Neutrosophc Sets ad Systems, Vol., 0 Exteded TOPSIS Method for the Multple ttrbute Decso Makg Problems Based o Iterval Neutrosophc Set Pgpg Ch,, ad Pede Lu,,* Cha-sea Iteratoal College, Dhurak Pudt versty, Bagkok 00, Thalad Isttute of No-govermetal Hgher Educato, Shadog Ygca versty, Ja 5004, P.R. of Cha School of Maagemet Scece ad Egeerg, Shadog versty of Face ad Ecoomcs, No.7, Erhuadog Road, Ja, 5004, P.R. of Cha. *E-mal: pede.lu@gmal.com Correspodg author bstract. The terval eutrosophc set INS ca be easer to express the complete, determate ad cosstet formato, ad TOPSIS s oe of the most commoly used ad effectve method for multple attrbute decso makg, however, geeral, t ca oly process the attrbute values wth crsp umbers. I ths paper, we have exteded TOPSIS to INS, ad wth respect to the multple attrbute decso makg problems whch the attrbute weghts are ukow ad the attrbute values take the form of INSs, we proposed a expaded TOPSIS method. Frstly, the defto of INS ad the operatoal laws are gve, ad dstace betwee INSs s defed. The, the attrbute weghts are determed based o the Maxmzg devato method ad a exteded TOPSIS method s developed to rak the alteratves. Fally, a llustratve example s gve to verfy the developed approach ad to demostrate ts practcalty ad effectveess. Keywords: terval eutrosophc set; TOPSIS; multple attrbute decso makg; Maxmzg devato method; Hammg dstace. Itroducto I real decso makg, there exst may mult-crtera decso-makg MCDM problems. Because of the ambguty of people's thkg ad the complexty of obectve thgs, the attrbute values of the MCDM problems caot always be expressed by crsp umbers, ad t maybe s easer to be descrbed by fuzzy formato. The fuzzy set FS theory, whch s proposed by Zadeh [], s oe of the most effectve tools for processg fuzzy formato; however, ts dsadvatage s that t oly has a membershp, ad s uable to express o-membershp. O the bass of FS, taassov [,] proposed the tutostc fuzzy set IFS by addg a omembershp fucto,.e., there are membershp or called truth-membershp T x ad o-membershp or called falsty-membershp F x tutostc fuzzy sets, ad they satsfy the codtos T x x [0,] ad 0 T x + F x. Further, taassov ad Gargov [4], taassov [5] proposed the terval-valued tutostc fuzzy set IVIFS by extedg the truth-membershp fucto ad falstymembershp fucto to terval umbers. IFSs ad IVIFSs ca oly hadle complete formato ot the determate formato ad cosstet formato. I IFSs, the determacy s - T x- F x by default. However, practce, the decso formato s ofte complete, determate ad cosstet formato. I order to process ths kd of formato, Smaradache [] further proposed the eutrosophc set NS by addg a depedet determacy-membershp o the bass of IFS, whch s a geeralzato of fuzzy set, terval valued fuzzy set, tutostc fuzzy set, ad so o. I NS, the determacy s quatfed explctly ad truthmembershp, determacy membershp, ad falsemembershp are completely depedet. Recetly, NSs have become a terestg research topc ad attracted wdely attetos. Wag et al. [7] proposed a sgle valued eutrosophc set SVNS from scetfc or egeerg pot of vew, whch s a stace of the eutrosophc set. Ye [8] proposed the correlato coeffcet ad weghted correlato coeffcet for SVNSs, ad he have proved that the cose smlarty degree s a specal case of the correlato coeffcet SVNS. Ye [8a] proposed Sgle valued eutrosophc crossetropy for multcrtera decso makg problems. Smlar to IVIFS, Wag et al. [9] proposed terval eutrosophc sets INSs whch the truth-membershp, determacy-membershp, ad false-membershp were exteded to terval umbers, ad dscussed some propertes ad comparg method of INSs. Ye [0] proposed the smlarty measures betwee INSs based o the Hammg ad Eucldea dstaces, ad developed a multcrtera decso-makg method based o the Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
