Distance and Similarity Measures for Multiple Attribute Decision Making with Single-Valued Neutrosophic Hesitant Fuzzy Information

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1 New Treds i Neutrosophic Theory ad Applicatios RIDVAN ŞAHIN, PEIDE LIU 2 Faculty of Educatio, Bayburt Uiversity, Bayburt, 69000, Turkey. mat.ridoe@gmail.com 2 School of Maagemet Sciece ad Egieerig, Shadog Uiversity of Fiace ad Ecoomics, Jia 25004, Shadog, Chia. peide.liu@gmail.com Distace ad Similarity Measures for Multiple Attribute Decisio Makig with Sigle-Valued Neutrosophic Hesitat Fuzzy Iformatio Abstract With respect to a combiatio of hesitat sets, ad sigle-valued eutrosophic sets which are a special case of eutrosophic sets, the sigle valued eutrosophic hesitat sets (SVNHFS have bee proposed as a ew theory set that allows the truth-membership degree, idetermiacy membership degree ad falsity-membership degree icludig a collectio of crisp values betwee zero ad oe, respectively. There is o cosesus o the best way to determie the order of a sequece of siglevalued eutrosophic hesitat fuzzy elemets. I this paper, we first develop a axiomatic system of distace ad similarity measures betwee sigle-valued eutrosophic hesitat fuzzy sets ad also propose a class of distace ad similarity measures based o three basic forms such that the geometric distace model, the set-theoretic approach, ad the matchig fuctios. The we utilize the distace measure betwee each alterative ad ideal alterative to establish a multiple attribute decisio makig method uder sigle-valued eutrosophic hesitat fuzzy eviromet. Fially, a umerical example of ivestmet alteratives is provided to show the effectiveess ad usefuless of the proposed approach. The advatages of the proposed distace measure over existig measures have bee discussed. Keywords Sigle-valued eutrosophic set, hesitat fuzzy set, sigle-valued eutrosophic hesitat fuzzy set, distace measure, similarity measure, multiple attribute decisio makig.. Itroductio Most of our traditioal tools for formal modelig, reasoig ad computig are crisp, determiistic ad precise i character. However, there are may complicated problems i ecoomics, egieerig, eviromet, social sciece, medical sciece, etc., that ivolve data which are ot always all crisp. Classical methods caot successfully hadle ucertaity, because the ucertaities appearig i these domais may be of various types. Zadeh (965 itroduced fuzzy 35

2 Floreti Smaradache, Surapati Pramaik (Editors sets (FS ad applied them i may fields icludig ucertaity. As a geeralizatio of the fuzzy sets, Ataassov (986 itroduced the cocept of ituitioistic fuzzy set (IFS. The Ataassov ad Gargov (989 exteded the cocept of IFS to iterval-valued ituitioistic fuzzy set (IVIFS. Curret literature has very large umber of distace ad similarity measures for FSs ad IFSs (Ataassov 986; Xuecheg 992; Che et al. 995; Liu et al. 205; Farhadiia 204; Wag 997; Szmidt ad Kacprzyk 2000; Khaleie ad Fasaghari 202; Grzegorzewski 2004; Wag ad Xi 2005; Xu 2007; Hug ad Yag 2007; Li 2007; Şahi 205; Ta 20. Torra ad Narukawa (2009 ad Torra (200 proposed the cocept of hesitat fuzzy set (HFS, discussed the relatioship betwee hesitat fuzzy set ad ituitioistic fuzzy set ad showed that the evelope of hesitat fuzzy set is a ituitioistic fuzzy set. The membership degree of a elemet i hesitat fuzzy set icludes a set of possible values betwee zero ad oe. Sice its appearace, the hesitat fuzzy iformatio has bee used to solve multiple attribute decisio makig problems. Xia ad Xu (20 defied some techiques for aggregatig hesitat fuzzy iformatio ad utilized their performaces i decisio makig. Based o the relatioship betwee HFS ad IFS, they proposed the set-theoretic laws of HFSs. Xu ad Xia (20 defied a collectio of distace measures for HFSs ad geerated the similarity measures associated with the proposed distace measures. Furthermore, Zhu et al. (202 itroduced dual hesitat fuzzy set (DHFS as a geeralizatio of FSs, IFSs, HFSs, ad fuzzy multisets (FMSs ad preseted some basic operatios of DHFSs. A DHFS are characterized by two class of possible values, the membership degrees ad omembership degrees. Therefore, DHFSs iclude FSs, IFSs, HFSs, ad FMSs uder certai coditios, ad so they have the desirable performaces ad advatages of its ow ad appear to be a more favorable method tha aforemetioed sets because of cosiderig much more iformatio give by decisio makers. Sigh (203 itroduced a comprehesive family of distace measures ad related similarity measures for DHFSs. As a ew brach of philosophy that combies the kowledge of logics, philosophy, set theory, ad probability, Smaradache (999, 2005 proposed the cocept of eutrosophic sets (NSs as a further geeralizatio of ucertaity modelig tools. Ulike the aforemetioed sets, a eutrosophic set cosists of three membership fuctios such that the truth-membership fuctio, the idetermiacy-membership fuctio ad the falsity membership fuctio. Additioally, the ucertaity preseted here, i.e. the idetermiacy factor, is idepedet o the truth ad falsity values, whereas the icorporated ucertaity is depedet o the degrees of belogigess ad o-belogigess of existig sets. The structure of NSs is ot appropriate to apply to real-life situatios. Therefore, Wag et al. (2005, 200 developed sigle-valued eutrosophic sets (SVNSs ad iterval eutrosophic sets (INSs, which are a extesio of NSs. Şahi (204 proposed a eutrosophic hierarchical clusterig algorithm based o relatioship betwee SVNSs. Şahi ad Küçük (204 defied a subsethood measure for SVNSs ad applied it i a decisio makig problem. The correlatio coefficiets of SVNSs as well as a decisio-makig method usig SVNSs were proposed by Ye (203. I additio, Ye (204b ivestigated the cocept of simplified eutrosophic sets (SNSs, which ca be expressed by three real umbers i the real uit iterval [0,], provided the set-theoretic operators of SNSs, ad developed a multi criteria decisio makig 36

