Article Single-Valued Neutrosophic Hybrid Arithmetic and Geometric Aggregation Operators and Their Decision-Making Method

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1 Article Sigle-Valued Neutrosophic Hybrid Arithmetic ad Geometric Aggregatio Operators ad Their Decisio-Makig Method Zhikag Lu * ad Ju Ye Departmet of Electrical ad Iformatio Egieerig, Shaoxig Uiversity, 508 Huacheg West Road, Shaoxig , Chia; yehu@aliyu.com * Correspodece: luzhikag@usx.edu.c; Tel.: Received: 25 Jue 2017; Accepted: 13 July 2017; Published: 15 July 2017 Abstract: Sigle-valued eutrosophic umbers (SVNNs ca express icomplete, idetermiate, ad icosistet iformatio i the real world. The, the commo weighted aggregatio operators of SVNNs may result i ureasoably aggregated results i some situatios. Based o the hybrid weighted arithmetic ad geometric aggregatio ad hybrid ordered weighted arithmetic ad geometric aggregatio ideas, this paper proposes SVNN hybrid weighted arithmetic ad geometric aggregatio (SVNNHWAGA ad SVNN hybrid ordered weighted arithmetic ad geometric aggregatio (SVNNHOWAGA operators ad ivestigates their ratioality ad effectiveess by umerical examples. The, we establish a multiple-attribute decisio-makig method based o the SVNNHWAGA or SVNNHOWAGA operator uder a SVNN eviromet. Fially, the multiple-attribute decisio-makig problem about the desig schemes of puchig machie is preseted as a case to show the applicatio ad ratioality of the proposed decisio-makig method. Keywords: multiple-attribute decisio-makig; sigle-valued eutrosophic umber; sigle-valued eutrosophic umber hybrid weighted arithmetic ad geometric aggregatio (SVNNHWAGA operator; sigle-valued eutrosophic umber hybrid ordered weighted arithmetic ad geometric aggregatio (SVNNHOWAGA operator 1. Itroductio Fuzzy decisio-makig theory has bee a importat research topic. May fuzzy theories, especially like fuzzy (liguistic sets [1,2], ituitioistic fuzzy sets [3], iterval-valued ituitioistic fuzzy sets [4] etc., have bee proposed ad applied to decisio-makig problems [5 8]. However, there exists icomplete, idetermiate, ad icosistet iformatio i the real life, which is ot expressed by the aforemetioed fuzzy sets. To represet icomplete, idetermiate, ad icosistet iformatio, Smaradache [9] firstly proposed eutrosophic sets as the geeralizatio of fuzzy sets, ituitioistic fuzzy sets, ad iterval-valued ituitioistic fuzzy sets. I the eutrosophic set, its truth, idetermiacy, ad falsity membership degrees lie i the real stadard/ostadard iterval [ 0, 1 + ]. To costrai them i the real stadard iterval [0, 1] for coveiet egieerig applicatios, sigle-valued eutrosophic sets [10], iterval eutrosophic sets [11], ad simplified eutrosophic sets (icludig sigle-valued eutrosophic sets ad iterval eutrosophic sets [12] were itroduced as the subclasses of the eutrosophic sets. Recetly, may researchers have developed various aggregatio operators of simplified eutrosophic umbers (icludig sigle-valued/iterval eutrosophic umbers to be used for multiple-attribute decisio-makig problems with simplified (sigle-valued/iterval eutrosophic iformatio. For example, they proposed simplified (sigle-valued/iterval weighted aggregatio operators [12], iterval eutrosophic umber-weighted arithmetic average ad iterval eutrosophic Iformatio 2017, 8, 84; doi: /ifo

2 Iformatio 2017, 8, 84 2 of 13 umber-weighted geometric average operators [13], sigle-valued eutrosophic umber ormalized weighted Boferroi mea operators [14], geeralized eutrosophic umber Hamacher aggregatio operators [15], iterval eutrosophic umber geeralized weighted aggregatio operator [16], iterval eutrosophic umber Choquet itegral operator [17], iterval eutrosophic umber ordered weighted arithmetic average ad iterval eutrosophic umber ordered weighted geometric average operators [18], iterval eutrosophic umber prioritized ordered weighted average operator [19], simplified eutrosophic umber prioritized aggregatio operator [20], sigle-valued eutrosophic umber ad iterval eutrosophic umber expoetial weighted aggregatio operators [21,22] etc. They have bee wildly used for