A Multi-criteria neutrosophic group decision making metod based TOPSIS for supplier selection

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1 A ulti-criteria eutrosophic group decisio maig metod based TOPSIS for supplier selectio Abstract Rıdva şahi* ad uhammed Yiğider** *Departmet of athematics, Faculty of Sciece, Atatur Uiversity, Erzurum **Departmet of athematics, Faculty of Sciece, Erzurum Techical Uiversity, Erzurum The process of multiple criteria decisio maig (CD is of determiig the best choice amog all of the probable alteratives. The problem of supplier selectio o which decisio maer has usually vague ad imprecise owledge is a typical example of multi criteria group decisio-maig problem. The covetioal crisp techiques has ot much effective for solvig CD problems because of imprecise or fuzziess ature of the liguistic assessmets. To fid the exact values for CD problems is both difficult ad impossible i more cases i real world. So, it is more reasoable to cosider the values of alteratives accordig to the criteria as sigle valued eutrosophic sets (SVNS. This paper deal with the techique for order preferece by similarity to ideal solutio (TOPSIS approach ad exted the TOPSIS method to CD problem with sigle valued eutrosophic iformatio. The value of each alterative ad the weight of each criterio are characterized by sigle valued eutrosophic umbers. ere, the importace of criteria ad alteratives is idetified by aggregatig idividual opiios of decisio maers (Ds via sigle valued eutrosophic weighted averagig (IFWA operator. The proposed method is, easy use, precise ad practical for solvig CD problem with sigle valued eutrosophic data. Fially, to show the applicability of the developed method, a umerical experimet for supplier choice is give as a applicatio of sigle valued eutrosophic TOPSIS method at ed of this paper. 1. Itroductio The cocept of eutrosophic set (NS developed by Smaradache ([1,] is a more geeral platform which exteds the cocepts of the classic set ad fuzzy set ([3], ituitioistic fuzzy set ([4] ad iterval valued ituitioistic fuzzy sets ([5]. I cotrast to ituitioistic fuzzy sets ad also iterval valued ituitioistic fuzzy sets, the idetermiacy is characterized explicitly i a eutrosophic set. A eutrosophic set has three basic compoets such that truth membership T, idetermiacy membership I ad falsity membership F, which are defied idepedetly of oe aother. But, a eutrosophic set will be more difficult to apply i real scietific ad egieerig fields. Therefore, Wag et al. ([6,7] proposed the cocepts of sigle valued eutrosophic set (SVNS ad iterval eutrosophic set (INS which are a istace of a eutrosophic set, ad provided the set-theoretic operators ad various properties of SVNSs ad INSs. SVNSs preset ucertaity, imprecise, icosistet ad icomplete iformatio existig i real world. Also, it would be more suitable to hadle idetermiate iformatio ad icosistet iformatio. We usually eed the decisio maig methods because of the ucertaity ad complex uder the physical ature of the problems. By the multi-criteria decisio maig methods, we ca determie the best alterative from multiple alteratives with respect to some criteria. Recetly, supplier selectio has become icreasigly importat i both academia ad idustry (see [8-1]. So there are may CD techiques developed for the supplier selectio problem. Some of these techiques are categorical method, weighted poit method ([13], matrix approach ([14], vedor performace matrix approach ([15] vedor profile aalysis (VPA ([16] aalytic hierarchy process (AP ([17-19], aalytic etwor process (ANP ([0], mathematical programmig ([1,] ad multiple objective programmig (OP ([3-5]. owever, most of these methods are developed with respect to crisp data ad so they have ot several ifluece factors such as imprecisio prefereces, additioal qualitative criteria ad icomplete iformatio. Therefore, fuzzy set theory (FST is more appropriate to overcome problems i decisio maig process. Li et al. ([6] ad olt ([7] proposed the applicatio of supplied selectio uder fuzzy data. Che et al. ([8] exteded the cocept of classic 1

