Multiple attribute decision making method based on some normal. neutrosophic Bonferroni mean operators

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1 Mauscrt Clc here to dowload Mauscrt: reaed_e.docx Clc here to vew led Refereces Multle attrbute decso ag ethod based o soe oral eutrosohc Boferro ea oerators Pede Lu ab * Hoggag L a a School of Maageet Scece ad Egeerg Shadog Uversty of Face ad Ecoocs Ja Shadog 00 Cha b School of Ecoocs ad Maageet Cvl vato Uversty of Cha Ta 0000 Cha bstract: Noral eutrosohc ubers (NNNs) are a sgfcat tool of descrbg the coleteess deteracy ad cosstecy of the decso-ag forato. I ths aer we frstly roose the defto ad the roertes of the NNNs ad the accuracy fucto the score fucto ad the oeratoal laws of the NNNs are develoed. The soe oerators are reseted cludg the oral eutrosohc Boferro ea (NNBM) oerator the oral eutrosohc weghted Boferro ea (NNWBM) oerator the oral eutrosohc geoetrc Boferro ea (NNGBM) oerator the oral eutrosohc weghted geoetrc Boferro ea (NNWGBM) oerator. We also study ther roertes ad secal cases. Further we ut forward a ultle attrbute decso ag ethod whch s based o the NNWBM ad NNWGBM oerators. Fally a llustratve exale s gve to verfy the ractcalty ad valdty of the roosed ethod. Keywords: Multle attrbute decso ag; Noral eutrosohc ubers; Noral eutrosohc Boferro ea aggregato oerator.. Itroducto s a ortat research brach of decso theory Multle attrbute decso ag (MDM) has a wde alcato ay areas. The ultle attrbute decso ag was frstly roosed ad aled to select the vestet olcy of the eterrses by Churcha et al. []. However because the fuzzess ad deteracy of the forato real decso ag are a coo heoeo uercal values are adeuate or suffcet to odel real-lfe decso robles. I sae occasos t ca be ore accurate to use the fuzzy ubers to descrbe the attrbute values fuzzy evroet. Because of the fuzzy deterate forato ost of the ultle attrbute decso ag robles Zadeh [] frstly roosed the fuzzy set (FS) ad used t to descrbe the fuzzy forato. taassov [] further reseted the tutostc fuzzy set (IFS) where he added the o-ebersh fucto to the FS. Now IFS has attracted ore ad ore attetos MDM ad ultle attrbute grou decso ag (MGDM). However due to the fact that the ebersh ad o-ebersh fucto IFS ca oly be show by the accurate ubers IFS s ot arorate to be used to solve the F-MDM robles. taassov [] Gargov ad taassov [] further roosed the terval-valued tutostc fuzzy set (IVIFS). Lu ad Zhag [] defed tragular tutostc fuzzy ubers (TIFNs). Wag [] roosed the traezodal tutostc fuzzy ubers (TrIFNs) ad the terval traezodal tutostc fuzzy ubers (ITrIFNs). But these extesos of IFS ca ust cosder the deteracy. Saradache [] roosed the eutrosohc set (NS) whch a deedet deteracy-ebersh fucto was added. I NS the truth-ebersh fucto s the sae fucto as the ebersh of IFS; lewse the falsty-ebersh s the sae fucto as the

