Bipolar Neutrosophic Projection Based Models for Multi-attribute Decision Making Problems

Transcription

1 Bpolar Neutrosophc Proecton Based Models for Mult-attrbute Decson Makng Probles urapat Praank* Partha Prat Dey** Bbhas C. Gr *** * Departent of Matheatcs Nandalal Ghosh B.. College Panpur P.O.-Narayanpur Dstrct North 4 Parganas Pn code-7436 West Bengal nda ** Departent of Matheatcs Jadavpur Unversty Kolkata West Bengal nda *** Departent of Matheatcs Jadavpur Unversty Kolkata West Bengal nda Eals: surapat.ath@gal.co parsur.fu@gal.co 3bcgr.u.ath@gal.co Abstract: Bpolar neutrosophc sets are the etenson of neutrosophc sets and are based on the dea of postve and negatve preferences of nforaton. Proecton easure s a useful apparatus for odelng real lfe decson akng probles. n the paper we have defned proecton bdrectonal proecton and hybrd proecton easures between bpolar neutrosophc sets and the proposed easures are then appled to ult-attrbute decson akng probles. he ratngs of perforance values of the alternatves wth respect to the attrbutes are epressed by bpolar neutrosophc values. We calculate proecton bdrectonal proecton and hybrd proecton easures between each alternatve and deal alternatve wth bpolar neutrosophc nforaton and then all the alternatves are ranked to dentfy best opton. nally a nuercal eaple s provded to deonstrate the applcablty and effectveness of the developed ethod. Coparson analyss wth other estng ethods s also provded. Keywords: Bpolar neutrosophc sets proecton easure bdrectonal proecton easure hybrd proecton easure ult-attrbute decson akng.. ntroducton or descrbng and anagng ndeternate and nconsstent nforaton arandache [] ntroduced neutrosophc sets whch has three ndependent coponents naely truth ebershp degree ndeternacy ebershp degree and falsty ebershp degree where le n] [. Later Wang et al. [] proposed sngle valued neutrosophc sets VNs to deal real decson akng probles where le n [0 ]. n 994 Zhang [3] [4] grounded the noton of bpolar fuy sets by etendng the concept of fuy sets. he value of ebershp degree of an eleent of bpolar fuy set belongs to [- ]. Wth reference to a bpolar fuy set the ebershp degree ero of an eleent reflects that the eleent s rrelevant to the correspondng property the ebershp degree belongs to 0] of an eleent reflects that the eleent soewhat satsfes the property and the ebershp degree belongs to [ 0 of an eleent reflects that the eleent soewhat satsfes the plct counter-property.

2 Del et al. [5] ntroduced the concept of bpolar neutrosophc sets BNs by cobnng the concept of bpolar fuy sets and neutrosophc sets. Wth reference to a bpolar neutrosophc set the postve ebershp degrees represent respectvely the truth ebershp ndeternate ebershp and falsty ebershp of an eleent X correspondng to the bpolar neutrosophc set and the negatve ebershp degree denotes the truth ebershp ndeternate ebershp and false ebershp of an eleent X to soe plct counter-property correspondng to the bpolar neutrosophc set. Proecton easure s a useful decson akng devce as t takes nto account the dstance as well as the ncluded angle for easurng the closeness degree between two obects [6 7]. Yue [6] and Zhang et al. [7] studed proecton based ult attrbute decson akng n crsp envronent.e. proectons are defned on ordnary nubers or crsp nubers. Yue [8] further nvestgated a new ult attrbute group decson akng MAGDM ethod based on deternng the weghts of the decson akers by eployng proecton technque wth nterval data. Yue and Ja [9] establshed a ethodology for MAGDM based on a new noraled proecton easure n whch the attrbute values are provded by decson akers n hybrd for wth crsp values and nterval data. Xu and Da [0] and Xu [] studed proecton ethod for decson akng n uncertan envronent wth preference nforaton. We [] dscussed a ult attrbute decson akng MADM ethod based on the proecton technque n whch the attrbute values are presented n ters of ntutonstc fuy nubers. Zhang et al. [3] proposed a grey relatonal proecton ethod for MADM based on ntutonstc trapeodal fuy nuber. Zeng et al. [4] nvestgated proectons on nterval valued ntutonstc fuy nubers and we developed odel and algorth to the MAGDM probles wth nterval-valued ntutonstc fuy nforaton. Xu and Hu [5] developed two proecton based odels for MADM n ntutonstc fuy envronent and nterval valued ntutonstc fuy envronent. un [6] presented a group decson akng ethod based on proecton ethod and score functon under nterval valued ntutonstc fuy envronent. sao and Chen [7] developed a novel proecton based coprosng ethod for ult crtera decson akng ethod n nterval valued ntutonstc fuy envronent. n neutrosophc envronent Chen and Ye [8] developed proecton based odel of neutrosophc nubers and presented MADM ethod to select clay-brcks n constructon feld. Bdrectonal proecton easure [9] consders the dstance and ncluded angle between two vectors y and also consders the bdrectonal proecton easure between two vectors y. Ye [9] defned bdrectonal proecton easure as an proveent of the general proecton easure of VNs to overcoe the drawback of the general proecton easure. n the sae study Ye [9] developed ult attrbute decson akng ethod for selectng probles of echancal desgn schees under a sngle-valued neutrosophc envronent. Ye [0] also presented bdrectonal proecton ethod for ultple attrbute group decson akng wth neutrosophc nubers.

