GRA Method of Multiple Attribute Decision Making with Single Valued Neutrosophic Hesitant Fuzzy Set Information

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1 New Trends n Neutrosophc Theory and Applcatons PRANAB BISWAS, SURAPATI PRAMANIK *, BIBHAS C. GIRI 3 Department of Mathematcs, Jadavpur Unversty, Kolkata, 70003, Inda. E-mal: paldam00@gmal.com * Department of Mathematcs, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpr, Dstrct-North 4 Parganas, West Bengal, PIN-7436, Inda. Correspondng author s E-mal: sura_pat@yahoo.co.n 3 Department of Mathematcs, Jadavpur Unversty, Kolkata, 70003, Inda.Emal: bcgr.jumath@gmal.com GRA Method of Multple Attrbute Decson Makng wth Sngle Valued Neutrosophc Hestant Fuzzy Set Informaton Abstract Sngle valued neutrosophc hestant fuzzy set has three ndependent parts, namely the truth membershp hestancy functon, ndetermnacy membershp hestancy functon, and falsty membershp hestancy functon, whch are n the form of sets that assume values n the unt nterval [0, ]. Sngle valued neutrosophc hestant fuzzy set s consdered as a powerful tool to express uncertan, ncomplete, ndetermnate and nconsstent nformaton n the process of mult attrbute decson makng problems. In ths paper we study mult attrbute decson makng problems n whch the ratng values are expressed wth sngle valued neutrosophc hestant fuzzy set nformaton. Frstly, we defne score value and accuracy value to compare sngle valued neutrosophc hestant fuzzy sets and then defne normalsed Hammng dstance between the sngle valued neutrosophc hestant fuzzy sets. Secondly, we propose the grey relatonal analyss method for mult attrbute decson makng under sngle valued neutrosophc hestant fuzzy set envronment. Fnally, we provde an llustratve example to demonstrate the valdty and effectveness of the proposed method. Keywords Hestant fuzzy sets, sngle-valued neutrosophc hestant fuzzy sets, score and accuracy functon, grey relatonal analyss method, mult-attrbute decson makng.. Introducton Mult-attrbute decson makng (MADM) used n human actvtes s a useful process for selectng the best alternatve that has the hghest degree of satsfacton from a set of feasble alternatves wth respect to the attrbutes. Because the real world s fuzzy rather than precse n nature, the ratng values of alternatve wth respect to attrbute consdered n MADM problems are often mprecse or ncomplete n nature. Ths has led to the development of the fuzzy set theory proposed by Zadeh []. Fuzzy set theory has been proved to be an effectve tool n MADM process [-6]. However, fuzzy set can represent mprecse nformaton wth membershp degree only. The ntutonstc fuzzy set (IFS) proposed by Attanasov [7], a generalsaton of fuzzy sets, s characterzed by membershp and non-membershp functons where non-membershp s 55

2 Florentn Smarandache, Surapat Pramank (Edtors) ndependent. Recently, IFS has been successfully appled n many decson makng problems, especally n MADM problems [8-]. However IFS can handle ncomplete nformaton and but t cannot express ndetermnate and nconsstent nformaton wth membershp and non-membershp functons. Smarandache [3] ntroduced the neutrosophc set (NS) from phlosophcal pont of vew to deal wth uncertan, mprecse, ncomplete and nconsstent nformaton that exst n real world. NS s charactersed wth truth membershp, ndetermnacy and falsty membershp degree, whch are ndependent n nature. Ths set generalses the concept of crsp set, fuzzy set, ntutonstc fuzzy set, paraconsstent set, dalethest set, paradoxst set, and tautologcal set. Snce the ntroducton of NS and sngle-valued neutrosophc set proposed by Wang et al. [4] n 00, the model of decson makng under neutrosophc envronment has been receved much attenton to the researchers. Many methods of MADM such as TOPSIS method [5, 6], grey relatonal analyss (GRA) method [7,8], dstance and smlarty measure method [9-3], and outrankng method [4] were developed under neutrosophc envronment. However, n a decson makng process sometmes decson maker may feel hestate to take decson among the set of possble values nstead of sngle value. Tora [5], Tora and Narukawa [6] ntroduced the hestant fuzzy set (HF), whch permts the membershp degree of an element to a gven set to be represented by the set of possble numercal values n [0,]. HF, an extenson of fuzzy set, s useful to deal uncertan nformaton n the process of MADM. Xa and Xu [7] proposed some aggregaton operators for hestant fuzzy nformaton and appled them to MADM problem n hestant fuzzy envronment. We [8] studed some models for hestant fuzzy MADM problem by developng some prortzed aggregaton operators for hestant fuzzy nformaton. Xu and Zhang [9] developed TOPSIS method for hestant fuzzy MADM wth ncomplete weght nformaton. Decson maker does not consder the non-membershp degrees of ratng values n hestant fuzzy MADM. However, non-membershp degrees play an mportant role to express ncomplete nformaton. Zhu et al. [30] gave the dea of the dual hestant fuzzy set (DHFS), n whch membershp degrees and non-membershp degrees are n the form of sets of values n [0,]. DHFS generalzes the HF sets and expresses ncomplete nformaton effectvely. Ye [3] and Chen et al.[3] proposed co-relaton method between DHFSs and appled the method to MADM wth hestant fuzzy nformaton. Sngh [33] defned and appled dstance and smlarty measure between DHFSs n MADM. However n a decson makng process, ndetermnate type nformaton cannot be captured wth DHFS. In 04, Ye [34] ntroduced sngle-valued neutrosophc hestant fuzzy set (SVNHFS) by coordnatng HFS and SVNS. SVNHFS generalses the FS, IFS, HFS, DHFS and SVNS, and can represent uncertan, mprecse, ncomplete and nconsstent nformaton. SVNHFSs are characterzed by truth hestancy, ndetermnacy hestancy and falsty-hestancy membershp functons whch are ndependent. Therefore SVNHFS can express the three knds of hestancy nformaton that exst n MADM n real stuatons. Ye [34] developed sngle valued neutrosophc hestant fuzzy weghted averagng and sngle valued neutrosophc hestant fuzzy weghted geometrc operators for SVNHFS nformaton and appled these two operators n MADM. Lu and Sh [35] proposed hybrd weghted average operator for nterval neutrosophc hestant fuzzy set n whch the truth hestancy, ndetermnacy hestancy and falsty-hestancy membershp functons are n the form of sets of nterval values contaned n [0, ]. Sahn and Lu [36] defned co-relaton coeffcent between SVNHFSs and used t for MADM. 56