4 Neutrosophc Sets ad Systems, Vol., 0 smlarty degree. However, so far, there has bee o research o extedg TOPSIS for INSs. TOPSIS The Order Performace techque based o Smlarty to Ideal Soluto, whch was proposed by Hwag ad Yoo [] s oe of popular decso makg methods. I last 0 years, may researchers have exteded ths method ad proposed dfferet modfcatos, ad t has bee appled usefully the practce to solve may problems dfferet felds for decso makers. Che [] exteded the TOPSIS for group decso makg problems whch the mportace weghts of varous crtera ad ratgs of alteratves wth respect to these crtera take the form of lgustc varables. The key of the proposed method s that these varables are trasformed to tragular fuzzy umbers. J et al. [] exteded TOPSIS method to MDM problems whch the attrbute values are the tutostc fuzzy sets, ad appled t to the evaluato of huma resources. We ad Lu [4] exteded TOPSIS method to the ucerta lgustc varables, ad appled t to the rsk evaluato of Hghtechology. Lu [5] proposed a exteded TOPSIS method to resolve the mult-attrbute decso-makg problems whch the attrbute weghts ad attrbute values are all terval vague value. Frstly, the deal ad egatve deal solutos are calculated based o the score fucto. The the dstace betwee the terval Vague values s defed, ad the dstaces betwee each alteratve ad the deal ad egatve deal solutos are calculated. The relatve closeess degree s calculated by TOPSIS method, ad the the orderg of the alteratves s cofrmed accordg to the relatve closeess degree. Lu ad Su [] proposed a exteded TOPSIS based o trapezod fuzzy lgustc varables, ad gave the method for determg attrbute weghts. Lu [7] proposed a exteded TOPSIS method for multple attrbute group decso makg based o geeralzed terval-valued trapezodal fuzzy umbers. Mohammad et al. [8] used fuzzy group TOPSIS method for selectg adequate securty mechasms e-busess processes. Verma et al. [9] proposed a terval-valued tutostc fuzzy TOPSIS method for solvg a faclty locato problem. Obvously, because TOPSIS s a mportat decso makg method, ad the terval eutrosophc set ca be easer to express the complete, determate ad cosstet formato, t s mportat to establsh a exteded TOPSIS method based o INS. I ths paper, we wll establsh a exteded TOPSIS method for the multple attrbute decso makg problems whch the attrbute weghts are ukow ad attrbute values take the form of INSs. I order to do so, the remader of ths paper s show as follows. I secto, we brefly revew some basc cocepts ad operatoal rules of INS ad propose the Hammg dstace ad the Euclda dstace betwee terval eutrosophc values INVs or terval eutrosophc sets, ad gve a proof of Hammg dstace ad a calcualto example. I Secto, we propose a method for determg the attrbute weghts based o the Maxmzg devato method ad exted the TOPSIS method to rak the alteratves, ad gve the detal decso steps. I Secto 4, we gve a example to llustrate the applcato of proposed method, ad compare the developed method wth the exstg method. I Secto 5, we coclude the paper. The Iterval Neutrosophc Set. The Defto of the Iterval Neutrosophc Set Defto []. Let X be a uverse of dscourse, wth a geerc elemet X deoted by x. eutrosophc set NS X s = { x T x, I x x x X} where, T, I ad F are the truth-membershp fucto, determacy-membershp fucto, ad the falsty-membershp fucto, respectvely. T x, I x ad F x are real stadard or ostadard + subsets