2 o-ebersh of IFS. The deteracy-ebersh fucto s the votal dfferece betwee NS ad IFS. The three arts cludg the truth-ebersh the deteracy ebersh ad the false-ebersh are coletely deedet. Based o ths Wag et al. [] exressed the values of deteracy-ebersh truth-ebersh ad false-ebersh wth the terval ubers whch has bee defed the terval eutrosohc set (INS).O the bass of the Hag ad Eucldea dstaces Ye [0] defed soe slarty easures betwee INS. The a ethod about the ultle attrbute decso ag based o the slarty degree was roosed. I real-lfe world the oral dstrbuto s wdely aled to a lot of felds. But both the IFS ad INS caot cosder the oral dstrbuto so the researches about the oral fuzzy forato are attractg ore ad ore attetos. Yag ad Ko[] frstly defed the oral fuzzy ubers(nfns) to exress the oral dstrbuto heoea. NFNs are ore reasoable ad realstc to exress the decso-ag forato tha FS a rado evroet. Based o the NFNs ad IFS Wag ad L [] roosed the oral tutostc fuzzy ubers (NIFNs) ad defed ts corresodg oeratos the stablty factor the score fucto ad so o. However there have ot bee researches about the cobato NFNs wth NNs. Now ore ad ore researchers ay atteto to the forato aggregato oerators whch have becoe a oular research toc [-0]. Boferro [] frstly roosed the Boferro ea (BM) oerator whch ca catch the terrelatosh betwee the ut arguets BM has bee aled ay alcato doas ad attracted ore ad ore attetos fro the researchers. Yager [] roosed soe geeralzatos about the BM such as the ordered weghted averagg (OW) oerator [] ad Chouet tegral []. Yager [] ad Belaov et al. [] defed aother geeralzed for of BM. Nevertheless Zhu et al. [] roosed the geoetrc Boferro ea (GBM) whch both the BM ad geoetrc ea (GM) are cosdered. U to ow there s o research o the oral eutrosohc decso-ag robles cosderg the terrelatosh betwee the ut oral eutrosohc arguets. Therefore t s ecessary to ay ore atteto to ths ssue. Because the BM oerator ca cosder the terrelatosh betwee the attrbutes ad the NNNs have the advatages of cosderg the oral rado forato ad the eutrosohc varables whch ca hadle the colete cosstet ad deterate forato. I ths aer we exted the Boferro ea to aggregate the oral eutrosohc varables by cobg BM aggregato oerator wth NNNs. We frstly roose two aggregato oerators called the oral eutrosohc Boferro ea (NNBM) oerator ad the oral eutrosohc geoetrc Boferro ea (NNGBM) oerator for aggregatg the oral eutrosohc ubers. The we study soe roertes of the ad dscuss ther soe secal cases. For the stuatos whch the ut arguets have dfferet weght we the develo the oral eutrosohc weghted Boferro ea (NNWBM) oerator ad the oral eutrosohc weghted geoetrc Boferro ea (NNWGBM) oerator ad the we roose two rocedures for ultle attrbute decso ag uder the evroets of the NNNs based o the roosed oerators. The reader of ths aer s costructed as follows. I the ext secto we troduce soe basc cocets of the NNNs soe oeratoal laws ad the roet characterstcs of NNNs. I secto soe aggregato oerators o the bass of the oral eutrosohc ubers are roosed such as the oral eutrosohc Boferro ea (NNBM) oerator the oral eutrosohc weghted Boferro ea (NNWBM) oerator the oral eutrosohc geoetrc Boferro ea (NNGBM) oerator the oral eutrosohc weghted geoetrc Boferro ea (NNWGBM) oerator ad ther roertes are dscussed. I secto a ultle attrbute decso ag ethod o the bass of