3 Ye [] defned credblty nduced nterval neutrosophc weghted arthetc averagng operator and credblty nduced nterval neutrosophc weghted geoetrc averagng operator and developed the proecton easure based rankng ethod for ult-attrbute decson akng MADM probles wth nterval neutrosophc nforaton and credblty nforaton. Dey et al. [] proposed a new approach to neutrosophc soft MADM usng grey relatonal proecton ethod. Dey et al. [3] defned weghted proecton easure wth nterval neutrosophc assessents and appled the proposed concept to solve MADM probles wth nterval valued neutrosophc nforaton. n the feld of bpolar neutrosophc envronent Del et al. [5] defned score accuracy and certanty functons n order to copare BNs and developed bpolar neutrosophc weghted average BNWA and bpolar neutrosophc weghted geoetrc BNWG operators to obtan collectve bpolar neutrosophc nforaton. n the sae study Del et al. [5] also proposed a ult-crtera decson akng MCDM approach on the bass of score accuracy and certanty functons and BNWA BNWG operators. Del and ubas [4] presented a sngle valued bpolar neutrosophc MCDM through correlaton coeffcent slarty easure. Şahn et al. [5] provded a MCDM ethod based on Jaccard slarty easure of BN. Uluçay et al. [6] defned Dce slarty weghted Dce slarty hybrd vector slarty weghted hybrd vector slarty easures under BNs and developed MCDM ethods based on the proposed slarty easures. Dey et al. [7] defned Hang and Eucldean dstance easures to copute the dstance between BNs and nvestgated a OP approach to derve the ost desrable alternatve. n ths study we defne proecton bdrectonal proecton and hybrd proecton easures under bpolar neutrosophc nforaton. hen we develop three algorths for solvng MADM probles wth bpolar neutrosophc assessents. We organe the rest of the paper n the followng way. n ecton we recall several useful defntons concernng VNs and BNs. ecton 3 defnes proecton bdrectonal proecton and hybrd proecton easures between BNs. ecton 4 s devoted to present three odels for solvng MADM under bpolar neutrosophc envronent. n ecton 5 we solve a decson akng proble wth bpolar neutrosophc nforaton on the bass of the proposed easures. Coparson analyss s provded to deonstrate the feasblty and fleblty of the proposed ethods n ecton 6. nally the last ecton provdes conclusons and future scope of work.. Basc concepts regardng VNs and BNs n ths ecton we provde soe basc defntons regardng VNs BNs whch are useful for the constructon of the paper.. ngle valued neutrosophc ets []

