--Manuscript Draft-- application in multiple attribute group decision making

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1 tertol Jourl of Mche Lerg d Cyberetcs he eutrosophc umber geerlzed weghted power vergg opertor d ts pplcto multple ttrbute group decso mg --Muscrpt Drft-- Muscrpt umber: Full tle: Artcle ype: Abstrct: JMLC-D--00 he eutrosophc umber geerlzed weghted power vergg opertor d ts pplcto multple ttrbute group decso mg Orgl Artcle eutrosophc umber s useful tool whch s used to overcome the dffculty of descrbg determte evluto formto. he purpose of the study s to propose some power ggregto opertors bsed o eutrosophc umber whch re used to del wth multple ttrbutes group decso mg problems more effectvely. Frstly the bsc cocepts d the opertol rules d the chrcterstcs of s re troduced. he some ggregto opertors bsed o eutrosophc umbers re developed cluded the eutrosophc umber weghted power vergg WPA opertor the eutrosophc umber weghted geometrc power vergg WGPA opertor the geerlzed eutrosophc umber weghted power vergg GWPAopertor. At the sme tme the propertes of bove opertors re studed such s dempotecy mootocty boudedess d so o. he the geerlzed eutrosophc umber weghted power vergg GWPA opertor s ppled to solve multple ttrbute group decso mg problems. Afterwrds umercl exmple s gve to demostrte the effectve of the ew developed method d some comprso re coducted to verfy the fluece of dfferet prmeters or to revel the dfferece wth other method. the ed the m cocluso of ths pper s summrzed. Powered by Edtorl Mger d ProduXo Mger from Ares Systems Corporto

2 Muscrpt Clc here to dowlod Muscrpt: X Lu-ew.pdf he eutrosophc umber geerlzed weghted power vergg opertor d ts pplcto multple ttrbute group decso mg Pede Lu b X Lu School of Mgemet Scece d Egeerg Shdog Uversty of Fce d EcoomcsJ Shdog 00 Ch b School of Ecoomcs d Mgemet Cvl Avto Uversty of Ch 0000 Ch he correspodg uthor: pede.lu@gml.com Abstrct: eutrosophc umber s useful tool whch s used to overcome the dffculty of descrbg determte evluto formto. he purpose of the study s to propose some power ggregto opertors bsed o eutrosophc umber whch re used to del wth multple ttrbutes group decso mg problems more effectvely. Frstly the bsc cocepts d the opertol rules d the chrcterstcs of s re troduced. he some ggregto opertors bsed o eutrosophc umbers re developed cluded the eutrosophc umber weghted power vergg WPA opertor the eutrosophc umber weghted geometrc power vergg WGPA opertor the geerlzed eutrosophc umber weghted power vergg GWPAopertor. At the sme tme the propertes of bove opertors re studed such s dempotecy mootocty boudedess d so o. he the geerlzed eutrosophc umber weghted power vergg GWPA opertor s ppled to solve multple ttrbute group decso mg problems. Afterwrds umercl exmple s gve to demostrte the effectve of the ew developed method d some comprso re coducted to verfy the fluece of dfferet prmeters or to revel the dfferece wth other method. the ed the m cocluso of ths pper s summrzed. Keywords: multple ttrbute group decso mg; eutrosophc umbers; power ggregto opertor; eutrosophc umbers power ggregto opertor.. troducto rel decso mg sce the fuzzess d complexty of decso mg problems sometmes the people s udgmets by crsp umbers hve dffculty coveyg ther opos thoroughly. Zdeh [] ovtvely proposed the fuzzy set FS to cope wth the fuzzy formto. Sce the fuzzy set hs oly the membershp degree d hs ot the o-membershp degree Atssov [] mde mprovemet to overcome ths shortcomg d proposed the tutostc fuzzy set FS whch s mde up wth membershp degree d o-membershp degree. However FS dd ot cosder the determcy-membershp degree. o fd more precse mesuremet Smrdche [] further proposed the eutrosophc umbers s d t c be dvded to determte prt d determte prt. he eutrosophc umber s the form of b. As we c see tht s the determte prt d b represets the determte prt. Obvously bout the determte prt the fewer t s the better t s. So the worst scero s b. Coversely the best cse s. o ths dy there s the lttle progress to cope wth determte problems by eutrosophc umbers felds of scetfc d egeerg techues. herefore t s ecessry to propose ew method bsed o eutrosophc umbers s to hdle group decso mg problems. Reserchers hve pd more d more ttetos o formto ggregto opertors. he OWA opertor c weght the puts ccordg to the rg posto of them the my extesos of the OWA opertor hve bee proposed such s ucert ggregto opertors [-] the duced