various decisio-makig problems i egieerig, ecoomics, ad maagemet areas. Furthermore, some scholars also proposed various multiple-attribute decisio-makig methods with sigle-valued eutrosophic umbers ad iterval eutrosophic umbers [23 27]. I the aforemetioed aggregatio operators, however, the iterval eutrosophic umber weighted arithmetic average ad iterval eutrosophic umber weighted geometric average operators ad the iterval eutrosophic umber ordered weighted arithmetic average ad iterval eutrosophic umber ordered weighted geometric average operators are commo aggregatio operatios i iformatio fusio ad decisio-makig areas. Especially whe iterval membership values i iterval eutrosophic umbers are degeerated to ay real umbers betwee 0 ad 1, the iterval eutrosophic umber weighted arithmetic average (INNWAA ad iterval eutrosophic umber weighted geometric average (INNWGA operators [13] ca be reduced to the sigle-valued eutrosophic umber weighted arithmetic average (SVNNWAA ad sigle-valued eutrosophic umber weighted geometric average (SVNNWGA operators, ad the iterval eutrosophic umber ordered weighted arithmetic average (INNOWAA ad iterval eutrosophic umber ordered weighted geometric average (INNOWGA operators [18] ca be reduced to the sigle-valued eutrosophic umber ordered weighted arithmetic average (SVNNOWAA ad sigle-valued eutrosophic umber ordered weighted geometric average (SVNNOWGA operators, respectively, as special cases of the existig iterval eutrosophic umber aggregatio operators [13,18]. However, they imply the drawbacks of their ureasoably aggregated results i some cases (see Sectio 2 i detail. For example, i the iformatio aggregatios of the SVNNWAA ad SVNNOWAA operators, their aggregated results may result i tedecy to the maximum value i some cases, while the aggregated results of the SVNNWGA ad SVNNOWGA operators may result i tedecy to the maximum weight value i some cases. Also, the SVNNWGA ad SVNNOWGA operators emphasize persoal maor poits [13,18] ad the SVNNWAA ad SVNNOWAA operators emphasize group s maor poits [13,18]. Motivated by the hybrid arithmetic ad geometric aggregatio operators of ituitioistic fuzzy umbers [28], this paper proposes the sigle-valued eutrosophic umber hybrid weighted arithmetic ad geometric aggregatio (SVNNHWAGA ad sigle-valued eutrosophic umber hybrid ordered weighted arithmetic ad geometric aggregatio (SVNNHOWAGA operators to realize more reasoable results i iformatio aggregatios of sigle-valued eutrosophic umbers, ad the idicates some properties of the SVNNHWAGA ad SVNNHOWAGA operators. Furthermore, a sigle-valued eutrosophic multiple-attribute decisio-makig method is established by usig the SVNNHWAGA or SVNNHOWAGA operator, ad the used for the decisio-makig problem of desig schemes of puchig machie uder a sigle-valued eutrosophic eviromet. The mai advatage of this study is that the proposed SVNNHWAGA ad SVNNHOWAGA operators ca overcome the drawbacks of the existig arithmetic/geometric average aggregatio operators of sigle-valued eutrosophic umbers i some situatios ad reach the moderate aggregatio values. The remaider of this paper is structured as the followig. I Sectio 2, we itroduce some basic cocepts ad operatios of sigle-valued eutrosophic umbers ad ivestigate some drawbacks of the SVNNWAA, SVNNOWAA, SVNNWGA, ad SVNNOWGA operators i some cases. I Sectio 3, we propose the SVNNHWAGA ad SVNNHOWAGA operators ad ivestigate their effectiveess ad ratioality based o umerical examples. Sectio 4 develops a sigle-valued eutrosophic multiple-attribute decisio-makig method based o the SVNNHWAGA or