3 New Treds i Neutrosophic Theory ad Applicatios (MCDM method based o the aggregatio operators of SNSs. But, Peg et al. (205 showed that some operatios of Ye (204b may also be urealistic i special cases, ad defied the ovel operatios ad aggregatio operators ad applied them to MCDM problems. Also, Ye (204a proposed the sigle valued eutrosophic cross-etropy for solvig multicriteria decisio makig (MCDM problems with sigle valued eutrosophic iformatio. Broumi ad Smaradache (203 exteded the correlatio coefficiet to INSs. Zhag et al. (204 developed a MCDM method based o aggregatio operators withi a iterval eutrosophic eviromet. Furthermore, Maumdar ad Samata (204 proposed the distace ad similarity measures betwee SVNSs. Ye (204d exteded these measures to INSs as based o the relatioship betwee similarity measures ad distaces. Liu ad Wag (204 discussed a sigle-valued eutrosophic ormalized weighted Boferroi mea (SVNNWBM operator based o Boferroi mea, the weighted Boferroi mea (WBM, ad the ormalized WBM. Peg et al. (205 itroduced the multi-valued eutrosophic sets (MVNSs ad developed the operatios of multi-valued eutrosophic umbers (MVNNs based o Eistei operatios. Recetly, Ye (205c proposed the cocept of sigle valued eutrosophic hesitat fuzzy set (SVNHFS as a geeralizatio of FSs, IFSs, HFSs, FMSs, ad also SVNSs ad discussed the basic operatios ad properties of SVNHFSs. SVNHFSs cosist of three parts, first is the truthmembership hesitacy fuctio, secod is the idetermiacy-membership hesitacy fuctio, ad third is the falsity-membership hesitacy fuctio. The curret sets, icludig FSs, IFSs, HFSs, FMSs, ad SVNSs ca be regarded as special cases of SVNHFSs. I a SVNHFS, the truthmembership hesitacy degrees, idetermiacy-membership hesitacy degrees ad falsitymembership hesitacy degrees are represeted by three sets of possible values betwee zero ad oe, respectively. Therefore, it is ot oly more geeral tha aforemetioed set but oly more suitable for solvig MADM problems due to cosiderig much more iformatio provided by decisio makers. From above aalysis, we caot utilize the curret measures for dealig with distace ad similarity measure betwee SVNHFSs. Therefore, we eed to develop ew distace ad similarity measures for SVNHFSs, because a SVNHFS cosists of three basic membership fuctio such that the truth-membership hesitacy fuctio ad idetermiacy-membership hesitacy fuctio ad falsity-membership hesitacy fuctio. I this paper, we first defie a compressive class of distace measures betwee SVNHFSs ad the proposed the similarity measures based o the geometric distace model, the set-theoretic approach ad the matchig fuctios. Also, we show that the proposed measures satisfies the axiom defiitio of distace ad similarity measures developed for SVNHFSs. Fially, we utilize the proposed distace measure to solve a MADM problem with sigle valued eutrosophic hesitat fuzzy iformatio. The rest of this paper is orgaized as follows. I sectio 2, we itroduce some basic cocepts related to HFS, SVNS ad SVNHFS, ad some operatioal ad theoretical laws. I Sectio 3, we propose a variety class of distace measures of SVNHFSs as a further geeralizatio of the existig distace measure for HFSs, DHFSs, IFSs, ad SVNSs. Based o the geometric distace model, the set-theoretic approach ad the matchig fuctios, we preset some similarity measures betwee SVNHFSs. Sectio 4 develops a MADM method with sigle valued eutrosophic hesitat fuzzy iformatio based o the proposed distace 37