3 ggregto opertors [] the lgustc ggregto opertors [-] the ucert lgustc ggregto opertors [] the fuzzy ggregto opertors [] the fuzzy lgustc ggregto opertors [] the duced lgustc ggregto opertors [] the duced ucert lgustc ggregto opertors [0] the fuzzy duced ggregto opertors [] d the tutostc fuzzy ggregto opertors []. Bsed o the opertors metoed bove Xu d Che [] proposed some tervl-vlued tutostc fuzzy rthmetc ggregto VFAA opertors such s the tervl-vlued tutostc fuzzy weghted ggregtovfwa opertor the tervl-vlued tutostc fuzzy ordered weghted ggregto VFOWA opertor d the tervl-vlued tutostc fuzzy hybrd ggregto VFHA opertor. Zho [] proposed the geerlzed tutostc fuzzy weghted ggregto GFWA opertor[] the geerlzed tutostc fuzzy ordered weghted GFOWA ggregto opertor d the geerlzed tutostc fuzzy hybrd ggregto GFHA opertor. However these opertors dd t cosder the reltoshp betwee the ttrbutes. So Yger [] developed power verge PA opertor to overcome ths shortcomg.e. t c cosder the reltoshp betwee the ttrbutes lrge mout of opertors bsed o PA hve bee developed to ggregte evluto formto order to dpt to vrous evromets [ -]. For stce power geometrc PG opertor geerlzed power verge GPA opertor lgustc geerlzed power verge LGPA opertor d so o. o ths dy there s ot the reserch o the combto the eutrosophc umbers wth power ggregto opertor. hus t s very ecessry to do the reserch bsed o eutrosophc umbers ggregto opertors. ths study we wll propose the geerlzed hybrd weghted power vergg opertor uder eutrosophc umbers evromet d the propose ew method for the multple ttrbute group decso problems whch hs two dvtges oe s tht t c cope wth the determcy of evluto formto precsely; other s tht t c te the reltoshp betwee the ttrbutes to cosderto. hs pper s wrtte s below: he secto s bout bsc cocepts the opertol rules d the chrcterstcs of s. secto some ggregto opertors bsed o eutrosophc umbers re developed such s the eutrosophc umber weghted power vergg WPA opertor the eutrosophc umber weghted geometrc power vergg WGPA opertor the geerlzed eutrosophc umber weghted power vergg GWPA opertor d the ther propertes re proved. secto we propose multple ttrbute group decso mg method bsed o the GWPA opertor d troduce the decso steps. secto umercl exmple s gve to demostrte the effectve of the ew developed method. secto the cocluso s mde.. Prelmres. Bsc cocepts of eutrosophc umbers d ther opertors he cocept of eutrosophc umber s frstly proposed by Smrdche eutrosophc probblty. t cludes two prts: determte prt d determte prt. Defto [0-]. Let [ ] be determte prt eutrosophc umber s deoted s: b where d b re both rel umbers d s the determte prt such tht 0 0 d / = udefed. Defto [0-]. Let b d b be two eutrosophc umbers the opertol

4 reltos of eutrosophc umbers re show s follows: bb b b b b bb b b b b b 0 b b b b b for 0 d b heorem. Let b be y eutrosophc umber 0 the opertol lws hve the followg chrcterstcs: 0 Proof. Obvously the euto s rght ccordg to the opertol rule expressed by. Obvously the euto 0 s rght ccordg to the opertol rule expressed by. For the left of the euto we hve b b b b Ad for the rght of the euto we hve b b b b b b b b So we c get euto s rght. For the euto we hve b b b b b So the euto s rght. For the left of the euto we hve b b b b b b b b b b b b b b