3 the oral eutrosohc weghted Boferro ea (NNWBM) oerator ad the oral eutrosohc weghted geoetrc Boferro ea (NNWGBM) oerator. I secto a uercal exale s gve to verfy the roosed aroach ad to rove ts effectveess ad ractcalty. I Secto we coclude the aer ad gve soe rears.. Prelares. The oral fuzzy set ad oral tutostc fuzzy set Defto []. Let X be a real uber set. s deoted as ( a ). If ts ebersh fucto satsfes: xa ( ) ( x) e ( 0) () the s called a oral fuzzy uber. The set of all oral fuzzy ubers s deoted as N. Defto []. Suose X s a ordary fte o-ety set ad ( a ) N ( a ) s a oral tutostc fuzzy uber (NIFN) whe ts ebersh fucto s exressed as: xa ( ) ad ts o-ebersh fucto s exressed as: ( x) e x X () xa ( ) ( x) ( ) e x X. () where 0 ( x) 0 ( x) ad 0. Whe ad 0 the NIFN wll becoe a NFN. Coared to NFNs the NIFN adds the o-ebersh fucto whch exresses the degree of ot belogg to ( a ). Moreover ( x) ( x) ( x) shows the degree of hestace. The set of NIFNs s deoted by NIFNS.. The eutrosohc set Defto []. Let X be a uverse of dscourse wth a geerc eleet X deoted by x. eutrosohc uber X s exressed as: ( x) x ( T ( x) I ( x) F ( x)) () where T ( x) s the truth-ebersh fucto I ( x) s the deteracy-ebersh fucto ad F x s the falsty-ebersh fucto. T x I x ad F ( ) subsets of 0. x are real stadard or ostadard There s o restrcto o the su of T ( x ) I ( x) ad F ( x ) so 0 T ( x) I ( x) F ( x). Defto []. Let X be a uverse of dscourse wth a geerc eleet X deoted by x. sgle valued eutrosohc uber X s ( x) x ( T ( x) I ( x) F ( x)) () where T ( x) s the truth-ebersh fucto I ( x) s the deteracy-ebersh fucto ad F ( ) x s the falsty-ebersh fucto. For each ot x X we have T ( x) I ( x) F ( x) [0] ad 0 T ( x) I ( x) F ( x).

4 The oral eutrosohc set Defto. Suose X s a uverse of dscourse wth a geerc eleet X deoted by x ad ( a ) N oral eutrosohc uber X s exressed as: Where the truth-ebersh fucto T ( x) satsfes: a ( T ( x) I ( x) F ( )) ( x) x x () T ( x) T e xa ( ) the deteracy-ebersh fucto I ( x ) satsfes: I ( x) ( I ad the falsty-ebersh fucto F ( x ) satsfes: F ( x) ( F ) e ) e x X xa ( ) xa ( ) x X. x X. For each ot x X we have T ( x) I ( x) F ( x) [0] ad 0 T ( x) I ( x) F ( x). The set of all oral eutrosohc ubers s deoted as R. Exale. The servce lfe of the la bulb obeys the oral dstrbuto the oral fuzzy uber s N (000 0). The exerts evaluate whether the servce lfe cofors to the oral dstrbuto. t last the exerts gve the evaluato values: the degree of result rage ( 000 0) s 0.; the degree of result ot rage ( 000 0) s 0.; ad the degree of hestace s 0.. So the fal evaluato result about the servce lfe of the la bulb s 0000 ( ). Defto. Let a a T I F ad a a T I F Eucldea dstace betwee a ad a s defed as follows: d( x y) L L L L L L T I F a T I F a L L L L L L T I F T I F be two NNNs the the ccordg to the oeratoal laws defed by Wag et al. [] we ca gve the followg defto. Defto.Let a a T I F ad a a T I F oeratoal rules are defed as follows. () a a a a T T TT I I F F () be two NNNs the the () () () a a ( a a a a ) T T I I I I F F F F a a () a a T I F ( ) ( ) 0 (0) / () a a a T I F ( ) ( ) ( ) 0 ()

5 Theore. Let a a T I F ad a a T I F the we have be two NNNs ad 0 () a a a a () () a a a a () ( a a ) a a () () () a a ( ) a () () a a ( a a ) () () a a a () Defto. Let a a T I F be a NNN ad the ts score fucto s s ( a ) a ( T I F ) s ( a ) (T I F ) () ad ts accuracy fucto s h ( a ) a ( T I F ) h ( a ) ( T I F ) () Defto. Let a a T I F ad a a T I F be two NNNs the values of score fuctos of a ad a are s ( a ) s ( a ) ad s ( a ) s ( a ) ad the values of accuracy fuctos of a ad a are h ( a ) h ( a ) ad h ( a ) h ( a ) resectvely. The there wll be: () If s ( ) ( a s a) the a a ; () If s ( ) ( a s a) the If h ( ) ( a h a) the a a ; If h ( ) ( a h a) the () If s ( ) ( a s a) the a a ; () If s ( ) ( a s a) the (a) If h a ) h ( ) the a a ; ( a (b)if h a ) h ( ) the a a. ( a. Noral eutrosohc Boferro ea oerators. NNBM ad NNWBM oerators Boferro [] frstly troduced the Boferro ea (BM) whch ca rovde the aggregato betwee the ax ad oerators ad the logcal or ad ad oerators. However the Boferro ea (BM) oerator [] has ostly bee used the stuato where the ut arguets are the o-egatve real ubers. I ths secto we wll study the BM oerator uder the evroets of NNNs. Based o Defto of BM [] we defe the Boferro ea oerator of NNNs as follows: Defto 0[]. Suose 0ada a... a s a set of NNNs. The Boferro ea oerator of NNNs s defed as NNBM a a a a a (0)

