Vector Similarity Measures between Refined Simplified Neutrosophic Sets and Their Multiple Attribute Decision-Making Method

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1 S S symmetry Artcle Vector Smlarty Measures betwee Refed Smplfed Neutrosophc Sets ad Ther Multple Attrbute Decso-Makg Method Jqa Che 1, Ju Ye 1,2, * ad Shgu Du 1 1 Key Laboratory of Rock Mechacs ad Geohazards, Shaoxg Uversty, 508 Huacheg West Road, Shaoxg , Cha; chejqa@yahoo.com J.C.; dsg@usx.edu.c S.D. 2 Departmet of Electrcal ad Iformato Egeerg, Shaoxg Uversty, 508 Huacheg West Road, Shaoxg , Cha * Correspodece: yeju@usx.edu.c Receved: 17 July 2017; Accepted: 8 August 2017; Publshed: 11 August 2017 Abstract: A refed sgle-valued/terval eutrosophc set s very sutable for the expresso ad applcato of decso-makg problems wth both attrbutes ad sub-attrbutes sce t s descrbed by ts refed truth, determacy, ad falsty degrees. However, exstg refed sgle-valued/terval eutrosophc smlarty measures ad ther decso-makg methods are scarcely studed exstg lterature ad caot deal wth ths decso-makg problem wth the weghts of both attrbutes ad sub-attrbutes a refed terval ad/or sgle-valued eutrosophc settg. To solve the ssue, ths paper frstly troduces a refed smplfed eutrosophc set RSNS, whch cotas the refed sgle-valued eutrosophc set RSVNS ad refed terval eutrosophc set RINS, ad the proposes vector smlarty measures of RSNSs based o the Jaccard, Dce, ad cose measures of smplfed eutrosophc sets vector space, ad the weghted Jaccard, Dce, ad cose measures of RSNSs by cosderg weghts of both basc elemets ad sub-elemets RSNS. Further, a decso-makg method wth the weghts of both attrbutes ad sub-attrbutes s developed based o the weghted Jaccard, Dce, ad cose measures of RSNSs uder RSNS RINS ad/or RSVNS evromets. The rakg order of all the alteratves ad the best oe ca be determed by oe of weghted vector smlarty measures betwee each alteratve ad the deal soluto deal alteratve. Fally, a actual example o the selectg problem of costructo projects llustrates the applcato ad effectveess of the proposed method. Keywords: refed smplfed eutrosophc set; refed sgle-valued eutrosophc set; refed terval eutrosophc set; vector smlarty measure; decso-makg 1. Itroducto Sce fuzzy set theory was troduced by Zadeh [1] 1965, t has bee successfully appled to decso-makg areas, ad fuzzy decso-makg has become a research focal pot sce the. Wth the creasg complexty of decso-makg problems actual applcatos, the fuzzy set s ot sutable for fuzzy expresso, whch volves the membershp degree ad o-membershp degree. Hece, a tutostc fuzzy set IFS [2] ad a terval-valued IFS [3] were troduced as the geeralzato of fuzzy set ad appled to decso-makg problems. However, the complete, determate, ad cosstet problems real lfe caot be explaed by meas of the IFS ad terval-valued IFS. Therefore, Smaradache [4] proposed the cocept of a eutrosophc set from a phlosophcal pot of vew, whch cossts of the truth, determacy, ad falsty memershp fuctos, deoted by T, I, F, to represet complete, determate, ad cosstet formato the real world. Sce the truth, determacy, ad falsty membershp degrees of T, I, F Symmetry 2017, 9, 153; do: /sym

2 Symmetry 2017, 9, of 13 the eutrosophc set le the real stadard/ostadard terval ] 0, 1 + [, Smaradache [4], Wag et al. [5,6], ad Ye [7,8] costraed the three membershp degrees the eutrosophc set to the sgle-valued membershp degrees ad the terval membershp degrees. These become a sgle-valued eutrosophc set SVNS, a terval eutrosophc set INS, ad a smplfed eutrosophc set SNS cludg SVNS ad INS, respectvely. Obvously, they are subclasses of the eutrosophc set for coveet applcatos scece ad egeerg felds, such as decso-makg [7 13] ad fault dagoss [14]. However, because there are both argumets ad sub-argumets/refed argumets the truth, determacy, ad falsty membershp degrees of T, I, F the eutrosophc set to express complex problems of the real world detal, oe eeds to refe truth, determacy, ad falsty formato. Hece, Smaradache [15] further exteded the eutrosophc logc to -valued refed eutrosophc logc, where he refed/splt the truth, determacy, ad falsty fuctos T, I, F to T 1, T 2,..., T r, I 1, I 2,..., I s, ad F 1, F 2,..., F t, respectvely, ad costructed them as a -valued refed eutrosophc set. Moreover, some researchers exteded the eutrosophc set to mult-valued eutrosophc set/eutrosophc multset/eutrosophc refed sets ad appled them to medcal dagoses [16 18] ad decso-makg [19 21]. I fact, the mult-valued eutrosophc sets/eutrosophc refed sets are eutrosophc multsets ther expressed forms [22,23]. Hece, these mult-valued eutrosophc sets/eutrosophc refed sets, that s, eutrosophc multsets, ad ther decso-makg methods caot express ad deal wth decso-makg problems wth both attrbutes ad sub-attrbutes. To solve the ssue, Ye ad Smaradache [22] proposed a refed sgle-valued eutrosophc set RSVNS, where the eutrosophc set {T, I, F} was refed to the RSVNS {T 1, T 2,..., T r, I 1, I 2,..., I r, F 1, F 2,..., F r }, ad proposed the smlarty measures