5 d the rght of the euto we hve b b b b b b b b bb b b So the euto s rght. For the euto we hve b b b b b b b b b So we c get the euto s rght. Defto []. Suppose tht b wth [ ] umber for b R where R s the set of rel umbers. o ormlze we get b b Defto []. Suppose tht b wth [ ] s y eutrosophc s y eutrosophc umber for b R where R s the set of rel umbers. We c gve the dstce betwee d s follow: [ b b ] [ b b ] d whch meets the followg crter: 0 d d 0 d d d d d 0 Defto []. Let b be set of eutrosophc umber [ ]... b R where R s the set of rel umbers the eutrosophc umber [ b b ]

6 so the possblty degree s b b P P 0 0 b b b b where P 0 P P d P 0.. he the vlue of... c be used for rg eutrosophc umber powered ggregto opertor s follows: order s follows: P herefore f the vlue of s bgger formto tht eutrosophc umbers represet s more precse. coseuece we r the eutrosophc umbers of scedg order order to get the best..... he Power Aggregto PA opertor Defto []. For rel umbers the power verge opertor s defed s where d PA sup sup mes the degree to whch supports. t stsfes the followg rules. sup sup sup 0 sup sup f - m - m. eutrosophc umber Aggregto Opertors A eutrosophc umber cludes two prts: determte prt d determte prt. hus t s good tool to express the determte d complete formto. At the sme tme the Power ggregto c te the reltoshp betwee the ttrbutes to cosderto. For ths reso we combe them together d develop some ds of eutrosophc umber ggregto opertors.. he eutrosophc umber Weghted Power Avergg Opertor Defto []. Let b be set of eutrosophc umbers the we defe PA

7 PA where sup d sup mes the support for from sup d. Obvously t stsfes the followg rules: sup sup ] [0 sup 0 sup sup m f m heorem. Let b be set of eutrosophc umbers d PA: S S. f b PA So the result of E. s stll. We use Mthemtcl ducto o to testfy the E. s follows: Proof. Whe t s cler tht the E. s rght. Suppose whe the E. s rght.e. b PA he whe we hve b PA b hus whe the E. s rght too

8 Accordgly we c get tht the E. s rght for ll. heorem. f Sup c the the power vergg opertor of s wll degrde to the rthmetc vergg opertor of s show s follows. heorem. dempotecy. Let ll b the Proof. Sce ll b we hve PA... PA b PA whch completes the proof of ths theorem. heorem. Mootocty. Let b d the Proof. Sce for ll So we c get b b b be two collectos of s whch meets PA PA. b b we c obt whch completes the proof of theorem. heorem. Boudedess. Let b... be set of s. f the Proof. Sce m bm b b b b PA PA... b m m m... m b. m PA the cse of ll we c obt b b b

9 m bm b b So we c get PA m m m PA PA Bsed o theorem we c ow So we c get heorem. Commuttvty. m m m PA PA m PA m. We ssume tht... s y permutto of... the Proof. PA PA Sce... s y permutto of... we hve the we c get So theorem s rght. Defto []. Let b b PA PA b be set of eutrosophc umbers d WPA:S S. f WPA where... s the weght vector of whch stsfes [0] d. WPA opertor s clled eutrosophc umber weghted power vergg opertor. heorem. Let b... be set of s... be the weght vector

10 of stsfyg... 0] [ d.he the result ggregted from Defto s stll eve b WPA where sup sup s the degree to whch supports. Prtculrly whe... the WPA opertor wll reduce to eutrosophc umber power vergg PA opertor: b WPA Obvously the result obted by E. s stll. he E. c be proved by Mthemtcl ducto o s follows: Proof. Obvously whe the E. s rght. Gve tht whe the E. s rght.e. b WPA he whe we hve WPA WPA b WPA b