of 0,. There s o restrcto o the sum of T x, I x ad F x, so + 0 T x + I x + F x. The NS was preseted from phlosophcal pot of vew. Obvously, t was dffcult to use the actual applcatos. Wag [7] further proposed the sgle valued eutrosophc set SVNS from scetfc or egeerg pot of vew, whch s a geeralzato of the exstg fuzzy sets, such as classcal set, fuzzy set, tutostc fuzzy set ad paracosstet sets etc., ad t was defed as follows. Defto [7]. Let X be a uverse of dscourse, wth a geerc elemet X deoted by x. sgle valued eutrosophc set X s = { x T x, I x x x X} where, T, I ad F are the truth-membershp fucto, determacy-membershp fucto, ad the falsty-membershp fucto, respectvely. For each pot x X, we have T x, I x x [0,], ad 0 T x + I x + F x. I the actual applcatos, sometmes, t s ot easy to express the truth-membershp, determacy-membershp ad falsty-membershp by crsp values, ad they may be easer to be expressed by terval umbers. Wag et al. [9] further defed terval eutrosophc sets INSs show as follows. Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
Neutrosophc Sets ad Systems, Vol., 0 5 Defto [7]. Let X be a uverse of dscourse, wth a geerc elemet X deoted by x. terval eutrosophc set X s = { x T x, I x x x X} where, T, I ad F are the truth-membershp fucto, determacy-membershp fucto, ad the falstymembershp fucto, respectvely. For each pot x X, we have T x, I x x [0,], ad 0 sup T x + sup I x + sup F x. For coveece, we ca L L L use x = [ T, T ],[ I, I ],[ F ] to represet a value INS, ad call terval eutrosophc value INV.. The Operatoal Rules of the Iterval Neutrosophc Values Defto 4. Let x = [ T L, T ],[ I L, I ],[ F L ] ad y = [ T L, T ],[ I L, I ],[ F L ] be two INVs, the the operatoal rules are defed as follows. The complemet of x s x = [ F ],[ I, I ],[ T, T ] 4 L L L x y = T + T T T, T + T T T, L L L L I I, I I, F F F L L L L L L x y = [ T T, T T ],[ I + I I I, I + I I I ], F + F F F + F F F L x = T, T, 4 L L I, I, F > 0 5 L L x = T, T, I, I, L F, F > 0. The Dstace betwee two INSs I the followg, we wll dscuss the dstace betwee two INSs. Defto 5. Let x = [ T L, T ],[ I L, I ],[ F L ], y = [ T, T ],[ I, I ],[ F ] ad L L L z = [ T, T ],[ I, I ],[ F ] be three INVs, S be a L L L collecto of all INVs, ad f be a mappg wth 5 7 8 f : Sˆ Sˆ R. If, codtos. 0 d x, y, d x, y = d y, x d x y meets the followg d x, x = 0 d x, y + d y, z d x, z The we ca call d x, y a dstace betwee twoinvs x ad y. Defto. Let x = [ T L, T ],[ I L, I ],[ F L ], y = [ T L, T ],[ I L, I ],[ F L ] be two INVs, the ad The Hammg dstace betwee x ad y s defed as follows, dh x y = T T + T T + I I + I I L L L L + F F + F F L L Proof. Obvously, 9 ca meet the above codtos ad Defato 5. I the followg, we wll prove 9 meets codto. For ay a INV z = [ T L, T ],[ I L, I ],[ F L ], we have, dh x z = T T + T T + I I + I I + F F + F F L L L L L L = T T + T T + T T + T T + I I + I I + I I + I I + F F + F F + F F + F F T T + T T + T T + T T + I I + I I + I I + I I ad + F F + F F + F F + F F 9 Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
Neutrosophc Sets ad Systems, Vol., 0 T T + T T + T T + T T + I I + I I + I I + I I + F F + F F + F F + F F = T T + T T + I I + I I L L L L + F F + F F L L + T T + T T + I I + I I L L L L + F F + F F L L = d x, y + d y, z H H.e., d x, y d y, z d x, z +. H H H The Euclda dstace betwee x ad s defed as follows. d E x, y = L L L L T T + T T + I I L L + I I + F F + F F 0 The proof s smlar to that of 9, t s omtted here. Further, we exted the dstace betwee two INVs x ad y to two INSs. Defto 7 Let L L L X = [ T, T ],[ I, I ],[ F ] =,,, ad Y = [ T