based o uo ad tersecto operatos of RSVNSs to solve decso-makg problems wth both attrbutes ad sub-attrbutes. The, Fa ad Ye [23] further preseted the cose measures of RSVNSs ad refed terval eutrosophc sets RINSs based the dstace ad cose fucto ad appled them to the decso-makg problems wth both attrbutes ad sub-attrbutes uder refed sgle-value/terval eutrosophc evromets. However, these cose measures caot hadle such a decso-makg problem wth the weghts of both attrbutes ad sub-attrbutes. I fact, RINSs ad/or RSVNSs are scarcely studed ad appled scece ad egeerg felds. Therefore, t s ecessary to develop ew smlarty measures ad ther decso-makg method refed terval ad/or sgle-value eutrosophc evromets. However, exstg lterature [22,23], the smlarty measures of RSVNSs ad RINSs ad ther decso-makg methods oly took to accout the basc elemet sgle-valued/terval eutrosophc umber RSVNS/RINS/attrbute weghts rather tha sub-elemet/sub-attrbute weghts weghts of refed elemets/refed attrbutes the measures of RSVNSs ad RINSs ad ther decso-makg methods. To overcome these drawbacks, ths paper frstly troduces a refed smplfed eutrosophc set RSNS, whch cludes the cocepts of RSVNS ad RINS, ad proposes the vector smlarty measures of RSNSs based o the Jaccard, Dce, ad cose measures betwee SNSs vector space [8]. Further, a decso-makg method s establshed based o the Jaccard/Dce/cose measures betwee RSNSs to solve multple attrbute decso-makg problems wth both attrbute weghts ad sub-attrbute weghts uder refed smplfed terval ad/or sgle-value eutrosophc evromets. The ma advatages of the proposed approach are that t ca solve decso-makg problems wth the weghts of both attrbutes ad sub-attrbutes ad exted exstg smlarty measures ad decso-makg methods [22,23], because the exstg smlarty measures ad decso-makg methods caot deal wth such a decso-makg problem wth the weghts of both attrbutes ad sub-attrbutes uder RSNS RINS ad/or RSVNS evromets. The rest of the paper s structured as follows. Secto 2 revews basc cocepts of SNSs ad vector smlarty measures of SNSs. I Secto 3, we troduces a RSNS cocept, cludg RSVNS ad RINS. Secto 4 proposes the Jaccard, Dce, ad cose smlarty measures three vector smlarty measures betwee RSNSs by cosderg weghts of elemets ad sub-elemets/refed elemets RSNSs. Secto 5 develops a multple attrbute decso-makg method wth both attrbute weghts ad

3 Symmetry 2017, 9, of 13 sub-attrbute weghts based o oe of three vector smlarty measures uder refed smplfed terval ad/or sgle-value eutrosophc evromets. I Secto 6, a actual example o the selecto problem of costructo projects s provded as the multple attrbute decso-makg problem wth both attrbute weghts ad sub-attrbute weghts to llustrate the applcato ad effectveess of the proposed method. Fally, coclusos ad future research are cotaed Secto Basc Cocepts of SNSs ad Vector Smlarty Measures of SNSs I 1995, Smaradache [4] proposed a cocept of eutrosophc sets from a phlosophcal pot of vew, whch s a part of eutrosophy ad exteds the cocepts of fuzzy sets, terval valued fuzzy sets, IFSs, ad terval valued IFSs. A eutrosophc set s characterzed depedetly by the truth, determacy ad falsty membershp fuctos, whch le a real stadard terval [0, 1] or a ostadard terval ] 0, 1 + [. For coveet scece ad egeerg applcatos, we eed to costra them the real stadard terval [0, 1] from a scece ad egeerg pot of vew. Thus, Ye [7,8] troduced the cocept of SNS as a smplfed form/subclass of the eutrosophc set. A SNS A a uverse of dscourse X s characterzed by ts truth, determacy, ad falsty membershp fuctos T A x, I A x, ad F A x, whch s deoted as A = { x, T A x, I A x, F A x x X}, where T A x, I A x ad F A x are sgleto subtervals/subsets the real stadard [0, 1], such that T A x: X [0, 1], I S x: X [0, 1], ad F S x: X [0, 1]. The, the SNS A cotas SVNS for T A x, I A x, F A x [0, 1] ad INS for T A x, I A x, F A x [0, 1]. For coveet expresso, a basc elemet x, T A x, I A x, F A x A s smply deoted as a smplfed eutrosophc umber SNN a = <T a, I a, F a >, where a cotas a sgle-value eutrosophc umber SVNN for T a, I a, F a [0, 1] ad a terval eutrosophc umber INN for T a, I a, F a [0, 1]. Assume that two SNSs are A ={a 1, a 2,..., a } ad B ={b 1, b 2,..., b }, where a j = <T aj, I aj, F aj > ad b j = <T bj, I bj, F bj > fo = 1, 2,..., are two collectos of SNNs. Based o the Jaccard, Dce, ad cose measures betwee two vectors, Ye [8] preseted the ther smlarty measures betwee SNSs SVNSs ad INSs A ad B vector space, respectvely, as follows: 1 Three vector smlarty measures betwee A ad B for SVNSs: M J A, B = 1 [ Taj 2 + I2 aj + F2 aj + M D A, B = 1 T aj T bj + I aj I bj + F aj F bj ] Tbj 2 + I2 bj + 1 F2 bj T aj T bj + I aj I bj + F aj F bj 2 T aj T bj + I aj I bj + F aj F aj T 2 aj + I2 aj + F2 aj + Tbj 2 + I2 bj + 2 F2 bj M C A, B = 1 T aj T bj + I aj I bj + F aj F bj Taj 2 + I2 aj + F2 aj Tbj 2 + I2 bj + F2 bj 3 2 Three vector smlarty measures betwee A ad B for INSs: M J A, B = 1 M D A, B = 1 ft aj ft bj + supt aj supt bj + fi aj fi bj + supi aj supi bj + ff aj ff bj + supf aj supf bj ft aj 2 + fi aj 2 + ff aj 2 + supt aj 2 + supi aj 2 + supf aj 2 +ft bj 2 + fi bj 2 + ff bj 2 + supt bj 2 + supi bj 2 + supf bj 2 ft aj ft bj + fi aj fi bj + ff aj ff bj supt aj supt bj + supi aj supi bj + supf aj supf bj 2 ft aj ft bj + fi aj fi bj + ff aj ff bj + supt aj supt bj + supi aj supi bj + supf aj supf bj ftaj 2 + fi aj 2 + ff aj 2 + supt aj 2 + supi aj 2 + supf aj 2 +ft bj 2 + fi bj 2 + ff bj 2 + supt bj 2 + supi bj 2 + supf bj 2 4 5