3 New Trends n Neutrosophc Theory and Applcatons Grey relatonal analyss (GRA)[37], a part of grey system theory, s successfully appled n solvng a varety of MADM problems n ntutonstc fuzzy envronment [38-4], neutrosophc envronment [43], nterval neutrosophc envronment [44, 45, 46], neutrosophc soft set envronment [47-49], rough neutrosophc envronment [50] respectvely. However, lterature revew reflects that GRA method of MADM wth SVNHFS has not been studed n the lterature. Therefore we need attenton for ths ssue. The am of the paper s to extend the concept of GRA method for solvng MADM problem n whch the ratng values of the alternatves over the attrbutes are consdered wth SVNHFSs. The rest of the paper s organsed as follows: Secton presents some basc concept related to SVNHFSs. In Secton 3, we propose GRA method for MADM problems, where ratng values are consdered wth SVNHFSs. In Secton 4, we llustrate our proposed method wth an example. Secton 5 presents concludng remarks of the study.. Prelmnares In ths secton we recall some basc defntons of hestant fuzzy set, sngle valued neutrosophc hestant fuzzy set, score functon accuracy functon of trangular fuzzy ntutonstc fuzzy numbers. Defnton. [5]Let X be a fxed set, then a hestant fuzzy set (HFS) A on X s n terms of a functon that when appled to X returns a subset of[0,],.e., A x, h ( x) x X, where, h ( x) s a set of some dfferent values n [0,], representng the A A possble membershp degrees of the element x X to A. For convenence, ha( x ) s called a hestant fuzzy element (HFE). Defnton. [34] Let X be fxed set, then a sngle valued hestant fuzzy element (SVHFE) N on X s defned as N x, t( x), ( x), f ( x) x X () where tx ( ), x ( ) and f( x ) represent three sets of values n 0,, denotng respectvely the possble truth, ndetermnacy and falsty membershp degree of the element x X to the set N. The membershp degrees tx ( ), x ( ) and f( x ) satsfy the followng condtons: 0,, ; 0 3 where, t( x), ( x), f ( x), () t ( x) max t( x), ( x) max ( x), f ( x) max f ( x) for t( x) t( x) t( x) all x X. For convenence, the trplet n( x) t( x), ( x), f ( x) s called a SVNHFE denoted by n t,, f. Note that the number of values for possble truth, ndetermnacy and falsty membershp degrees of the element n dfferent SVNHFEs may be dfferent. Defnton 3. [34] Let n t,, f and n t,, f be two SVNHFEs, the followng operatonal rules are defned as follows: 7. n n t t t t f f { },{ },{, } ; t,, f, t,, f 8. n n t t f f f f { },{ },{ } ; t,, f, t,, f 9. n t f { ( ) },{ },{ }, 0 ; t,, f 0. n t f { },{ ( ) },{ ( ) }, 0. t,, f 57