L, ],[ L, ],[ L, T I I F F ] =,,, be two INSs, the The Hammg dstace betwee X ad Y s defed as follows L L L L L L, = + + + + + d X Y T T T T I I I I F F F F H = The Euclda dstace betwee X ad Y s defed as follows L L L L L L d X, Y = T T + T T + I I + I I + F F + F F E = For example, f two INSs X ad Y are [0.5,0.],[0.,0.],[0.9,0.9], [0.8,0.9],[0.4,0.4],[0.,0.], [0.,0.4],[0.8,0.9],[0.7,0.8] ad [0.7,0.8], [0.4,0.5],[0.,0.],[0.5,0.],[0.5,0.5],[0.,0.4],[0.,0.],[ 0.,0.4],[0.,0.4], the the dstaces of Hammg ad Euclda betwee X ad Y ca be calculated as follows. dh X, Y = 0.5 0.7 + 0. 0.8 + 0. 0.4 + 0. 0.5 + 0.9 0. + 0.9 0. + 0.8 0.5 + 0.9 0. + 0.4 0.5 + 0.4 0.5 + 0. 0. + 0. 0.4 + 0. 0. + 0.4 0. + 0.8 0. + 0.9 0.4 + 0.7 0. + 0.8 0.4 = 0. de X, Y = SQRT 0.5 0.7 + 0. 0.8 + 0. 0.4 + 0. 0.5 + 0.9 0. + 0.9 0. + 0.8 0.5 + 0.9 0. + 0.4 0.5 + 0.4 0.5 + 0. 0. + 0. 0.4 + 0. 0. + 0.4 0. + 0.8 0. + 0.9 0.4 + 0.7 0. + 0.8 0.4 = 0. exteded TOPSIS Method for multple attrbute decso makg based o INSs For a multple attrbute decso problem, let =,,, m be a dscrete set of alteratves, C = C, C,, C be the set of attrbutes, W = w, w,, w T be the weghtg vector of the attrbutes, ad meets w =, w 0. where w s ukow. Suppose that = X = x s the decso m L L L matrx, where x = [ T, T ],[ I, I ],[ F ] takes the form of the INVs for alteratve wth respect to attrbute C. The steps of the rakg the alteratves based o these codtos are show as follows Step. Stadardzed decso matrx I geeral, there are two types attrbutes, the more the attbute value s, the better the alteratve s, ths type s called beft type; o the cotrary, the more the attbute value s, the worse the alteratve s, ths type s called cost type. Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
Neutrosophc Sets ad Systems, Vol., 0 7 I order to to elmate the fluece of the attrbute types, we eed covert the cost type to beft type. Suppose the stadardzed matrx s expressed by R = r, where m L L L r = [ T, T ],[ I, I ],[ F ], the we have r = x f the attrbute s beft type r = x f the attrbute s cos t type Where, x s the complemet of x. Step. Calculate attrbute weghts Because the attrbute weghts are completely ukow, we eed to determe the attrbute weghts. The maxmzg devato method, whch s proposed by Wag [0], s a good tool to calculate the attrbute weghts for MDM problems wth umercal formato. The prcple of ths method s descrbed as follows. For a MDM problem, f the attrbute values for all alteratves have lttle dffereces, such a attrbute wll play a small mportat role rakg the alteratves, especally, for a attrbute, f the attrbute values for all alteratves are equal, the attrbute has o effect o the rakg results. Cotrarwse, f attrbute values for all alteratves uder a attrbute have obvous dffereces, such a attrbute wll play a mportat role rakg the alteratves. Based o ths vew, f the attrbute values of all alteratves for a gve attrbute have a lttle devatos, we ca assg a lttle weght for ths attrbute; otherwse, the attrbute whch makes larger devatos should be set a bgger weght. Especally, f the attrbute values of all alteratves are all equal wth respect to a gve attrbute, the the weght of such a attrbute may be set to 0. For a MDM problem, the devato values of alteratve to all the other alteratves uder the m attrbute C ca be defed as D w = d r, r w, the m m m D w = D w = d r, r w l = = l= l l = represets the total devato values of all alteratves to the other alteratves for the attrbute C. m m represets the D w = D w = d r, r w l = = = l= devato of all attrbutes for all alteratves to the other alteratves. The optmze model s