4 Let X be a unversal space of ponts wth a generc eleent of X denoted by then a VN P s charactered by a truth ebershp functon an ndeternate ebershp functon P and a falsty ebershp functon P. A VN P s epressed n the followng way. P = { P P P X} where P P P : X [0 ] and 0 P + P + 3 for each pont X.. Bpolar Neutrosophc et [5] Consder X be a unversal space of obects then a BN n X s presented as follows: = { X} where postve ebershp degrees : X [0 ] and P P : X [- 0].he denote the truth ebershp ndeternate ebershp and falsty ebershp functons of an eleent X correspondng to a BN and the negatve ebershp degrees denote the truth ebershp ndeternate ebershp and falsty ebershp of an eleent X to several plct counter property assocated wth a BN. or convenence a bpolar neutrosophc value BNV s presented as q ~ = < >. Defnton [5]. Let = { X} and = { BNs. hen f and only f for all X. ; X} be two Defnton [5]. Let = { X} and = { BNs. hen = f and only f = = = for all X. = ; = X} be two Defnton 3 [5]. Let = { = X} be a BN. he copleent of s represented by c and s defned as follows: c = { + } - c = { + } - c = { + } - ; c = { - } - c = { - } - c = { - } -.

5 Defnton 4. Let = { X} and = { BNs. her unon s defned as follows: = {Ma Mn Mn Ma her ntersecton s defned as follows: = {Mn Ma Ma Mn X} be two } X. Ma Ma Mn Mn } X. Defnton 5 [5]. Let ~q = < > and ~q = < > be two BNNs then.. ~q = < - α >;. ~q = < >;. ~q + ~q = < >; v. ~q. ~q = < > where > Proecton bdrectonal proecton and hybrd proecton easures of BNs hs ecton proposes a general proecton a bdrectonal proecton and a hybrd proecton easures for BNs. Defnton 6. Consder X = be a fnte unverse of dscourse and be a BN n X then odulus of s defned as follows: = α = [ where α = =. Defnton 7 [0 8]. Consder u = u u u and v = v v v be two vectors then the proecton of vector u onto vector v can be defned as follows: ]

6 Pro u v = u Cos u v = u v u v u = v v u where Pro u v represents that the closeness of u and v n agntude. Defnton 8. Consder X = be a fnte unverse of dscourse and be two BNs n X then Pro = Cos =. s called the proecton of on where = ] [ = ] [ and. =. ] [ Eaple. = < > = < > be two BNs n X then the proecton of on s obtaned as follows: Pro =. = = he bgger value of Pro reflects that and are closer to each other. However n sngle valued neutrosophc envronent Ye [0] observed that for two vectors and the general proecton easure cannot descrbe accurately the degree of close to. We also notce that the general proecton ncorporated by Xu [] s not reasonable n several cases under bpolar neutrosophc settng for eaple let = = < a a a -a -a -a > and = < a a a -a -a -a > then Pro = a and Pro = a. hs shows that s uch closer to than whch s not true because =. Ye [0] opned that s equal to whenever Pro and Pro should be equal to. herefore Ye [0] proposed an alternatve ethod called bdrectonal proecton easure to overcoe the ltaton of general proecton easure as gven below. Defnton 9 [0]. Consder and y be two vectors then the bdrectonal proecton between and y s defned as follows: B-pro y =.. y y y = y y y y.

7 where y denote the odulus of and y respectvely and. y s the nner product between and y. Here B-Pro y = f and only f = y and 0 B-Pro y.e. bdrectonal proecton s a noraled easure. Defnton 0. Consder = and = be two BNs n X = then the bdrectonal proecton easure between and s defned as follows: B-Pro =.. =. where = = [ ] [ ] and [. = ] Proposton. Let B-Pro be a bdrectonal proecton easure between BNs and then we have. 0 B-Pro ;. B-Pro = B-Pro ; 3. B-Pro = for =. Proof.. B-Pro = 0 f and only f ether = 0 or = 0.e. when ether = or = whch s trval case. or two non-ero vectors and + -. obvously B-Pro.. ro defnton. =. therefore B- Pro = = = B-Pro Obvously B-Pro = only when =. e. when = = = = = =. hs copletes the proof.