6 Theore. Let a a T I F ( ) be a set of NNNs the the result aggregated fro Defto 0 wll be stll a NNN ad eve a a a a a a NNBM a a a a a T T I I F F Proof. By the oeratoal rules of the NNNs we have a a a T I F a a a T I F ad a a a a a a T T I I F F aa the a a aa a a a a ad T T I I F F a a aa a a a a ()

7 / / T T I I / F F The / a a a a a a a a a a T T I I F F whch coletes the roof of the theore. I the followg we wll dscuss soe roertes of NNBM oerator as follows. Theore. (Ideotecy). the Leta a... a be a set of NNNs f all a (... ) are eual.e. a a (... ) NNBM a a... a a Proof. sce a a (... ) the accordg to defto 0 a a NNBM a a a a a a a whch coletes the roof of theore. a Theore. (Coutatvty). Let a (... ) s ay erutato of a (... ).The

8 Proof. Let NNBM a a a NNBM a a a NNBM a a a a a NNBM a a a a a Scea a... a a s ay erutato ofa a... a a a a the we have Thus NNBM a a... a NNBM a a... a.. whch coletes the roof of the theore. Now we dscuss soe secal cases of the NNBM by assgg dfferet values to the araeters : () If 0 the 0 NNBM a a a a () whch we call t the oral eutrosohc geeralzed ea (NNGM) oerator. () If ad 0 the 0 NNBM a a a a () whch we call t the oral eutrosohc ea (NNM) oerator. () If ad 0 the 0 NNBM a a a a whch we call t the oral eutrosohc suare ea (NNSM) oerator. () If ad the NNBM a a a aa whch we call t the Noral eutrosohc terrelated suare ea (NNISM) oerator. () () The NNBM oerator ust cosders the relatosh of the aggregated arguets but gores the ortace of ther weghts. I the followg we wll defe aother Boferro ea oerator the oral eutrosohc weghted Boferro ea (NNWBM) oerator to overcoe the shortcog. Defto. Leta a... a be a set of NNNs. The weghted Boferro ea oerator of NNNs s defed as

9 NNWBM a a a wa T w w w w Where... ad w. wa s the weght vector of NNNs. a (... ) 0 w... Theore. Let a a T I F ( ) be a set of the NNNs the the result aggregated based o the Defto wll be stll a NNN ad eve wa w a wa w a wa w a wa w a w a w wa w w w T T w w w w I I F F The NNWBM oerator has the followg roertes: Theore. (Ideotecy). Leta a... a be a collecto of NNNs f all a (... ) are eual.e. a (... ) for all the NNWBM a a... a a The roof of the Theore ca be easly coleted wth the sae way as the Theore. Theore. (Coutatvty). Let a (... ) s ay erutato of a (... ).The NNWBM a a a NNWBM a a a The roof of the Theore ca be easly coleted wth the sae way as the Theore. a () ()

10 NNGBM ad NNWGBM oerators Defto. Suose 0ada a... a be a set of NNNs. The geoetrc Boferro ea oerator of the NNNs s defed as NNGBM a a... a a a Theore. Let a a T I F ( ) be a set of the NNNs the the result aggregated based o the Defto wll be stll a NNN ad eve NNGBM a a... a a a a a a a () T T I I F F Proof. By the oeratoal laws of the NNNs we have a ( a ) ( T ) I F a ( a ) ( T ) I F ad T T a a a a I I F F the a a a a a a a a T T I I F F ad ()