costructed as follows: m m max D w = d r, r w s t w = w = l = = l=., = 0,, The we ca get m m dr, r = l= l = m m d = = l= l w r, r 4 5 Furthermore, we ca get the ormalzed attrbute weght based o ths model: w = m m = l= m m = = l= dr, r l dr, r l Step. se the exteded TOPSIS method to rak the alteratves The basc prcple of TOPSIS s that the best alteratve should have the shortest dstace to the postve deal soluto ad the farthest dstace to the egatve deal soluto. The postve deal soluto marked as V + s a best soluto whch each attrbute value s the best oe of all alteratves, ad the egatve deal soluto marked as V - s aother worst soluto whch each attrbute value s the worst value of all alteratves. The steps of rakg the alteratves by the exteded TOPSIS are show as follows. calculate the weghted matrx wr wr wr wr wr wr Y wr wr wr Where y = [ T L, T ],[ I L, I ],[ F L ] = y m = m m m 7 Determe the postve deal soluto ad egatve deal soluto: ccordg to the defto of INV, we ca defe the absolute postve deal soluto ad egatve deal soluto show as follows. + y = [,],[0,0],[0,0] y = [0,0],[,],[,] =,,, 8 Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
8 Neutrosophc Sets ad Systems, Vol., 0 or we ca select the vrtual postve deal soluto ad egatve deal soluto by selectg the best values for each attrbute from all alteratves. + L L y = [max T,max T ],[m I,m I ], L [m F, m F ] L L y = [m T,m T ],[max I,max I ], L [max F,max F ] =,,, 9 Calculate the dstace betwee the alteratve ad postve deal soluto/ Negatve deal soluto The dstace betwee the alteratve ad postve deal soluto/ egatve deal soluto s: + + d = d y, y = d = d y, y = =,,, m 4 Calculate the relatve closeess coeffcet 0 + d RCC =. =,,, m + d + d 5 Rak the alteratves tlze the relatve closeess coeffcet to rak the alteratves. The smaller RCC s, the better alteratve s. 4 applcato example I order to demostrate the applcato of the proposed method, we wll cte a example about the vestmet selecto of a compay adapted from [0]. There s a compay, whch wats to vest a sum of moey to a dustry. There are 4 alteratves whch ca be cosdered by a pael, cludg: s a car compay; s a food compay; s a computer compay; 4 4 s a arms compay. The evaluato o the alteratves s based o three crtera: C s the rsk; C s the growth; C s the evrometal mpact. where C ad C are beeft crtera, ad C s a cost crtero. Suppose the crtera weghts are ukow. The fal decso formato ca be obtaed by the INVs, ad show table. Table The evaluato values of four possble alteratves wth respect to the three crtera C C C [0.4,0.5],[0.,0.],[0.,0.4] [0.4,0.],[0.,0.],[0.,0.4] [0.7,0.9],[0.,0.],[0.4,0.5] [0.,0.7],[0.,0.],[0.,0.] [0.,0.7],[0.,0.],[0.,0.] [0.,0.],[0.,0.5],[0.8,0.9] [0.,0.],[0.,0.],[0.,0.4] [0.5,0.],[0.,0.],[0.,0.4] [0.4,0.5],[0.,0.4],[0.7,0.9] [0.7,0.8],[0.0,0.],[0.,0.] [0.,0.7],[0.,0.],[0.,0.] [0.,0.7],[0.,0.4],[0.8,0.9] 4 4. Rakg the alteratves ths example We adopt the proposed method to rak the alteratves. To get the best alteratves, the followg steps are volved: Covert the cost crtero to beeft crtero. Sce C s a cost crtero, we ca replace x,,,4 = wth x,,,4 =, ad get the decso matrx R : [0.4,0.5],[0.,0.],[0.,0.4] [0.4,0.],[0.,0.],[0.,0.4] [0.,0.7],[0.,0.],[0.,0.] [0.,0.7],[0.,0.],[0.,0.] R = [0.,0.],[0.,0.],[0.,0.4] [0.5,0.],[0.,0.],[0.,0.4] [0.7, 0.8],[0.0,0.],[0.,0.] [0.,0.7],[0.,0.],[0.,0.] [0.4,0.5],[0.7,0.8],[0.7,0.9] [0.8,0.9],[0.5,0.7],[0.,0.] [0.7,0.9],[0.,0.8],[0.4,0.5] [0.8,0.9],[0.,0.7],[0.,0.7] Calculate attrbute weghts bout the dstace formula, we ca use the Hammg dstace defed 9, ad get dr, r l, =,,,4; =,,. l dr, r = dr, r = dr, r = 0 dr, r = 0., dr, r = 0.08, dr, r = 0.00 dr, r = 0.0, dr, r = 0.050, dr, r = 0.50 dr, r = 0., dr, r = 0.00, dr, r = 0.7 4 4 4 dr, r = 0., dr, r = 0.08, dr, r = 0.00 dr, r = dr, r = dr, r = 0 Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