4 Floreti Smaradache, Surapati Pramaik (Editors measure for SVNHFSs. I Sectio 6, a illustrative example is provided to demostrate the applicatio ad effectiveess of the developed method. Sectio 7 gives related comparative aalysis. Fially, coclusios ad future work are give i Sectio Prelimiaries I this subsectio, we give some cocepts related to NSs ad SVNSs. 2. Neutrosophic set Defiitio. (Smaradache 2005 Let X be a uiverse of discourse, the a eutrosophic set is defied as: 38 A = { x, T A (x, I A (x, F A (x : x X}, ( which is characterized by a truth-membership fuctio T A : X ]0, + [, a idetermiacymembership fuctio I A : X ]0, + [ad a falsity-membership fuctio F A : X ]0, + [. 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 represetatios t A (x, i A (x ad f A (x istead of T A (x, I A (x ad F A (x, respectively. Wag et al. (200 defied the sigle valued eutrosophic set which is a istace of eutrosophic set. 2.2.Sigle valued eutrosophic sets Defiitio 2. Wag et al. (200 Let X be a uiverse of discourse, the a sigle valued eutrosophic set is defied as: A = { x, t A (x, i A (x, f A (x : x X}, (2 where t A : X [0,], i A : X [0,] ad f A : X [0,] with 0 t A (x + i A (x + f A (x 3 for all x X. The values t A (x, i(x ad f A (x deote the truth-membership degree, the idetermiacymembership degree ad the falsity membership degree of x to A, respectively. 2.3.Hesitat fuzzy sets Defiitio 3. (Torra 200 A hesitat fuzzy set M o X is defied i terms of a fuctio h M whe applied to X, which returs a fiite subset of [0,], i.e., M = { x, h M (x : x X}, (3 where h M (x is a set of some differet values i [0,], represetig the possible membership degrees of the elemet x X to M. 2.4.Sigle-valued eutrosophic hesitat sets Defiitio 4. (Ye 204c Let X be a fixed set, the a sigle-valued eutrosophic hesitat fuzzy set A o X is defied as, A = { x, (t A (x, i A(x, f A(x : x X} (4 i which t A (x, i A(x, ad f A(x are three sets of some differet values i [0,], deotig the truth-membership hesitat degrees, idetermiacy-membership hesitat degrees, ad falsitymembership hesitat degrees of the elemet x X to A, respectively, with the coditios 0 γ, δ, η ad 0 γ + + δ + + η + 3, where γ t A (x, δ i (x, η f A(x, γ + t A + (x =

5 New Treds i Neutrosophic Theory ad Applicatios γ t A(x max{γ}, δ + i A + (x = max{δ} δ i A (x, ad η + f A+ (x = η f A(x max{η} for x X. For coveiece, the three tuple A = { (t A (x, i A (x, f A(x} is called a sigle-valued eutrosophic hesitat fuzzy elemet (SVNHFE or a triple hesitat fuzzy elemet, which is deoted by the simplified symbol A = { (t A, i A, f A}. Now, we give the followig defiitios to propose the distace ad similarity measures betwee SVNHFSs. Defiitio 5 Let A, B ad C be three SVNHSs o X = {x, x 2,, x }, the the distace measure betwee A ad B is defied as d (A, B, which satisfies the followig properties: ( 0 d (A, B ; (2 d (A, B = 0 if ad oly if A = B; (3 d (A, B = d (B, A. (4 d (A, B d (A, C ad d (B, C d (A, C, if A B C. Defiitio 6. Let A, B ad C be three SVNHSs o X = {x, x 2,, x }, the the similarity measure betwee A ad B is defied as s (A, B, which satisfies the followig properties: ( 0 s (A, B ; (2 s (A, B = if ad oly if A = B; (3 s (A, B = s (B, A. (4 s (A, B s (A, C ad s (B, C s (A, C, if A B C. From Defiitios 5 ad 6, it is oted that s (A, B = d (A, B. Similar to HFS, i most of the cases, the umber of values i differet SVNHFEs might be differet, i.e., l t A(x i l t B(x i, l i A (x i l i B (x i ad l f A(x i l f B(x i. Let = max{l t A(x i, l t B(x i }, = max{l i A (x i, l i B (x i} ad = max{l f A(x i, l f B(x i } for each x i X. We ca make them have the same umber of elemets through addig some elemets to the SVNHFE which has less umber of elemets. The selectio of this operatio maily depeds o the decisio makers risk prefereces. Pessimists expect ufavorable outcomes ad may add the miimum of the truth-membership degree ad maximum value of idetermiacy-membership degree ad falsity-membership degree. Optimists aticipate desirable outcomes ad may add the maximum of the truth-membership degree ad miimum value of idetermiacy-membership degree ad falsity-membership degree That is, accordig to the pessimistic priciple, if l t A(x i < l t B(x i, the the least value of t A (x i or t B (x i will be added to t A (x i. Moreover, if l i A (x i < l i B (x i, the the largest value of l t A(x i or l t B(x i will be iserted i i A(x i for x i X. Similarity, if l f A(x i < l f B(x i, the the largest value of l f A(x i or l f B(x i will be iserted i f A(x i for x i X. 3. Some distace measures for SVNHFSs I this sectio, we give some distace measures betwee two SVNHFSs. Based o the geometric distace model for SVNHFSs, we defie the followig distace measures. ( Geeralized sigle valued eutrosophic hesitat ormalized distace (GN, for > 0; 39