4 Symmetry 2017, 9, of 13 M C A, B = 1 ft aj ft bj + fi aj fi bj + ff aj ff bj + supt aj supt bj + supi aj supi bj + supf aj supf bj ft aj 2 + fi aj 2 + ff aj 2 + supt aj 2 + supi aj 2 + supf aj 2 ft bj 2 + fi bj 2 + ff bj 2 + supt bj 2 + supi bj 2 + supf bj 2 6 Clearly, Equatos 1 3 are specal cases of Equatos 4 6 whe the upper ad lower lmts of the terval umbers for T aj = [f T aj, sup T aj ] I aj = [f I aj, sup I aj ], F aj = [f F aj, sup F aj ], T bj = [f T bj, sup T bj ], I bj = [f I bj, sup I bj ], ad F bj = [f F bj, sup F bj ] are equal. The, the Jaccard, Dce, ad cose measures M k A, B k = J, D, C cotas the followg propertes [8]: P1 0 M k A, B 1; P2 M k A, B = M k B, A; P3 M k A, B = 1 f A = B,.e., T aj, = T bj, I aj = I bj, ad F aj = F bj fo = 1, 2,...,. 3. Refed Smplfed Neutrosophc Sets As the cocept of SNS [7,8], a SNS A a uverse of dscourse X s deoted as A = { x, T A x, I A x, F A x x X}, where the values of ts truth, determacy, ad falsty membershp fuctos T A x, I A x ad F A x for x X are sgle-value ad/or terval values [0, 1]. The, SNS cota INS ad/or SVNS. If the compoets T A x, I A x, F A x SNS are refed splt to T A x 1, T A x 2,..., T A x r, I A x 1, I A x 2,..., I A x r, ad F A x 1, F A x 2,..., F A x r, respectvely, for x X, x = {x 1, x 2,..., x r }, ad a postve teger r, the they ca be costructed as RSNS by the refemet of SNS, whch s defed below. Defto 1. Let X be a uverse of dscourse, the a RSNS A X ca be defed as A = { x, T A x 1, T A x 2,..., T A x r, I A x 1, I A x 2,..., I A x r, F A x 1, F A x 2,..., F A x r x X, xj x }, where T A x 1, T A x 2,..., T A x r, I A x 1, I A x 2,..., I A x r, F A x 1, F A x 2,..., F A x r for x X, x j x = {x 1, x 2,..., x r } j = 1, 2,..., r, ad a postve teger r are subtervals/subsets the real stadard terval [0, 1], such that T A x 1, T A x 2,..., T A x r : X [0, 1], I A x 1, I A x 2,..., I A x r : X [0, 1], ad F A x 1, F A x 2,..., F A x r : X [0, 1]. The, the RSNS A cotas the followg two cocepts: 1 If T A x 1, T A x 2,..., T A x r [0, 1], I A x 1, I A x 2,..., I A x r [0, 1], ad F A x 1, F A x 2,..., F A x r [0, 1] A for x X ad x j x j =1, 2,..., r are cosdered as sgle/exact values [0, 1], the A reduces to RSVNS [22], whch satsfes the codto 0 T A x j + I A x j + F A x j 3 fo = 1, 2,..., r; 2 If T A x 1, T A x 2,..., T A x r [0, 1], I A x 1, I A x 2,..., I A x r [0, 1], ad F A x 1, F A x 2,..., F A x r [0, 1] A for x X ad x j x j =1, 2,..., r are cosdered as terval values [0, 1], the A reduces to RINS [23], whch satsfes the codto 0 supt A x j + supi A x j + supf A x j 3 fo = 1, 2,..., r. Partcularly whe the lower ad upper lmts of T A x j = [f T A x j, sup T A x j ], I A x j = [f I A x j, sup I A x j ] ad F A x j = [f F A x j, sup F A x j ] A for x X ad x j x j = 1, 2,..., r are equal, the RINS A reduces to the RSVNS A. Clearly, RSVNS s a specal case of RINS. If some lower ad upper lmts of T A x j = [f T A x j, sup T A x j ]/I A x j = [f I A x j, sup I A x j ]/F A x j = [f F A x j, sup F A x j ] RINS are equal, the t ca be deoted as a specal terval equal terval of the lower ad upper lmts T A x j = [T A x j, T A x j ]/I A x j = [I A x j, I A x j ]/F A x j = [F A x j, F A x j ]. Hece, RINS ca cota RINS ad/or SVNS formato hybrd formato of both.

5 Symmetry 2017, 9, of 13 For coveet expresso, a basc elemet <x, T A x 1, T A x 2,..., T A x r, I A x 1, I A x 2,..., I A x r, F A x 1, F A x 2,..., F A x r > A s smply deoted as a = <T a1, T a2,..., T ar, I a1, I a2,..., I ar, F a1, F a2,..., F ar >, whch s called a refed smplfed eutrosophc umber RSNN. Let two RSNNs be a = <T a1, T a2,..., T ar, I a1, I a2,..., I ar, F a1, F a2,..., F ar > ad b = <T b1, T b2,..., T br, I b1, I b2,..., I br, F b1, F b2,..., F br > for T aj, T bj, I aj, I bj, F aj, F bj [0, 1] j = 1, 2,..., r. The, there are the followg relatos betwee a ad b: 1 Cotamet: a b, f ad oly f T aj T bj, I aj I bj, F aj F bj fo = 1, 2,..., r; 2 Equalty: a = b, f ad oly f a b ad b a,.e., T aj = T bj, I aj = I bj, F aj = F bj fo = 1, 2,..., r; 3 Uo: a b = T a1 T b1, T a2 T b2,..., T ar T br, I a1 I b1, I a2 I b2,..., I ar I br, F a1 F b1, F a2 F b2,..., F ar F br ; 4 Itersecto: a b = T a1 T b1, T a2 T b2,..., T ar T br, I a1 I b1, I a2 I b2,..., I ar I br, F a1 F b1, F a2 F b2,..., F ar F br. Let two RSNNs be a = T a1, T a2,..., T ar, I a1, I a2,..., I ar, F a1, F a2,..., F ar ad b = T b1, T b2,..., T br, I b1, I b2,..., I br, F b1, F b2,..., F br for T aj, T bj, I aj, I bj, F aj, F bj [0, 1] j = 1, 2,..., r. The, there are the followg relatos of a ad b: 1 Cotamet: a b, f ad oly