Neutrosophc Sets ad Systems, Vol., 0 9 dr, r = 0., dr, r = 0.00, dr, r = 0.08 dr, r = 0.00, dr, r = 0.07, dr, r = 0.08 4 4 4 dr, r = 0.0, dr, r = 0.050, dr, r = 0.50 dr, r = 0., dr, r = 0.00, dr, r = 0.08 dr, r = dr, r = dr, r = 0 dr, r = 0., dr, r = 0.7, dr, r = 0.00 4 4 4 dr, r = 0., dr, r = 0.00, dr, r = 0.7 4 4 4 dr, r = 0.00, dr, r = 0.07, dr, r = 0.08 4 4 4 dr, r = 0., dr, r = 0.7, dr, r = 0.00 4 4 4 dr, r = dr, r = dr, r = 0 4 4 4 4 4 4 The accordg to, we ca get the attrbute weghts show as follows. w = 0., w = 0.97, w = 0.47 se the exteded TOPSIS method to rak the alteratves calculate the weghted matrx I formula 7, we ca calculate wr =,,,4; =,, by formula 7. For example, we ca calculate wr = 0. 0. 0. 0. [ 0.4, 0.5 ],[0.,0. ] 0. 0.,[0.,0.4 ] = [0.7,0.4],[0.555,0.4],[0.4,0.75] The we ca get the weghted matrx Y [0.7,0.4],[0.555,0.4],[0.4,0.75] [0.85,0.57],[0.40,0.555],[0.555,0.4] Y = [0.,0.85],[0.555,0.4],[0.4,0.75] [0.57,0.445],[0.000,0.40],[0.40,0.555] [0.09,0.5],[0.5,0.789],[0.78,0.85] [0.5,0.],[0.5,0.78],[0.78,0.789] [0.8,0.5],[0.78,0.789],[0.789,0.85] [0.5,0.],[0.5,0.78],[0.5,0.789] [0.00,0.],[0.85,0.907],[0.85,0.955] [0.505,0.4],[0.79,0.85],[0.59,0.800] [0.409,0.4],[0.800,0.907],[0.70,0.79] [0.505,0.4],[0.800,0.85],[0.800,0.85] Determe the postve deal soluto ad egatve deal soluto. ccordg to 9, we ca get the vrtual postve deal soluto ad egatve deal soluto show asa follows. y + = [0.57,0.445],[0.000,0.40],[0.40,0.555] [0.5,0.],[0.5,0.78],[0.5,0.789] [0.505,0.4],[0.79,0.85],[0.59,0.79] y = [0.,0.4],[0.555,0.4],[0.4,0.75] [0.09,0.5],[0.78,0.789],[0.789,0.85] [0.00,0.],[0.85,0.907],[0.85,0.955] Calculate the dstace betwee the alteratve ad postve deal soluto/ Negatve deal soluto ccordg to 9, we ca get the dstace betwee the alteratve ad postve deal soluto/ egatve deal soluto show as follows. d = 0.5, d = 0.80, d = 0.77, d = 0.05 + + + + 4 d = 0.04, d = 0.85, d = 0.89, d = 0.50 4 v Calculate the relatve closeess coeffcet ccordg to, we ca calculate the the relatve closeess coeffcet show as follows. RCC = 0.94, RCC = 0.9, RCC = 0., RCC = 0.4 4 v Rak the alteratves ccordg to the relatve closeess coeffcet, we ca get the rakg from the best to worst. 4 4. Compare wth the exstg method I order to further llustrate the effectveess of the proposed method ths paper, we compare wth method proposed by Ye [0]. However, because the attrbute weghts ad postve deal soluto/ Negatve deal soluto are dfferet from Ye [0], the rakg result s dfferet; addto, Ye [0] oly cosder the smlarty measure betwee each alteratve ad postve deal soluto. If we adopt the same attrbute weghts ad postve deal soluto deal soluto, ad oly cosder the dstace betwee each alteratve ad postve deal soluto, we ca get the same rakg result from these two methods. Comparg wth the method proposed by Ye [0], the method proposed ths paper ca solve the multple attrbute problems wth ukow weghts, ad ca provde a compromse soluto whch cosders the dstaces to postve deal soluto ad Negatve deal soluto. I addto, t s smpler calculato process tha Ye [0]. 5 Coclusos The terval eutrosophc set ca be easer to express the complete, determate ad cosstet formato, ad t s a geeralzato of fuzzy set, terval valued fuzzy set, tutostc fuzzy set, ad so o. Ths paper proposed the operatoal laws of the terval eutrosophc set, ad defed the Hammg dstace ad the Euclda dstace. The Maxmzg devato method s used to determe the attrbute weghts ad the TOPSIS Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set