2 TOPSIS method to solve supplier selectio problems i fuzzy set theory. TOPSIS (Techique for order preferece by similarity to a ideal solutio method which is oe of the most used classical CD methods has developed by wag ad Yoo ([9]. The the proposed set theories have provided the differet multi-criteria decisio maig methods. Some authors ([30-41] studied o multi-criteria decisiomaig methods based fuzzy data. Bora et al. ([41] proposed the TOPSIS method to select appropriate supplier uder ituitioistic fuzzy eviromet. The the TOPSIS method for CD problem has exteded i iterval valued ituitioistic fuzzy sets by Ye ([4]. As metioed above, the sigle valued eutrosophic iformatio is a geeralizatio of ituitioistic fuzzy iformatio, while ituitioistic fuzzy iformatio is geeralizes the fuzzy iformatio. O oe had, a sigle valued eutrosophic set is a istace of eutrosophic set, which give us a additioal possibility to represet ucertaity, imprecise, icomplete, ad icosistet iformatio existig i real world. It ca describe ad hadle idetermiate iformatio ad icosistet iformatio. owever, the coector i the fuzzy set is defied with respect to T, i.e. membership oly, hece the iformatio of idetermiacy ad o-membership is lost. The coectors i the ituitioistic fuzzy set are defied with respect to T ad F, i.e. membership ad omembership oly, hece the idetermiacy is what is left from 1, while i the eutrosophic set, they ca be defied by ay of them (o restrictio ([1]. For example, whe we as the opiio of a expert about certai statemet, oe may say that the possibility i which the statemet is true is 0.6, the statemet is false is 0.5 ad the statemet is ot sure is 0.. For eutrosophic otatio, it ca be expressed as x(0.6,0.,0.5. For further example, suppose there are 10 voters durig a votig process. Five vote aye, two vote blacball ad three vote are udecided. For eutrosophic otatio, it ca be characterized as x(0.5,0.3,0.. owever, the expressio are beyod the scope of the ituitioistic fuzzy set. Therefore, the cocept of sigle valued eutrosophic set is more geeral structure ad very suitable to overcome the metioed issues. The we say that the TOPSIS method uder sigle valued eutrosophic eviromet is suitable for decisio maig. oreover, the sigle valued eutrosophic TOPSIS ot oly use for sigle valued eutrosophic iformatio, but also exteds the ituitioistic fuzzy TOPSIS ad the fuzzy TOPSIS. But, util ow there have bee o may studies o multi criteria decisio maig methods which are criterio values for alteratives are sigle valued eutrosophic sets. Ye ([43] preseted the correlatio coefficiet of SVNSs ad the cross-etropy measure of SVNSs ad applied them to sigle valued eutrosophic decisio-maig problems. Also Ye ([44] defied sigle valued eutrosophic cross etropy which is proposed as a extesio of the cross etropy of fuzzy sets, Recetly, Zhag et al. ([46] established two iterval eutrosophic aggregatio operators such as iterval eutrosophic weighted arithmetic operator ad iterval eutrosophic weighted geometric operator ad preseted a method for multi criteria decisio maig problems based o the aggregatio operators. The mai purposes of this paper were (1 to defie oe equatio to calculate performace weights of decisio maers expressed by sigle valued eutrosophic umbers ( to establish a multi criteria decisio maig method based o TOPSIS method uder sigle