6 Floreti Smaradache, Surapati Pramaik (Editors d GN = ( 3 ( t A + i A + f A σ( (x i f B, (5 where t A σ( (x i, t B σ( (x i ; i A σ( (x i, i B σ( (x i ad f A σ( (x i, f B σ( (x i are the th largest values of truth-membership hesitat degrees, idetermiacy-membership hesitat degrees, ad falsitymembership hesitat degrees of A ad B, respectively. i. If =, Eq. (5 reduces a sigle valued eutrosophic hesitat ormalized Hammig distace (NH: d NH = 3 ( t A + i A + f A σ( (x i f B. (6 ii. If = 2, Eq. (5 reduces a sigle valued eutrosophic hesitat ormalized Euclidea distace (NE d NE = ( 3 ( t A 2 + i A 2 + f A σ( (x i 2 f B 2. (7 Equatio (5 ca be viewed as a most geeralized case of distace measures. We ca see that if there is o idetermiacy i SVNHFS, the the idetermiacy-membership value of SVNHFS will disappear, hece, Eqs. (5, (6, ad (7 are reduced to a geeralized dual hesitat ormalized distace, a dual hesitat ormalized Hammig distace ad a dual hesitat ormalized Euclidea distace, respectively (i.e., the distace measures proposed by Sigh 203. I additio, if there is o both idetermiacy ad omembership i SVNHFS, the both idetermiacy-membership value ad falsity-membership value of SVNHFS will disappear, hece, Eqs. (5, (6, ad (7 are reduced to a geeralized hesitat ormalized distace, a hesitat ormalized Hammig distace 40

7 New Treds i Neutrosophic Theory ad Applicatios ad a hesitat ormalized Euclidea distace, respectively (i.e., the distace measure proposed by Xu ad Xia 20. If we apply the Hausdorff metric to the distace measure, we obtai that (2 Geeralized sigle valued eutrosophic hesitat ormalized Hausdorff distace (GNH: d GNH = ( 3 max ( t A, i A, f A i. If =, Eq. (6 reduces a sigle valued eutrosophic hesitat ormalized Hammig Hausdorff distace (NHH:. (8 d NHH = ( 3 max ( t A, i A, f A σ( (x i f σ( B (x i. (9 ii. If = 2, Eq. (6 reduces a sigle valued eutrosophic hesitat ormalized Euclidea Hausdorff distace (NEH: d NEH = ( 3 max ( t A 2, i A 2, f A σ( (x i f B 2 2. (0 I may practical situatios, the weight of each elemet x i X should be take ito accout. For istace, i MADM problems, the cosidered attribute usually has differet importace, thus eeds to be assiged with differet weights. Sice i SVNHFSs, we have three types of degree, oe is truth-membership degree, other is idetermiacy-membership ad fial is falsity-membership degree. Sice three degrees may have differet importace, accordig to decisio maker, differet weights ca be assiged to each elemet i each degree. Assume that the weights ω = (ω, ω 2, ω T with ω [0,], ω i = ; ψ = (ψ, ψ 2, ψ T with ψ i [0,], ψ i = ad φ = (φ, φ 2, φ T with φ i [0,], φ i = deote the weights assiged to truthmembership degree, idetermiacy-membership degree ad falsity-membership degree, respectively, of SVNHFS. Now, we preset the followig weighted distace measures for SVNHFSs. (3 Geeralized sigle valued eutrosophic hesitat weighted distace (GW: d GW = 3 ω i ( t A + ψ i ( l i (x i i A ( ( + φ i ( f A σ( (x i f B. ( 4

8 Floreti Smaradache, Surapati Pramaik (Editors i. If =, the we get a sigle valued eutrosophic hesitat weighted Hammig distace (WH: d WH = 3 ( ω ( t A + ψ ( i A + φ ( f A σ( (x i f B. (2 ii. If = 2, the we get a sigle valued eutrosophic hesitat weighted Euclidea distace (WE: d WE = 3 ω ( t A 2 + ψ ( l i (x i i A 2 ( ( + φ ( f A σ( (x i 2 f B 2. (3 (4 Geeralized sigle valued eutrosophic hesitat weighted Hausdorff distace (GWH, for > 0; d GWH = ( 3 max (ω i t A, ψ i A, φ f A σ( (x i f B. (4 i. =, the we get a sigle valued eutrosophic hesitat weighted Hammig Hausdorff distace (WHH: d WHH = ( 3 max (ω i t A, ψ i i A, φ i f A. (5 ii. = 2, the we get a sigle valued eutrosophic hesitat weighted Euclidea Hausdorff distace (WEH: 42

9 New Treds i Neutrosophic Theory ad Applicatios d WEH = ( 3 max (ω i ( t A 2, ψ i ( i A 2, φ i ( f A σ( (x i f B 2 2. (6 Next, we shall show that the proposed distace measures satisfy axiom defiitio of distace measure. Theorem 7. Let A, B ad C be ay SVNHFSs, the d NH (A, B is a distace measure. Proof. We should prove that d NH (A, B satisfies axioms (D-(D4. (D Suppose that A ad B are two SVNHFSs with attributes, the t A 0, i A 0 ad f A 0 ad so t A 0, Thus, we have that i A 0 ad f A 0. 3 ( t A + i A ad d NH (A, B 0. O the other had, sice + f A 0 t A, i A ad i A, we get 3 ( t A + i A ad so d NH (A, B. + f A The it implies that 0 d NH (A, B. 43

10 Floreti Smaradache, Surapati Pramaik (Editors D2 44 d NH (A, B = 3 ( t A + f A = 3 ( t B σ( (x i t A + f B σ( (x i f A = d NH (B, A. (D3 Cosider A = B, the + i A + i B σ( (x i i A A = B t A σ( (x i + i A σ( (x i + f A σ( (x i = t B σ( (x i + i B σ( (x i + f B σ( (x i t A + i A + f A = 0 3 ( t B σ( (x i t σ( A (x i + i B σ( (x i i A d NH (A, B = 0. (D4 Sice A B C, we have + f B σ( (x i f A = 0 t A σ( (x i t B σ( (x i t C σ( (x i, i C σ( (x i i B σ( (x i i A σ( (x i, f C σ( (x i f B σ( (x i f A σ( (x i.