11 So whe the E. s rght too. Accordg to d we c get tht the E. s rght for ll. heorem. f Sup c c [ 0] the the weghted power vergg opertor of s wll reduce to the weghted rthmetc vergg opertor of swaa s follows: heorem 0. dempotecy. Let ll b the Proof: Sce ll WPA... WPA WPA b b the we hve whch completes the proof of theorem 0. heorem. Mootocty. Let Proof. Sce 0 b b b b d b be two sets of s whch stsfes b b for ll the b b for ll we c get So we c get WPA WPA whch completes the proof of theorem. b heorem. Boudedess. Let b... be set of s. f b WPA WPA. b

12 he Proof. Sce for ll we c get So we c get m m m... b m WPA m b b bm b m b b WPA m m m WPA WPA Accordg to theorem So we c get WPA m m m WPA m WPA m whch complete the proof of the theorem.. he eutrosophc umber Weghted Geometrc Power Avergg Opertor Defto []. Let b be set of s d GPA:S S. he eutrosophc umber geometrc power vergg opertor s defed s: where GPA Sup the weght of s opertor s oler weghted-geometrc ggregto opertor. heorem. dempotecy. Let b be set of s. f for ll b the heorem.mootocty. Let GPA b. m. Obvously the GPA b d b be two collectos of s stsfyg b b for ll

13 the GPA GPA. heorem. Boudedess. Let b... be set of s f b d b the heorem. Commuttvty. m GPA m Let... be y permutto of... the GPA GPA Defto. Let b be set of s d GPA:S S. We defe WGPAeutrosophc umber weghted geometrc power opertor s follows: WGPA Where s the weghtg vector of the d... s the weght vector of whch stsfes [0] w [0] w. Speclly whe the WGPA opertor wll reduce to eutrosophc umber geometrc power vergg GPA opertor. heorem. Let b be set of s d he the result obted usg E. s stll d WGPA b he proof process s smlr to theorem so we c omt t here. Let the the euto turs to: u u u WGPA b heorem. dempotecy. Let b... be set of s f 0 b... the u WGPA. 0 m m

14 he geerlzed eutrosophc umber weghted power vergg opertor Defto 0. Let b... be set of s d GPA:S S f GPA 0 where... s the weght vector of stsfyg [ 0]... d 0. he GPA s clled geerlzed eutrosophc umber power opertor. Defto. Let where u / b... be set of s d GWPA:S S f / u GWPA... s the weght vector of stsfyg [ 0] d 0. he GWPA s clled geerlzed eutrosophc umber weghted power opertor. heorem. Let by E. s stll d b... be set of s d 0. he the result obted / / GWPA u u b / u he proof s smlr to the theorem t s omtted here. Obvously there re some propertes for the GWPA opertor s follows. Whe 0 GWPA / u u u b So the GWPA opertor s degrted to the WGPA opertor. Whe GWPA u / So the GWPA opertor s degrded to the WPA opertor. u u u b heorem 0. dempotecy. Let b... be set of s f 0 b... the GWPA 0 u u

15 Multple ttrbute group decso-mg method bsed o GWPA opertor ths secto we wll provde llustrtve exmple by pplyg the power opertor uder eutrosophc umbers. Suppose tht A A A... A m s set of ltertves C C C... C set of ttrbutes d D D D... s the set of decso mers. D s We use eutrosophc umber s b b R... s;... ;... m to express evluto vlue cme from the th... s decso mer for the ltertve A... m uder the ttrbute C... by usg scle from less ft to 0 more ft wth determcy. hus we c get the th eutrosophc umber decso mtrx : m m Becuse ech ttrbute hs dfferet mportce the ttrbute weght vector C s wth [ 0] d. Smlrly the weghts of decso mers represet the dfferet mportce of ech decso mer D.. s d the weghtg vector of decso mers s w w w w wth w 0 w. he method of the decso mg method volves the followg steps: Step : ormlze decso mtrx wth euto we hve b b f Step : Clculte d U wth euto we hve m [ b b ] [ b b ] d Step : Utlze the GWPA opertor we hve sup u m b GWPA