f f T aj f T bj, sup T aj sup T bj, f I aj f I bj, sup I aj sup I bj, f F aj f F bj, ad sup F aj sup F bj fo = 1, 2,..., r; 2 Equalty: a = b, f ad oly f a b ad b a,.e., f T aj = f T bj, sup T aj = sup T bj, f I aj = f I bj, sup I aj = sup I bj, f F aj = f F bj, ad sup F aj = sup F bj fo = 1, 2,..., r; 3 Uo: a b = [fta1 ft b1, supt a1 supt b1 ], [ft a2 ft b2, supt a2 supt b2 ],..., [ft ar ft br, supt ar supt br ], [fi a1 fi b1, supi a1 supi b1 ], [fi a2 fi b2, supi a2 supi b2 ],..., [fi ar fi br, supi ar supi br ], [ff a1 ff b1, supf a1 supf b1 ], [ff a2 ff b2, supf a2 supf b2 ],..., [ff ar ff br, supf ar supf br ] 4 Itersecto: a b = [fta1 ft b1, supt a1 supt b1 ], [ft a2 ft b2, supt a2 supt b2 ],..., [ft ar ft br, supt ar supt br ], [fi a1 fi b1, supi a1 supi b1 ], [fi a2 fi b2, supi a2 supi b2 ],..., [fi ar fi br, supi ar supi br ], [ff a1 ff b1, supf a1 supf b1 ], [ff a2 ff b2, supf a2 supf b2 ],..., [ff ar ff br, supf ar supf br ] 4. Vector Smlarty Measures of RSNSs Based o the Jaccard, Dce, ad cose measures betwee SNSs vector space [8], ths secto proposes the three vector smlarty measures betwee RSNSs. Defto 2. Let two RSNSs be A ={a 1, a 2,..., a } ad B ={b 1, b 2,..., b }, where a j = T aj 1, T aj 2,..., T aj, I aj 1, I aj 2,..., I aj, F aj 1, F aj 2,..., F aj ad b j = T bj 1, T bj 2,..., T bj, I bj 1, I bj 2,..., I bj, F bj 1, F bj 2,..., F bj fo = 1, 2,..., are two collectos of RSNNs for T aj k, I aj k, F aj k, T bj k, I bj k, F bj k [0, 1] or T aj k, I aj k, F aj k, T bj k, I bj k, F bj k [0, 1] j = 1, 2,..., ; k = 1, 2,...,. The, the Jaccard, Dce, ad cose measures betwee A ad B are defed, respectvely, as follows: 1 Three vector smlarty measures betwee A ad B for RSVNSs: R J A, B = 1 1 [ Ta 2 j k +I2 a j k +F2 a j k + T aj kt bj k+i aj k I bj k+f aj kf bj k Tb 2 j k +I2 b j k +F2 b j k T aj kt bj k+i aj k I bj k+f aj kf bj k ] 7

6 Symmetry 2017, 9, of 13 R D A, B = 1 r 1 j r j 2 T aj kt bj k + I aj k I bj k + F aj kf bj k + T 2 a j k + I2 a j k + F2 a j k T 2 b j k + I2 b j k + F2 b j k 8 R C A, B = 1 r 1 j r j T aj kt bj k + I aj k I bj k + F aj kf bj k T 2 + ajk I2 ajk + F2 ajk T 2 + bjk I2 bjk + F2 bjk 9 2 Three vector smlarty measures betwee A ad B for RINSs: R J A, B = 1 1 ft a j kft bj k + supt aj ksupt bj k + fi aj kfi bj k +supi aj ksupi bj k + ff aj kff bj k + supf aj ksupf bj k ft aj k 2 + fi aj k 2 + ff aj k 2 + supt aj k 2 + supi aj k 2 + supf aj k 2 +ft bj k 2 + fi bj k 2 + ff bj k 2 + supt bj k 2 + supi bj k 2 + supf bj k 2 ft aj kft bj k + fi aj kfi bj k + ff aj kff bj k supt aj ksupt bj k + supi aj ksupi bj k + supf aj ksupf bj k, 10 R D A, B = ft a j kft bj k + fi aj kfi bj k + ff aj kff bj k +supt aj ksupt bj k + supi aj ksupi bj k + supf aj ksupf bj k ft aj k 2 + fi aj k 2 + ff aj k 2 + supt aj k 2 + supi aj k 2 + supf aj k 2 +ft bj k 2 + fi bj k 2 + ff bj k 2 + supt bj k 2 + supi bj k 2 + supf bj k 2, 11 R C A, B = 1 1 ft a j kft bj k + fi aj kfi bj k + ff aj kff bj k +supt aj ksupt bj k + supi aj ksupi bj k + supf aj ksupf bj k ft aj k 2 + fi aj k 2 + ff aj k 2 + supt aj k 2 + supi aj k 2 + supf aj k 2 ft bj k 2 + fi bj k 2 + ff bj k 2 + supt bj k 2 + supi bj k 2 + supf bj k 2 12 Clearly, Equatos 7 9 are specal cases of Equatos whe the upper ad lower lmts of the terval umbers for T aj k, I aj k, F aj k,t bj k, I bj k, F bj k [0, 1] j = 1, 2,..., ; k = 1, 2,..., are equal. Especally whe k = 1, Equatos 7 12 are reduced to Equatos 1 6. Based o the propertes of the Jaccard, Dce, ad cose measures of SNSs [8], t s obvous that the Jaccard, Dce, ad cose measures of RSNSs for R s A, B s = J, D, C also cota the followg propertes P1 P3: P1 0 R s A, B 1; P2 R s A, B = R s B, A; P3 R s A, B = 1 f A = B,.e., T aj k = T bj k, I aj k = I bj k, F aj k = F bj k fo = 1, 2,..., ad k = 1, 2,...,. Whe we cosder the weghts of dfferet elemets ad sub-elemets RSNS, the weght of elemets a j ad b j j = 1, 2,..., the RSNSs A ad B s gve as w j [0, 1] wth w j = 1 ad the weght of the refed compoets sub-elemets T aj k, I aj k, F aj k ad T bj k, I bj k, F bj k k = 1, 2,..., a j = T aj 1, T aj 2,..., T aj, I aj 1, I aj 2,..., I aj, F aj 1, F aj 2,..., F aj ad b j = T bj 1, T bj 2,..., T bj, I bj 1, I bj 2,..., I bj, F bj 1, F bj 2,..., F bj j = 1, 2,..., s cosdered as ω k [0, 1] wth ω k = 1, the weghted Jaccard, Dce, ad cose measures betwee A ad B are preseted, respectvely, as follows: 1 Three weghted vector smlarty measures betwee A ad B for RSVNSs: R W J A, B = w j T aj kt bj k+i aj k I bj k+f aj kf bj k ω k [ Ta 2 j k +I2 a j k +F2 a j k + Tb 2 j k +I2 b j k +F2 b j k T aj kt bj k+i aj k I bj k+f aj kf bj k ] 13