8 Eaple. Assue that = < > = < > be two BNs n the unverse of dscourse X then the bdrectonal proecton easure between on s coputed as gven below. B-Pro = = Defnton. Let = and = be two BNs n X = then the hybrd proecton easure s defned as the cobnaton of proecton easure and bdrectonal proecton easure. he hybrd proecton easure between and s represented as follows: Hyb-Pro = Pro + - B-Pro = where = ] [ = ] [ and. = ] [ where 0. Eaple 3. Assue that = < > = < > be two BNs n the unverse of dscourse X then the hybrd proecton easure between on wth = 0.7 s calculated as gven below. Hyb-Pro = = Proecton b-drectonal proecton and hybrd proecton based decson akng ethods for MADM probles wth bpolar neutrosophc nforaton n ths ecton we develop proecton based decson akng odels to MADM probles wth bpolar neutrosophc assessents. Consder E = {E E E } be a dscrete set of feasble alternatves = { n} n be a set of attrbutes under consderaton and w = w w w n be the weght vector of the attrbutes such that 0 w and n w =. Now we provde three algorths for MADM probles nvolvng bpolar neutrosophc nforaton.

9 4.. Algorth. tep. he ratng of evaluaton value of alternatve E = for the predefned attrbute = n s presented by the decson aker n ters of BNVs and the bpolar neutrosophc decson atr s constructed as gven below. q q... q q q... q n q = n q q... qn where q = < > wth [0 ] and for = ; = n. tep. We forulate bpolar weghted decson atr by ultplyng weghts w of the attrbutes as follows:... w q = n where = w. q = < - w n... n = n n w w - - w - - w - - w > = < > wth [0 ] and for = ; = n. tep 3. We dentfy the bpolar neutrosophc postve deal soluton BNP [5 6] as follows: P e g = < [{ Ma }; { Mn }] f g e f [{ Mn }; { Ma }] [{ Mn }; { Ma }] [{ Mn }; { Ma }] [{ Ma }; { Mn }][{ Ma }; { Mn }] > = n where and are beneft and cost type attrbutes respectvely. tep 4. Deterne the proecton easure between ; = n by usng the followng Eq. n P and Z = n for all =

10 Pro Z = P n n [ e f g e f g [ e f g e f g tep 5. ank the alternatves n a descendng order based on the proecton easure Pro Z P for = and bgger value of Pro Z P deternes the best alternatve. 4.. Algorth. tep. Gve the bpolar neutrosophc decson atr n. n tep. Construct weghted bpolar neutrosophc decson atr ; = n. tep 3. Deterne P e g ; = n. f g e f ] q = ; = ] = P tep 4. Copute the bdrectonal proecton easure between and Z = for all = ; = n by usng the Eq. as gven below. n P B-Pro Z P = Z P P Z Z Z. P n where Z = [ ] =. n P = [ e f g e f g ] and n P Z. = [ f g e f g e ] =. tep 5. Accordng to the bdrectonal proecton easure B-Pro Z P for = alternatves are ranked and bgger value of B-Pro Z P reflects the best opton Algorth 3. tep. Construct the bpolar neutrosophc decson atr = n. n tep. orulate the weghted bpolar neutrosophc decson atr ; = n. P tep 3. dentfy e g = n. f g e f n q = ; = n

11 tep 4. By cobnng proecton easure Pro Z P and bdrectonal proecton easure B-Pro Z between Hyb-Pro Z P and Z = n P P = Pro P we calculate the hybrd proecton easure for all = ; = n as follows. Z + - B-Pro Z P P P Z. Z = + - P P P P Z Z Z. n where Z = [ ] = n P = [ e f g e f g ] Z. P n = 0. [ e f g e f g ] = wth tep 5. We rank all the alternatves n accordance wth the hybrd proecton easure Hyb-Pro Z P and greater value of Hyb-Pro Z P ples the better alternatve. 5. A nuercal eaple Consder the proble studed n [5 7] where a custoer desres to purchase a car. uppose four types of car alternatves E = 3 4 are taken nto consderaton n the decson akng stuaton. our attrbutes naely uel econoy Aerod Cofort 3 afety 4 s consdered to evaluate the alternatves. Assue the weght vector [5] of the attrbute s gven by w = w w w 3 w 4 = Method. he proposed proecton easure based decson akng wth bpolar neutrosophc nforaton for car selecton s presented n the followng steps: tep : Construct the bpolar neutrosophc decson atr he bpolar neutrosophc decson atr q presented by the decson aker as gven below see able. able. he bpolar neutrosophc decson atr n 3 4 E < > < > < > < >