3 Iformatio 2017, 8, 84 3 of 13 SVNNHOWAGA operator. Sectio 5 presets a multiple-attribute decisio-makig problem about the desig schemes of puchig machie as a case to illustrate the applicatio ad effectiveess of the preseted decisio-makig method. Sectio 6 gives some coclusios ad further research. 2. Some Cocepts ad Operatios of Sigle-Valued Neutrosophic Numbers Defiitio 1 [1]. Let X be a uiversal of discourse. A sigle-valued eutrosophic set N i X is characterized by truth, idetermiacy, ad falsity membership fuctios TN(x, UN(x, ad VN(x, respectively, where the values of the three fuctios TN(x, UN(x, ad VN(x are real umbers betwee 0 ad 1, satisfyig TN(x, UN(x, VN(x [0, 1] ad 0 TN(x + UN(x + VN(x 3 for x X. Thus, the sigle-valued eutrosophic set N is deoted as the followig form: N x, T ( x, U ( x, V ( x x X N N For coveiece, a basic elemet <x, TN(x, UN(x, VN(x> i the sigle-valued eutrosophic set N is deoted by z = <T, U, V> for short, which is called a sigle-valued eutrosophic umber. Sice a sigle-valued eutrosophic umber is a special case of a iterval eutrosophic umber, the cocepts ad operatios of iterval eutrosophic umbers ca be itroduced to sigle-valued eutrosophic umbers. Let two sigle-valued eutrosophic umbers be z1 = <T1, U1, V1> ad z2 = <T2, U2, V2>. The, there are the followig relatios [9 13]: (1 (z1 c = <V1, 1 U1, T1> (complemet of z1; (2 z1 z2 if ad oly if T1 T2, U1 U2 ad V1 V2; (3 z1 = z2 if ad oly if z1 z2 ad z2 z1. z z T T TT, U U, VV ; ( z z TT, U U U U, V V VV ; ( z 1 (1 T, U, V for α > 0; ( z z,1 (1 U,1 (1 V for α > 0. ( For ay sigle-valued eutrosophic umber z = <T, U, V>, its score ad accuracy fuctios [13] are defied, respectively, as follows: N E( z (2 T U V / 3, E( z [0,1] (1 H( z T V, H( z [ 1,1] (2 Defiitio 2 [13]. Let two sigle-valued eutrosophic umbers be z1 = <T1, U1, V1> ad z2 = <T2, U2, V2>, the the rakig method based o both the score values of E(z1 ad E(z2 ad the accuracy degrees of H(z1 ad H(z2 has the followig relatios: (1 If E(z1 > E(z2, the z1 z2; (2 If E(z1 = E(z2 ad H(z1 > H(z2, the z1 z2; (3 If E(z1 = E(z2 ad H(z1 = H(z2, the z1 = z2. Let z = <T, U, V> ( = 1, 2,, be a collectio of sigle-valued eutrosophic umbers. The, the SVNNWAA ad SVNNWGA operators [13] are itroduced, respectively, as follows: 1 2 w w w (3 SVNNWAA( z, z,..., z w z 1 (1 T, ( U, ( V

4 Iformatio 2017, 8, 84 4 of w w w w (4 SVNNWGA( z, z,..., z z ( T,1 (1 U,1 (1 V 1 where w ( = 1, 2,, is the weight of z ( = 1, 2,,, satisfyig w [0, 1] ad w 1. Whe the orders of all the argumets are cosidered by importat positios i the aggregatio process of sigle-valued eutrosophic umbers, the SVNNOWAA ad SVNNOWGA operators [18] are itroduced, respectively, as follows: 1 2 p( p( p( p( (5 SVNNOWAA( z, z,..., z z 1 (1 T, ( U, ( V 1 2 p( p( p( p( (6 SVNNOWGA( z, z,..., z z ( T,1 (1 U,1 (1 V where (p(1, p(2,, p( is a permutatio of (1, 2,, based o p( 1 p( for = 2,, ; (ζ1, ζ2,, ζ is a associated weight vector, satisfyig ζ [0, 1] ad 1 1. The, the SVNNOWAA ad SVNNOWGA operators ca reflect the importat degrees of the ordered positios of argumets. Although the above four aggregatio operators are commo aggregatio operatios i iformatio fusio ad decisio-makig areas, they imply some drawbacks, which result i tedecy to the maximum argumets or weight values of their aggregated values. For example, some drawbacks are show by the followig two umerical examples. Example 1. Let two sigle-valued eutrosophic umbers be z1 = <0.001, 0, 0> ad z2 = <1, 0, 0> with their weights ad associated weights w1 = ζ1 = 0.9 ad w2 = ζ2 = 0.1, respectively. The, by usig Equatios (3 (6 we ca yield SVNNWAA(z1, z2 = <1, 0, 0>, SVNNWGA(z1, z2 = <0.002, 0, 0>, SVNNOWAA(z1, z2 = <1, 0, 0>, ad SVNNOWGA(z1, z2 = <0.5012, 0, 0>. Example 2. Also take two sigle-valued eutrosophic umbers z1 = <0.001, 0, 0> ad z2 = <1, 0, 0> with their weights w1 = ζ1 = 0.1 ad w2 = ζ2 = 0.9, respectively. The, we ca obtai SVNNWAA(z1, z2 = <1, 0, 0>, SVNNWGA(z1, z2 = <0.5012, 0, 0>, SVNNOWAA(z1, z2 = <1, 0, 0>, ad SVNNOWGA(z1, z2 = <0.002, 0, 0>. From the aggregated results of the two examples, it is obvious that the aggregated values of the SVNNWAA ad SVNNOWAA operators idicate tedecy to the maximum value, ad the the aggregated values of the SVNNWGA ad SVNNOWGA operators idicate tedecy to the maximum weight value. Therefore, the SVNNWAA, SVNNOWAA, SVNNWGA ad SVNNOWGA operators may result i ureasoably aggregated results of sigle-valued eutrosophic umbers i some cases. To overcome these drawbacks, we eed to improve these aggregatio operators ad to propose hybrid arithmetic ad geometric aggregatio operators of sigle-valued eutrosophic umbers as the extesio of hybrid arithmetic ad geometric aggregatio operators of ituitioistic fuzzy umbers i [28]. 3. Hybrid Arithmetic ad Geometric Aggregatio Operators of Sigle-Valued Neutrosophic Numbers 3.1. SVNNHWAGA Operator Defiitio 3. Let z = <T, U, V> ( = 1, 2,, be a collectio of sigle-valued eutrosophic umbers. The, the SVNNHWAGA operator is defied by