7 Symmetry 2017, 9, of 13 R WD A, B = w j 2 T aj kt bj k + I aj k I bj k + F aj kf bj k ω k Ta 2 j k + I2 a j k + F2 a j k + Tb 2 j k + I2 b j k + 14 F2 b j k R WC A, B = w j ω k T aj kt bj k + I aj k I bj k + F aj kf bj k T 2 + ajk I2 ajk + F2 ajk T 2 + bjk I2 bjk + F2 bjk 15 2 Three weghted vector smlarty measures betwee A ad B for RINSs: R W J A, B = w j ω k ft a j kft bj k + supt aj ksupt bj k + fi aj kfi bj k +supi aj ksupi bj k + ff aj kff bj k + supf aj ksupf bj k ft aj k 2 + fi aj k 2 + ff aj k 2 + supt aj k 2 + supi aj k 2 + supf aj k 2 +ft bj k 2 + fi bj k 2 + ff bj k 2 + supt bj k 2 + supi bj k 2 + supf bj k 2 ft aj kft bj k + fi aj kfi bj k + ff aj kff bj k supt aj ksupt bj k + supi aj ksupi bj k + supf aj ksupf bj k, 16 R WD A, B = w j R WC A, B = w j ft 2 a j kft bj k + fi aj kfi bj k + ff aj kff bj k +supt aj ksupt bj k + supi aj ksupi bj k + supf aj ksupf bj k ω k ft aj k 2 + fi aj k 2 + ff aj k 2 + supt aj k 2 + supi aj k 2 + supf aj k 2 ω k +ft bj k 2 + fi bj k 2 + ff bj k 2 + supt bj k 2 + supi bj k 2 + supf bj k 2 ft a j kft bj k + fi aj kfi bj k + ff aj kff bj k +supt aj ksupt bj k + supi aj ksupi bj k + supf aj ksupf bj k ft aj k 2 + fi aj k 2 + ff aj k 2 + supt aj k 2 + supi aj k 2 + supf aj k 2 ft bj k 2 + fi bj k 2 + ff bj k 2 + supt bj k 2 + supi bj k 2 + supf bj k 2, Clearly, Equatos are specal cases of Equatos whe the upper ad lower lmts of the terval umbers for T aj k, I aj k, F aj k,t bj k, I bj k, F bj k [0, 1] j = 1, 2,..., ; k = 1, 2,..., are equal. Especally whe each w j = 1/ ad ω k = 1/ j = 1, 2,..., ; k = 1, 2,...,, Equatos are reduced to Equatos Obvously, the weghted Jaccard, Dce, ad cose measures of RSNSs for R Ws A, B s = J, D, C also satsfes the followg propertes P1 P3: P1 0 R Ws A, B 1; P2 R Ws A, B = R Ws B, A; P3 R Ws A, B = 1 f A = B,.e., T aj k = T bj k, I aj k = I bj k, F aj k = F bj k fo = 1, 2,..., ad k = 1, 2,...,. 5. Decso-Makg Method Usg the Vector Smlarty Measures I a decso-makg problem wth multple attrbutes ad sub-attrbutes, assume that A = {A 1, A 2,..., A m } s a set of m alteratves, whch eeds to satsfes a set of attrbutes B = {b 1, b 2,..., b }}, where b j j = 1, 2,..., may be refed/splt to a set of sub-attrbutes b j = {b j1, b j2,..., b jrj j = 1, 2,...,. If } the decso-maker provdes the sutablty evaluato values of attrbutes b j = {b j1, b j2,..., b jrj j =1, 2,..., o the alteratve A = 1, 2,..., m by usg RSNS: A = { } bj b j,t A b j1,t A b j2,...,t A b jrj,i A b j1, I A b j2,..., I A b jrj,f A b j1,f A b j2,...,f A b jrj B,b jk b j. For coveet expresso, each basc elemet the RSNS A s represeted by RSNN: a j = T aj 1,T aj 2,...,T aj,i aj 1, I aj 2,..., I aj,f aj 1,F aj 2,...,F aj for = 1, 2,..., m ad j = 1, 2,...,. Hece, we ca costruct the refed smplfed eutrosophc decso matrx Ma j m, as show Table 1.

8 Symmetry 2017, 9, of 13 Table 1. Refed smplfed eutrosophc decso matrx Ma j m. b 1 b 2... b {b 11, b 12,..., b 1r1 } {b 21, b 22,..., b 2r2 } {b 1, b 2,..., b r } A 1 a 11 a a 1 A 2 a 21 a a A m a m1 a m2... a m Whe the weghts of each attrbute b j j = 1, 2,..., ad ts sub-attrbutes are cosdered as havg dfferet mportace, the weght vector of the attrbutes s gve by W = w 1, w 2,... }, w wth w j [0, 1] ad w j = 1 ad the weght vector for each sub-attrbute set {b j1, b j2,..., b jrj s gve } as ω j = {ω j1, ω j2,..., ω jrj j = 1, 2,..., wth ω jk [0, 1] ad ω jk = 1. Thus, the decso steps are descrbed as follows: Step 1: We determe the deal soluto deal RSNN from the refed smplfed eutrosophc decso matrx Ma j m as follows: a j = max T j1, max..., m T j2,..., maxt jrj, mi j1, mi j2, I jrj, mf j1, mf j2,..., mf jrj for RSVNN 19 or a j = [max [m [m ft j1, maxsupt j1 ], [maxft j2, maxsupt j2 ],..., [maxft jrj, maxsupt jrj ], fi j1, msupi j1 ], [mfi j2, msupi j2 ],..., [mfi jrj, msupi jrj ], for RINN, 20 ff j1, msupf j1 ], [mff j2, msupf j2 ],..., [mff jrj, msupf jrj ] whch s costructed as the deal alteratve A = { a 1, a 2,..., a }. Step 2: The smlarty measure betwee each alteratve A = 1, 2,..., m ad the deal alteratve A * ca be calculated by usg oe of Equatos or Equatos 16 18, ad obtaed as the values of R Ws A, A * for = 1, 2,..., m ad s = J or D or C. Step 3: Accordg to the values of R Ws A, A * for = 1, 2,..., m ad s = J or D or C, the alteratves are raked a descedg order. The greater value of R Ws A, A * meas the best alteratve. Step 4: Ed. 6. Illustratve Example o the Selecto of Costructo Projects I ths secto, we apply the proposed decso-makg method to the selecto of costructo projects adapted from [23]. Some costructo compay wats to select oe of potetal costructo projects. The compay provdes four potetal costructo projects as ther set A = {A 1, A 2, A 3, A 4 }. To select the best oe of them, experts or decso-makers eed to make a decso of these costructo projects correspodg to three attrbutes ad ther seve sub-attrbutes, whch are descrbed as follows: 1 Facal state b 1 cotas two sub-attrbutes: budget cotrol b 11 ad rsk/retur rato b 12 ; 2 Evrometal protecto b 2 cotas three sub-attrbutes: publc relato b 21, geographcal locato b 22, ad health ad safety b 23 ; 3 Techology b 3 cotas tow sub-attrbutes: techcal kow-how b 31, techologcal capablty b 32. The, the weght vector of the three attrbutes s gve by W = 0.4, 0.3, 0.3 ad the weght vectors of the three sub-attrbute sets {b 11, b 12 }, {b 21, b 22, b 23 }, ad {b 31, b 32 } are gve, respectvely, by ω 1 = 0.6, 0.4, ω 2 = 0.25, 0.4, 0.35, ad ω 3 = 0.45, 0.55.