11 New Treds i Neutrosophic Theory ad Applicatios The it follows that ad so t A t A σ( (x i t C, i A i A σ( (x i i C, f A f A σ( (x i f C d NH (A, B = 3 ( t A + f A 3 ( t A σ( (x i t C + f A σ( (x i f C = d NH (A, C. Similarly, we ca prove d NH (B, C d NH (A, C. + i A + i A σ( (x i i C Theorem 8. Let A ad B be two SVNHSs, the d GH (A, B ad d NE (A, B are two distace measures. Proof. By the similar proof maer of Theorem 7, we ca also give the proof of Theorem 8 (omitted. Theorem 9. Let A, B ad C be ay SVNHFSs, the d GNH (A, B is the distace measure. Proof. We should prove that d GNH (A, B satisfies axioms (D-(D4. (D Suppose that A ad B are two SVNHFSs with attributes, the t A 0, i A 0 ad f A 0 ad so ( 3 max ( t A, i A, f A σ( (x i f σ( B (x i Thus, we have d GNH (A, B

12 Floreti Smaradache, Surapati Pramaik (Editors (D2 46 d GNH (A, B = ( 3 max ( t A, i A, f A = ( 3 max ( t B σ( (x i t σ( A (x i, i B σ( (x i i A, f B σ( (x i f A = d GNH (B, A (D3 Let d GNH (A, B = 0, the ( 3 max ( t A, i A, f A t A σ( (x i = t B σ( (x i, A = B. (D4 Sice A B C, we have i A σ( (x i = i B σ( (x i ad f A σ( (x i = f B σ( (x i t A σ( (x i t B σ( (x i t C σ( (x i, i C σ( (x i i B σ( (x i i A σ( (x i ad f C σ( (x i f B σ( (x i f A σ( (x i. The it follows that ad so max ( t A, i A, f A max ( t A σ( (x i t C, i A σ( (x i i C, f A σ( (x i f C d GNH (A, B = ( 3 max ( t A, i A, f A ( 3 max ( t A σ( (x i t C, i A σ( (x i i C, f A σ( (x i f C = d GNH (A, C. Similarly, we ca prove d GNH (B, C d GNH (A, C. Theorem 0. Let A ad B be two SVNHSs, the d NHH (A, B ad d NEH (A, B are two distace measures. Proof. By the similar proof maer of Theorem 9, we ca also give the proof of Theorem 0 (omitted. 4. Some similarity measures for SVNHFSs I this sectio, we preset some similarity measures based o the proposed distace measures betwee SVNHFSs. 4.. The similarity measures based o geometric distace model for SVNHFSs With respect to Eq. (5, the similarity measure ca be defied as follows: ( Similarity measure based o geeralized sigle valued eutrosophic hesitat ormalized distace: = 0

13 New Treds i Neutrosophic Theory ad Applicatios s GN (A, B = d GN (A, B = d GN = ( 3 ( t A + i A + f A Similarly, we give aother similarity measures based o distace measure as follows: (i Similarity measure based o sigle valued eutrosophic hesitat ormalized Hammig distace: s NH (A, B = d NH (A, B (8 (ii Similarity measure based o sigle valued eutrosophic hesitat ormalized Euclidia distace: s NE (A, B = d NE (A, B (9 (2 Similarity measure based o geeralized sigle valued eutrosophic hesitat ormalized Hausdorff distace: s GNH(A, B = d GNH (A, B (20 (i Similarity measure based o sigle valued eutrosophic hesitat ormalized Hammig Hausdorff distace: s NHH (A, B = d NHH (A, B (2 (ii Similarity measure based o sigle valued eutrosophic hesitat ormalized Euclidia Hausdorff distace: s NEH (A, B = d NEH (A, B (22 (3 Similarity measure based o geeralized sigle valued eutrosophic hesitat weighted Hausdorff distace: s GW (A, B = d GW (A, B (23 (i Similarity measure based o sigle valued eutrosophic hesitat weighted Hammig distace: s WH (A, B = d WH (A, B (24 (ii Similarity measure based o sigle valued eutrosophic hesitat weighted Euclidia distace: s WE (A, B = d WE (A, B ( Similarity measure based o the set-theoretic approach Let A ad B be two SVNHFSs, the we defie a similarity measure from the poit of settheoretic view as follows: s ST (A, B = 3 (mi t AB (x i + (mi i AB(x i + (mi f AB (x i, (26 l l (max t AB(x i + i (x i l (max i AB(x i + f (x i t (x i (max f AB (x i where t AB(x i = (t A σ( (x i, t B σ( (x i, i AB(x i = (i A σ( (x i, f AB (x i = (f A σ( (x i (7 By takig ito accout the weight of each elemet x i X for truth-membership fuctio, idetermiacy-membership fuctio ad falsity membership fuctio, we defie a similarity measure as: 47