16 to obt the comprehesve vlues of ech decso mer:... m;... s. Step : Utlzed the GWPA opertor we hve b GWPA s to obt the collectve overll vlues of ech ltertves:... m. Step : Clculte the possblty degree P P we hve P P b b b b b Step : Clculte the vlues of... m for rg the orders we hve P 00 b Step : R the vlues of... m descedg order ccordg d the the best ltertve s obted.. A umercl exmple We use the geerlzed eutrosophc umber weghted power vergg opertor to del wth multple ttrbute group decso mg problems. A vestmet compy wts to choose best vestmet proect from four possble ltertves: A s cr compy; A s food compy; A s computer compy; A s rms compy. here re three ttrbutes tht the vestmet compy wts to te to cosderto: C s the rs fctor; C s the growth fctor; C s the evrometl fctor. he weghtg vector of the ttrbutes s he compy vtes three expertsd D D to evlute the four ltertves. he expert weght vector s w he th expert evlutes these four potetl ltertves terms of these three ttrbutes by the form of eutrosophc umber d the evluto vlues re show tbles -. he we c me the best ltertve for ths vestmet. b for b R ble he evluto vlues of four ltertves wth respect to the three ttrbutes by the expert D C C C A + + A A + A + ble he evluto vlues of four ltertves wth respect to the three ttrbutes by the expert D C C C A

17 A + A + A + ble he evluto vlues of four ltertves wth respect to the three ttrbutes by the expert D C C C A + A + A + A +. he evluto steps of the ew MAGDM method bsed o GWPA opertor ormlze the decso mtrx by euto we c get the ormlzed decso mtrx show s follows bles -. ble he evluto vlues of four ltertves wth respect to the three ttrbutes by experts D. D C C C A A 0. A A ble he evluto vlues of four ltertves wth respect to the three ttrbutes by experts D. D C C C A A 0.+ A 0. + A + 0. ble he evluto vlues of four ltertves wth respect to the three ttrbutes by experts D. D C C C A A A A f Clculte d U d Clculte d by euto we hve the results show tbles -. f ble Results from clcultg d f d d d

18 ble Results from clcultg d f d d d ble Results from clcultg d f d d d Clculte U by eutos d we hve U Clculte the comprehesve vlues ; of ech expert D by the euto suppose we hve:

19 Clculte the overll vlues we c get: Clculte the possblty degree P P usg euto suppose [ ] we c get P Clculte the vlues of... m usg euto we c get R the four ltertves. Sce the rg order of the four ltertves s A A A A. So the best choce s A.. he fluece of the prmeter d the determte rge for o the orderg of the ltertves Dfferet vlues of prmeter re used to express dfferet level of the metlty of decso mers becuse the bgger s more optmstc decso mers re. ths secto order to chec to whch degree dfferet prmeter flueces decso mg results dfferet vlues of re used to lyze the orderg results show tble. suppose [ ] ble Orderg of the ltertves by utlzg the dfferet GWPA opertor Rg A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A

20 A A A A From ble we c get the dfferet vlues of my led to dfferet seuece GWPA opertor. Whe 0 the order of the ltertves s A A A A d the best choce s A. Whe the order of the ltertves s A A A A d the best choce s A. Whe. 0 the order of the ltertves s A A A A d the best choce s A. Smlr to the prmeter wth the purpose of checg to whch degree dfferet prmeter flueces decso mg results the dfferet rges of re used to clculte the orderg results show tble. suppose ble Orderg of the ltertves by dfferet determte rges for GWPA opertor Rg 0 / A A A A [ 0 0.] A A A A [00.] A A A A [00.] A A A A [00.] A A A A [0] A A A A From ble we c get the dfferet vlues of my led to dfferet seuece GWPA opertor. Whe 0 the order of the ltertves s A A A A so the best choce s A. Whe [ 0 0.] the order of the ltertves s A A A A so the best choce s A. Whe [ 0 0.] [0 0.] the order of the ltertves s A A A A so the best choce s A. Whe 0 the order of the ltertves s A A A A d the best ltertve s A. order to demostrte the effectveess of the ew method ths pper we compre the orderg results of the ew method wth the orderg results of the method proposed by []. From the tble d the tble we c fd tht the two methods produce dfferet rg results. Wht s more the best choce s dfferet too. ble he orderg results produced by the old method proposed by Ye[]. Rg 0 / A A A A [00.] A A A A [00.] A A A A