5 Iformatio 2017, 8, 84 5 of 13 (1 w 1 1 SVNNHWAGA( z1, z2,..., z w z z (7 1 where w ( = 1, 2,, is the weight of z ( = 1, 2,,, satisfyig w [0, 1] ad w 1, ad α [0, 1]. Theorem 1. Let z = <T, U, V> ( = 1, 2,, be a collectio of sigle-valued eutrosophic umbers ad α be ay real umber i [0,1]. Thus, the aggregated value of the SVNNHWAGA operator is also a sigle-valued eutrosophic umber, which is calculated by w T 1 1 (1 w 1 (1 T, (1 (1 w w w SVNNHWAGA( z1, z2,..., z w z z 1 1 U (1 U, w V 1 1 w 1 1 (1 V (1 (1 1 where w is the weight of z ( = 1, 2,,, satisfyig w [0, 1] ad w 1. Proof: Correspodig to the operatioal laws of sigle-valued eutrosophic umbers i Sectio 2 ad the SVNNWAA ad SVNNWGA operators, we have the followig result:

6 Iformatio 2017, 8, 84 6 of 13 w 1 1 SVNNHWAGA( z1, z2,..., z w z z (1 w w w w w w T U V T U V (1,,,1 (1,1 (1 (1 w w w T U V (1,1 1,1 1 (1 (1 (1 w w w T,1 (1 U,1 (1 V (1, (1 (1 (1 w w w w T T U U (1 w w 1 1U 1 (1 U, 1 1 (1 (1 w w w w 1 1V 1 (1 V 1 1V 1 (1 V w w w w 1 (1 T T, 1 1U 1 (1 U (1 (1 (1 (1 w w w w 1 (1 U 1U 1 U (1 U, w 1 1V 1 (1 V 1 1 w (1 1 (1 1 1 (1 (1 (1 w w w w V V V V w T 1 1 w U 1 1 (1 w 1 (1 T, w 1 1 (1 U w V 1 1 w 1 1 (1 V (1 (1,. Therefore, this completes the proof of Equatio (8. Based o the properties of the INNWAA ad INNWGA operators [13], the SVNNHWAGA operator also cotais these properties: (1 Idempotecy: If z = z for = 1, 2,,, the there is SVNNHWAGA ( z1, z2,, z z. (2 Boudedess: If zmi mi( z1, z2,..., z ad zmax max( z1, z2,..., z for = 1, 2,,, the there z SVNNHWAGA ( z, z,, z z. exists mi 1 2 max

7 Iformatio 2017, 8, 84 7 of 13 (3 Mootoicity: If z z * for = 1, 2,,, the SVNNHWAGA ( z1, z2,, z * * * SVNNHWAGA ( z1, z2,, z holds. Whe the some values of α [0,1] are specified as the special cases, we ca ivestigate the families of the SVNNHWAGA operator below: (1 The SVNNHWAGA operator reduces to the SVNNWAA operator if α = 1; (2 The SVNNHWAGA operator reduces to the SVNNWGA operator if α = 0; (3 The SVNNHWAGA operator is the mea of the SVNNWAA ad SVNNWGA operators if α = SVNNHOWAGA Operator Defiitio 4. Let z = <T, U, V> ( = 1, 2,, be a collectio of sigle-valued eutrosophic umbers. The, the SVNNHOWAGA operator is defied by 1 2 p( p( 1 1 SVNNHOWAGA ( z, z,..., z z z (1 (9 where (p(1, p(2,, p( is a permutatio of (1, 2,, based o p( 1 p( for = 2,, ; (ζ1, ζ2,, ζ is a associated weight vector, satisfyig ζ [0, 1] ad 1 1 ; α is ay real umber betwee 0 ad 1. The, the SVNNHOWAGA operator ca reflect the importat degrees of the ordered positios of argumets i the aggregated process of sigle-valued eutrosophic umbers. Theorem 2. Let z = <T, U, V> ( = 1, 2,, be a collectio of sigle-valued eutrosophic umbers ad α [0, 1]. Thus, the aggregated value of the SVNNHOWAGA operator is also a sigle-valued eutrosophic umber, which is calculated by Tp( Tp( 1 1 (1 1 (1, (1 (1 1 2 p( p( p( p( SVNNHOWAGA( z, z,..., z z z 1 1 U (1 U, (1 Vp( Vp( (1 (10 where (p(1, p(2,, p( is a permutatio of (1, 2,, based o p( 1 p( for = 2,, ; (ζ1, ζ2,, ζ is a associated weight vector with 1 for ζ [0,1]. 1 Based o the similar proof maer of Theorem 1, it is obvious that Equatio (10 ca hold. Here, we omit the proof. Correspodig to the properties of the INNOWAA ad INNOWGA operators i [18], the SVNNHOWAGA operator also cotais the followig properties: (1 Idempotecy: If z = z for = 1, 2,,, the there exists SVNNHOWAGA ( z1, z2,..., z z. (2 Boudedess: If zmi mi( z1, z2,..., z ad zmax max( z1, z2,..., z for = 1, 2,,, the there z SVNNHOWAGA ( z, z,, z z. exists mi 1 2 max (3 Mootoicity: If z z * for = 1, 2,,, the SVNNHOWAGA ( z1, z2,, z * * * SVNNHOWAGA ( z1, z2,, z holds. ' ' ' ( z1, z2,, z is ay permutatio of ( z1, z2,, z, the ' ' ' SVNNHOWAGA ( z, z,, z SVNNHOWAGA ( z, z,, z holds. (4 Commutativity: If Whe some values of α [0, 1] are specified as the special cases, we ca ivestigate the families of the SVNNHOWAGA operator below:

8 Iformatio 2017, 8, 84 8 of 13 (1 The SVNNHOWAGA operator reduces to the SVNNOWAA operator ifα= 1; (2 The SVNNHOWAGA operator reduces to the SVNNOWGA operator ifα= 0; (3 The SVNNHOWAGA operator is the mea of the SVNNOWAA ad SVNNOWGA operators if α = Numerical Examples We still cosider the aforemetioed two umerical examples to illustrate the effectiveess ad ratioality of the aggregated values of the SVNNHWAGA ad SVNNHOWAGA operators. Geerally takig α = 0.5, we apply the SVNNHWAGA ad SVNNHOWAGA operators to calculate the two umerical examples i Sectio 2. For Example 1, by usig Equatios (8 ad (10 we ca obtai SVNNHWAGA(z1, z2 = <0.0447, 0, 0>, which is the moderate value betwee SVNNWAA(z1, z2 = <1, 0, 0> ad SVNNWGA(z1, z2 = <0.002, 0, 0>, ad SVNNHOWAGA(z1, z2 = <0.7079, 0, 0>, which is the moderate value betwee SVNNOWAA(z1, z2 = <1, 0, 0> ad SVNNOWGA(z1, z2 = <0.5012, 0, 0>. For Example 2, by usig Equatios (8 ad (10 we get SVNNHWAGA(z1, z2 = <0.7079, 0, 0>, which is the moderate value betwee SVNNWAA(z1, z2 = <1, 0, 0> ad SVNNWGA(z1, z2 = <0.5012, 0, 0>, ad SVNNHOWAGA(z1, z2 = <0.0447, 0, 0>, which is the moderate value betwee SVNNOWAA(z1, z2 = <1, 0, 0> ad SVNNOWGA(z1, z2 = <0.002, 0, 0>. From the above aggregated results of the two umerical examples, the SVNNHWAGA ad SVNNHOWAGA operators idicate their moderate values. Obviously, the SVNNHWAGA ad SVNNHOWAGA operators demostrate their effectiveess ad ratioality i the iformatio aggregatios. 4. Decisio-Makig Method Usig the SVNNHWAGA or SVNNHOWAGA Operator This sectio develops a multiple-attribute decisio-makig method by usig the SVNNHWAGA or SVNNHOWAGA operator. I a multiple-attribute decisio-makig problem, suppose that Z = {Z1, Z2,, Zm} is a set of alteratives ad A = {A1, A2,, A} is a set of attributes. By decisio-makers suitability evaluatio for each attribute A over each alterative Zi, all the evaluatio values are expressed by sigle-valued eutrosophic umbers zi = <Ti, Ui, Vi>, where Ti, Ui, Vi [0,1] ad 0 Ti + Ui + Vi 3 ( = 1, 2,, ; i = 1, 2,, m. I the sigle-valued eutrosophic umber zi, Ti idicates the degree that the alterative Zi is suitable for the attribute A, Ui idicates the degree that the alterative Zi is usure/idetermiate for the attribute A, ad Vi idicates the degree that the alterative Zi is usuitable for the attribute A. Thus, all the evaluatio values ca be costructed as a sigle-valued eutrosophic decisio matrix D = (zim. Hece, we ca apply the proposed decisio-makig method based o the SVNNHWAGA or SVNNHOWAGA operator to the multiple-attribute decisio-makig problem ad give the followig decisio procedures: 1 Step 1. Suppose that the weight vector of attributes is w = (w1, w2,, w ad satisfies w 1 for w [0, 1]. The, the aggregated value of zi (i = 1, 2,, m for each alterative Zi (i = 1, 2,, m is calculated by the followig SVNNHWAGA operator: w w Ti Ti 1 1 (1 (1 w w w i i1 i2 i i i i i w Vi 1 1 (1 1 (1, z SVNNHWAGA( z, z,..., z w z z 1 1 U (1 U, w 1 1 (1 Vi (1 (11

9 Iformatio 2017, 8, 84 9 of 13 O the other had, suppose that the ordered importat positios of all the argumets are give by the associated weight vector ζ = (ζ1, ζ2,, ζ, satisfyig 1 for ζ [0,1]. Thus, the 1 aggregated value of zi (i = 1, 2,, m for each alterative Zi (i = 1, 2,, m is calculated by the followig SVNNHOWAGA operator: Tip( Tip( 1 1 (1 (1 i i1 i2 i ip( ip( ip( ip( Vip( 1 1 (1 1 (1, z SVNNHOWAGA( z, z,..., z z z 1 1 U (1 U, 11 (1 V ( ip (1 (12 Step 2. By Equatio (1 (Equatio (2 if ecessary, we calculate the score values of E(zi (accuracy degrees of H(zi if ecessary (i = 1, 2,, m. Step 3. Correspodig to the score values (accuracy degrees, we rak all the alteratives i a descedig order ad determie the best choice based o the alterative with the largest value. Step 4. Ed. 5. MADM Problem of the Desig Schemes of Puchig Machie I this sectio, a applied example about the multiple-attribute decisio-makig (MADM problem of the desig schemes (alteratives of puchig machie is itroduced