11 a a a a a a a a T T I I F F the a a a a a a a a T T I I F F whch coletes the roof of the theore. The geoetrc Boferro ea oerator of the NNNs has soe roertes as follows: Theore. (Ideotecy). Leta a... a be a set of the NNNs. If all a (... ) are eual.e. a a (... ) for all the NNGBM a a... a a The roof of the Theore ca be easly coleted slar to the Theore. Theore 0. (Coutatvty). Suose a (... ) s ay erutato of a (... ).The NNGBM a a a NNGBM a a a The roof of the Theore 0 ca be easly coleted wth the sae way as the Theore. Now we dscuss soe secal cases of the NNGBM by assgg dfferet values to the araeters : () If 0 the

12 NNGBM 0 a a... a a whch we call t the Noral eutrosohc geeralzed geoetrc ea (NNGGM) oerator. () If ad 0 the NNGBM 0 a a... a a () whch we call t the Noral eutrosohc geoetrc ea (NNGM) oerator. () If ad 0 the NNGBM 0 a a... a a (0) () whch we call t the Noral eutrosohc suare geoetrc ea (NNSGM) oerator. () If ad the NNGBM a a... a a a whch we call t the Noral eutrosohc terrelated suare geoetrc ea (NNISGM) oerator. Slar to the NNBM oerator the NNGBM oerator also ust cosders the terrelatosh of the ut arguets ad gores ther ow ortace. I the followg we wll exted the NNGBM to the oral eutrosohc weghted Boferro ea (NNWGBM) oerator whch ca ot oly cosders the terrelatosh but also taes the weghts to accout. Defto. Leta a... a be a set of NNNs. The weghted geoetrc Boferro ea oerator of the NNNs wll be defed as: NNWGBM T w w w w Where... ad w. w w a a... a a a s the weght vector of NNNs (... ) Theore. Let a a T I F a 0 w... () () ( ) be a set of the NNNs the the result aggregated based o the Defto wll be stll a NNN ad eve w w NNWGBM a a... a a a w w w a w a w w a a w w a a ()

13 w w w w T T I I w w F F The weghted geoetrc Boferro ea of the NNNs has soe roertes as follow. Theore. (Ideotecy). Leta a... a be a set of NNNs f all a (... ) are eual.e. a a (... ) for all the NNWPG a a... a a The roof of the Theore ca be easly coleted wth the sae way as the Theore. Theore (Coutatvty). Let a (... ) be ay erutato of a (... ). The NNGBM a a a NNGBM a a a The roof of the Theore ca be easly coleted slar to Theore.. ultle attrbute decso ag ethod o the bass of NNWBM ad NNWGBM oerator I ths secto we wll aly the oral eutrosohc weghted geoetrc Boferro ea(nnwbm) oerator (or NNWGBM) to solve the ultle attrbute decso ag robles o the bass of the NNNs. For a ultle attrbute decso ag roble suose alteratves ad C C C C ad the evaluato value of the alteratve s the set of the s the set of the attrbutes. Suose each attrbutes are deedet o the codto of the attrbute C a ( a )( T I F ) whch s reseted by the for of the NNN where T I F [0] ad T I F. The weght vector of the attrbute s w ( w w w ) whch w [0] w. The we use the oral eutrosohc weghted geoetrc Boferro ea(nnwbm) oerator (or NNWGBM ) to develo a ethod to deal wth the ultle attrbute decso ag robles as follows. Ste. Noralze the decso atrx. Because there are two tyes of attrbute.e. the beeft tye ad the cost tye we frstly covert the dfferet tyes to the sae oe. So the decso atrx of oral eutrosohc varables D ) ( a wll be coverted to the stadardzed atrx D ( a ) s

14 () a ax ( a ) ax ( a ) a For the beeft tye: a T I F For the cost tye: a a ( ) F I T a ax ( a) a Ste. Calculate the corehesve evaluato values of the alteratves based o the NNWBM oerator (or NNWGBM). (geerally we ca tae ) a NNWBM a a a w a wa w a wa w a wa wa w w a w w w T T w w w w I I F F or w w a NNWGBM a a... a a a w w w a w a w w a a w w a a w w T T w w I I () () ()