70 Neutrosophc Sets ad Systems, Vol., 0 method s exteded to terval eutrosophc set. Fally, a llustratve example has bee gve to show the steps of the developed method. It shows that ths method s smple ad easy to use ad t costatly erches ad develops the theory ad method of multple attrbute decso makg, ad proposed a ew dea for solvg the MDM problems. I the future, we shall cotue workg the exteso ad applcato of the proposed method. ckowledgemets Ths paper s supported by the Natoal Natural Scece Foudato of Cha No. 774, the Humates ad Socal Sceces Research Proect of Mstry of Educato of Cha No. YJC004, the Natural Scece Foudato of Shadog Provce No. ZR0GQ0, No. ZR0FM0, Socal Scece Plag Proect of Shadog Provce No. BGLJ0, ad Graduate educato ovato proects Shadog Provce SDYY05. Refereces [] L.. Zadehuzzy sets, Iformato ad Cotrol 8958-5. [] K.T. taassov, Itutostc fuzzy setsuzzy Sets ad Systems 0 98 87-9. [] K.T. taassov, More o tutostc fuzzy setsuzzy Sets ad Systems 9897-4. [4] K.T. taassov, G. Gargov, Iterval-valued tutostc fuzzy setsuzzy Sets ad Systems 9894-49. [5] K.T. taassov, Operators over terval-valued tutostc fuzzy setsuzzy Sets ad Systems 499459-74. [] F Smaradache, ufyg feld logcs. eutrosophy: Neutrosophc probablty, set ad logc, merca Research Press, Rehoboth, 999. [7] H.Wag. Smaradache, Y. Zhag R. Suderrama, Sgle valued eutrosophc sets, Proc Of 0th 47 It Cof o Fuzzy Theory ad Techology, Salt Lake Cty, 477 tah, 005. [8] J. Ye, Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet, Iteratoal Joural of Geeral Systems 44 0 8-94. [8a] J. Ye, Sgle valued eutrosophc cross-etropy for multcrtera decso makg problems, ppled Mathematcal Modellg, 0 do: 0.0/.apm.0.07.00. [9] H. Wag. Smaradache, Y.Q. Zhag, et al., Iterval eutrosophc sets ad logc: Theory ad applcatos computg, Hexs, Phoex, Z, 005. [0] J. Ye, Smlarty measures betwee terval eutrosophc sets ad ther applcatos multcrtera decsomakg, Joural of Itellget & Fuzzy Systems, DOI: 0./IFS-074 0. [] C.L.Hwag ad K. Yoo, Multple ttrbute Decso Makg ad pplcatos, Sprger, New York, NY,98. [] T.C. Che, Extesos of the TOPSIS for group decsomakg uder fuzzy evrometuzzy Sets ad Systems 4000-9. [] F. J, P.D., Lu, X. Zhag, Evaluato Study of Huma Resources Based o Itutostc Fuzzy Set ad TOPSIS Method, Joural of Iformato ad Computatoal Scece 4007 0-08. [4] Y.Q. We, P.D. Lu, Rsk Evaluato Method of Hghtechology Based o certa Lgustc Varable ad TOPSIS Method, Joural of Computers 4 009 7-8. [5] P.D. Lu, Mult-ttrbute Decso-Makg Method Research Based o Iterval Vague Set ad TOPSIS Method, Techologcal ad Ecoomc Developmet of Ecoomy 5 009 45 4. [] P.D. Lu, Y. Su, The exteded TOPSIS based o trapezod fuzzy lgustc varables, Joural of Covergece Iformato Techology 5 00 8 5. [7] P.D. Lu, Exteded TOPSIS Method for Multple ttrbute Group Decso Makg based o Geeralzed Iterval-valued Trapezodal Fuzzy Numbers, Iformatca 5 0 85-9. [8] S. Mohammad, S. Golara, ad N. Mousav, Selectg adequate securty mechasms e-busess processes usg fuzzy TOPSIS, Iteratoal Joural of Fuzzy System pplcato 0 5-5. [9].K. Verma, R. Verma, ad N.C. Mahataclty locato selecto: a terval valued tutostc fuzzy TOPSIS approach, Joural of Moder Mathematcs ad Statstcs 4 00 8-7. [0] Y.M.Wag, sg the method of maxmzg devatos to make decso for mult-dces, System Egeerg ad Electrocs 7998 4,. Receved: November 0, 0. ccepted: December 8, 0. Pgpg Ch, ad Pede Lu, Exteded TOPSIS Method Based o Iterval Neutrosophc Set