4 Florentn Smarandache, Surapat Pramank (Edtors) Defnton 4. Let n t,, f (,,..., n) be a collecton of SVNHFEs, then the score functon Sn ( ), and accuracy functon An ( ) of n (,,..., n) can be defned as follows:. Sn ( ) (3) l l l An ( ) ; l l t t t t f f f f (4) where, l t, l, and l f, are the numbers of values of t,, and f respectvely n n. Defnton 5. Let n t,, f and n t,, f be two SVNHFEs, the followng rules can be defned for comparson purposes:. If S( n) S( n), then n s greater than n and denoted by n n ;. If S( n) S( n) and A( n) A( n), then n n ; 3. If S( n) S( n) and A( n) A( n), then n n. Defnton 6. Let n t,, f and n t,, f dstance s defned as D( n, n) 3 lt t l t t l l l f f l f f be two SVNHFEs, the normalsed Hammng (5) where l t k, l k, and l are the possble membershp values n f n k k for k,, respectvely. The dstance functon D( n, n ) of two SVNHFEs n and n satsfes the followng propertes:. D n n 0 (, ) ;. D( n, n) 0f and only f n n ; 3. D n n D n n (, ) (, ); 4. If n n n3, and n 3 s an SVNHFE on X, then D( n, n) D( n, n3) and D n n3 D n n3 (, ) (, ). 3. GRA method for mult-attrbute decson makng wth SVNHFS nformaton In ths secton, we propose GRA based approach to fnd out the best alternatve n multattrbute decson makng problem n SVNHFS envronment. Assume that A A, A,..., Am be the dscrete set of m alternatves and C C, C,..., Cn be the set of n attrbutes for a mult-attrbute decson makng problem. Suppose that the ratng values of the th alternatve A (,,..., m) over the attrbute C ( j,,..., n) are expressed n terms of SVNHFSs x t,, f, where j t { t,0 }, {,0 }, and f { f,0 } are the possble truth, ndetermnacy and falsty membershp degrees, respectvely. Wth these ratng values, we can construct a decson matrx X ( x ) mn, where the entres of ths matrx are SVNHFSs. The decson matrx can be presented as follows: x x... x x X x... x x x... x n n m m mn We develop the GRA method usng the followng steps by consderng the weght vector (,,..., ) T of attrbutes where w j [0,] and w. j j W w w w n n (6)

5 New Trends n Neutrosophc Theory and Applcatons Step. Determne the sngle valued neutrosophc hestant fuzzy postve deal soluton (SVNHFPIS) A and the sngle valued neutrosophc hestant fuzzy negatve deal soluton (SVNHFNIS) A of alternatves n the decson matrx X by the followng equatons, respectvely: max ( x),max ( x ),...,max ( xn)for beneft typeattrbute; m m m A mn ( x ),mn ( x ),...,mn ( x ) forcost typeattrbute n m m m A, A,..., A n mn ( x),mn ( x ),...,mn ( xn )for beneft typeattrbute; m m m A max ( x ),max ( x ),...,max ( x ) forcost typeattrbute n (8) m m m A, A,..., An The ratng values x can be compared by the score functon S( x ) and accuracy functon Ax ( ) defned n Defnton 3. Step. Determne the grey relatonal co-effcent of each alternatve from A and A by the followng equatons: A mn mn D( x, A ) max max D( x, A ) j j m m m m D( x, Aj ) max max D( x, Aj ) m m mn mn D( x, A ) max max D( x, A ) j j m m m m D( x, Aj ) max max D( x, Aj ) m m where the dentfcaton co-effcent s consdered as 0.5. Step 3.Calculate the degree of grey relatonal coeffcent of each alternatve A (,,..., m) from and A by the followng equatons: n wj j () n wj j () Step 4.Calculate the relatve closeness co-effcent for each alternatve A (,,.., m) wth respect to the postve deal soluton A as for,,.., m (7) (9) (0). (3) Step 5.Rank the alternatve accordng the relatve closeness co-effcent (,,.., m). 4. A Numercal Example In ths secton we consder the example adopted from Ye [34] to llustrate the applcaton of the proposed GRA method for MADM proposed n Secton 4. Consder an nvestment company that wants to nvest a sum of money n the best opton. The followng four possble alternatves are consdered to nvest the money:. A s the car company;. A s the food company; 3. A 3 s the computer company; 4. A 4 s the arms company. The nvestment company must take a decson accordng to the followng three attrbutes: 59