14 Floreti Smaradache, Surapati Pramaik (Editors s WST 48 (A, B = 3 ω (mi t AB(x i + ψ (mi i AB(x i + φ (mi f AB (x i l l ω (max t AB (x i + i (x i l ψ (max i AB (x i + f (x i t (x i φ (max f AB (x i 4.3. Similarity measure based o matchig fuctio The cocept of similarity betwee FSs based o a matchig fuctio was defied by Che et al. (995. The Xu (2007 exteded the matchig fuctio to deal with the similarity measures for IFSs. I the followig, we propose the similarity measure for SVNHFSs based o the matchig fuctio. Suppose that A ad B are two SVNHFSs, the we defie a similarity measure based o the matchig fuctio as follows: s MF (A, B = 3 t AB(x i + i AB(x i + f AB (x i l max ( t (x i l t AB (x i, t (x i l i AB(x i, t (x i f AB (x i where t AB(x i = (t A σ( (x i t B σ( (x i, i AB(x i = (i A σ( (x i i B σ( (x i ad f AB (x i = (f A σ( (x i f B σ( (x i, ad t AB(x i = (t A σ( (x i 2 + (t B σ( (x i 2, i AB (x i = (i A σ( (x i 2 + (i B σ( (x i 2 ad f AB (x i = (f A σ( (x i 2 + (f B σ( (x i 2. If we cosider weight of each x X, the we get s WMF (A, B = 3 ω ( t AB(x i + ψ ( i AB(x i + φ ( f AB (x i l max ( t (x i l ω ( t AB(x i, t (x i l ψ ( i AB(x i, t (x i φ ( f AB (x i It is clear that s WMF (A, B satisfies all the properties described i Defiitio Decisio-makig method based o the sigle-valued eutrosophic hesitat fuzzy iformatio I this sectio, we apply the developed distace ad similarity measures to a MADM problem with sigle-valued eutrosophic hesitat fuzzy iformatio. For the MADM problem, let A = {A, A 2,, A m } be a set of alteratives, C = {C, C 2,, C } be a set of attributes. Suppose that ω = (ω, ω 2, ω T, ψ = (ψ, ψ 2, ψ T ad φ = (φ, φ 2, φ T are the potetial weightig vector assiged to the truth-membership, the idetermiacy-membership ad the falsity-membership, respectively, i each alterative, where ω 0, ψ 0 ad φ 0, =,2,,, ω =, ψ = ad φ =. If the decisio makers provide several values for the alterative A i (i =,2,, m uder the attribute C ( =,2,,, these values ca be characterized as a SVNHFN e i = {t i, i i, f i } ( =,2,, ; i =,2,, m. Assume that E = [e i ] m is the decisio matrix, where e i is expressed by a sigle-valued eutrosophic hesitat fuzzy elemet. I multiple attribute decisio-makig eviromets, we ca utilize the cocept of ideal poit to determie the best alterative i the decisio set. Although the ideal alterative does ot exist i real world, it does provide a useful theoretical costruct agaist which to evaluate alteratives. Therefore, we propose each ideal SVNHFN i the ideal alterative A = { C, e : C C} as e = {t, i, f } = {{}, {0}, {0}} ( =,2,,. (27 (28 (29

15 New Treds i Neutrosophic Theory ad Applicatios Thus, we ca develop a procedure for the decisio maker to select the best choice with sigle valued eutrosophic hesitat fuzzy iformatio, which ca be give as follows: Step. Compute the distace (similarity measure betwee a alterative A i (i =,2,..., m ad the ideal alterative A by usig proposed distace (similarity measure. Step 2. Rak all of the alterative with respect to the values of distace (similarity measure. Step 3. Choose the best alterative with respect to the miimum value of distace (maximum value of similarity. Step4. Ed. 6. Practical example Here, a example for the multicriteria decisio-makig problem of alteratives is used as the demostratio of the applicatio of the proposed decisio-makig method, as well as the effectiveess of the proposed method. We take the example adopted from Ye (204c to illustrate the utility of the proposed distace ad similarity measures. Also, we show that the results obtaied usig the proposed distace measure are more reasoable tha the results obtaied usig Ye s (204c cosie similarity measure. Example. Suppose that a ivestmet compay that wats to ivest a sum of moey i the best optio. There is a pael with four possible alteratives i which to ivest the moey: ( A is a car compay, (2 A 2 is a food compay, (3 A 3 is a computer compay, ad (4 A 4 is a arms compay. The ivestmet compay must make a decisio accordig to the three attributes: ( C is the risk aalysis; (2 C 2 is the growth aalysis, ad (3 C 3 is the evirometal impact aalysis. Suppose that ω = (0.35, 0.25, 0.40, ψ = (0.35, 0.40, 0.25, ad φ = (0.30, 0.40, 0.30 are the attribute weight vector for truth-membership degree, the idetermiacy-membership degree ad the falsity membership degree, respectively. The four possible alteratives are to be evaluated uder these three attributes ad are preseted i the form of sigle valued eutrosophic hesitat fuzzy iformatio by decisio maker accordig to three attributes C ( =,2,3, as expressed i the followig sigle valued eutrosophic hesitat fuzzy decisio matrix E: Table : Decisio matrix E { { 0.3, 0.4, 0.5 }, { 0. }, { 0.3, 0.4 } } { { 0.5, 0.6 }, { 0.2, 0.3 }, { 0.3,0.4 } } { { 0.2, 0.3 }, { 0., 0.2 }, { 0.5, 0.6 } } { { 0.6, 0.7 }, { 0., 0.2 }, { 0.2, 0.3 } } { { 0.6, 0.7 }, { 0. }, { 0.3 } } { { 0.6, 0.7 }, { 0., 0.2 }, { 0., 0.2 } } E = ( { { 0.5, 0.6 }, { 0.4 }, { 0.2, 0.3 } } { { 0.6 }, { 0.3 }, { 0.4 } } { { 0.5, 0.6 }, { 0. }, { 0.3 } } { { 0.7, 0.8 }, { 0. }, { 0., 0.2 } } { { 0.6, 0.7 }, { 0. }, { 0.2 } } { { 0.3, 0.5 }, { 0.2 }, { 0., 0.2, 0.3 } } To get the best alterative(s, the followig steps are ivolved: Step. Usig Eq. (2, we ca compute the sigle valued eutrosophic hesitat weighted Hammig distace betwee the alteratives ad the ideal alterative as: d NH (A, A = , d NH (A 2, A = , d NH (A 3, A = , d NH (A 4, A = Step 2. With respect to the values of weighted Hammig distace, we ca rak the alteratives as A 2 A 4 A 3 A. Step 3. The alterative A 2 is the optimal choice accordig to the miimum value amog weighted Hammig distaces, which is ot i agreemet with the oe obtaied i Ye (204c. 49