from [29] to illustrate the applicatio ad ratioality of the proposed decisio-makig method by the actual case. The coceptual desig stage maily cotais two tasks, i which desigers firstly provide differet desig schemes (potetial alteratives accordig to their kowledge ad experiece, ad the decisio makers (desigers give their suitability evaluatio (decisio makig of all the potetial desig schemes. Some maufacturig compay wats to desig the puchig machie to develop a ew mechaical product. Geerally speakig, the puchig machie cosists of the reducig mechaism, puchig mechaism, ad feed itermittet mechaism to costruct its motio scheme. Based o the motio scheme, a group of desigers provides a set of four potetial desig schemes (alteratives Z = {Z1, Z2, Z3, Z4} by their kowledge ad experieces, which are show i Table 1. The desigers (decisio makers must make a decisio depedig o the requiremets of the four attributes: (a A1 is the maufacturig cost; (b A2 is the structure complexity; (c A3 is the trasmissio effectiveess; (d A4 is the reliability; (e A5 is the maitaiability. By the decisio makers suitability evaluatio for the alteratives of Zi (i = 1, 2, 3, 4 with respect to the attributes of A ( = 1, 2, 3, 4, 5, their evaluatio values are expressed by the form of sigle-valued eutrosophic umbers. Thus, the sigle-valued eutrosophic umber decisio matrix D = (zi4 5 ca be costructed as follows: (0.75, 0.1, 0.4 (0.8, 0.1, 0.3 (0.85, 0.1, 0.2 (0.85, 0.1, 0.3 (0.9, 0.1, 0.2 (0.7, 0.1, 0.5 (0.75, 0.1, 0.1 (0.75, 0.2, 0.1 (0.8, 0.1, 0.1 (0.8, 0.2, 0.3 D (0.8, 0.2, 0.3 (0.78, 0.1, 0.2 (0.8, 0.1, 0.2 (0.8, 0.2, 0.2 (0.75, 0.1, 0.3 (0.9, 0.1, 0.2 (0.85, 0.1, 0.1 (0.9, 0.1, 0.2 (0.85, 0.1, 0.3 (0.85, 0.2, 0.3 Table 1. Four desig schemes (potetial alteratives of puchig machie [29]. Potetial Alterative Z1 Z2 Z3 Z4 Reducig mechaism Gear reducer Gear head motor Gear reducer Gear head motor Puchig mechaism Crak-slider mechaism Six-bar puchig mechaism Six-bar puchig mechaism Crak-slider mechaism

10 Iformatio 2017, 8, of 13 Dial feed itermittet mechaism Sheave mechaism Ratchet feed mechaism The, we ca apply the developed decisio-makig method based o the SVNNHWAGA or SVNNHOWAGA operator to the multiple-attribute decisio-makig problem of the desig schemes of puchig machie. If the weight vector of the five attributes is cosidered as w = (0.25, 0.2, 0.25, 0.15, 0.15 i the multiple-attribute decisio-makig problem, the the decisio steps are preseted as follows: Step 1. By Equatio (11 (geerally take α = 0.5, we calculate the aggregated values of zi (i = 1, 2, 3, 4 for each alterative Zi (i = 1, 2, 3, 4 as the followig results: z1 = <0.8255, , >, z2 = <0.7534, , >, z3 = <0.7888, , >, ad z4 = <0.8761, , >. Step 2. By Equatio (1, we calculate the score values of E(zi for each alterative Zi (i = 1, 2, 3, 4 as the followig values: E(z1 = , E(z2 = , E(z3 = , ad E(z4 = Step 3. Accordig to E(z4 > E(z1 > E(z3 > E(z2, the rakig of the four desig schemes is Z4 Z1 Z3 Z2. So, the best desig scheme is Z4. These results are the same as i [29]. If the associated weight vector ζ = (0.4, 0.3, 0.1, 0.1, 0.1 is cosidered as the ordered importat positios of all the give argumets i the multiple-attribute decisio-makig problem, the decisio steps are described as the follows: Step 1. By Equatio (12 (i geeral take α = 0.5, we calculate the aggregated values of zi (i = 1, 2, 3, 4 for each alterative Zi (i = 1, 2, 3, 4 as the followig results: z1 = <0.8577, , >, z2 = <0.7708, , >, z3 = <0.7892, , >, ad z4 = <0.8711, , >. Step 2. By Equatio (1, we calculate the score values of E(zi for each scheme Zi (i = 1, 2, 3, 4 as the followig values: E(z1 = , E(z2 = , E(z3 = , ad E(z4 = Step 3. Accordig to E(z4 > E(z1 > E(z2 > E(z3, the rakig of the four desig schemes is Z4 Z1 Z2 Z3. Thus, the best desig scheme is also Z4. Although the above two rakig orders show little differece, the best scheme Z4 is idetical. For comparative coveiece, all the results of the proposed decisio-makig approach ad the related decisio-makig methods based o the SVNNWAA, SVNNWGA, SVNNOWAA, ad SVNNOWGA operators are summarized i Table 2.

