patter Recogtos, Patter Recogto Letters, 00), -5 [0] H.W. Lu ad G.. Wag, ult-crtera decso-makg meods based o tutostc fuzzy sets., Europea oural of Operatoal Research, 79007), 0. [] Lu Hua-We, New smlarty measures betwee tutostc fuzzy sets ad betwee elemets, aematcal ad Computer odelg, Volume 4, Issues -005), 6-70 [] H.B. tchell, Rakg-Itutostc Fuzzy Numbers, Iteratoal oural of Ucertaty, Fuzzess ad Kowledge-Based Systems, ) 004), [] H. B. tchell, O e Degfeg Chuta smlarty measure ad ts applcato to patter recogto, Patter Recogto Letters, 46)00), [4]. OL h EL geartagh, fuzzy trasportato algorm, Fuzzy Sets ad Systems, 898), 5 4. [5]. Pakowska ad. Wygralak, Geeral IF-sets w tragular orms ad er applcatos to group decso makg, Iformato Sceces, 76006), [6]. Sakawa, I. Nshzak ad Y. Uemura, Iteractve fuzzy programmg for two-level lear ad lear fractoal producto ad assgmet problems: a case study, Europea. Oper. Res., 5 00), [7] D. Stoyaova, ore o Cartesa product over tutostc fuzzy sets, BUSEFL, 5499), 9-. [8] E. Szmdt ad. Kacprzyk, Itutostc fuzzy sets group decso makg., NIFS, ) 996), 5-.

18 It.. Pure ppl. Sc. Techol., ) 0), [9] E. Szmdt ad. Kacprzyk, Remarks o some applcatos of tutostc fuzzy sets decso makg. NIFS, )996), -. [40] E. Szmdt ad. Kacprzyk, Group decso makg va tutostc fuzzy sets..fubest'96, October 9-, Sofa, Bulgara, 996), 07-. [4] E. Szmdt ad. Kacprzyk, Itutostc fuzzy sets for more realstc group decso makg. Iteratoal Coferece o Trasto to dvaced arket Isttutos ad Ecoomes, Warsaw, ue 8-, 997), [4] E. Szmdt ad. Kacprzyk, Usg tutostc fuzzy sets group decso makg, Cotrol ad Cyberetcs, 00), [4]. Tada ad H. Ish, teger fuzzy trasportato problem, Comput. a. ppl., 996), [44] X. Wag, Fuzzy optmal assgmet problem, Fuzzy a., 987), [45] Z. S Xu ad R. R. Yager, Some geometrc aggregato operators based o tutostc fuzzy sets, Iteratoal oural of Geeral Systems, 5006), [46] Z. S. Xu, Itutostc preferece relatos ad er applcato group decso makg. Iformato Sceces, 77007), [47] Z. S Xu, Some smlarty measures of tutostc fuzzy sets ad er applcatos to multple attrbute decso makg., Fuzzy Optmzato ad Decso akg, 6007), 09-. [48] Z. S. Xu, odels for multple attrbute decso makg w tutostc fuzzy formato, Iteratoal oural of Ucertaty, Fuzzess ad Kowledge-Based Systems, 5007), [49] Z. S. Xu, Itutostc fuzzy aggregato operators., IEEE Trasactos o Fuzzy Systems, 5 007), [50] Yahog L, Davd L. Olso, Zheg Q, Smlarty measures betwee tutostc fuzzy vague) sets: comparatve aalyss, Patter Recogto Letters, 8)007), [5] Zadeh L., Fuzzy sets, Iformato ad Cotrol, 8965), 8 5. [5] Zhzhe Lag, Pegfe Sh, Smlarty measures o tutostc fuzzy sets, Patter Recogto Letters, 45)00),