16 Floreti Smaradache, Surapati Pramaik (Editors Above example clearly shows that the developed method is effective ad applicable uder siglevalued eutrosophic hesitat fuzzy eviromet. 7. Related comparative aalysis Case. Ye (204c proposed a method based o sigle-valued eutrosophic hesitat fuzzy aggregatio operators ad cosie measure fuctio to fid the best alterative. This method lacks the decisio makers risk factor, which causes the distortio of similarity betwee a alterative ad the ideal alterative ad makes the proposed method more realistic. Therefore, the results of the proposed method do t coicide with the existig method Ye (204c. I proposed method, we ot oly cosider the decisio makers risk case but also the idividual weightig vectors of truthmembership, idetermiacy-membership, ad falsity-membership degrees of each elemet i decisio space, separately. From Table 2, we ca see that the rakigs are chaged accordig to differet parameters, cosequetly, the proposed distace measure ca provide a more flexible decisio ad more choice for decisio makers because of the decisio maker risk factor ad the idividual weightig vector of membership degrees. Combiig the aalyses above, our method is more precise ad reliable tha the result produced i Ye (204c. Table 2: Results obtaied by Eq. (5 correspodig differet values A A 2 A 3 A 4 Rakig = A 2 A 4 A 3 A = A 2 A 4 A 3 A = A 2 A 3 A 4 A = A 2 A 3 A 4 A Case 2. I order to validate the feasibility of the proposed decisio makig method, we give aother comparative study betwee our method ad existig methods ad use the cocept of weighted Euclidia distace. Xu ad Xia s method (20 is used to rak the HFSs which are oly characterized by a set of the membership degrees, whereas Sigh s method (203 is applied to DHFSs which are take ito accout both the membership hesitat degree ad the o-membership hesitat degree i decisio makig process. 50 Table 3: Relatioships betwee existig methods ad proposed method Methods Rakigs The best alterative(s The worst alterative(s Xu ad Xia s method A 2 A 3 A 4 A 2 A (20 A A 2 A 4 A 3 A 2 A Sigh s method (205 A A 4 A 2 A 3 A 4 A Ye s method (2005 A Our method A 2 A 4 A 3 A A 2 A O the other had, we also utilize the weighted Euclidia distace to determie the fial rakig order of all the alteratives associated with SVNHFS, which are expressed by the truth-membership

17 New Treds i Neutrosophic Theory ad Applicatios hesitat degree, idetermiacy-membership hesitat degree, ad falsity-membership hesitat degree, to calculate the distace measures ad to rak all of the alteratives accordig to these values. Usig the MCDM problem i Example, the results with differet methods are show i Table 3. Accordig to the results preseted i Table 3, if the distace methods i Xu ad Xia (20 ad the Sigh (203 are used, the the best alteratives are A 2 ad the worst oe is A, respectively. Ye s method (204c say that the best oes are A 4 ad the worst oe is A. With respect to proposed method i this paper, the best oe is A 2 ad A is the worst oe. But, there are some small differeces i the rakig of the alteratives due to defiitio of set theories. Additioally, from results of Table 3, we ca say that the cocept of distace measure is more remarkable ad more useable tha cosie measure to determie the order of the alteratives. As metioed above, the sigle valued eutrosophic hesitat fuzzy set is a geeralizatio of FSs, IFSs, HFSs, FMSs, DHFSs ad also SVNSs. Therefore a SVNHFS (truth-membership hesitat degree, idetermiacy-membership hesitat degree, ad falsity-membership hesitat degree cotais more iformatio tha the HFS (membership hesitat degree, the IFS (both membership degree ad omembership degree, the DHFS (membership hesitat degree ad omembership hesitat degree, ad also SVNS (truth-membership degree, idetermiacy-membership degree, ad falsity-membership degree. The, the proposed distace ad similarity measures of SVNHFSs is a further geeralizatio of the distace ad similarity measures of FSs, IFSs, HFSs, FMSs, DHFSs ad also SVNSs. I other words, the distace ad similarity measures of FSs, IFSs, HFSs, FMSs, DHFSs ad also SVNSs are special cases of the distace ad similarity measures of SVNHFSs proposed i this paper. Therefore, the discrimiatio measures for SVNHFSs ca be used to solve ot oly distace ad similarity measures with SVNHFSs but also the problems of fuzzy eviromet, hesitat fuzzy eviromet, ituitioistic fuzzy eviromet, dual hesitat fuzzy eviromet ad sigle valued eutrosophic eviromet, whereas the methods i Xu ad Xia (20, Xu (2007, Sigh (203 ad Maumdar ad Samata (204 are oly sustaiable for problems with HFSs, IFSs, DHFSs, ad SVNSs, respectively. Moreover, sice SVNHFSs iclude the aforemetioed fuzzy sets, the decisio-makig method usig the proposed distace ad similarity measures is more geeral ad more feasibility tha existig decisio-makig methods i fuzzy settig, ituitioistic fuzzy settig, hesitat fuzzy settig, dual hesitat fuzzy settig, ad sigle-valued eutrosophic settig. 8. Coclusios Based o the combiatio of both HFSs ad SVNSs as a further geeralizatio of fuzzy cocepts, the SVNHFS cotais more iformatio because it takes ito accout the iformatio of its truthmembership hesitat degree, idetermiacy-membership hesitat degree, ad falsity-membership hesitat degree, whereas the HFS oly cotais the iformatio of its membership hesitat degrees ad DHFS cotais the iformatio of its membership hesitat degree ad omembership hesitat degree. Therefore, it has the desirable characteristics ad advatages of its ow, appears to be a more flexible method tha the existig methods ad iclude much more iformatio give by decisio makers. Based o the geometric distace model, the set-theoretic approach, ad the 5