11 Iformatio 2017, 8, of 13 Table 2. Decisio-makig results of differet aggregatio operators. Aggregatio Operator Aggregated Result Score Value Rakig SVNNWAA (α = 1 SVNNWGA (α = 0 SVNNHWAGA (α = 0.5 SVNNOWAA (α = 1 SVNNOWGA (α = 0 SVNNHOWAGA (α = 0.5 z1 = <0.8301, , >, z2 = <0.7553, , >, z3 = <0.7892, , >, z4 = <0.8775, , > z1 = <0.8209, , >, z2 = <0.7516, , >, z3 = <0.7883, , >, z4 = <0.8746, , > z1 = <0.8255, , >, z2 = <0.7534, , >, z3 = <0.7888, , >, z4 = <0.8761, , > z1 = <0.8619, , >, z2 = <0.7723, , >, z3 = <0.7896, , >, z4 = <0.8725, , > z1 = <0.8536, , >, z2 = <0.7693, , >, z3 = <0.7888, , >, z4 = <0.8697, , > z1 = <0.8577, , >, z2 = <0.7708, , >, z3 = <0.7892, , >, z4 = <0.8711, , > E(z1 = , E(z2 = , E(z3 = , E(z4 = E(z1 = , E(z2 = , E(z3 = , E(z4 = E(z1 = , E(z2 = , E(z3 = , E(z4 = E(z1 = , E(z2 = , E(z3 = , E(z4 = E(z1 = , E(z2 = , E(z3 = , E(z4 = E(z1 = , E(z2 = , E(z3 = , E(z4 = Z4 Z1 Z2 Z3 Z4 Z1 Z3 Z2 Z4 Z1 Z3 Z2 Z4 Z1 Z2 Z3 Z4 Z1 Z2 Z3 Z4 Z1 Z2 Z3 The results of Table 2 show that all the aggregated results of the SVNNHWAGA ad SVNNHOWAGA operators ted toward moderate values betwee the aggregated values of the SVNNWAA ad SVNNWGA operators or the SVNNOWAA ad SVNNOWGA operators. Hece, the SVNNHWAGA ad SVNNHOWAGA operators are suitable ad effective ad ca overcome the drawbacks of the SVNNWAA, SVNNOWAA, SVNNWGA ad SVNNOWGA operators. Furthermore, the differet aggregatio operators may show differet rakig orders. The, the differet values of α may result i differet rakig orders. Compared with the decisio-makig method based o the hybrid arithmetic ad geometric aggregatio operators of ituitioistic fuzzy umbers itroduced by Ye [28], Ye s method [28] uses the hybrid arithmetic ad geometric aggregatio operators of ituitioistic fuzzy umbers to aggregate ituitioistic fuzzy umbers ad applied them to multiple-attribute decisio-makig problems uder a ituitioistic fuzzy umber eviromet; while the multiple-attribute decisio-makig method proposed i this paper uses the hybrid arithmetic ad geometric aggregatio operators of sigle-valued eutrosophic umbers to aggregate sigle-valued eutrosophic iformatio as the extesio of the hybrid arithmetic ad geometric aggregatio operators of ituitioistic fuzzy umbers ad exteds the decisio-makig method based o the hybrid arithmetic ad geometric aggregatio operators of ituitioistic fuzzy umbers [28], because a ituitioistic fuzzy umber is a special case of a sigle-valued eutrosophic umber ad caot express ad hadle idetermiate ad icosistet iformatio. Thus, it is obvious that the SVNNHWAGA ad SVNNHOWAGA operators ad their decisio-makig method proposed i this paper are superior to the previous hybrid arithmetic ad geometric aggregatio operators ad decisio-makig method with ituitioistic fuzzy umbers [28], because the latter caot express ad deal with the decisio-makig problems with idetermiate ad icosistet iformatio. Hece, the proposed SVNNHWAGA ad SVNNHOWAGA operators ad their decisio-makig method i this study are more geeral ad more suitable uder idetermiate ad icosistet eviromets tha the previous Ye s method [28]. However, the preseted decisio-makig method based o the SVNNHWAGA or SVNNHOWAGA operator demostrates its suitability ad effectiveess i some decisio-makig situatios sice it ca overcome the drawbacks of the ureasoably aggregated results i some cases (as metioed i Sectio 2. Sice some value of α is specified by the preferece ad actual