12 E < > < > < > < > E3 < > < > < > < > E4 < > < > < > < > tep. Constructon of weghted bpolar neutrosophc decson atr he weghted decson atr s obtaned by ultplyng weghts of the n attrbutes to the bpolar neutrosophc decson atr as follows see able. able. he weghted bpolar neutrosophc decson atr 3 4 E < > < > < > < > E < > < > < > < > E 3 < > < > < > < > E 4 < > < > < > < > tep 3. electon of BNP P he BNP = e f g e f g = 3 4 s coputed fro the weghted decson atr as follows: e f g e f g = < >; f g e f e g = < >; 3 f 3 g3 e3 f 3 e g = < >; 4 f 4 g 4 e4 f 4 3 e g = < >. 4 tep 4. Deternaton of weghted proecton easure

13 he proecton easure between postve deal bpolar neutrosophc soluton and each weghted decson atr can be obtaned as follows: Pro Z P = 3.44 Pro Z P 4 Z P = tep 5. ank the alternatves We observe that Pro Z P n > Pro Z P 3 = Pro Z P 4 > Pro Z P P = 3.8 Pro 3 > Pro Z P herefore the rankng order of the cars s E E E 4 E 3 and hence E s the best alternatve for the custoer. Method. he proposed bdrectonal proecton easure based decson akng for car selecton s presented as follows: tep. ae as Method tep. ae as Method tep 3. ae as Method tep 4. Calculaton of bdrectonal proecton easure he bdrectonal proecton easure between postve deal bpolar neutrosophc P soluton and each weghted decson atr can be deterned as gven below. B-Pro Z P = B-Pro Z P = 0.80 B-Pro Z 3 P = B- Pro Z 4 P = tep 5. ankng the alternatves Here we notce that B-Pro Z 3 P > B-Pro Z 4 P > B-Pro Z P > B- Pro Z P and therefore the rankng order of the alternatves s obtaned as E 3 E 4 E E. Hence E 3 s the best choce aong the alternatves. Method 3. he proposed hybrd proecton easure based MADM wth bpolar neutrosophc nforaton s provded as follows: tep. ae as Method tep. ae as Method tep 3. ae as Method tep 4. Coputaton of hybrd proecton easure he hybrd proecton easures for dfferent values of [0 ] and the rankng order are shown n the followng able 3 n.

14 able 3. esults of hybrd proecton easure for dfferent valus of larty easure Measure values ankng order Hyb-Pro Z P 0.5 Hyb-Pro Z P =.4970 Hyb-Pro Z P =.489 Hyb-Pro Z 3 P =.508 Hyb-Pro Z 4 P =.503 Hyb-Pro Z P Hyb-Pro Z P =.385 Hyb-Pro Z P =.536 Hyb-Pro Z 3 P =.066 Hyb-Pro Z 4 P =.436 E4 > E3 > E > E E > E4 > E > E3 Hyb-Pro Z P Hyb-Pro Z P Hyb-Pro Z P =.7800 Hyb-Pro Z P =.854 Hyb-Pro Z 3 P =.64 Hyb-Pro Z 4 P =.7670 Hyb-Pro Z P = Hyb-Pro Z P = 3.85 Hyb-Pro Z 3 P =.9589 Hyb-Pro Z 4 P = 3.40 E > E > E4 > E E > E > E4 > E3 6. Coparatve analyss n the ecton we copare the results obtaned fro the proposed ethods wth the results derved fro other estng ethods under bpolar neutrosophc envronent to show the effectveness of the developed ethods. Dey et al. [7] assue that the weghts of the attrbutes are not dentcal and weghts are fully unknown to the decson aker. Dey et al. [7] forulated ang devaton odel under bpolar neutrosophc assessent to copute unknown weghts of the attrbutes as w = By consderng w = the proposed proecton easure are shown as follows: Pro 4 Z P Z P = nce Pro = Pro Z P Z P > Pro Z P 3 = Pro Z P 4 > Pro Z P 3 > Pro Z P = 3.65 Pro therefore the rankng order of the four alternatves s gven by E E E 4 E 3. hus E s the best choce for the custoer.