valued eutrosophic values for supplier selectio, (3 to show the applicatio ad effectiveess of the proposed method with a example, ad (4 to preset performaces of alteratives accordig to each criterio via graphics visualizig the relatioships amog alteratives, SVN positive ideal solutio ad SVN egative ideal solutio. We orgaize the rest of the paper as follows: i the followig sectio, we give prelimiary defiitios of eutrosophic sets ad sigle valued eutrosophic sets ad propose a score fuctio for raig SVN umbers. I Sectio 3, we preset a techical to exted TOPSIS method i sigle valued eutrosophic eviromet. I Sectio 4, we illustrate our developed method by a example. This paper is termiated i Sectio 5.. Prelimiaries.1 Neutrosophic set I the followig, we give a brief review of some prelimiaries. Defiitio 1. [1] Let X be a space of poits (objects ad x X. A eutrosophic set A i X is defied by a truth-membership fuctio T A (x, a idetermiacymembership fuctio I A (x ad a falsity-membership

3 fuctio F A (x. T A (x, I A (x ad F A (x are real stadard or real ostadard subsets of ]0, 1 + [. That is T A (x: X ]0, 1 + [, T A (x: X ]0, 1 + [ ad T A (x: X ]0, 1 + [. There is ot restrictio o the sum of T A (x, I A (x ad F A (x, so 0 sup T A (x sup I A (x sup F A (x 3 +. I the followig, we adopt the otatios u A (x, r A (x ad v A (x istead of T A (x, I A (x ad F A (x, respectively. Also we write SVN umbers istead of sigle valued eutrosophic umbers.. Sigle valued eutrosophic sets A sigle valued eutrosophic set (SVNS has bee defied i ([6] as follows: Defiitio. Let X be a uiverse of discourse. A sigle valued eutrosophic set A over X is a object havig the form A = { x, u A (x, r A (x, v A (x : x X} (1 where u A (x: X [0,1], r A (x: X [0,1] ad v A (x: X [0,1] with 0 u A (x + r A (x + v A (x 3 for all x X. The itervals u A (x, r A (x ad v A (x deote the truth- membership degree, the idetermiacy-membership degree ad the falsity membership degree of x to A, respectively. For coveiece, a SVN umber is deoted by A = (a, b, c, where a, b, c [0,1] ad a + b + c 3. Defiitio 3. [46] Let A 1 = (a 1, b 1, c 1 ad A = (a, b, c be two SVN umbers, the summatio betwee A 1 ad A is defied as follows: A 1 A = (a 1 + a a 1 a, b 1 b, c 1 c ( Defiitio 4. [46] Let A 1 = (a 1, b 1, c 1 ad A = (a, b, c be two SVN umbers, the multiplicatio betwee A 1 ad A is defied as follows: A 1 A = (a 1 a, b 1 + b b 1 b, c 1 + c c 1 c. (3 Defiitio 5. [46] Let A = (a, b, c be a SVN umber ad λ R a arbitrary positive real umber, the λa = (1 (1 a λ, b λ, c λ, λ > 0. (4 Based o the study give i ([46], we defie the weighted aggregatio operators related to SVNSs as follows: Defiitio 6. Let {A 1, A,, A } be the set of SVN umbers, where A j = (a j, b j, c j (j = 1,,,. The sigle valued eutrosophic weighted average operator o them is defied by λ j A j = (1 (1 a j λ j, ( (b j λ j, (c j λ j (5 where λ j is the weight of A j (j = 1,,,, λ j [0,1] ad λ j = 1. Defiitio 7. [45] Let A = (A 1, A,.., A be a vector of SVN umbers such that A j = (a j, b j, c j (j = 1,,, ad B i = ( B i1, B i,, B im (i = 1,,, m be m vectors of SVN umbers such that B ij = (a ij, b ij, c ij (i = 1,,, m, (j = 1,,,. The the separatio measure betwee B i s ad A based o Euclidia distace is defied as follows: s i = ( 1 3 {( a ij a j + ( b ij b j +( c ij c j } (i = 1,,, m. (6 Next, we proposed a score fuctio for raig SVN umbers as follows: Defiitio 8. Let A = (a, b, c be a sigle valued eutrosophic umber, a score fuctio S of a sigle valued eutrosophic value, based o the truthmembership degree, idetermiacy-membership degree ad falsity membership degree is defied by S(A = where S(A [ 1,1]. 1 + a b c 1 (7 The score fuctio S is reduced the score fuctio proposed by Li ([47] if b = 0 ad a + c 1. Example 9. Let A 1 = (0.5,0.,0.6 ad A = (0.6,0.3,0. be two sigle valued eutrosophic umbers for two alteratives. The, by applyig Defiitio 8, we ca obtai S(A 1 = = S(A = = 0.4. I this case, we ca say that alterative A is better tha A 1. 3