15 w w F F where. Ste. Calculate the score value of each corehesve evaluato value by euato (). Ste. Ra all the alteratves ad select the ost desrable oe(s) accordg to the defto. Ste. Ed. The uercal exale I ths secto based o NNWBM oerator (NNWGBM) a uercal exale s gve to verfy the roosed aroach. There s a coay whch s lag to vest soe oey to a dustry (cted fro [0]). There are four alteratve coaes to be chose cludg: () s a car coay; () s a food coay; () s a couter coay; () s a ars coay. There are three evaluato attrbutes cludg: () C s the rs; () C s the growth; () C s the evroet. We ca ow the attrbutes C ad C are beeft crtera ad the tye of C s cost. The weght vector of the attrbutes s (0.0.0.). The fal evaluato outcoes are exressed by the NNNs ad show table. Table The evaluato values of four alteratves wth resect to the three attrbutes C C C <(0.)(0.0.0.)> <(0.)(0.0.0.)> <(0.)(0.0.0.)> <(0.)(0.0.0.)> <(0.)(0.0.0.)> <(0.)(0.0.0.)> <(.0.)(0.0.0.)> <(0.)(0.0.0.)> <(.0.)(0.0.0.)> <(0.)(0.0.0.)> <(0.)(0.0.0.)> <(.0.)(0.0.0.)>. Procedure of decso ag ethod based o the NNWBM oerator. () Noralze the decso atrx Sce C ad C are beeft attrbutes ad C s a cost attrbute we utlze the forulas () ad () to obta the stadardzed decso atrx whch s show Table. Table. The stadardzed decso atrx C C C <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0)(0.0.0.)> <(0.0.)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0)(0.0.0.)> () Calculate the corehesve evaluato value of each alteratve by forula ().(suose ==). a a a

16 a () Calculate the score fucto by forula (). s a 0. s a 0. s a 0. s a 0. () Ra all of the alteratves ad choose the ost desrable oe by the score fucto. ccordg to the score fucto s ( a ) the rag s. Thus the best alteratve s.. Procedure of decso ag ethod based o the NNWGBM oerator. ()Noralze the decso atrx Sce C ad C are beeft attrbutes ad C s a cost crtero we use the forulas () ad () to get the stadardzed decso atrx whch ca be show Table. Table. The stadardzed decso atrx C C C <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0)(0.0.0.)> <(0.0.)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.)(0.0.0.)> <(0.0.0)(0.0.0.)> <(0.0)(0.0.0.)> () Calculate the corehesve evaluato value of each alteratve by forula (). (suose ==). a a a a () Calculate the score fucto by forula (). s a. s a. s a. s a. () Ra all of the alteratves ad choose the ost desrable oe by the score fucto. ccordg to the score fucto s ( a ) the rag s. Thus the best alteratve s.. alyss the effect of the factor I order to deostrate the fluece of the araeter o decso ag results of ths exale we use the dfferet values NNWBM or NNWGBM oerator ste to ra the alteratves. The rag results are show Table ad Table. Table Orderg of the alteratves by utlzg the dfferet NNWBM oerator Score values s ( a ) Rag

17 s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) 0. 0 s ( a ) 0. s ( a ) 0. s ( a ) 0.0 s ( a ) s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) s ( a ) 0.0 s ( a ) 0. s ( a ) 0.00 s ( a ) 0. 0 s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) s ( a ) 0. s ( a) 0. s ( a ) 0. s ( a ) 0. s ( a ) 0.0 s ( a ) 0. s ( a ) 0. s ( a ) s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) 0.0 s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) 0.0 s ( a ) 0. s ( a ) 0.0 s ( a ) 0. 0 s ( a ) 0.0 s ( a ) 0. s ( a ) 0. s ( a ) 0. Table Orderg of the alteratves by utlzg the dfferet NNWGBM oerator Score values s ( a ) Rag 0 s ( a ). s ( a ).0 s ( a ). s ( a ). 0 s ( a ). s ( a).0 s ( a ).0 s ( a ).