21 [00.] [00.] [0] A A A A A A A A A A A A he method proposed by Ye[] s bsed o de-eutrosophcto process t does t relze the mportce of the rules of powerg operto. he ew method proposed ths pper s bsed o the eutrosophc umber geerlzed weghted power vergg opertors. Eve the vlue of s sme whe we chge the vlue of the result s dfferet. he exmple detfes the vldty of the multple ttrbute group decso mg mesure d t provdes the more geerl d flexble fetures s d re ssged dfferet vlues.. Coclusos ths pper we frstly use eutrosophc umber to express ucert or ccurte evluto formto. he we propose geerlzed eutrosophc umber weghted power vergg GWPA opertor s ew method to del wth multple ttrbute group decso mg problems whch c te the reltoshp betwee the decso rgumets d the metlty of the decso mers to cosderto. Sce the decso mers hve ther terest d the ctul eed they c ssg the dfferet vlue whch mes the result more flexble d relble. Flly we use the possblty degree rg method to choose the best choce. Afterwrd we gve umercl exmple to revel the prctcblty of the ew method. Especlly we use the dfferet vlues of d dfferet determte rges for to lyze the effectveess. he sgfcce of the pper s tht we combe eutrosophc umber wth power ggregto opertors to cope wth multple ttrbute group decso mg problems. For further reserch other ggregto opertors c be ppled to combe wth eutrosophc umber to obt the best ltertve. Acowledgmet hs pper s supported by the tol turl Scece Foudto of Ch os. d the Humtes d Socl Sceces Reserch Proect of Mstry of Educto of Ch o. YJC0 Shdog Provcl Socl Scece Plg Proect o. BGLJ0 the turl Scece Foudto of Shdog Provce o.zr0fm0 d Grdute educto ovto proects Shdog Provce SDYY0. Refereces [] L. A. Zdeh Fuzzy collectos formto d Cotrol. [] K.. Atssov tutostc fuzzy collectos Fuzzy collectos d Systems 0. [] F. Smrdche A ufyg feld logcs. eutrosophy: eutrosophc probblty setd logc Amerc Reserch Press Rehoboth. [] Z.S. Xu Ucert Multple Attrbute Decso Mg: Methods d Applctos sghu Uversty Press Beg 00. [] Z.S. Xu Group decso mg bsed o multple types of lgustc preferece reltos formto Sceces 00.