18 Floreti Smaradache, Surapati Pramaik (Editors matchig fuctios, this paper proposed some distace ad similarity measures betwee SVHNSs as a ew extesio of discrimiatio measures betwee fuzzy sets, hesitat fuzzy sets, dual hesitat fuzzy sets ad the sigle-valued eutrosophic sets. I a multiple attribute decisio makig process with sigle-valued eutrosophic hesitat fuzzy iformatio, the proposed distace measure betwee each alterative ad the ideal alterative was used to rak the alteratives ad determie the best oe(s accordig to the measure values. Fially, a umeric example was give to verify the proposed approach ad to show its practicality ad favorable. The developed method has useable ad effective calculatio, ad presets a ew model for hadlig decisio-makig problems uder the sigle-valued eutrosophic hesitat fuzzy eviromet. I the future, we shall further develop more discrimiatio measures such as correlatio coefficiet, etropy ad crossetropy for SVNHFSs ad apply them to solve practical applicatios i these areas, such as group decisio makig, expert system, clusterig, iformatio fusio system, fault diagoses, ad medical diagoses. Refereces. K. Ataassov, Ituitioistic fuzzy sets. Fuzzy Sets ad Systems, (20 ( K. Ataassov ad G. Gargov (989 Iterval-valued ituitioistic fuzzy sets Fuzzy Sets ad Systems, 3( S. Broumi, F. Smaradache (203 Correlatio coefficiet of iterval eutrosophic set, Applied Mechaics ad Materials, S.M. Che, S.M. Yeh ad P.H. Hsiao (995 A compariso of similarity measures of fuzzy values. Fuzzy Sets ad Systems, 72: P. Grzegorzewski (2004 Distaces betwee ituitioistic fuzzy sets ad/or iterval-valued fuzzy sets based o the Hausdorff metric. Fuzzy Sets ad Systems, 48, W.L. Hug ad M.S. Yag (2007 Similarity measures of ituitioistic fuzzy sets based o L p metric. Iteratioal Joural of Approximate Reasoig, 46: W.L. Hug ad M.S. Yag (2004 Similarity measures betwee type-2 fuzzy sets. Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems, 2, P. Liu ad Y. Wag, Multiple attribute decisio-makig method based o sigle-valued eutrosophic ormalized weighted Boferroi mea, Neural Computig ad Applicatios, 25 (7-8 ( Y.H. Li, D.L. Olso ad Z. Qi (2007 Similarity measures betwee ituitioistic fuzzy (vague sets: a comparative aalysis. Patter Recogitio Letters, 28: Z.Z. Liag, P.F. Shi (2003 Similarity measures o ituitioistic fuzzy sets. Patter Recogitio Letters, 24: R. Şahi (205 Fuzzy multicriteria decisio makig method based o the improved accuracy fuctio for iterval valued ituitioistic fuzzy sets. Soft Computig, DOI: 0.007/s x 2. R. Şahi (204, Neutrosophic hierarchical clusterig algorithms, Neutrosophic Sets ad Systems, 2, R Şahi ad A. Küçük (204 Subsethood measures for sigle valued eutrosophic sets, Joural of Itelliget & Fuzzy Systems DOI: /IFS F. Smaradache (2005 A geeralizatio of the ituitioistic fuzzy set. Iteratioal oural of Pure ad Applied Mathematics, F. Smaradache, A uifyig field i logics. eutrosophy: Neutrosophic probability, set ad logic, America Research Press, Rehoboth, H. Wag, F. Smaradache, Y.Q. Zhag ad R. Suderrama, (2005 Iterval eutrosophic sets ad logic: Theory ad applicatios i computig, Hexis, Phoeix, AZ. 52

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