4 .3 TOPSIS ethod ad Liguistic Variables I the sectio, we briefly summarize the TOPSIS method ad its applicatios. The we discuss the usig TOPSIS method i solvig CD problems. We give the relatioships betwee liguistic variables ad sigle valued eutrosophic umbers. The TOPSIS (Techique for Order Preferece by Similarity to Ideal Solutio method was iitiated by wag ad Yoo ([9]. It is very suitable practical method which is oe of the methods of the multicriteria decisio maig. I practice, the TOPSIS method is a process of determiig the alterative which is closest to the ideal solutio, i.e. raig the alteratives with respect to their distaces from the ideal ad the egative ideal solutio ad has applied to may areas relyig o computer support to overcome evaluatio problems uder a fiite umber of alteratives. I this method, the grades of optios are determied accordig to ideal solutio similarity. If the similarity rate of a optio is more close to a ideal solutio which is the best from ay aspect that does ot exist practically, it has a higher grade ad also is the optimal choice. A liguistic variable is a variable whose values are characterized with words or seteces istead of umbers i a atural or artificial laguage. The value of a liguistic variable is expressed as a elemet of its term set. The cocept of a liguistic variable is very useful for solvig decisio maig problems with complex cotet. For example, we ca express the performace ratigs of alteratives o qualitative attributes by liguistic variables such as very importat, importat, medium, uimportat, very uimportat, etc. Such liguistic values ca be represeted usig sigle valued eutrosophic umbers. For example, importat ad very importat ca be expressed by sigle valued eutrosophic umbers (0., 0.3, 0.5 ad (0.6, 0.9, 1.0, respectively. Fudametally, liguistic terms are idividual variatios for a liguistic variable. That is, liguistic terms do ot meet precise meaig ad it may be iterpreted differetly by differet people. The cover of a determied term are pretty subjective ad it may vary as the case. Therefore, liguistic terms caot be idicated by classic set theory ad also each liguistic term is associated with a sigle valued eutrosophic set. The followig example illustrates that situatio. Example 10. Let X = {x 1, x, x 3, x 4, x 5 } be five alteratives i the uiverse of cars. Suppose that quality of the cars is a liguistic variable ad T(price = {extremely high, very high, medium, very low} is set of liguistic terms for this variable. Sice each liguistic term is characterized with its ow sigle valued eutrosophic set, two of them might be defied as follows: T very high = { (x 1, 0.5,0.7,0,4, (x, 0.1,0.3,0,4, (x 4, 0.5,0.4,0,1, (x 5, 0.3,0.3,0,5 } T medium = { (x 1, 0.,0.4,0,6, (x 3, 0.3,0.1,0,5, (x 4, 0.,0.4,0,7, (x 5, 0.4,0.1,0,6 } Supplier selectio has a very importat place i multi criteria decisio maig. I our supplier selectio approach, we firstly collect the idividual evaluatios of multiple decisio maers ad the we decide for a fial select. I the method, there are -decisio maers, m-alteratives ad -criteria. -decisio maers evaluate the importace of the m alteratives uder criteria ad ra the performace of the criteria with respect to liguistic statemets coverted ito sigle valued eutrosophic umbers. ere, the decisio maers utilize ofte a set of weights such that W ={very importat, importat, medium, uimportat, very uimportat} ad the importace weights based o sigle valued eutrosophic values of the