15 Now by takng w = the bdrectonal proecton easure are calculated as gven below. B-Pro Z P = 0.83 B-Pro Z P = 0.8 B-Pro Z 3 P = B- Pro Z 4 P = nce B-Pro Z 4 P > B-Pro Z 3 P > B-Pro Z P > B-Pro Z P consequently the rankng order of the four alternatves s gven by E 4 E 3 E E and hence E 4 s obvously the best opton for the custoer. Also by takng w = the proposed hybrd proecton easures for dfferent values of [0 ] and the rankng order are revealed n the followng able 4. able 4. esults of hybrd proecton easure for dfferent valus of larty easure Measure values ankng order Hyb-Pro Z P 0.5 Hyb-Pro Z P =.4573 Hyb-Pro Z P =.455 Hyb-Pro Z 3 P =.597 Hyb-Pro Z 4 P =.56 Hyb-Pro Z P Hyb-Pro Z P =.034 Hyb-Pro Z P =.099 Hyb-Pro Z 3 P =.0740 Hyb-Pro Z 4 P =.70 E4 > E3 > E > E E4 > E > E > E3 Hyb-Pro Z P Hyb-Pro Z P Hyb-Pro Z P =.4940 Hyb-Pro Z P =.743 Hyb-Pro Z 3 P =.68 Hyb-Pro Z 4 P =.699 Hyb-Pro Z P = Hyb-Pro Z P = 3.96 Hyb-Pro Z 3 P =.9448 Hyb-Pro Z 4 P = E > E4 > E3 > E E > E > E4 > E3 Del et al. [5] assue the weght vector of the attrbutes as w = and the rankng order based on score values s presented as follows: E 3 E 4 E E hus E 3 was the ost desrable alternatve. Dey et al. [7] eployed ang devaton ethod to fnd unknown attrbute weghts as w = he rankng order of the alternatves s presented based on the relatve closeness coeffcent as gven below. E 3 E E 4 E Obvously E 3 was the ost sutable opton for the custoer.

16 Dey et al. [7] also consder the weght vector of the attrbutes as w = then by usng OP ethod the rankng order of the cars s represented as follows: E 4 E E 3 E. o E 4 would be the ost preferable alternatve for the buyer. We observe that dfferent proecton easures provde dfferent rankng results and the proecton easure s weght senstve. herefore decson aker should choose the proecton easure and weghts of the attrbutes n the decson akng contet accordng to hs/her needs desres and practcal condton 7. Concluson n ths paper we have defned proecton bdrectonal proecton easures between bpolar neutrosophc sets. urther we have defned a hybrd proecton easure by cobnng proecton and bdrectonal proecton easures. hrough these proecton easures we have developed three algorths for ult-attrbute decson akng odels under bpolar neutrosophc envronent for choosng the best alternatve. nally a car selecton proble has been provded to show the fleblty and applcablty of the proposed ethods. urtherore coparson analyss of the proposed ethods wth the other estng ethods has also been deonstrated. he proposed algorths can be etended to nterval bpolar neutrosophc envronent. n future we shall apply proecton bdrectonal proecton and hybrd proecton easures of nterval bpolar neutrosophc sets for group decson akng edcal dagnoss weaver selecton pattern recognton probles. e f e r e n c e s. arandache. A unfyng feld of logcs. Neutrosophy: Neutrosophc probablty set and logc. Aercan esearch Press ehoboth Wang H.. arandache Y. Zhang. underraan. ngle valued neutrosophc sets. Multspace and Mult-tructure Vol pp Zhang W.. Bpolar fuy sets and relatons: a coputatonal fraework for cogntve odelng and ultagent decson analyss. n: Proc. of EEE Conf. 994 pp DO: 0.09/JC Zhang W. Bpolar fuy sets. n: Proc. of the EEE World Congress on Coputatonal cence ueee Anchorage Alaska 998 pp DO: 0.09/UZZY Del. M. Al. arandache. Bpolar neutrosophc sets and ther applcaton based on ultcrtera decson akng probles. n: nternatonal conference on advanced echatronc systes CAMech EEE August pp Yue Z. L. Approach to group decson akng based on deternng the weghts of eperts by usng proecton ethod. Appled Matheatcal Modellng Vol No 7 pp Zhang G. Y. Jng H. Huang Y. Gao. Applcaton of proved grey relatonal proecton ethod to evaluate sustanable buldng envelope perforance. Appled Energy Vol No pp Yue Z. Applcaton of the proecton ethod to deterne weghts of decson akers for group decson akng. centa ranca Vol. 9 0 No