18 s ( a ) 0.0 s ( a ) 0.0 s ( a ) 0. s ( a ) 0. 0 s ( a ). s ( a ).0 s ( a ). s ( a ). 0 s ( a ). s ( a).0 s ( a ).0 s ( a ). 0 0 s ( a ) 0. s ( a) 0.0 s ( a ) 0. s ( a ) 0. s ( a ). s ( a ).0 s ( a ). s ( a ). 0 s ( a ) 0.0 s ( a) 0.00 s ( a ) 0. s ( a ) 0. s ( a ). s ( a). s ( a ). s ( a ). s ( a ).0 s ( a). s ( a ).0 s ( a ).0 0 s ( a ) 0. s ( a ) 0. s ( a ) 0. s ( a ) 0.0 s we ca see fro Table the orderg of the alteratves ay be dfferet for the dfferet values of NNWB oerator. But the best alteratve s the sae oe. I the Table the orderg of the alteratves also ay be dfferet for the dfferet values of. The best alteratve s or.i ractcal alcatos we geerally adot the values of the two araeters as whch are ot oly easy ad tutve but also fully cature the correlatos betwee crtera.. Coclusos The ultle attrbute decso ag ethod o the bass of oral eutrosohc varables has a wder alcato ay doas. The oral eutrosohc set (NNS) wll be ore arorate to deal wth the coleteess deteracy ad cosstecy of the decso-ag forato ad the Boferro ea (BM) oerator ca cosder the terrelatoshs betwee the ut arguets. So ths aer we roosed two aggregato oerators called the oral eutrosohc Boferro ea (NNBM) oerator ad the oral eutrosohc geoetrc Boferro ea (NNGBM) oerator for aggregatg the forato exressed by the oral eutrosohc ubers.

19 We studed soe roertes of the ad dscuss ther soe secal cases. For the stuatos whch the ut arguets have dfferet weght we the develoed the oral eutrosohc weghted Boferro ea (NNWBM) oerator ad the oral eutrosohc weghted geoetrc Boferro ea (NNWGBM) oerator o the bass of whch we roose two rocedures for ultle attrbute decso ag uder the evroets where the forato s exressed by the NNNs. Moreover we use the NNWBM oerator ad NNWGBM oerator to aggregate the evaluato forato of alteratves so the decso aers ca get the desrable alteratve accordg wth ther terest ad the ractcal eed by chagg the values of whch aes the results of the roosed ultle attrbute decso ag ethod ore flexble ad relable. I the further research the study about the alcatos of the ew decso ag ethod s ecessary ad sgfcatve because the alcatos of the oral dstrbuto s wdely dstrbuted ay doa the ucerta evroet. cowledget Ths aer s suorted by the Natoal Natural Scece Foudato of Cha (Nos. ad ) the Huates ad Socal Sceces Research Proect of Mstry of Educato of Cha (No. YJC00) Shadog Provcal Socal Scece Plag Proect (No. BGLJ0) the Natural Scece Foudato of Shadog Provce (No.ZR0FM0) ad Graduate educato ovato roects Shadog Provce (SDYY0). Refereces [] C.W. Churcha R.L. coff E.L. roff Itroducto to Oeratos Research New Yor: Wley. [] L.. Zadeh Fuzzy sets Iforato ad Cotrol ()-. [] K.T. taassov Itutostc fuzzy sets Fuzzy Sets ad Systes 0 (). [] K.T. taassov Oerators over terval-valued tutostc fuzzy sets Fuzzy Sets ad Systes ()-. [] K.T. taassov G. Gargov Iterval-valued tutostc fuzzy sets Fuzzy Sets ad Systes (). [] X. Zhag P.D. Lu Method for aggregatg tragular tutostc fuzzy forato ad ts alcato to decso ag Techologcal ad Ecooc Develoet of Ecooy (00) 0 0. [] J.Q. Wag Overvew o fuzzy ult-crtera decso-ag aroach Cotrol ad Decso (00) 0 0. [] F. Saradache ufyg feld logcs. eutrosohy: Neutrosohc robablty set ad logc erca Research Press Rehoboth. [] H. Wag F. Saradache Y.Q. Zhag et al. Iterval eutrosohc sets ad logc: Theory ad alcatos coutg Hexs Phoex Z 00. [0] J. Ye Slarty easures betwee terval eutrosohc sets ad ther alcatos ultcrtera decso-ag Joural of Itellget & Fuzzy Systes (0). [] M.S. Yag Ko. CH O a class of fuzzy c-ubers clusterg rocedures for fuzzy data Fuzzy Sets Syst () : 0. [] J.Q. Wag H.B. LMult-crtera decso-ag ethod based o aggregato oerators for tutostc lgustc fuzzy uberscotrol Decs (00)( ):. [] C. Boferro Sulle ede ultle d oteze Bolleto Mateatca Italaa (0)-0.