liguistic terms is give as Table 1. Table 1. Importace weight as liguistic variables Liguistic terms SVNSs Very importat (VI (0.90,0.10,0.10 Importat (I (0.75,0.5,0.0 edium ( (0.50,0.50,0.50 Uimportat (UI (0.35,0.75,0.80 Very uimportat (VUI (0.10,0.90,0.90 oreover, i Table, we give the set of liguistic terms used to rate the importace of alteratives accordig to decisio maers. Table. Liguistic terms to rate the importace of alteratives Liguistic terms SVNSs Extremely good (E / extremely (1.00,0.00,0.00 high (E Very very good (VV / very very (0.90,0.10,0.10 high (VV Very good (V / (0.80,0.15,0.0 very high (V ood ( / high ( (0.70,0.5,0.30 edium good ( / medium high (0.60,0.35,0.40 ( edium ( / fair (F (0.50,0.50,0.50 edium bad (B / medium low (0.40,0.65,0.60 (L Bad (B / low (L (0.30,0.75,0.70 Very bad (VB / (0.0,0.85,0.80 very low (VL 4

5 Very very bad (VVB / very very low (VVL Extremely bad (EB / extremely low (EL 3. Sigle Valued Neutrosophic TOPSIS (0.10,0.90,0.90 (0.00,1.00,1.00 ere, we exted the TOPSIS method i sigle valued eutrosophic sets. Suppose that A = {ρ 1, ρ,, ρ m } is a set of alteratives ad = {β 1, β,, β } is a set of criteria. We costruct the procedure of sigle valued eutrosophic TOPSIS process, which is as follows: Step 1: Determie the weight of decisio maers. I the step, we idetify the importace of decisiomaers usig the liguistic set give i Table 1. Assume that our decisio group process has decisio maers ad A t = (a t, b t, c t is a SVN umber expressig tth decisio maer. The we obtai the weight of tth decisio maer as follows: a t a t + b t ( a δ t = t + c t a a t + b t ( t a t + c t δ t 0 ad δ t = 1. (8 ere, the weight of each decisio maer is calculated taig ito accout the truth-membership value, the idetermiacy-membership value ad the falsitymembership value from them. Step : Costructio of aggregated sigle valued eutrosophic decisio matrix with respect to decisio maers. To costruct oe group decisio by aggregatig all the idividual decisios, we eed to obtai aggregated sigle valued eutrosophic decisio matrix D. ere, it is defied by D = δ t D t, where D = d ij = (u ij, r ij, v ij ad d ij = (1 (1 u (t ij δ t, (r (t ij δ t, (v (t ij δ t The aggregated sigle valued eutrosophic decisio matrix D of decisio maers ca be expressed as ρ 11 ρ 1 ρ 1 ρ 1 ρ ρ D = ( ρ m1 ρ m ρ m (9 where ρ ij (i = 1,,, m; j = 1,,, deotes a SVN value. Step 3: Determie the weights of criterio. Each criteria accordig to decisio maers i decisio maig process may have differet importace. By aggregatig the criteria values ad the weight values of decisio maers for the importace of each criteria, we ca obtai the weights of the criteria. Assume that weights of criteria is deoted by W = (w 1, w,, w where w j idicates the relative importace of criterio β j. Let w j (t = (aj (t, bj (t, cj (t be a SVN umber expressig the criteria β j (j = 1,,, by the tth decisio maer. The the weight vector of criteria are obtaied by formula (5 as follows: w j = δ 1 w j (1, δ w j (,, δ w j ( = (1 (1 a (t j δ t, (b (t j δ t, (c (t j δ t (10 Step 4: Costructio of aggregated weighted sigle valued eutrosophic decisio matrix with respect to criteria. By usig the weight of criteria (W ad the aggregated weighted sigle valued decisio matrix (D, we obtai the aggregated weighted sigle valued eutrosophic decisio matrix. Let us assume that D = (d ij. The it is defied by where d ij D = D W, (11 = w j d ij = (a ij, b ij, c ij. Thus, the aggregated sigle valued eutrosophic matrix of criteria ca be expressed as ρw 11 ρw 1 ρw 1 D ρw 1 ρ ρw = ( ρw m1 ρw m ρw m Step 5: Calculatio sigle valued positive-ideal solutio (SVN-PIS ad sigle valued egative-ideal solutio (SVN-NIS. I TOPSIS method, the evaluatio criteria ca be categorized ito two categories, beefit ad cost. Let 1 be a collectio of beefit criteria ad be a collectio of cost criteria. Accordig to sigle valued eutrosophic set theory ad the priciple of classical TOPSIS method, SVN-PIS ad SVN-NIS ca be defied as follows, respectively; 5