9 Symmetry 2017, 9, of 13 I the followg, we use the proposed decso-makg method for solvg the decso-makg problem of costructo projects uder RSVNN ad/or RINN evromets to show the applcatos ad effectveess of the proposed decso-makg method. Uder RSVNN evromet, experts or decso-makers are requred to evaluate the four possble alteratves uder the above three attrbutes cludg seve sub-attrbutes by sutablty judgmets, whch are descrbed by RSVNN a j = T aj 1, T aj 2,..., T aj, I aj 1, I aj 2,..., I aj, F aj 1, F aj 2,..., F aj for T aj 1, T aj 2,..., T aj [0, 1], I aj 1, I aj 2,..., I aj [0, 1], ad F aj 1, F aj 2,..., F aj [0, 1] = 1, 2, 3, 4; j = 1, 2, 3; r 1 = 2, r 2 = 3, r 3 = 2. Thus, we ca costruct the followg refed smplfed eutrosophc decso matrx Ma j 4 3, whch s show Table 2. Table 2. Defed smplfed eutrosophc decso matrx Ma j 4 3 uder refed sgle-valued eutrosophc set RSVNS evromet. b 1 b 2 b 3 {b 11, b 12 } {b 21, b 22, b 23 } {b 31, b 32 } A 1 <0.6, 0.7, 0.2, 0.1, 0.2, 0.3> <0.9, 0.7, 0.8, 0.1, 0.3, 0.2, 0.2, 0.2, 0.1> <0.6, 0.8, 0.3, 0.2, 0.3, 0.4> A 2 <0.8, 0.7, 0.1, 0.2, 0.3, 0.2> <0.7, 0.8, 0.7, 0.2, 0.4, 0.3, 0.1, 0.2, 0.1> <0.8, 0.8, 0.1, 0.2, 0.1, 0.2> A 3 <0.6, 0.8, 0.1, 0.3, 0.3, 0.4> <0.8, 0.6, 0.7, 0.3, 0.1, 0.1, 0.2, 0.1, 0.2> <0.8, 0.7, 0.4, 0.3, 0.2, 0.1> A 4 <0.7, 0.6, 0.1, 0.2, 0.2, 0.3> <0.7, 0.8, 0.7, 0.2, 0.2, 0.1, 0.1, 0.2, 0.2> <0.7, 0.7, 0.2, 0.3, 0.2, 0.3> Uder RSVNS evromet, the proposed decso-makg method s appled to the selecto problem of the costructo projects. The decso steps are descrbed as follows: Step 1: By Equato 19, the deal soluto deal RSVNS ca be determed as the followg deal alteratve: A * = {<0.8, 0.8, 0.1, 0.1, 0.2, 0.2>, <0.9, 0.8, 0.8, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1>, <0.8, 0.8, 0.1, 0.2, 0.1, 0.1>}. Step 2: Accordg to oe of Equatos 13 15, the weghted smlarty measure values betwee each alteratve A = 1, 2, 3, 4 ad the deal alteratve A * ca be obtaed ad all the results are show Table 3. Step 3: I Table 3, sce all the measure values are R Ws A 2, A * > R Ws A 4, A * > R Ws A 3, A * > R Ws A 1, A * for s = J, D, C, all the rakg orders of the four alteratves are A 2 A 4 A 3 A 1. Hece, the alteratve A 2 s the best choce amog all the costructo projects. Table 3. All the measure values betwee A = 1, 2, 3, 4 ad A * for RSVNSs ad rakg orders of the four alteratves. Measure Method Measure Value Rakg Order The Best Choce WA, A * [23] R WJ A, A * R WD A, A * R WC A, A * WA 1, A * = , WA 2, A * = , WA 3, A * = , WA 4, A * = R WJ A 1, A * = , R WJ A 2, A * = , R WJ A 3, A * = , R WJ A 4, A * = R WD A 1, A * = , R WD A 2, A * = , R WD A 3, A * = , R WD A 4, A * = R WC A 1, A * = , R WC A 2, A * = , R WC A 3, A * = , R WC A 4, A * = Uder RINS evromet, o the other had, experts or decso-makers are requred to evaluate the four possble alteratves uder the above three attrbutes cludg seve sub-attrbutes by sutablty judgmets, whch are descrbed by RINN a j = T aj 1, T aj 2,..., T aj, I aj 1, I aj 2,..., I aj, F aj 1, F aj 2,..., F aj for T aj 1, T aj 2,..., T aj [0, 1], I aj 1, I aj 2,..., I aj [0, 1], ad F aj 1, F aj 2,..., F aj [0, 1] = 1, 2, 3, 4; j = 1, 2, 3; r 1 = 2, r 2 = 3,

10 Symmetry 2017, 9, of 13 r 3 = 2. Thus, we ca costruct the followg refed smplfed eutrosophc decso matrx Ma j 4 3, whch s show Table 4. Table 4. Defed smplfed eutrosophc decso matrx Ma j 4 3 uder refed terval eutrosophc set RINS evromet. A 1 <[0.6,0.7],[ 0.7,0.8],[0.2,0.3], [ 0.1,0.2],[0.2,0.3],[ 0.3,0.4]> A 2 <[0.8,0.9],[0.7,0.8],[0.1,0.2], [ 0.2,0.3],[0.3,0.4],[ 0.2,0.3]> A 3 <[0.6,0.7],[0.8,0.9],[0.1,0.2], [ 0.3,0.4],[0.3,0.4],[0.4,0.5]> A 4 <[0.7,0.8],[0.6,0.7],[0.1,0.2], [0.2,0.3],[0.2,0.3],[0.3,0.4]> b 1 b 2 b 3 {b 11, b 12 } {b 21, b 22, b 23 } {b 31, b 32 } <[0.8,0.9],[0.7,0.8],[0.8,0.9], [0.1,0.2],[0.3,0.4],[0.2,0.3], [0.2,0.3],[0.2,0.3],[0.1,0.2]> <[0.7,0.8],[0.8,0.9],[0.7,0.8], [0.2,0.3],[0.4,0.5],[0.3,0.4], [0.1,0.2],[0.2,0.3],[0.1,0.2]> <[0.8,0.9],[0.6,0.7],[0.7,0.8], [0.3,0.4],[0.1,0.2],[0.1,0.2], [0.2,0.3],[0.1,0.2],[0.2,0.3]> <[0.7,0.8],[0.8,0.9],[0.7,0.8], [0.2,0.3],[0.2,0.3],[0.1,0.2], [0.1,0.2],[0.2,0.3],[0.2,0.3]> <[0.6,0.7],[0.8,0.9],[0.3,0.4], [0.2,0.3],[0.3,0.4],[0.4,0.5]> <[0.8,0.9],[0.8,0.9],[0.1,0.2], [0.2,0.3],[0.1,0.2],[0.2,0.3]> <[0.8,0.9],[0.7,0.8],[0.4,0.5], [0.3,0.4],[0.2,0.3],[0.1,0.2]> <[0.7,0.8],[0.7,0.8],[0.2,0.3], [0.3,0.4],[0.2,0.3],[0.3,0.4]> Uder RINS evromet, the proposed decso-makg method s appled to the selecto problem of