17 9. Yue Z. Y. Ja. A drect proecton-based group decson-akng ethodology wth crsp values and nterval data. oft Coputng 05. DO 0.007/s Xu Z... Da. Proecton ethod for uncertan ult-attrbute decson akng wth preference nforaton on alternatves. nternatonal Journal of nforaton & Decson Makng Vol pp Xu Z. On ethod for uncertan ultple attrbute decson akng probles wth uncertan ultplcatve preference nforaton on alternatves. uy optaton and Decson Makng Vol pp We G. W. Decson-akng based on proecton for ntutonstc fuy ultple attrbutes. Chnese Journal of Manageent Vol No 9 pp Zhang X.. Jn P. Lu. A grey relatonal proecton ethod for ult-attrbute decson akng based on ntutonstc trapeodal fuy nuber. Appled Matheatcal Modellng Vol No 5 pp Zeng.. Baležents J. Chen G. Luo. A proecton ethod for ultple attrbute group decson akng wth ntutonstc fuy nforaton. nforatca Vol No 3 pp Xu Z. H. Hu. Proecton odels for ntutonstc fuy ultple attrbute decson akng. nternatonal Journal of echnology & Decson Makng Vol No pp un G. A group decson akng ethod based on proecton ethod and score functon under V envronent. Brtsh Journal of Matheatcs & Coputer cence Vol No pp DO: /BJMC/05/ sao C. Y.. Y. Chen. A proecton-based coprosng ethod for ultple crtera decson analyss wth nterval-valued ntutonstc fuy nforaton. Appled oft Coputng Volue pp Chen J. J. Ye. A proecton odel of neutrosophc nubers for ultple attrbute decson akng of clay-brck selecton. Neutrosophc ets and ystes Vol. 06 pp Ye J. Proecton and bdrectonal proecton easures of sngle-valued neutrosophc sets and ther decson-akng ethod for echancal desgn schees. Journal of Eperental & heoretcal Artfcal ntellgence 06. DO: 0.080/09583X Ye J. Bdrectonal proecton ethod for ultple attrbute group decson akng wth neutrosophc nubers. Neural Coputng and Applcatons 05. DO: 0.007/s Ye J. nterval neutrosophc ultple attrbute decson-akng ethod wth credblty nforaton. nternatonal Journal of uy ystes 06. DO: 0.007/s Dey P. P.. Praank B. C. Gr. Neutrosophc soft ult-attrbute decson akng based on grey relatonal proecton ethod. Neutrosophc ets and ystes Vol. 06 pp Dey P. P.. Praank B. C. Gr Etended proecton based odels for solvng ultple attrbute decson akng probles wth nterval valued neutrosophc nforaton. n:. arandache. Praank ed. New rends n Neutrosophc heory and Applcatons Pons asbl Brussells 06 pp Del. Y. A. ubas. Multple crtera decson akng ethod on sngle valued bpolar neutrosophc set based on correlaton coeffcent slarty easure. n: nternatonal Conference on atheatcs and atheatcs educaton CMME-06 rat Unversty Elag urkey May Şahn M.. Del V. Uluçay. Jaccard vector slarty easure of bpolar neutrosophc set based on ult-crtera decson akng. n: nternatonal Conference on Natural cence and Engneerng CNAE 6 Klls March Uluçay V.. Del M. Şahn. larty easures of bpolar neutrosophc sets and ther applcaton to ultple crtera decson akng. Neural Coputng and Applcatons 06. DO: 0.007/s Dey P. P.. Praank B. C. Gr. OP for solvng ult-attrbute decson akng probles under b-polar neutrosophc envronent. n:. arandache. Praank ed. New rends n Neutrosophc heory and Applcatons Pons asbl Brussells 06 pp

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