6 ρ + = (a ρ + w(β j, b ρ + w(β j, c ρ + w(β j (1 ρ = (a ρ w(β j, b ρ w(β j, c ρ w(β j (13 where max a ρi w(β j, if j 1 a ρ + i w(β j = ( mi a ρi i w(β j, if j mi b ρi b ρ + w(β j = ( i w(β j, if j 1 max b ρi w(β j, if j i mi c ρi c ρ + w(β j = ( i w(β j, if j 1 max c ρi w(β j, if j i ad mi a ρi a ρ w(β j = ( i w(β j, if j 1 max a ρi w(β j, if j i max b ρi w(β j, if j 1 b ρ i w(β j = ( mi b ρi i w(β j, if j max c ρi w(β j, if j 1 c ρ i w(β j = ( mi c ρi i w(β j, if j Step 6: Calculate of distace measures from SVN-PIS ad SVNNIS. To measure distace of each alterative ρ i from SVN- PIS ad SVN-NIS, we use the distace measure give by Eq. (6. s + i = ( 1 3 {( a ij a + j + ( b ij b + j +( c ij c + j } ad (i = 1,,, m, (14 s i = ( 1 3 {( a ij a j + ( b ij b j +( c ij c j } (i = 1,,, m. (15 Step 7: Calculate the closeess coefficiet (CC Fially, we compute relative closeess coefficiet of each alterative with respect to sigle valued eutrosophic ideal solutios by usig ρ j = s s + +s, where 0 ρ j 1. (16 Step 8: Determie the ra of alteratives. Accordig to descedig order of relative closeess coefficiet we ca ra all alteratives Numerical example Assume that for supplier selectio i a productio idustry, four decisio maers (D has bee appoited to evaluate 5 supplier alteratives (ρ i ; 1,,..., 5 with respect to five performace criteria such that delivery, quality, flexibility, service ad price. The decisio-maers utilize a liguistic set of weights to determie the performace of each criterio. The iformatio of weights provided to the five criteria by the four decisio maers are preseted i Table 3. Table 3. The importace weights of the decisio criteria Criteria D(1 D( D(3 D(4 Delivery VI VI VI I Quality I I Flexibility VI VI I VI Service I I UI Price VI VI We assume that the decisio maers use the liguistic variables ad ratigs to state the suitability of the supplier alteratives uder each of the subjective criteria. The results are show i Table (4-8. Table 4. The ratigs of the alteratives for delivery criterio Delivery Supp. D(1 D( D(3 D(4 ρ 1 ρ ρ 3 ρ 4 ρ 5 V V V V V Table 5. The ratigs of the alteratives for quality criterio Quality Supp. D(1 D( D(3 D(4 ρ 1 ρ ρ 3 ρ 4 ρ 5 V V V V Table 6. The ratigs of the alteratives for flexibility criterio Flexibility Supp. D(1 D( D(3 D(4 ρ 1 ρ ρ 3 ρ 4 ρ 5 V V V V Table 7. The ratigs of the alteratives for service criterio Service Supp. D(1 D( D(3 D(4 ρ 1 ρ V V

7 ρ 3 ρ 4 ρ 5 B V V Table 8. The ratigs of the alteratives for price criterio Price Supp. D(1 D( D(3 D(4 ρ 1 ρ ρ 3 ρ 4 ρ 5 V V V V V Next, we apply the procedure of sigle valued eutrosophic TOPSIS process, which is as follows: Step 1. Determie the weights of the decisio maers. By usig Eq. (8, we obtai the weights of the decisio maers (Table 9. Table 9. The importace of decisio maers ad their weights. D(1 D( D(3 D(4 Table 10. Aggregated SVN decisio matrix Lig. T. VI I UI Weight The we deotes the weight vector of the decisio maers by δ = [δ 1, δ, δ 3, δ 4 ]. Step. Costructio of aggregated sigle valued eutrosophic decisio matrix with respect to decisio maers. The ratigs assiged by the decisio maers to each alterative were give i Table (4-8, respectively. The the aggregated SVN decisio matrix obtaied by aggregatig of opiios of decisio maers is costructed by Eq. (9. The result is give i Table 10. Step 3. Determie the weights of criterio. We calculate the weights of each criterio by usig Eq. (10. I order to do that, we use the iformatio