the costructo projects. The decso steps are descrbed as follows: Step 1: By Equato 20, the deal soluto deal RINS ca be determed as the followg deal alteratve: A * = {<[0.8, 0.9], [0.8, 0.9], [0.1, 0.2], [0.1, 0.2], [0.2, 0.3], [0.2, 0.3]>, <[0.8, 0.9], [0.8, 0.9], [0.8, 0.9], [0.1, 0.2], [0.1, 0.2], [0.1, 0.2], [0.1, 0.2], [0.1, 0.2], [0.1, 0.2]>, <[0.8, 0.9], [0.8, 0.9], [0.1, 0.2], [0.2, 0.3], [0.1, 0.2], [0.1, 0.2]>}. Step 2: By usg oe of Equatos 16 18, the weghted smlarty measure values betwee each alteratve A = 1, 2, 3, 4 ad the deal alteratve A * ca be calculated, ad the all the results are show Table 5. Step 3: I Table 5, sce all the measure values are R Ws A 2, A * > R Ws A 4, A * > R Ws A 3, A * > R Ws A 1, A * for s = J, D, C, all the rakg orders of the four alteratves are A 2 A 4 A 3 A 1. Hece, the alteratve A 2 s the best choce amog all the costructo projects. Table 5. All the measure values betwee A = 1, 2, 3, 4 ad A * for RINSs ad rakg orders of the four alteratves. Measure Method Measure Value Rakg Order The Best Choce WA, A * [23] R WJ A, A * R WD A, A * R WC A, A * WA 1, A * = , WA 2, A * = , WA 3, A * = , WA 4, A * = R WJ A 1, A * = , R WJ A 2, A * = , R WJ A 3, A * = , R WJ A 4, A * = R WD A 1, A * = , R WD A 2, A * = , R WD A 3, A * = , R WD A 4, A * = R WC A 1, A * = , R WC A 2, A * = , R WC A 3, A * = , R WC A 4, A * = For coveet comparso wth exstg related method [23], the decso results based o the cose fucto wthout cosderg sub-attrbute weghts the lterature [23] are also dcated Tables 3 ad 5. Obvously, all the rakg orders are detcal, whch dcate the feasblty ad effectveess of the developed decso-makg method based o the proposed measures R Ws for

11 Symmetry 2017, 9, of 13 s = J, D, C. However, the exstg related decso-makg methods wth RSVNSs ad RINSs [22,23] caot deal wth such a decso-makg problem wth both attrbute weghts ad sub-attrbute weghts ths paper. Although the same computatoal complexty decso-makg algorthms s show by comparso of the method of ths study wth the related methods troduced [22,23], the developed method ths study exteds the methods [22,23] ad s more feasble ad more geeral tha the exstg related decso-makg methods [22,23]. It s obvous that the ew developed decso-makg method a RSNS RINS ad/or SVNS settg s superor to the exstg related methods a RINS or SVNS settg [22,23]. Compared wth tradtoal decso-makg approaches wthout sub-attrbutes [7 13,19 21], the decso-makg approach proposed ths study ca deal wth decso-makg problems wth both attrbutes ad sub-attrbutes; whle tradtoal decso-makg approaches [7 13,19 21] caot deal wth such a decso-makg problem wth both attrbutes ad sub-attrbutes. Hece, the proposed decso-makg approach s superor to tradtoal oes [7 13,19 21]. However, the study ths paper provdes ew three vector measures ad ther decso-makg method as the ma cotrbutos due to o study of exstg lterature o the vector smlarty measures ad decso-makg methods wth RSNSs RSVNSs ad/or RINSs. Clearly, the ma advatages of ths study are that t ca solve decso-makg problems wth the weghts of both attrbutes ad sub-attrbutes, whch all exstg methods caot deal wth, ad exted exstg smlarty measures ad decso-makg methods. To aalyze the sestvtes of the proposed approach, let us chage the RINS of the alteratve A 4 to the RSNS A 4 = {<[0.7,0.7], [0.6,0.6], [0.2,0.2], [0.2,0.2], [0.3,0.3], [0.3,0.3]>, <[0.7,0.8], [0.8,0.9], [0.7,0.8], [0.2,0.3], [0.2,0.3], [0.1,0.2], [0.1,0.2], [0.2,0.3], [0.2,0.3]>, <[0.7,0.8], [0.7,0.8], [0.2,0.3], [0.3,0.4], [0.2,0.3], [0.3,0.4]> wth hybrd formato of both RSVNNs ad RINNs. The, by above smlar computato steps, we ca obta all the measure values, whch are show Table 6. Table 6. All the measure values betwee A = 1, 2, 3, 4 ad A * for RSNSs ad rakg orders of the four alteratves. Measure Method Measure Value Rakg Order The Best Choce R WJ A, A * R WD A, A * R WC A, A * R WJ A 1, A * = , R WJ A 2, A * = , R WJ A 3, A * = , R WJ A 4, A * = R WD A 1, A * = , R WD A 2, A * = , R WD A 3, A * = , R WD A 4, A * = R WC A 1, A * = , R WC A 2, A * = , R WC A 3, A * = , R WC A 4, A * = A 2 A 3 A 4 A 1 A 2 A 2 A 3 A 4 A 1 A 2 The results of Table 6 demostrate the rakg orders based o R WJ A, A * ad R WD A, A * are the same, but ther decso-makg method ca chage the prevous rakg orders ad show some dfferece betwee two alteratves A 3 ad A 4 ; whle the best oe s stll A 2. Clearly, the decso-makg approach based o the Jaccard ad Dce measures shows some sestvty ths case. However, the rakg order based o R WC A, A * stll keeps the prevous rakg order, ad the the decso-makg approach based o the cose measure shows some robustess/sestvty ths case. I actual decso-makg problems, decso-makers ca select oe of three vector measures of RSNSs to apply t to multple attrbute decso-makg problems wth weghts of attrbutes ad sub-attrbutes accordg to ther preferece ad actual requremets. 