6 Florentn Smarandache, Surapat Pramank (Edtors) 60. C s the rsk analyss;. C s the growth analyss; 3. C 3 s the envronmental mpact analyss. The attrbute weght vector s gven as W (0.35, ) T. The four possble alternatves { A, A, A3, A 4} are evaluated usng SVNHFEs under three attrbutes Cj( j,,3). We can arrange the ratng values n a matrx form called a SVNHF decson matrx X ( x) (see Table-). 43 Table. Sngle valued neutrosophc hestant fuzzy decson matrx C C C 3 0.3,0.4,0.5, 0., 0.3, ,0.6, 0.,0.3, 0.3, ,0.4,0.5, 0., 0.3, ,0.7, 0.,0., 0., ,0.7, 0., ,0.4,0.5, 0., 0.3, ,0.6, 0.4, 0., , 0.3, ,0.6, 0., ,0.8, 0., 0.,0. 0.6,0.7, 0., ,0.5, 0., 0.,0.,0.3 Now we apply the proposed method to fnd out the best alternatve, whch can be descrbed as follows: Step. Comparng the attrbute values by score functon and accuracy functon of SVNHFEs, we can determne the neutrosophc hestant fuzzy postve deal soluton (SVNHFPIS) A by the Eq.(7) as follows: A 0.7,0.8, 0., 0.,0., 0.6,0.7, 0., 0., 0.6,0.7, 0.,0., 0.,0. (4) A, A, A 3 Smlarly, we can determne the neutrosophc hestant fuzzy negatve deal soluton (SVNHFPIS) A by the Eq.(8) as follows: A 0.5,0.6, 0.4, 0.,0.3, 0.6, 0.3, 0.4, 0.,0.3, 0.,0., 0.5,0.6 (5) A, A, A 3 Step. Calculate the grey relatonal co-effcent of each alternatve from postve deal solutons A and negatve deal solutons A by equatons (9) and (0) for 0.5, respectvely (7) Here, we consder,,3,4 and j,,3. Step 3.Calculate the degree of grey relatonal co-effcent of each alternatve from A and A by Eqs. () and (), respectvely. (6)

7 New Trends n Neutrosophc Theory and Applcatons (8) (9) 3 4 Step 4.Calculate the relatve closeness coeffcent for each alternatve A (,,3,4) by Eq.(3) , , , and Step 5. Rank the alternatve accordng to the relatve closeness coeffcent (,,3,4). Therefore A4 A A3 A ndcates that the most desrable alternatve s A. 4 We notce that the rankng order obtaned by the proposed method s ndfferent wth the rankng of the alternatve obtaned by Ye s method [34]. 5. Conclusons In general, the nformaton of ratng values consdered n MADM problems s mprecse, ndetermnate, ncomplete and nconsstent n nature. SVNHFS s a useful tool that can capture all these type of nformaton n MADM process. In ths paper we nvestgate MADM problem n whch ratng values are consdered wth SVNHFSs. To extend the GRA method for MADM, we frst defne score value, accuracy value, certanty value, and normalsed Hammng dstance of SVNHFS. Havng defned the postve deal soluton (PIS) and the negatve deal soluton (NIS) by score value and accuracy value, we calculate the grey relatonal degree between each alternatve and deal alternatves (PIS and NIS). Then we determne a relatve relatonal degree to obtan the rankng order of all alternatves by calculatng the degree of grey relaton to both the postve and negatve deal soluton smultaneously. Fnally, we provde an llustratve example to show the valdty and effectveness of the proposed approach. The proposed approach s compared wth other exstng methods to show that our approach s straghtforward and can be appled effectvely wth other decson makng problems under SVNHF envronment. In future, we wll extend the proposed approach to MADM under SVNHFS envronment wth unknown weght nformaton and MADM wth nterval valued neutrosophc hestant fuzzy envronment. References. L.A. Zadeh, Fuzzy sets, Informaton Control, 8(965) R. Bellman, L.A. Zadeh, Decson makng n a fuzzy envronment, Management Scence 7B (4)(970) C.L Hwang, K. Yoon, Multple attrbute decson makng: Methods and Applcatons, Sprnger-Verlag, Berln, S.J. Chen, C.L Hwang, Fuzzy multple attrbute decson makng: Methods and Applcatons, Sprnger- Verlag, Berln, L. Zeng, Expected value method for fuzzy multple attrbute decson makng, Tsnghua Scence and Technology (006) C.T. Chen, Extenson of TOPSIS for group decson-makng under fuzzy envronment, Fuzzy Sets and Systems 4(000) K.T. Atanassov, Intutonstc fuzzy sets, Fuzzy Sets and Systems 0(986) E. Szmdt, J. Kacprzyk, Usng ntutonstc fuzzy sets n group decson makng, Control and Cybernetcs 3(00) Z. Xu, Intutonstc preference relatons and ther applcatons n group decson makng, Informaton Scences 7(007) DF, L, YC, Wang, S, Lu, F, Shan. Fractonal programmng methodology for mult-attrbute group decson makng usng IFS, Appled Soft Computng 9(009) G.W. We, Gray relatonal analyss method for ntutonstc fuzzy multple attrbute decson makng, Expert Systems and Applcatons 38(0)

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