from Table 3 ad preset it i Table 1. β 1 β β 3 β 4 β 5 ρ 1 (0.717,0.8,0.8 (0.658,0.90,0.341 (0.541,0.45,0.458 (0.57,0.394,0.47 (0.599,0.36,0.400 ρ (0.679,0.66,0.30 (0.637,0.314,0.36 (0.755,0.191,0.44 (0.714,0.35,0.85 (0.671,0.81,0.38 ρ 3 (0.548,0.49,0.451 (0.651,0.30,0.348 (0.585,0.374,0.414 (0.579,0.370,0.40 (0.64,0.348,0.375 ρ 4 (0.636,0.313,0.363 (0.600,0.361,0.399 (0.636,0.313,0.363 (0.553,0.4,0.446 (0.545,0.48,0.454 ρ 5 (0.70,0.44,0.97 (0.681,0.66,0.318 (0.681,0.65,0.318 (0.681,0.79,0.318 (0.754,0.19,0.45 Table 11. Aggregated weighted SVN decisio matrix β 1 β β 3 β 4 β 5 ρ 1 (0.6,0.330,0.374 (0.571,0.384,0.45 (0.469,0.474,0.500 (0.496,0.474,0.500 (0.50,0.447,0.476 ρ (0.41,0.544,0.553 (0.396,0.574,0.580 (0.469,0.55,0.530 (0.443,0.55,0.530 (0.416,0.554,0.558 ρ 3 (0.47,0.508,0.53 (0.561,0.399,0.434 (0.504,0.457,0.497 (0.499,0.457,0.497 (0.537,0.438,0.458 ρ 4 (0.40,0.571,0.577 (0.379,0.601,0.601 (0.40,0.640,0.63 (0.349,0.640,0.63 (0.344,0.643,0.637 ρ 5 (0.505,0.455,0.497 (0.490,0.471,0.509 (0.490,0.481,0.509 (0.490,0.481,0.509 (0.543,0.418,0.456 Criteria β 1 β β 3 β 4 β 5 Table 1. The weights of criteria. Weight (0.867,0.13,0.17 (0.60,0.379,0.34 (0.861,0.138,0.131 (0.63,0.367,0.336 (0.70,0.79,0.79 β 1 β β 3 β 4 β 5 Table 13. SVN- PIS ad SVN-NIS values. SVN PIS (0.6,0.330,0.374 (0.571,0.384,0.45 (0.504,0.457,0.497 (0.499,0.457,0.497 (0.344,0.643,0.637 SVN NIS (0.40,0.571,0.577 (0.379,0.601,0.601 (0.40,0.640,0.63 (0.349,0.640,0.63 (0.543,0.418,

8 Step 4: Costructio of aggregated weighted sigle valued eutrosophic decisio matrix with respect to criteria. To costruct the aggregated weighted SVN decisio matrix, we use the Eq. (11 ad give it i Table 11. Step 5. Calculatio SVN positive-ideal solutio ad SVN egative-ideal solutio. By usig Eq. (1 ad Eq. (13, SVN positive-ideal solutio ad SVN egative-ideal solutio were calculated as Table 13. Step 6. Calculate the separatio measures. Separatio measure of each alterative from the positive-ideal solutio ad egative ideal solutio are calculated usig Eq. (14 ad Eq. (15 ad are give by Table 14. Table 14. Separatio measures ad the relative closeess coefficiet of each alterative. Alter. d d + CC Raig ρ ρ ρ ρ ρ Step 7: Calculate the closeess coefficiet (CC We determie the closeess coefficiet of all alterative by Eq. (16. The last colum of Table 14 presets the result. Step 8. Ra the alteratives. Accordig to descedig order of relative closeess coefficiets values, four alteratives are raed as ρ 1 > ρ 3 > ρ 5 > ρ > ρ 4 as i Table 14. The, the alterative ρ 1 is also the most desirable alterative. From the example, we ca see that the proposed eutrosophic decisio-maig method is more suitable for real scietific ad egieerig applicatios because it ca hadle ot oly icomplete iformatio but also the icosistet iformatio ad idetermiate iformatio existig i real world. The techique proposed i this paper exteds existig decisio maig methods ad provides a ew viewpoit for multi criteria group decisio maig. The TOPSIS method is a very importat techical for the process of multi criteria decisio maig. There are may TOPSIS methods for solvig multi criteria decisio maig problems with the fuzzy iformatio ad its extesio, the ituitioistic fuzzy iformatio ad the iterval valued ituitioistic fuzzy iformatio. Sice the sigle valued eutrosophic sets geeralize the cocepts of fuzzy sets ad ituitioistic fuzzy sets, the existig TOPSIS methods is ot suitable for hadlig the sigle valued eutrosophic iformatio icludig uow weights of decisio maers ad the criteria values for alteratives. Therefore, we eed to exted the method to eutrosophic eviromet. The developed decisio maig method ca utilize the proposed score fuctio of sigle valued eutrosophic umbers to ra alteratives i the process of multi criteria decisio maig. The performace ratigs of decisio maers ad criteria o alteratives are characterized by liguistic variables. The weights of decisio maers are calculated via a developed equatio while the weights of the criteria are obtaied by aggregatig the criteria values provided by decisio maers ad the weight values of decisio maers for the importace of each criteria. The method proposed i this paper are geeral ad more flexible tha existig decisio maig methods. 