22 [] Z.S. Xu Depedet ucert ordered weghted ggregto opertors formto Fuso [] J.M. Mergó M. Csovs Decso-mg wth dstce mesures d duced ggregto opertors Computer & dustrl Egeerg 0. [] J.M. Mergó M. Csovs duced d ucert hevy OWA opertorscomputers & dustrl Egeerg 0 0. [] H. Herrer E. Herrer-Vedm J.L. Verdegy A seuetl selecto process group decso mg wth lgustc ssessmet pproch formto Sceces. [0] F. Herrer E. Herrer-Vedm Aggregto opertors for lgustc weghted formto EEE rsctos o Systems M d Cyberetcs- Prt B:Cyberetcs. [] P.D. Lu Some geerlzed depedet ggregto opertors wth tutostc lgustc umbers d ther pplcto to group decso mg J.Comput. Syst. Sc. 0. [] J.M. Mergó M. Csovs L. Mrtı ez Lgustc ggregto opertors for lgustc decso mg bsed o the Dempster Shfer theory of evdece tertol Jourl of Ucertty Fuzzess d Kowledge-Bsed Systems [] P.D. Lu X. Zhg A Approch to Group Decso Mg Bsed o -dmeso Ucert Lgustc Assessmet formto echologcl d Ecoomc Developmet of Ecoomy 0. [] Pede. Lu Fe eg A exteded ODM method for multple ttrbute group decso-mg bsed o -dmeso ucert lgustc Vrble Complexty 0 dol:0.00/cplx. [] F. Chcl F. Herrer E. Herrer-Vedm. he ordered weghted geometrc opertor: propertes d pplctos. : Proceedgs of th tertol coferece o formto processg d mgemet of ucertty owledge-bsed systems Mdrd Sp: 000 pp.. [] J.M. Mergó M. Csovs he fuzzy geerlzed OWA opertor d ts pplcto strtegc decso mg Cyberetcs d Systems [] F. Herrer E. Herrer-Vedm E.L. Mrtı ez A fuzzy lgustc methodology to del wth ublced lgustc term collectos EEE rsctos o Fuzzy Systems [] J.M. Mergó A.M. Gl-Lfuete L.G. Zhou H.Y. Che duced d lgustc geerlzed ggregto opertors d ther pplcto lgustc group decso mg Group Decso d egotto 0 dol:0.00/s0 00. [] Z.S. Xu duced ucert lgustc OWA opertors ppled to group decso mg formto Fuso 00. [0] Z.S. Xu A pproch bsed o the ucert LOWG d duced ucert LOWG opertors to group decso mg wth ucert multplctve lgustc preferece reltos Decso Support Systems 00. [] J.M. Mergó A.M. Gl-Lfuete Fuzzy duced geerlzed ggregto opertors d ts pplcto mult-perso decso mg Expert Systems wth Applctos 0. [] Z.S. Xu M.M. X duced geerlzed tutostc fuzzy opertors Kowledge-Bsed Systems 0 0. [] Z.S. Xu J. Che Approch to group decso mg bsed o tervl-vlued tutostc udgmet mtrces Systems Egeerg heory d Prctce 00. [] H. Zho Z.S. Xu M.F. S.S. Lu Geerlzed ggregto opertors for tutostc fuzzy collectos tertol Jourl of tellget Systems 00 0.

23 [] R.R. Yger O ordered weghted vergg ggregto opertors multcrter decso mg EEE rsctos o Systems M d Cyberetcs B 0. [] Pede Lu Fe egmultple crter decso mg method bsed o orml tervl-vlued tutostc fuzzy geerlzed ggregto opertorcomplexty 0 do:0.00/ cplx. [] G.W. We Ucert lgustc hybrd geometrc me opertor d ts pplcto to group decso mg uder ucert lgustc evromet tertol Jourl of Ucertty Fuzzess Kowledge-Bsed Systems 00. [] P.D. Lu Y. Su he multple-ttrbute decso mg method bsed o the FLHOWA opertor Computers d Mthemtcs wth Applctos 00. [] L.G. Zhou H.Y. Che Geerlzed ordered weghted logrthm ggregto opertors d ther pplctos to group decso mg tertol Jourl of tellget Systems [0] F. Smrdche eutrosophy: eutrosophc probblty collecto d logc Amerc Reserch Press Rehoboth USA. [] F. Smrdche troducto to eutrosophc mesure eutrosophc tegrl d eutrosophc Probblty Stech & Educto Publsher Crov Columbus 0. [] F. Smrdche troducto to eutrosophc sttstcs Stech & Educto Publshg 0. [] J. Ye Smlrty mesures betwee tervl eutrosophc collectos d ther pplctos multcrter decso-mg Jourl of tellget d Fuzzy Systems 0. [] Z. S. Xu d Q. L. D he ucert OWA opertor tertol Jourl of tellget Systems 00. [] P.D. Lu d Y. M. Wg Multple ttrbute group decso mg methods bsed o tutostc lgustc power geerlzed ggregto opertors Appled Soft Computg [] P.D. Lu d L. L. Sh he geerlzed hybrd weghted verge opertor bsed o tervl eutrosophc hestt set d ts pplcto to multple ttrbute decso mg eurl Computg d Applctos 0 -.