20 [] R. R. Yager O geeralzed Boferro ea oerators for ult-crtera aggregatoiteratoal Joural of roxate Reasog 0(00) -. [] R. R. Yager O ordered weghted averagg aggregato oerators ult crtera decso ag. IEEE Trasactos o Systes Ma ad Cyberetcs (). -0. [] G. Chouet Theory of caactes. ales de l'isttut Fourer ()-. [] R. R. Yager G. Belaov S. Jaes O geeralzed Boferro eas I: Proceedgs of the Eurofuse Worsho Preferece Modellg ad Decso alyss - Seteber 00 [] G. Belaov S. Jaes J. Mordelova T. Rucschlossova R. R. Yager Geeralzed Boferro ea oerators ult-crtera aggregato Fuzzy Sets ad Systes (00) - [] B. Zhu Z. S. Xu M.M. Xa Hestat fuzzy geoetrc Boferro eas Iforato Sceces 0()(00) -. [0] J.Q. Wag K.J. L H.Y. Zhag X.H. Che score fucto based o relatve etroy ad ts alcato tutostc oral fuzzy ultle crtera decso ag Joural of Itellget & Fuzzy Systes (0). [] J.Q. Wag K.J. L Mult-crtera decso-ag ethod based o duced tutostc oral fuzzy related aggregato oerators It J Ucerta Fuzzess Kowl Based Syst 0(0). [] J.Q. Wag K.J. L Mult-crtera decso-ag ethod based o tutostc oral fuzzy aggregato oerators Systes Eg Theory Pract (0) 0 0. [] O. Holder Uber ee Mttelwertsatz Gottge Nachrchte (). [] J. L. Jese Srurles foctos covexesetles egualtes etre les valeurs oyees cta Math 0 (0). [] J.Q. Wag P. Zhou K.J. L H.Y. Zhag X.H. Che Mult-crtera decso-ag ethod based o oral tutostc fuzzy duced geeralzed aggregato oerator TOP (0) 0-. [] Lu PD Lu ZM. Zhag X.Soe Itutostc Ucerta Lgustc Heroa ea Oerators ad Ther lcato to Grou Decso Mag led Matheatcs ad Coutato 0(0) 0. [] P.D. Lu Soe Geeralzed Deedet ggregato Oerators wth Itutostc Lgustc Nubers ad Ther lcato to Grou Decso Mag Joural of Couter ad Syste Sceces () (0) [] P.D. Lu F. J Methods for ggregatg Itutostc Ucerta Lgustc varables ad Ther lcato to Grou Decso Mag Iforato Sceces0 (0) [] P.D. Lu Y.B. Che Y.C. Chu Itutostc Ucerta Lgustc Weghted Boferro OW Oerator ad Its lcato to Multle ttrbute Decso Mag Cyberetcs ad Systes ()(0) - [0] P.D. Lu Soe Haacher aggregato oerators based o the terval-valued tutostc fuzzy ubers ad ther alcato to Grou Decso Mag IEEE Trasactos o Fuzzy systes ()(0) -