5. Coclusios I this paper, we exteded TOPSIS method that is oe of the familiar methods i multi-attribute decisiomaig problem i sigle valued eutrosophic sets ad proposed a multi-criteria group decisio maig based o sigle valued eutrosophic TOPSIS for evaluatio of supplier. Sice to solve a decisio maig problem expressed by crisp data is more difficult uder ucertai eviromet, sigle valued eutrosophic sets are more useful to overcome such situatios. I the evaluatio process, weights of decisio maers, the aggregatio of the criteria ad the impact of alteratives o criteria with respect to decisio maers is very importat to appropriately perform evaluatio process. I order to do that, the ratigs of each alterative accordig to each criterio ad the weights of each criterio were provided as liguistic terms expressed by sigle valued eutrosophic umbers. Also SVNWA operator is utilized to aggregate all idividual decisio maers opiios for determiig the importace of criteria ad the alteratives. Firstly, sigle valued eutrosophic positive-ideal solutio ad sigle valued eutrosophic egative-ideal solutio were obtaied usig the Euclidea distace. The the relative closeess coefficiets of alteratives were calculated ad fially raig the alteratives was doe. 8

9 TOPSIS method based o sigle valued eutrosophic set is more useful for solvig multi-criteria decisiomaig problems because of cosiderig order of importace of decisio maers. So, the sigle valued eutrosophic TOPSIS ca be preferable for dealig with icomplete, idetermie ad icosistet iformatio i CD problems such as selectig project ad persoel, selectig a flexible maufacturig system ad may further areas of maretig research problems ad maagemet decisio problems. 0,4 0,3 0, 0,1 0-0,1-0, -0,3-0, Alterative 3 PIS NIS The relatioships amog the alteratives ad their positive-ideal solutio ad egative ideal solutio is preseted i Fig. (1-5. 0,4 0,3 0, 0,1 0-0,1-0, -0,3-0, Alterative 1 PIS NIS Figure 1: The relatioships amog A 1 ad its PIS ad NIS 0,4 0,3 0, 0,1 0-0,1-0, -0,3-0, Alterative PIS NIS Figure 1: The relatioships amog A 3 ad its PIS ad NIS 0,4 0,3 0, 0,1 0-0, , -0,3-0,4 Alterative 4 PIS NIS Figure 4: The relatioships amog A 4 ad its PIS ad NIS 0,4 0,3 0, 0,1 0-0, , -0,3-0,4 Alterative 5 PIS NIS Figure 5: The relatioships amog A 5 ad its PIS ad NIS Figure : The relatioships amog A ad its PIS ad NIS Refereces [1] Smaradache, F. (1999. A uifyig field i logics. Neutrosophy: Neutrosophic probability, set ad logic, America Research Press, Rehoboth. [] Smaradache, F. (005. A geeralizatio of the ituitioistic fuzzy set. Iteratioal joural of Pure ad Applied athematics, 4, [3] L. A. Zadeh, Fuzzy sets, Iformatio ad Cotrol, vol. 8, o.3, pp , [4] K. T. Ataassov, Ituitioistic fuzzy sets, Fuzzy Sets ad Systems, vol. 0, o. 1, pp , [5] K. T. Ataassov ad. argov, Iterval valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems, vol. 31, o. 3, pp , 1989 [6] Wag,., Smaradache, F., Zhag, Y. Q., ad Suderrama, R., (010. Sigle valued eutrosophic sets, ultispace ad ultistructure ( [7] Wag,., Smaradache, F., Zhag, Y. Q. ad Suderrama. (005. Iterval eutrosophic sets ad logic: Theory ad applicatios i computig, exis, Phoeix, AZ. 9

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