12 Iformatio 2017, 8, of 13 requiremets of decisio makers, the decisio-makig method proposed i this study appears to be more flexible ad more reasoable tha the decisio-makig method based o oe of the SVNNWAA, SVNNOWAA, SVNNWGA, ad SVNNOWGA operators. 6. Coclusios This paper preseted the SVNNHWAGA ad SVNNHOWAGA operators to overcome some drawbacks of the SVNNWAA, SVNNOWAA, SVNNWGA ad SVNNOWGA operators to aggregate sigle-valued eutrosophic umbers i some cases, ad the ivestigated their some properties ad ratioality. Furthermore, we established a multiple-attribute decisio-makig method based o the SVNNHWAGA or SVNNHOWAGA operator. Fially, a multiple-attribute decisio-makig problem about desig schemes of puchig machie is preseted as a case to show the applicatio ad ratioality of the proposed decisio-makig method. However, the proposed decisio-makig method based o the SVNNHWAGA or SVNNHOWAGA operator provides a effective ad reasoable way for multiple-attribute decisio-makig problems sice the SVNNWAA, SVNNOWAA, SVNNWGA ad SVNNOWGA operators are special cases of the SVNNHWAGA ad SVNNHOWAGA operators uder a sigle-valued eutrosophic eviromet. Similarly, this study will be also further exteded to iterval eutrosophic sets ad applicatios, like group decisio makig, patter recogitio, fault diagosis, ad medical diagosis, ad so o. Ackowledgmets: This paper was supported by the Natioal Natural Sciece Foudatio of Chia (No Author Cotributios: Zhikag Lu proposed the SVNNHWAGA ad SVNNHOWAGA operators ad their decisio-makig method; Ju Ye preseted a multiple-attribute decisio-makig problem about desig schemes of puchig machie ad gave comparative aalysis; both authors wrote the paper together. Coflicts of Iterest: The authors declare o coflicts of iterest. Refereces 1. Zadeh, L.A. Fuzzy Sets. If. Cotrol 1965, 8, Zadeh, L.A. The cocept of a liguistic variable ad its applicatio to approximate reasoig Part I. If. Sci. 1975, 8, Ataassov, K. Ituitioistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, Ataassov, K.; Gargov, G. Iterval-valued ituitioistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, Ye, J. Multicriteria decisio-makig method usig the Dice similarity measure based o the reduct ituitioistic fuzzy sets of iterval-valued ituitioistic fuzzy sets. Appl. Math. Model. 2012, 36, De Maio, C.; Feza, G.; Loia, V.; Orciuoli, F.; Herrera-Viedma, E. A cotext-aware fuzzy liguistic cosesus model supportig iovatio processes. I Proceedigs of the 2016 IEEE Iteratioal Coferece o Fuzzy Systems, Vacouver, BC, Caada, July 2016; pp De Maio, C.; Feza, G.; Loia, V.; Orciuoli, F.; Herrera-Viedma, E. A framework for cotext-aware heterogeeous group decisio makig i busiess processes. Kowl.-Based Syst. 2016, 102, Wa, S.P.; Xu, J.; Dog, J.Y. Aggregatig decisio iformatio ito iterval-valued ituitioistic fuzzy umbers for heterogeeous multi-attribute group decisio makig. Kowl.-Based Syst. 2016, 113, Smaradache, F. Neutrosophy: Neutrosophic Probability, Set, ad Logic; America Research Press: Rehoboth, DE, USA, Wag, H.; Smaradache, F.; Zhag, Y.Q.; Suderrama, R. Sigle valued eutrosophic sets. Multisp. Multistruct. 2010, 4, Wag, H.; Smaradache, F.; Zhag, Y.Q.; Suderrama, R. Iterval Neutrosophic Sets ad Logic: Theory ad Applicatios i Computig; Hexis: Phoeix, AZ, USA, Ye, J. A multicriteria decisio-makig method usig aggregatio operators for simplified eutrosophic sets. J. Itell. Fuzzy Syst. 2014, 26, Zhag, H.Y.; Wag, J.Q.; Che, X.H. Iterval eutrosophic sets ad their applicatio i multicriteria decisio makig problems. Sci. World J. 2014, 2014, Liu, P.D.; Wag, Y.M. Multiple attribute decisio makig method based o sigle-valued eutrosophic ormalized weighted Boferroi mea. Neural Comput. Appl. 2014, 25,

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