7. Coclusos Ths paper troduced RSNSs, cludg the cocepts of RSVNSs ad RINSs, ad proposed the vector smlarty measures of RSNSs, cludg the Jaccard, Dce, ad cose measures betwee RSNSs RSVNSs ad RINSs vector space. It the preseted the weghted Jaccard, Dce, ad cose measures betwee RSNSs RSVNSs ad RINSs by cosderg the weghts of basc elemets

12 Symmetry 2017, 9, of 13 RSNSs ad the weghts of sub-elemets the refed weghts each RSNN. Further, we establshed a decso-makg method based o the weghted Jaccard/Dce/cose measures of RSNSs RSVNSs ad RINSs to deal wth multple attrbute decso-makg problems wth both attrbute weghts ad sub-attrbute weghts uder RSNS RINS ad/or RSVNS evromets. I the decso-makg process, through the Jaccard/Dce/cose measures betwee each alteratve ad the deal alteratve, the rakg order of all alteratves ad the best oe ca be determed based o the measure values. Fally, a actual example o the decso-makg problem of costructo projects wth RSNS RSVNS ad/or RINS formato s provded to demostrate the applcato ad effectveess of the proposed method. The proposed approach s very sutable for actual applcatos decso-makg problems wth weghts of both attrbutes ad sub-attrbutes uder RSNS RINS ad/or RSVNS evromets, ad provdes a ew decso-makg method. I the future, we shall further exted the proposed method to group decso-makg, clusterg aalyss, medcal dagoss, fault dagoss, ad so forth. Ackowledgmets: Ths paper was supported by the Natoal Natural Scece Foudato of Cha Nos , Author Cotrbutos: Ju Ye proposed the vector smlarty measures of RSNSs, cludg the weghted Jaccard, Dce, ad cose measures betwee RSNSs RSVNSs ad RINSs vector space; Jqa Che ad Ju Ye establshed a decso-makg method based o the weghted Jaccard/Dce/cose measures of RSNSs RSVNSs ad RINSs to deal wth multple attrbute decso-makg problems wth both attrbute weghts ad sub-attrbute weghts uder RSNS RINS ad/or RSVNS evromets; Jqa Che ad Shgu Du gave a actual example o the decso-makg problem of costructo projects wth RSNS RSVNS ad/or RINS formato ad ts calculato ad comparatve aalyss; we wrote the paper together. Coflcts of Iterest: The authors declare o coflcts of terest. Refereces 1. Zadeh, L.A. Fuzzy sets. If. Cotrol. 1965, 8, [CrossRef] 2. Ataassov, K. Itutostc fuzzy sets. Fuzzy Sets Syst. 1986, 20, [CrossRef] 3. Ataassov, K.; Gargov, G. Iterval-valued tutostc fuzzy sets. Fuzzy Sets Syst. 1989, 31, [CrossRef] 4. Smaradache, F. Neutrosophy: Neutrosophc Probablty, Set, ad Logc; Amerca Research Press: Rehoboth, IL, USA, Wag, H.; Smaradache, F.; Zhag, Y.Q.; Suderrama, R. Iterval Neutrosophc Sets ad Logc: Theory ad Applcatos Computg; Hexs: Phoex, AZ, USA, Wag, H.; Smaradache, F.; Zhag, Y.Q.; Suderrama, R. Sgle valued eutrosophc sets. Multspace Multstruct. 2010, 4, Ye, J. A multcrtera decso-makg method usg aggregato operators for smplfed eutrosophc sets. J. Itell. Fuzzy Syst. 2014, 26, Ye, J. Vector smlarty measures of smplfed eutrosophc sets ad ther applcato multcrtera decso makg. It. J. Fuzzy Syst. 2014, 16, Zavadskas, E.K.; Bausys, R.; Lazauskas, M. Sustaable assessmet of alteratve stes for the costructo of a waste cerato plat by applyg WASPAS method wth sgle-valued eutrosophc set. Sustaablty 2015, 7, [CrossRef] 10. Zavadskas, E.K.; Bausys, R.; Kaklauskas, A.; Ubartė, I.; Kuzmskė, A.; Gudeė, N. Sustaable market valuato of buldgs by the sgle-valued eutrosophc MAMVA method. Appl. Soft Comput. 2017, 57, [CrossRef] 11. Lu, Z.K.; Ye, J. Cose measures of eutrosophc cubc sets for multple attrbute decso-makg. Symmetry 2017, 9, 121. [CrossRef] 12. Lu, Z.K.; Ye, J. Sgle-valued eutrosophc hybrd arthmetc ad geometrc aggregato operators ad ther decso-makg method. Iformato 2017, 8, 84. [CrossRef] 13. Che, J.Q.; Ye, J. Some sgle-valued eutrosophc Domb weghted aggregato operators for multple attrbute decso-makg. Symmetry 2017, 9, 82. [CrossRef] 14. Ye, J. Sgle valued eutrosophc smlarty measures based o cotaget fucto ad ther applcato the fault dagoss of steam turbe. Soft Comput. 2017, 21, [CrossRef]

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